Trends in Mathematics

Quaternion and Clifford Fourier Transforms and Wavelets

Eckhard HitzerStephen J. Sangwine Editors

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Editors

Eckhard HitzerStephen J. Sangwine

Quaternion and Clifford FourierTransforms and Wavelets

Editors Eckhard Hitzer Stephen J. Sangwine Department of Material Science School of Computer Science International Christian University Tokyo, Japan

ISBN 978-3-0348-0602-2 ISBN 978-3-0348-0603-9 (eBook) DOI 10.1007/978-3-0348-0603-9 Springer Basel Heidelberg New York Dordrecht London Library of Congress Control Number: 2013939066 Mathematics Subject Classification (2010): 11R52, 15A66, 42A38, 65T60, 42C40, 68U10, 94A08, 94A12 © Springer Basel 2013 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. Exempted from this legal reservation are brief excerpts in connection with reviews or scholarly analysis or material supplied specifically for the purpose of being entered and executed on a computer system, for exclusive use by the purchaser of the work. Duplication of this publication or parts thereof is permitted only under the provisions of the Copyright Law of the Publisher’s location, in its current version, and permission for use must always be obtained from Springer. Permissions for use may be obtained through RightsLink at the Copyright Clearance Center. Violations are liable to prosecution under the respective Copyright Law. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. While the advice and information in this book are believed to be true and accurate at the date of publication, neither the authors nor the editors nor the publisher can accept any legal responsibility for any errors or omissions that may be made. The publisher makes no warranty, express or implied, with respect to the material contained herein. Printed on acid-free paper Springer Basel is part of Springer Science+Business Media (www.springer.com)

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Contents

Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii

F. Brackx, E. Hitzer and S.J. SangwineHistory of Quaternion and Clifford–Fourier Transformsand Wavelets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xi

Part I: Quaternions

1 T.A. EllQuaternion Fourier Transform: Re-tooling Image andSignal Processing Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

2 E. Hitzer and S.J. SangwineThe Orthogonal 2D Planes Split of Quaternions andSteerable Quaternion Fourier Transformations . . . . . . . . . . . . . . . . . . . . . . . 15

3 N. Le Bihan and S.J. SangwineQuaternionic Spectral Analysis of Non-Stationary ImproperComplex Signals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

4 E.U. Moya-Sanchez and E. Bayro-CorrochanoQuaternionic Local Phase for Low-level Image ProcessingUsing Atomic Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

5 S. Georgiev and J. MoraisBochner’s Theorems in the Framework of Quaternion Analysis . . . . . . 85

6 S. Georgiev, J. Morais, K.I. Kou and W. SproßigBochner–Minlos Theorem and Quaternion Fourier Transform . . . . . . . . 105

Part II: Clifford Algebra

7 E. Hitzer, J. Helmstetter and R. AblamowiczSquare Roots of −1 in Real Clifford Algebras . . . . . . . . . . . . . . . . . . . . . . . 123

8 R. Bujack, G. Scheuermann and E. HitzerA General Geometric Fourier Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155

vi Contents

9 T. Batard and M. BerthierClifford–Fourier Transform and Spinor Representationof Images . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177

10 P.R. Girard, R. Pujol, P. Clarysse, A. Marion,R. Goutte and P. DelachartreAnalytic Video (2D + 𝑡) Signals Using Clifford–Fourier Transformsin Multiquaternion Grassmann–Hamilton–Clifford Algebras . . . . . . . . . 197

11 S. Bernstein, J.-L. Bouchot, M. Reinhardt and B. HeiseGeneralized Analytic Signals in Image Processing: Comparison,Theory and Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221

12 R. Soulard and P. CarreColour Extension of Monogenic Wavelets with Geometric Algebra:Application to Color Image Denoising . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 247

13 S. BernsteinSeeing the Invisible and Maxwell’s Equations . . . . . . . . . . . . . . . . . . . . . . . . 269

14 M. BahriA Generalized Windowed Fourier Transform inReal Clifford Algebra 𝐶ℓ0,𝑛 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 285

15 Y. Fu, U. Kahler and P. CerejeirasThe Balian–Low Theorem for the WindowedClifford–Fourier Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 299

16 S. Li and T. QianSparse Representation of Signals in Hardy Space . . . . . . . . . . . . . . . . . . . . 321

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 333

Quaternion and Clifford–Fourier Transforms and Wavelets

Trends in Mathematics, vii–xc⃝ 2013 Springer Basel

Preface

One hundred and seventy years ago (in 1843) W.R. Hamilton formally introducedthe four-dimensional quaternions, perceiving them as one of the major discoveriesof his life. One year later, in 1844, H. Grassmann published the first version of hisAusdehnungslehre, now known as Grassmann algebra, without any dimensionallimitations. Circa thirty years later (in 1876) W.K. Clifford supplemented theGrassmann product of vectors with an inner product, which fundamentally unifiedthe preceding works of Hamilton and Grassmann in the form of Clifford’s geometricalgebras or Clifford algebras. A Clifford algebra is a complete algebra of a vectorspace and all its subspaces, including the measurement of volumes and dihedralangles between any pair of subspaces.

To work in higher dimensions with quaternion and Clifford algebras allows usto systematically generalize known concepts of symmetry, phase, analytic signaland holomorphic function to higher dimensions. And as demonstrated in the cur-rent proceedings, it successfully generalizes Fourier and wavelet transformationsto higher dimensions. This is interesting both for the development of analysis inhigher dimensions, as well as for a broad range of applications in multi-dimensionalsignal, image and color image processing. Therefore a wide variety of readers frompure mathematicians, keen to learn about the latest developments in quaternionand Clifford analysis, to physicists and engineers in search of dimensionally ap-propriate and efficient tools in concrete applications, will find many interestingcontributions in this book.

The contributions in this volume originated as papers in a session on Quater-nion and Clifford–Fourier transforms and wavelets of the 9th International Con-ference on Clifford Algebras and their Applications (ICCA9), which took placefrom 15th to 20th July 2011 at the Bauhaus-University in Weimar, Germany. Thesession was organized by the editors of this volume.

After the conference we asked the contributors to prepare expanded versionsof their works for this volume, and many of them agreed to participate. The ex-panded submissions were subjected to a further round of reviews (in addition tothe original reviews for the ICCA9 itself) in order to ensure that each contributionwas clearly presented and worthy of publication. We are very grateful to all thosereviewers whose efforts contributed significantly to the quality of the final chaptersby asking the authors to revise, clarify or to expand on points in their drafts.

viii Preface

The contributions have been edited to achieve as much uniformity in presen-tation and notation as can reasonably be achieved across the somewhat differenttraditions that have arisen in the quaternion and Clifford communities. We hopethat this volume will contribute to a growing unification of ideas across the ex-panding field of hypercomplex Fourier transforms and wavelets.

The book is divided into two parts: Chapters 1 to 6 deal exclusively withquaternions ℍ, while Chapters 7 to 16 mainly deal with Clifford algebras 𝐶ℓ𝑝,𝑞,but sometimes include high-dimensional complex as well as quaternionic resultsin several subsections. This is natural, since complex numbers (ℂ ∼= 𝐶ℓ0,1) andquaternions (ℍ ∼= 𝐶ℓ0,2) are low-dimensional Clifford algebras, and often appearas subalgebras, e.g., ℂ ∼= 𝐶𝑙+2,0, ℍ

∼= 𝐶𝑙+3,0, etc. The first chapter was writtenespecially for this volume to provide some background on the history of the subject,and to show how the contributions that follow relate to each other and to priorwork. We especially thank Fred Brackx (Ghent/Belgium) for agreeing to contributeto this chapter at a late stage in the preparation of the book.

The quaternionic part begins with an exploration by Ell (Chapter 1) of theevolution of quaternion Fourier transform (QFT) definitions as a framework forproblems in vector-image and vector-signal processing, ranging from NMR prob-lems to applications in colour image processing. Next, follows an investigation byHitzer and Sangwine (Chapter 2) into a steerable quaternion algebra split, whichleads to: a local phase rotation interpretation of the classical two-sided QFT, effi-cient fast numerical implementations and the design of new steerable QFTs.

Then Le Bihan and Sangwine (Chapter 3) perform a quaternionic spectralanalysis of non-stationary improper complex signals with possible correlation ofreal and imaginary signal parts. With a one-dimensional QFT they introduce ahyperanalytic signal closely linked to the geometric features of improper com-plex signals. In the field of low level image processing Moya-Sanchez and Bayro-Corrochano (Chapter 4) employ quaternionic atomic functions to enhance geo-metric image features and to analytically express image processing operations likelow-pass, steerable and multiscale filtering, derivatives, and local phase computa-tion.

In the next two chapters on quaternion analysis Georgiev and Morais (Chap-ter 5) characterize a class of quaternion Bochner functions generated via a quater-nion Fourier–Stieltjes transform and generalize Bochner’s theorem to quaternionfunctions. In Chapter 6 Georgiev, Morais, Kou and Sproßig study the asymptoticbehavior of the QFT, apply the QFT to probability measures, including positivedefinite measures, and extend the classical Bochner–Minlos theorem to the frame-work of quaternion analysis.

The Clifford algebra part begins with Chapter 7 by Hitzer, Helmstetter andAblamowicz, who establish a detailed algebraic characterization of the continuousmanifolds of (multivector) square roots of −1 in all real Clifford algebras 𝐶ℓ𝑝,𝑞,including as examples detailed computer generated tables of representative squareroots of −1 in dimensions 𝑛 = 𝑝+ 𝑞 = 5, 7 with signature 𝑠 = 𝑝− 𝑞 = 3(mod 4).

Preface ix

Their work is fundamental for any form of Clifford–Fourier transform (CFT) usingmultivector square roots of −1 instead of the complex imaginary unit. Based onthis Bujack, Scheuermann and Hitzer (Chapter 8) introduce a general (Clifford)geometric Fourier transform covering most CFTs in the literature. They prove arange of standard properties and specify the necessary conditions in the transformdesign.

A series of four chapters on image processing begins with Batard and Ber-thier’s (Chapter 9) on spinorial representation of images focusing on edge- andtexture detection based on a special CFT for spinor fields, that takes into accountthe Riemannian geometry of the image surface. Then Girard, Pujol, Clarysse, Mar-ion, Goutte and Delachartre (Chapter 10) investigate analytic signals in Cliffordalgebras of 𝑛-dimensional quadratic spaces, and especially for three-dimensionalvideo (2D + 𝑇 ) signals in (complex) biquaternions (∼= 𝐶ℓ3,0). Generalizing fromthe right-sided QFT to a rotor CFT in 𝐶ℓ3,0, which allows a complex fast Fouriertransform (FFT) decomposition, they investigate the corresponding analytic videosignal including its generalized six biquaternionic phases. Next, Bernstein, Bou-chot, Reinhardt and Heise (Chapter 11) undertake a mathematical overview of gen-eralizations of analytic signals to higher-dimensional complex and Clifford analysistogether with applications (and comparisons) for artificial and real-world imagesamples.

Soulard and Carre (Chapter 12) define a novel colour monogenic wavelettransform, leading to a non-marginal multiresolution colour geometric analysis ofimages. They show a first application through the definition of a full colour imagedenoising scheme based on statistical modeling of coefficients.

Motivated by applications in optical coherence tomography, Bernstein (Chap-ter 13) studies inverse scattering for Dirac operators with scalar, vector and quater-nionic potentials, by writing Maxwell’s equations as Dirac equations in Cliffordalgebra (i.e., complex biquaternions). For that she considers factorizations of theHelmholtz equation and related fundamental solutions; standard- and Faddeev’sGreen functions.

In Chapter 14 Bahri introduces a windowed CFT for signal functions 𝑓 :ℝ𝑛 → 𝐶ℓ0,𝑛, and investigates some of its properties. For a different type of win-dowed CFT for signal functions 𝑓 : ℝ𝑛 → 𝐶ℓ𝑛,0, 𝑛 = 2, 3(mod 4), Fu, Kahler andCerejeiras establish in Chapter 15 a Balian–Low theorem, a strong form of Heisen-berg’s classical uncertainty principle. They make essential use of Clifford framesand the Clifford–Zak transform.

Finally, Li and Qian (Chapter 16) employ a compressed sensing technique inorder to introduce a new kind of sparse representation of signals in a Hardy spacedictionary (of elementary wave forms) over a unit disk, together with examplesillustrating the new algorithm.

We thank all the authors for their enthusiastic participation in the projectand their enormous patience with the review and editing process. We further thankthe organizer of the ICCA9 conference K. Guerlebeck and his dedicated team for

x Preface

their strong support in organizing the ICCA9 session on Quaternion and Clifford–Fourier Transforms and Wavelets. We finally thank T. Hempfling and B. Hellriegelof Birkhauser Springer Basel AG for venturing to accept and skillfully accompanythis proceedings with a still rather unconventional theme, thus going one morestep in fulfilling the 170 year old visions of Hamilton and Grassmann.

Eckhard HitzerTokyo, Japan

Stephen SangwineColchester, United Kingdom

October 2012

Quaternion and Clifford–Fourier Transforms and Wavelets

Trends in Mathematics, xi–xxviic⃝ 2013 Springer Basel

History of Quaternion and Clifford–FourierTransforms and Wavelets

Fred Brackx, Eckhard Hitzer and Stephen J. Sangwine

Abstract. We survey the historical development of quaternion and Clifford–Fourier transforms and wavelets.

Mathematics Subject Classification (2010). Primary 42B10; secondary 15A66,16H05, 42C40, 16-03.

Keywords. Quaternions, Clifford algebra, Fourier transforms, wavelet trans-forms.

The development of hypercomplex Fourier transforms and wavelets has taken placein several different threads, reflected in the overview of the subject presented inthis chapter. We present in Section 1 an overview of the development of quaternionFourier transforms, then in Section 2 the development of Clifford–Fourier trans-forms. Finally, since wavelets are a more recent development, and the distinctionbetween their quaternion and Clifford algebra approach has been much less pro-nounced than in the case of Fourier transforms, Section 3 reviews the history ofboth quaternion and Clifford wavelets.

We recognise that the history we present here may be incomplete, and thatwork by some authors may have been overlooked, for which we can only offer ourhumble apologies.

1. Quaternion Fourier Transforms (QFT)

1.1. Major Developments in the History of the Quaternion Fourier Transform

Quaternions [51] were first applied to Fourier transforms by Ernst [49, § 6.4.2]and Delsuc [41, Eqn. 20] in the late 1980s, seemingly without knowledge of theearlier work of Sommen [90, 91] on Clifford–Fourier and Laplace transforms fur-ther explained in Section 2.2. Ernst and Delsuc’s quaternion transforms were two-dimensional (that is they had two independent variables) and proposed for ap-plication to nuclear magnetic resonance (NMR) imaging. Written in terms of two

xii F. Brackx, E. Hitzer and S.J. Sangwine

independent time variables1 𝑡1 and 𝑡2, the forward transforms were of the followingform2:

ℱ(𝜔1, 𝜔2) =

∞∫−∞

∞∫−∞

𝑓(𝑡1, 𝑡2)𝑒𝒊𝜔1𝑡1𝑒𝒋𝜔2𝑡2d𝑡1d𝑡2 . (1.1)

Notice the use of different quaternion basis units 𝒊 and 𝒋 in each of the two ex-ponentials, a feature that was essential to maintain the separation between thetwo dimensions (the prime motivation for using a quaternion Fourier transforma-tion was to avoid the mixing of information that occurred when using a complexFourier transform – something that now seems obvious, but must have been lessso in the 1980s). The signal waveforms/samples measured in NMR are complex, sothe quaternion aspect of this transform was essential only for maintaining the sep-aration between the two dimensions. As we will see below, there was some unusedpotential here.

The fact that exponentials in the above formulation do not commute (witheach other, or with the ‘signal’ function 𝑓), means that other formulations arepossible3, and indeed Ell in 1992 [45, 46] formulated a transform with the twoexponentials positioned either side of the signal function:

ℱ(𝜔1, 𝜔2) =

∞∫−∞

∞∫−∞

𝑒𝒊𝜔1𝑡1𝑓(𝑡1, 𝑡2) 𝑒𝒋𝜔2𝑡2 d𝑡1d𝑡2 . (1.2)

Ell’s transform was a theoretical development, but it was soon applied to thepractical problem of computing a holistic Fourier transform of a colour image [84]in which the signal samples (discrete image pixels) had three-dimensional values(represented as quaternions with zero scalar parts). This was a major changefrom the previously intended application in nuclear magnetic resonance, becausenow the two-dimensional nature of the transform mirrored the two-dimensionalnature of the image, and the four-dimensional nature of the algebra used followednaturally from the three-dimensional nature of the image pixels.

Other researchers in signal and image processing have followed Ell’s formu-lation (with trivial changes of basis units in the exponentials) [27, 24, 25], but aswith the NMR transforms, the quaternion nature of the transforms was appliedessentially to separation of the two independent dimensions of an image (Bulow’swork [24, 25] was based on greyscale images, that is with one-dimensional pixelvalues). Two new ideas emerged in 1998 in a paper by Sangwine and Ell [86].These were, firstly, the choice of a general root 𝜇 of −1 (a unit quaternion withzero scalar part) rather than a basis unit (𝒊, 𝒋 or 𝒌) of the quaternion algebra,

1The two independent time variables arise naturally from the formulation of two-dimensionalNMR spectroscopy.2Note, that Georgiev et al. use this form of the quaternion Fourier transform (QFT) in Chapter 6to extend the Bochner–Minlos theorem to quaternion analysis. Moreover, the same form of QFTis extended by Georgiev and Morais in Chapter 5 to a quaternion Fourier–Stieltjes transform.3See Chapter 1 by Ell in this volume with a systematic review of possible forms of quaternionFourier transformations.

Quaternion, Clifford–Fourier & Wavelet Transforms History xiii

and secondly, the choice of a single exponential rather than two (giving a choiceof ordering relative to the quaternionic signal function):

ℱ(𝜔1, 𝜔2) =

∞∫−∞

∞∫−∞

𝑒𝜇(𝜔1𝑡1+𝜔2𝑡2)𝑓(𝑡1, 𝑡2)d𝑡1d𝑡2 . (1.3)

This made possible a quaternion Fourier transform of a one-dimensional signal:

ℱ(𝜔) =∞∫

−∞𝑒𝜇𝜔𝑡𝑓(𝑡)d𝑡 . (1.4)

Such a transform makes sense only if the signal function has quaternion values,suggesting applications where the signal has three or four independent components.(An example is vibrations in a solid, such as rock, detected by a sensor with threemutually orthogonal transducers, such as a vector geophone.)

Very little has appeared in print about the interpretation of the Fourier coef-ficients resulting from a quaternion Fourier transform. One interpretation is com-ponents of different symmetry, as explained by Ell in Chapter 1. Sangwine and Ellin 2007 published a paper about quaternion Fourier transforms applied to colourimages, with a detailed explanation of the Fourier coefficients in terms of ellipticalpaths in colour space (the 𝑛-dimensional space of the values of the image pixels ina colour image) [48].

1.2. Splitting Quaternions and the QFT

Following the earlier works of Ernst, Ell, Sangwine (see Section 1.1), and Bulow[24, 25], Hitzer thoroughly studied the quaternion Fourier transform (QFT) appliedto quaternion-valued functions in [54]. As part of this work a quaternion split

𝑞± =1

2(𝑞 ± 𝒊𝑞𝒋), 𝑞 ∈ ℍ, (1.5)

was devised and applied, which led to a better understanding of 𝐺𝐿(ℝ2) trans-formation properties of the QFT spectrum of two-dimensional images, includingcolour images, and opened the way to a generalization of the QFT concept to afull spacetime Fourier transformation (SFT) for spacetime algebra 𝐶ℓ3,1-valuedsignals.

This was followed up by the establishment of a fully directional (opposedto componentwise) uncertainty principle for the QFT and the SFT [58]. Indepen-dently Mawardi et al. [77] established a componentwise uncertainty principle forthe QFT.

The QFT with a Gabor window was treated by Bulow [24], a study whichhas been continued by Mawardi et al. in [1].

Hitzer reports in [59] initial results (obtained in co-operation with Sangwine)about a further generalization of the QFT to a general form of orthogonal 2Dplanes split (OPS-) QFT, where the split (1.5) with respect to two orthogonalpure quaternion units 𝒊, 𝒋 is generalized to a steerable split with respect to any two

xiv F. Brackx, E. Hitzer and S.J. Sangwine

pure unit quaternions 𝑓, 𝑔 ∈ ℍ, 𝑓2 = 𝑔2 = −1. This approach is fully elaboratedupon in a contribution to the current volume (see Chapter 2). Note that theCayley–Dickson form [87] of quaternions and the related simplex/perplex split[47] are obtained for 𝑓 = 𝑔 = 𝒊 (or more general 𝑓 = 𝑔 = 𝜇), which is employed inChapter 3 for a novel spectral analysis of non-stationary improper complex signals.

2. Clifford–Fourier Transformations in Clifford’sGeometric Algebra

W.K. Clifford introduced (Clifford) geometric algebras in 1876 [28]. An introduc-tion to the vector and multivector calculus, with functions taking values in Cliffordalgebras, used in the field of Clifford–Fourier transforms (CFT) can be found in[53, 52]. A tutorial introduction to CFTs and Clifford wavelet transforms can befound in [55]. The Clifford algebra application survey [65] contains an up to datesection on applications of Clifford algebra integral transforms, including CFTs,QFTs and wavelet transforms4.

2.1. How Clifford Algebra Square Roots of −1 Lead to Clifford–FourierTransformations

In 1990 Jancewicz defined a trivector Fourier transformation

ℱ3{𝑔}(𝝎) =∫ℝ3

𝑔(x)𝑒−𝑖3x⋅𝝎𝑑3x, 𝑖3 = 𝒆1𝒆2𝒆3, 𝑔 : ℝ3 → 𝐶ℓ3,0, (2.1)

for the electromagnetic field5 replacing the imaginary unit 𝑖 ∈ ℂ by the centraltrivector 𝑖3, 𝑖

23 = −1, of the geometric algebra 𝐶ℓ3,0 of three-dimensional Euclidean

space ℝ3 = ℝ3,0 with orthonormal vector basis {𝒆1, 𝒆2, 𝒆3}.In [50] Felsberg makes use of signal embeddings in low-dimensional Clifford

algebras ℝ2,0 and ℝ3,0 to define his Clifford–Fourier transform (CFT) for one-dimensional signals as

ℱ𝑓𝑒1 [𝑓 ](𝑢) =

∫ℝ

exp (−2𝜋𝑖2𝑢𝑥) 𝑓(𝑥) 𝑑𝑥, 𝑖2 = 𝒆1𝒆2, 𝑓 : ℝ→ ℝ, (2.2)

where he uses the pseudoscalar 𝑖2 ∈ 𝐶ℓ2,0, 𝑖22 = −1. For two-dimensional signals6

he defines the CFT as

ℱ𝑓𝑒2 [𝑓 ](𝑢) =

∫ℝ2

exp (−2𝜋𝑖3 < 𝑢, 𝑥 >) 𝑓(𝑥) 𝑑𝑥, 𝑓 : ℝ2 → ℝ2, (2.3)

4Fourier and wavelet transforms provide alternative signal and image representations. See Chap-

ter 9 for a spinorial representation and Chapter 16 by Li and Qian for a sparse representation ofsignals in a Hardy space dictionary (of elementary wave forms) over a unit disk.5Note also Chapter 11 in this volume, in which Bernstein considers optical coherence tomography,formulating the Maxwell equations with the Dirac operator and Clifford algebra.6Note in this context the spinor representation of images by Batard and Berthier in Chapter 9of this volume. The authors apply a CFT in 𝐶ℓ3,0 to the spinor represenation, which uses inthe exponential kernel an adapted choice of bivector, that belongs to the orthonormal frame

of the tangent bundle of an oriented two-dimensional Riemannian manifold, isometrically im-mersed in ℝ3.

Quaternion, Clifford–Fourier & Wavelet Transforms History xv

where he uses the pseudoscalar 𝑖3 ∈ 𝐶ℓ3,0. It is used amongst others to introducea concept of two-dimensional analytic signal. Together with Bulow and Sommer,Felsberg applied these CFTs to image stucture processing (key-notion: structuremultivector) [50, 24].

Ebling and Scheuermann [44, 43] consequently applied to vector signal pro-cessing in two- and three dimensions, respectively, the following two-dimensionalCFT

ℱ2{𝑓}(𝝎) =∫ℝ2

𝑓(x)𝑒−𝑖2x⋅𝝎𝑑2x, 𝑓 : ℝ2 → ℝ2, (2.4)

with Clifford–Fourier kernel

exp (−𝒆1𝒆2(𝜔1𝑥1 + 𝜔2𝑥2)), (2.5)

and the three-dimensional CFT (2.1) of Jancewicz with Clifford–Fourier kernel

exp (−𝒆1𝒆2𝒆3(𝜔1𝑥1 + 𝜔2𝑥2 + 𝜔3𝑥3)). (2.6)

An important integral operation defined and applied in this context by Eblingand Scheuermann was the Clifford convolution. These Clifford–Fourier transformsand the corresponding convolution theorems allow Ebling and Scheuermann foramongst others the analysis of vector-valued patterns in the frequency domain.

Note that the latter Fourier kernel (2.6) has also been used by Mawardi andHitzer in [78, 63, 78] to define their Clifford–Fourier transform of three-dimensionalmultivector signals: that means, they researched the properties of ℱ3{𝑔}(𝝎) of(2.1) in detail when applied to full multivector signals 𝑔 : ℝ3 → 𝐶ℓ3,0. This includedan investigation of the uncertainty inequality for this type of CFT. They subse-quently generalized ℱ3{𝑔}(𝝎) to dimensions 𝑛 = 3(mod 4), i.e., 𝑛 = 3, 7, 11, . . .,

ℱ𝑛{𝑔}(𝝎) =∫ℝ𝑛

𝑔(x)𝑒−𝑖𝑛x⋅𝝎𝑑𝑛x, 𝑔 : ℝ𝑛 → 𝐶ℓ𝑛,0, (2.7)

which is straightforward, since for these dimensions the pseudoscalar 𝑖𝑛 = 𝒆1 . . . 𝒆𝑛squares to −1 and is central [64], i.e., it commutes with every other multivectorbelonging to 𝐶ℓ𝑛,0. A little less trivial is the generalization of ℱ2{𝑓}(𝝎) of (2.4) to

ℱ𝑛{𝑓}(𝝎) =∫ℝ𝑛

𝑓(x)𝑒−𝑖𝑛x⋅𝝎𝑑𝑛x, 𝑓 : ℝ𝑛 → 𝐶ℓ𝑛,0, (2.8)

with 𝑛 = 2(mod 4), i.e., 𝑛 = 2, 6, 10 . . ., because in these dimensions the pseu-doscalar 𝑖𝑛 = 𝒆1 . . .𝒆𝑛 squares to −1, but it ceases to be central. So the relativeorder of the factors in ℱ𝑛{𝑓}(𝝎) becomes important, see [66] for a systematicinvestigation and comparison.

In the context of generalizing quaternion Fourier transforms (QFT) via alge-bra isomorphisms to higher-dimensional Clifford algebras, Hitzer [54] constructeda spacetime Fourier transform (SFT) in the full algebra of spacetime 𝐶ℓ3,1, whichincludes the CFT (2.1) as a partial transform of space. Implemented analogously(isomorphicaly) to the orthogonal 2D planes split of quaternions, the SFT permitsa natural spacetime split, which algebraically splits the SFT into right and leftpropagating multivector wave packets. This analysis allows to compute the effect

xvi F. Brackx, E. Hitzer and S.J. Sangwine

of Lorentz transformations on the spectra of these wavepackets, as well as a 4Ddirectional spacetime uncertainty formula [58] for spacetime signals.

Mawardi et al. extended the CFT ℱ2{𝑓}(𝝎) of (2.4) to a windowed CFT in[76]. Fu et al. establish in Chapter 15 a strong version of Heisenberg’s uncertaintyprinciple for Gabor-windowed CFTs.

In Chapter 8 in this volume, Bujack, Scheuermann, and Hitzer, expand thenotion of Clifford–Fourier transform to include multiple left and right exponentialkernel factors, in which commuting (or anticommuting) blades, that square to−1, replace the complex unit 𝑖 ∈ ℂ, thus managing to include most practicallyused CFTs in a single comprehensive framework. Based on this they have alsoconstructed a general CFT convolution theorem [23].

Spurred by the systematic investigation of (complex quaternion) biquater-nion square roots of −1 in 𝐶ℓ3,0 by Sangwine [85], Hitzer and Ablamowicz [62]systematically investigated the explicit equations and solutions for square roots of−1 in all real Clifford algebras 𝐶ℓ𝑝,𝑞, 𝑝+ 𝑞 ≤ 4. This investigation is continued inthe present volume in Chapter 7 by Hitzer, Helmstetter and Ablamowicz for allsquare roots of −1 in all real Clifford algebras 𝐶ℓ𝑝,𝑞 without restricting the valueof 𝑛 = 𝑝 + 𝑞. One important motivation for this is the relevance of the Cliffordalgebra square roots of −1 for the general construction of CFTs, where the imagi-nary unit 𝑖 ∈ ℂ is replaced by a

√−1 ∈ 𝐶ℓ𝑝,𝑞, without restriction to pseudoscalarsor blades.

Based on the knowledge of square roots of −1 in real Clifford algebras 𝐶ℓ𝑝,𝑞,[60] develops a general CFT in 𝐶ℓ𝑝,𝑞, wherein the complex unit 𝑖 ∈ ℂ is replacedby any square root of −1 chosen from any component and (or) conjugation classof the submanifold of square roots of −1 in 𝐶ℓ𝑝,𝑞, and details its properties, in-cluding a convolution theorem. A similar general approach is taken in [61] for theconstruction of two-sided CFTs in real Clifford algebras 𝐶ℓ𝑝,𝑞, freely choosing twosquare roots from any one or two components and (or) conjugation classes of thesubmanifold of square roots of −1 in 𝐶ℓ𝑝,𝑞. These transformations are thereforegenerically steerable.

This algebraically motivated approach may in the future be favorably com-bined with group theoretic, operator theoretic and spinorial approaches, to bediscussed in the following.

2.2. The Clifford–Fourier Transform in the Light of Clifford Analysis

Two robust tools used in image processing and computer vision for the analysisof scalar fields are convolution and Fourier transformation. Several attempts havebeen made to extend these methods to two- and three-dimensional vector fieldsand even multi-vector fields. Let us give an overview of those generalized Fouriertransforms.

In [25] Bulow and Sommer define a so-called quaternionic Fourier transformof two-dimensional signals 𝑓(𝑥1, 𝑥2) taking their values in the algebra ℍ of realquaternions. Note that the quaternion algebra ℍ is nothing else but (isomorphicto) the Clifford algebra 𝐶ℓ0,2 where, traditionally, the basis vectors are denoted

Quaternion, Clifford–Fourier & Wavelet Transforms History xvii

by 𝒊 and 𝒋, with 𝒊2 = 𝒋2 = −1, and the bivector by 𝒌 = 𝒊𝒋. In terms of these basisvectors this quaternionic Fourier transform takes the form

ℱ𝑞[𝑓 ](𝑢1, 𝑢2) =

∫ℝ2

exp (−2𝜋𝒊𝑢1𝑥1) 𝑓(𝑥1, 𝑥2) exp (−2𝜋𝒋𝑢2𝑥2) 𝑑𝑥. (2.9)

Due to the non-commutativity of the multiplication in ℍ, the convolution theoremfor this quaternionic Fourier transform is rather complicated, see also [23].

This is also the case for its higher-dimensional analogue, the so-called Clif-ford–Fourier transform7 in 𝐶ℓ0,𝑚 given by

ℱ𝑐𝑙[𝑓 ](𝑢) =

∫ℝ𝑚

𝑓(𝑥) exp (−2𝜋𝒆1𝑢1𝑥1) . . . exp (−2𝜋𝒆𝑚𝑢𝑚𝑥𝑚) 𝑑𝑥. (2.10)

Note that for 𝑚 = 1 and interpreting the Clifford basis vector 𝒆1 as the imaginaryunit 𝑖, the Clifford–Fourier transform (2.10) reduces to the standard Fourier trans-form on the real line, while for 𝑚 = 2 the quaternionic Fourier transform (2.9) isrecovered when restricting to real signals.

Finally Bulow and Sommer also introduce a so-called commutative hyper-complex Fourier transform given by

ℱℎ[𝑓 ](𝑢) =

∫ℝ𝑚

𝑓(𝑥) exp(−2𝜋∑𝑚

𝑗=1��𝑗𝑢𝑗𝑥𝑗

)𝑑𝑥 (2.11)

where the basis vectors (��1, . . . , ��𝑚) obey the commutative multiplication rules��𝑗 ��𝑘 = ��𝑘��𝑗 , 𝑗, 𝑘 = 1, . . . ,𝑚, while still retaining ��2

𝑗 = −1, 𝑗 = 1, . . . ,𝑚. Thiscommutative hypercomplex Fourier transform offers the advantage of a simpleconvolution theorem.

The hypercomplex Fourier transforms ℱ𝑞, ℱ𝑐𝑙 and ℱℎ enable Bulow andSommer to establish a theory of multi-dimensional signal analysis and in partic-ular to introduce the notions of multi-dimensional analytic signal8, Gabor filter,instantaneous and local amplitude and phase, etc.

In this context the Clifford–Fourier transformations by Felsberg [50] for one-and two-dimensional signals, by Ebling and Scheuermann for two- and three-dimensional vector signal processing [44, 43], and by Mawardi and Hitzer for gen-eral multivector signals in 𝐶ℓ3,0 [78, 63, 78], and their respective kernels, as alreadyreviewed in Section 2.1, should also be considered.

The above-mentioned Clifford–Fourier kernel of Bulow and Sommer

exp (−2𝜋𝒆1𝑢1𝑥1) ⋅ ⋅ ⋅ exp (−2𝜋𝒆𝑚𝑢𝑚𝑥𝑚) (2.12)

was in fact already introduced in [19] and [89] as a theoretical concept in theframework of Clifford analysis. This generalized Fourier transform was furtherelaborated by Sommen in [90, 91] in connection with similar generalizations ofthe Cauchy, Hilbert and Laplace transforms. In this context also the work of Li,McIntosh and Qian should be mentioned; in [72] they generalize the standard

7Note that in this volume Mawardi establishes in Chapter 14 a windowed version of the CFT

(2.10).8See also Chapter 10 by Girard et al. and Chapter 11 by Bernstein et al. in this volume.

xviii F. Brackx, E. Hitzer and S.J. Sangwine

multi-dimensional Fourier transform of a function in ℝ𝑚, by extending the Fourierkernel exp

(𝑖⟨𝜉, 𝑥⟩)

to a function which is holomorphic in ℂ𝑚 and monogenic9

in ℝ𝑚+1.In [15, 16, 18] Brackx, De Schepper and Sommen follow another philoso-

phy in their construction of a Clifford–Fourier transform. One of the most funda-mental features of Clifford analysis is the factorization of the Laplace operator.Indeed, whereas in general the square root of the Laplace operator is only a pseudo-differential operator, by embedding Euclidean space into a Clifford algebra, onecan realize

√−Δ𝑚 as the Dirac operator ∂𝑥. In this way Clifford analysis sponta-neously refines harmonic analysis. In the same order of ideas, Brackx et al. decidedto not replace nor to improve the classical Fourier transform by a Clifford analysisalternative, since a refinement of it automatically appears within the language ofClifford analysis. The key step to making this refinement apparent is to interpretthe standard Fourier transform as an operator exponential:

ℱ = exp(−𝑖

𝜋

2ℋ)=

∞∑𝑘=0

1

𝑘!

(−𝑖

𝜋

2

)𝑘ℋ𝑘 , (2.13)

where ℋ is the scalar operator

ℋ =1

2

(−Δ𝑚 + 𝑟2 −𝑚). (2.14)

This expression links the Fourier transform with the Lie algebra 𝔰𝔩2 generated

by Δ𝑚 and 𝑟2 = ∣𝑥∣2 and with the theory of the quantum harmonic oscillatordetermined by the Hamiltonian − 1

2

(Δ𝑚 − 𝑟2

). Splitting the operator ℋ into a

sum of Clifford algebra-valued second-order operators containing the angular Diracoperator Γ, one is led, in a natural way, to a pair of transforms ℱℋ± , the harmonicaverage of which is precisely the standard Fourier transform:

ℱℋ± = exp

(𝑖𝜋𝑚

4

)exp

(∓ 𝑖𝜋Γ

2

)exp

(𝑖𝜋

4

(Δ𝑚 − 𝑟2

)). (2.15)

For the special case of dimension two, Brackx et al. obtain a closed form forthe kernel of the integral representation of this Clifford–Fourier transform leadingto its internal representation

ℱℋ± [𝑓 ](𝜉) = ℱℋ± [𝑓 ](𝜉1, 𝜉2) =1

2𝜋

∫ℝ2

exp(±𝒆12(𝜉1𝑥2 − 𝜉2𝑥1)

)𝑓(𝑥) 𝑑𝑥 , (2.16)

which enables the generalization of the calculation rules for the standard Fouriertransform both in the 𝐿1 and in the 𝐿2 context. Moreover, the Clifford–Fouriertransform of Ebling and Scheuermann

ℱ𝑒[𝑓 ](𝜉) =

∫ℝ2

exp (−𝒆12(𝑥1𝜉1 + 𝑥2𝜉2)) 𝑓(𝑥) 𝑑𝑥 , (2.17)

9See also in this volume Chapter 4 by Moya-Sanchez and Bayro-Corrochano on the applicationof atomic function based monogenic signals.

Quaternion, Clifford–Fourier & Wavelet Transforms History xix

can be expressed in terms of the Clifford–Fourier transform:

ℱ𝑒[𝑓 ](𝜉) = 2𝜋ℱℋ± [𝑓 ](∓𝜉2,±𝜉1) = 2𝜋ℱℋ± [𝑓 ](±𝒆12𝜉) , (2.18)

taking into account that, under the isomorphism between the Clifford algebras𝐶ℓ2,0 and 𝐶ℓ0,2, both pseudoscalars are isomorphic images of each other.

The question whether ℱℋ± can be written as an integral transform is an-swered positively in the case of even dimension by De Bie and Xu in [39]. Theintegral kernel of this transform is not easy to obtain and looks quite complicated.In the case of odd dimension the problem is still open.

Recently, in [35], De Bie and De Schepper have studied the fractional Clifford–Fourier transform as a generalization of both the standard fractional Fourier trans-form and the Clifford–Fourier transform. It is given as an operator exponential by

ℱ𝛼,𝛽 = exp

(𝑖𝛼𝑚

2

)exp (𝑖𝛽Γ) exp

(𝑖𝛼

2

(Δ𝑚 − 𝑟2

)). (2.19)

For the corresponding integral kernel a series expansion is obtained, and, in thecase of dimension two, an explicit expression in terms of Bessel functions.

The above, more or less chronological, overview of generalized Fourier trans-forms in the framework of quaternionic and Clifford analysis, gives the impressionof a medley of ad hoc constructions. However there is a structure behind some ofthese generalizations, which becomes apparent when, as already slightly touchedupon above, the Fourier transform is linked to group representation theory, inparticular the Lie algebras 𝔰𝔩2 and 𝔬𝔰𝔭(1∣2). This unifying character is beautifullydemonstrated by De Bie in the overview paper [34], where, next to an extensivebibliography, also new results on some of the transformations mentioned belowcan be found. It is shown that using realizations of the Lie algebra 𝔰𝔩2 one is leadto scalar generalizations of the Fourier transform, such as:

(i) the fractional Fourier transform, which is, as the standard Fourier transform,invariant under the orthogonal group; this transform has been reinventedseveral times as well in mathematics as in physics, and is attributed to Namias[81], Condon [30], Bargmann [2], Collins [29], Moshinsky and Quesne [80];for a detailed overview of the theory and recent applications of the fractionalFourier transform we refer the reader to [82];

(ii) the Dunkl transform, see, e.g., [42], where the symmetry is reduced to thatof a finite reflection group;

(iii) the radially deformed Fourier transform, see, e.g., [71], which encompassesboth the fractional Fourier and the Dunkl transform;

(iv) the super Fourier transform, see, e.g., [33, 31], which is defined in the contextof superspaces and is invariant under the product of the orthogonal with thesymplectic group.

Realizations of the Lie algebra 𝔬𝔰𝔭(1∣2), on the contrary, need the framework ofClifford analysis, and lead to:

xx F. Brackx, E. Hitzer and S.J. Sangwine

(v) the Clifford–Fourier transform and the fractional Clifford–Fourier transform,both already mentioned above; meanwhile an entire class of Clifford–Fouriertransforms has been thoroughly studied in [36];

(vi) the radially deformed hypercomplex Fourier transform, which appears as aspecial case in the theory of radial deformations of the Lie algebra 𝔬𝔰𝔭(1∣2),see [38, 37], and is a topic of current research, see [32].

3. Quaternion and Clifford Wavelets

3.1. Clifford Wavelets in Clifford Analysis

The interest of the Ghent Clifford Research Group for generalizations of the Fouriertransform in the framework of Clifford analysis, grew out from the study of the mul-tidimensional Continuous Wavelet Transform in this particular setting. Clifford-wavelet theory, however restricted to the continuous wavelet transform, was initi-ated by Brackx and Sommen in [20] and further developed by N. De Schepper in herPhD thesis [40]. The Clifford-wavelets originate from a mother wavelet not only bytranslation and dilation, but also by rotation, making the Clifford-wavelets appro-priate for detecting directional phenomena. Rotations are implemented as specificactions on the variable by a spin element, since, indeed, the special orthogonalgroup SO(𝑚) is doubly covered by the spin group Spin(𝑚) of the real Cliffordalgebra 𝐶ℓ0,𝑚. The mother wavelets themselves are derived from intentionally de-vised orthogonal polynomials in Euclidean space. It should be noted that theseorthogonal polynomials are not tensor products of one-dimensional ones, but gen-uine multidimensional ones satisfying the usual properties such as a Rodriguesformula, recurrence relations, and differential equations. In this way multidimen-sional Clifford wavelets were constructed grafted on the Hermite polynomials [21],Laguerre polynomials [14], Gegenbauer polynomials [13], Jacobi polynomials [17],and Bessel functions [22].

Taking the dimension 𝑚 to be even, say𝑚 = 2𝑛, introducing a complex struc-ture, i.e., an SO(2𝑛)-element squaring up to −1, and considering functions withvalues in the complex Clifford algebra ℂ2𝑛, so-called Hermitian Clifford analysisoriginates as a refinement of standard or Euclidean Clifford analysis. It should benoticed that the traditional holomorphic functions of several complex variables ap-pear as a special case of Hermitian Clifford analysis, when the function values arerestricted to a specific homogeneous part of spinor space. In this Hermitian settingthe standard Dirac operator, which is invariant under the orthogonal group O(𝑚),is split into two Hermitian Dirac operators, which are now invariant under theunitary group U(𝑛). Also in this Hermitian Clifford analysis framework, multidi-mensional wavelets have been introduced by Brackx, H. De Schepper and Sommen[11, 12], as kernels for a Hermitian Continuous Wavelet Transform, and (general-ized) Hermitian Clifford–Hermite polynomials have been devised to generate thecorresponding Hermitian wavelets [9, 10].

Quaternion, Clifford–Fourier & Wavelet Transforms History xxi

3.2. Further Developments in Quaternion and Clifford Wavelet Theory

Clifford algebra multiresolution analysis (MRA) has been pioneered by M. Mitrea[79]. Important are also the electromagnetic signal application oriented develop-ments of Clifford algebra wavelets by G. Kaiser [70, 67, 68, 69].

Quaternion MRA Wavelets with applications to image analysis have beendeveloped in [92] by Traversoni. Clifford algebra multiresolution analysis has beenapplied by Bayro-Corrochano [5, 3, 4] to: Clifford wavelet neural networks (infor-mation processing), also considering quaternionic MRA, a quaternionic waveletphase concept, as well as applications to (e.g., robotic) motion estimation andimage processing.

Beyond this Zhao and Peng [94] established a theory of quaternion-valuedadmissible wavelets. Zhao [93] studied Clifford algebra-valued admissible (continu-ous) wavelets using the complex Fourier transform for the spectral representation.Mawardi and Hitzer [74, 75] extended this to continuous Clifford and Clifford–Gabor wavelets in 𝐶ℓ3,0 using the CFT of (2.1) for the spectral representation.They also studied a corresponding Clifford wavelet transform uncertainty prin-ciple. Hitzer [56, 57] generalized this approach to continous admissible Cliffordalgebra wavelets in real Clifford algebras 𝐶ℓ𝑛,0 of dimensions 𝑛 = 2, 3(mod 4),i.e., 𝑛 = 2, 3, 6, 7, 10, 11, . . .. Restricted to 𝐶ℓ𝑛,0 of dimensions 𝑛 = 2(mod 4) thisapproach has also been taken up in [73].

Kahler et al. [26] treated monogenic (Clifford) wavelets over the unit ball.Bernstein studied Clifford continuous wavelet transforms in 𝐿0,2 and 𝐿0,3 [6], aswell as monogenic kernels and wavelets on the three-dimensional sphere [7]. Bern-stein et al. [8] further studied Clifford diffusion wavelets on conformally flat cylin-ders and tori. In the current volume Soulard and Carre extend in Chapter 12 thetheory and application of monogenic wavelets to colour image denoising.

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[46] T.A. Ell. Quaternion-Fourier transforms for analysis of 2-dimensional linear time-invariant partial-differential systems. In Proceedings of the 32nd Conference on De-cision and Control, pages 1830–1841, San Antonio, Texas, USA, 15–17 December1993. IEEE Control Systems Society.

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[60] E. Hitzer. The Clifford Fourier transform in real Clifford algebras. In K. Guerlebeck,T. Lahmer, and F. Werner, editors, Proceedings 19th International Conference onthe Application of Computer Science and Mathematics in Architecture and CivilEngineering, Weimar, Germany, 4–6 July 2012.

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[84] S.J. Sangwine. Fourier transforms of colour images using quaternion, or hypercom-plex, numbers. Electronics Letters, 32(21):1979–1980, 10 Oct. 1996.

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[86] S.J. Sangwine and T.A. Ell. The discrete Fourier transform of a colour image. In J.M.Blackledge and M.J. Turner, editors, Image Processing II Mathematical Methods, Al-gorithms and Applications, pages 430–441, Chichester, 2000. Horwood Publishing forInstitute of Mathematics and its Applications. Proceedings Second IMA Conferenceon Image Processing, De Montfort University, Leicester, UK, September 1998.

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[88] T.E. Simos, G. Psihoyios, and C. Tsitouras, editors. NUMERICAL ANALYSIS ANDAPPLIED MATHEMATICS: International Conference on Numerical Analysis andApplied Mathematics 2009, volume 1168 of AIP Conference Proceedings, Rethymno,Crete (Greece), 18–22 September 2009.

[89] F. Sommen. A product and an exponential function in hypercomplex function theory.Applicable Analysis, 12:13–26, 1981.

[90] F. Sommen. Hypercomplex Fourier and Laplace transforms I. Illinois Journal ofMathematics, 26(2):332–352, 1982.

[91] F. Sommen. Hypercomplex Fourier and Laplace transforms II. Complex Variables,1(2–3):209–238, 1983.

[92] L. Traversoni. Quaternion wavelet problems. In Proceedings of 8th International Sym-posium on Approximation Theory, Texas A& M University, Jan. 1995.

[93] J. Zhao. Clifford algebra-valued admissible wavelets associated with admissiblegroup. Acta Scientarium Naturalium Universitatis Pekinensis, 41(5):667–670, 2005.

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Fred BrackxDepartment of Mathematical AnalysisFaculty of EngineeringGhent UniversityGalglaan 2B-9000 Gent, Belgiume-mail: Freddy.Brackx@ugent.be

Eckhard HitzerCollege of Liberal ArtsDepartment of Material ScienceInternational Christian University181-8585 Tokyo, Japane-mail: hitzer@icu.ac.jp

Stephen J. SangwineSchool of Computer Science

and Electronic EngineeringUniversity of EssexWivenhoe ParkColchester, CO4 3SQ, UKe-mail: sjs@essex.ac.uk

Part I

Quaternions

Quaternion and Clifford–Fourier Transforms and Wavelets

Trends in Mathematics, 3–14c⃝ 2013 Springer Basel

1 Quaternion Fourier Transform: Re-toolingImage and Signal Processing Analysis

Todd Anthony Ell

‘Did you ask a good question today?’ – Janet Teig

Abstract. Quaternion Fourier transforms (QFT’s) provide expressive powerand elegance in the analysis of higher-dimensional linear invariant systems.But, this power comes at a cost – an overwhelming number of choices in theQFT definition, each with consequences. This chapter explores the evolutionof QFT definitions as a framework from which to solve specific problems invector-image and vector-signal processing.

Mathematics Subject Classification (2010). Primary 11R52; secondary 42B10.

Keywords. Quaternion, Fourier transform.

1. Introduction

In recent years there has been an increasing recognition on the part of engineersand investigators in image and signal processing of holistic vector approaches tospectral analysis. Generally speaking, this type of spectral analysis treats the vec-tor components of a system not in an iterated, channel-wise fashion but insteadin a holistic, gestalt fashion. The Quaternion Fourier transform (QFT) is one suchanalysis tool.

One of the earliest documented attempts (1987) at describing this type ofspectral analysis was in the area of two-dimensional nuclear magnet resonance.Ernst, et al. [6, pp. 307–308] briefly discusses using a hypercomplex Fourier trans-form as a method to independently adjust phase angles with respect to two fre-quency variables in two-dimensional spectroscopy. After introducing the conceptthey immediately fall back to an iterated approach leaving the idea unexplored.For similar reasons, Ell [2] in 1992 independently explored the use of QFTs as atool in the analysis of linear time-invariant systems of partial differential equations(PDEs). Ell specifically ‘designed’ a quaternion Fourier transform whose spectral

4 T.A. Ell

operators allowed him to disambiguate partial derivatives with respect to two dif-ferent independent variables. Ell’s original QFT was given by

𝐻 [𝒋𝜔,𝒌𝜈] =

∫ℝ2

𝑒−𝒋𝜔𝑡ℎ (𝑡, 𝜏) 𝑒−𝒌𝜈𝜏𝑑𝑡𝑑𝜏 , (1.1)

where 𝐻 [𝒋𝜔,𝒌𝜈] ∈ ℍ (the set of quaternions), 𝒋 and 𝒌 are Hamilton’s hypercom-plex operators, and ℎ (𝑡, 𝜏) : ℝ× ℝ→ ℝ (the set of reals). The partial-differentialequivalent spectral operators for this transform are given by

∂

∂𝑡ℎ (𝑡, 𝜏)⇔ 𝒋𝜔𝐻 [𝒋𝜔,𝒌𝜈] ,

∂

∂𝜏ℎ (𝑡, 𝜏)⇔ 𝐻 [𝒋𝜔,𝒌𝜈]𝒌𝜈. (1.2)

These two differentials have clearly different spectral signatures in contrast to thetwo-dimensional iterated complex Fourier transform where

∂

∂𝑡ℎ (𝑡, 𝜏 )⇔ 𝒋𝜔𝐻 [𝒋𝜔, 𝒋𝜈] ,

∂

∂𝜏ℎ (𝑡, 𝜏 )⇔ 𝒋𝜈𝐻 [𝒋𝜔, 𝒋𝜈] , (1.3)

especially when 𝜔 = 𝜈, at which point the complex spectral domain responses areindistinguishable. This was the first step towards stability analysis in designingcontrollers for systems described by PDEs.

The slow adoption of QFTs at the present time by the engineering communityis due in part to their lack of practical understanding of its properties. This slowadoption is further exacerbated by the variety of transform definitions available.But, in the middle of difficulty lies opportunity. Instead of attempting to find thesingle best QFT (which cannot meet every design engineer’s needs) we provideinstead the means to allow the designer to select the definition most appropriateto his specific problem. That means, allow him to re-tool for the analysis problemat hand.

For example, when QFTs were later applied to colour-image processing [4],where each colour pixel in an image is treated as a 3-vector with basis {𝒊, 𝒋,𝒌} ∈ ℍ,it became apparent that there was no preferential association of colour-space axeswith either the basis or the QFT’s exponential-kernel axis. This lead to the nextgeneralization of the QFT defined as

ℱ+ [𝜔, 𝜈] =

∫ℝ2

𝑒−𝝁(𝜔𝑡+𝜈𝜏)𝑓 (𝑡, 𝜏 ) 𝑑𝑡𝑑𝜏 , (1.4)

where the transform kernel axis 𝝁 is any pure unit quaternion, i.e.,

𝝁 ∈ {𝒊𝑥+ 𝒋𝑦 + 𝒌𝑧 ∈ ℍ ∣ 𝑥2 + 𝑦2 + 𝑧2 = 1}

so that 𝝁2 = −1. Still later it was realized [9] that since there is no preferreddirection of indexing the image’s pixels then the sign on the transform kernel isalso arbitrary, so that a forward QFT could also be defined as

ℱ− [𝜔, 𝜈] =

∫ℝ2

𝑒+𝝁(𝜔𝑡+𝜈𝜏)𝑓 (𝑡, 𝜏 ) 𝑑𝑡𝑑𝜏 , (1.5)

and the two definitions could be intermixed without concern of creating a non-causal set of image processing filters. This led to several simplifications of a spectralform of the vector correlation operation on two images [8].

1. Quaternion Fourier Transform 5

Bearing in mind such diverse application of various QFTs, the focus of thiswork is to detail as broad a set of QFT definitions as possible, and where known,some of the issues associated with applying them to problems in signal and im-age processing. It also includes a review of approaches taken to define the inter-relations between the various QFT definitions.

2. Preliminaries

To provide a basis for discussion this section gives nomenclature, basic facts onquaternions, and some useful subsets and algebraic equations.

2.1. Just the Facts

The quaternion algebra over the reals ℝ, denoted by

ℍ = {𝑞 = 𝑟0 + 𝒊𝑟1 + 𝒋𝑟2 + 𝒌𝑟3 ∣ 𝑟0, 𝑟1, 𝑟2, 𝑟3 ∈ ℝ} , (2.1)

is an associative non-commutative four-dimensional algebra, which obeys Hamil-ton’s multiplication rules

𝒊𝒋 = 𝒌 = −𝒋𝒌, 𝒋𝒌 = 𝒊 = −𝒌𝒋, 𝒌𝒊 = 𝒋 = −𝒊𝒌, (2.2)

𝒊2 = 𝒋2 = 𝒌2 = 𝒊𝒋𝒌 = −1. (2.3)

The quaternion conjugate is defined by

𝑞 = 𝑟0 − 𝒊𝑟1 − 𝒋𝑟2 − 𝒌𝑟3 , (2.4)

which is an anti-involution, i.e., 𝑞 = 𝑞, 𝑝+ 𝑞 = 𝑝 + 𝑞, and 𝑞𝑝 = 𝑝 𝑞. The norm ofa quaternion is defined as

∣𝑞∣ = √𝑞𝑞 =√

𝑟20 + 𝑟21 + 𝑟22 + 𝑟23 . (2.5)

Using the conjugate and norm of 𝑞, one can define the inverse of 𝑞 ∈ ℍ ∖ {0} as

𝑞−1 = 𝑞/∣𝑞∣2. (2.6)

Two classical operators on quaternions are the vector- and scalar-part, 𝑉 [.] and𝑆[.], respectively; these are defined as

𝑉 [𝑞] = 𝒊𝑟1 + 𝒋𝑟2 + 𝒌𝑟3, 𝑆 [𝑞] = 𝑟0. (2.7)

2.2. Useful Subsets

Various subsets of the quaternions are of interest and used repeatedly throughoutthis work. The 3-vector subset of ℍ is the set of pure quaternions defined as

𝑉 [ℍ] = {𝑞 = 𝒊𝑟1 + 𝒋𝑟2 + 𝒌𝑟3 ∈ ℍ } . (2.8)

The set of pure, unit length quaternions is denoted 𝕊3ℍ, i.e.,

𝕊3ℍ =

{𝝁 = 𝒊𝑟1 + 𝒋𝑟2 + 𝒌𝑟3 ∈ ℍ ∣ 𝑟21 + 𝑟22 + 𝑟23 = 1

}. (2.9)

6 T.A. Ell

Each element of 𝕊3ℍcreates a distinct copy of the complex numbers because 𝝁2 =

−1, that is, each creates an injective ring homomorphism from ℂ to ℍ. So, foreach 𝝁 ∈ 𝕊3

ℍ, we associate a complex sub-field of ℍ denoted

ℂ𝝁 ={𝛼+ 𝛽𝝁; ∣ 𝛼, 𝛽 ∈ ℝ,𝝁 ∈ 𝕊3

ℍ

}. (2.10)

2.3. Useful Algebraic Equations

In various quaternion equations the non-commutativity of the multiplication causesdifficulty, however, there are algebraic forms which assist in making simplifications.The following three defined forms appear to be the most useful.

Definition 2.1 (Even-Odd Form). Every 𝑓 : ℝ2 → ℍ can be split into even andodd parts along the 𝑥- and 𝑦-axis as

𝑓 (𝑥, 𝑦) = 𝑓ee (𝑥, 𝑦) + 𝑓eo (𝑥, 𝑦) + 𝑓oe (𝑥, 𝑦) + 𝑓oo (𝑥, 𝑦) (2.11)

where 𝑓eo denotes the part of 𝑓 that is even with respect to 𝑥 and odd with respectto 𝑦, etc., given as

𝑓ee (𝑥, 𝑦) =14 (𝑓 (𝑥, 𝑦) + 𝑓 (−𝑥, 𝑦) + 𝑓 (𝑥,−𝑦) + 𝑓 (−𝑥,−𝑦)) ,

𝑓eo (𝑥, 𝑦) =14 (𝑓 (𝑥, 𝑦) + 𝑓 (−𝑥, 𝑦)− 𝑓 (𝑥,−𝑦)− 𝑓 (−𝑥,−𝑦)) ,

𝑓oe (𝑥, 𝑦) =14 (𝑓 (𝑥, 𝑦)− 𝑓 (−𝑥, 𝑦) + 𝑓 (𝑥,−𝑦)− 𝑓 (−𝑥,−𝑦)) ,

𝑓oo (𝑥, 𝑦) =14 (𝑓 (𝑥, 𝑦)− 𝑓 (−𝑥, 𝑦)− 𝑓 (𝑥,−𝑦) + 𝑓 (−𝑥,−𝑦)) .

(2.12)

Definition 2.2 (Symplectic Form [3]). Every 𝑞 = 𝑟0 + 𝒊𝑟1 + 𝒋𝑟2 + 𝒌𝑟3 ∈ ℍ can berewritten in terms of a new basis of operators {𝝁1,𝝁2,𝝁3} as

𝑞 = 𝑟′0 + 𝝁1𝑟′1 + 𝝁2𝑟

′2 + 𝝁3𝑟

′3 = (𝑟′0 + 𝝁1𝑟

′1) + (𝑟′2 + 𝝁1𝑟

′3)𝝁2 , (2.13)

where 𝝁1𝝁2 = 𝝁3, 𝝁1,2,3 ∈ 𝕊3ℍ, hence they form an orthogonal triad.

Remark 2.3. The mapping {𝑟1, 𝑟2, 𝑟3} → {𝑟′1, 𝑟′2, 𝑟′3} is a change in basis from{𝒊, 𝒋,𝒌} to {𝝁1,𝝁2,𝝁3} via

𝑟′0 = 𝑟0, 𝑟′𝑛 = − 12 (𝑉 [𝑞]𝝁𝑛 + 𝝁𝑛𝑉 [𝑞]) , 𝑛 = {1, 2, 3} . (2.14)

Remark 2.4. The symplectic form essentially decomposes a quaternion with re-spect to a specific complex sub-field. That is

𝑞 = (𝑟′0 + 𝝁1𝑟′1) + (𝑟′2 + 𝝁1𝑟

′3)𝝁2 = 𝑐1 + 𝑐2𝝁2 , (2.15)

where 𝑐1,2 ∈ ℂ𝝁1. The author coined the terms simplex and perplex parts of 𝑞, for

𝑐1 and 𝑐2, respectively.

Remark 2.5. The symplectic form works for any permutation of the basis{𝝁1,𝝁2,𝝁3} so that the simplex and complex parts can be taken from any complexsub-field ℂ𝝁𝑛

Further, the swap rule applies to the last term, i.e., 𝑐2𝝁2 = 𝝁2𝑐2,where the over bar denotes both quaternion and complex sub-field conjugation.

1. Quaternion Fourier Transform 7

Definition 2.6 (Split Form [7]). Every 𝑞 ∈ ℍ can be split as

𝑞 = 𝑞+ + 𝑞−, 𝑞± = 12 (𝑞 ± 𝝁1𝑞𝝁2) , (2.16)

where 𝝁1𝝁2 = 𝝁3, and 𝝁1,2,3 ∈ 𝕊3ℍ.

Remark 2.7. The split form allows for the explicit ordering of factors with respectto the operators. So, for example, 𝑞 = 𝑟0 + 𝝁1𝑟1 + 𝝁2𝑟2 + 𝝁3𝑟3 becomes

𝑞± = {(𝑟0 ± 𝑟3) + 𝝁1 (𝑟1 ± 𝑟2)} 1± 𝝁3

2

=1± 𝝁3

2{(𝑟0 ± 𝑟3)− 𝝁2 (𝑟1 ± 𝑟2)} . (2.17)

Euler’s formula holds for quaternions, so any unit length quaternion can bewritten as cos 𝑎+𝝁 sin 𝑎 = 𝑒𝝁𝑎, for 𝑎 ∈ ℝ and 𝝁 ∈ 𝕊3

ℍ. Here 𝝁 is referred to as the

(Eigen-) axis and 𝑎 as the (Eigen-) phase angle. Although in general 𝑒𝑞 𝑒𝑝 ∕= 𝑒𝑞+𝑝

for 𝑝, 𝑞 ∈ ℍ, their exponential product is a linear combination of exponentials ofthe sum and difference of their phase angles. This can be written in two ways asshown in the following two propositions.

Proposition 2.8 (Exponential Split). Let 𝝁1,2 ∈ 𝕊3ℍand 𝑎, 𝑏 ∈ ℝ, then

𝑒𝝁1𝑎𝑒𝝁2𝑏 = 𝑒𝝁1(𝑎−𝑏) 1 + 𝝁3

2+ 𝑒𝝁1(𝑎+𝑏) 1− 𝝁3

2(2.18)

and

𝑒𝝁1𝑎𝑒𝝁2𝑏 =1 + 𝝁3

2𝑒𝝁2(𝑏−𝑎) +

1− 𝝁3

2𝑒𝝁2(𝑏+𝑎) , (2.19)

where 𝝁1𝝁2 = 𝝁3 and 𝝁3 ∈ 𝕊3ℍ.

Proof. Application of split form to the exponential product. □

Proposition 2.9 (Exponential Modulation). Let 𝝁1,2 ∈ 𝕊3ℍand 𝑎, 𝑏 ∈ ℝ, then

𝑒𝝁1𝑎𝑒𝝁2𝑏 = 12

(𝑒𝝁1(𝑎+𝑏) + 𝑒𝝁1(𝑎−𝑏)

)− 1

2𝝁1

(𝑒𝝁1(𝑎+𝑏) − 𝑒𝝁1(𝑎−𝑏)

)𝝁2 (2.20)

and

𝑒𝝁1𝑎𝑒𝝁2𝑏 = 12

(𝑒𝝁2(𝑏+𝑎) + 𝑒𝝁2(𝑏−𝑎)

)− 1

2𝝁1

(𝑒𝝁2(𝑏+𝑎) − 𝑒𝝁2(𝑏−𝑎)

)𝝁2 . (2.21)

Proof. Direct application of Euler’s formula and trigonometric identities. □

Remark 2.10. The sandwich terms (i.e., 𝝁1(.)𝝁2) in the exponential equationsintroduce 4-space rotations into the interpretation of the product [1]. For if 𝑝 =𝝁1𝑞𝝁2, then 𝑝 is a rotated version of 𝑞 about the (𝝁1,𝝁2)-plane by 𝜋

2 .

8 T.A. Ell

3. Quaternion Fourier Transforms

The purpose of this section is to enumerate a list of possible definitions for a quater-nion Fourier transform. This is followed by a discussion regarding various operatorproperties used in the engineering fields that require simple Fourier transform pairsbetween the non-transformed operation and the equivalent Fourier domain oper-ation, i.e., the so-called operator pairs as seen in most engineering textbooks onFourier analysis. Finally, a discussion on how the inter-relationship between QFTdefinitions are explored, not so as to reduce them to a single canonical form, butto provide the investigator a tool to cross between definitions when necessary soas to gain insight into operator properties.

3.1. Transform Definitions

Although there has been much use of the QFT forms currently in circulation, thereare however more available. Not all ‘degrees-of-freedom’ have been exploited. Thenon-commutativity of the quaternion multiplication gave rise to the left- and right-handed QFT kernels. The infinite number of square-roots of −1 (the cardinality of𝕊ℍ) gave rise to the two-sided, or sandwiched kernel. One concept left unexploredis the implication of the exponential product of two quaternions, i.e., 𝑒𝑝𝑒𝑞 ∕= 𝑒𝑝+𝑞.When this is also taken into account, the list expands to eight distinct QFTs asenumerated in Table 1.

Table 1. QFT kernel definitions for 𝑓 : ℝ2 → ℍ.

Left Right Sandwich

Single-axis 𝑒−𝝁1(𝜔𝑥+𝝂𝑦) 𝑓(.) 𝑓(.) 𝑒−𝝁1(𝜔𝑥+𝝂𝑦) 𝑒−𝝁1𝜔𝑥 𝑓(.) 𝑒−𝝁1𝝂𝑦

Dual-axis 𝑒−(𝝁1𝜔𝑥+𝝁2𝝂𝑦) 𝑓(.) 𝑓(.) 𝑒−(𝝁1𝜔𝑥+𝝁2𝝂𝑦) –

Factored 𝑒−𝝁1𝜔𝑥𝑒−𝝁2𝝂𝑦 𝑓(.) 𝑓(.) 𝑒−𝝁1𝜔𝑥𝑒−𝝁2𝝂𝑦 𝑒−𝝁1𝜔𝑥 𝑓(.) 𝑒−𝝁2𝝂𝑦

Depending on the value space of 𝑓(𝑥, 𝑦), the available number of distinctQFT forms changes. Table 1 shows the options when 𝑓 : ℝ2 → ℍ. However, if𝑓 : ℝ2 → ℝ, then all chirality options (left, right, and sandwiched) collapse to thesame form, leaving three distinct choices: single-axis and factored and un-factoreddual-axis forms.

If neither 𝑥 nor 𝑦 are time-like, so that causality of the solution is not afactor, then the number of given QFTs doubles. Variations created by conjugat-ing the quaternion-exponential kernel of both the forward and inverse transformare usually a matter of convention – the signs must be opposites. For non-causalsystems, however, the sign on the kernel can be taken both ways; each definingits own forward transform from the spatial to spatial-frequency domain. To dis-tinguish the two versions of the forward transform, one is called forward the other

1. Quaternion Fourier Transform 9

reverse. Of course, the inverse transform is still obtained by conjugating the cor-responding forward (or reverse) kernel. Hence, one may define the single-axis, left-and right-sided, forward and reverse transforms as follows1.

Definition 3.1 (Single-axis, Left-sided QFT). The single-axis, left-sided, forward(ℱ+𝐿) and reverse (ℱ−𝐿) QFTs are defined as

ℱ±𝐿 [𝑓 (𝑥, 𝑦)] =

∫∫ℝ2

𝑒∓𝝁1(𝜔𝑥+𝜈𝑦)𝑓 (𝑥, 𝑦) 𝑑𝑥𝑑𝑦 =𝐹±𝐿 [𝜔, 𝜈] . (3.1)

Definition 3.2 (Single-axis, Right-sided QFT). The single-axis, right-sided, forward(ℱ+𝑅) and reverse (ℱ−𝑅) QFTs are defined as

ℱ±𝑅 [𝑓 (𝑥, 𝑦)] =

∫∫ℝ2

𝑓 (𝑥, 𝑦) 𝑒∓𝝁1(𝜔𝑥+𝜈𝑦)𝑑𝑥𝑑𝑦 =𝐹±𝑅 [𝜔, 𝜈] . (3.2)

All of the entries in Table 1 exploit the fact that unit length complex numbersact as rotation operators within the complex plane. There is, however, anotherrotation operator – the 3-space rotation operator for which quaternions are famous.Table 2 lists additional definitions under the provision that 𝑓 takes on valuesrestricted to 𝑉 [ℍ]. Note the factor of 1

2 in the kernel exponent, this is included sothat the frequency scales between the various definitions align.

Table 2. QFT kernel definitions exclusively for 𝑓 : ℝ2 → 𝑉 [ℍ].

3-Space Rotator

Single-axis 𝑒−𝝁1𝜔𝑥/2 𝑓(.) 𝑒+𝝁1𝜈𝑦/2

Dual-axis 𝑒−(𝝁1𝜔𝑥+𝝁2𝜈𝑦)/2 𝑓(.) 𝑒+(𝝁1𝜔𝑥+𝝁2𝜈𝑦)/2

Dual-axis, factored 𝑒−𝝁1𝜔𝑥/2𝑒−𝝁2𝜈𝑦/2 𝑓(.) 𝑒+𝝁2𝜈𝑦/2𝑒+𝝁1𝜔𝑥/2

Taking all these permutations in mind, one arrives at 22 unique QFT defini-tions.

3.2. Functional Relationships

There are several properties used in complex Fourier transform (CFT) analysis thatone hopes will carry over to the QFT in some fashion. These are listed in Table3 from which we will discuss the challenges which arise in QFT analysis. In whatfollows let 𝑓 (𝑥, 𝑦)⇔ 𝐹 [𝜔, 𝜈] denote transform pairs, i.e., ℱ [𝑓 (𝑥, 𝑦)] = 𝐹 [𝜔, 𝜈] isthe forward (or reverse) transform and ℱ−1 [𝐹 (𝜔, 𝜈)] = 𝑓 (𝑥, 𝑦) is its inversion.

Inversion. Every transform should be invertible. Although this seems obvious,there are instances where a given transform is not. For example, if the re-striction of 𝑓 : ℝ2 → 𝑉 [ℍ] were not imposed on the inputs of Table 2, then

1Note, that in (3.1) and (3.2) the arguments 𝑥 and 𝑦 of 𝑓 in ℱ±𝐿,𝑅 [𝑓 (𝑥, 𝑦)] are shown for clarity,

but are actually dummy arguments, which are integrated out. A more mathematical notationwould be ℱ±𝐿,𝑅{𝑓}(𝜔, 𝜈) = 𝐹±𝐿,𝑅(𝜔, 𝜈).

10 T.A. Ell

Table 3. Fourier transform ℱ properties. [𝛼, 𝛽, 𝛾, 𝛿 ∈ ℝ]

Property Definition

Inversion ℱ−1 [ℱ [𝑓 (𝑥, 𝑦)]] = 𝑓 (𝑥, 𝑦)

Linearity 𝛼𝑓(𝑥, 𝑦) + 𝛽𝑔(𝑥, 𝑦)

Complex Degenerate (𝝁1 = 𝒊 and 𝑓 : ℝ2 → ℂ𝒊) → (QFT∼=CFT)

Convolution 𝑓 ∘ 𝑔(𝑥, 𝑦) = (?)

Correlation 𝑓 ★ 𝑔(𝑥, 𝑦) = (?)

Modulation 𝑒𝝁1𝜔0𝑥𝑓(.), 𝑒𝝁2𝜔0𝑥𝑓(.), 𝑓(.)𝑒𝝁1𝜈0𝑦, 𝑓(.)𝑒𝝁2𝜈0𝑦, etc.

Scaling 𝑓(𝑥/𝛼, 𝑦/𝛽)

Translation 𝑓(𝑥− 𝑥0, 𝑦 − 𝑦0)

Rotation 𝑓(𝑥 cos𝛼− 𝑦 sin𝛼, 𝑥 sin𝛼+ 𝑦 cos𝛼)

Axis-reversal 𝑓(−𝑥, 𝑦), 𝑓(𝑥,−𝑦), 𝑓(−𝑥,−𝑦)

Re-coordinate 𝑓(𝛼𝑥+ 𝛽𝑦, 𝛾𝑥+ 𝛿𝑦)

Conjugation 𝑓(𝑥, 𝑦)

Differentials ∂∂𝑥 ,

∂∂𝑦 ,

∂2

∂𝑥∂𝑦 , etc.

every transform of that table would cease to be invertible. This is because anyreal-valued function, or the scalar part of full quaternion valued functions,commute with the kernel factors which then vanish from under the integral.

Linearity. 𝛼𝑓(𝑥, 𝑦)+𝛽𝑔(𝑥, 𝑦)⇔ 𝛼𝐹 [𝜔, 𝜈]+𝛽𝐺[𝜔, 𝜈] where 𝛼, 𝛽 ∈ ℝ. A quick checkverifies that this property holds for all proposed QFT definitions.

Complex Degenerate. For the single-axis transforms, if 𝝁1 = 𝒊 and 𝑓 : ℝ2 → ℂ𝒊,then the QFT should ideally degenerate to the twice iterated complex Fouriertransform. This degenerate property cannot apply to the dual axis, factoredforms of Tables 1 and 2 since if 𝝁1 = 𝝁2 = 𝒊, these forms reduce to theirsingle-axis versions.

Convolution (Faltung) theorem. Rarely does the standard, complex transform typepair 𝑓 ∘ 𝑔 (𝑥, 𝑦) ⇔ 𝐹 [𝜔, 𝜈]𝐺 [𝜔, 𝜈] exist in such a simple form for the QFT.Even the very definition of convolution needs an update since 𝑓 ∘ 𝑔 ∕= 𝑔 ∘ 𝑓when 𝑓 and 𝑔 are ℍ-valued. The definition is altered again based on which ofthe two functions is translated, i.e., is the integrand 𝑓(𝑥 − 𝑥′, 𝑦 − 𝑦′)𝑔(𝑥, 𝑦)or 𝑓(𝑥, 𝑦)𝑔(𝑥−𝑥′, 𝑦− 𝑦′). This gives rise to at least four distinct convolutiondefinitions. This will also alter the spectral operator pair.

Further, if 𝑓 is an input function, then 𝑔 is typically related to theimpulse response of a system. But, if 𝑔 : ℝ2 → 𝑉 [ℍ], is a single impulseresponse sufficient to describe such a system? Or does it take at least twoorthogonally oriented impulses, say 𝝁1𝛿(𝑥, 𝑦) and 𝝁2𝛿(𝑥, 𝑦), where 𝝁1 ⊥ 𝝁2

and 𝛿(𝑥, 𝑦) is the Dirac delta function?

1. Quaternion Fourier Transform 11

Correlation. Consider the correlation definition of two ℝ-valued functions 𝑓 and𝑔 (let 𝑓, 𝑔 : ℝ→ ℝ for simplicity of discussion)

𝑓 ★ 𝑔 (𝑡) =

∫ℝ

𝑓 (𝜏) 𝑔 (𝜏 − 𝑡) 𝑑𝜏 =

∫ℝ

𝑓 (𝜏 + 𝑡) 𝑔 (𝜏) 𝑑𝜏, (3.3)

where substituting 𝜏 − 𝑡 = 𝜏 ′ yields the second form. For the correlation ofreal-valued functions this is entirely sufficient.

However, for ℂ-valued functions a conjugation operation is required toensure the relation of the autocorrelation functions (𝑓 ★ 𝑓) to the powerspectrum as required by the Wiener-Khintchine theorem. This effectivelyensures that the power spectrum of a complex auto-correlation is ℝ-valued.The complex extension to the cross-correlation function can then be given as

𝑓 ★ 𝑔 (𝑡) =

∫ℝ

𝑓 (𝜏) 𝑔 (𝜏 − 𝑡) 𝑑𝜏 , (3.4)

or alternatively as

𝑓 ★ 𝑔 (𝑡) =

∫ℝ

𝑓 (𝜏) 𝑔 (𝜏 + 𝑡) 𝑑𝜏 . (3.5)

In general, the literature does not give significance to the direction of theshifted signal (𝜏 ± 𝑡). However, in the case of vector correlation matchingproblems, such as colour image registration, direction is fundamental.

Taking this into consideration, for ℍ-valued functions the equivalentcorrelation could be either

𝑓 ★ 𝑔 (𝑥, 𝑦) =

∫ℝ2

𝑓 (𝑥′, 𝑦′) 𝑔 (𝑥′ − 𝑥, 𝑦′ − 𝑦)𝑑𝑥′𝑑𝑦′, (3.6)

or

𝑓 ★ 𝑔 (𝑥, 𝑦) =

∫ℝ2

𝑓 (𝑥′ + 𝑥, 𝑦′ + 𝑦) 𝑔 (𝑥′, 𝑦′)𝑑𝑥′𝑑𝑦′, (3.7)

depending on which shift direction is required. For more details see [9, 4].Note that the correlation result is not necessarily ℝ-valued.

Modulation. There are multiple types of frequency modulation that need to beaddressed. The modulating exponential can be applied from the left or right,can be driven as a function of either input parameter (i.e., 𝑥 or 𝑦), andbe pointing along one of the kernel axes (i.e., 𝝁1 or 𝝁2). Some options aredetailed in the Karnaugh map of Table 4.

In summary, when addressing the QFT operator properties one often needsto regress back to the basic operator definitions and their underlying assumptions,then verify they are still valid for generalization to quaternion forms. Either theoperator definition itself needs to be modified (as in the case of correlation) orthe number of permutations on the definition increases (as in convolution andmodulation).

12 T.A. Ell

Table 4. Frequency Modulations

Left Right

𝑥𝑒𝝁2𝜔0𝑥𝑓(.) 𝑓(.)𝑒𝝁2𝜔0𝑥 𝝁2

𝑒𝝁1𝜔0𝑥𝑓(.) 𝑓(.)𝑒𝝁1𝜔0𝑥

𝝁1

𝑦𝑒𝝁1𝝂0𝑦𝑓(.) 𝑓(.)𝑒𝝁1𝝂0𝑦

𝑒𝝁2𝝂0𝑦𝑓(.) 𝑓(.)𝑒𝝁2𝝂0𝑦 𝝁2

3.3. Relationships between Transforms

At the heart of all methods for determining inter-relationships between variousQFTs is a decomposing process, of either the input function 𝑓(.) or the exponential-kernel, so that their parts can be commuted into an alternate QFT form. Ell andSangwine [5] used the symplectic form to link the single-axis, left and right, forwardand reverse forms of the QFT via simplex and perplex complex sub-fields. Yeh[10] reworked these relationships and made further connections to the dual-axis,factored form QFT, but instead used even-odd decomposition of the input function.This approach essentially split each QFT into cosine and sine QFTs. Hitzer’s [7]approach was to use the split form to factor the input function and kernel intofactors with respect to the hypercomplex operators, so as to manipulate the resultto an alternate QFT.

The inter-relationships between the various transform definitions not onlygive insight into the subsequent Fourier analysis, they are also used to simplifyoperator pairs. For example, the inter-relationships between the single-axis formsas given in Definitions 3.2 and 3.1 were used with the symplectic form (Def. 2.2)by Ell and Sangwine [5] to arrive at operator pairs for the convolution operator.Let the single-sided convolutions be defined as follows.

Definition 3.3 (Convolution [5]). The left- and right-sided convolution are defined,respectively, as

ℎ𝐿 ∘ 𝑓 (𝑥, 𝑦) =

∫∫ℝ2

ℎ𝐿 (𝑥′, 𝑦′) 𝑓 (𝑥− 𝑥′, 𝑦 − 𝑦′) 𝑑𝑥′𝑑𝑦′,

𝑓 ∘ ℎ𝑅 (𝑥, 𝑦) =

∫∫ℝ2

𝑓 (𝑥− 𝑥′, 𝑦 − 𝑦′)ℎ𝑅 (𝑥′, 𝑦′) 𝑑𝑥′𝑑𝑦′.(3.8)

Now, let the QFT of the input function 𝑓 be symplectically decomposed withrespect to 𝝁1 as

ℱ±𝐿 [𝑓 (𝑥, 𝑦)] = 𝐹±𝐿1 [𝜔, 𝜈] + 𝐹±𝐿2 [𝜔, 𝜈]𝝁2

and

ℱ±(𝐿,𝑅) [ℎ𝑅 (𝑥, 𝑦)] = 𝐻±(𝐿,𝑅)𝑅 [𝜔, 𝜈] ,

1. Quaternion Fourier Transform 13

then the right-convolution operator can be written as

ℱ±𝐿 [𝑓 ∘ ℎ𝑅] (𝜔, 𝜈) = 𝐹±𝐿1 [𝜔, 𝜈]𝐻±𝐿𝑅 [𝜔, 𝜈] + 𝐹±𝐿2 [𝜔, 𝜈]𝝁2𝐻

∓𝐿𝑅 [𝜔, 𝜈] .

Note the use of both forward and reverse QFT transforms. Such a compact oper-ator formula would not be possible without the intermixing of QFT definitions.

4. Conclusions

The three currently defined quaternion Fourier transforms have been shown tobe incomplete. By careful consideration of the underlying reasons for those threeforms, this list has been extended to no less than twenty-two unique definitions.Future work may show that some of these definitions hold little of practical valueor, without loss of generality, they may be reduced to but a few. The shift fromiterated, channel-wise vector analysis to gestalt vector-image and vector-signalanalysis shows promise. This promise raises several challenges:

1. Are there other, more suitable quaternion Fourier transform definitions?2. Can these transforms be reduced to a salient few?3. Are there additional decomposition methods, like the even-odd, split, and

symplectic discussed herein, which can be used?4. All the decomposition methods used to simplify the operator formulas are

at odds with the very gestalt, holistic approach espoused, can this be doneotherwise?

These questions will be the focus of future efforts.

Acknowledgment

Many thanks to Iesus propheta a Nazareth Galilaeae.

References

[1] H.S.M. Coxeter. Quaternions and reflections. The American Mathematical Monthly,53(3):136–146, Mar. 1946.

[2] T.A. Ell. Hypercomplex Spectral Transformations. PhD thesis, University of Min-nesota, June 1992.

[3] T.A. Ell and S.J. Sangwine. Decomposition of 2D hypercomplex Fourier transformsinto pairs of complex Fourier transforms. In M. Gabbouj and P. Kuosmanen, editors,Proceedings of EUSIPCO 2000, Tenth European Signal Processing Conference, vol-ume II, pages 1061–1064, Tampere, Finland, 5–8 Sept. 2000. European Associationfor Signal Processing.

[4] T.A. Ell and S.J. Sangwine. Hypercomplex Wiener-Khintchine theorem with ap-plication to color image correlation. In IEEE International Conference on ImageProcessing (ICIP 2000), volume II, pages 792–795, Vancouver, Canada, 11–14 Sept.2000. IEEE.

[5] T.A. Ell and S.J. Sangwine. Hypercomplex Fourier transforms of color images. IEEETransactions on Image Processing, 16(1):22–35, Jan. 2007.

14 T.A. Ell

[6] R.R. Ernst, G. Bodenhausen, and A. Wokaun. Principles of Nuclear Magnetic Res-onance in One and Two Dimensions. International Series of Monographs on Chem-istry. Oxford University Press, 1987.

[7] E. Hitzer. Quaternion Fourier transform on quaternion fields and generalizations.Advances in Applied Clifford Algebras, 17(3):497–517, May 2007.

[8] C.E. Moxey, S.J. Sangwine, and T.A. Ell. Vector phase correlation. Electronics Let-ters, 37(25):1513–1515, Dec. 2001.

[9] C.E. Moxey, S.J. Sangwine, and T.A. Ell. Hypercomplex correlation techniques forvector images. IEEE Transactions on Signal Processing, 51(7):1941–1953, July 2003.

[10] M.-H. Yeh. Relationships among various 2-D quaternion Fourier transforms. IEEESignal Processing Letters, 15:669–672, Nov. 2008.

Todd Anthony EllEngineering FellowGoodrich, Sensors and Integrated Systems14300 Judicial RoadBurnsville, MN 55306, USAe-mail: t.ell@ieee.org

Quaternion and Clifford–Fourier Transforms and Wavelets

Trends in Mathematics, 15–39c⃝ 2013 Springer Basel

2 The Orthogonal 2D Planes Split ofQuaternions and Steerable QuaternionFourier Transformations

Eckhard Hitzer and Stephen J. Sangwine

Abstract. The two-sided quaternionic Fourier transformation (QFT) was in-troduced in [2] for the analysis of 2D linear time-invariant partial-differentialsystems. In further theoretical investigations [4, 5] a special split of quater-nions was introduced, then called ±split. In the current chapter we analyzethis split further, interpret it geometrically as an orthogonal 2D planes split(OPS), and generalize it to a freely steerable split of ℍ into two orthogonal2D analysis planes. The new general form of the OPS split allows us to findnew geometric interpretations for the action of the QFT on the signal. Thesecond major result of this work is a variety of new steerable forms of theQFT, their geometric interpretation, and for each form, OPS split theorems,which allow fast and efficient numerical implementation with standard FFTsoftware.

Mathematics Subject Classification (2010). Primary 16H05; secondary 42B10,94A12, 94A08, 65R10.

Keywords. Quaternion signals, orthogonal 2D planes split, quaternion Fouriertransformations, steerable transforms, geometric interpretation, fast imple-mentations.

1. Introduction

The two-sided quaternionic Fourier transformation (QFT) was introduced in [2] forthe analysis of 2D linear time-invariant partial-differential systems. Subsequentlyit has been applied in many fields, including colour image processing [8]. This ledto further theoretical investigations [4, 5], where a special split of quaternions wasintroduced, then called the ±split. An interesting physical consequence was thatthis split resulted in a left and right travelling multivector wave packet analysis,when generalizing the QFT to a full spacetime Fourier transform (SFT). In thecurrent chapter we investigate this split further, interpret it geometrically and

16 E. Hitzer and S.J. Sangwine

generalize it to a freely steerable1 split of ℍ into two orthogonal 2D analysis planes.For reasons to become obvious we prefer to call it from now on the orthogonal 2Dplanes split (OPS).

The general form of the OPS split allows us to find new geometric interpre-tations for the action of the QFT on the signal. The second major result of thiswork is a variety of new forms of the QFT, their detailed geometric interpretation,and for each form, an OPS split theorem, which allows fast and efficient numericalimplementation with standard FFT software. A preliminary formal investigationof these new OPS-QFTs can be found in [6].

The chapter is organized as follows. We first introduce in Section 2 severalproperties of quaternions together with a brief review of the ±-split of [4, 5].In Section 3 we generalize this split to a freely steerable orthogonal 2D planessplit (OPS) of quaternions ℍ. In Section 4 we use the general OPS of Section 3 togeneralize the two-sided QFT to a new two-sided QFT with freely steerable analysisplanes, complete with a detailed local geometric transformation interpretation. Thegeometric interpretation of the OPS in Section 3 further allows the construction ofa new type of steerable QFT with a direct phase angle interpretation. In Section 5we finally investigate new steerable QFTs involving quaternion conjugation. Theirlocal geometric interpretation crucially relies on the notion of 4D rotary reflections.

2. Orthogonal Planes Split of Quaternions withTwo Orthonormal Pure Unit Quaternions

Gauss, Rodrigues and Hamilton’s four-dimensional (4D) quaternion algebra ℍ isdefined over ℝ with three imaginary units:

𝒊𝒋 = −𝒋𝒊 = 𝒌, 𝒋𝒌 = −𝒌𝒋 = 𝒊, 𝒌𝒊 = −𝒊𝒌 = 𝒋,

𝒊2 = 𝒋2 = 𝒌2 = 𝒊𝒋𝒌 = −1. (2.1)

Every quaternion can be written explicitly as

𝑞 = 𝑞𝑟 + 𝑞𝑖𝒊+ 𝑞𝑗𝒋 + 𝑞𝑘𝒌 ∈ ℍ, 𝑞𝑟, 𝑞𝑖, 𝑞𝑗, 𝑞𝑘 ∈ ℝ, (2.2)

and has a quaternion conjugate (equivalent2 to Clifford conjugation in 𝐶ℓ+3,0 and

𝐶ℓ0,2)

𝑞 = 𝑞𝑟 − 𝑞𝑖𝒊− 𝑞𝑗𝒋 − 𝑞𝑘𝒌, 𝑝𝑞 = 𝑞 𝑝, (2.3)

which leaves the scalar part 𝑞𝑟 unchanged. This leads to the norm of 𝑞 ∈ ℍ

∣𝑞∣ =√

𝑞𝑞 =√

𝑞2𝑟 + 𝑞2𝑖 + 𝑞2𝑗 + 𝑞2𝑘, ∣𝑝𝑞∣ = ∣𝑝∣ ∣𝑞∣ . (2.4)

The part V(𝑞) = 𝑞 − 𝑞𝑟 = 12 (𝑞 − 𝑞) = 𝑞𝑖𝒊+ 𝑞𝑗𝒋 + 𝑞𝑘𝒌 is called a pure quaternion,

and it squares to the negative number −(𝑞2𝑖 +𝑞2𝑗 +𝑞2𝑘). Every unit quaternion (i.e.,

1Compare Section 3.4, in particular Theorem 3.5.2This may be important in generalisations of the QFT, such as to a space-time Fourier transformin [4], or a general two-sided Clifford–Fourier transform in [7].

2. Orthogonal 2D Planes Split 17

∣𝑞∣ = 1) can be written as:

𝑞 = 𝑞𝑟 + 𝑞𝑖𝒊+ 𝑞𝑗𝒋 + 𝑞𝑘𝒌 = 𝑞𝑟 +√

𝑞2𝑖 + 𝑞2𝑗 + 𝑞2𝑘 𝝁(𝑞)

= cos𝛼+ 𝝁(𝑞) sin𝛼 = 𝑒𝛼𝝁(𝑞),(2.5)

where

cos𝛼 = 𝑞𝑟, sin𝛼 =√

𝑞2𝑖 + 𝑞2𝑗 + 𝑞2𝑘,

𝝁(𝑞) =V(𝑞)

∣𝑞∣ =𝑞𝑖𝒊+ 𝑞𝑗𝒋 + 𝑞𝑘𝒌√

𝑞2𝑖 + 𝑞2𝑗 + 𝑞2𝑘

, and 𝝁(𝑞)2= −1. (2.6)

The inverse of a non-zero quaternion is

𝑞−1 =𝑞

∣𝑞∣2 =𝑞

𝑞𝑞. (2.7)

The scalar part of a quaternion is defined as

S(𝑞) = 𝑞𝑟 =1

2(𝑞 + 𝑞), (2.8)

with symmetries

S(𝑝𝑞) = S(𝑞𝑝) = 𝑝𝑟𝑞𝑟 − 𝑝𝑖𝑞𝑖 − 𝑝𝑗𝑞𝑗 − 𝑝𝑘𝑞𝑘, S(𝑞) = S(𝑞) , ∀𝑝, 𝑞 ∈ ℍ, (2.9)

and linearity

S(𝛼𝑝+ 𝛽𝑞) = 𝛼 S(𝑝) + 𝛽 S(𝑞) = 𝛼𝑝𝑟 + 𝛽𝑞𝑟 , ∀𝑝, 𝑞 ∈ ℍ, 𝛼, 𝛽 ∈ ℝ. (2.10)

The scalar part and the quaternion conjugate allow the definition of the ℝ4 innerproduct3 of two quaternions 𝑝, 𝑞 as

S(𝑝𝑞) = 𝑝𝑟𝑞𝑟 + 𝑝𝑖𝑞𝑖 + 𝑝𝑗𝑞𝑗 + 𝑝𝑘𝑞𝑘 ∈ ℝ. (2.11)

Definition 2.1 (Orthogonality of quaternions). Two quaternions 𝑝, 𝑞 ∈ ℍ are or-thogonal 𝑝 ⊥ 𝑞, if and only if the inner product S(𝑝𝑞) = 0.

The orthogonal4 2D planes split (OPS) of quaternions with respect to theorthonormal pure unit quaternions 𝒊, 𝒋 [4, 5] is defined by

𝑞 = 𝑞+ + 𝑞−, 𝑞± =1

2(𝑞 ± 𝒊𝑞𝒋). (2.12)

Explicitly in real components 𝑞𝑟, 𝑞𝑖, 𝑞𝑗 , 𝑞𝑘 ∈ ℝ using (2.1) we get

𝑞± = {𝑞𝑟 ± 𝑞𝑘 + 𝒊(𝑞𝑖 ∓ 𝑞𝑗)}1± 𝒌

2=

1± 𝒌

2{𝑞𝑟 ± 𝑞𝑘 + 𝒋(𝑞𝑗 ∓ 𝑞𝑖)}. (2.13)

This leads to the following new Pythagorean modulus identity [5]

Lemma 2.2 (Modulus identity). For 𝑞 ∈ ℍ

∣𝑞∣2 = ∣𝑞−∣2 + ∣𝑞+∣2 . (2.14)

3Note that we do not use the notation 𝑝 ⋅ 𝑞, which is unconventional for full quaternions.4Compare Lemma 2.3.

18 E. Hitzer and S.J. Sangwine

Lemma 2.3 (Orthogonality of OPS split parts). Given any two quaternions 𝑝, 𝑞 ∈ ℍ

and applying the OPS of (2.12) the resulting parts are orthogonal

S(𝑝+𝑞−) = 0, S(𝑝−𝑞+) = 0, (2.15)

i.e., 𝑝+ ⊥ 𝑞− and 𝑝− ⊥ 𝑞+.

In Lemma 2.3 (proved in [5]) the second identity follows from the first byS(𝑥) = S(𝑥) , ∀𝑥 ∈ ℍ, and 𝑝−𝑞+ = 𝑞+𝑝−.

It is evident, that instead of 𝒊, 𝒋, any pair of orthonormal pure quaternionscan be used to produce an analogous split. This is a first indication, that the OPS of(2.12) is in fact steerable. We observe, that 𝒊𝑞𝒋 = 𝑞+−𝑞−, i.e., under the map 𝒊( )𝒋the 𝑞+ part is invariant, the 𝑞− part changes sign. Both parts are according to (2.13)two-dimensional, and by Lemma 2.3 they span two completely orthogonal planes.The 𝑞+-plane is spanned by the orthogonal quaternions {𝒊 − 𝒋, 1 + 𝒊𝒋 = 1 + 𝒌},whereas the 𝑞−-plane is, e.g., spanned by {𝒊+ 𝒋, 1− 𝒊𝒋 = 1− 𝒌}, i.e., we have thetwo 2D subspace bases

𝑞+-basis: {𝒊− 𝒋, 1 + 𝒊𝒋 = 1 + 𝒌}, 𝑞−-basis: {𝒊+ 𝒋, 1− 𝒊𝒋 = 1− 𝒌}. (2.16)

Note that all basis vectors of (2.16)

{𝒊− 𝒋, 1 + 𝒊𝒋, 𝒊+ 𝒋, 1− 𝒊𝒋} (2.17)

together form an orthogonal basis of ℍ interpreted as ℝ4.The map 𝒊( )𝒋 rotates the 𝑞−-plane by 180∘ around the 2D 𝑞+ axis plane. Note

that in agreement with its geometric interpretation, the map 𝒊( )𝒋 is an involution,because applying it twice leads to identity

𝒊(𝒊𝑞𝒋)𝒋 = 𝒊2𝑞𝒋2 = (−1)2𝑞 = 𝑞. (2.18)

3. General Orthogonal 2D Planes Split

We will study generalizations of the OPS split by replacing 𝒊, 𝒋 by arbitrary unitquaternions 𝑓, 𝑔. Even with this generalization, the map 𝑓( )𝑔 continues to be aninvolution, because 𝑓2𝑞𝑔2 = (−1)2𝑞 = 𝑞. For clarity we study the cases 𝑓 ∕= ±𝑔,and 𝑓 = 𝑔 separately, though they have a lot in common, and do not always needto be distinguished in specific applications.

3.1. Orthogonal 2D Planes Split Using Two Linearly IndependentPure Unit Quaternions

Our result is now, that all these properties hold, even if in the above considerationsthe pair 𝒊, 𝒋 is replaced by an arbitrary pair of linearly independent nonorthogonalpure quaternions 𝑓, 𝑔, 𝑓2 = 𝑔2 = −1, 𝑓 ∕= ±𝑔. The OPS is then re-defined withrespect to the linearly independent pure unit quaternions 𝑓, 𝑔 as

𝑞± =1

2(𝑞 ± 𝑓𝑞𝑔). (3.1)

2. Orthogonal 2D Planes Split 19

Equation (2.12) is a special case with 𝑓 = 𝒊, 𝑔 = 𝒋. We observe from (3.1), that𝑓𝑞𝑔 = 𝑞+ − 𝑞−, i.e., under the map 𝑓( )𝑔 the 𝑞+ part is invariant, but the 𝑞− partchanges sign

𝑓𝑞±𝑔 =1

2(𝑓𝑞𝑔 ± 𝑓2𝑞𝑔2) =

1

2(𝑓𝑞𝑔 ± 𝑞) = ±1

2(𝑞 ± 𝑓𝑞𝑔) = ±𝑞±. (3.2)

We now show that even for (3.1) both parts are two-dimensional, and span twocompletely orthogonal planes. The 𝑞+-plane is spanned

5 by the orthogonal pair ofquaternions {𝑓 − 𝑔, 1 + 𝑓𝑔}:

S((𝑓 − 𝑔)(1 + 𝑓𝑔)

)= S((𝑓 − 𝑔)(1 + (−𝑔)(−𝑓)))

= S(𝑓 + 𝑓𝑔𝑓 − 𝑔 − 𝑔2𝑓

) (2.9)= S

(𝑓 + 𝑓2𝑔 − 𝑔 + 𝑓

)= 2S(𝑓 − 𝑔) = 0, (3.3)

whereas the 𝑞−-plane is, e.g., spanned by {𝑓 + 𝑔, 1 − 𝑓𝑔}. The quaternions 𝑓 +𝑔, 1− 𝑓𝑔 can be proved to be mutually orthogonal by simply replacing 𝑔 → −𝑔 in(3.3). Note that we have

𝑓(𝑓 − 𝑔)𝑔 = 𝑓2𝑔 − 𝑓𝑔2 = −𝑔 + 𝑓 = 𝑓 − 𝑔,

𝑓(1 + 𝑓𝑔)𝑔 = 𝑓𝑔 + 𝑓2𝑔2 = 𝑓𝑔 + 1 = 1 + 𝑓𝑔,(3.4)

as well as𝑓(𝑓 + 𝑔)𝑔 = 𝑓2𝑔 + 𝑓𝑔2 = −𝑔 − 𝑓 = −(𝑓 + 𝑔),

𝑓(1− 𝑓𝑔)𝑔 = 𝑓𝑔 − 𝑓2𝑔2 = 𝑓𝑔 − 1 = −(1− 𝑓𝑔).(3.5)

We now want to generalize Lemma 2.3.

Lemma 3.1 (Orthogonality of two OPS planes). Given any two quaternions 𝑞, 𝑝 ∈ℍ and applying the OPS (3.1) with respect to two linearly independent pure unitquaternions 𝑓, 𝑔 we get zero for the scalar part of the mixed products

S(𝑝+𝑞−) = 0, S(𝑝−𝑞+) = 0. (3.6)

We prove the first identity, the second follows from S(𝑥) = S(𝑥).

S(𝑝+𝑞−) =1

4S((𝑝+ 𝑓𝑝𝑔)(𝑞 − 𝑔𝑞𝑓)) =

1

4S(𝑝𝑞 − 𝑓𝑝𝑔𝑔𝑞𝑓 + 𝑓𝑝𝑔𝑞 − 𝑝𝑔𝑞𝑓)

(2.10), (2.9)=

1

4S(𝑝𝑞 − 𝑝𝑞 + 𝑝𝑔𝑞𝑓 − 𝑝𝑔𝑞𝑓) = 0. (3.7)

Thus the set

{𝑓 − 𝑔, 1 + 𝑓𝑔, 𝑓 + 𝑔, 1− 𝑓𝑔} (3.8)

forms a 4D orthogonal basis of ℍ interpreted by (2.11) as ℝ4, where we have forthe orthogonal 2D planes the subspace bases:

𝑞+-basis: {𝑓 − 𝑔, 1 + 𝑓𝑔}, 𝑞−-basis: {𝑓 + 𝑔, 1− 𝑓𝑔}. (3.9)

5For 𝑓 = 𝒊, 𝑔 = 𝒋 this is in agreement with (2.13) and (2.16)!

20 E. Hitzer and S.J. Sangwine

We can therefore use the following representation for every 𝑞 ∈ ℍ by means of fourreal coefficients 𝑞1, 𝑞2, 𝑞3, 𝑞4 ∈ ℝ

𝑞 = 𝑞1(1 + 𝑓𝑔) + 𝑞2(𝑓 − 𝑔) + 𝑞3(1 − 𝑓𝑔) + 𝑞4(𝑓 + 𝑔), (3.10)

where

𝑞1 = S(𝑞(1 + 𝑓𝑔)−1

), 𝑞2 = S

(𝑞(𝑓 − 𝑔)−1

),

𝑞3 = S(𝑞(1 − 𝑓𝑔)−1

), 𝑞4 = S

(𝑞(𝑓 + 𝑔)−1

).

(3.11)

As an example we have for 𝑓 = 𝒊, 𝑔 = 𝒋 according to (2.13) the coefficients for thedecomposition with respect to the orthogonal basis (3.8)

𝑞1 =1

2(𝑞𝑟 + 𝑞𝑘), 𝑞2 =

1

2(𝑞𝑖 − 𝑞𝑗), 𝑞3 =

1

2(𝑞𝑟 − 𝑞𝑘), 𝑞4 =

1

2(𝑞𝑖 + 𝑞𝑗). (3.12)

Moreover, using

𝑓 − 𝑔 = 𝑓(1 + 𝑓𝑔) = (1 + 𝑓𝑔)(−𝑔), 𝑓 + 𝑔 = 𝑓(1− 𝑓𝑔) = (1− 𝑓𝑔)𝑔, (3.13)

we have the following left and right factoring properties

𝑞+ = 𝑞1(1 + 𝑓𝑔) + 𝑞2(𝑓 − 𝑔) = (𝑞1 + 𝑞2𝑓)(1 + 𝑓𝑔)(3.14)

= (1 + 𝑓𝑔)(𝑞1 − 𝑞2𝑔),

𝑞− = 𝑞3(1− 𝑓𝑔) + 𝑞4(𝑓 + 𝑔) = (𝑞3 + 𝑞4𝑓)(1− 𝑓𝑔)(3.15)

= (1− 𝑓𝑔)(𝑞3 + 𝑞4𝑔).

Equations (3.4) and (3.5) further show that the map 𝑓( )𝑔 rotates the 𝑞−-plane by 180∘ around the 𝑞+ axis plane. We found that our interpretation of themap 𝑓( )𝑔 is in perfect agreement with Coxeter’s notion of half-turn in [1]. Thisopens the way for new types of QFTs, where the pair of square roots of −1 involveddoes not necessarily need to be orthogonal.

Before suggesting a generalization of the QFT, we will establish a new set ofvery useful algebraic identities. Based on (3.14) and (3.15) we get for 𝛼, 𝛽 ∈ ℝ

𝑒𝛼𝑓𝑞𝑒𝛽𝑔 = 𝑒𝛼𝑓𝑞+𝑒𝛽𝑔 + 𝑒𝛼𝑓𝑞−𝑒𝛽𝑔,

𝑒𝛼𝑓𝑞+𝑒𝛽𝑔 = (𝑞1 + 𝑞2𝑓)𝑒𝛼𝑓 (1 + 𝑓𝑔)𝑒𝛽𝑔 = 𝑒𝛼𝑓 (1 + 𝑓𝑔)𝑒𝛽𝑔(𝑞1 − 𝑞2𝑔), (3.16)

𝑒𝛼𝑓𝑞−𝑒𝛽𝑔 = (𝑞3 + 𝑞4𝑓)𝑒𝛼𝑓 (1− 𝑓𝑔)𝑒𝛽𝑔 = 𝑒𝛼𝑓 (1− 𝑓𝑔)𝑒𝛽𝑔(𝑞3 + 𝑞4𝑔).

Using (3.14) again we obtain

𝑒𝛼𝑓 (1 + 𝑓𝑔) = (cos𝛼+ 𝑓 sin𝛼)(1 + 𝑓𝑔)

(3.14)= (1 + 𝑓𝑔)(cos𝛼− 𝑔 sin𝛼) = (1 + 𝑓𝑔)𝑒−𝛼𝑔,

(3.17)

where we set 𝑞1 = cos𝛼, 𝑞2 = sin𝛼 for applying (3.14). Replacing in (3.17)−𝛼→ 𝛽we get

𝑒−𝛽𝑓(1 + 𝑓𝑔) = (1 + 𝑓𝑔)𝑒𝛽𝑔, (3.18)

2. Orthogonal 2D Planes Split 21

Furthermore, replacing in (3.17) 𝑔 → −𝑔 and subsequently 𝛼→ 𝛽 we get

𝑒𝛼𝑓 (1− 𝑓𝑔) = (1 − 𝑓𝑔)𝑒𝛼𝑔,

𝑒𝛽𝑓(1− 𝑓𝑔) = (1 − 𝑓𝑔)𝑒𝛽𝑔.(3.19)

Applying (3.14), (3.16), (3.17) and (3.18) we can rewrite

𝑒𝛼𝑓𝑞+𝑒𝛽𝑔(3.16)= (𝑞1 + 𝑞2𝑓)𝑒

𝛼𝑓 (1 + 𝑓𝑔)𝑒𝛽𝑔(3.17)= (𝑞1 + 𝑞2𝑓)(1 + 𝑓𝑔)𝑒(𝛽−𝛼)𝑔

(3.14)= 𝑞+𝑒(𝛽−𝛼)𝑔, (3.20)

or equivalently as

𝑒𝛼𝑓𝑞+𝑒𝛽𝑔(3.16)= 𝑒𝛼𝑓 (1 + 𝑓𝑔)𝑒𝛽𝑔(𝑞1 − 𝑞2𝑔)

(3.18)= 𝑒(𝛼−𝛽)𝑓(1 + 𝑓𝑔)(𝑞1 − 𝑞2𝑔)

(3.14)= 𝑒(𝛼−𝛽)𝑓𝑞+. (3.21)

In the same way by changing 𝑔 → −𝑔, 𝛽 → −𝛽 in (3.20) and (3.21) we can rewrite

𝑒𝛼𝑓𝑞−𝑒𝛽𝑔 = 𝑒(𝛼+𝛽)𝑓𝑞− = 𝑞−𝑒(𝛼+𝛽)𝑔. (3.22)

The result is therefore

𝑒𝛼𝑓𝑞±𝑒𝛽𝑔 = 𝑞±𝑒(𝛽∓𝛼)𝑔 = 𝑒(𝛼∓𝛽)𝑓𝑞±. (3.23)

3.2. Orthogonal 2D Planes Split Using One Pure Unit Quaternion

We now treat the case for 𝑔 = 𝑓, 𝑓2 = −1. We then have the map 𝑓( )𝑓 , and theOPS split with respect to 𝑓 ∈ ℍ, 𝑓2 = −1,

𝑞± =1

2(𝑞 ± 𝑓𝑞𝑓). (3.24)

The pure quaternion 𝒊 can be rotated by 𝑅 = 𝒊(𝒊+ 𝑓), see (3.27), into the quater-nion unit 𝑓 and back. Therefore studying the map 𝒊( )𝒊 is, up to the constantrotation between 𝒊 and 𝑓 , the same as studying 𝑓( )𝑓 . This gives

𝒊𝑞𝒊 = 𝒊(𝑞𝑟 + 𝑞𝑖𝒊+ 𝑞𝑗𝒋 + 𝑞𝑘𝒌)𝒊 = −𝑞𝑟 − 𝑞𝑖𝒊+ 𝑞𝑗𝒋 + 𝑞𝑘𝒌. (3.25)

The OPS with respect to 𝑓 = 𝑔 = 𝒊 gives

𝑞± =1

2(𝑞 ± 𝒊𝑞𝒊), 𝑞+ = 𝑞𝑗𝒋 + 𝑞𝑘𝒌 = (𝑞𝑗 + 𝑞𝑘𝒊)𝒋, 𝑞− = 𝑞𝑟 + 𝑞𝑖𝒊, (3.26)

where the 𝑞+-plane is two-dimensional and manifestly orthogonal to the 2D 𝑞−-plane. This form (3.26) of the OPS is therefore identical to the quaternionic sim-plex/perplex split of [3].

For 𝑔 = 𝑓 the 𝑞−-plane is always spanned by {1, 𝑓}. The rotation operator𝑅 = 𝒊(𝒊+𝑓), with squared norm ∣𝑅∣2 = ∣𝒊(𝒊+𝑓)∣2 = ∣(𝒊+𝑓)∣2 = −(𝒊+𝑓)2, rotates𝒊 into 𝑓 according to

𝑅−1𝒊𝑅 =𝑅

∣𝑅∣2 𝒊𝑅 =(𝒊+ 𝑓)𝒊𝒊𝒊(𝒊+ 𝑓)

−(𝒊+ 𝑓)2=

(𝒊+ 𝑓)𝒊(𝒊(−𝑓) + 1)𝑓

(𝒊+ 𝑓)2

=(𝒊+ 𝑓)(𝑓 + 𝒊)𝑓

(𝒊+ 𝑓)2= 𝑓.

(3.27)

22 E. Hitzer and S.J. Sangwine

The rotation 𝑅 leaves 1 invariant and thus rotates the whole {1, 𝒊} plane into the𝑞−-plane spanned by {1, 𝑓}. Consequently 𝑅 also rotates the {𝒋,𝒌} plane into the𝑞+-plane spanned by {𝒋′ = 𝑅−1𝒋𝑅, 𝒌′ = 𝑅−1𝒌𝑅}. We thus constructively obtainthe fully orthonormal 4D basis of ℍ as

{1, 𝑓, 𝒋′,𝒌′} = 𝑅−1{1, 𝒊, 𝒋,𝒌}𝑅, 𝑅 = 𝒊(𝒊+ 𝑓), (3.28)

for any chosen pure unit quaternion 𝑓 . We further have, for the orthogonal 2Dplanes created in (3.24) the subspace bases:

𝑞+-basis: {𝒋′,𝒌′}, 𝑞−-basis: {1, 𝑓}. (3.29)

The rotation 𝑅 (an orthogonal transformation!) of (3.27) preserves the fun-damental quaternionic orthonormality and the anticommutation relations

𝑓𝒋′ = 𝒌′ = −𝒋′𝑓, 𝒌′𝑓 = 𝒋′ = −𝑓𝒌′ 𝒋′𝒌′ = 𝑓 = −𝒌′𝒋′. (3.30)

Hence

𝑓𝑞𝑓 = 𝑓(𝑞+ + 𝑞−)𝑓 = 𝑞+ − 𝑞−, i.e., 𝑓𝑞±𝑓 = ±𝑞±, (3.31)

which represents again a half-turn by 180∘ in the 2D 𝑞−-plane around the 2D𝑞+-plane (as axis).

Figures 1 and 2 illustrate this decomposition for the case where 𝑓 is theunit pure quaternion 1√

3(𝒊 + 𝒋 + 𝒌). This decomposition corresponds (for pure

quaternions) to the classical luminance-chrominance decomposition used in colourimage processing, as illustrated, for example, in [3, Figure 2]. Three hundredunit quaternions randomly oriented in 4-space were decomposed. Figure 1 showsthe three hundred points in 4-space, projected onto the six orthogonal planes{𝑒, 𝒊′}, {𝑒, 𝒋′}, {𝑒,𝒌′}, {𝒊′, 𝒋′}, {𝒋′,𝒌′}, {𝒌′, 𝒊′} where 𝑒 = 1 and 𝒊′ = 𝑓 , as given in(3.28). The six views at the top show the 𝑞+-plane, and the six below show the𝑞−-plane.

Figure 2 shows the vector parts of the decomposed quaternions. The basisfor the plot is {𝒊′, 𝒋 ′,𝒌′}, where 𝒊′ = 𝑓 as given in (3.28). The green circles showthe components in the {1, 𝑓} plane, which intersects the 3-space of the vector partonly along the line 𝑓 (which is the luminance or grey line of colour image pixels).The red line on the figure corresponds to 𝑓 . The blue circles show the componentsin the {𝒋′,𝒌′} plane, which is entirely within the 3-space. It is orthogonal to 𝑓 andcorresponds to the chrominance plane of colour image processing.

The next question is the influence the current OPS (3.24) has for left andright exponential factors of the form

𝑒𝛼𝑓𝑞± 𝑒𝛽𝑓 . (3.32)

We learn from (3.30) that

𝑒𝛼𝑓𝑞±𝑒𝛽𝑓 = 𝑒(𝛼∓𝛽)𝑓𝑞± = 𝑞±𝑒(𝛽∓𝛼)𝑓 , (3.33)

which is identical to (3.23), if we insert 𝑔 = 𝑓 in (3.23).

2. Orthogonal 2D Planes Split 23

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Figure 1. 4D scatter plot of quaternions decomposed using the or-thogonal planes split of (3.24) with one unit pure quaternion 𝑓 = 𝒊′ =1√3(𝒊+ 𝒋 + 𝒌) = 𝑔.

24 E. Hitzer and S.J. Sangwine

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Figure 2. Scatter plot of vector parts of quaternions decomposed usingthe orthogonal planes split of (3.24) with one pure unit quaternion 𝑓 =𝒊′ = 1√

3(𝒊+ 𝒋 + 𝒌) = 𝑔. The red line corresponds to the direction of 𝑓 .

Next, we consider 𝑔 = −𝑓, 𝑓2 = −1. We then have the map 𝑓( )(−𝑓), andthe OPS split with respect to 𝑓,−𝑓 ∈ ℍ, 𝑓2 = −1,

𝑞± =1

2(𝑞 ± 𝑓𝑞(−𝑓)) =

1

2(𝑞 ∓ 𝑓𝑞𝑓). (3.34)

Again we can study 𝑓 = 𝒊 first, because for general pure unit quaternions 𝑓 theunit quaternion 𝒊 can be rotated by (3.27) into the quaternion unit 𝑓 and back.Therefore studying the map 𝒊( )(−𝒊) is up to the constant rotation 𝑅 of (3.27) thesame as studying 𝑓( )(−𝑓). This gives the map

𝒊𝑞(−𝒊) = 𝒊(𝑞𝑟 + 𝑞𝑖𝒊+ 𝑞𝑗𝒋 + 𝑞𝑘𝒌)(−𝒊) = 𝑞𝑟 + 𝑞𝑖𝒊− 𝑞𝑗𝒋 − 𝑞𝑘𝒌. (3.35)

The OPS with respect to 𝑓 = 𝒊, 𝑔 = −𝒊 gives

𝑞± =1

2(𝑞 ± 𝒊𝑞(−𝒊)), 𝑞− = 𝑞𝑗𝒋 + 𝑞𝑘𝒌 = (𝑞𝑗 + 𝑞𝑘𝒊)𝒋, 𝑞+ = 𝑞𝑟 + 𝑞𝑖𝒊, (3.36)

where, compared to 𝑓 = 𝑔 = 𝒊, the 2D 𝑞+-plane and the 2D 𝑞−-planes appear in-terchanged. The form (3.36) of the OPS is again identical to the quaternionic sim-plex/perplex split of [3], but the simplex and perplex parts appear interchanged.

For 𝑔 = −𝑓 the 𝑞+-plane is always spanned by {1, 𝑓}. The rotation 𝑅 of(3.27) rotates 𝒊 into 𝑓 and leaves 1 invariant and thus rotates the whole {1, 𝒊}

2. Orthogonal 2D Planes Split 25

plane into the 𝑞+-plane spanned by {1, 𝑓}. Consequently, 𝑅 of (3.27) also rotatesthe {𝒋,𝒌} plane into the 𝑞−-plane spanned by {𝒋′ = 𝑅−1𝒋𝑅, 𝒌′ = 𝑅−1𝒌𝑅}.

We therefore have for the orthogonal 2D planes created in (3.34) the subspacebases:

𝑞+-basis: {1, 𝑓}, 𝑞−-basis: {𝒋′,𝒌′}. (3.37)

We again obtain the fully orthonormal 4D basis (3.28) of ℍ, preservingthe fundamental quaternionic orthonormality and the anticommutation relations(3.30).

Hence for (3.34)

𝑓𝑞(−𝑓) = 𝑓(𝑞+ + 𝑞−)(−𝑓) = 𝑞+ − 𝑞−, i.e., 𝑓𝑞±(−𝑓) = ±𝑞±, (3.38)

which represents again a half-turn by 180∘ in the 2D 𝑞−-plane around the 2D𝑞+-plane (as axis).

The remaining question is the influence the current OPS (3.34) has for leftand right exponential factors of the form

𝑒𝛼𝑓𝑞±𝑒−𝛽𝑓 . (3.39)

We learn from (3.30) that

𝑒𝛼𝑓𝑞±𝑒−𝛽𝑓 = 𝑒(𝛼∓𝛽)𝑓𝑞± = 𝑞±𝑒−(𝛽∓𝛼)𝑓 , (3.40)

which is identical to (3.23), if we insert 𝑔 = −𝑓 in (3.23).

For (3.23) therefore, we do not any longer need to distinguish the cases 𝑓 ∕=±𝑔 and 𝑓 = ±𝑔. This motivates us to a general OPS definition for any pair of purequaternions 𝑓, 𝑔, and we get a general lemma.

Definition 3.2 (General orthogonal 2D planes split). Let 𝑓, 𝑔 ∈ ℍ be an arbitrarypair of pure quaternions 𝑓, 𝑔, 𝑓2 = 𝑔2 = −1, including the cases 𝑓 = ±𝑔. Thegeneral OPS is then defined with respect to the two pure unit quaternions 𝑓, 𝑔 as

𝑞± =1

2(𝑞 ± 𝑓𝑞𝑔). (3.41)

Remark 3.3. The three generalized OPS (3.1), (3.24), and (3.34) are formallyidentical and are now subsumed in (3.41) of Definition 3.2, where the values 𝑔 = ±𝑓are explicitly included, i.e., any pair of pure unit quaternions 𝑓, 𝑔 ∈ ℍ, 𝑓2 = 𝑔2 =−1, is admissible.

Lemma 3.4. With respect to the general OPS of Definition 3.2 we have for left andright exponential factors the identity

𝑒𝛼𝑓𝑞±𝑒𝛽𝑔 = 𝑞±𝑒(𝛽∓𝛼)𝑔 = 𝑒(𝛼∓𝛽)𝑓𝑞±. (3.42)

26 E. Hitzer and S.J. Sangwine

3.3. Geometric Interpretation of Left and Right Exponential Factors in 𝒇 , 𝒈

We obtain the following general geometric interpretation. The map 𝑓( )𝑔 alwaysrepresents a rotation by angle 𝜋 in the 𝑞−-plane (around the 𝑞+-plane), the map𝑓 𝑡( )𝑔𝑡, 𝑡 ∈ ℝ, similarly represents a rotation by angle 𝑡𝜋 in the 𝑞−-plane (aroundthe 𝑞+-plane as axis). Replacing6 𝑔 → −𝑔 in the map 𝑓( )𝑔 we further find that

𝑓𝑞±(−𝑔) = ∓𝑞±. (3.43)

Therefore the map 𝑓( )(−𝑔) = 𝑓( )𝑔−1, because 𝑔−1 = −𝑔, represents a rotationby angle 𝜋 in the 𝑞+-plane (around the 𝑞−-plane), exchanging the roles of 2Drotation plane and 2D rotation axis. Similarly, the map 𝑓𝑠( )𝑔−𝑠, 𝑠 ∈ ℝ, representsa rotation by angle 𝑠𝜋 in the 𝑞+-plane (around the 𝑞−-plane as axis).

The product of these two rotations gives

𝑓 𝑡+𝑠𝑞𝑔𝑡−𝑠 = 𝑒(𝑡+𝑠)𝜋2 𝑓𝑞𝑒(𝑡−𝑠)

𝜋2 𝑔 = 𝑒𝛼𝑓𝑞𝑒𝛽𝑔,

𝛼 = (𝑡+ 𝑠)𝜋

2, 𝛽 = (𝑡− 𝑠)

𝜋

2,

(3.44)

where based on (2.5) we used the identities 𝑓 = 𝑒𝜋2 𝑓 and 𝑔 = 𝑒

𝜋2 𝑔.

The geometric interpretation of (3.44) is a rotation by angle 𝛼 + 𝛽 in the𝑞−-plane (around the 𝑞+-plane), and a second rotation by angle 𝛼− 𝛽 in the 𝑞+-plane (around the 𝑞−-plane). For 𝛼 = 𝛽 = 𝜋/2 we recover the map 𝑓( )𝑔, and for𝛼 = −𝛽 = 𝜋/2 we recover the map 𝑓( )𝑔−1.

3.4. Determination of 𝒇, 𝒈 for Given Steerable Pair of Orthogonal 2D Planes

Equations (3.9), (3.29), and (3.37) tell us how the pair of pure unit quaternions𝑓, 𝑔 ∈ ℍ used in the general OPS of Definition 3.2, leads to an explicit basis forthe resulting two orthogonal 2D planes, the 𝑞+-plane and the 𝑞−-plane. We nowask the opposite question: how can we determine from a given steerable pair oforthogonal 2D planes in ℍ the pair of pure unit quaternions 𝑓, 𝑔 ∈ ℍ, which splitsℍ exactly into this given pair of orthogonal 2D planes?

To answer this question, we first observe that in a 4D space it is sufficientto know only one 2D plane explicitly, specified, e.g., by a pair of orthogonal unitquaternions 𝑎, 𝑏 ∈ ℍ, ∣𝑎∣ = ∣𝑏∣ = 1, and without restriction of generality 𝑏2 = −1,i.e., 𝑏 can be a pure unit quaternion 𝑏 = 𝝁(𝑏). But for 𝑎 = cos𝛼 + 𝝁(𝑎) sin𝛼,compare (2.6), we must distinguish S(𝑎) = cos𝛼 ∕= 0 and S(𝑎) = cos𝛼 = 0, i.e., of𝑎 also being a pure quaternion with 𝑎2 = −1. The second orthogonal 2D plane isthen simply the orthogonal complement in ℍ to the 𝑎, 𝑏-plane.

Let us first treat the case S(𝑎) = cos𝛼 ∕= 0. We set

𝑓 := 𝑎𝑏, 𝑔 := 𝑎𝑏. (3.45)

6Alternatively and equivalently we could replace 𝑓 → −𝑓 instead of 𝑔 → −𝑔.

2. Orthogonal 2D Planes Split 27

With this setting we get for the basis of the 𝑞−-plane

𝑓 + 𝑔 = 𝑎𝑏+ 𝑎𝑏 = 2S(𝑎) 𝑏,

1− 𝑓𝑔 = 1− 𝑎𝑏𝑎𝑏 = 1− 𝑎2𝑏2 = 1 + 𝑎2

= 1 + cos2 𝛼− sin2 𝛼+ 2𝝁(𝑎) cos𝛼 sin𝛼

= 2 cos𝛼(cos𝛼+ 𝝁(𝑎) sin𝛼) = 2 S(𝑎) 𝑎.

(3.46)

For the equality 𝑎𝑏𝑎𝑏 = 𝑎2𝑏2 we used the orthogonality of 𝑎, 𝑏, which means thatthe vector part of 𝑎 must be orthogonal to the pure unit quaternion 𝑏, i.e., it mustanticommute with 𝑏

𝑎𝑏 = 𝑏𝑎, 𝑏𝑎 = 𝑎𝑏. (3.47)

Comparing (3.9) and (3.46), the plane spanned by the two orthogonal unit quater-nions 𝑎, 𝑏 ∈ ℍ is indeed the 𝑞−-plane for S(𝑎) = cos𝛼 ∕= 0. The orthogonal 𝑞+-planeis simply given by its basis vectors (3.9), inserting (3.45). This leads to the pair oforthogonal unit quaternions 𝑐, 𝑑 for the 𝑞+-plane as

𝑐 =𝑓 − 𝑔

∣𝑓 − 𝑔∣ =𝑎𝑏− 𝑎𝑏

∣(𝑎− 𝑎)𝑏∣ =𝑎− 𝑎

∣𝑎− 𝑎∣𝑏 = 𝝁(𝑎)𝑏, (3.48)

𝑑 =1 + 𝑓𝑔

∣1 + 𝑓𝑔∣ =𝑓 − 𝑔

∣𝑓 − 𝑔∣𝑔 = 𝑐𝑔 = 𝝁(𝑎)𝑏𝑔 = 𝝁(𝑎)𝑏𝑎𝑏 = 𝝁(𝑎)𝑎𝑏2 = −𝝁(𝑎)𝑎, (3.49)

where we have used (3.13) for the second, and (3.47) for the sixth equality in(3.49).

Let us also verify that 𝑓, 𝑔 of (3.45) are both pure unit quaternions using(3.47)

𝑓2 = 𝑎𝑏𝑎𝑏 = (𝑎𝑎)𝑏𝑏 = −1, 𝑔2 = 𝑎𝑏𝑎𝑏 = (𝑎𝑎)𝑏𝑏 = −1. (3.50)

Note, that if we would set in (3.45) for 𝑔 := −𝑎𝑏, then the 𝑎, 𝑏-plane wouldhave become the 𝑞+-plane instead of the 𝑞−-plane. We can therefore determine bythe sign in the definition of 𝑔, which of the two OPS planes the 𝑎, 𝑏-plane is torepresent.

For both 𝑎 and 𝑏 being two pure orthogonal quaternions, we can again set

𝑓 := 𝑎𝑏⇒ 𝑓2 = 𝑎𝑏𝑎𝑏 = −𝑎2𝑏2 = −1, 𝑔 := 𝑎𝑏 = −𝑎𝑏 = −𝑓, (3.51)

where due to the orthogonality of the pure unit quaternions 𝑎, 𝑏 we were able touse 𝑏𝑎 = −𝑎𝑏. In this case 𝑓 = 𝑎𝑏 is thus also shown to be a pure unit quaternion.Now the 𝑞−-plane of the corresponding OPS (3.34) is spanned by {𝑎, 𝑏}, whereasthe 𝑞+-plane is spanned by {1, 𝑓}. Setting instead 𝑔 := −𝑎𝑏 = 𝑎𝑏 = 𝑓 , the 𝑞−-plane of the corresponding OPS (3.24) is spanned by {1, 𝑓}, wheras the 𝑞+-planeis spanned by {𝑎, 𝑏}.

We summarize our results in the following theorem.

Theorem 3.5. (Determination of 𝒇, 𝒈 from given steerable 2D plane) Given any2D plane in ℍ in terms of two unit quaternions 𝑎, 𝑏, where 𝑏 is without restriction

28 E. Hitzer and S.J. Sangwine

of generality pure, i.e., 𝑏2 = −1, we can make the given plane the 𝑞−-plane of theOPS 𝑞± = 1

2 (𝑞 ± 𝑓𝑞𝑔), by setting

𝑓 := 𝑎𝑏, 𝑔 := 𝑎𝑏. (3.52)

For S(𝑎) ∕= 0 the orthogonal 𝑞+-plane is fully determined by the orthogonal unitquaternions

𝑐 = 𝝁(𝑎)𝑏, 𝑑 = −𝝁(𝑎)𝑎. (3.53)

where 𝝁(𝑎) is as defined in (2.6). For S(𝑎) = 0 the orthogonal 𝑞+-plane with basis{1, 𝑓} is instead fully determined by 𝑓 = −𝑔 = 𝑎𝑏.

Setting alternatively

𝑓 := 𝑎𝑏, 𝑔 := −𝑎𝑏. (3.54)

makes the given 𝑎, 𝑏-plane the 𝑞+-plane instead. For S(𝑎) ∕= 0 the orthogonal 𝑞−-plane is then fully determined by (3.54) and (3.9), with the same orthogonal unitquaternions 𝑐 = 𝝁(𝑎)𝑏, 𝑑 = −𝝁(𝑎)𝑎 as in (3.53). For S(𝑎) = 0 the orthogonal𝑞−-plane with basis {1, 𝑓} is then instead fully determined by 𝑓 = 𝑔 = 𝑎𝑏.

An illustration of the decomposition is given in Figures 3 and 4. Again, threehundred unit pure quaternions randomly oriented in 4-space have been decom-posed into two sets using the decomposition of Definition 3.2 and two unit purequaternions 𝑓 and 𝑔 computed as in Theorem 3.5. 𝑏 was the pure unit quaternion1√3(𝒊 + 𝒋 + 𝒌) and 𝑎 was the full unit quaternion 1√

2+ 1

2 (𝒊 − 𝒋). 𝑐 and 𝑑 were

computed by (3.53) as 𝑐 = 𝝁(𝑎)𝑏 and 𝑑 = −𝝁(𝑎)𝑎.Figure 3 shows the three hundred points in 4-space, projected onto the six

orthogonal planes {𝑐, 𝑑}, {𝑐, 𝑏}, {𝑐, 𝑎}, {𝑑, 𝑏}, {𝑏, 𝑎}, {𝑎, 𝑑} where the orthonormal4-space basis {𝑐, 𝑑, 𝑏, 𝑎} = {(𝑓 − 𝑔)/∣𝑓 − 𝑔∣, (1 + 𝑓𝑔)/∣1 + 𝑓𝑔∣, (𝑓 + 𝑔)/∣𝑓 + 𝑔∣, (1−𝑓𝑔)/∣1− 𝑓𝑔∣}. The six views at the top show the 𝑞+-plane, and the six below showthe 𝑞−-plane. Figure 4 shows the vector parts of the decomposed quaternions.

4. New QFT Forms: OPS-QFTs with Two PureUnit Quaternions 𝒇, 𝒈

4.1. Generalized OPS Leads to New Steerable Type of QFT

We begin with a straightforward generalization of the (double-sided form of the)QFT [4, 5] in ℍ by replacing 𝒊 with 𝑓 and 𝒋 with 𝑔 defined as

Definition 4.1. (QFT with respect to two pure unit quaternions 𝒇, 𝒈)Let 𝑓, 𝑔 ∈ ℍ, 𝑓2 = 𝑔2 = −1, be any two pure unit quaternions. The quaternionFourier transform with respect to 𝑓, 𝑔 is

ℱ𝑓,𝑔{ℎ}(𝝎) =∫ℝ2

𝑒−𝑓𝑥1𝜔1ℎ(𝒙) 𝑒−𝑔𝑥2𝜔2𝑑2𝒙, (4.1)

where ℎ ∈ 𝐿1(ℝ2,ℍ), 𝑑2𝒙 = 𝑑𝑥1𝑑𝑥2 and 𝒙,𝝎 ∈ ℝ2.

2. Orthogonal 2D Planes Split 29

j´

0.5 0 0.5

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Figure 3. 4D scatter plot of quaternions decomposed using the orthog-onal planes split of Definition 3.2.

30 E. Hitzer and S.J. Sangwine

0.6 0.4 0.2 0 0.2 0.4 0.6

0.5

0

0.5

0.8

0.6

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0.2

0

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Figure 4. Scatter plot of vector parts of quaternions decomposed usingthe orthogonal planes split of Definition 3.2.

Note, that the pure unit quaternions 𝑓, 𝑔 in Definition 4.1 do not need to beorthogonal, and that the cases 𝑓 = ±𝑔 are fully included.

Linearity of the integral (4.1) allows us to use the OPS split ℎ = ℎ− + ℎ+

ℱ𝑓,𝑔{ℎ}(𝝎) = ℱ𝑓,𝑔{ℎ−}(𝝎) + ℱ𝑓,𝑔{ℎ+}(𝝎)= ℱ𝑓,𝑔

− {ℎ}(𝝎) + ℱ𝑓,𝑔+ {ℎ}(𝝎),

(4.2)

since by their construction the operators of the Fourier transformation ℱ𝑓,𝑔, andof the OPS with respect to 𝑓, 𝑔 commute. From Lemma 3.4 follows

Theorem 4.2 (QFT of ℎ±). The QFT of the ℎ± OPS split parts, with respect totwo unit quaternions 𝑓, 𝑔, of a quaternion module function ℎ ∈ 𝐿1(ℝ2,ℍ) have thequasi-complex forms

ℱ𝑓,𝑔± {ℎ} = ℱ𝑓,𝑔{ℎ±} =

∫ℝ2

ℎ±𝑒−𝑔(𝑥2𝜔2∓𝑥1𝜔1)𝑑2𝑥

=

∫ℝ2

𝑒−𝑓(𝑥1𝜔1∓𝑥2𝜔2)ℎ±𝑑2𝑥 .

(4.3)

Remark 4.3. The quasi-complex forms in Theorem 4.2 allow us to establish dis-cretized and fast versions of the QFT of Definition 4.1 as sums of two complexdiscretized and fast Fourier transformations (FFT), respectively.

2. Orthogonal 2D Planes Split 31

−−

+

−

−

+

+

−

Figure 5. Geometric interpretation of integrand of QFT𝑓,𝑔 in Defini-tion 4.1 in terms of local phase rotations in 𝑞±-planes.

We can now give a geometric interpretation of the integrand of the QFT𝑓,𝑔 inDefinition 4.1 in terms of local phase rotations, compare Section 3.3. The integrandproduct

𝑒−𝑓𝑥1𝜔1ℎ(𝒙) 𝑒−𝑔𝑥2𝜔2 (4.4)

represents a local rotation by the phase angle −(𝑥1𝜔1 + 𝑥2𝜔2) in the 𝑞−-plane,and by the phase angle −(𝑥1𝜔1 − 𝑥2𝜔2) = 𝑥2𝜔2 − 𝑥1𝜔1 in the orthogonal 𝑞+-plane, compare Figure 5, which depicts two completely orthogonal planes in fourdimensions.

Based on Theorem 3.5 the two phase rotation planes (analysis planes) canbe freely steered by defining the two pure unit quaternions 𝑓, 𝑔 used in Definition4.1 according to (3.52) or (3.54).

4.2. Two Phase Angle Version of QFT

The above newly gained geometric understanding motivates us to propose a furthernew version of the QFT𝑓,𝑔, with a straightforward two phase angle interpretation.

Definition 4.4. (Phase angle QFT with respect to 𝒇, 𝒈)Let 𝑓, 𝑔 ∈ ℍ, 𝑓2 = 𝑔2 = −1, be any two pure unit quaternions. The phase anglequaternion Fourier transform with respect to 𝑓, 𝑔 is

ℱ𝑓,𝑔𝐷 {ℎ}(𝝎) =

∫ℝ2

𝑒−𝑓12 (𝑥1𝜔1+𝑥2𝜔2)ℎ(𝒙) 𝑒−𝑔

12 (𝑥1𝜔1−𝑥2𝜔2)𝑑2𝒙. (4.5)

where again ℎ ∈ 𝐿1(ℝ2,ℍ), 𝑑2𝒙 = 𝑑𝑥1𝑑𝑥2 and 𝒙,𝝎 ∈ ℝ2.

The geometric interpretation of the integrand of (4.5) is a local phase rotationby angle −(𝑥1𝜔1 + 𝑥2𝜔2)/2 − (𝑥1𝜔1 − 𝑥2𝜔2)/2 = −𝑥1𝜔1 in the 𝑞−-plane, and asecond local phase rotation by angle −(𝑥1𝜔1+𝑥2𝜔2)/2+(𝑥1𝜔1−𝑥2𝜔2)/2 = −𝑥2𝜔2

in the 𝑞+-plane, compare Section 3.3.If we apply the OPS𝑓,𝑔 split to (4.5) we obtain the following theorem.

32 E. Hitzer and S.J. Sangwine

Theorem 4.5 (Phase angle QFT of 𝒉±). The phase angle QFT of Definition 4.4applied to the ℎ± OPS split parts, with respect to two pure unit quaternions 𝑓, 𝑔, ofa quaternion module function ℎ ∈ 𝐿1(ℝ2,ℍ) leads to the quasi-complex expressions

ℱ𝑓,𝑔𝐷+{ℎ} = ℱ𝑓,𝑔

𝐷 {ℎ+} =∫ℝ2

ℎ+𝑒+𝑔𝑥2𝜔2𝑑2𝑥 =

∫ℝ2

𝑒−𝑓𝑥2𝜔2ℎ+𝑑2𝑥 , (4.6)

ℱ𝑓,𝑔𝐷−{ℎ} = ℱ𝑓,𝑔

𝐷 {ℎ−} =∫ℝ2

ℎ−𝑒−𝑔𝑥1𝜔1𝑑2𝑥 =

∫ℝ2

𝑒−𝑓𝑥1𝜔1ℎ−𝑑2𝑥. (4.7)

Note that based on Theorem 3.5 the two phase rotation planes (analysisplanes) are again freely steerable.

Theorem 4.5 allows us to establish discretized and fast versions of the phaseangle QFT of Definition 4.4 as sums of two complex discretized and fast Fouriertransformations (FFT), respectively.

The maps 𝑓( )𝑔 considered so far did not involve quaternion conjugation 𝑞 →𝑞. In the following we investigate maps which additionally conjugate the argument,

i.e., of type 𝑓( )𝑔, which are also involutions.

5. Involutions and QFTs Involving Quaternion Conjugation

5.1. Involutions Involving Quaternion Conjugations

The simplest case is quaternion conjugation itself

𝑞 → 𝑞 = 𝑞𝑟 − 𝑞𝑖𝒊− 𝑞𝑗𝒋 − 𝑞𝑘𝒌, (5.1)

which can be interpreted as a reflection at the real line 𝑞𝑟. The real line through theorigin remains pointwise invariant, while every other point in the 3D hyperplaneof pure quaternions is reflected to the opposite side of the real line. The relatedinvolution

𝑞 → −𝑞 = −𝑞𝑟 + 𝑞𝑖𝒊+ 𝑞𝑗𝒋 + 𝑞𝑘𝒌, (5.2)

is the reflection at the 3D hyperplane of pure quaternions (which stay invariant),i.e., only the real line is changed into its negative 𝑞𝑟 → −𝑞𝑟.

Similarly any pure unit quaternion factor like 𝒊 in the map

𝑞 → 𝒊 𝑞𝒊 = −𝑞𝑟 + 𝑞𝑖𝒊− 𝑞𝑗𝒋 − 𝑞𝑘𝒌, (5.3)

leads to a reflection at the (pointwise invariant) line through the origin with di-rection 𝒊, while the map

𝑞 → −𝒊 𝑞𝒊 = 𝑞𝑟 − 𝑞𝑖𝒊+ 𝑞𝑗𝒋 + 𝑞𝑘𝒌, (5.4)

leads to a reflection at the invariant 3D hyperplane orthogonal to the line throughthe origin with direction 𝒊. The map

𝑞 → 𝑓 𝑞𝑓, (5.5)

leads to a reflection at the (pointwise invariant) line with direction 𝑓 through theorigin, while the map

𝑞 → −𝑓 𝑞𝑓, (5.6)

2. Orthogonal 2D Planes Split 33

leads to a reflection at the invariant 3D hyperplane orthogonal to the line withdirection 𝑓 through the origin.

Next we turn to a map of the type

𝑞 → −𝑒𝛼𝑓𝑞𝑒𝛼𝑓 . (5.7)

Its set of pointwise invariants is given by

𝑞 = −𝑒𝛼𝑓𝑞𝑒𝛼𝑓 ⇔ 𝑒−𝛼𝑓𝑞 = −𝑞𝑒𝛼𝑓 ⇔ 𝑒−𝛼𝑓𝑞 + 𝑞𝑒𝛼𝑓 = 0

⇔ S(𝑞𝑒𝛼𝑓

)= 0 ⇔ 𝑞 ⊥ 𝑒𝛼𝑓 .

(5.8)

We further observe that

𝑒𝛼𝑓 → −𝑒𝛼𝑓𝑒−𝛼𝑓𝑒𝛼𝑓 = −𝑒𝛼𝑓 . (5.9)

The map −𝑎( )𝑎, with unit quaternion 𝑎 = 𝑒𝛼𝑓 , therefore represents a reflectionat the invariant 3D hyperplane orthogonal to the line through the origin withdirection 𝑎.

Similarly, the map 𝑎( )𝑎, with unit quaternion 𝑎 = 𝑒𝛼𝑓 , then represents areflection at the (pointwise invariant) line with direction 𝑎 through the origin.

The combination of two such reflections (both at 3D hyperplanes, or both atlines), given by unit quaternions 𝑎, 𝑏, leads to a rotation

−𝑏−𝑎𝑞𝑎𝑏 = 𝑏𝑎𝑞𝑎𝑏 = 𝑏𝑎𝑞𝑎𝑏 = 𝑟𝑞𝑠,

𝑟 = 𝑏𝑎, 𝑠 = 𝑎𝑏, ∣𝑟∣ = ∣𝑏∣ ∣𝑎∣ = 1 = ∣𝑠∣ , (5.10)

in two orthogonal planes, exactly as studied in Section 3.3.

The combination of three reflections at 3D hyperplanes, given by unit quater-nions 𝑎, 𝑏, 𝑐, leads to

−𝑐[−𝑏−𝑎𝑞𝑎𝑏]𝑐 = 𝑑 𝑞𝑡, 𝑑 = −𝑐𝑏𝑎, 𝑡 = 𝑎𝑏𝑐, ∣𝑑∣ = ∣𝑐∣ ∣𝑏∣ ∣𝑎∣ = ∣𝑡∣ = 1. (5.11)

The product of the reflection map −𝑞 of (5.2) with 𝑑 𝑞𝑡 leads to −𝑑𝑞𝑡, a double

rotation as studied in Section 4. Therefore 𝑑 ( ) 𝑡 represents a rotary reflection(rotation reflection). The three reflections −𝑎𝑞𝑎, −𝑏𝑞𝑏, −𝑐𝑞𝑐 have the intersectionof the three 3D hyperplanes as a resulting common pointwise invariant line, whichis 𝑑+ 𝑡, because

𝑑 (𝑑+ 𝑡) 𝑡 = 𝑑 𝑡𝑡+ 𝑑 𝑑𝑡 = 𝑑+ 𝑡. (5.12)

In the remaining 3D hyperplane, orthogonal to the pointwise invariant line throughthe origin in direction 𝑑+ 𝑡, the axis of the rotary reflection is

𝑑 (𝑑− 𝑡) 𝑡 = −𝑑 𝑡𝑡+ 𝑑 𝑑𝑡 = −𝑑+ 𝑡 = −(𝑑− 𝑡). (5.13)

We now also understand that a sign change of 𝑑 → −𝑑 (compare three reflec-

tions at three 3D hyperplanes −𝑐[−𝑏(−𝑎𝑞𝑎)𝑏]𝑐 with three reflections at three lines

+𝑐[+𝑏(+𝑎𝑞𝑎)𝑏]𝑐) simply exchanges the roles of pointwise invariant line 𝑑 + 𝑡 androtary reflection axis 𝑑− 𝑡.

34 E. Hitzer and S.J. Sangwine

Next, we seek for an explicit description of the rotation plane of the rotary

reflection 𝑑 ( ) 𝑡. We find that for the unit quaternions 𝑑 = 𝑒𝛼𝑔, 𝑡 = 𝑒𝛽𝑓 the com-mutator

[𝑑, 𝑡] = 𝑑𝑡− 𝑡𝑑 = 𝑒𝛼𝑔𝑒𝛽𝑓 − 𝑒𝛽𝑓𝑒𝛼𝑔 = (𝑔𝑓 − 𝑓𝑔) sin𝛼 sin𝛽, (5.14)

is a pure quaternion, because

𝑔𝑓 − 𝑓𝑔 = 𝑓𝑔 − 𝑔𝑓 = −(𝑔𝑓 − 𝑓𝑔). (5.15)

Moreover, [𝑑, 𝑡] is orthogonal to 𝑑 and 𝑡, and therefore orthogonal to the planespanned by the pointwise invariant line 𝑑+ 𝑡 and the rotary reflection axis 𝑑− 𝑡,because

S([𝑑, 𝑡]𝑑

)= S(𝑑𝑡𝑑− 𝑡𝑑 𝑑

)= 0, S

([𝑑, 𝑡]𝑡

)= 0. (5.16)

We obtain a second quaternion in the plane orthogonal to 𝑑 + 𝑡, and 𝑑 − 𝑡, byapplying the rotary reflection to [𝑑, 𝑡]

𝑑 [𝑑, 𝑡] 𝑡 = −𝑑[𝑑, 𝑡]𝑡 = −[𝑑, 𝑡]𝑑𝑡, (5.17)

because 𝑑 is orthogonal to the pure quaternion [𝑑, 𝑡]. We can construct an or-

thogonal basis of the plane of the rotary reflection 𝑑 ( ) 𝑡 by computing the pair oforthogonal quaternions

𝑣1,2 = [𝑑, 𝑡]∓ 𝑑 [𝑑, 𝑡] 𝑡 = [𝑑, 𝑡]± [𝑑, 𝑡]𝑑𝑡 = [𝑑, 𝑡](1 ± 𝑑𝑡). (5.18)

For finally computing the rotation angle, we need to know the relative length ofthe two orthogonal quaternions 𝑣1, 𝑣2 of (5.18). For this it helps to represent theunit quaternion 𝑑𝑡 as

𝑑𝑡 = 𝑒𝛾𝑢, 𝛾 ∈ ℝ, 𝑢 ∈ ℍ, 𝑢2 = −1. (5.19)

We then obtain for the length ratio

𝑟2 =∣𝑣1∣2∣𝑣2∣2 =

∣1 + 𝑑𝑡∣2∣1− 𝑑𝑡∣2 =

(1 + 𝑑𝑡)(1 + 𝑡𝑑)

(1 − 𝑑𝑡)(1− 𝑡𝑑)=

1 + 𝑑𝑡𝑡𝑑+ 𝑑𝑡+ 𝑡𝑑

1 + 𝑑𝑡𝑡𝑑− 𝑑𝑡− 𝑡𝑑

=2 + 2 cos 𝛾

2− 2 cos 𝛾=

1 + cos 𝛾

1− cos 𝛾.

(5.20)

By applying the rotary reflection 𝑑 ( ) 𝑡 to 𝑣1 and decomposing the result withrespect to the pair of orthogonal quaternions in the rotary reflection plane (5.18)we can compute the rotation angle. Applying the rotary reflection to 𝑣1 gives

𝑑 𝑣1 𝑡 = 𝑑 [𝑑, 𝑡]− 𝑑[𝑑, 𝑡]𝑡 𝑡 = 𝑑[𝑑, 𝑡](1 + 𝑑𝑡)𝑡 = 𝑑(1 + 𝑡𝑑)[𝑑, 𝑡]𝑡

= 𝑑(1 + 𝑡𝑑)(−[𝑑, 𝑡])𝑡 = −[𝑑, 𝑡](𝑑𝑡+ 𝑑𝑡𝑑𝑡).(5.21)

The square of 𝑑𝑡 is

(𝑑𝑡)2 = (cos 𝛾 + 𝑢 sin 𝛾)2 = −1 + 2 cos 𝛾 [cos 𝛾 + 𝑢 sin 𝛾]

= −1 + 2 cos 𝛾 𝑑𝑡.(5.22)

2. Orthogonal 2D Planes Split 35

We therefore get

𝑑 𝑣1 𝑡 = −[𝑑, 𝑡](𝑑𝑡− 1 + 2 cos𝛾𝑑𝑡) = [𝑑, 𝑡](1− (1 + 2 cos𝛾)𝑑𝑡)

= 𝑎1𝑣1 + 𝑎2𝑟𝑣2,(5.23)

and need to solve

1− (1 + 2 cos 𝛾)𝑑𝑡 = 𝑎1(1 + 𝑑𝑡) + 𝑎2𝑟(1 − 𝑑𝑡), (5.24)

which leads to

𝑑 𝑣1 𝑡 = − cos𝛾𝑣1 + sin 𝛾𝑟𝑣2 = cos(𝜋 − 𝛾)𝑣1 + sin(𝜋 − 𝛾)𝑟𝑣2. (5.25)

The rotation angle of the rotary reflection 𝑑 ( ) 𝑡 in its rotation plane 𝑣1, 𝑣2 istherefore

Γ = 𝜋 − 𝛾, 𝛾 = arccosS(𝑑𝑡). (5.26)

In terms of 𝑑 = 𝑒𝛼𝑔, 𝑡 = 𝑒𝛽𝑓 we get

𝑑𝑡 = cos𝛼 cos𝛽 − 𝑔 sin𝛼 cos𝛽 + 𝑓 cos𝛼 sin𝛽 − 𝑔𝑓 sin𝛼 sin𝛽. (5.27)

And with the angle 𝜔 between 𝑔 and 𝑓

𝑔𝑓 =1

2(𝑔𝑓 + 𝑓𝑔) +

1

2(𝑔𝑓 − 𝑓𝑔) = S(𝑔𝑓) +

1

2[𝑔, 𝑓 ]

= − cos𝜔 − sin𝜔[𝑔, 𝑓 ]

∣[𝑔, 𝑓 ]∣ ,(5.28)

we finally obtain for 𝛾 the scalar part S(𝑑𝑡)as

S(𝑑𝑡)= cos 𝛾 = cos𝛼 cos𝛽 + cos𝜔 sin𝛼 sin𝛽

= cos𝛼 cos𝛽 − S(𝑔𝑓) sin𝛼 sin𝛽.(5.29)

In the special case of 𝑔 = ±𝑓 , S(𝑔𝑓) = ∓1, i.e., for 𝜔 = 0, 𝜋, we get from(5.29) that

S(𝑑𝑡)= cos𝛼 cos𝛽 ± sin𝛼 sin𝛽 = cos𝛼 cos𝛽 + sin(±𝛼) sin𝛽

= cos(±𝛼− 𝛽), (5.30)

and thus using (5.26) the rotation angle would become

Γ = 𝜋 − (±𝛼− 𝛽) = 𝜋 ∓ 𝛼+ 𝛽. (5.31)

Yet (5.26) was derived assuming [𝑑, 𝑡] ∕= 0. But direct inspection shows that (5.31)is indeed correct: For 𝑔 = ±𝑓 the plane 𝑑 + 𝑡, 𝑑 − 𝑡 is identical to the 1, 𝑓 plane.The rotation plane is thus a plane of pure quaternions orthogonal to the 1, 𝑓 plane.The quaternion conjugation in 𝑞 �→ 𝑑 𝑞 𝑡 leads to a rotation by 𝜋 and the left andright factors lead to further rotations by ∓𝛼 and 𝛽, respectively. Thus (5.31) isverified as a special case of (5.26) for 𝑔 = ±𝑓 .

By substituting in Lemma 3.4 (𝛼, 𝛽)→ (−𝛽,−𝛼), and by taking the quater-nion conjugate we obtain the following lemma.

36 E. Hitzer and S.J. Sangwine

Lemma 5.1. Let 𝑞± = 12 (𝑞 ± 𝑓𝑞𝑔) be the OPS of Definition 3.2. For left and right

exponential factors we have the identity

𝑒𝛼𝑔 𝑞±𝑒𝛽𝑓 = 𝑞±𝑒(𝛽∓𝛼)𝑓 = 𝑒(𝛼∓𝛽)𝑔 𝑞±. (5.32)

5.2. New Steerable QFTs with Quaternion Conjugation andTwo Pure Unit Quaternions 𝒇, 𝒈

We therefore consider now the following new variant of the (double-sided form ofthe) QFT [4, 5] in ℍ (replacing both 𝒊 with 𝑔 and 𝒋 with 𝑓 , and using quaternionconjugation). It is essentially the quaternion conjugate of the new QFT of Def-inition 4.1, but because of its distinct local transformation geometry it deservesseparate treatment.

Definition 5.2. (QFT with respect to 𝒇, 𝒈, including quaternion conjugation)Let 𝑓, 𝑔 ∈ ℍ, 𝑓2 = 𝑔2 = −1, be any two pure unit quaternions. The quaternionFourier transform with respect to 𝑓, 𝑔, involving quaternion conjugation, is

ℱ𝑔,𝑓𝑐 {ℎ}(𝝎) = ℱ𝑓,𝑔{ℎ}(−𝝎) =

∫ℝ2

𝑒−𝑔𝑥1𝜔1ℎ(𝒙) 𝑒−𝑓𝑥2𝜔2𝑑2𝒙, (5.33)

where ℎ ∈ 𝐿1(ℝ2,ℍ), 𝑑2𝒙 = 𝑑𝑥1𝑑𝑥2 and 𝒙,𝝎 ∈ ℝ2.

Linearity of the integral in (5.33) of Definition 5.2 leads to the followingcorollary to Theorem 4.2.

Corollary 5.3 (QFT ℱ𝑔,𝑓𝑐 of ℎ±). The QFT ℱ𝑔,𝑓

𝑐 (5.33) of the ℎ± = 12 (ℎ ± 𝑓ℎ𝑔)

OPS split parts, with respect to any two unit quaternions 𝑓, 𝑔, of a quaternionmodule function ℎ ∈ 𝐿1(ℝ2,ℍ) have the quasi-complex forms

ℱ𝑔,𝑓𝑐 {ℎ±}(𝝎) = ℱ𝑓,𝑔{ℎ±}(−𝝎) =

∫ℝ2

ℎ±𝑒−𝑓(𝑥2𝜔2∓𝑥1𝜔1)𝑑2𝑥

=

∫ℝ2

𝑒−𝑔(𝑥1𝜔1∓𝑥2𝜔2)ℎ±𝑑2𝑥 .

(5.34)

Note, that the pure unit quaternions 𝑓, 𝑔 in Definition 5.2 and Corollary5.3 do not need to be orthogonal, and that the cases 𝑓 = ±𝑔 are fully included.Corollary 5.3 leads to discretized and fast versions of the QFT with quaternionconjugation of Definition 5.2.

It is important to note that the roles (sides) of 𝑓, 𝑔 appear exchanged in(5.33) of Definition 5.2 and in Corollary 5.3, although the same OPS of Definition3.2 is applied to the signal ℎ as in Sections 3 and 4. This role change is due tothe presence of quaternion conjugation in Definition 5.2. Note that it is possibleto first apply (5.33) to ℎ, and subsequently split the integral with the OPS𝑔,𝑓

ℱ𝑐,± = 12 (ℱ𝑐± 𝑔ℱ𝑐𝑓), where the particular order of 𝑔 from the left and 𝑓 from the

right is due to the application of conjugation in (5.34) to ℎ± after ℎ is split with(3.41) into ℎ+ and ℎ−.

2. Orthogonal 2D Planes Split 37

5.3. Local Geometric Interpretation of the QFT with Quaternion Conjugation

Regarding the local geometric interpretation of the QFT with quaternion conju-gation of Definition 5.2 we need to distinguish the following cases, depending on[𝑑, 𝑡] and on whether the left and right phase factors

𝑑 = 𝑒−𝑔𝑥1𝜔1 , 𝑡 = 𝑒−𝑓𝑥2𝜔2 , (5.35)

attain scalar values ±1 or not.Let us first assume that [𝑑, 𝑡] ∕= 0, which by (5.14) is equivalent to 𝑔 ∕= ±𝑓 ,

and sin(𝑥1𝜔1) ∕= 0, and sin(𝑥2𝜔2) ∕= 0. Then we have the generic case of a localrotary reflection with pointwise invariant line of direction

𝑑+ 𝑡 = 𝑒−𝑔𝑥1𝜔1 + 𝑒−𝑓𝑥2𝜔2 , (5.36)

rotation axis in direction

𝑑− 𝑡 = 𝑒−𝑔𝑥1𝜔1 − 𝑒−𝑓𝑥2𝜔2 , (5.37)

rotation plane with basis (5.18), and by (5.26) and (5.29) the general rotationangle

Γ = 𝜋 − arccosS(𝑑𝑡),

S(𝑑𝑡)= cos(𝑥1𝜔1) cos(𝑥2𝜔2)− S(𝑔𝑓) sin(𝑥1𝜔1) sin(𝑥2𝜔2). (5.38)

Whenever 𝑔 = ±𝑓 , or when sin(𝑥1𝜔1) = 0 (𝑥1𝜔1 = 0, 𝜋[mod 2𝜋], i.e., 𝑑 =±1), we get for the pointwise invariant line in direction 𝑑 + 𝑡 the simpler unit

quaternion direction expression 𝑒−12 (±𝑥1𝜔1+𝑥2𝜔2)𝑓 , because we can apply

𝑒𝛼𝑓 + 𝑒𝛽𝑓 = 𝑒12 (𝛼+𝛽)𝑓(𝑒

12 (𝛼−𝛽)𝑓 + 𝑒

12 (𝛽−𝛼)𝑓 ) = 𝑒

12 (𝛼+𝛽)𝑓2 cos

𝛼− 𝛽

2, (5.39)

and similarly for the rotation axis 𝑑− 𝑡 we obtain the direction expression

𝑒−12 (±𝑥1𝜔1+𝑥2𝜔2+𝜋)𝑓 ,

whereas the rotation angle is by (5.31) simply

Γ = 𝜋 ± 𝑥1𝜔1 − 𝑥2𝜔2. (5.40)

For sin(𝑥2𝜔2) = 0 (𝑥2𝜔2 = 0, 𝜋[mod2𝜋], i.e., 𝑡 = ±1), the pointwise invariantline in direction 𝑑+ 𝑡 simplifies by (5.39) to 𝑒−

12 (𝑥1𝜔1+𝑥2𝜔2)𝑔, and the rotation axis

with direction 𝑑− 𝑡 simplifies to 𝑒−12 (𝑥1𝜔1+𝑥2𝜔2+𝜋)𝑔, whereas the angle of rotation

is by (5.31) simplyΓ = 𝜋 + 𝑥1𝜔1 − 𝑥2𝜔2. (5.41)

5.4. Phase Angle QFT with Respect to 𝒇, 𝒈, Including Quaternion Conjugation

Even when quaternion conjugation is applied to the signal ℎ we can propose afurther new version of the QFT𝑔,𝑓

𝑐 , with a straightforward two phase angle inter-pretation. The following definition to some degree ignores the resulting local rotaryreflection effect of combining quaternion conjugation and left and right phase fac-tors of Section 5.3, but depending on the application context, it may neverthelessbe of interest in its own right.

38 E. Hitzer and S.J. Sangwine

Definition 5.4 (Phase angle QFTwith respect to 𝒇, 𝒈, including quaternion con-jugation). Let 𝑓, 𝑔 ∈ ℍ, 𝑓2 = 𝑔2 = −1, be any two pure unit quaternions. Thephase angle quaternion Fourier transform with respect to 𝑓, 𝑔, involving quaternionconjugation, is

ℱ𝑔,𝑓𝑐𝐷 {ℎ}(𝝎) = ℱ𝑓,𝑔

𝐷 {ℎ}(−𝜔1, 𝜔2) =

∫ℝ2

𝑒−𝑔12 (𝑥1𝜔1+𝑥2𝜔2)ℎ(𝒙) 𝑒−𝑓

12 (𝑥1𝜔1−𝑥2𝜔2)𝑑2𝒙.

(5.42)where again ℎ ∈ 𝐿1(ℝ2,ℍ), 𝑑2𝒙 = 𝑑𝑥1𝑑𝑥2 and 𝒙,𝝎 ∈ ℝ2.

Based on Lemma 5.1, one possible geometric interpretation of the integrandof (5.42)) is a local phase rotation of ℎ+ by angle −(𝑥1𝜔1 − 𝑥2𝜔2)/2 + (𝑥1𝜔1 +

𝑥2𝜔2)/2 = +𝑥2𝜔2 in the 𝑞+ plane, and a second local phase rotation of ℎ− by angle−(𝑥1𝜔1 − 𝑥2𝜔2)/2− (𝑥1𝜔1 + 𝑥2𝜔2)/2 = −𝑥1𝜔1 in the 𝑞− plane. This is expressedin the following corollary to Theorem 4.5.

Corollary 5.5 (Phase angle QFT of 𝒉±, involving quaternion conjugation). Thephase angle QFT with quaternion conjugation of Definition 5.4 applied to the ℎ±OPS split parts, with respect to any two pure unit quaternions 𝑓, 𝑔, of a quaternionmodule function ℎ ∈ 𝐿1(ℝ2,ℍ) leads to the quasi-complex expressions

ℱ𝑔,𝑓𝑐𝐷 {ℎ+}(𝝎) = ℱ𝑔,𝑓

𝐷 {ℎ+}(−𝜔1, 𝜔2) =

∫ℝ2

ℎ+𝑒+𝑓𝑥2𝜔2𝑑2𝑥 =

∫ℝ2

𝑒−𝑔𝑥2𝜔2ℎ+𝑑2𝑥 ,

(5.43)

ℱ𝑔,𝑓𝑐𝐷 {ℎ−}(𝝎) = ℱ𝑔,𝑓

𝐷 {ℎ−}(−𝜔1, 𝜔2) =

∫ℝ2

ℎ−𝑒−𝑓𝑥1𝜔1𝑑2𝑥 =

∫ℝ2

𝑒−𝑔𝑥1𝜔1ℎ−𝑑2𝑥 .

(5.44)

Note that based on Theorem 3.5 the two phase rotation planes (analysisplanes) are again freely steerable. Corollary 5.5 leads to discretized and fast ver-sions of the phase angle QFT with quaternion conjugation of Definition 5.4.

6. Conclusion

The involution maps 𝒊( )𝒋 and 𝑓( )𝑔 have led us to explore a range of similar quater-nionic maps 𝑞 �→ 𝑎𝑞𝑏 and 𝑞 �→ 𝑎𝑞𝑏, where 𝑎, 𝑏 are taken to be unit quaternions.Geometric interpretations of these maps as reflections, rotations, and rotary reflec-tions in 4D can mostly be found in [1]. We have further developed these geometricinterpretations to gain a complete local transformation geometric understandingof the integrands of the proposed new quaternion Fourier transformations (QFTs)applied to general quaternionic signals ℎ ∈ 𝐿1(ℝ2,ℍ). This new geometric under-standing is also valid for the special cases of the hitherto well-known left-sided,right-sided, and left- and right-sided (two-sided) QFTs of [2, 4, 8, 3] and numerousother references.

Our newly gained geometric understanding itself motivated us to proposenew types of QFTs with specific geometric properties. The investigation of these

2. Orthogonal 2D Planes Split 39

new types of QFTs with the generalized form of the orthogonal 2D planes splitof Definition 3.2 lead to important QFT split theorems, which allow the use ofdiscrete and (complex) Fourier transform software for efficient discretized and fastnumerical implementations.

Finally, we are convinced that our geometric interpretation of old and newQFTs paves the way for new applications, e.g., regarding steerable filter design forspecific tasks in image, colour image and signal processing, etc.

Acknowledgement. E.H. wishes to thank God for the joy of doing this research,his family, and S.J.S. for his great cooperation and hospitality.

References

[1] H.S.M. Coxeter. Quaternions and reflections. The American Mathematical Monthly,53(3):136–146, Mar. 1946.

[2] T.A. Ell. Quaternion-Fourier transforms for analysis of 2-dimensional linear time-invariant partial-differential systems. In Proceedings of the 32nd Conference on De-cision and Control, pages 1830–1841, San Antonio, Texas, USA, 15–17 December1993. IEEE Control Systems Society.

[3] T.A. Ell and S.J. Sangwine. Hypercomplex Fourier transforms of color images. IEEETransactions on Image Processing, 16(1):22–35, Jan. 2007.

[4] E. Hitzer. Quaternion Fourier transform on quaternion fields and generalizations.Advances in Applied Clifford Algebras, 17(3):497–517, May 2007.

[5] E. Hitzer. Directional uncertainty principle for quaternion Fourier transforms. Ad-vances in Applied Clifford Algebras, 20(2):271–284, 2010.

[6] E. Hitzer. OPS-QFTs: A new type of quaternion Fourier transform based on theorthogonal planes split with one or two general pure quaternions. In InternationalConference on Numerical Analysis and Applied Mathematics, volume 1389 of AIPConference Proceedings, pages 280–283, Halkidiki, Greece, 19–25 September 2011.American Institute of Physics.

[7] E. Hitzer. Two-sided Clifford Fourier transform with two square roots of −1 in 𝐶ℓ𝑝,𝑞.In Proceedings of the 5th Conference on Applied Geometric Algebras in ComputerScience and Engineering (AGACSE 2012), La Rochelle, France, 2–4 July 2012.

[8] S.J. Sangwine. Fourier transforms of colour images using quaternion, or hypercom-plex, numbers. Electronics Letters, 32(21):1979–1980, 10 Oct. 1996.

Eckhard HitzerCollege of Liberal Arts, Department of Material ScienceInternational Christian University, 181-8585 Tokyo, Japane-mail: hitzer@icu.ac.jp

Stephen J. SangwineSchool of Computer Science and Electronic EngineeringUniversity of Essex, Wivenhoe Park, Colchester, CO4 3SQ, UKe-mail: sjs@essex.ac.uk

Quaternion and Clifford–Fourier Transforms and Wavelets

Trends in Mathematics, 41–56c⃝ 2013 Springer Basel

3 Quaternionic Spectral Analysis ofNon-Stationary Improper Complex Signals

Nicolas Le Bihan and Stephen J. Sangwine

Abstract. We consider the problem of the spectral content of a complex im-proper signal and the time-varying behaviour of this spectral content. Thesignals considered are one-dimensional (1D), complex-valued, with possiblecorrelation between the real and imaginary parts, i.e., improper complex sig-nals. As a consequence, it is well known that the ‘classical’ (complex-valued)Fourier transform does not exhibit Hermitian symmetry and also that it isnecessary to consider simultaneously the spectrum and the pseudo-spectrumto completely characterize such signals. Hence, an ‘augmented’ representa-tion is necessary. However, this does not provide a geometric analysis of thecomplex improper signal.

We propose another approach for the analysis of improper complex sig-nals based on the use of a 1D Quaternion Fourier Transform (QFT). In thecase where complex signals are non-stationary, we investigate the extension ofthe well-known ‘analytic signal’ and introduce the quaternion-valued ‘hyper-analytic signal’. As with the hypercomplex two-dimensional (2D) extensionof the analytic signal proposed by Bulow in 2001, our extension of analyticsignals for complex-valued signals can be obtained by the inverse QFT of thequaternion-valued spectrum after suppressing negative frequencies.

Analysis of the hyperanalytic signal reveals the time-varying frequencycontent of the corresponding complex signal. Using two different represen-tations of quaternions, we show how modulus and quaternion angles of thehyperanalytic signal are linked to geometric features of the complex signal.This allows the definition of the angular velocity and the complex envelope ofa complex signal. These concepts are illustrated on synthetic signal examples.

The hyperanalytic signal can be seen as the exact counterpart of theclassical analytic signal, and should be thought of as the very first and sim-plest quaternionic time-frequency representation for improper non-stationarycomplex-valued signals.

Mathematics Subject Classification (2010). 65T50, 11R52.

Keywords. Quaternions, complex signals, Fourier transform, analytic signal.

42 N. Le Bihan and S.J. Sangwine

1. Introduction

The analytic signal has been known since 1948 from the work of Ville [21] andGabor [7]. It can be easily described, even though its theoretical ramifications aredeep. Its use in non-stationary signal analysis is routine and it has been used innumerous applications. Simply put, given a real-valued signal 𝑓(𝑡), its analyticsignal 𝑎(𝑡) is a complex signal with real part equal to 𝑓(𝑡), and imaginary partorthogonal to 𝑓(𝑡). The imaginary part is sometimes known as the quadraturesignal – in the case where 𝑓(𝑡) is a sinusoid, the imaginary part of the analyticsignal is in quadrature, that is with a phase difference of −𝜋/2. The orthogonalsignal is related to 𝑓(𝑡) by the Hilbert transform [9, 10]. The analytic signal hasthe interesting property that its modulus ∣𝑎(𝑡)∣ is an envelope of the signal 𝑓(𝑡).The envelope is also known as the instantaneous amplitude. Thus if 𝑓(𝑡) is anamplitude-modulated sinusoid, the envelope ∣𝑎(𝑡)∣, subject to certain conditions,is the modulating signal. The argument of the analytic signal, ∠ 𝑎(𝑡) is known asthe instantaneous phase. The analytic signal has a third interesting property: ithas a one-sided Fourier transform. Thus a simple algorithm for constructing theanalytic signal (algebraically or numerically) is to compute the Fourier transformof 𝑓(𝑡), multiply the Fourier transform by a unit step which is zero for negativefrequencies, and then construct the inverse Fourier transform.

In this chapter we extend the concept of the analytic signal from the case ofa real signal 𝑓(𝑡) with a complex analytic signal 𝑎(𝑡), to a complex signal 𝑧(𝑡) witha hypercomplex analytic signal ℎ(𝑡), which we call the hyperanalytic signal. Just asthe classical complex analytic signal contains both the original real signal (in thereal part) and a real orthogonal signal (in the imaginary part), a hyperanalyticsignal contains two complex signals: the original signal and an orthogonal signal.We have previously published partial results on this topic [19, 15, 14]. Here wedevelop a clear idea of how to generalise the classic case of amplitude modulationto a complex signal, and show for the first time that this leads to a correctlyextended analytic signal concept in which the (complex ) envelope and phase haveclear interpretations.

The construction of an orthogonal signal alone would not constitute a fullgeneralisation of the classical analytic signal to the complex case: it is also neces-sary to generalise the envelope and phase concepts, and this can only be done byinterpreting the original and orthogonal complex signals as a pair. In this chapterwe have only one way to do this: by representing the pair of complex signals asa quaternion signal. This arises naturally from the method above for creating anorthogonal signal, but also from the Cayley–Dickson construction of a quaternionas a complex number with complex real and imaginary parts (with different rootsof −1 used in each of the two levels of complex number).

Extension of the analytic signal concept to 2D signals, that is images, withreal, complex or quaternion-valued pixels is of interest, but outside the scopeof this chapter. Some work has been done on this, notably by Bulow, Felsberg,Sommer and Hahn [1, 2, 6, 8]. The principal issue to be solved in the 2D case

3. ℍ-spectral analysis 43

is to generalise the concept of a single-sided spectrum. Hahn considered a singlequadrant or orthant spectrum, Sommer et al. considered a spectrum with supportlimited to half the complex plane, not necessarily confined to two quadrants, butstill with real sample or pixel values.

Recently, Lilly and Olhede [16] have published a paper on bivariate analyticsignal concepts without explicitly considering the complex signal case which wecover here. Their approach is linked to a specific signal model, the modulatedelliptical signal, which they illustrate with the example of a drifting oceanographicfloat. The approach taken in the present chapter is more general and withoutreference to a specific signal model.

2. 1D Quaternion Fourier Transform

In this section, we will be concerned with the definition and properties of thequaternion Fourier transform (QFT) of complex-valued signals. Before introducingthe main definitions, we give some prerequisites on quaternion-valued signals.

2.1. Preliminary Remarks

Quaternions were discovered by Sir W.R. Hamilton in 1843 [11]. Quaternions are4D hypercomplex numbers that form a noncommutative division ring denotedℍ. A quaternion 𝑞 ∈ ℍ has a Cartesian form: 𝑞 = 𝑞0 + 𝑞1𝒊 + 𝑞2𝒋 + 𝑞3𝒌, with𝑞0, 𝑞1, 𝑞2, 𝑞3 ∈ ℝ and 𝒊, 𝒋,𝒌 roots of −1 satisfying 𝒊2 = 𝒋2 = 𝒌2 = 𝒊𝒋𝒌 = −1.The scalar part of 𝑞 is: S(𝑞) = 𝑞0. The vector part of 𝑞 is: V(𝑞) = 𝑞 − S(𝑞).Quaternion multiplication is not commutative, so that in general 𝑞𝑝 ∕= 𝑝𝑞 for𝑝, 𝑞 ∈ ℍ. The conjugate of 𝑞 is 𝑞 = 𝑞0 − 𝑞1𝒊 − 𝑞2𝒋 − 𝑞3𝒌. The norm of 𝑞 is

∥𝑞∥ = ∣𝑞∣2 = (𝑞20 + 𝑞21 + 𝑞22 + 𝑞23) = 𝑞𝑞. A quaternion with 𝑞0 = 0 is called pure. If∣𝑞∣ = 1, then 𝑞 is called a unit quaternion. The inverse of 𝑞 is 𝑞−1 = 𝑞/ ∥𝑞∥. Pureunit quaternions are special quaternions, among which are 𝒊, 𝒋 and 𝒌. Togetherwith the identity of ℍ, they form a quaternion basis : {1, 𝒊, 𝒋,𝒌}. In fact, givenany two unit pure quaternions, 𝜇 and 𝜉, which are orthogonal to each other (i.e.,S(𝜇𝜉) = 0), then {1, 𝜇, 𝜉, 𝜇𝜉} is a quaternion basis.

Quaternions can also be viewed as complex numbers with complex compo-nents, i.e., one can write 𝑞 = 𝑧1 + 𝒋𝑧2 in the basis {1, 𝒊, 𝒋,𝒌} with 𝑧1, 𝑧2 ∈ ℂ𝒊, i.e.,𝑧𝛼 = ℜ(𝑧𝛼) + 𝒊ℑ(𝑧𝛼) for 𝛼 = 1, 21. This is called the Cayley–Dickson form.

Among the possible representations of 𝑞, two of them will be used in thischapter: the polar (Euler) form and the polar Cayley–Dickson form [20].

Polar form. Any non-zero quaternion 𝑞 can be written:

𝑞 = ∣𝑞∣ 𝑒𝜇𝑞𝜙𝑞 = ∣𝑞∣ (cos𝜙𝑞 + 𝜇𝑞 sin𝜙𝑞) ,

1In the sequel, we denote by ℂ𝜇 the set of complex numbers with root of −1 = 𝜇. Note thatthese are degenerate quaternions, where all vector parts point in the same direction.

44 N. Le Bihan and S.J. Sangwine

where 𝜇𝑞 is the axis of 𝑞 and 𝜙𝑞 is the angle of 𝑞. For future use in this chapter,we give here their explicit expressions:{

𝜇𝑞 = V(𝑞) / ∣V(𝑞)∣ ,𝜙𝑞 = arctan (∣V(𝑞)∣ / S(𝑞)) . (2.1)

The axis and angle are used in interpreting the hyperanalytic signal in Section 4.

Polar Cayley–Dickson form. Any quaternion 𝑞 also has a unique polar Cayley–Dickson form [20] given by:

𝑞 = 𝐴𝑞 exp(𝐵𝑞𝒋) = (𝑎0 + 𝑎1𝒊) exp((𝑏0 + 𝑏1𝒊)𝒋), (2.2)

where 𝐴𝑞 = 𝑎0 + 𝑎1𝒊 is the complex modulus of 𝑞 and 𝐵𝑞 = 𝑏0 + 𝑏1𝒊 its complexphase. It is proven in [20] that given a quaternion 𝑝 = (𝑏0 + 𝑏1𝒊)𝒋 = 𝑏0𝒋 + 𝑏1𝒌, itsexponential is given as:

𝑒𝑝 = cos ∣𝑝∣+ 𝑝

∣𝑝∣ sin ∣𝑝∣ (2.3)

= cos(√

𝑏20 + 𝑏21) + 𝒋𝑏0√

𝑏20 + 𝑏21sin(√

𝑏20 + 𝑏21)

+ 𝒌𝑏1√

𝑏20 + 𝑏21sin(√

𝑏20 + 𝑏21). (2.4)

As a consequence, using the right-hand side expression in (2.2), the Cartesiancoordinates of 𝑞 can be linked to the polar Cayley–Dickson ones in the followingway:⎧⎨⎩

𝐴𝑞 =𝑞0 + 𝒊𝑞1

cos(√

𝑞22 + 𝑞23)

𝐵𝑞 = arctan(√

𝑞22 + 𝑞23)√

𝑞22 + 𝑞23

(𝑞0𝑞3 + 𝑞1𝑞2𝑞20 + 𝑞21

+ 𝒊𝑞0𝑞3 − 𝑞1𝑞2𝑞20 + 𝑞21

).

(2.5)

In Section 4, we will make use of the complex modulus 𝐴𝑞 for interpretationof the hyperanalytic signal.

2.2. 1D Quaternion Fourier Transform

In this chapter, we use a 1D version of the (right-side) QFT first defined in discrete-time form in [17]. Thus, it is necessary to specify the axis (a pure unit quaternion)of the transform. So, we will denote by ℱ𝜇 [ ] a QFT with transformation axis 𝜇.For convenience below, we refer to this as a 𝑄𝐹𝑇𝜇. In order to work with theclassical quaternion basis, in the sequel we will use 𝒋 as the transform axis. Theonly restriction on the transform axis is that it must be orthogonal to the originalbasis of the signal (here {1, 𝒊}). We now present the definition and some propertiesof the transform used here.

3. ℍ-spectral analysis 45

Definition 1. Given a complex-valued signal 𝑧(𝑡) = 𝑧𝑟(𝑡) + 𝒊𝑧𝑖(𝑡), its quaternionFourier transform with respect to axis 𝒋 is:

𝑍𝒋(𝜈) = ℱ𝒋 [𝑧(𝑡)] =

+∞∫−∞

𝑧(𝑡)𝑒−𝒋2𝜋𝜈𝑡d𝑡, (2.6)

and the inverse transform is:

𝑧(𝑡) = ℱ−1𝒋 [𝑍𝒋(𝜈)] =

+∞∫−∞

𝑍𝒋(𝜈)𝑒𝒋2𝜋𝜈𝑡d𝜈. (2.7)

Property 1. Given a complex signal 𝑧(𝑡) = 𝑧𝑟(𝑡)+𝒊𝑧𝑖(𝑡) and its quaternion Fouriertransform denoted 𝑍𝒋(𝜈), then the following properties hold:

∙ The even part of 𝑧𝑟(𝑡) is in ℜ(𝑍𝒋(𝜈)),∙ The odd part of 𝑧𝑟(𝑡) is in ℑ𝒋(𝑍𝒋(𝜈)),∙ The even part of 𝑧𝑖(𝑡) is in ℑ𝒊(𝑍𝒋(𝜈)),∙ The odd part of 𝑧𝑖(𝑡) is in ℑ𝒌(𝑍𝒋(𝜈)).

Proof. Expand (2.6) into real and imaginary parts with respect to 𝒊, and expandthe quaternion exponential into cosine and sine components:

𝑍𝒋(𝜈) =

+∞∫−∞

[𝑧𝑟(𝑡) + 𝒊𝑧𝑖(𝑡)] [cos(2𝜋𝜈𝑡)− 𝒋 sin(2𝜋𝜈𝑡)] d𝑡

=

+∞∫−∞

𝑧𝑟(𝑡) cos(2𝜋𝜈𝑡)d𝑡− 𝒋

+∞∫−∞

𝑧𝑟(𝑡) sin(2𝜋𝜈𝑡)d𝑡

+ 𝒊

+∞∫−∞

𝑧𝑖(𝑡) cos(2𝜋𝜈𝑡)d𝑡− 𝒌

+∞∫−∞

𝑧𝑖(𝑡) sin(2𝜋𝜈𝑡)d𝑡,

from which the stated properties are evident. □

These properties are central to the justification of the use of the QFT toanalyze a complex-valued signal carrying complementary but different informationin its real and imaginary parts. Using the QFT, it is possible to have the oddand even parts of the real and imaginary parts of the signal in four differentcomponents in the transform domain. This idea was also the initial motivation ofBulow, Sommer and Felsberg when they developed the monogenic signal for images[1, 2, 6]. Note that the use of hypercomplex Fourier transforms was originallyintroduced in 2D Nuclear Magnetic Resonance image analysis [5, 3].

We now turn to the link between a complex signal and the quaternion signalthat can be uniquely associated to it.

46 N. Le Bihan and S.J. Sangwine

Property 2. A one-sided 𝑄𝐹𝑇 𝑋(𝜈), obtained from a complex signal as in Defi-nition 1 by suppressing the negative frequencies, that is with 𝑋(𝜈) = 0 for 𝜈 < 0,corresponds to a full quaternion-valued signal in the time domain.

Proof. The proof is based on the symmetry properties, listed in Property 1, fulfilledby the 𝑄𝐹𝑇𝒋 of complex-valued signals. From this, it is easily verified that, giventhe 𝑄𝐹𝑇𝒋 of a complex signal 𝑢(𝑡) denoted by 𝑈𝒋(𝜈), then (1 + sign(𝜈))𝑈𝒋(𝜈)is right-sided (i.e., it vanishes for all 𝜈 < 0). Here sign(𝜈) is the classical signfunction, which is equal to 1 for 𝜈 > 0 and equal to −1 for 𝜈 < 0. Second, theinverse transform, or 𝐼𝑄𝐹𝑇𝒋 of (1 + sign(𝜈))𝑈𝒋(𝜈) can be decomposed as follows:

𝐼𝑄𝐹𝑇𝒋 [(1 + sign(𝜈))𝑈𝒋(𝜈)] = 𝐼𝑄𝐹𝑇𝒋 [𝑈𝒋(𝜈)] + 𝐼𝑄𝐹𝑇𝒋 [sign(𝜈)𝑈𝒋(𝜈)]

By definition, 𝐼𝑄𝐹𝑇𝒋 [𝑈𝒋(𝜈)] is a complex-valued signal with non-zero real and𝒊-imaginary parts, and null 𝒋-imaginary and 𝒌-imaginary parts. To complete theproof we show that 𝐼𝑄𝐹𝑇𝒋 [sign(𝜈)𝑈𝒋(𝜈)] has null real and 𝒊-imaginary parts, andnon-zero 𝒋-imaginary and 𝒌-imaginary parts. Consider the original complex signal𝑢(𝑡) = 𝑢𝑟(𝑡)+𝑢𝑖(𝑡)𝒊, as composed of odd and even, real and imaginary parts (fourcomponents in total). Property 1 shows how these four components map to thefour Cartesian components of 𝑈𝒋(𝜈), namely:

∙ The even part of 𝑢𝑟(𝑡) �→ ℜ(𝑈𝒋(𝜈)), which is even,∙ The even part of 𝑢𝑖(𝑡) �→ ℑ𝒊(𝑈𝒋(𝜈)), which is even,∙ The odd part of 𝑢𝑟(𝑡) �→ ℑ𝒋(𝑈𝒋(𝜈)), which is odd.∙ The odd part of 𝑢𝑖(𝑡) �→ ℑ𝒌(𝑈𝒋(𝜈)), which is odd.

Multiplication by sign(𝜈) changes the parity to the following:

∙ sign(𝜈)ℜ(𝑈𝒋(𝜈)) is odd,∙ sign(𝜈)ℑ𝒊(𝑈𝒋(𝜈)), is odd,∙ sign(𝜈)ℑ𝒋(𝑈𝒋(𝜈)), is even.∙ sign(𝜈)ℑ𝒌(𝑈𝒋(𝜈)), is even.

and the inverse transform 𝐼𝑄𝐹𝑇𝒋 maps these components to:

∙ sign(𝜈)ℜ(𝑈𝒋(𝜈)) �→ 𝒋𝑢1(𝑡),∙ sign(𝜈)ℑ𝒊(𝑈𝒋(𝜈)) �→ 𝒊𝒋𝑢2(𝑡) = 𝒌𝑢2(𝑡),∙ sign(𝜈)ℑ𝒋(𝑈𝒋(𝜈)) �→ 𝒋𝑢3(𝑡),∙ sign(𝜈)ℑ𝒌(𝑈𝒋(𝜈)) �→ 𝒌𝑢4(𝑡),

where 𝑢𝑥(𝑡), 𝑥 = 1, 2, 3, 4 are real functions of 𝑡. Hence 𝐼𝑄𝐹𝑇𝒋 [sign(𝜈)𝑈𝒋(𝜈)] hasnull real and null 𝒊-imaginary parts, but non-zero 𝒋-imaginary and 𝒌-imaginaryparts as previously stated. □

Property 3. Given a complex signal 𝑥(𝑡), one can associate to it a unique canonicalpair corresponding to a modulus and phase. These modulus and phase are uniquelydefined through the hyperanalytic signal, which is quaternion valued.

Proof. Cancelling the negative frequencies of the QFT leads to a quaternion signalin the time domain. Then, any quaternion signal has a modulus and phase definedusing its Cayley–Dickson polar form. □

3. ℍ-spectral analysis 47

2.3. Convolution

We consider the special case of convolution of a complex signal by a real signal.Consider 𝑔 and 𝑓 such that: 𝑔 : ℝ+ → ℂ and 𝑓 : ℝ+ → ℝ. Now, consider theQFT𝒋 of their convolution:

ℱ𝒋 [(𝑔 ∗ 𝑓)(𝑡)] =+∞∫−∞

+∞∫−∞

𝑔(𝜏)𝑓(𝑡 − 𝜏)d𝜏𝑒−2𝒋𝜋𝜈𝑡d𝑡

=

+∞∫−∞

+∞∫−∞

𝑔(𝜏)𝑒−𝒋2𝜋𝜈(𝑡′+𝜏)𝑓(𝑡′)𝑑𝜏d𝑡′

=

+∞∫−∞

𝑔(𝜏)𝑒−𝒋2𝜋𝜈𝜏d𝜏

+∞∫−∞

𝑓(𝑡′)𝑒−𝒋2𝜋𝜈𝑡′d𝑡′

= ℱ𝒋 [𝑔(𝑡)]ℱ𝒋 [𝑓(𝑡)] .

(2.8)

Thus, the definition used for the QFT here verifies the convolution theorem in theconsidered case. This specific case will be of use in our definition of the hyperan-alytic signal.

2.4. The Quaternion Fourier Transform of the Hilbert Transform

It is straightforward to verify that the quaternion Fourier transform of a real signal𝑥(𝑡) = 1/𝜋𝑡 is −𝒋 sign(𝜈), where 𝒋 is the axis of the transform. Substituting 𝑥(𝑡)into (2.6), we get:

ℱ𝒋

[1

𝜋𝑡

]=

1

𝜋

+∞∫−∞

𝑒−𝒋2𝜋𝜈𝑡

𝑡d𝑡,

and this is clearly isomorphic to the classical complex case. The solution in theclassical case is −𝑖 sign(𝜈), and hence in the quaternion case must be as statedabove.

It is also straightforward to see that, given an arbitrary real signal 𝑦(𝑡),subject only to the constraint that its classical Hilbert transform ℋ [𝑦(𝑡)] exists,then one can easily show that the classical Hilbert transform of the signal may beobtained using a quaternion Fourier transform as follows:

ℋ [𝑦(𝑡)] = ℱ−1𝒋 [−𝒋 sign(𝜈)𝑌𝒋(𝜈)] (2.9)

where 𝑌𝒋(𝜈) = ℱ𝒋 [𝑦(𝑡)]. This result follows from the isomorphism between thequaternion and complex Fourier transforms when operating on a real signal, andit may be seen to be the result of a convolution between the signal 𝑦(𝑡) and thequaternion Fourier transform of 𝑥(𝑡) = 1/𝜋𝑡. Note that 𝒋 and 𝑌𝒋(𝜈) commute asa consequence of 𝑦(𝑡) being real.

48 N. Le Bihan and S.J. Sangwine

3. The Hypercomplex Analytic Signal

We define the hyperanalytic signal 𝑧+(𝑡) by a similar approach to that originallydeveloped by Ville [21]. The following definitions give the details of the construc-tion of this signal. Note that the signal 𝑧(𝑡) is considered to be non-analytic, orimproper, in the classical (complex) sense, that is its real and imaginary parts arenot orthogonal. However, the following definitions are valid if 𝑧(𝑡) is analytic, asit can be considered as a degenerate case of the more general non-analytic case.

Definition 2. Consider a complex signal 𝑧(𝑡) = 𝑧𝑟(𝑡) + 𝒊𝑧𝑖(𝑡) and its quaternionFourier transform 𝑍𝒋(𝜈) as defined in Definition 1. Then, the hypercomplex ana-logue of the Hilbert transform of 𝑧(𝑡), is as follows:

ℋ𝒋 [𝑧(𝑡)] = ℱ−1𝒋 [−𝒋 sign(𝜈)𝑍𝒋(𝜈)] , (3.1)

where the Hilbert transform is thought of as: ℋ𝒋 [𝑧(𝑡)] = 𝑝.𝑣. (𝑧 ∗ (1/𝜋𝑡)), wherethe principal value (p.v.) is understood in its classical way. This result follows fromequation (2.9) and the linearity of the quaternion Fourier transform. To extractthe imaginary part, the vector part of the quaternion signal must be multiplied by−𝒊. An alternative is to take the scalar or inner product of the vector part with𝒊. Note that 𝒋 and 𝑍𝒋(𝜈) anticommute because 𝒋 is orthogonal to 𝒊, the axis of𝑍𝒋(𝜈). Therefore the ordering is not arbitrary, but changing it simply conjugatesthe result.

Definition 3. Given a complex-valued signal 𝑧(𝑡) that can be expressed in the formof a quaternion as 𝑧(𝑡) = 𝑧𝑟(𝑡) + 𝒊𝑧𝑖(𝑡), then the hypercomplex analytic signal of𝑧(𝑡), denoted 𝑧+(𝑡) is given by:

𝑧+(𝑡) = 𝑧(𝑡) + 𝒋ℋ𝒋 [𝑧(𝑡)] , (3.2)

where ℋ𝒋 [𝑧(𝑡)] is the hypercomplex analogue of the Hilbert transform of 𝑧(𝑡)defined in the preceding definition. The quaternion Fourier transform of this hy-percomplex analytic signal is thus:

𝑍+(𝜈) = 𝑍𝒋(𝜈)− 𝒋2 sign(𝜈)𝑍𝒋(𝜈)

= [1 + sign(𝜈)]𝑍𝒋(𝜈)

= 2𝑈(𝜈)𝑍𝒋(𝜈),

where 𝑈(𝜈) is the classical unit step function.

This result is unique to the quaternion Fourier transform representation ofthe hypercomplex analytic signal – the hypercomplex analytic signal has a one-sided quaternion Fourier spectrum. This means that the hypercomplex analyticsignal may be constructed from a complex signal 𝑧(𝑡) in exactly the same waythat an analytic signal may be constructed from a real signal 𝑥(𝑡), by suppressionof negative frequencies in the Fourier spectrum. The only difference is that in thehypercomplex analytic case, a quaternion rather than a complex Fourier transformmust be used, and of course the complex signal must be put in the form 𝑧(𝑡) =

3. ℍ-spectral analysis 49

𝑧𝑟(𝑡) + 𝒊𝑧𝑖(𝑡) which, although a quaternion signal, is isomorphic to the originalcomplex signal.

A second important property of the hypercomplex analytic signal is that itmaintains a separation between the different even and odd parts of the originalsignal.

Property 4. The original signal 𝑧(𝑡) is the simplex part [4, Theorem 1], [18, § 13.1.3]of its corresponding hypercomplex analytic signal. The perplex part is the orthog-onal or ‘quadrature’ component, 𝑜(𝑡). They are obtained by:

𝑧(𝑡) =1

2(𝑧+(𝑡)− 𝒊𝑧+(𝑡)𝒊) , (3.3)

𝑜(𝑡) =1

2(𝑧+(𝑡) + 𝒊𝑧+(𝑡)𝒊) . (3.4)

Proof. This follows from equation (3.2). Writing this in full by substituting theorthogonal signal for ℋ𝒋 [𝑧(𝑡)]:

𝑧+(𝑡) = 𝑧(𝑡) + 𝒋𝑜(𝑡) = 𝑧𝑟(𝑡) + 𝒊𝑧𝑖(𝑡) + 𝒋𝑜𝑟(𝑡)− 𝒌𝑜𝑖(𝑡),

and substituting this into equation (3.3), we get:

𝑧(𝑡) =1

2

(𝑧𝑟(𝑡) + 𝒊𝑧𝑖(𝑡) + 𝒋𝑜𝑟(𝑡)− 𝒌𝑜𝑖(𝑡)

−𝒊 [𝑧𝑟(𝑡) + 𝒊𝑧𝑖(𝑡) + 𝒋𝑜𝑟(𝑡)− 𝒌𝑜𝑖(𝑡)] 𝒊

),

and since 𝒊 and 𝒋 are orthogonal unit pure quaternions, 𝒊𝒋 = −𝒋𝒊:

=1

2

(𝑧𝑟(𝑡) + 𝒊𝑧𝑖(𝑡) + 𝒋𝑜𝑟(𝑡)− 𝒌𝑜𝑖(𝑡)

+ 𝑧𝑟(𝑡) + 𝒊𝑧𝑖(𝑡)− 𝒋𝑜𝑟(𝑡) + 𝒌𝑜𝑖(𝑡)

),

from which the first part of the result follows. Equation (3.4) differs only in thesign of the second term, and it is straightforward to see that if 𝑧+(𝑡) is substituted,𝑧(𝑡) cancels out, leaving 𝑜(𝑡). □

4. Geometric Instantaneous Amplitude and Phase

Using the hypercomplex analytic signal, we now present the geometric featuresthat can be obtained thanks to two representations of 𝑧+(𝑡). First, we notice thatthe hypercomplex analytic signal has a polar form given as:

𝑧+(𝑡) = 𝜌+R(𝑡)𝑒𝜇

+(𝑡)𝜙+(𝑡), (4.1)

where 𝜌+R = ∣𝑧+(𝑡)∣ is the real modulus of 𝑧+(𝑡), 𝜙+(𝑡) is its argument and 𝜇+(𝑡) its

axis. The real modulus, or real envelope, is not very informative, as it is real valuedand so does not provide a 2D description of the slowly varying part (envelope) of𝑧(𝑡). Nonetheless, a complex modulus can be defined using the modulus of thepolar Cayley–Dickson representation that is more informative on the geometricfeatures of the original improper signal 𝑧(𝑡).

50 N. Le Bihan and S.J. Sangwine

4.1. Complex Envelope

The complex modulus of the hypercomplex analytic signal has properties verysimilar to the ‘classical’ case. It is defined as the modulus of the Cayley–Dicksonpolar representation of 𝑧+(𝑡). Considering the Cayley–Dickson polar form of thehypercomplex analytic signal given as:

𝑧+(𝑡) = 𝜌+C(𝑡)𝑒

Φ+(𝑡)𝒋 , (4.2)

then the complex envelope of 𝑧(𝑡) is simply 𝜌+C(𝑡). An illustration of a complex

envelope of a complex improper signal is given in Figures 1 and 3. In Figure 1,an improper complex signal made of a low frequency envelope modulating a highfrequency linear frequency sweep is presented. The complex envelope obtainedthrough the hyperanalytic signal is displayed in red (and in blue a negated versionto show how the envelope encompass the signal). In Figure 3, a similar signal, butwith a quadratic sweep, is displayed. Again, the complex envelope fits the slowlyvarying pattern of the signal.

Note that we do not make use of the phase of the Cayley–Dickson polar formin the sequel. It was demonstrated in [15] that this phase can reveal informationon the ‘modulation’ part of the improper complex signal. Here we are looking forgeometric descriptors of the improper complex signal through the hyperanalyticsignal. For this purpose, we now show how the phase of the polar form of 𝑧+(𝑡)can be linked with the angular velocity concept.

4.2. Angular Velocity

The phase of a hyperanalytic signal is slightly different from the well-known con-cept of phase for a complex-valued signal. First, it is a three-dimensional quantitymade of an axis and a phase (i.e., a pure unit quaternion and a scalar). Using thepolar form of 𝑧+(𝑡), its normalized part, denoted 𝑧+(𝑡), is simply:

𝑧+(𝑡) =𝑧+(𝑡)

𝜌+R(𝑡)

, (4.3)

where it is assumed that 𝜌+R(𝑡) ∕= 0 for all 𝑡. In the case where 𝜌+

R(𝑡) = 0, it simplymeans that the original signal 𝑧(𝑡) = 0 and so is 𝑧+(𝑡). If not, then 𝑧+(𝑡) is a unitquaternion, i.e., an element of 𝒮3. Without loss of generality, one can write 𝑧+(𝑡)as:

𝑧+(𝑡) = exp [𝜃(𝑡)v(𝑡)] , (4.4)

where 𝜃(𝑡) is scalar valued and v(𝑡) is a pure unit quaternion. Now, it is wellknown [13, 12] from quaternion formulations of kinematics that the instantaneousfrequency is given by:

𝜙(𝑡) =d𝜃(𝑡)

d𝑡= arg

(d𝑧+(𝑡)

d𝑡

), (4.5)

where the time derivative of 𝑧+(𝑡) is understood as:

d𝑧+(𝑡)

d𝑡= lim

Δ𝑡→0

𝑧+(𝑡+Δ𝑡)𝑧+(𝑡)−1

Δ𝑡. (4.6)

3. ℍ-spectral analysis 51

1

0.5

0

0.5

1

00.10.20.30.40.50.60.70.80.91

1.5

1

0.5

0

0.5

1

1.5

TimeReal

Imag

Figure 1. Improper complex signal consisting of a modulating lowfrequency envelope and a high frequency linear frequency sweep. Theenvelope 𝜌+

C(𝑡) is indicated by the arrow at left and −𝜌+C(𝑡) by the arrow

at right.

This is in fact the multiplicative increment (a unit pure quaternion) between thehyperanalytic signal at time 𝑡 and time 𝑡+Δ𝑡. The argument in (4.5) is thus theamount of angle per unit of time by which the signal has been rotated (in rad/s).The axis of this increment gives the direction of this rotation. This is an angularvelocity, which corresponds, for a complex improper signal, to its instantaneousfrequency. This angular velocity is a geometrical and spectral local informationon the signal 𝑧(𝑡), as it gives locally the frequency content and the geometricalorientation and behaviour of the signal with time.

In Figures 1 to 4, we illustrate this concept of instantaneous frequency forimproper complex signals. In Figure 1, an improper complex signal made of alinear sweep (single frequency signal with frequency linearly changing with time)and a complex envelope is presented. Its complex envelope is presented in red (theinverse of the envelope is also plotted, in blue, for visualization purposes), whichfits the ‘low frequency’ behaviour of the signal. In Figure 2, the angular velocity ofthe hyperanalytic signal is compared to the frequency sweep used to generate the

52 N. Le Bihan and S.J. Sangwine

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

5

10

15

20

25

30

35

40

45

50

Time

Freq

uenc

y

Figure 2. Linear frequency sweep estimated as the angular velocity ofthe hyperanalytic signal (with ripples and discontinuity), and originallinear frequency sweep used to generate the original improper complexsignal (without ripples). The discontinuity in the estimate is due to thefact that the angular velocity obtained from the hyperanalytic signal isnot differentiable at 𝑡 = 0.5.

signal, showing good agreement, except at a singularity point (where the phase ofthe hyperanalytic signal is not differentiable). A similar procedure is carried outin Figures 3 and 4, where the ‘high frequency’ content is now a quadratic sweep(frequency varying quadratically with time). The same conclusions can be drawnabout the complex envelope and the angular velocity.

The complex envelope and the angular velocity are the extensions, for com-plex improper signals, of the well-known envelope and instantaneous frequencyfor real signals. They allow a local description of the geometrical and spectral be-haviour of the signal and thus consist in the simplest time-frequency representationfor improper complex signals. As in the classical case, these local descriptors havetheir limitations. Here, the number of frequency components must be kept to onefor the angular velocity to be able to recover the instantaneous frequency. Thislimitation makes impossible the identification of local spectral contents in the caseof mixtures of signals or wide-band signals. Such signals require more sophisticated

3. ℍ-spectral analysis 53

1

0.5

0

0.5

1

00.2

0.40.6

0.81

0.25

0.2

0.15

0.1

0.05

0

0.05

0.1

0.15

0.2

0.25

Time

Real

Imag

Figure 3. Improper complex signal consisting in a modulating lowfrequency envelope and a high frequency quadratic frequency sweep.

time-frequency representations that could be developed, based on the quaternionFT. The study of such representations is a natural step after the work presented.

5. Conclusions

We have shown in this chapter how the classical analytic signal concept may beextended to the case of the hyperanalytic signal of an original complex signal. Thequaternion based approach yields an interpretation of the hyperanalytic signal as aquaternion signal which leads naturally to the definition of the complex envelope.Also, the use of the polar form of the hyperanalytic signal allows the derivationof an angular velocity, which is indeed an instantaneous frequency. Both the enve-lope and the angular velocity allow a local and spectral description of a compleximproper signal, leading to the first geometrical time-frequency representation forcomplex improper signals.

Future work will consist in developing other time-frequency representationsfor improper complex signals with more diverse spectral behaviour.

54 N. Le Bihan and S.J. Sangwine

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

5

10

15

20

25

Time

Freq

uenc

y

Figure 4. Quadratic frequency sweep estimated as the angular veloc-ity of the hyperanalytic signal (with ripples), and original quadraticfrequency sweep used to generate the original improper complex sig-nal (without ripples). The discontinuity at 𝑡 = 0.5 is due to the factthat the angular velocity obtained from the hyperanalytic signal is notdifferentiable at this point.

Acknowledgement

We thank Eckhard Hitzer for suggesting the proof of Property 2.

References

[1] T. Bulow. Hypercomplex Spectral Signal Representations for the Processing and Anal-ysis of Images. PhD thesis, University of Kiel, Germany, Institut fur Informatik undPraktische Mathematik, Aug. 1999.

[2] T. Bulow and G. Sommer. Hypercomplex signals – a novel extension of the ana-lytic signal to the multidimensional case. IEEE Transactions on Signal Processing,49(11):2844–2852, Nov. 2001.

[3] M.A. Delsuc. Spectral representation of 2D NMR spectra by hypercomplex numbers.Journal of magnetic resonance, 77(1):119–124, Mar. 1988.

3. ℍ-spectral analysis 55

[4] T.A. Ell and S.J. Sangwine. Hypercomplex Wiener-Khintchine theorem with ap-plication to color image correlation. In IEEE International Conference on ImageProcessing (ICIP 2000), volume II, pages 792–795, Vancouver, Canada, 11–14 Sept.2000. IEEE.

[5] R.R. Ernst, G. Bodenhausen, and A. Wokaun. Principles of Nuclear Magnetic Res-onance in One and Two Dimensions. International Series of Monographs on Chem-istry. Oxford University Press, 1987.

[6] M. Felsberg and G. Sommer. The monogenic signal. IEEE Transactions on SignalProcessing, 49(12):3136–3144, Dec. 2001.

[7] D. Gabor. Theory of communication. Journal of the Institution of Electrical Engi-neers, 93(26):429–457, 1946. Part III.

[8] S.L. Hahn. Multidimensional complex signals with single-orthant spectra. Proceed-ings of the IEEE, 80(8):1287–1300, Aug. 1992.

[9] S.L. Hahn. Hilbert transforms. In A.D. Poularikas, editor, The transforms and appli-cations handbook, chapter 7, pages 463–629. CRC Press, Boca Raton, 1996. A CRChandbook published in cooperation with IEEE press.

[10] S.L. Hahn. Hilbert transforms in signal processing. Artech House signal processinglibrary. Artech House, Boston, London, 1996.

[11] W.R. Hamilton. Lectures on Quaternions. Hodges and Smith, Dublin, 1853. Avail-able online at Cornell University Library: http://historical.library.cornell.edu/math/.

[12] A.J. Hanson. Visualizing Quaternions. The Morgan Kaufmann Series in Interactive3D Technology. Elsevier/Morgan Kaufmann, San Francisco, 2006.

[13] J.B. Kuipers. Quaternions and Rotation Sequences. Princeton University Press,Princeton, New Jersey, 1999.

[14] N. Le Bihan and S.J. Sangwine. About the extension of the 1D analytic signal toimproper complex valued signals. In Eighth International Conference on Mathemat-ics in Signal Processing, page 45, The Royal Agricultural College, Cirencester, UK,16–18 December 2008.

[15] N. Le Bihan and S.J. Sangwine. The H-analytic signal. In Proceedings of EUSIPCO2008, 16th European Signal Processing Conference, page 5, Lausanne, Switzerland,25–29 Aug. 2008. European Association for Signal Processing.

[16] J.M. Lilly and S.C. Olhede. Bivariate instantaneous frequency and bandwidth. IEEETransactions on Signal Processing, 58(2):591–603, Feb. 2010.

[17] S.J. Sangwine and T.A. Ell. The discrete Fourier transform of a colour image. In J.M.Blackledge and M.J. Turner, editors, Image Processing II Mathematical Methods, Al-gorithms and Applications, pages 430–441, Chichester, 2000. Horwood Publishing forInstitute of Mathematics and its Applications. Proceedings Second IMA Conferenceon Image Processing, De Montfort University, Leicester, UK, September 1998.

[18] S.J. Sangwine, T.A. Ell, and N. Le Bihan. Hypercomplex models and processing ofvector images. In C. Collet, J. Chanussot, and K. Chehdi, editors,Multivariate ImageProcessing, Digital Signal and Image Processing Series, chapter 13, pages 407–436.ISTE Ltd, and John Wiley, London, and Hoboken, NJ, 2010.

[19] S.J. Sangwine and N. Le Bihan. Hypercomplex analytic signals: Extension of theanalytic signal concept to complex signals. In Proceedings of EUSIPCO 2007, 15th

56 N. Le Bihan and S.J. Sangwine

European Signal Processing Conference, pages 621–4, Poznan, Poland, 3–7 Sept.2007. European Association for Signal Processing.

[20] S.J. Sangwine and N. Le Bihan. Quaternion polar representation with a complexmodulus and complex argument inspired by the Cayley–Dickson form. Advances inApplied Clifford Algebras, 20(1):111–120, Mar. 2010. Published online 22 August2008.

[21] J. Ville. Theorie et applications de la notion de signal analytique. Cables et Trans-mission, 2A:61–74, 1948.

Nicolas Le BihanGIPSA-Lab/CNRS11 Rue des mathematiquesF-38402 Saint Martin d’Heres, Francee-mail: nicolas.le-bihan@gipsa-lab.grenoble-inp.fr

Stephen J. SangwineSchool of Computer Science and Electronic EngineeringUniversity of EssexWivenhoe ParkColchester CO4 3SQ, UKe-mail: sjs@essex.ac.uk

Quaternion and Clifford–Fourier Transforms and Wavelets

Trends in Mathematics, 57–83c⃝ 2013 Springer Basel

4 Quaternionic Local Phase for Low-levelImage Processing Using Atomic Functions

E. Ulises Moya-Sanchez and E. Bayro-Corrochano

Abstract. In this work we address the topic of image processing using anatomic function (AF) in a representation of quaternionic algebra. Our ap-proach is based on the most important AF, the up(𝑥) function. The mainreason to use the atomic function up(𝑥) is that this function can expressanalytically multiple operations commonly used in image processing such aslow-pass filtering, derivatives, local phase, and multiscale and steering filters.Therefore, the modelling process in low level-processing becomes easy usingthis function. The quaternionic algebra can be used in image analysis becauselines (even), edges (odd) and the symmetry of some geometric objects in ℝ2

are enhanced. The applications show an example of how up(𝑥) can be ap-plied in some basic operations in image processing and for quaternionic phasecomputation.

Mathematics Subject Classification (2010). Primary 11R52; secondary 65D18.

Keywords. Quaternionic phase.

1. Introduction

The visual system is the most advanced of our senses. Therefore, it is easy tounderstand that the processing of images plays an important role in human per-ception and computer vision [3, 9]. In this chapter we address the topic of imageprocessing using geometric algebra (GA) for computer vision applications in low-level and mid-level (geometric feature extraction and analysis) processing, whichbelong to the first layers of a bottom-up computer vision system.

Complex and hypercomplex numbers play an important role in signal pro-cessing [9, 5], especially in order to obtain local features such as the frequency andphase. In 1D signal analysis, it is usual to compute via the FFT a local magni-tude and phase using the analytic signal. The entire potential of the local phase

This work has been supported by CINVESTAV and CONACYT.

58 E.U. Moya-Sanchez and E. Bayro-Corrochano

information of the images is shown when constraints or invariants are required tobe found. In other words, we use the quaternionic (local) phase because the localphase can be used to link the low-level image processing with the upper layers.The phase information is invariant to illumination changes and can be used todetect low-level geometric characteristics of lines or edges [9, 3, 13]. The phasecan also be used to measure the local decomposition of the image according to itssymmetries [6, 9].

This work has two basic goals: to present a new approach based on an atomicfunction (AF) up(𝑥) in a representation of geometric algebra (GA) and to applythe phase information to reduce the gap between low-level processing and imageanalysis. Our approach is based on the most important AF, the up(𝑥) function[12], quaternionic algebra, and multiscale and steering filters. The up(𝑥) functionhas good locality properties (compact support), the derivative of any order canbe expressed easily, the up(𝑥) and dup(𝑥) can mimic the simple cells of the mam-malian visual processing system [14], and the approximation to other functions(polynomial) is relatively simple [12]. As we show in this work, the atomic func-tion up(𝑥) can be used as a building block to build multiple operations commonlyused in image processing, such as low-pass filtering, 𝑛th-order derivatives, localphase, etc.

The applications presented show an example of how the AF can be applied toquaternionic phase analysis. It is based on line and edge detection and symmetrymeasurement using the phase. As a result, we can reduce the gap between thelow-level processing and the computer vision applications without abandoning thegeometric algebra framework.

2. Atomic Functions

The atomic functions were first developed in the 1970s, jointly by V.L. and V.A.Rvachev. By definition, the AF are compactly supported, infinitely differentiablesolutions of differential functional equations (see (2.1)) with a shifted argument[12], that is

𝐿𝑓(𝑥) = 𝜆

𝑀∑𝑘=1

𝑐(𝑘)𝑓(𝑎𝑥− 𝑏(𝑘)), ∣𝑎∣ > 1, 𝑏, 𝑐, 𝜆 ∈ 𝑁, (2.1)

where 𝐿 = d𝑛

d𝑥𝑛+𝑎1

d𝑛−1

d𝑥𝑛−1 + ⋅ ⋅ ⋅+𝑎𝑛 is a linear differential operator with constant

coefficients. In the AF class, the function up(𝑥) is the simplest and, at the sametime, the most useful primitive function to generate other kinds of AFs [12].

2.1. Mother Atomic Function up(𝒙)

In general, the atomic function up(𝑥) is generated by infinite convolutions of rect-angular impulses. The function up(𝑥) has the following representation in terms of

4. Quaternionic Local Phase 59

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0.1

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0.8

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1

space (x)150 100 50 0 50 100 150

0

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1

frequency ( )

Figure 1. Left: atomic function up(𝑥); right: Fourier transform up(𝜈).

the Fourier transform [12, 8]:

up(𝑥) =1

2𝜋

∫ ∞

−∞

∞∏𝑘=1

sin(𝜈2−𝑘)𝜈2−𝑘

𝑒𝒊𝜈𝑥d𝜈. (2.2)

=1

2𝜋

∫ ∞

−∞up(𝜈)𝑒𝒊𝜈𝑥d𝜈 (2.3)

Figure 1 shows the up(𝑥) and its Fourier transform up(𝜈).

The AF windows were compared with classic windows by means of parameterssuch as the equivalent noise bandwidth, the 50% overlapping region correlation, theparasitic modulation amplitude, the maximum conversion losses (in decibels), themaximum side lobe level (in decibels), the asymptotic decay rate of the side lobes(in decibels per octave), the window width at the six- decibel level, the coherentgain, etc. All atomic windows exceed classic ones in terms of the asymptotic decayrate [12].

However the main reasons that we found to use the AF up(𝑥) instead ofother kernels such as Gauss, Gabor, and log-Gabor is that many operations com-monly used in image processing can be expressed analytically, in contrast usingthe Gauss or the log-Gabor this could be impossible. Additionally the followinguseful properties of the AF have been reported in [12, 10]

∙ There are explicit equations for the values of the moments and Fourier trans-form (see (2.3)). The even moments of up(𝑥) are

𝑎𝑛 =

1∫−1

𝑥𝑛 up(𝑥)d𝑥 (2.4)

𝑎2𝑛 =(2𝑛)!

22𝑛 − 1

𝑛∑𝑘=1

𝑎2𝑛−2𝑘

(2𝑛− 2𝑘)!(2𝑘 + 1)!(2.5)

60 E.U. Moya-Sanchez and E. Bayro-Corrochano

where 𝑎0 = 1 and 𝑎2𝑛+1 = 0. The odd-order moments are

𝑏𝑛 =

1∫0

𝑥𝑛 up(𝑥)d𝑥 (2.6)

𝑏2𝑛+1 =1

(𝑛+ 1)22𝑛+1

𝑛+1∑𝑘=1

𝑎2𝑛−2𝑘+2

(2𝑘

2𝑛+ 2𝑘

)(2.7)

where 𝑏2𝑛 = 𝑎2𝑛. A similar relation for Gauss or Log-Gabor does not exist.∙ The up(𝑥) function is a compactly supported function in the spatial do-main. Therefore, we can obtain good local characteristics. In addition tocompactly supported functions and integrable functions, functions that havea sufficiently rapid decay at infinity can also be convolved. The Gauss andLog-Gabor functions do not have such compact support.

∙ Translations with smaller steps yield polynomials (𝑥𝑛) of any degree, i.e.,∞∑

𝑘=−∞𝑐𝑘 up(𝑥− (𝑘2𝑛)) ≡ 𝑥𝑛 𝑐𝑘, 𝑥 ∈ ℝ. (2.8)

∙ Since derivatives of any order can be represented in terms of simple shifts,we can easily represent any derivative operator or 𝑛th-order derivative:

d(𝑛) up(𝑥) = 2𝑛(𝑛+1)/22𝑛∑𝑘=1

𝛿𝑘 up(2𝑛𝑥+ 2𝑛 + 1− 2𝑘), (2.9)

where 𝛿2𝑘 = −𝛿𝑘, 𝛿2𝑘−1 = 𝛿𝑘, 𝛿2𝑘 = 1.∙ The AFs are infinitely differentiable (𝐶∞). As a result, the AFs and theirFourier transforms are rapidly decreasing functions. (Their Fourier trans-forms decrease on the real axis faster than any power function.)

A natural extension of the up(𝑥) function to the case of many variables is basedon the usual tensor product of 1D up(𝑥) [10]. As a result, we have

up(𝑥, 𝑦) = up(𝑥) up(𝑦). (2.10)

In Figure 2, we show a 2D atomic function in the spatial and frequency domains.

2.2. The dup(𝒙) Function

There are some mask operators, including the Sobel, Prewitt, and Kirsh, that areused to extract edges from images. A common drawback of these operators is thatit is impossible to ensure that they adapt to the intrinsic signal parameters over awide range of the working band, i.e., they are truncated and discrete versions of adifferential operator. [8]. This means that adaptation of the differential operator tothe behavior of the input signal by broadening or narrowing its band is desirable,in order to ensure a maximum signal-to-noise ratio [8].

This problem reduces to the synthesis of infinitely differentiable finite func-tions with a small wide bandwidth that are used for constructing the weightingwindows [8]. One of the most effective solutions is obtained with the help of the

4. Quaternionic Local Phase 61

Figure 2. Top: Two views of up(𝑥, 𝑦). Bottom: Fourier transform up(𝜈, 𝜐).

atomic functions [8]. The AF can be used in two ways: construction of a windowin the frequency region to obtain the required improvement in properties of thepulse characteristic; or direct synthesis based on (2.1) [8]. Therefore, the functionup(𝑥) satisfies (2.1) as follows:

dup(𝑥) = 2 up(2𝑥+ 1)− 2 up(2𝑥− 1) (2.11)

Figure 3 shows the dup(𝑥) function and its Fourier transform 𝑑𝑢𝑝(𝜈). If we computethe Fourier transform of (2.11), we obtain

𝒊𝜈𝐹 [up(2𝑥)] =(𝑒𝒊𝜈 − 𝑒−𝒊𝜈

)𝐹 [up(2𝑥)]) (2.12)

𝐹 (dup(𝑥)) = 2𝒊 sin(𝜈)𝐹 (up(2𝑥)) (2.13)

By differentiating (2.1) term by term, we obtain [8]

d(𝑛) up(𝑥) = 2𝑛(𝑛+1)/22𝑛∑𝑘=1

𝛿𝑘 up(2𝑛𝑥+ 2𝑛 + 1− 2𝑘), (2.14)

where 𝛿2𝑘 = −𝛿𝑘, 𝛿2𝑘−1 = 𝛿𝑘, and 𝛿2𝑘 = 1. The function dup(𝑥) provides a goodwindow in the spatial frequency regions because the side lobe has been completely

62 E.U. Moya-Sanchez and E. Bayro-Corrochano

150 100 50 0 50 100 150

1

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1

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0

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Frequency ( )

Figure 3. Left: derivative of atomic function dup(𝑥); Right: Fourier

transform dup(𝜈).

eliminated [8]. Similarly to (2.10), we can get a 2D expression of each derivative:

dup(𝑥, 𝑦)𝑥 = dup(𝑥) up(𝑦) (2.15)

dup(𝑥, 𝑦)𝑦 = up(𝑥) dup(𝑦) (2.16)

dup(𝑥, 𝑦)𝑥,𝑦 = dup(𝑥) dup(𝑦) (2.17)

Figure 4 shows two graphics of dup(𝑥, 𝑦)𝑦, dup(𝑥, 𝑦)𝑥,𝑦 in the spatial domain.

Figure 4. Left: dup(𝑥, 𝑦)𝑥,𝑦; right: dup(𝑥, 𝑦)𝑦.

3. Why the Use of the Atomic Function?

In this section we justify our introduction of the atomic function up(𝑥) as a kernel.In this regard, we discuss the role of the Gauss, LogGauss and the up(𝑥) kernelsin low-level image processing. First, we analyse their characteristics from a math-ematical perspective and then we carry out an experimental test to show theirstrengths in the processing of real images. We have to stress that we are interestedin the application of the up(𝑥) kernel not just for filtering in scale space. This

4. Quaternionic Local Phase 63

up(𝑥) kernel is promising because it has a range of multiple possible applicationswhich is wider than the possible applications of the Gauss or LogGauss kernels.

3.1. Dyadic Shifts and the Riesz Transform

As was explained in Section 2, atomic functions are compactly supported, in-finitely differentiable solutions of differential equations with a shifted argument.Consequently an atomic function can be seen as an appropriate building block oflinear shift invariant (LSI) operators to implement complex operators for imageprocessing. In contrast the Gauss and LogGauss functions require derivatives andconvolutions with a function of series of impulses for building complex operators.We believe that atomic functions are promising to develop operators based onlinear combination of shifted atomic functions. Section 2.2 illustrates the differ-entiator atomic function dup(𝑥). In 2002 Petermichl [16] showed that the Hilberttransform lies in the closed convex hull of dyadic singular operators, thus theHilbert transform can be represented as an average of dyadic shifts. Petermichl etal. [17] show that the same is true for ℝ𝑛, therefore the Riesz transforms can alsobe obtained as the results of averaging of dyadic shifts. Thus we claim that theRiesz transforms can be obtained by averaging dyadic shifts of atomic functions.

3.2. Monogenic Signals and the Atomic Function

The monogenic signal was introduced by Felsberg, Bulow, and Sommer [7]. Weoutline briefly the concept of the monogenic signal and then explain the use ofthe atomic-function based monogenic signal. If we embed ℝ3 into a subspace of ℍspanned by just {1, 𝒊, 𝒋} according to

𝑞 = (𝒊, 𝒋, 1)𝒙 = 𝑥3 + 𝑥1𝒊+ 𝑥2𝒋, (3.1)

and further embed the vector field 𝒈 as follows:

𝑔ℍ = (−𝒊,−𝒋, 1)𝒈 = 𝑔3 − 𝑔1𝒊− 𝑔2𝒋 (3.2)

then ∇× 𝑔(𝒙) = 0 and ∇ ⋅ 𝑔(𝒙) = ⟨∇, 𝑔(𝒙)⟩ = 0 are equivalent to the generalizedCauchy–Riemann equations from Clifford analysis [4, 15]. All functions that fulfillthese equations are known as 𝑚𝑜𝑛𝑜𝑔𝑒𝑛𝑖𝑐 functions. Using the same embedding,the monogenic signal can be defined in the frequency domain as follows:

𝐹𝑀 (𝑢) = 𝐺3(𝑢1, 𝑢2, 0)− 𝒊𝐺1(𝑢1, 𝑢2, 0)− 𝒋𝐺2(𝑢1, 𝑢2, 0)

= 𝐹 (𝒖)− (𝒊, 𝒋)𝑭𝑅(𝒖) =∣𝒖∣+ (1,𝒌)𝒖

∣𝒖∣ 𝐹 (𝒖), (3.3)

where the inverse Fourier transform of 𝐹𝑀 (𝑢) is given by

𝑓𝑀 (𝒙) = 𝑓(𝒙)− (𝒊, 𝒋)𝒇𝑅(𝒙) = 𝑓(𝒙) + ℎ𝑅 ★ 𝑓(𝒙), (3.4)

64 E.U. Moya-Sanchez and E. Bayro-Corrochano

where 𝒇𝑅(𝒙) stands for the Riesz transform obtained by taking the inverse trans-form of 𝑭𝑅(𝒖) as follows:

𝑭𝑅(𝒖) =𝒊𝒖

∣𝒖∣𝐹 (𝒖) = 𝑯𝑅(𝒖)𝐹 (𝒖)←→ 𝒇𝑅(𝒙) = −𝒙

2𝜋 ∣𝒙∣2 ★ 𝑓(𝒙)

= ℎ𝑅(𝒙) ★ 𝑓(𝒙), (3.5)

where ★ stands for the convolution operation. Note that here the Riesz transformis the generalization of the 1D Hilbert transform. Using the fundamental solutionof the 3D Laplace equation restricted to the open half-space 𝑧 > 0 with boundarycondition, the solution is defined as

𝑓𝑀 (𝑥, 𝑦, 𝑧) = ℎ𝑃 ★ 𝑓(𝑥, 𝑦, 𝑧) + ℎ𝑃 ★ ℎ𝑅 ★ 𝑓(𝑥, 𝑦, 𝑧)

= ℎ𝑃 ★ (1 + ℎ𝑅★)𝑓(𝑥, 𝑦, 𝑧), (3.6)

where ℎ𝑃 stands for the 2D Poisson kernel. Setting in 𝑓𝑀 (𝑥, 𝑦, 𝑧) the variable 𝑧equal to zero, we obtain the so-called monogenic signal. Some authors have used theGauss kernel instead of the Poisson kernel, because the Poisson kernel establishesa linear scale space similar to the Gaussian scale space.

The atomic function is also an LSI operator; therefore, it appears that its useensures a computation in a linear scale space as well.

The monogenic functions are the solutions of the generalized Cauchy–Rie-mann equations or Laplace-type equations. The atomic function can be used tocompute compactly supported solutions of functional differential equations, forexample, (2.1). Conditions under which the type of equations (2.1) have solu-tions with compact support and an explicit form were obtained by Ravachev [12].Compactly supported solutions of equations of the type (2.1) are called atomicfunctions .

Now, for the case of 2D signal processing, we can apply the wavelet steerabil-ity and the Riesz transform. In this regard we will utilize the quaternion waveletatomic function, which will be discussed in more depth below. For the scale spacefiltering, authors use the Gauss or Poisson kernels for the Riesz transformation[6, 7]. It is known that the Poisson kernel is the fundamental solution of the 3DLaplace equation, however there are authors who use instead the Poisson, Gaussor LogGauss function. Felsberg [6] showed that the spatial extent of the Poissonkernel is greater than that of the Gaussian kernel. In addition the uncertainty ofthe Poisson kernel for 1D signals is slightly worse than that of the Gaussian kernel(by a factor of

√2). Nowadays many authors are now using the LogGauss kernel

for implementing monogenic signals, because it has better properties than boththe Gauss and Poisson kernels. Thus, we can infer that, in contrast, the atomicfunction as a spatially-compact kernel guarantees an analytic and closed solutionof Laplace type equations. In addition, as we said above, the Riesz transformscan be obtained by averaging dyadic operators, thus the use of a compact atomic

4. Quaternionic Local Phase 65

function avoids increased truncation errors. However, by the case of the noncom-pact Poisson, Gauss and LogGauss kernels, in practice, their larger spatial extentsrequire either larger filter masks otherwise they cause increased truncation errors.

4. Quaternion Algebra ℍ

The even subalgebra 𝒢+3,0,0 (bivector basis) is isomorphic to the quaternion alge-

bra ℍ, which is an associative, non-commutative, four-dimensional algebra thatconsists of one real element and three imaginary elements.

𝑞 = 𝑎+ 𝑏𝒊+ 𝑐𝒋 + 𝑑𝒌, 𝑎, 𝑏, 𝑐, 𝑑 ∈ ℝ (4.1)

The units 𝒊, 𝒋 obey the relations 𝒊2 = 𝒋2 = −1, 𝒊𝒋 = 𝒌. ℍ is geometrically inspired,and the imaginary components can be described in terms of the basis of ℝ3 space,𝒊→ 𝒆23, 𝒋 → 𝒆12,𝒌→ 𝒆31. Another important property of ℍ is the phase concept.A polar representation of 𝑞 is

𝑞 = ∣𝑞∣ 𝑒𝒊𝜙𝑒𝒌𝜓𝑒𝒋𝜃, (4.2)

where ∣𝑞∣ = √𝑞𝑞 where 𝑞 is a conjugate of 𝑞 = 𝑎 − 𝑏𝒊 − 𝑐𝒋 − 𝑑𝒌 and the angles

(𝜙, 𝜃, 𝜓) represent the three quaternionic phases [5].

4.1. Quaternionic Atomic Function qup(𝒙, 𝒚)

Since a 2D signal can be split into even (e) and odd (o) parts [5],

𝑓(𝑥, 𝑦) = 𝑓ee(𝑥, 𝑦) + 𝑓oe(𝑥, 𝑦) + 𝑓eo(𝑥, 𝑦) + 𝑓oo(𝑥, 𝑦), (4.3)

we can separate the four components of up(𝑥, 𝑦) and represent it as a quaternionas follows [13, 14]:

qup(𝑥, 𝑦) = up(𝑥, 𝑦)[cos(𝑤𝑥) cos(𝑤𝑦)

+ 𝒊 sin(𝑤𝑥) cos(𝑤𝑦) + 𝒋 cos(𝑤𝑥) sin(𝑤𝑦) + 𝒌 sin(𝑤𝑥) sin(𝑤𝑦)]

= qupee(𝑥, 𝑦) + 𝒊 qupoe(𝑥, 𝑦) + 𝒋 qupeo(𝑥, 𝑦) + 𝒌 qupoo(𝑥, 𝑦). (4.4)

Figure 5 shows the quaternion atomic function qup(𝑥, 𝑦) in the spatial domain withits four components: the real part qupee(𝑥, 𝑦) and the imaginary parts qupeo(𝑥, 𝑦),qupoe(𝑥, 𝑦), and qupoo(𝑥, 𝑦). We can see even and odd symmetries in the horizon-tal, vertical, and diagonal axes.

Figure 5. Atomic function qup(𝑥, 𝑦). From left to right: qupee(𝑥, 𝑦),qupoe(𝑥, 𝑦), qupeo(𝑥, 𝑦), and qupoo(𝑥, 𝑦).

66 E.U. Moya-Sanchez and E. Bayro-Corrochano

4.2. Quaternionic Atomic Wavelet

The importance of wavelet transforms (real, complex, hypercomplex) has beendiscussed in many references [1, 18]. In this work, we use the quaternionic wavelettransform (𝑄𝑊𝑇 ). The 𝑄𝑊𝑇 can be seen as an extension of the complex wavelettransform (𝐶𝑊𝑇 ) [1]. The multiresolution analysis applied in 𝑅𝑊𝑇 and 𝐶𝑊𝑇can be straightforwardly extended to the 𝑄𝑊𝑇 [1].

Our approach is based on the quaternionic Fourier transform (𝑄𝐹𝑇 ), (4.5),which is also the basis for the quaternionic analytic function defined by Bulow [5].The kernel of the 2D 𝑄𝐹𝑇 is given by

𝑒−𝒊2𝜋𝜐𝑥𝑓(𝑥, 𝑦)𝑒−𝒋2𝜋𝜈𝑦, (4.5)

where the real part is cos(2𝜋𝜈𝑥) cos(2𝜋𝜐𝑦); the imaginary parts (𝒊, 𝒋,𝒌) are its par-tial Hilbert transform cos(2𝜋𝜈𝑥) sin(2𝜋𝜐𝑦), sin(2𝜋𝜈𝑥) cos(2𝜋𝜐𝑦) (horizontal andvertical); and total Hilbert transform (diagonal) sin(2𝜋𝜈𝑥) sin(2𝜋𝜐𝑦). Using the𝑄𝐹𝑇 basis, we can obtain the 2D phase information that satisfies the definitionof the quaternionic analytic signal. To compute the multiscale approach, we use a2D separable implementation. We independently apply two sets of ℎ and 𝑔 waveletfilters.

ℎ = exp

(𝒊𝑐1𝑢1𝑥

𝜎1

)up(𝑥, 𝑦, 𝜎1) exp

(𝒋𝑠1𝜔𝑣1𝑦

𝜎1

)= ℎee + 𝒊ℎoe + 𝒋ℎeo + 𝒌ℎoo (4.6)

𝑔 = exp

(𝒊𝑐2𝑤2𝑥

𝜎2

)up(𝑥, 𝑦, 𝜎1) exp

(𝒋𝑠2𝜔𝑣2𝑦

𝜎2

)= 𝑔ee + 𝒊𝑔oe + 𝒋𝑔eo + 𝒌𝑔oo, (4.7)

where the extra 𝜏 parameter in the up function stands for the filter width. Theprocedure for quaternionic wavelet multiresolution analysis depicted partially inFigure 6 is as follows [1]:

1. Convolve the image (2D signal) at level 𝑛 with the scale and wave filtersℎ and 𝑔 along the rows of the 2D signal. The latter filters are the discreteversions of those filters given in (4.6) and (4.7).

2. The ℎ and 𝑔 filters are convolved with the columns of the previous responsesof the filters ℎ and 𝑔.

3. Subsample the responses of these filters by a factor of 2 (↑ 2).4. The real part of the approximation at level 𝑛 + 1 is taken as input at the

next level. This process continues through all levels, repeating the steps juststated.

As can be seen in Figure 6, the low level of the pyramid is the highest level ofresolution, and as you move up, the resolution decreases.

4. Quaternionic Local Phase 67

Image

Level 1

h1

g1

Rows

h1

g1

Columns

h1

g1

2

22

2

2

2

Approximation

Horizontal

Vertical

Diagonal2=subsampling

Image2

Level 2

h2

g2

Rows

h2

g2

Colums

h2

g2

2

22

2

2

2

Approximation

Horizontal

Vertical

Diagonal2=subsampling

Figure 6. Multiresolution approach.

4.3. Steerable Quaternionic Filter

Due to our approach not being invariant to rotations [6], we require a filter bankin order to get different line or edge orientations. Figure 7 shows different orien-tation filters and real and imaginary parts. We steer the mother atomic functionwavelet through a multiresolution pyramid in order to detect quaternionic phasechanges, which can be used for the feature detection of lines, edges, centroids, andorientation in geometric structures.

5. Quaternionic Local Phase Information

In this chapter, we refer to the phase information as the local phase in order toseparate the structure or geometric information and the amplitude in a certainpart of the signal. Moreover, the phase information permits us to obtain invariantor equivariant1 response. For instance, it has been shown that the phase has aninvariant response to changes in image brightness and the phase can be used tomeasure the symmetry or asymmetry of objects [11, 2, 9]. These invariant andequivariant responses are the key part to link the low-level processing with theimage analysis and the upper layers in computer vision applications.

The local phase means the computation of the phase at a certain positionin a real signal. In 1D signals, the analytic signal based on the Hilbert transform(𝑓𝑯(𝑥)) [5] is given by

𝑓𝐴(𝑓(𝑥)) = 𝑓(𝑥) + 𝒊𝑓𝑯(𝑥), (5.1)

𝑓𝐴(𝑓(𝑥)) = ∣𝐴∣ 𝑒𝒊𝜃, (5.2)

where ∣𝐴∣ =√

𝑓(𝑥)2 + 𝑓𝑯(𝑥)2 and 𝜃 = arctan

(𝑓(𝑥)𝑓𝑯(𝑥)

)permits us to extract the

magnitude and phase independently. In 2D signals, the Hilbert transform is notenough to compute the magnitude and phase independently in any direction [6]. Inorder to solve this, the quaternionic analytic (see (5.3)) signal and the monogenic

1Equivariance: monotonic dependency of value or parameter under some transformation.

68 E.U. Moya-Sanchez and E. Bayro-Corrochano

Figure 7. From left to right: qup(15∘), qup(40∘), qup(90∘), qup(140∘),and qup(155∘)

signal have been proposed by Bulow [5] and Felsberg [6], respectively. Until now,we have used an approximation of the quaternionic analytic signal based on thebasis of 𝑄𝐹𝑇 to extract some oriented axis symmetries.

Figure 8 contains, at the top, an image with lines and edges, at the centrean image profile, and at the bottom the profiles of the three quaternionic phases.In the phase profiles, we can distinguish between a line (even) and an edge (odd)using the phase (𝜃). These results are similar to the results reported by Granlund[9] using only a complex phase, because they used an image that changes in onedirection.

5.1. Quaternionic Analytic Signal

The quaternionic analytic signal in the spatial domain is defined as [5]:

𝑓 𝑞𝐴(𝑥, 𝑦) = 𝑓(𝑥, 𝑦) + 𝒊𝑓𝑯𝒊(𝑥, 𝑦) + 𝒋𝑓𝑯𝒋(𝑥, 𝑦) + 𝒌𝑓𝑯𝒌(𝑥, 𝑦), (5.3)

where 𝑓𝑯𝒊(𝑥, 𝑦) = 𝑓(𝑥, 𝑦) ∗ (𝛿(𝑦)/𝜋𝑥) and 𝑓𝑯𝒋(𝑥, 𝑦) = 𝑓(𝑥, 𝑦) ∗ (𝛿(𝑥)/𝜋𝑦) are thepartial Hilbert transforms and 𝑓𝑯𝒌(𝑥, 𝑦) = 𝑓(𝑥, 𝑦) ∗ (𝛿(𝑥, 𝑦)/𝜋2𝑥𝑦

)is the total

Hilbert transform. Bulow has shown that the 𝑄𝐹𝑇 kernel is expressed in terms ofthe Hilbert transforms. The phases can be computed easily using a 3D rotation ma-trix ℳ, which can be factored into three rotations, 𝑅 = 𝑅𝑥(2𝜙), 𝑅𝑧(2𝜓), 𝑅𝑦(2𝜃),

4. Quaternionic Local Phase 69

0 20 40 60 80 100 120

20

40

60

80

100

120

0 20 40 60 80 100 120

0

0.2

0.4

0.6

0.8

1

20 40 60 80 100 120

1

0.5

0

0.5

1

1.5

Figure 8. Image (top left), image profile (top right), and three quater-nionic phases: profile, line, and edge (bottom).

in the coordinate axes [5], i.e.,

ℳ(𝑞) =ℳ(𝑞1)ℳ(𝑞2)ℳ(𝑞3) (5.4)

𝑞1 = 𝑒𝒊𝜙, 𝑞2 = 𝑒𝒋𝜃, 𝑞3 = 𝑒𝒌𝜓 (5.5)

ℳ(𝑞) =

⎛⎝𝑎2 + 𝑏2 − 𝑐2 − 𝑑2 2(𝑏𝑐− 𝑎𝑑) 2(𝑏𝑑+ 𝑎𝑐)2(𝑏𝑐+ 𝑎𝑑) 𝑎2 − 𝑏2 + 𝑐2 − 𝑑2 2(𝑐𝑑− 𝑎𝑏)2(𝑏𝑑− 𝑎𝑐) 2(𝑐𝑑+ 𝑎𝑏) 𝑎2 − 𝑏2 − 𝑐2 + 𝑑2

⎞⎠ (5.6)

𝑅 =

⎛⎜⎜⎜⎜⎜⎜⎜⎝

cos(2𝜓) cos(2𝜃) − sin(2𝜓) cos(2𝜓) sin(2𝜃)

cos(2𝜙) sin(2𝜓) cos(2𝜃)+ sin(2𝜙) sin(2𝜃) cos(2𝜙) cos(2𝜓)

cos(2𝜙) sin(2𝜓) sin(2𝜃)− sin(2𝜙) cos(2𝜃)

sin(2𝜙) sin(2𝜓) cos(2𝜃)− cos(2𝜙) sin(2𝜃) sin(2𝜙) cos(2𝜓)

sin(2𝜙) sin(2𝜓) sin(2𝜃)+ cos(2𝜙) cos(2𝜃)

⎞⎟⎟⎟⎟⎟⎟⎟⎠.

(5.7)

The quaternionic phases are expressed by the following rules:

𝜓 = −1

2arcsin(2(𝑏𝑐− 𝑎𝑑)). (5.8)

70 E.U. Moya-Sanchez and E. Bayro-Corrochano

∙ If 𝜓 ∈ ]−𝜋4 ,

𝜋4

[, then 𝜙 = 1

2 arg𝒊 (𝑞𝒯𝒋 (𝑞)) and 𝜃 = 12 arg𝒋 (𝒯𝒊 (𝑞) 𝑞).

∙ If 𝜓 = ±𝜋4 , then select either 𝜙 = 0 and 𝜃 = 1

2 arg𝒋 (𝒯𝒌 (𝑞) 𝑞) or 𝜃 = 0 and

𝜙 = 12 arg𝒊 (𝑞𝒯𝒌 (𝑞)).

∙ If 𝑒𝒊𝜙𝑒𝒌𝜓𝑒𝒋𝜃 = −𝑞 and 𝜙 ≥ 0, then 𝜙→ 𝜙− 𝜋.∙ If 𝑒𝒊𝜙𝑒𝒌𝜓𝑒𝒋𝜃 = −𝑞 and 𝜙 < 0, then 𝜙→ 𝜙+ 𝜋.

The phase ranges are (𝜙, 𝜃, 𝜓) ∈ [−𝜋, 𝜋[× [−𝜋2 ,

𝜋2

[× [−𝜋4 ,

𝜋4

]. The applications of

the quaternionic analytic signal in image processing have to be limited in a narrowband, and Bulow used a Gauss window with the 𝑄𝐹𝑇 kernel (which can be seenas Gabor) to approximate the quaternionic analytic function. In our work, insteadof a Gauss window, we use a compact support window, the up(𝑥, 𝑦) function.

5.2. Quaternionic Phase Analysis

There are many points of view to see the phase information. The local phase canbe used as a measure of the symmetry in the 2D signals of the object [11, 9].The symmetry is related to middle-level properties if it remains invariant undersome transformation. A symmetric analysis related to the phase was proposedby P. Kovesi, using the even (𝑒𝑛(𝑥)) and odd (𝑜𝑛(𝑥)) responses of wavelets atscale 𝑛 [11]:

Sym(𝑥) =

∑𝑛

∣𝑒𝑛(𝑥)∣ − ∣𝑜𝑛(𝑥)∣∑𝑛

𝐴𝑛(𝑥)(5.9)

Asym(𝑥) =

∑𝑛

∣𝑜𝑛(𝑥)∣ − ∣𝑒𝑛(𝑥)∣∑𝑛

𝐴𝑛(𝑥)(5.10)

where 𝐴𝑛(𝑥) =√

𝑒𝑛(𝑥)2 + 𝑜𝑛(𝑥)2. Since the phase information has the capabilityto decode the geometrical information into even or odd symmetries (see Figure 8),we use the phase information to do a geometric analysis of the image. Note thatthe quaternionic phase analysis helps to reduce the gap between the low-level andthe middle-level processing.

5.3. Hilbert Transform Using AF

The Hilbert transform and the derivative are closely related, and the Hilbert trans-form can actually be computed using a derivative and some convolution proper-ties [19]:

𝑓 ★ (𝑔 ★ ℎ) = (𝑓 ★ 𝑔) ★ ℎ (5.11)

∇ (𝑓 ★ 𝑔) = ∇𝑓 ★ 𝑔 = 𝑓 ★∇𝑔, (5.12)

4. Quaternionic Local Phase 71

where

𝑓(𝑥, 𝑦), 𝑔(𝑥, 𝑦), ℎ(𝑥, 𝑦) ∈ ℝ2 and ∇ = 𝒆1∂

∂𝑥+ 𝒆2

∂

∂𝑦.

If 𝑔(𝑥, 𝑦) = −(1/𝜋) log ∣𝑥∣ log ∣𝑦∣ 2, and if we use the convolution distribution prop-erties, we can express the Hilbert transform and the partial Hilbert transform (see(5.3)) as

𝑓𝑯𝒊(𝑥, 𝑦) =∂𝑓(𝑥, 𝑦)

∂𝑥★− 1

𝜋log ∣𝑥∣ (5.13)

𝑓𝑯𝒋 (𝑥, 𝑦) =∂𝑓(𝑥, 𝑦)

∂𝑦★− 1

𝜋log ∣𝑦∣ (5.14)

𝑓𝑯𝒌(𝑥, 𝑦) =∂2𝑓(𝑥, 𝑦)

∂𝑥∂𝑦★− 1

𝜋2log ∣𝑥∣ log ∣𝑦∣ (5.15)

and we can use the convolution association property to get the equation of a certainpart of the signal in terms of dup(𝑥):

𝑓𝑯𝒊(𝑥, 𝑦) = 𝑓(𝑥, 𝑦) ★

(dup(𝑥, 𝑦)𝑥 ★− 1

𝜋log ∣𝑥∣

)(5.16)

𝑓𝑯𝒋 (𝑥, 𝑦) = 𝑓(𝑥, 𝑦) ★

(dup(𝑥, 𝑦)𝑦 ★− 1

𝜋log ∣𝑦∣

)(5.17)

𝑓𝑯𝒌(𝑥, 𝑦) = 𝑓(𝑥, 𝑦) ★

(dup(𝑥, 𝑦)𝑥𝑦 ★− 1

𝜋2log ∣𝑥∣ log ∣𝑦∣

). (5.18)

Moreover, it has been shown by Petermichl et al. [17] that the Hilbert andRiesz transforms can be implemented with the average of dyadic shifts, and thedyadic shift operations appear naturally using the AF and its derivatives.

6. Applications

6.1. Convolution 𝑰 ★ qup(𝒙, 𝒚)

Figure 9 shows the convolution of the qup (real and imaginary parts) filter on ashadow chessboard image. The real part corresponds to a low-pass filtering of theimage. In addition, we can see a selective detection of lines in the horizontal and

2log ∣𝑥∣ is the fundamental solution of the Laplace equation.

Figure 9. Convolution of qup(𝑥, 𝑦) with chessboard image. From leftto right: original image, real part, 𝒊-part, 𝒋-part, and 𝒌-part.

72 E.U. Moya-Sanchez and E. Bayro-Corrochano

Figure 10. Convolution of qup(𝑥, 𝑦) with image (letter H). From leftto right: original image, real part, 𝒊-part, 𝒋-part, and 𝒌-part.

Figure 11. Convolution of qup(𝑥, 𝑦) with image (letter L). From leftto right: original image, real part, 𝒊-part, 𝒋-part, and 𝒌-part are shown.

vertical orientations, particularly in the 𝒊-part and 𝒋-part. On the other hand, the𝒌-part can be used to detect the corners as a diagonal response. We can see howthe response in the light part is more intensive than that in the shadowed part.We show in Figures 10 and 11 how the qup(𝑥, 𝑦) filters respond to other images.We can see a similar behaviour in Figure 9. We can also notice that the directconvolution of qup(𝑥, 𝑦) with the image is sensitive to the contrast.

6.2. Derivative of up(𝒙)

Figure 12 illustrates the convolution of the first derivative, dup(𝑥, 𝑦), and a shadowchessboard. The convolution with dup can be used as an oriented change detec-tor with a simple rotation. Similarly to Figure 9, the convolved image has a lowresponse in the shadow part.

Figure 12. Convolution of dup(𝑥, 𝑦) with the chess image. (a) Testimage; (b) result of the convolution of the image with dup(𝑥, 𝑦, 0∘); (c)dup(𝑥, 𝑦, 45∘); (d) dup(𝑥, 𝑦, 135∘).

4. Quaternionic Local Phase 73

6.3. Image Processing Using Monogenic Signals

We use these three kernels to implement monogenic signals whereby the apertureof the filters are varied at three scales of a multiresolution pyramid. The samereference frequency and the same aperture rate variation are used for the threemonogenic signals. In Figures 13, 14 and 15, you can appreciate at these scales thebetter consistency of the up(𝑥) kernel particularly to detect via the phase concept

(a) Upper row: Gauss, Hx and Hy frequency; lower row: energy, 𝜃, 𝜓

(b) Upper row: LogGabor, Hx and Hy frequency; lower row: energy, 𝜃, 𝜓

Figure 13. Monogenic signal and response at the first level of filtering(𝑠 = 1). (a) Gabor monogenic; (b) LogGabor monogenic signal.

74 E.U. Moya-Sanchez and E. Bayro-Corrochano

(c) Upper row: up, Hx and Hy frequency; lower row: energy, 𝜃, 𝜓

Figure 13. Monogenic signal and response at the first level of filtering(𝑠 = 1). (c) up monogenic.

the corners and borders diminishing the expected blurring by a checkerboard im-age. We suspect that this an effect of the compactness in space of the up(𝑥) kernel,the Moire effects are milder especially if you observe the images at scales 2 and3, whereas in both the Gauss and LogGauss kernel-based monogenic phases thereare noticeable artifacts in the form of bubbles or circular spots. These are often aresult of Moire effects. In contrast the phases of the up(𝑥) based monogenic signalstill preserves the scales and edges of the checkerboard.

6.4. Quaternionic Local Phase

As we have mentioned before, the (local) phase information can be used to extractthe structure information (line and edges) independently of illumination changes,and we can measure the symmetry (or asymmetry) with the phase information asa middle-level property. The performance of qup to detect edges or lines using thequaternionic phases in different images is shown in this section.

In Figure 17, we present different geometric figures with lines that representthe profiles (top, centre, and bottom). Similarly to Figure 8, the profiles of theimages are shown as well as the quaternionic phase profile. This figure illustrateshow the three phases have a different symmetric or antisymmetric response on thetop, center, and bottom profile. These results motivate us to use the quaternionicphases as a measure of symmetry using the even or odd response of the quaternionicphases in a horizontal or vertical direction. In others words, the quaternionic phaseinformation decomposes the image into oriented edges or lines.

4. Quaternionic Local Phase 75

As an example of other possible applications of the phase information inimage analysis of geometric figures, we show Figure 18. The original image, themagnitude, and the three phases are shown. The first row of Figure 18 shows foursquares with a different illumination and area; one of the squares is rotated, andthis is the only square that can be seen in the 𝜓 phase. In the second row, thefour squares have been rotated 15∘, and the 𝜓 phase responds to the four squares,whereas in the third row, the four figures are rotated 45∘, and the 𝜓 phase onlydetects three of the four squares. In this example, the 𝜓 phase only responds tothe orientation of the square (a geometric transformation), independently of thesize, contrast, or position of the image.

The magnitude and the quaternionic phases for a circle with lines are shownin Figure 19. In this image, we tuned the filter parameters in order to obtain aresponse (positive and negative) to diagonal lines. Again, in this image, the shadowor the size of the lines in the image does not change the phase response. The blacklines correspond to a cone beam with a 45∘ orientation (±15∘) and the white linesare oriented 135∘ (±15∘).

In Figures 19 and 18, lines or geometric objects have been shown separately.Figure 20 illustrates an image with lines and geometric objects. In this case, we canuse the quaternionic phases to extract different oriented edges of geometric objectsor different oriented line textures. The 𝜃 and 𝜙 phases detect lines or edges in somevertical and horizontal directions, while the 𝜓 phase detects the diagonal responseand the corners in squares or geometric figures. Even if the geometric objects haveinternal lines, different illuminations, or positions, the edges of each square can

(a) Upper row: Gauss, Hx and Hy frequency; lower row: energy, 𝜃, 𝜓

Figure 14. Monogenic signal and response at the second level of fil-tering (𝑠 = 2). (a) Gabor filters.

76 E.U. Moya-Sanchez and E. Bayro-Corrochano

(b) Upper row: LogGabor, Hx and Hy frequency; lower row: energy, 𝜃, 𝜓

(c) Upper row: up, Hx and Hy frequency; lower row: energy, 𝜃, 𝜓

Figure 14. Monogenic signal and response at the second level of fil-tering (𝑠 = 2). (b) LogGabor and (c) up filters.

be detected in the 𝜙 and 𝜃 images. Furthermore, in the 𝜙 phase, the horizontallines are highlighted, whereas the vertical lines do not appear. The 𝜃 phases showa similar result, but in this case the vertical edges and lines are highlighted. In the𝜓 phase, the vertical lines or edges are highlighted.

4. Quaternionic Local Phase 77

(a) Upper row: Gauss, Hx and Hy frequency; lower row: energy, 𝜃, 𝜓

(b) Upper row: LogGabor, Hx and Hy frequency; lower row: energy, 𝜃, 𝜓

Figure 15. Monogenic signal and response at the third level of filtering(𝑠 = 3). (a) Gabor and (b) LogGabor filters.

6.5. Phase Symmetry

As an example of many possible applications, we show the computation of themain axis in one orientation using the 𝜃 phase. We use the information of Fig-ure 21 to determine a threshold. A reflection operation can be used to measurethe symmetry.

78 E.U. Moya-Sanchez and E. Bayro-Corrochano

(c) Upper row: up, Hx and Hy frequency; lower row: energy, 𝜃, 𝜓

Figure 15. Monogenic signal and response at the third level of filtering(𝑠 = 3). (c) up filters.

Figure 16. From left to right: circle; edges.

6.6. Multiresolution and Steerable Filters: qup(𝒙, 𝒚)

In this section, we present multiscale and steerable filter results. Figure 22 shows acircle and its multiscale processing. In the left column, we see a multiscale filtering.In this figure, the 𝜓 phase responds better to diagonal lines, and we can see coarseto fine details. In the right group of the figure, we can see the effect of differentorientations of our steerable filters.

7. Conclusions

In this work, we have presented image processing and analysis using an atomicfunction and the quaternionic phase concept. As we have shown, the atomic func-tion up(𝑥) has shown potential in image processing as a building block to build

4. Quaternionic Local Phase 79

100 200 300 400 500 600 7000.2

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0.4

Figure 17. Left: quaternionic phase profiles of a circle. Top, centreand bottom phase profiles are shown. Right: quaternionic phase profilesof edge images. Top, centre and bottom phase profiles are shown.

multiple operations that can be done analytically such as low-pass filtering, deriva-tives, local phase, and multiscale and steering filters. We have shown that thefunction qup(𝑥, 𝑦) is useful to detect lines or edges in a specific orientation usingthe quaternionic phase concept. Additionally, an oriented texture can be chosen

80 E.U. Moya-Sanchez and E. Bayro-Corrochano

Figure 18. From left to right: image, magnitude, and the quaternionicphases 𝜙, 𝜃 and 𝜓. We can see how the 𝜓 phase responds to the rotatedobject.

Figure 19. From left to right: image, magnitude, and the quaternionicphases. We can see how the 𝜓 phase responds to the rotated object.

using the quaternionic phases. As an initial step, we have shown how to do the im-age analysis of geometric objects in ℝ2 using the symmetry response of the phase.As in other applications of geometric algebra we can take advantage of the con-straints. Since the information from the three phases is independent of illuminationchanges, algorithms using the quaternionic atomic function can be less sensitivethan other methods based on the illumination changes. These results motivatedus to find other invariants such as rotation invariants using the Riesz transform.In future work, we expect to develop a complete computer vision approach basedon geometric algebra.

4. Quaternionic Local Phase 81

Figure 20. Image (left) and 2D quaternionic phases (left to right: 𝜙,𝜃, 𝜓). A texture based on lines can be detected or discriminated, andat the same time, the phase information can highlight the edges.

Figure 21. Circle image. The symmetry behaviour of the phase is shown.

References

[1] E. Bayro-Corrochano. The theory and use of the quaternion wavelet transform. Jour-nal of Mathematical Imaging and Vision, 24:19–35, 2006.

[2] J. Bernd. Digital Image Processing. Springer-Verlag, New York, 1993.

[3] J. Bigun. Vision with Direction. Springer, 2006.

[4] F. Brackx, R. Delanghe, and F. Sommen. Clifford Analysis, volume 76. Pitman,Boston, 1982.

[5] T. Bulow. Hypercomplex Spectral Signal Representations for the Processing and Anal-ysis of Images. PhD thesis, University of Kiel, Germany, Institut fur Informatik undPraktische Mathematik, Aug. 1999.

[6] M. Felsberg. Low-Level Image Processing with the Structure Multivector. PhD thesis,Christian-Albrechts-Universitat, Institut fur Informatik und Praktische Mathematik,Kiel, 2002.

[7] M. Felsberg and G. Sommer. The monogenic signal. IEEE Transactions on SignalProcessing, 49(12):3136–3144, Dec. 2001.

[8] A.S. Gorshkov, V.F. Kravchenko, and V.A. Rvachev. Estimation of the discrete de-rivative of a signal on the basis of atomic functions. Izmeritelnaya Tekhnika, 1(8):10,1992.

[9] G.H. Granlund and H. Knutsson. Signal Processing for Computer Vision. Kluwer,Dordrecht, 1995.

82 E.U. Moya-Sanchez and E. Bayro-Corrochano

(a)

(b)

Figure 22. Multiresolution and steerable filters. Columns left to right:image, magnitude, quaternionic phases 𝜙, 𝜃, 𝜓. (a) multiscale approach.(b) steering filters top to bottom 0∘, 45∘, 90∘ convolved with the image.

4. Quaternionic Local Phase 83

[10] V.M. Kolodyazhnya and V.A. Rvachov. Atomic functions: Generalization to themultivariable case and promising applications. Cybernetics and Systems Analysis,46(6), 2007.

[11] P. Kovesi. Invariant Measures of Image Features from Phase Information. PhD the-sis, University of Western Australia, Australia, 1996.

[12] V. Kravchenko, V. Ponomaryov, and H. Perez-Meana. Adaptive digital processing ofmultidimensional signals with applications. Moscow Fizmatlit, Moscow, 2010.

[13] E. Moya-Sanchez and E. Bayro-Corrochano. Quaternion atomic function waveletfor applications in image processing. In I. Bloch and R. Cesar, editors, Progress inPattern Recognition, Image Analysis, Computer Vision, and Applications, volume6419 of Lecture Notes in Computer Science, pages 346–353. Springer, 2010.

[14] E. Moya-Sanchez and E. Vazquez-Santacruz. A geometric bio-inspired model forrecognition of low-level structures. In T. Honkela, W. Duch, M. Girolami, andS. Kaski, editors, Artificial Neural Networks and Machine Learning – ICANN 2011,volume 6792 of Lecture Notes in Computer Science, pages 429–436. Springer, 2011.

[15] M.N. Nabighian. Toward a three-dimensional automatic interpretation of potentialfield data via generalized Hilbert transforms: Fundamental relations. Geophysics,49(6):780–786, June 1982.

[16] S. Petermichl. Dyadic shifts and logarithmic estimate for Hankel operators withmatrix symbol. Comptes Rendus de l’Academie des Sciences, 330(6):455–460, 2000.

[17] S. Petermichl, S. Treil, and A. Volberg. Why are the Riesz transforms averages of thedyadic shifts? Publicacions matematiques, 46(Extra 1):209–228, 2002. Proceedingsof the 6th International Conference on Harmonic Analysis and Partial DifferentialEquations, El Escorial (Madrid), 2002.

[18] I.W. Selesnick, R.G. Baraniuk, and N.G. Kingsbury. The dual-tree complex wavelettransform. IEEE Signal Processing Magazine, 22(6):123–151, Nov. 2005.

[19] B. Svensson. A Multidimensional Filtering Framework with Applications to Lo-cal Structure Analysis and Image Enhancement. PhD thesis, Linkoping University,Linkooping, Sweden, 2008.

E. Ulises Moya-Sanchez and E. Bayro-CorrochanoCINVESTAVCampus Guadalajara, Av. del Bosque 1145Colonia El Bajio, CP 45019Zapopan, Jalisco, Mexicoe-mail: emoya@gdl.cinvestav.mx

edb@gdl.cinvestav.mx

Quaternion and Clifford–Fourier Transforms and Wavelets

Trends in Mathematics, 85–104c⃝ 2013 Springer Basel

5 Bochner’s Theorems in the Frameworkof Quaternion Analysis

S. Georgiev and J. Morais

Abstract. Let 𝜎(𝑥) be a nondecreasing function, such that 𝜎(−∞) = 0,𝜎(∞) = 1 and let us denote by ℬ the class of functions which can be repre-sented by a Fourier–Stieltjes integral 𝑓(𝑡) =

∫∞−∞ 𝑒𝑖𝑡𝑥𝑑𝜎(𝑥). The purpose of

this chapter is to give a characterization of the class ℬ and to give a gener-alization of the classical theorem of Bochner in the framework of quaternionanalysis.

Mathematics Subject Classification (2010). Primary 30G35; secondary 42A38.

Keywords. Quaternion analysis, quaternion Fourier transform, quaternionFourier–Stieltjes integral, Bochner theorem.

1. Introduction

In a recent paper [5], we discussed special properties of the asymptotic behaviourof the quaternion Fourier transform (QFT) and provided a straightforward gener-alization of the classical Bochner–Minlos theorem to the framework of quaternionanalysis. The main objective of the present chapter is to extend, using similar tech-niques, the theorem of Bochner on Fourier integral transforms of complex-valuedfunctions of positive type to functions with values in the Hamiltonian quater-nion algebra in which the exponential function is replaced by a (noncommutative)quaternion exponential product. The Fourier–Stieltjes transform (FST) is a well-known generalization of the standard Fourier transform, and is frequently appliedin certain areas of theoretical and applied probability and stochastic processes con-texts. The present study aims to develop further numerical integration methodsfor solving partial differential equations in the quaternion analysis setting.

The chapter is organized as follows. Section 2 recalls the classical Bochnertheorem on the Fourier integral transforms of functions of positive type and col-lects some basic concepts in quaternion analysis. Section 3 defines and analyzesdifferent types of quaternion Fourier–Stieltjes transforms (QFST) and establishes

86 S. Georgiev and J. Morais

a number of their important properties. The underlying signals are continuousfunctions of bounded variation defined in ℝ2 and taking values on the quaternionalgebra. We proceed by proving the uniform continuity on this transform. Then,we describe the interplay between uniform continuity and quaternion distribution.Section 4 contains the main result of the chapter – the counterpart of the Bochnertheorem for the noncommutative structure of quaternion functions (see Theorem4.3 below). To prove our Bochner theorem, we introduce the notion of positive-type quaternion function and deduce some of its characteristics, and rely on theasymptotic behaviour and other general properties of the QFST. To the best ofour knowledge this is done here for the first time. In the interests of simplicity ofpresentation, we have not extended this work to its most general form. Furtherinvestigations are in progress and will be reported in full in a forthcoming paper.

2. Preliminaries

2.1. Bochner Theorem

In this subsection, we review the classical Bochner theorem on Fourier integraltransforms, which can be found, e.g., in [1, 2].

A complex-valued function 𝑓(𝑡) defined on the interval (−∞,∞) is said tobe positive definite if it satisfies the following conditions:

1. 𝑓 is bounded and continuous on (−∞,∞);

2. 𝑓(−𝑡) = 𝑓(𝑡), for all 𝑡;3. for any set of real numbers 𝑡1, . . . , 𝑡𝑠, complex numbers 𝑎1, . . . , 𝑎𝑠, and any

positive integer 𝑠, the inequality

𝑠∑𝑚=1

𝑠∑𝑛=1

𝑎𝑚𝑎𝑛𝑓(𝑡𝑚 − 𝑡𝑛) ≥ 0 is satisfied.

The following theorem is due to Bochner [1].

Theorem 2.1 (Bochner). If 𝜎(𝑥) is a nondecreasing bounded function on the in-terval (−∞,∞), and if 𝑓(𝑡) is defined by the Stieltjes integral

𝑓(𝑡) =

∫ ∞

−∞𝑒𝑖𝑡𝑥𝑑𝜎(𝑥), −∞ < 𝑡 <∞, (2.1)

then 𝑓(𝑡) is a continuous function of the positive type.

It is of interest to remark at this point that the Fourier–Stieltjes transformof nondecreasing bounded functions can be easily seen continuous functions ofpositive type. Conversely, if 𝑓(𝑡) is measurable on (−∞,∞), and 𝑓 is of the positivetype, then there exists a nondecreasing bounded function 𝜎(𝑥) such that 𝑓(𝑡) isgiven by (2.1) for almost all 𝑥, −∞ < 𝑥 < ∞. In the converse part, Bochnerassumed 𝑓(𝑡) to be continuous, and showed that 𝜎(𝑥) is such that (2.1) is true forall 𝑡. Riesz, on the other hand, succeeded to prove that the measurability of 𝑓 wassufficient in the converse.

5. Bochner’s Theorems 87

2.2. Quaternion Analysis

The present subsection collects some basic facts about quaternions and the (left-sided) QFT, which will be needed throughout the text.

In all that follows let

ℍ := {𝑧 = 𝑎+ 𝑏𝒊+ 𝑐𝒋 + 𝑑𝒌 : 𝑎, 𝑏, 𝑐, 𝑑 ∈ ℝ} (2.2)

denote the Hamiltonian skew field, i.e., quaternions, where the imaginary units 𝒊,𝒋, and 𝒌 are subject to the multiplication rules:

𝒊2 = 𝒋2 = 𝒌2 = −1,𝒊𝒋 = 𝒌 = −𝒋𝒊, 𝒋𝒌 = 𝒊 = −𝒌𝒋, 𝒌𝒊 = 𝒋 = −𝒊𝒌. (2.3)

Like in the complex case, S(𝑧) = 𝑎 and V(𝑧) = 𝑏𝒊+ 𝑐𝒋 + 𝑑𝒌 define the scalar andvector parts of 𝑧. The conjugate of 𝑧 is 𝑧 = 𝑎− 𝑏𝒊− 𝑐𝒋 − 𝑑𝒌, and the norm of 𝑧 isdefined by

∣𝑧∣ = √𝑧𝑧 =√𝑧𝑧 =

√𝑎2 + 𝑏2 + 𝑐2 + 𝑑2, (2.4)

which coincides with the corresponding Euclidean norm of 𝑧 as a vector in ℝ4. Forx := (𝑥1, 𝑥2) ∈ ℝ2 we consider ℍ-valued functions defined in ℝ2, i.e., functions ofthe form

𝑓(x) := [𝑓(x)]0 + [𝑓(x)]1𝒊+ [𝑓(x)]2𝒋 + [𝑓(x)]3𝒌, (2.5)

[𝑓 ]𝑙 : ℝ2 −→ ℝ (𝑙 = 0, 1, 2, 3).

Properties (like integrability, continuity or differentiability) that are ascribed to 𝑓have to be fulfilled by all its components [𝑓 ]𝑙.

Let 𝐿1(ℝ2;ℍ) denote the linear space of integrableℍ-valued functions definedin ℝ2. The left-sided QFT of 𝑓 ∈ 𝐿1(ℝ2;ℍ) is given by [6]

ℱ(𝑓) : ℝ2 → ℍ, ℱ(𝑓)(𝝎) :=

∫ℝ2

e(𝝎,x) 𝑓(x) 𝑑2x, (2.6)

where the kernel function

e : ℝ2 × ℝ2 → ℍ, e(𝝎,x) := 𝑒−𝒋𝜔2𝑥2𝑒−𝒊𝜔1𝑥1 , (2.7)

is called the (left-sided) quaternion Fourier kernel. For 𝑖 = 1, 2, 𝑥𝑖 will denote thespace coordinates and 𝜔𝑖 the angular frequencies. The previous definition of theQFT varies from the original one only in the fact that we use 2D vectors insteadof scalars and that it is defined to be 2D. It is of interest to remark at this pointthat the product in (2.6) has to be performed in a fixed order since, in general,e(𝝎,x) does not commute with every element of the algebra.

Under suitable conditions, the original signal 𝑓 can be reconstructed fromℱ(𝑓) by the inverse transform. The (left-sided) inverse QFT of 𝑔 ∈ 𝐿1(ℝ2;ℍ) isgiven by

ℱ−1(𝑔) : ℝ2 → ℍ, ℱ−1(𝑔)(x) =1

(2𝜋)2

∫ℝ2

e(𝝎,x) 𝑔(𝝎) 𝑑2𝝎 , (2.8)

where e(𝝎,x) is called the inverse (left-sided) quaternion Fourier kernel.

88 S. Georgiev and J. Morais

3. Quaternion Fourier–Stieltjes Transform and its Properties

This section generalizes the classical FST to Hamilton’s quaternion algebra. Usingthe 4D analogue of the (complex-valued) function 𝜎(𝑥) described before, we extendthe FST to the QFST. As we shall see later, some properties of the FST can beextended in this context.

3.1. Quaternion Fourier–Stieltjes Transform

Partially motivated by the results from Bulow [3], the idea behind the constructionof a quaternion counterpart of the Stieltjes integral is to replace the exponentialfunction in (2.1) by a suitable (noncommutative) quaternion exponential product.

In the sequel, we consider the functions 𝜎1, 𝜎2 : ℝ → ℍ of the quaternionform (2.5), such that ∣𝜎𝑖∣ ≤ 𝛿𝑖 for real numbers 𝛿𝑖 <∞ (𝑖 = 1, 2). Due to the non-commutativity of the quaternions, we shall define three different types of QFST.

Definition 3.1. The QFST ℱ𝒮(𝜎1, 𝜎2) : ℝ2 → ℍ of 𝜎1(𝑥1) and 𝜎2(𝑥2) is definedas the Stieltjes integrals:

1. Right-sided QFST:

ℱ𝒮𝑟(𝜎1, 𝜎2)(𝜔1, 𝜔2) :=

∫ℝ2

𝑑𝜎1(𝑥1)𝑑𝜎2(𝑥2)𝑒

𝒊𝜔1𝑥1𝑒𝒋𝜔2𝑥2 . (3.1)

2. Left-sided QFST:

ℱ𝒮𝑙(𝜎1, 𝜎2)(𝜔1, 𝜔2) :=

∫ℝ2

𝑒𝒋𝜔2𝑥2𝑒𝒊𝜔1𝑥1𝑑𝜎1(𝑥1)𝑑𝜎2(𝑥2). (3.2)

3. Two-sided QFST:

ℱ𝒮𝑠(𝜎1, 𝜎2)(𝜔1, 𝜔2) :=

∫ℝ2

𝑒𝒊𝜔1𝑥1𝑑𝜎1(𝑥1)𝑑𝜎2(𝑥2)𝑒

𝒋𝜔2𝑥2 . (3.3)

Remark 3.2. We remind the reader that, the order of the exponentials in (3.1)–(3.3) is fixed because of the noncommutativity of the quaternion product. It is ofinterest to remark at this point that in case that 𝑑𝜎1𝑑𝜎2 = 𝑓(x)𝑑2x, the formulaeabove reduce to the usual definitions for the right-, left- and two-sided QFT [4, 7, 6],which only differ by the signs in the exponential kernel terms.

It is significant to note that, in practice, the integrals (3.1)–(3.3) will alwaysexist. For example, for any two real variables 𝜔1, 𝜔2, and for real constants 𝑎, 𝑏 itholds that

∣ℱ𝒮𝑟(𝜎1, 𝜎2)∣ ≤

∣∣∣∫ 𝑎−∞ (∫ 𝑏−∞+∫∞𝑏

)𝑑𝜎1(𝑥1)𝑑𝜎

2(𝑥2)𝑒𝒊𝜔1𝑥1𝑒𝒋𝜔2𝑥2

∣∣∣+∣∣∣∫∞𝑎 (∫ 𝑏

−∞+∫∞𝑏

)𝑑𝜎1(𝑥1)𝑑𝜎

2(𝑥2)𝑒𝒊𝜔1𝑥1𝑒𝒋𝜔2𝑥2

∣∣∣≤ (∣𝜎1(∞)− 𝜎1(𝑏)∣+ ∣𝜎1(𝑏)− 𝜎1(−∞)∣)× (∣𝜎2(∞)− 𝜎2(𝑎)∣+ ∣𝜎2(𝑎)− 𝜎2(−∞)∣) . (3.4)

5. Bochner’s Theorems 89

Remark 3.3. Throughout this text we will only investigate the integral (3.1), whichwe denote for simplicity by ℱ𝒮(𝜎1, 𝜎2). Nevertheless, all computations can beeasily adapted to (3.2) and (3.3). In view of (3.1) and (3.2), a straightforwardcalculation shows that

ℱ𝒮𝑟(𝜎1, 𝜎2)(𝜔1, 𝜔2) =

∫ℝ2

𝑒−𝒋𝜔2𝑥2𝑒−𝒊𝜔1𝑥1𝑑𝜎1(𝑥1)𝑑𝜎2(𝑥2)

= ℱ𝒮 𝑙(𝜎2, 𝜎1)(−𝜔1,−𝜔2). (3.5)

For the sake of further simplicity, in the considerations to follow we will often omitthe subscript and, additionally, write only ℱ𝒮(𝜔1,𝜔2) instead of ℱ𝒮(𝜎1,𝜎2)(𝜔1,𝜔2).We can restate the definition of right-sided QFST (3.1) in equivalent terms as fol-lows.

Lemma 3.4. The (right-sided ) QFST has the closed-form representation

ℱ𝒮(𝜔1, 𝜔2) := [Φ(𝜔1, 𝜔2)]0 + [Φ(𝜔1, 𝜔2)]1 + [Φ(𝜔1, 𝜔2)]2 + [Φ(𝜔1, 𝜔2)]3 , (3.6)

where we used the integrals

[Φ(𝜔1, 𝜔2)]0 =

∫ℝ2

𝑑𝜎1(𝑥1)𝑑𝜎2(𝑥2) cos(𝜔1𝑥1) cos(𝜔2𝑥2), (3.7)

[Φ(𝜔1, 𝜔2)]1 =

∫ℝ2

𝑑𝜎1(𝑥1)𝑑𝜎2(𝑥2)𝒊 sin(𝜔1𝑥1) cos(𝜔2𝑥2), (3.8)

[Φ(𝜔1, 𝜔2)]2 =

∫ℝ2

𝑑𝜎1(𝑥1)𝑑𝜎2(𝑥2)𝒋 cos(𝜔1𝑥1) sin(𝜔2𝑥2), (3.9)

[Φ(𝜔1, 𝜔2)]3 =

∫ℝ2

𝑑𝜎1(𝑥1)𝑑𝜎2(𝑥2)𝒌 sin(𝜔1𝑥1) sin(𝜔2𝑥2). (3.10)

Corollary 3.5. The (right-sided) QFST satisfies the following identities:

ℱ𝒮(𝜔1, 𝜔2) + ℱ𝒮(𝜔1,−𝜔2) = 2 (Φ0(𝜔1, 𝜔2) + Φ1(𝜔1, 𝜔2)) , (3.11)

ℱ𝒮(𝜔1, 𝜔2)−ℱ𝒮(𝜔1,−𝜔2) = 2 (Φ2(𝜔1, 𝜔2) + Φ3(𝜔1, 𝜔2)) , (3.12)

ℱ𝒮(𝜔1, 𝜔2) + ℱ𝒮(−𝜔1, 𝜔2) = 2 (Φ0((𝜔1, 𝜔2) + Φ2(𝜔1, 𝜔2)) , (3.13)

ℱ𝒮(𝜔1, 𝜔2)−ℱ𝒮(−𝜔1, 𝜔2) = 2 (Φ1(𝜔1, 𝜔2) + Φ3(𝜔1, 𝜔2)) , (3.14)

ℱ𝒮(𝜔1, 𝜔2) + ℱ𝒮(−𝜔1,−𝜔2) = 2 (Φ0(𝜔1, 𝜔2) + Φ3(𝜔1, 𝜔2)) , (3.15)

ℱ𝒮(𝜔1, 𝜔2)−ℱ𝒮(−𝜔1,−𝜔2) = 2 (Φ1(𝜔1, 𝜔2) + Φ2(𝜔1, 𝜔2)) , (3.16)

ℱ𝒮(𝜔1,−𝜔2) + ℱ𝒮(−𝜔1,−𝜔2) = 2 (Φ0(𝜔1, 𝜔2)− Φ2(𝜔1, 𝜔2)) , (3.17)

ℱ𝒮(𝜔1,−𝜔2)−ℱ𝒮(−𝜔1,−𝜔2) = 2 (Φ1(𝜔1, 𝜔2)− Φ3(𝜔1, 𝜔2)) , (3.18)

ℱ𝒮(−𝜔1, 𝜔2) + ℱ𝒮(−𝜔1,−𝜔2) = 2 (Φ0(𝜔1, 𝜔2)− Φ1(𝜔1, 𝜔2)) , (3.19)

ℱ𝒮(−𝜔1, 𝜔2)−ℱ𝒮(−𝜔1,−𝜔2) = 2 (Φ2(𝜔1, 𝜔2)− Φ3(𝜔1, 𝜔2)) . (3.20)

3.2. Properties

This subsection presents certain properties of the asymptotic behaviour of theQFST, and establishes some of its basic properties.

90 S. Georgiev and J. Morais

We begin with the notions of monotonic increasing, bounded variation anddistribution in the context of quaternion analysis.

Definition 3.6. A function 𝜎 : ℝ → ℍ is called monotonic increasing, if all of itscomponents [𝜎]𝑙 (𝑙 = 0, 1, 2, 3) are monotonic increasing functions.

Definition 3.7. A function 𝜎 : ℝ → ℍ is called with bounded variation on ℝ, ifthere exists a real number 𝑀 <∞ such that

∫ℝ∣𝑑𝜎(𝑥)∣ < 𝑀 .

Definition 3.8. A function 𝜎 : ℝ → ℍ is said to be a quaternion distribution, if itis of bounded variation and monotonic increasing, and if the following limits exist

lim𝑥−→𝑦+

𝜎(𝑥) = 𝜎(𝑦 + 0), and lim𝑥−→𝑦−𝜎(𝑥) = 𝜎(𝑦 − 0), (3.21)

(taken over all directions) for which

𝜎(𝑦) =1

2[𝜎(𝑦 + 0) + 𝜎(𝑦 − 0)] (3.22)

holds almost everywhere on ℝ.

To proceed with, it is significant to note that, for every two functions 𝜎1, 𝜎2 :ℝ→ ℍ of the quaternion form (2.5), the study of the properties of the distribution

ℱ𝒮(𝜔1, 𝜔2) :=

∫ℝ2

𝑑𝜎1(𝑥1)𝑑𝜎2(𝑥2)𝑒

𝒊𝜔1𝑥1𝑒𝒋𝜔2𝑥2 (3.23)

is reduced to the separate study of each component∫ℝ2

[𝑑𝜎1(𝑥1)

]𝑙

[𝑑𝜎2(𝑥2)

]𝑚

𝑒𝒊𝜔1𝑥1𝑒𝒋𝜔2𝑥2 (𝑙,𝑚 = 0, 1, 2, 3) . (3.24)

From now on, we denote the class of functions which can be represented as (3.23)by ℬ. Functions in ℬ are called (right) quaternion Bochner functions and ℬ willbe referred to as the (right) quaternion Bochner set. It follows that all membersof ℬ are entire functions of the real variables 𝜔1, 𝜔2.

Firstly, we shall note the following property of ℬ:Proposition 3.9. ℬ is a linear space.

Proof. Let 𝑓, 𝑔 ∈ ℬ and 𝑧1, 𝑧2, 𝑧3, 𝑧4 be quaternion numbers. It is easily seen that

𝑧1𝑓𝑧2 + 𝑧3𝑔𝑧4 ∈ ℬ. (3.25)

□The foregoing discussion suggests the computation of certain equalities for

quaternion Bochner functions. If 𝑓 ∈ ℬ is given, then

𝑓(0, 𝜔2) =(𝜎1(∞)− 𝜎1(−∞)

) ∫ℝ

𝑑𝜎2(𝑥2)𝑒𝒋𝜔2𝑥2 , (3.26)

𝑓(𝜔1, 0) =

∫ℝ

𝑑𝜎1(𝑥1)(𝜎2(∞)− 𝜎2(−∞)

)𝑒𝒊𝜔1𝑥1 , (3.27)

𝑓(0, 0) =(𝜎1(∞)− 𝜎1(−∞)

) (𝜎2(∞)− 𝜎2(−∞)

). (3.28)

5. Bochner’s Theorems 91

In particular, a simple argument gives

𝑓(−𝜔1,−𝜔2) =

∫ℝ2

𝑑𝜎1(𝑥1)𝑑𝜎2(𝑥2)𝑒

−𝒊𝜔1𝑥1𝑒−𝒋𝜔2𝑥2

=

∫ℝ2

𝑒𝒋𝜔2𝑥2𝑒𝒊𝜔1𝑥1𝑑𝜎1(𝑥1)𝑑𝜎2(𝑥2)

= 𝑔(𝜔1, 𝜔2), (3.29)

where 𝑔 is any function which can be represented as ℱ𝒮 𝑙(𝜎2, 𝜎1)(𝜔1, 𝜔2). We nowhave the following property:

Proposition 3.10. Every element of ℬ is a continuous bounded function.

Proof. Let 𝑓 be any function in ℬ. The continuity of 𝑓 is obvious. For the bound-edness a direct computation shows that

∣𝑓(𝜔1, 𝜔2)∣ ≤∫ℝ2

∣𝑑𝜎1(𝑥1)𝑑𝜎2(𝑥2)∣ ∣𝑒𝒊𝜔1𝑥1𝑒𝒋𝜔2𝑥2 ∣ (3.30)

≤ ∣𝜎1(∞)− 𝜎1(−∞)∣ ∣𝜎2(∞)− 𝜎2(−∞)∣ . □

Suppose now that 𝜎1, 𝜎2 : ℝ → ℍ are fixed, and 𝑓 ∈ ℬ does not vanishidentically. Question: Depending on the parity of 𝜎1(𝑥1) and 𝜎2(𝑥2), what can wesay about the representation of 𝑓? If there is any similarity or difference in theparity of these functions, does this lead to some particular functions 𝑓 ∈ ℬ? Theanswers are given by the following proposition:

Proposition 3.11. Let 𝜎1, 𝜎2 : ℝ→ ℍ, and 𝑓 ∈ ℬ be given.

1. If 𝜎1(𝑥1) is an odd function then

𝑓(𝜔1, 𝜔2) = 2

∫ ∞−∞

∫ ∞

0

𝑑𝜎1(𝑥1)𝑑𝜎2(𝑥2)𝒊 sin(𝜔1𝑥1)𝑒

𝒋𝜔2𝑥2 ,

𝑓(0, 𝜔2) = 0. (3.31)

2. If 𝜎1(𝑥1) is an even function then

𝑓(𝜔1, 𝜔2) = 2

∫ ∞−∞

∫ ∞

0

𝑑𝜎1(𝑥1)𝑑𝜎2(𝑥2) cos(𝜔1𝑥1)𝑒

𝒋𝜔2𝑥2 ,

𝑓(0, 0) = 2(𝜎1(∞)− 𝜎1(0)

) (𝜎2(∞)− 𝜎2(−∞)

). (3.32)

3. If 𝜎2(𝑥2) is an odd function then

𝑓(𝜔1, 𝜔2) = 2

∫ ∞0

∫ ∞

−∞𝑑𝜎1(𝑥1)𝑑𝜎

2(𝑥2)𝑒𝒊𝜔1𝑥1 sin(𝜔2𝑥2)𝒋,

𝑓(𝜔1, 0) = 0. (3.33)

92 S. Georgiev and J. Morais

4. If 𝜎2(𝑥2) is an even function then

𝑓(𝜔1, 𝜔2) = 2

∫ ∞

0

∫ ∞

−∞𝑑𝜎1(𝑥1)𝑑𝜎

2(𝑥2)𝑒𝒊𝜔1𝑥1 cos(𝜔2𝑥2),

𝑓(0, 0) = 2(𝜎1(∞) − 𝜎1(−∞)

) (𝜎2(∞)− 𝜎2(0)

). (3.34)

5. If 𝜎1(𝑥1) and 𝜎2(𝑥2) are both odd functions then

𝑓(𝜔1, 𝜔2) = 4

∫ ∞0

∫ ∞

0

𝑑𝜎1(𝑥1)𝑑𝜎2(𝑥2)𝒌 sin(𝜔1𝑥1) sin(𝜔2𝑥2),

𝑓(𝜔1, 0) = 𝑓(0, 𝜔2) = 0. (3.35)

6. If 𝜎1(𝑥1) is an odd function and 𝜎2(𝑥2) an even function then

𝑓(𝜔1, 𝜔2) = 4

∫ ∞

0

∫ ∞

0

𝑑𝜎1(𝑥1)𝑑𝜎2(𝑥2)𝒊 sin(𝜔1𝑥1) cos(𝜔2𝑥2),

𝑓(0, 𝜔2) = 0. (3.36)

7. If 𝜎1(𝑥1) is an even function and 𝜎2(𝑥2) an odd function then

𝑓(𝜔1, 𝜔2) = 4

∫ ∞0

∫ ∞

0

𝑑𝜎1(𝑥1)𝑑𝜎2(𝑥2)𝒋 cos(𝜔1𝑥1) sin(𝜔2𝑥2),

𝑓(𝜔1, 0) = 0. (3.37)

8. If 𝜎1(𝑥1) and 𝜎2(𝑥2) are both even functions then

𝑓(𝜔1, 𝜔2) = 4

∫ ∞

0

∫ ∞

0

𝑑𝜎1(𝑥1)𝑑𝜎2(𝑥2) cos(𝜔1𝑥1) cos(𝜔2𝑥2),

𝑓(0, 0) = 4(𝜎1(∞)− 𝜎1(0)

) (𝜎2(∞)− 𝜎2(0)

). (3.38)

Proof. For sake of brevity we prove only the first statement, the proof of theremaining statements is similar. Let 𝜎1(𝑥1) be an odd function. Using the trigono-metric representation of the function 𝑒𝒊𝜔1𝑥1 , a straightforward computation showsthat

𝑓(𝜔1, 𝜔2) =

∫ ∞

−∞

∫ ∞

−∞𝑑𝜎1(𝑥1)𝑑𝜎

2(𝑥2) 𝑒𝒊𝜔1𝑥1 𝑒𝒋𝜔2𝑥2

=

∫ ∞

−∞

(∫ ∞

−∞𝑑𝜎1(𝑥1) cos(𝜔1𝑥1)

)𝑑𝜎2(𝑥2) 𝑒

𝒋𝜔2𝑥2

+

∫ ∞

−∞

(∫ ∞

−∞𝑑𝜎1(𝑥1) sin(𝜔1𝑥1)

)𝑑𝜎2(𝑥2) 𝒊 𝑒

𝒋𝜔2𝑥2 (3.39)

= 2

∫ ∞

−∞

∫ ∞

0

𝑑𝜎1(𝑥1)𝑑𝜎2(𝑥2) 𝒊 sin(𝜔1𝑥1) 𝑒

𝒋𝜔2𝑥2 . □

3.3. Uniform Continuity

In this subsection we discuss uniform continuity and its relationship to the QFST.We begin by defining uniform continuity.

5. Bochner’s Theorems 93

Definition 3.12. A quaternion function 𝑓 : Ω ⊂ ℝ2 → ℍ is uniformly continuouson Ω, if and only if for all 𝜖 > 0 there exists a 𝛿 > 0, such that ∣𝑓(𝜔1)− 𝑓(𝜔2)∣ < 𝜖for all 𝜔1, 𝜔2 ∈ Ω whenever ∣𝜔1 − 𝜔2∣ < 𝛿.

We now prove some results related to the asymptotic behaviour of the QFST.

Proposition 3.13. Let 𝑓 be an element of ℬ. For any natural number 𝑛, let 𝑓𝑛 :ℝ× [−𝑛, 𝑛]→ ℍ be the function given by

𝑓𝑛(𝜔1, 𝜔2) =

∫ 𝑛

−𝑛

∫ ∞

−∞𝑑𝜎1(𝑥1)𝑑𝜎

2(𝑥2)𝑒𝒊𝜔1𝑥1𝑒𝒋𝜔2𝑥2 .

Then 𝑓𝑛(𝜔1, 𝜔2) −→𝑛−→∞ 𝑓(𝜔1, 𝜔2) uniformly. Also, if the 𝑓𝑛 are uniformly con-tinuous functions, then 𝑓 is also a uniformly continuous function.

Proof. A first straightforward computation shows that

∣𝑓(𝜔1, 𝜔2)− 𝑓𝑛(𝜔1, 𝜔2)∣

=

∣∣∣∣ ∞∫−∞∞∫−∞

𝑑𝜎1(𝑥1)𝑑𝜎2(𝑥2)𝑒

𝒊𝜔1𝑥1𝑒𝒋𝜔2𝑥2 −𝑛∫−𝑛

∞∫−∞

𝑑𝜎1(𝑥1)𝑑𝜎2(𝑥2)𝑒

𝒊𝜔1𝑥1𝑒𝒋𝜔2𝑥2

∣∣∣∣=

∣∣∣∣ −𝑛∫−∞∞∫−∞

𝑑𝜎1(𝑥1)𝑑𝜎2(𝑥2)𝑒

𝒊𝜔1𝑥1𝑒𝒋𝜔2𝑥2 +∞∫𝑛

∞∫−∞

𝑑𝜎1(𝑥1)𝑑𝜎2(𝑥2)𝑒

𝒊𝜔1𝑥1𝑒𝒋𝜔2𝑥2

∣∣∣∣≤ ∣𝜎1(∞)− 𝜎1(−∞)∣ ∣𝜎2(−𝑛)− 𝜎2(−∞)∣+ ∣𝜎1(∞)− 𝜎1(−∞)∣ ∣𝜎2(∞)− 𝜎2(𝑛)∣ .

(3.40)

Moreover, bearing in mind that

𝜎2(−𝑛)− 𝜎2(−∞) −→𝑛−→∞ 0, 𝜎2(∞) − 𝜎2(𝑛) −→𝑛−→∞ 0, (3.41)

and hence, 𝑓𝑛(𝜔1, 𝜔2) −→𝑛−→∞ 𝑓(𝜔1, 𝜔2) uniformly. In addition, we claim that ifall the 𝑓𝑛(𝜔1, 𝜔2) are uniformly continuous functions, it follows that 𝑓(𝜔1, 𝜔2) isalso a uniformly continuous function. □

Likewise, we have the following analogous result.

Proposition 3.14. Let 𝑓 be an element of ℬ. For any natural number 𝑛, let 𝑓𝑛 :[−𝑛, 𝑛]× ℝ→ ℍ be the function given by

𝑓𝑛(𝜔1, 𝜔2) =

∫ ∞

−∞

∫ 𝑛

−𝑛𝑑𝜎1(𝑥1)𝑑𝜎

2(𝑥2) 𝑒𝒊𝜔1𝑥1 𝑒𝒋𝜔2𝑥2 .

Then 𝑓𝑛(𝜔1, 𝜔2) −→𝑛−→∞ 𝑓(𝜔1, 𝜔2) uniformly. Also, if all the 𝑓𝑛 are uniformlycontinuous functions then 𝑓 is also a uniformly continuous function.

94 S. Georgiev and J. Morais

Proposition 3.15. Let 𝑓 be an element of ℬ. For any natural number 𝑛, let 𝑓𝑛 :[−𝑛, 𝑛]× [−𝑛, 𝑛]→ ℍ be the function given by

𝑓𝑛(𝜔1, 𝜔2) =

∫ 𝑛

−𝑛

∫ 𝑛

−𝑛𝑑𝜎1(𝑥1)𝑑𝜎

2(𝑥2) 𝑒𝒊𝜔1𝑥1 𝑒𝒋𝜔2𝑥2 .

Then 𝑓𝑛(𝜔1, 𝜔2) −→𝑛−→∞ 𝑓(𝜔1, 𝜔2) uniformly. Also, if all the 𝑓𝑛 are uniformlycontinuous functions then 𝑓 is also a uniformly continuous function.

Proof. We set 𝐴 := 𝑑𝜎1(𝑥1)𝑑𝜎2(𝑥2) 𝑒

𝒊𝜔1𝑥1 𝑒𝒋𝜔2𝑥2 . The proof follows from a simpleobservation:∫ ∞

−∞

∫ ∞

−∞𝐴−

∫ 𝑛

−𝑛

∫ 𝑛

−𝑛𝐴 =

∫ −𝑛

−∞

∫ −𝑛

−∞𝐴+

∫ −𝑛

−∞

∫ 𝑛

−𝑛𝐴+

∫ −𝑛

−∞

∫ ∞

𝑛

𝐴

+

∫ 𝑛

−𝑛

∫ −𝑛

−∞𝐴+

∫ 𝑛

−𝑛

∫ ∞

𝑛

𝐴+

∫ ∞

𝑛

∫ −𝑛

−∞𝐴 (3.42)

+

∫ ∞

𝑛

∫ 𝑛

−𝑛𝐴+

∫ ∞

𝑛

∫ ∞

𝑛

𝐴. □

We come now to the main theorem of this section.

Theorem 3.16. Let 𝑓 ∈ ℬ be given, and 𝑔 : ℝ2 −→ ℍ be a continuous and absolutelyintegrable function. For any 𝜎1, 𝜎2 : ℝ −→ ℍ the following relations hold:

1.

∫ℝ2

𝑓(𝑡1, 𝑡2)𝑔(𝜔1 − 𝑡1, 𝜂 − 𝜔2)𝑑𝑡1𝑑𝑡2

=

∫ℝ2

𝑑𝜎1(𝑥1)𝑑𝜎2(𝑥2)

∫ℝ2

𝑒𝒊𝑡1𝑥1𝑒𝒋𝑡2𝑥2𝑔(𝜔1 − 𝑡1, 𝜂 − 𝜔2)𝑑𝑡1𝑑𝑡2; (3.43)

2.

∫ℝ2

𝑓(𝑡1, 𝑡2)𝑔(𝑡1, 𝑡2)𝑑𝑡1𝑑𝑡2

= (2𝜋)2∫ℝ2

𝑑𝜎1(𝑥1)𝑑𝜎2(𝑥2)ℱ−1(𝑔)(𝑥1, 𝑥2). (3.44)

Proof. Assume 𝑔 : ℝ2 −→ ℍ to be a continuous and absolutely integrable function.For any real variables 𝜌 and 𝜒 we define the function

𝑅(𝑥1, 𝑥2, 𝜌, 𝜒) :=

∫ 𝜒

−𝜒

∫ 𝜌

−𝜌𝑒𝒊𝑡1𝑥1𝑒𝒋𝑡2𝑥2𝑔(𝑡1, 𝑡2)𝑑𝑡1𝑑𝑡2. (3.45)

We set 𝑓𝑛(𝑡1, 𝑡2) =∫ 𝑛−𝑛∫ 𝑛−𝑛 𝑑𝜎1(𝑥1)𝑑𝜎

2(𝑥2)𝑒𝒊𝑡1𝑥1𝑒𝒋𝑡2𝑥2 . Using the fact that 𝑔 is an

absolutely integrable function, it follows that∫ 𝜒

−𝜒

∫ 𝜌

−𝜌𝑓𝑛(𝑡1, 𝑡2)𝑔(𝑡1, 𝑡2)𝑑𝑡1𝑑𝑡2

=

∫ 𝜒

−𝜒

∫ 𝜌

−𝜌

(∫ 𝑛

−𝑛

∫ 𝑛

−𝑛𝑑𝜎1(𝑥1)𝑑𝜎

2(𝑥2)𝑒𝒊𝑡1𝑥1𝑒𝒋𝑡2𝑥2

)𝑔(𝑡1, 𝑡2)𝑑𝑡1𝑑𝑡2

5. Bochner’s Theorems 95

=

∫ 𝑛

−𝑛

∫ 𝑛

−𝑛𝑑𝜎1(𝑥1)𝑑𝜎

2(𝑥2)

∫ 𝜒

−𝜒

∫ 𝜌

𝜌

𝑒𝒊𝑡1𝑥1𝑒𝒋𝑡2𝑥2𝑔(𝑡1, 𝑡2)𝑑𝑡1𝑑𝑡2

=

∫ 𝑛

−𝑛

∫ 𝑛

−𝑛𝑑𝜎1(𝑥1)𝑑𝜎

2(𝑥2)𝑅(𝑥1, 𝑥2, 𝜌, 𝜂). (3.46)

From the last proposition we know that lim𝑛−→∞ 𝑓𝑛(𝜔1, 𝜔2) = 𝑓(𝜔1, 𝜔2) converges

uniformly. Moreover, since 𝑔(𝑡1, 𝑡2) is an absolutely integrable function, it followsthat 𝑅(𝑥1, 𝑥2, 𝜌, 𝜒) is also a uniformly continuous function. Hence

lim𝑛−→∞

𝜒∫−𝜒

𝜌∫−𝜌

𝑓𝑛(𝑡1, 𝑡2)𝑔(𝑡1, 𝑡2)𝑑𝑡1𝑑𝑡2 =

𝜒∫−𝜒

𝜌∫−𝜌

𝑓(𝑡1, 𝑡2)𝑔(𝑡1, 𝑡2)𝑑𝑡1𝑑𝑡2. (3.47)

With this argument in hand, and based on (3.46) we conclude that

lim𝑛−→∞

∫ 𝜒

−𝜒

∫ 𝜌

−𝜌𝑓𝑛(𝑡1, 𝑡2)𝑔(𝑡1, 𝑡2)𝑑𝑡1𝑑𝑡2 =

∫ℝ2

𝑑𝜎1(𝑥1)𝑑𝜎2(𝑥2)𝑅(𝑥1, 𝑥2, 𝜌, 𝜂).

(3.48)From the last equality and from (3.47) we obtain∫

ℝ2

𝑑𝜎1(𝑥1)𝑑𝜎2(𝑥2)𝑅(𝑥1, 𝑥2, 𝜌, 𝜂) =

∫ 𝜒

−𝜒

∫ 𝜌

𝜌

𝑓(𝑡1, 𝑡2)𝑔(𝑡1, 𝑡2)𝑑𝑡1𝑑𝑡2. (3.49)

In addition, we have

𝑅(𝑥1, 𝑥2, 𝜌, 𝜒) −→ 𝜌 −→∞𝜒 −→∞

∫ℝ2

𝑒𝒊𝑡1𝑥1 𝑒𝒋𝑡2𝑥2𝑔(𝑡1, 𝑡2)𝑑𝑡1𝑑𝑡2, (3.50)

and hence, for any fixed 𝑎, 𝑏 > 0 it follows that∫ 𝑎

−𝑎

∫ 𝑏

−𝑏𝑑𝜎1(𝑥1)𝑑𝜎

2(𝑥2)𝑅(𝑥1, 𝑥2, 𝜌, 𝜂) (3.51)

−→ 𝜌 −→∞𝜒 −→∞

∫ 𝑎

−𝑎

∫ 𝑏

−𝑏𝑑𝜎1(𝑥1)𝑑𝜎

2(𝑥2)

∫ℝ2

𝑒𝒊𝑡1𝑥1 𝑒𝒋𝑡2𝑥2𝑔(𝑡1, 𝑡2)𝑑𝑡1𝑑𝑡2.

For the sake of brevity, in the considerations to follow we will often omit the argu-ment and write simply 𝑅 instead of 𝑅(𝑥1, 𝑥2, 𝜌, 𝜂). Since 𝑅 is uniformly bounded,there exists a positive constant 𝑀 so that ∣𝑅∣ ≤ 𝑀 for all 𝑥1, 𝑥2, 𝜌, 𝜂 ∈ ℝ. Wedefine

𝐼 :=

∣∣∣∣∣∫ −𝑎

−∞

∫ −𝑏

−∞𝑑𝜎1(𝑥1)𝑑𝜎

2(𝑥2)𝑅 +

∫ 𝑎

−𝑎

∫ −𝑏

−∞𝑑𝜎1(𝑥1)𝑑𝜎

2(𝑥2)𝑅

+

∫ ∞

𝑎

∫ −𝑏

−∞𝑑𝜎1(𝑥1)𝑑𝜎

2(𝑥2)𝑅 +

∫ −𝑎

−∞

∫ 𝑏

−𝑏𝑑𝜎1(𝑥1)𝑑𝜎

2(𝑥2)𝑅

96 S. Georgiev and J. Morais

+

∫ ∞

𝑎

∫ 𝑏

−𝑏𝑑𝜎1(𝑥1)𝑑𝜎

2(𝑥2)𝑅 +

∫ −𝑎

−∞

∫ ∞

𝑏

𝑑𝜎1(𝑥1)𝑑𝜎2(𝑥2)𝑅

+

∫ 𝑎

−𝑎

∫ ∞

𝑏

𝑑𝜎1(𝑥1)𝑑𝜎2(𝑥2)𝑅+

∫ ∞

𝑎

∫ ∞

𝑏

𝑑𝜎1(𝑥1)𝑑𝜎2(𝑥2)𝑅

∣∣∣∣ . (3.52)

Therefore, we obtain

𝐼 ≤𝑀 [∣𝜎1(−𝑏)− 𝜎1(−∞)∣ ∣𝜎2(−𝑎)− 𝜎2(−∞)∣+ ∣𝜎1(−𝑏)− 𝜎1(−∞)∣ ∣𝜎2(𝑎)− 𝜎2(−𝑎)∣+ ∣𝜎1(−𝑏)− 𝜎1(−∞)∣ ∣𝜎2(∞)− 𝜎2(𝑎)∣+ ∣𝜎1(𝑏)− 𝜎1(−𝑏)∣ ∣𝜎2(−𝑎)− 𝜎2(−∞)∣+ ∣𝜎1(𝑏)− 𝜎1(−𝑏)∣ ∣𝜎2(∞)− 𝜎2(𝑎)∣+ ∣𝜎1(∞)− 𝜎1(𝑏)∣ ∣𝜎2(−𝑎)− 𝜎2(−∞)∣+ ∣𝜎1(∞)− 𝜎1(𝑏)∣ ∣𝜎2(𝑎)− 𝜎2(−𝑎)∣+ ∣𝜎1(∞)− 𝜎1(𝑏)∣ ∣𝜎2(∞)− 𝜎2(𝑎)∣] −→𝑎,𝑏−→∞ 0. (3.53)

Using the last inequality and (3.51) we get

lim𝑎,𝑏−→∞

∫ 𝑎

−𝑎

∫ 𝑏

−𝑏𝑑𝜎1(𝑥1)𝑑𝜎

2(𝑥2)𝑅(𝑥1, 𝑥2, 𝜌, 𝜂) (3.54)

=

∫ℝ2

𝑑𝜎1(𝑥1)𝑑𝜎2(𝑥2)

∫ℝ2

𝑒𝒊𝑡1𝑥1 𝑒𝒋𝑡2𝑥2𝑔(𝑡1, 𝑡2)𝑑𝑡1𝑑𝑡2.

Based on this and with (3.48) we find∫ℝ2

𝑓(𝑡1, 𝑡2)𝑔(𝑡1, 𝑡2)𝑑𝑡1𝑑𝑡2

=

∫ℝ2

𝑑𝜎1(𝑥1)𝑑𝜎2(𝑥2)

∫ℝ2

𝑒𝒊𝑡1𝑥1 𝑒𝒋𝑡2𝑥2𝑔(𝑡1, 𝑡2)𝑑𝑡1𝑑𝑡2

= (2𝜋)2∫ℝ2

𝑑𝜎1(𝑥1)𝑑𝜎2(𝑥2)ℱ−1(𝑔)(𝑥1, 𝑥2). (3.55)

Making the change of variables 𝑡1 −→ 𝜔1− 𝑡1, 𝑡2 −→ 𝜂−𝜔2 in the definition of 𝑔,we finally find∫

ℝ2

𝑓(𝑡1, 𝑡2)𝑔(𝜔1 − 𝑡1, 𝜂 − 𝜔2)𝑑𝑡1𝑑𝑡2 (3.56)

=

∫ℝ2

𝑑𝜎1(𝑥1)𝑑𝜎2(𝑥2)

∫ℝ2

𝑒𝒊𝑡1𝑥1 𝑒𝒋𝑡2𝑥2𝑔(𝜔1 − 𝑡1, 𝜂 − 𝜔2)𝑑𝑡1𝑑𝑡2. □

Theorem 3.17. For any 𝜎1, 𝜎2 : ℝ −→ ℍ, consider the functions

𝑔(𝜔1) =

∫ℝ

𝑑𝜎1(𝑥1) 𝑒𝒊𝜔1𝑥1 , ℎ(𝜔2) =

∫ℝ

𝑑𝜎2(𝑥2) 𝑒𝒋𝜔2𝑥2 . (3.57)

5. Bochner’s Theorems 97

Then for any real number 𝜌 the following equalities hold:

𝜎1(𝜌)− 𝜎1(0) =1

2𝜋

∫ℝ

𝑔(𝜔1)𝑒−𝒊𝜌𝜔1 − 1

−𝒊𝜔1𝑑𝜔1, (3.58)

𝜎2(𝜌)− 𝜎2(0) =1

2𝜋

∫ℝ

ℎ(𝜔2)𝑒−𝒋𝜌𝜔2 − 1

−𝒋𝜔2𝑑𝜔2. (3.59)

Proof. We begin with the following observation:

𝑔(𝜔1)(𝑒−𝒊𝜌𝜔1 − 1

)=

∫ℝ

𝑑𝜎1(𝑥1) 𝑒𝒊𝜔1𝑥1

(𝑒−𝒊𝜌𝜔1 − 1

)=

∫ℝ

𝑑𝜎1(𝑥1) 𝑒𝒊𝜔1(𝑥1−𝜌) −

∫ℝ

𝑑𝜎1(𝑥1) 𝑒𝒊𝜔1𝑥1

=

∫ℝ

𝑑𝜎1(𝑥1 + 𝜌) 𝑒𝒊𝜔1𝑥1 −∫ℝ

𝑑𝜎1(𝑥1) 𝑒𝒊𝜔1𝑥1

= lim𝑛−→∞

(∫ 𝑛

−𝑛𝑑𝜎1(𝑥1 + 𝜌) 𝑒𝒊𝜔1𝑥1 −

∫ 𝑛

−𝑛𝑑𝜎1(𝑥1) 𝑒

𝒊𝜔1𝑥1

)= lim

𝑛−→∞

[𝜎1(𝑛+ 𝜌)𝑒𝒊𝑛𝜔1 − 𝜎1(−𝑛+ 𝜌)𝑒−𝒊𝑛𝜔1 − 𝜎1(𝑛)𝑒𝒊𝑛𝜔1

+ 𝜎1(−𝑛)𝑒−𝒊𝑛𝜔1 −∫ 𝑛

−𝑛

(𝜎1(𝑥1 + 𝜌)− 𝜎1(𝑥1)

)𝑑𝑥1 𝑒

𝒊𝜔1𝑥1𝒊𝜔1

]= −

∫ℝ

(𝜎1(𝑥1 + 𝜌)− 𝜎1(𝑥1)

)𝑑𝑥1 𝑒

𝒊𝜔1𝑥1𝒊𝜔1.

(3.60)

Therefore, it is easy to see that

𝑔(𝜔1)(𝑒−𝒊𝜌𝜔1 − 1

) 1

−𝒊𝜔1=

∫ℝ

(𝜎1(𝑥1 + 𝜌)− 𝜎1(𝑥1)

)𝑑𝑥1 𝑒

𝒊𝜔1𝑥1 . (3.61)

From the last equality and from the inverse Fourier transform formula we find that

𝜎1(𝑥1 + 𝜌)− 𝜎1(𝑥1) =1

2𝜋

∫ℝ

𝑔(𝜔1)𝑒−𝒊𝜌𝜔1 − 1

−𝒊𝜔1𝑒−𝒊𝑥1𝜔1𝑑𝜔1, (3.62)

and, in particular for 𝑥1 = 0 we find that

𝜎1(𝜌)− 𝜎1(0) =1

2𝜋

∫ℝ

𝑔(𝜔1)𝑒−𝒊𝜌𝜔1 − 1

−𝒊𝜔1𝑑𝜔1. (3.63)

In a similar way we may deduce that

𝜎2(𝜌)− 𝜎2(0) =1

2𝜋

∫ℝ

ℎ(𝜔1)𝑒−𝒋𝜌𝜔1 − 1

−𝒋𝜔1𝑑𝜔1 , (3.64)

which completes the proof. □

98 S. Georgiev and J. Morais

3.4. Connecting Uniform Continuity to Quaternion Distributions

In this subsection we describe the connection between uniform continuity andquaternion distributions.

Theorem 3.18. Let the sequences of quaternion distributions

𝜎1(𝑥1), 𝜎2(𝑥1), . . . , 𝜎

𝑛(𝑥1), . . . , 𝜏1(𝑥2), 𝜏2(𝑥2), . . . , 𝜏

𝑛(𝑥2), . . . (3.65)

be convergent to the distributions 𝜎0(𝑥1) and 𝜏0(𝑥2), respectively. Assume that

lim𝑛−→∞𝜎𝑛(±∞) = 𝜎0(±∞), lim

𝑛−→∞ 𝜏𝑛(±∞) = 𝜏0(±∞), (3.66)

where 𝑓𝑛(𝜔1, 𝜔2) ∈ ℬ corresponds to the distributions 𝜎𝑛(𝑥1) and 𝜏𝑛(𝑥2), and𝑓0(𝜔1, 𝜔2) corresponds to the distributions 𝜎0(𝑥1) and 𝜏0(𝑥2), respectively. Then

lim𝑛−→∞ 𝑓𝑛(𝜔1, 𝜔2) = 𝑓0(𝜔1, 𝜔2).

Proof. To begin with, we choose 𝑎 > 0 so that 𝜎0(𝑥1) and 𝜏0(𝑥2) are continuousfunctions at ±𝑎. Then

lim𝑛−→∞𝜎𝑛(±𝑎) = 𝜎0(±𝑎), lim

𝑛−→∞ 𝜏𝑛(±𝑎) = 𝜏0(±𝑎). (3.67)

For brevity of exposition, in the sequel we omit the arguments and write simply𝑑𝜎𝑘 and 𝑑𝜏𝑘 instead of 𝑑𝜎𝑘(𝑥1) and 𝑑𝜏𝑘(𝑥2), respectively. Now, we make use ofthe following representations for 𝑓𝑛(𝜔1, 𝜔2) and 𝑓0(𝜔1, 𝜔2):

𝑓𝑛(𝜔1, 𝜔2) =

∫ℝ2

𝑑𝜎𝑛𝑑𝜏𝑛 𝑒𝒊𝜔1𝑥1 𝑒𝒋𝜔2𝑥2

=

∫ 𝑎

−𝑎

∫ 𝑎

−𝑎𝑑𝜎𝑛𝑑𝜏𝑛 𝑒𝒊𝜔1𝑥1 𝑒𝒋𝜔2𝑥2 +

∫ −𝑎

−∞

∫ −𝑎

−∞𝑑𝜎𝑛𝑑𝜏𝑛 𝑒𝒊𝜔1𝑥1 𝑒𝒋𝜔2𝑥2

+

∫ −𝑎

−∞

∫ 𝑎

−𝑎𝑑𝜎𝑛𝑑𝜏𝑛 𝑒𝒊𝜔1𝑥1 𝑒𝒋𝜔2𝑥2 +

∫ −𝑎

−∞

∫ ∞

𝑎

𝑑𝜎𝑛𝑑𝜏𝑛 𝑒𝒊𝜔1𝑥1 𝑒𝒋𝜔2𝑥2

+

∫ 𝑎

−𝑎

∫ −𝑎

−∞𝑑𝜎𝑛𝑑𝜏𝑛 𝑒𝒊𝜔1𝑥1 𝑒𝒋𝜔2𝑥2 +

∫ 𝑎

−𝑎

∫ ∞

𝑎

𝑑𝜎𝑛𝑑𝜏𝑛 𝑒𝒊𝜔1𝑥1 𝑒𝒋𝜔2𝑥2

+

∫ ∞

𝑎

∫ −𝑎

−∞𝑑𝜎𝑛𝑑𝜏𝑛 𝑒𝒊𝜔1𝑥1 𝑒𝒋𝜔2𝑥2 +

∫ ∞

𝑎

∫ 𝑎

−𝑎𝑑𝜎𝑛𝑑𝜏𝑛 𝑒𝒊𝜔1𝑥1 𝑒𝒋𝜔2𝑥2

+

∫ ∞

𝑎

∫ ∞

𝑎

𝑑𝜎𝑛𝑑𝜏𝑛 𝑒𝒊𝜔1𝑥1 𝑒𝒋𝜔2𝑥2 , (3.68)

and,

𝑓0(𝜔1, 𝜔2) =

∫ℝ2

𝑑𝜎0𝑑𝜏0 𝑒𝒊𝜔1𝑥1 𝑒𝒋𝜔2𝑥2

=

∫ 𝑎

−𝑎

∫ 𝑎

−𝑎𝑑𝜎0𝑑𝜏0 𝑒𝒊𝜔1𝑥1 𝑒𝒋𝜔2𝑥2 +

∫ −𝑎

−∞

∫ −𝑎

−∞𝑑𝜎0𝑑𝜏0 𝑒𝒊𝜔1𝑥1 𝑒𝒋𝜔2𝑥2

+

∫ −𝑎

−∞

∫ 𝑎

−𝑎𝑑𝜎0𝑑𝜏0 𝑒𝒊𝜔1𝑥1 𝑒𝒋𝜔2𝑥2 +

∫ −𝑎

−∞

∫ ∞

𝑎

𝑑𝜎0𝑑𝜏0 𝑒𝒊𝜔1𝑥1 𝑒𝒋𝜔2𝑥2

5. Bochner’s Theorems 99

+

∫ 𝑎

−𝑎

∫ −𝑎

−∞𝑑𝜎0𝑑𝜏0 𝑒𝒊𝜔1𝑥1 𝑒𝒋𝜔2𝑥2 +

∫ 𝑎

−𝑎

∫ ∞

𝑎

𝑑𝜎0𝑑𝜏0 𝑒𝒊𝜔1𝑥1 𝑒𝒋𝜔2𝑥2

+

∫ ∞

𝑎

∫ −𝑎

−∞𝑑𝜎0𝑑𝜏0 𝑒𝒊𝜔1𝑥1 𝑒𝒋𝜔2𝑥2 +

∫ ∞

𝑎

∫ 𝑎

−𝑎𝑑𝜎0𝑑𝜏0 𝑒𝒊𝜔1𝑥1 𝑒𝒋𝜔2𝑥2

+

∫ ∞

𝑎

∫ ∞

𝑎

𝑑𝜎0𝑑𝜏0 𝑒𝒊𝜔1𝑥1 𝑒𝒋𝜔2𝑥2 . (3.69)

Furthermore, we set

𝐼1(𝜔1, 𝜔2, 𝑎)

=

∫ 𝑎

−𝑎

∫ 𝑎

−𝑎𝑑𝜎0𝑑𝜏0 𝑒𝒊𝜔1𝑥1 𝑒𝒋𝜔2𝑥2 −

∫ 𝑎

−𝑎

∫ 𝑎

−𝑎𝑑𝜎𝑛𝑑𝜏𝑛 𝑒𝒊𝜔1𝑥1 𝑒𝒋𝜔2𝑥2 , (3.70)

𝐼2(𝜔1, 𝜔2, 𝑎)

=

∫ −𝑎

−∞

∫ −𝑎

−∞𝑑𝜎0𝑑𝜏0 𝑒𝒊𝜔1𝑥1 𝑒𝒋𝜔2𝑥2 −

∫ −𝑎

−∞

∫ −𝑎

−∞𝑑𝜎𝑛𝑑𝜏𝑛 𝑒𝒊𝜔1𝑥1 𝑒𝒋𝜔2𝑥2 , (3.71)

𝐼3(𝜔1, 𝜔2, 𝑎)

=

∫ −𝑎

−∞

∫ 𝑎

−𝑎𝑑𝜎0𝑑𝜏0 𝑒𝒊𝜔1𝑥1 𝑒𝒋𝜔2𝑥2 −

∫ −𝑎

−∞

∫ 𝑎

−𝑎𝑑𝜎𝑛𝑑𝜏𝑛 𝑒𝒊𝜔1𝑥1 𝑒𝒋𝜔2𝑥2 , (3.72)

𝐼4(𝜔1, 𝜔2, 𝑎)

=

∫ −𝑎

−∞

∫ ∞

𝑎

𝑑𝜎0𝑑𝜏0 𝑒𝒊𝜔1𝑥1 𝑒𝒋𝜔2𝑥2 −∫ −𝑎

−∞

∫ ∞

𝑎

𝑑𝜎𝑛𝑑𝜏𝑛 𝑒𝒊𝜔1𝑥1 𝑒𝒋𝜔2𝑥2 , (3.73)

𝐼5(𝜔1, 𝜔2, 𝑎)

=

∫ 𝑎

−𝑎

∫ −𝑎

−∞𝑑𝜎0𝑑𝜏0 𝑒𝒊𝜔1𝑥1 𝑒𝒋𝜔2𝑥2 −

∫ 𝑎

−𝑎

∫ −𝑎

−∞𝑑𝜎𝑛𝑑𝜏𝑛 𝑒𝒊𝜔1𝑥1 𝑒𝒋𝜔2𝑥2 , (3.74)

𝐼6(𝜔1, 𝜔2, 𝑎)

=

∫ 𝑎

−𝑎

∫ ∞

𝑎

𝑑𝜎0𝑑𝜏0 𝑒𝒊𝜔1𝑥1 𝑒𝒋𝜔2𝑥2 −∫ 𝑎

−𝑎

∫ ∞

𝑎

𝑑𝜎𝑛𝑑𝜏𝑛 𝑒𝒊𝜔1𝑥1 𝑒𝒋𝜔2𝑥2 , (3.75)

𝐼7(𝜔1, 𝜔2, 𝑎)

=

∫ ∞

𝑎

∫ −𝑎

−∞𝑑𝜎0𝑑𝜏0 𝑒𝒊𝜔1𝑥1 𝑒𝒋𝜔2𝑥2 −

∫ ∞

𝑎

∫ −𝑎

−∞𝑑𝜎𝑛𝑑𝜏𝑛 𝑒𝒊𝜔1𝑥1 𝑒𝒋𝜔2𝑥2 , (3.76)

𝐼8(𝜔1, 𝜔2, 𝑎)

=

∫ ∞

𝑎

∫ 𝑎

−𝑎𝑑𝜎0𝑑𝜏0 𝑒𝒊𝜔1𝑥1 𝑒𝒋𝜔2𝑥2 −

∫ ∞

𝑎

∫ 𝑎

−𝑎𝑑𝜎𝑛𝑑𝜏𝑛 𝑒𝒊𝜔1𝑥1 𝑒𝒋𝜔2𝑥2 , (3.77)

𝐼9(𝜔1, 𝜔2, 𝑎)

=

∫ ∞

𝑎

∫ ∞

𝑎

𝑑𝜎0𝑑𝜏0 𝑒𝒊𝜔1𝑥1 𝑒𝒋𝜔2𝑥2 −∫ ∞

𝑎

∫ ∞

𝑎

𝑑𝜎𝑛𝑑𝜏𝑛 𝑒𝒊𝜔1𝑥1 𝑒𝒋𝜔2𝑥2 . (3.78)

We further define 𝐴𝑙(𝜔1, 𝜔2, 𝑎) = lim𝑛−→∞ ∣𝐼 𝑙(𝜔1, 𝑎)∣ (𝑙 = 1, 2, 3, 4, 5, 6, 7, 8).

100 S. Georgiev and J. Morais

For 𝐼1(𝜔1, 𝑎) a straightforward computation shows that

∣𝐼1(𝜔1, 𝜔2, 𝑎)∣

=

∣∣∣∣∣(𝜎𝑛(𝑎)𝑒𝒊𝜔1𝑎 − 𝜎𝑛(−𝑎)𝑒−𝒊𝜔1𝑎 −

∫ 𝑎

−𝑎𝜎𝑛(𝑥1)𝒊𝜔1𝑒

𝒊𝜔1𝑥1𝑑𝑥1

)(𝜏𝑛(𝑎)𝑒𝒋𝜔2𝑎 − 𝜏𝑛(−𝑎)𝑒−𝒋𝜔2𝑎 −

∫ 𝑎

−𝑎𝜏𝑛(𝑥2)𝒊𝜔2𝑒

𝒋𝜔2𝑥2𝑑𝑥2

)−(𝜎0(𝑎)𝑒𝒊𝜔1𝑎 − 𝜎0(−𝑎)𝑒−𝒊𝜔1𝑎 −

∫ 𝑎

−𝑎𝜎0(𝑥1)𝒊𝜔1𝑒

𝒊𝜔1𝑥1𝑑𝑥1

)(𝜏0(𝑎)𝑒𝒋𝜔2𝑎 − 𝜏𝑛(−𝑎)𝑒−𝒋𝜔2𝑎 −

∫ 𝑎

−𝑎𝜏0(𝑥2)𝒋𝜔2𝑒

𝒋𝜔2𝑥2𝑑𝑥2

)∣∣∣∣∣. (3.79)

Based on these results and on (3.67), and the convergence of the sequences 𝜎𝑛(𝑥1)and 𝜏𝑛(𝑥2), it follows that 𝐴1(𝜔1, 𝜔2, 𝑎) = 0. Hence

∣𝐼2(𝜔1, 𝑎)∣ ≤∫ −𝑎

−∞

∫ −𝑎

−∞𝑑𝜎0(𝑥1)𝑑𝜏

0(𝑥2) ∣𝑒𝒊𝜔1𝑥1𝑒𝒋𝜔2𝑥2 ∣

+

∫ −𝑎

−∞

∫ −𝑎

−∞𝑑𝜎𝑛(𝑥1)𝑑𝜏

𝑛(𝑥2) ∣𝑒𝒊𝜔1𝑥1𝑒𝒋𝜔2𝑥2 ∣

≤ ∣𝜎0(−𝑎)− 𝜎0(−∞)∣ ∣𝜏0(−𝑎)− 𝜏0(−∞)∣+ ∣𝜎𝑛(−𝑎)− 𝜎𝑛(−∞)∣ ∣𝜏𝑛(−𝑎)− 𝜏𝑛(−∞)∣ . (3.80)

From the last inequality and from (3.66) it follows that for any 𝜖 > 0 we can choose𝑎 > 0 large enough so that 𝐴2(𝜔1, 𝜔2, 𝑎) < 𝜖. In a similar way, we have

𝐴𝑙(𝜔1, 𝜔2, 𝑎) < 𝜖, 𝑙 = 3, 4, 5, 6, 7, 8, 9. (3.81)

By combining these arguments, we finally obtain

lim𝑛−→∞ 𝑓𝑛(𝜔1, 𝜔2) = 𝑓0(𝜔1, 𝜔2), (3.82)

which completes the proof. □

4. The Main Theorem

In this section we shall extend Bochner’s result to the noncommutative structureof quaternion functions. We first define a few general properties of quaternionpositive-type functions.

4.1. Positive Functions

Let us define the notion of positive definite measure in the context of quaternionanalysis.

Definition 4.1. A function 𝑓 : ℝ2 −→ ℍ is called positive definite, if it satisfies thefollowing conditions:

5. Bochner’s Theorems 101

1. 𝑓 is bounded and continuous on ℝ2.

2. 𝑓(−𝝎) = 𝑔(𝝎) for all 𝝎 ∈ ℝ2, where 𝑔 is any function which can be repre-

sented as ℱ𝒮 𝑙(𝜎2, 𝜎1)(𝜔1, 𝜔2).3. For any 𝜆𝑚 = (𝜆1𝑚, 𝜆2𝑚) ∈ ℝ2, quaternion numbers 𝑧𝑚 (𝑚 = 1, 2, . . . , 𝑛),

and any positive integer 𝑛 the following inequality is satisfied:𝑛∑

𝑚,𝑙=1𝑚≥𝑙

𝑓(𝜆𝑚 − 𝜆𝑙) 𝑧𝑚𝑧𝑙 +𝑛∑

𝑚,𝑙=1𝑚>𝑙

𝑧𝑚𝑧𝑙𝑓(𝜆𝑙 − 𝜆𝑚) ≥ 0. (4.1)

These parameters are measured such that:

(i) When 𝜆𝑚 = 𝜆𝑙 for 𝑚 ∕= 𝑙 (𝑚, 𝑙 = 1, 2, . . . , 𝑛) it follows𝑛∑

𝑚,𝑙=1𝑚≤𝑙

𝑓(0, 0)𝑧𝑚𝑧𝑙 +

𝑛∑𝑚,𝑙=1𝑚>𝑙

𝑧𝑚𝑧𝑙 𝑓(0, 0) =

𝑛∑𝑚,𝑙=1𝑚≤𝑙

(𝑓(0, 0)𝑧𝑚𝑧𝑙 + 𝑧𝑙𝑧𝑚𝑓(0, 0)

)

=∑𝑚,𝑙=1𝑚≤𝑙

(𝑓(0, 0)𝑧𝑚𝑧𝑙 + 𝑓(0, 0)𝑧𝑚𝑧𝑙

)

= 2

𝑛∑𝑚,𝑙=1𝑚≤𝑙

S(𝑓(0, 0)𝑧𝑚𝑧𝑙) ≥ 0. (4.2)

(ii) When 𝜆𝑚 = 𝜆𝑙, 𝑧𝑚 = 𝑧𝑙 for 𝑚 ∕= 𝑙 (𝑚, 𝑙 = 1, 2, . . . , 𝑛), and by using theprevious observation we get

2 S(𝑓(0, 0))

𝑛∑𝑚=1

∣𝑧𝑚∣2 ≥ 0 ⇔ S(𝑓(0, 0)) ≥ 0. (4.3)

(iii) When 𝑛 = 2, 𝜆1 = 𝝎, and 𝜆2 = 0, it follows that

𝑓(𝝎)𝑧1𝑧2 + 𝑓(0, 0)(∣𝑧1∣2 + ∣𝑧2∣2

)+ 𝑧2𝑧1𝑓(𝝎)

= 2 S(𝑓(𝝎)𝑧1𝑧2) +(∣𝑧1∣2 + ∣𝑧2∣2

)𝑓(0, 0). (4.4)

In particular, when 𝑧1 = 𝑧2 we obtain

2 ∣𝑧1∣2 (S(𝑓(𝝎)) + 𝑓(0, 0)) ≥ 0 ⇔ S(𝑓(𝝎)) + 𝑓(0, 0) ≥ 0. (4.5)

Remark 4.2. It should be noted that not all functions in ℬ are positive definite.To verify this claim take, for example, the function 𝑓(𝜔1, 𝜔2) = −𝑒−𝒊𝜔1 . It is easyto see that expression (4.1) is not satisfied.

4.2. Bochner Theorem

Before proving the generalization of Bochner’s theorem to quaternion functions,we consider the following set

𝒞 :=

{𝑓 : ℝ2 −→ ℍ : 𝑓 =

∫ℝ2

𝑑∣∣𝜎1(𝑥1)

∣∣ 𝑑 ∣∣𝜎2(𝑥2)∣∣ 𝑒𝒊𝜔1𝑥1𝑒𝒋𝜔2𝑥2

}⊂ ℬ. (4.6)

102 S. Georgiev and J. Morais

We are now ready for the classification theorem.

Theorem 4.3. If 𝑓 ∈ 𝒞 then 𝑓 is positive definite.

Proof. Statement 1 of Definition 4.1 is proved in Proposition 3.10, and Statement2 follows from (3.29). Let 𝜆𝑚 = (𝜆1𝑚, 𝜆2𝑚) ∈ ℝ2, 𝑧𝑚 ∈ ℍ (𝑚 = 1, 2, . . . , 𝑛). Forany 𝜎1, 𝜎2 : ℝ→ ℍ, straightforward computations show that

𝑛∑𝑚,𝑙=1𝑚≤𝑙

𝑓(𝜆𝑚 − 𝜆𝑙)𝑧𝑚𝑧𝑙 +

𝑛∑𝑚,𝑙=1𝑚>𝑙

𝑧𝑚𝑧𝑙𝑓(𝜆𝑙 − 𝜆𝑚)

=𝑛∑

𝑚,𝑙=1𝑚≤𝑙

(∫ℝ2

𝑑∣∣𝜎1(𝑥1)

∣∣ 𝑑 ∣∣𝜎2(𝑥2)∣∣ 𝑒𝒊(𝜆1𝑚−𝜆1𝑙)𝑥1𝑒𝒋(𝜆2𝑚−𝜆2𝑙)𝑥2

)𝑧𝑚𝑧𝑙

+

𝑛∑𝑚,𝑙=1𝑚>𝑙

𝑧𝑚𝑧𝑙

∫ℝ2

𝑑 ∣𝜎1(𝑥1)∣ 𝑑 ∣𝜎2(𝑥2)∣ 𝑒𝒊(𝜆1𝑙−𝜆1𝑚)𝑥1𝑒𝒋(𝜆2𝑚−𝜆2𝑙)𝑥2

=

𝑛∑𝑚,𝑙=1𝑚≤𝑙

(∫ℝ2

𝑑∣∣𝜎1(𝑥1)

∣∣ 𝑑 ∣∣𝜎2(𝑥2)∣∣ 𝑒𝒊(𝜆1𝑚−𝜆1𝑙)𝑥1𝑒𝒋(𝜆2𝑚−𝜆2𝑙)𝑥2

)𝑧𝑚𝑧𝑙

+

𝑛∑𝑚,𝑙=1𝑚>𝑙

𝑧𝑚𝑧𝑙

∫ℝ2

𝑒−𝒋(𝜆2𝑚−𝜆2𝑙)𝑥2𝑒−𝒊(𝜆1𝑙−𝜆1𝑚)𝑥1𝑑∣∣𝜎1(𝑥1)

∣∣ 𝑑 ∣∣𝜎2(𝑥2)∣∣

=

𝑛∑𝑚=1

∣𝑧𝑚∣2(∣∣𝜎1(∞)

∣∣− ∣∣𝜎1(−∞)∣∣) (∣∣𝜎2(∞)

∣∣− ∣∣𝜎2(−∞)∣∣)

+

𝑛∑𝑚,𝑙=1𝑚<𝑙

∫ℝ2

𝑑∣∣𝜎1(𝑥1)

∣∣ 𝑑 ∣∣𝜎2(𝑥2)∣∣⎛⎝ 𝑒𝒊(𝜆1𝑚−𝜆1𝑙)𝑥1𝑒𝒋(𝜆2𝑚−𝜆2𝑙)𝛽𝑧𝑚𝑧𝑙

+

𝑧𝑙𝑧𝑚𝑒−𝒋(𝜆2𝑚−𝜆2𝑙)𝑥2𝑒−𝒊(𝜆1𝑚−𝜆1𝑙)𝑥1

⎞⎠=

𝑛∑𝑚=1

∣𝑧𝑚∣2(∣∣𝜎1(∞)

∣∣− ∣∣𝜎1(−∞)∣∣) (∣∣𝜎2(∞)

∣∣− ∣∣𝜎2(−∞)∣∣)

+ 2

𝑛∑𝑚,𝑙=1𝑚<𝑙

∫ℝ2

𝑑∣∣𝜎1(𝑥1)

∣∣ 𝑑 ∣∣𝜎2(𝑥2)∣∣S(𝑒𝒊(𝜆1𝑚−𝜆1𝑙)𝑥1𝑒𝒋(𝜆2𝑚−𝜆2𝑙)𝑥2𝑧𝑚𝑧𝑙

)

≥𝑛∑

𝑚=1

∣𝑧𝑚∣2(∣∣𝜎1(∞)

∣∣− ∣∣𝜎1(−∞)∣∣) (∣∣𝜎2(∞)

∣∣ − ∣∣𝜎2(−∞)∣∣)

− 2

𝑛∑𝑚,𝑙=1𝑚<𝑙

∫ℝ2

∣𝑧𝑚∣∣𝑧𝑙∣ 𝑑∣∣𝜎1(𝑥1)

∣∣ 𝑑 ∣∣𝜎2(𝑥2)∣∣

5. Bochner’s Theorems 103

=

⎛⎜⎝ 𝑛∑𝑚=1

∣𝑧𝑚∣2 − 2

𝑛∑𝑚,𝑙=1𝑚<𝑙

∣𝑧𝑚∣∣𝑧𝑙∣

⎞⎟⎠(∣∣𝜎1(∞)∣∣− ∣∣𝜎1(−∞)

∣∣) (∣∣𝜎2(∞)∣∣− ∣∣𝜎2(−∞)

∣∣)

=

(𝑛∑

𝑚=1

∣𝑧𝑚∣)2 (∣∣𝜎1(∞)

∣∣− ∣∣𝜎1(−∞)∣∣) (∣∣𝜎2(∞)

∣∣− ∣∣𝜎2(−∞)∣∣) ≥ 0. (4.7)

This proves that 𝑓 is positive definite. □

The extension of Bochner’s result to a much larger class of quaternion func-tions remains a challenge to future research.

Acknowledgement

Partial support from the Foundation for Science and Technology (FCT) via thegrant DD-VU-02/90, Bulgaria, is acknowledged by the first named author. Thesecond named author acknowledges financial support from the Foundation for Sci-ence and Technology (FCT) via the post-doctoral grant SFRH/ BPD/66342/2009.This work was supported by FEDER funds through the COMPETE – Opera-tional Programme Factors of Competitiveness (‘Programa Operacional Factoresde Competitividade’) and by Portuguese funds through the Center for Researchand Development in Mathematics and Applications (University of Aveiro) andthe Portuguese Foundation for Science and Technology (‘FCT – Fundacao paraa Ciencia e a Tecnologia’), within project PEst-C/MAT/UI4106/2011 with theCOMPETE number FCOMP-01-0124-FEDER-022690.

References

[1] S. Bochner. Monotone funktionen, Stieltjessche integrate, und harmonische analyse.Mathematische Annalen, 108:378–410, 1933.

[2] S. Bochner. Lectures on Fourier Integrals. Princeton University Press, Princeton, NewJersey, 1959.

[3] T. Bulow. Hypercomplex Spectral Signal Representations for the Processing and Anal-ysis of Images. PhD thesis, University of Kiel, Germany, Institut fur Informatik undPraktische Mathematik, Aug. 1999.

[4] T. Bulow, M. Felsberg, and G. Sommer. Non-commutative hypercomplex Fouriertransforms of multidimensional signals. In G. Sommer, editor, Geometric computingwith Clifford Algebras: Theoretical Foundations and Applications in Computer Visionand Robotics, pages 187–207, Berlin, 2001. Springer.

[5] S. Georgiev, J. Morais, K.I. Kou, and W. Sproßig. Bochner–Minlos theorem andquaternion Fourier transform. To appear.

[6] B. Mawardi, E. Hitzer, A. Hayashi, and R. Ashino. An uncertainty principle for quater-nion Fourier transform. Computers and Mathematics with Applications, 56(9):2411–2417, 2008.

104 S. Georgiev and J. Morais

[7] B. Mawardi and E.M.S. Hitzer. Clifford Fourier transformation and uncertainty prin-ciple for the Clifford algebra 𝐶ℓ3,0. Advances in Applied Clifford Algebras, 16(1):41–61,2006.

S. GeorgievDepartment of Differential EquationsUniversity of SofiaSofia, Bulgariae-mail: sgg2000bg@yahoo.com

J. MoraisCentro de Investigacao e Desenvolvimento

em Matematica e Aplicacoes (CIDMA)Universidade de AveiroP3810-193 Aveiro, Portugale-mail: joao.pedro.morais@ua.pt

Quaternion and Clifford–Fourier Transforms and Wavelets

Trends in Mathematics, 105–120c⃝ 2013 Springer Basel

6 Bochner–Minlos Theorem andQuaternion Fourier Transform

S. Georgiev, J. Morais, K.I. Kou and W. Sproßig

Abstract. There have been several attempts in the literature to generalizethe classical Fourier transform by making use of the Hamiltonian quaternionalgebra. The first part of this chapter features certain properties of the as-ymptotic behaviour of the quaternion Fourier transform. In the second partwe introduce the quaternion Fourier transform of a probability measure, andwe establish some of its basic properties. In the final analysis, we introducethe notion of positive definite measure, and we set out to extend the classicalBochner–Minlos theorem to the framework of quaternion analysis.

Mathematics Subject Classification (2010). Primary 30G35; secondary 42A38;tertiary 42A82.

Keywords. Quaternion analysis, quaternion Fourier transform, asymptotic be-haviour, positive definitely measure, Bochner–Minlos theorem.

1. Introduction and Statement of Results

As is well known, the classical Fourier transform (FT) has wide applications inengineering, computer sciences, physics and applied mathematics. For instance,the FT can be used to provide signal analysis techniques where the signal fromthe original time domain is transformed to the frequency domain. Therefore thereexists great interest and considerable effort to extend the FT to higher dimensions,and study its properties and interdependencies (see, e.g., [1–7, 9–11, 17, 18, 20–23] and elsewhere). In view of numerous applications in physics and engineeringproblems, one is particularly interested in higher-dimensional analogues to ℝ𝑛, inparticular, for 𝑛 = 4. To this end, so far quaternion analysis offers the possibilityof generalizing the underlying function theory in 2D to 4D, with the advantageof meeting exactly these goals. To aid the reader, see [15, 16, 19, 24, 25] for morecomplete accounts of this subject and related topics.

The first part of the present work is devoted to the study of the asymp-totic behaviour of the quaternion Fourier transform (QFT). The QFT was first

106 S. Georgiev, J. Morais, K.I. Kou and W. Sproßig

introduced by Ell in [10]. He proposed a two-sided QFT and studied some of itsapplications and properties. Later, Bulow [7] (cf. [8] and [1,20]) has also conducteda generalization of the real and complex FT using the quaternion algebra basedon two complex variables, but, to the best of our knowledge, a detailed study ofits asymptotic behaviour has not been carried out yet. The main motivation of thepresent study is to develop further general numerical methods for partial differen-tial equations and to extend localization theorems for summation of Fourier seriesin the quaternion analysis setting. In a forthcoming article we shall describe theseconnections in more detail and illustrate them by some typical examples.

Due to the noncommutativity of the quaternions, there are three differenttypes of QFT: a right-sided QFT, a left-sided QFT, and a two-sided QFT [23].We will carry out the investigation of the following finite integral (defined fromthe time domain to the frequency domain)

ℱ𝑟(𝑓)(𝜔1, 𝜔2) :=

∫ 𝑏

𝑎

∫ 𝑏

𝑎

𝑓(𝑥1, 𝑥2) 𝑒−𝒊𝜔1𝑥1 𝑒−𝒋𝜔2𝑥2 𝑑𝑥1𝑑𝑥2, (1.1)

where the signal 𝑓 : [𝑎, 𝑏]× [𝑎, 𝑏] ⊂ ℝ2 −→ ℍ will be taken to be

𝑓(𝑥1, 𝑥2) := [𝑓(𝑥1, 𝑥2)]0 + [𝑓(𝑥1, 𝑥2)]1𝒊+ [𝑓(𝑥1, 𝑥2)]2𝒋 + [𝑓(𝑥1, 𝑥2)]3𝒌,

[𝑓 ]𝑙 : [𝑎, 𝑏]× [𝑎, 𝑏] −→ ℝ (𝑙 = 0, 1, 2, 3)

satisfying certain conditions, guaranteeing the convergence of the above integral.ℱ𝑟(𝑓)(𝜔1, 𝜔2) is the (finite) right-sided Fourier transform [21] of the quaternionfunction 𝑓(𝑥1, 𝑥2), and it may be interpreted as a quaternionic extension of theclassical FT; the exponential product 𝑒−𝒊𝜔1𝑥1𝑒−𝒋𝜔2𝑥2 is called the (right-sided)quaternion Fourier kernel, and for 𝑖 = 1, 2; 𝑥𝑖 will denote the space and 𝜔𝑖 theangular frequency variables. The previous definition of the QFT varies from theoriginal one only in the fact that we use 2D vectors instead of scalars and that itis defined to be two-dimensional. Here 𝒊, 𝒋 and 𝒌 are unit pure quaternions (i.e.,the quaternions with unit magnitude having no scalar part) that are orthogonalto each other. We point out that the product in (1.1) has to be written in a fixedorder since, in general, 𝑒−𝒊𝜔1𝑥1𝑒−𝒋𝜔2𝑥2 does not commute with every element ofthe algebra.

Remark 1.1. Throughout this text we investigate the integral (1.1) only that, forsimplicity, we denote by ℱ(𝑓). Nevertheless, all results can be easily performedfrom the left-hand side:

ℱ𝑙(𝑓)(𝜔1, 𝜔2) :=

∫ 𝑏

𝑎

∫ 𝑏

𝑎

𝑒−𝒋𝜔2𝑥2 𝑒−𝒊𝜔1𝑥1𝑓(𝑥1, 𝑥2) 𝑑𝑥1𝑑𝑥2,

since

ℱ𝑟(𝑓)(−𝜔1,−𝜔2) =

∫ 𝑏

𝑎

∫ 𝑏

𝑎

𝑓(𝑥1, 𝑥2)𝑒𝒊𝜔1𝑥1𝑒𝒋𝜔2𝑥2𝑑𝑥1𝑑𝑥2

6. Bochner–Minlos Theorem and Quaternion Fourier Transform 107

=

∫ 𝑏

𝑎

∫ 𝑏

𝑎

𝑓(𝑥1, 𝑥2)𝑒−𝒋𝜔2𝑥2𝑒−𝒊𝜔1𝑥1𝑑𝑥1𝑑𝑥2

=

∫ 𝑏

𝑎

∫ 𝑏

𝑎

𝑒−𝒋𝜔2𝑥2𝑒−𝒊𝜔1𝑥1𝑓(𝑥1, 𝑥2)𝑑𝑥1𝑑𝑥2 = ℱ𝑙(𝑓)(𝜔1, 𝜔2).

Lemma 1.2. The QFT of a 2D signal 𝑓 ∈ 𝐿1([𝑎, 𝑏]× [𝑎, 𝑏];ℍ) has the closed-formrepresentation:

ℱ(𝑓)(𝜔1, 𝜔2) := Φ0(𝜔1, 𝜔2) + Φ1(𝜔1, 𝜔2) + Φ2(𝜔1, 𝜔2) + Φ3(𝜔1, 𝜔2),

where the integrals are

Φ0(𝜔1, 𝜔2) =

∫ 𝑏

𝑎

∫ 𝑏

𝑎

𝑓(𝑥1, 𝑥2) cos(𝜔1𝑥1) cos(𝜔2𝑥2)𝑑𝑥1𝑑𝑥2,

Φ1(𝜔1, 𝜔2) = −∫ 𝑏

𝑎

∫ 𝑏

𝑎

𝑓(𝑥1, 𝑥2)𝒊 sin(𝜔1𝑥1) cos(𝜔2𝑥2)𝑑𝑥1𝑑𝑥2,

Φ2(𝜔1, 𝜔2) = −∫ 𝑏

𝑎

∫ 𝑏

𝑎

𝑓(𝑥1, 𝑥2)𝒋 cos(𝜔1𝑥1) sin(𝜔2𝑥2)𝑑𝑥1𝑑𝑥2,

Φ3(𝜔1, 𝜔2) =

∫ 𝑏

𝑎

∫ 𝑏

𝑎

𝑓(𝑥1, 𝑥2)𝒌 sin(𝜔1𝑥1) sin(𝜔2𝑥2)𝑑𝑥1𝑑𝑥2.

For illustrative purposes, we have the following identities:

Corollary 1.3. The QFT of a 2D signal 𝑓 ∈ 𝐿1([𝑎, 𝑏] × [𝑎, 𝑏];ℍ) satisfies the fol-lowing relations:

ℱ(𝑓)(𝜔1, 𝜔2) + ℱ(𝑓)(𝜔1,−𝜔2) = 2 (Φ0(𝜔1, 𝜔2) + Φ1(𝜔1, 𝜔2)) ,

ℱ(𝑓)(𝜔1, 𝜔2)−ℱ(𝑓)(𝜔1,−𝜔2) = 2 (Φ2(𝜔1, 𝜔2) + Φ3(𝜔1, 𝜔2)) ,

ℱ(𝑓)(𝜔1, 𝜔2) + ℱ(𝑓)(−𝜔1, 𝜔2) = 2 (Φ0(𝜔1, 𝜔2) + Φ2(𝜔1, 𝜔2)) ,

ℱ(𝑓)(𝜔1, 𝜔2)−ℱ(𝑓)(−𝜔1, 𝜔2) = 2 (Φ1(𝜔1, 𝜔2) + Φ3(𝜔1, 𝜔2)) ,

ℱ(𝑓)(𝜔1, 𝜔2) + ℱ(𝑓)(−𝜔1,−𝜔2) = 2 (Φ0(𝜔1, 𝜔2) + Φ3(𝜔1, 𝜔2)) ,

ℱ(𝑓)(𝜔1, 𝜔2)−ℱ(𝑓)(−𝜔1,−𝜔2) = 2 (Φ1(𝜔1, 𝜔2) + Φ2(𝜔1, 𝜔2)) ,

ℱ(𝑓)(𝜔1,−𝜔2) + ℱ(𝑓)(−𝜔1,−𝜔2) = 2 (Φ0(𝜔1, 𝜔2)− Φ2(𝜔1, 𝜔2)) ,

ℱ(𝑓)(𝜔1,−𝜔2)−ℱ(𝑓)(−𝜔1,−𝜔2) = 2 (Φ1(𝜔1, 𝜔2)− Φ3(𝜔1, 𝜔2)) ,

ℱ(𝑓)(−𝜔1, 𝜔2) + ℱ(𝑓)(−𝜔1,−𝜔2) = 2 (Φ0(𝜔1, 𝜔2)− Φ1(𝜔1, 𝜔2)) ,

ℱ(𝑓)(−𝜔1, 𝜔2)−ℱ(𝑓)(−𝜔1,−𝜔2) = 2 (Φ2(𝜔1, 𝜔2)− Φ3(𝜔1, 𝜔2)) .

Under suitable conditions, the original signal 𝑓 can be reconstructed fromℱ(𝑓) by the inverse transform (frequency to time domains).

Definition 1.4. The (right-sided) inverse QFT of 𝑔 ∈ 𝐿1(ℝ2;ℍ) is given by

ℱ−1𝑟 (𝑔) : ℝ2 −→ ℍ,

ℱ−1𝑟 (𝑔)(𝑥1, 𝑥2) :=

1

(2𝜋)2

∫ℝ2

𝑔(𝜔1, 𝜔2) 𝑒𝒋𝜔2𝑥2 𝑒𝒊𝜔1𝑥1 𝑑𝜔1𝑑𝜔2.

108 S. Georgiev, J. Morais, K.I. Kou and W. Sproßig

It has the closed-form representation:

ℱ−1𝑟 (𝑔)(𝑥1, 𝑥2) := Φ0(𝑥1, 𝑥2) + Φ1(𝑥1, 𝑥2) + Φ2(𝑥1, 𝑥2) + Φ3(𝑥1, 𝑥2),

where the integrals are

Φ0(𝑥1, 𝑥2) =1

(2𝜋)2

∫ 𝑏

𝑎

∫ 𝑏

𝑎

𝑔(𝜔1, 𝜔2) cos(𝜔1𝑥1) cos(𝜔2𝑥2)𝑑𝜔1𝑑𝜔2,

Φ1(𝑥1, 𝑥2) =1

(2𝜋)2

∫ 𝑏

𝑎

∫ 𝑏

𝑎

𝑔(𝜔1, 𝜔2)𝒊 sin(𝜔1𝑥1) cos(𝜔2𝑥2)𝑑𝜔1𝑑𝜔2,

Φ2(𝑥1, 𝑥2) =1

(2𝜋)2

∫ 𝑏

𝑎

∫ 𝑏

𝑎

𝑔(𝜔1, 𝜔2)𝒋 cos(𝜔1𝑥1) sin(𝜔2𝑥2)𝑑𝜔1𝑑𝜔2,

Φ3(𝑥1, 𝑥2) = − 1

(2𝜋)2

∫ 𝑏

𝑎

∫ 𝑏

𝑎

𝑔(𝜔1, 𝜔2)𝒌 sin(𝜔1𝑥1) sin(𝜔2𝑥2)𝑑𝜔1𝑑𝜔2.

The quaternion exponential product 𝑒𝒋𝑥2𝜔2 𝑒𝒊𝜔1𝑥1 is called the inverse (right-sided)quaternion Fourier kernel.

Remark 1.5. Again, all computations can easily be converted to other conventions,since

ℱ−1𝑟 (𝑔)(−𝑥1,−𝑥2) =

1

(2𝜋)2

∫ 𝑏

𝑎

∫ 𝑏

𝑎

𝑔(𝜔1, 𝜔2)𝑒𝒊𝜔1𝑥1 𝑒𝒋𝜔2𝑥2𝑑𝜔1𝑑𝜔2

=1

(2𝜋)2

∫ 𝑏

𝑎

∫ 𝑏

𝑎

𝑒𝒊𝜔1𝑥1 𝑒𝒋𝜔2𝑥2𝑔(𝜔1, 𝜔2)𝑑𝜔1𝑑𝜔2

:= ℱ−1𝑙 (𝑔)(𝑥1, 𝑥2).

For convenience, below we will denote ℱ−1𝑟 as ℱ−1.

The present chapter has two main goals. The first consists in studying theasymptotic behaviour of the integral (1.1) under the assumption that 𝑓 belongsto 𝐿1((𝑎, 𝑏) × (𝑐, 𝑑);ℍ) where 𝑎, 𝑏, 𝑐, 𝑑 can be both finite and infinite points. Weshall be interested in the connection between the function 𝑓(𝑥1, 𝑥2), and the be-haviour of its Fourier transform ℱ(𝑓)(𝜔1, 𝜔2) at infinity. These properties have aninterest on their own for further applications to number theory, combinatorics, sig-nal processing, imaging, computer vision and numerical analysis. The complexityof the underlying computations will need some attention. Central to this view-point are certain Fourier transform techniques, which as in the complex case,would be familiar to the reader. Our second goal consists in extending the clas-sical Bochner–Minlos theorem to a noncommutative structure as in the case ofquaternion functions. The resulting theorem guarantees the existence and unique-ness of the corresponding probability measure defined on a dual space. This willbe done using the concept of quaternion Fourier transform of a probability mea-sure. For the reader’s convenience and for the sake of easy reference, the chapteris motivated by the results presented in [12, 13].

6. Bochner–Minlos Theorem and Quaternion Fourier Transform 109

Although the presented results can be extended to generalized Clifford alge-bras as well, we will focus the discussion on the QFT for conciseness here. Theproof of its generalization to higher dimensions is possible but needs more com-plicated calculations, which exceed the scope of this manuscript. These results arestill under further investigation and will be reported in a forthcoming paper.

2. Preliminaries

At this stage we briefly recall basic algebraic facts about quaternions necessary forthe sequel. Let ℍ := {𝑝 = 𝑎+ 𝑏𝒊+ 𝑐𝒋 + 𝑑𝒌 : 𝑎, 𝑏, 𝑐, 𝑑 ∈ ℝ} be a four-dimensionalassociative and noncommutative algebra, where the imaginary units 𝒊, 𝒋, and 𝒌are subject to the Hamiltonian multiplication rules

𝒊2 = 𝒋2 = 𝒌2 = −1;𝒊𝒋 = 𝒌 = −𝒋𝒊, 𝒋𝒌 = 𝒊 = −𝒌𝒋, 𝒌𝒊 = 𝒋 = −𝒊𝒌.

The scalar and vector parts of 𝑝, S(𝑝) andV(𝑝), are defined as the 𝑎 and 𝑏𝒊+𝑐𝒋+𝑑𝒌terms, respectively. For the scalar part the cyclic product rule S(𝑝𝑞𝑟) = S(𝑞𝑟𝑝) isvalid. Further step is quaternion conjugation introduced similarly to that of thecomplex numbers 𝑝 = 𝑎−𝑏𝒊−𝑐𝒋−𝑑𝒌. The quaternion conjugation is an anti-linearinvolution

𝑝 = 𝑝, 𝑝+ 𝑞 = 𝑝+ 𝑞, 𝑞𝑝 = 𝑝𝑞, 𝜆𝑝 = 𝜆𝑝 (∀𝜆 ∈ ℝ).

The norm of 𝑝 is defined by ∣𝑝∣ = √𝑝𝑝 =

√𝑝𝑝 =

√𝑎2 + 𝑏2 + 𝑐2 + 𝑑2, and it

coincides with its corresponding Euclidean norm as a vector in ℝ4. For everytwo quaternions 𝑝 and 𝑞 the triangle inequalities hold ∣𝑝+ 𝑞∣ ≤ ∣𝑝∣ + ∣𝑞∣, and∣ ∣𝑝∣ − ∣𝑞∣ ∣ ≤ ∣𝑝± 𝑞∣. Also, we have ∣S(𝑝)∣ ≤ ∣𝑝∣ and ∣V(𝑝)∣ ≤ ∣𝑝∣.

In the sequel, a quaternion sequence is a collection of real quaternions 𝑝0, 𝑝1,𝑝2, . . . ‘labelled’ by nonnegative integers. We shall denote such a sequence by {𝑝𝑛}where 𝑛 = 0, 1, 2, . . . and 𝑝𝑛 = 𝑝0,𝑛 + 𝑝1,𝑛𝒊+ 𝑝2,𝑛𝒋 + 𝑝3,𝑛𝒌 are the elements of thesequence, 𝑝𝑙,𝑛 ∈ ℝ (𝑙 = 0, 1, 2, 3). To supplement our investigations, we recall thekey notion of convergence of a quaternion sequence.

Definition 2.1. The quaternion sequence {𝑝𝑛} is called convergent to the quater-nion 𝑝 = 𝑎+𝑏𝒊+𝑐𝒋+𝑑𝒌 if lim

𝑛−→∞ ∣𝑝𝑛−𝑝∣ = 0. We will use the traditional notation:

lim𝑛−→∞ 𝑝𝑛 = 𝑝.

Lemma 2.2. Let {𝑝𝑛} and {𝑞𝑛} be two quaternion sequences for which lim𝑛−→∞ 𝑝𝑛 = 𝑝

and lim𝑛−→∞ 𝑞𝑛 = 𝑞, for 𝑝, 𝑞 ∈ ℍ. Then

1. lim𝑛−→∞(𝑝𝑛 ± 𝑞𝑛) = 𝑝± 𝑞;

2. lim𝑛−→∞(𝑝𝑛𝑞𝑛) = 𝑝𝑞;

3. lim𝑛−→∞(𝛼𝑝𝑛) = 𝛼𝑝, 𝛼 ∈ ℍ.

110 S. Georgiev, J. Morais, K.I. Kou and W. Sproßig

Now, let 𝑠 be the space of sequences

𝑠 := {{𝑝𝑛} : lim𝑛−→∞𝑛𝑡𝑝𝑛 = 0, ∀ 𝑡 ∈ ℕ0},

where 𝑝𝑛 ∈ ℍ, and let

𝑠𝑚 := {{𝑝𝑛} : ∥𝑝∥2𝑚 :=∞∑𝑛=0

(1 + 𝑛2)𝑚 ∣𝑝𝑛∣2 < +∞}, 𝑚 ∈ ℤ.

Proposition 2.3. We have 𝑠 = ∩𝑚∈ℤ𝑠𝑚.

Proof. Let 𝑝 ∈ 𝑠 be arbitrarily chosen and fixed, and 𝑡 ∈ ℕ0. From the definitionof the space 𝑠 we have that lim

𝑛−→∞𝑛𝑡𝑝𝑛 = 0. Then for 𝐶 > 0 a natural number

𝑁 = 𝑁(𝐶) can be found so that for every 𝑛 > 𝑁 the following holds

𝑛𝑡∣𝑝𝑛∣ ≤ 𝐶.

Hence, it follows∞∑𝑛=0

(1 + 𝑛2

)[ 𝑡2 ] ∣𝑝𝑛∣2 ≤ ∣𝑝0∣2 + 𝐶2∞∑𝑛=1

(1 + 𝑛2

)[ 𝑡4 ] 1

𝑛2𝑡< +∞,

and therefore 𝑝 ∈ 𝑠[ 𝑡4 ]. Since 𝑡 ∈ ℕ0 is arbitrary we conclude that 𝑝 ∈ 𝑠𝑚 for every

𝑚 ∈ ℕ0, from where 𝑝 ∈ ∩𝑚∈ℕ𝑠𝑚. Since 𝑝 ∈ 𝑠 was arbitrarily chosen it followsthat 𝑠 ⊂ ∩𝑚∈ℕ𝑠𝑚. Now, let 𝑝 ∈ ∩𝑚∈ℤ𝑠𝑚. Then for every 𝑚 ∈ ℕ0 we have that

∞∑𝑛=0

(1 + 𝑛2

)𝑚 ∣𝑝𝑛∣2 < +∞,

from where lim𝑛−→∞

(1 + 𝑛2

)𝑚 ∣𝑝𝑛∣2 = 0. Therefore lim𝑛−→∞𝑛𝑡𝑝𝑛 = 0 for every 𝑡 ≤ 2𝑚.

Since 𝑚 was arbitrarily chosen then lim𝑛−→∞𝑛𝑡𝑝𝑛 = 0 for every 𝑡 ∈ ℕ0. Conse-

quently, 𝑝 ∈ 𝑠 and since 𝑝 ∈ ∩𝑚∈ℕ𝑠𝑚 was arbitrarily chosen we conclude that∩𝑚∈ℕ𝑠𝑚 ⊂ 𝑠. □

In the sequel, consider the countable family of semi-norms on 𝑠

∥𝑝∥2𝑚 =

∞∑𝑛=0

(1 + 𝑛2

)𝑚 ∣𝑝𝑛∣2 .Lemma 2.4. 𝑠 is completely a Hausdorff space.

Proof. If 𝑝 ∈ 𝑠 and ∥𝑝∥𝑚 = 0 we get∑∞𝑛=0

(1 + 𝑛2)𝑚 ∣𝑝𝑛∣2 = 0.

Whence, 𝑝𝑛 = 0 for every natural 𝑛 (including zero), and consequently 𝑝 = 0. It

follows that 𝑠 is a Hausdorff space. Now, let {𝑝𝑘𝑛} −→𝑘−→∞ {𝑝𝑛}, i.e., lim𝑘−→∞

𝑝𝑘𝑛 =

𝑝𝑛 for every 𝑛 ∈ ℕ0. Then ∥∥𝑝𝑘 − 𝑝∥∥𝑚−→𝑘−→∞ 0.

It follows that 𝑠 is completely a space. □

6. Bochner–Minlos Theorem and Quaternion Fourier Transform 111

Let us define the metric

𝜌(𝑝, 𝑞) =

∞∑𝑚=0

∥𝑝− 𝑞∥𝑚1 + ∥𝑝− 𝑞∥𝑚

2−𝑚

for 𝑝, 𝑞 ∈ 𝑠. Evidently, the above metric has the translation property. In otherwords, from the countable family of seminorms we can define a metric with thetranslation property. From this follows the result that

Lemma 2.5. 𝑠 is a Frechet space.

Now, let 𝑠′ denote the topological dual space to 𝑠 given by 𝑠′ = ∪𝑚∈ℤ𝑠𝑚.We will denote the set of all sequences by ℝℕ0 , and we equip the space 𝑠′ withcylindrical topology. Every element 𝑝′ ∈ 𝑠′ acts on each element 𝑝 ∈ 𝑠 as follows

⟨𝑝′, 𝑝⟩ =∞∑𝑛=0

𝑝′𝑛𝑝𝑛; lim𝑛−→∞ 𝑝′𝑛 = 𝑝′, lim

𝑛−→∞ 𝑝𝑛 = 𝑝.

3. Asymptotic Behaviour of the QFT

There are numerous questions that remain untouched about the behaviour of theQFT. We will fill in some of these gaps and discuss some rudiments of the asymp-totic behaviour of (1.1).

We begin by proving the following result, which provides an interesting andefficient convergence characterization of the QFT.

Theorem 3.1. Let 𝑎, 𝑏, 𝑐, 𝑑 ∈ ℝ and 𝑓 ∈ 𝐿1((𝑎, 𝑏)×(𝑐, 𝑑);ℍ), then ℱ(𝑓) is uniformlycontinuous and bounded. Moreover, it satisfies

lim𝜔1−→±∞

ℱ(𝑓)(𝜔1, 𝜔2) = 0

uniformly in 𝜔2, and

lim𝜔2−→±∞

ℱ(𝑓)(𝜔1, 𝜔2) = 0

uniformly in 𝜔1. The results hold with the same proof when the region (𝑎, 𝑏)×(𝑐, 𝑑)

is replaced by its topological closure (𝑎, 𝑏)× (𝑐, 𝑑) (𝑎, 𝑏, 𝑐, 𝑑 can be ±∞) or anyregion 𝐺 (or 𝐺) in the space ℝ2.

Proof. It is easy to show that

∣ℱ(𝑓)(𝜔1, 𝜔2)∣ ≤∫ 𝑏

𝑎

∫ 𝑑

𝑐

∣𝑓(𝑥1, 𝑥2)∣ 𝑑𝑥1𝑑𝑥2

= ∥𝑓∥𝐿1((𝑎,𝑏)×(𝑐,𝑑);ℍ) . (3.1)

So, the transform ℱ(𝑓) is bounded.To prove the limit assertions, we use a density argument as in the classical

case (Riemann–Lebesgue Lemma). We first assume that both 𝑓 and ∂𝑥1𝑓 arecontinuous with compact support. Obviously, such functions form a dense subspace

112 S. Georgiev, J. Morais, K.I. Kou and W. Sproßig

in 𝐿1((𝑎, 𝑏)× (𝑐, 𝑑);ℍ). By using integration by parts to the 𝑥1-variable, with theiterated integration, we have

ℱ(𝑓)(𝜔1, 𝜔2) =

∫ 𝑏

𝑎

(𝑓(𝑏, 𝑥2)

1

i𝜔1𝑒−i𝜔1𝑏 𝑒−j𝜔2𝑥2 − 𝑓(𝑎, 𝑥2)

1

i𝜔1𝑒−i𝜔1𝑎 𝑒−j𝜔2𝑥2

)𝑑𝑥2

−∫ 𝑏

𝑎

∫ 𝑏

𝑎

∂𝑥1𝑓(𝑥1, 𝑥2)1

i𝜔1𝑒−i𝜔1𝑥1 𝑒−j𝜔2𝑥2 𝑑𝑥1𝑑𝑥2

= −∫ 𝑏

𝑎

∫ 𝑑

𝑐

∂𝑥1𝑓(𝑥1, 𝑥2)1

i𝜔1𝑒−i𝜔1𝑥1 𝑒−j𝜔2𝑥2 𝑑𝑥1𝑑𝑥2.

Therefore,

∣ℱ(𝑓)(𝜔1, 𝜔2)∣ ≤ 1

∣𝜔1∣ ∥∂𝑥1𝑓∥𝐿1((𝑎,𝑏)×(𝑐,𝑑);ℍ) . (3.2)

So,

lim𝜔1−→±∞

ℱ(𝑓)(𝜔1, 𝜔2) = 0

uniformly in 𝜔2. Similarly we can prove

lim𝜔2−→±∞

ℱ(𝑓)(𝜔1, 𝜔2) = 0

uniformly in 𝜔1.

Now assume 𝑓 ∈ 𝐿1((𝑎, 𝑏) × (𝑐, 𝑑);ℍ). For any given 𝜀 > 0, there exists afunction 𝑓𝜀 in the above-mentioned dense class, such that

∥𝑓 − 𝑓𝜀∥𝐿1((𝑎,𝑏)×(𝑐,𝑑);ℍ) < 𝜀.

By (3.1), we have

∣ℱ(𝑓)(𝜔1, 𝜔2)∣ ≤ ∣ℱ(𝑓𝜀)(𝜔1, 𝜔2)∣+ ∥𝑓 − 𝑓𝜀∥𝐿1((𝑎,𝑏)×(𝑐,𝑑);ℍ)

≤ ∣ℱ(𝑓𝜀)(𝜔1, 𝜔2)∣+ 𝜀.

By using the result just proved for the density class, we have

lim𝜔1→±∞

∣ℱ(𝑓)(𝜔1, 𝜔2)∣ ≤ 𝜀,

uniformly in 𝜔2. Since 𝜀 is arbitrary, we have

lim𝜔1→±∞

ℱ(𝑓)(𝜔1, 𝜔2) = 0 (3.3)

uniformly in 𝜔2. Similarly,

lim𝜔2→±∞

ℱ(𝑓)(𝜔1, 𝜔2) = 0, (3.4)

uniformly in 𝜔1.

Now we show the uniform continuity of ℱ(𝑓)(𝜔1, 𝜔2). Given (3.3) and (3.4),since continuous functions are uniform continuous in compact sets, it suffices to

6. Bochner–Minlos Theorem and Quaternion Fourier Transform 113

show that ℱ(𝑓) is continuous at every point (𝜔1, 𝜔2). In fact,

ℱ(𝑓)(𝜔1 + 𝜌1, 𝜔2 + 𝜌2)−ℱ(𝑓)(𝜔1, 𝜔2)

=

∫ 𝑏

𝑎

∫ 𝑑

𝑐

𝑓(𝑥1, 𝑥2)[𝑒−𝒊𝑥1(𝜔1+𝜌1)𝑒−𝒋𝑥2(𝜔2+𝜌2) − 𝑒−𝒊𝑥2𝜔1𝑒−𝒋𝑥2𝜔2

]𝑑𝑥1𝑑𝑥2.

For any 𝜌1, 𝜌2 > 0, the integrand is dominated by a constant multiple of ∣𝑓(𝑥1, 𝑥2)∣.Since the factor in the straight brackets tends to zero, by the Lebesgue DominatedConvergence Theorem, we have

lim𝜌1,𝜌2→0

ℱ(𝑓)(𝜔1 + 𝜌1, 𝜔2 + 𝜌2)−ℱ(𝑓)(𝜔1, 𝜔2) = 0,

proving the desired continuity. □

4. Bochner–Minlos Theorem

In this section we extend the classical Bochner–Minlos theorem (named afterBochner and Adol’fovich Minlos) to the framework of quaternion analysis. Theresulting theorem guarantees the existence of the corresponding probability mea-sure defined on a dual space. In particular, some interesting properties of theunderlying measure are extended in this setting.

Definition 4.1. Let 𝜇 be a finite positive measure on ℝ2. The QFT of 𝜇 is thefunction ℱ𝒊𝒋(𝜇) : ℝ

2 −→ ℍ given by

ℱ𝒊𝒋(𝜇)(𝜔1, 𝜔2) :=

∫ℝ2

𝑒−𝒊𝜔1𝑥1𝑒−𝒋𝜔2𝑥2𝑑𝜇(𝑥1, 𝑥2),

or the function ℱ𝒋𝒊(𝜇) : ℝ2 −→ ℍ given by

ℱ𝒋𝒊(𝜇)(𝜔1, 𝜔2) :=

∫ℝ2

𝑒−𝒋𝜔2𝑥2𝑒−𝒊𝜔1𝑥1𝑑𝜇(𝑥1, 𝑥2).

Proposition 4.2. Let 𝜇 be a finite positive measure on ℝ2. The functionals ℱ𝒊𝒋(𝜇)and ℱ𝒋𝒊(𝜇) satisfy the following basic properties:

1. ℱ𝒊𝒋(𝜇)(0, 0) = ℱ𝒋𝒊(𝜇)(0, 0) = 1;

2. ℱ𝒊𝒋(𝜇)(−𝜔1,−𝜔2) = ℱ𝒋𝒊(𝜇)(𝜔1, 𝜔2);

3. ℱ𝒋𝒊(𝜇)(−𝜔1,−𝜔2) = ℱ𝒊𝒋(𝜇)(𝜔1, 𝜔2);4. ℱ𝒊𝒋(𝜇)(−𝜔1,−𝜔2) + ℱ𝒊𝒋(𝜇)(𝜔1, 𝜔2)

= 2

∫ℝ2

(cos(𝜔1𝑥1) cos(𝜔2𝑥2) + 𝒌 sin(𝜔1𝑥1) sin(𝜔2𝑥2)) 𝑑𝜇(𝑥1, 𝑥2);

5. ℱ𝒊𝒋(𝜇)(𝜔1, 𝜔2) + ℱ𝒋𝒊(𝜇)(−𝜔1,−𝜔2) = 2

∫ℝ2

cos(𝜔1𝑥1) cos(𝜔2𝑥2)𝑑𝜇(𝑥1, 𝑥2).

Proof. For the first statement, note that

ℱ𝒊𝒋(𝜇)(0, 0) =

∫ℝ2

𝑑𝜇(𝑥1, 𝑥2) = 𝜇(ℝ2).

114 S. Georgiev, J. Morais, K.I. Kou and W. Sproßig

For Statement 2 a straightforward computation shows that

ℱ𝒊𝒋(𝜇)(−𝜔1,−𝜔2) =

∫ℝ2

𝑒𝒊𝜔1𝑥1𝑒𝒋𝜔2𝑥2𝑑𝜇(𝑥1, 𝑥2) =

∫ℝ2

𝑒−𝒋𝜔2𝑥2𝑒−𝒊𝜔1𝑥1𝑑𝜇(𝑥1, 𝑥2)

=

∫ℝ2

𝑒−𝒋𝜔2𝑥2𝑒−𝒊𝜔1𝑥1𝑑𝜇(𝑥1, 𝑥2) = ℱ𝒋𝒊(𝜇)(𝜔1, 𝜔2).

The proof of Statement 3 will be omitted, being similar to the previous one. Now,taking into account that

𝑒−𝒊𝜔1𝑥1𝑒−𝒋𝜔2𝑥2 = cos(𝜔1𝑥1) cos(𝜔2𝑥2)− 𝒊 sin(𝜔1𝑥1) cos(𝜔2𝑥2)

− 𝒋 cos(𝜔1𝑥1) sin(𝜔2𝑥2) + 𝒌 sin(𝜔1𝑥1) sin(𝜔2𝑥2),

and

𝑒𝒊𝜔1𝑥1𝑒𝒋𝜔2𝑥2 = cos(𝜔1𝑥1) cos(𝜔2𝑥2) + 𝒊 sin(𝜔1𝑥1) cos(𝜔2𝑥2)

+ 𝒋 cos(𝜔1𝑥1) sin(𝜔2𝑥2) + 𝒌 sin(𝜔1𝑥1) sin(𝜔2𝑥2),

we obtain

ℱ𝒊𝒋(𝜇)(𝜔1, 𝜔2) + ℱ𝒊𝒋(𝜇)(−𝜔1,−𝜔2)

=

∫ℝ2

(𝑒−𝒊𝜔1𝑥1𝑒−𝒋𝜔2𝑥2 + 𝑒𝒊𝜔1𝑥1𝑒𝒋𝜔2𝑥2

)𝑑𝜇(𝑥1, 𝑥2)

= 2

∫ℝ2

(cos(𝜔1𝑥1) cos(𝜔2𝑥2) + 𝒌 sin(𝜔1𝑥1) sin(𝜔2𝑥2)) 𝑑𝜇(𝑥1, 𝑥2).

For the last statement we use the relation

𝑒𝒋𝑤2𝑥2𝑒𝒊𝜔1𝑥1 = cos(𝜔1𝑥1) cos(𝜔2𝑥2) + 𝒊 sin(𝜔1𝑥1) cos(𝜔2𝑥2)

+ 𝒋 cos(𝜔1𝑥1) sin(𝜔2𝑥2)− 𝒌 sin(𝜔1𝑥1) sin(𝜔2𝑥2).

Therefore, it follows

ℱ𝒊𝒋(𝜇)(𝜔1, 𝜔2) + ℱ𝒋𝒊(𝜇)(−𝜔1,−𝜔2)

=

∫ℝ2

(𝑒−𝒊𝜔1𝑥1𝑒−𝒋𝜔2𝑥2 + 𝑒𝒋𝜔2𝑥2𝑒𝒊𝜔1𝑥1

)𝑑𝜇(𝑥1, 𝑥2)

= 2

∫ℝ2

cos(𝜔1𝑥1) cos(𝜔2𝑥2)𝑑𝜇(𝑥1, 𝑥2). □

In the sequel, let us denote by 𝕊(ℝ2) the Schwartz space of smooth quaternionfunctions on ℝ2. We formulate a first result.

Proposition 4.3. Let 𝜙 ∈ 𝕊(ℝ2) and 𝜇 be a finite positive measure on ℝ2. Then

1.

∫ℝ2

ℱ𝒊𝒋(𝜇)(𝜔1, 𝜔2)𝜙(𝜔1, 𝜔2)𝑑𝜔1𝑑𝜔2 =

∫ℝ2

ℱ𝒊𝒋(𝜙)(𝑥1, 𝑥2)𝑑𝜇(𝑥1, 𝑥2);

2.

∫ℝ2

ℱ𝒋𝒊(𝜇)(𝜔1, 𝜔2)𝜙(𝜔1, 𝜔2)𝑑𝜔1𝑑𝜔2 =

∫ℝ2

ℱ𝒋𝒊(𝜙)(𝑥1, 𝑥2)𝑑𝜇(𝑥1, 𝑥2).

6. Bochner–Minlos Theorem and Quaternion Fourier Transform 115

Proof. For simplicity we just present the computations of the first equality. Theproof of the second one is similar. A direct computation shows that∫

ℝ2

ℱ𝒊𝒋(𝜇)(𝜔1, 𝜔2)𝜙(𝜔1, 𝜔2)𝑑𝜔1𝑑𝜔2

=

∫ℝ2

∫ℝ2

𝑒−𝒊𝜔1𝑥1𝑒−𝒋𝜔2𝑥2𝑑𝜇(𝑥1, 𝑥2)𝜙(𝜔1, 𝜔2)𝑑𝜔1𝑑𝜔2

=

∫ℝ2

∫ℝ2

𝑒−𝒊𝜔1𝑥1𝑒−𝒋𝜔2𝑥2𝜙(𝜔1, 𝜔2)𝑑𝜔1𝑑𝜔2𝑑𝜇(𝑥1, 𝑥2)

=

∫ℝ2

ℱ𝒊𝒋(𝜙)(𝑥1, 𝑥2)𝑑𝜇(𝑥1, 𝑥2). □

We now analyze some key properties of the above-mentioned functionals.

Proposition 4.4. Let 𝜇 and 𝜈 be finite positive measures on ℝ2. The functionalsℱ𝒊𝒋(𝜇) and ℱ𝒋𝒊(𝜇) are linear, i.e., for every 𝑐, 𝑑 ∈ ℍ it holds:

ℱ𝒊𝒋(𝜇𝑐+ 𝜈𝑑) = ℱ𝒊𝒋(𝜇)𝑐+ ℱ𝒊𝒋(𝜈)𝑑,

ℱ𝒋𝒊(𝜇𝑐+ 𝜈𝑑) = ℱ𝒋𝒊(𝜇)𝑐+ ℱ𝒋𝒊(𝜈)𝑑.

Proposition 4.5. Let 𝜇 be a finite positive measure on ℝ2. For any 𝑎𝑖 ∈ ℝ ∖ {0}(𝑖 = 1, 2) the following conditions hold:

1. ℱ𝒊𝒋(𝜇(𝑎1𝑥1, 𝑎2𝑥2)) = ℱ𝒊𝒋(𝜇(𝑥1, 𝑥2))(𝜔1𝑎1

, 𝜔2𝑎2

);

2. ℱ𝒋𝒊(𝜇(𝑎1𝑥1, 𝑎2𝑥2)) = ℱ𝒋𝒊(𝜇(𝑥1, 𝑥2))(𝜔1𝑎1

, 𝜔2𝑎2

).

Proof. For simplicity we just present the proof of the first condition. A straight-forward computation shows that

ℱ𝒊𝒋(𝜇(𝑎1𝑥1, 𝑎2𝑥2)) =

∫ℝ2

𝑒−𝒊𝜔1𝑥1𝑒−𝒋𝜔2𝑥2𝑑𝜇(𝑎1𝑥1, 𝑎2𝑥2)

=

∫ℝ2

𝑒−𝒊𝜔1𝑎1

(𝑎1𝑥1)𝑒−𝒋 𝜔2𝑎2

(𝑎2𝑥2)𝑑𝜇(𝑎1𝑥1, 𝑎2𝑥2)

=

∫ℝ2

𝑒−𝒊𝜔1𝑎1

𝑦1𝑒−𝒋 𝜔2𝑎2

𝑦2𝑑𝜇(𝑦1, 𝑦2)

= ℱ𝒊𝒋(𝜇(𝑥1, 𝑥2))

(𝜔1

𝑎1,𝜔2

𝑎2

). □

We proceed to define the notion of positive definitely function in the contextof quaternion analysis.

Definition 4.6. Let 𝑓 be a quaternion function on ℝ2 that is continuous andbounded. For every finite positive measure 𝜇 on ℝ2 the function 𝑓 is said tobe positive definite if

𝑁∑𝑘,𝑙=1,𝑘<𝑙

𝑧𝑘𝑧𝑙𝑓(𝜆𝑘 − 𝜆𝑙) +𝑁∑

𝑘,𝑙=1,𝑘<𝑙

𝑓(𝜆𝑘 − 𝜆𝑙)𝑧𝑙𝑧𝑘 +𝑁∑𝑘=1

∣𝑧𝑘∣2 𝜇(ℝ2) ≥ 0

116 S. Georgiev, J. Morais, K.I. Kou and W. Sproßig

for every 𝜆1, 𝜆2, . . . , 𝜆𝑁 ∈ ℝ2, 𝑧1, 𝑧2, . . . , 𝑧𝑁 ∈ ℍ. These parameters are measuredsuch that:

1. When 𝜆1 = 𝜆2 = ⋅ ⋅ ⋅ = 𝜆𝑁 , and 𝑧1 = 𝑧2 = ⋅ ⋅ ⋅ = 𝑧𝑁 it follows

2𝑓(0, 0) + 𝜇(ℝ2) ≥ 0;

2. When 𝑁 = 2, 𝜆1 = (𝑥1, 𝑥2), and 𝜆2 = (0, 0), we have

𝑧1𝑧2𝑓(𝑥1, 𝑥2) + 𝑓(−𝑥1,−𝑥2)𝑧2𝑧1 +(∣𝑧1∣2 + ∣𝑧2∣2

)𝜇(ℝ2) ≥ 0,

which is valid if 𝑓(−𝑥1,−𝑥2) = 𝑓(𝑥1, 𝑥2).

Proposition 4.7. The functional ℱ𝒊𝒋(𝜇) is positive definite and bounded.

Proof. Let 𝜆1, 𝜆2, . . . , 𝜆𝑁 ∈ ℝ2 such that 𝜆𝑖 = (𝜆1𝑘, 𝜆2𝑘), and 𝑧1, 𝑧2, . . . , 𝑧𝑁 ∈ ℍ.Direct computations show that

𝑁∑𝑘,𝑙,𝑘<𝑙

𝑧𝑘𝑧𝑙ℱ𝒊𝒋(𝜇)(𝜆𝑘 − 𝜆𝑙) +

𝑁∑𝑘,𝑙=1,𝑘<𝑙

ℱij(𝜇)(𝜆𝑘 − 𝜆𝑙)𝑧𝑙𝑧𝑘 +

𝑁∑𝑘=1

∣𝑧𝑘∣2 𝜇 (ℝ2)

=

∫ℝ2

𝑁∑𝑘,𝑙=1,𝑘<𝑙

𝑧𝑘𝑧𝑙𝑒−𝒊(𝜆1𝑘−𝜆1𝑙)𝑥1𝑒−𝒋(𝜆2𝑘−𝜆2𝑙)𝑥2𝑑𝜇(𝑥1, 𝑥2)

+

∫ℝ2

𝑁∑𝑘,𝑙=1,𝑘<𝑙

𝑒−𝒋(𝜆2𝑘−𝜆2𝑙)𝑥2𝑒−𝒊(𝜆1𝑘−𝜆1𝑙)𝑥1𝑧𝑙𝑧𝑘𝑑𝜇(𝑥1, 𝑥2)

+

∫ℝ2

𝑁∑𝑘=1

∣𝑧𝑘∣2 𝑑𝜇(𝑥1, 𝑥2)

=

∫ℝ2

𝑁∑𝑘,𝑙=1,𝑘<𝑙

𝑧𝑘𝑧𝑙𝑒−𝒊(𝜆1𝑘−𝜆1𝑙)𝑥1𝑒−𝒋(𝜆2𝑘−𝜆2𝑙)𝑥2𝑑𝜇(𝑥1, 𝑥2)

+

∫ℝ2

𝑁∑𝑘,𝑙=1,𝑘<𝑙

𝑧𝑘𝑧𝑙𝑒−𝒊(𝜆1𝑘−𝜆1𝑙)𝑥1𝑒−𝒋(𝜆2𝑘−𝜆2𝑙)𝑥2𝑑𝜇(𝑥1, 𝑥2)

+

∫ℝ2

𝑁∑𝑘=1

∣𝑧𝑘∣2 𝑑𝜇(𝑥1, 𝑥2)

=

∫ℝ2

⎡⎣ 𝑁∑𝑘=1

∣𝑧𝑘∣2 + 2

𝑁∑𝑘,𝑙=1,𝑘<𝑙

S(𝑧𝑘𝑧𝑙𝑒

−𝒊(𝜆1𝑘−𝜆1𝑙)𝑥1𝑒−𝒋(𝜆2𝑘−𝜆2𝑙)𝑥2

)⎤⎦ 𝑑𝜇(𝑥1, 𝑥2)

≥∫ℝ2

⎛⎝ 𝑁∑𝑘=1

∣𝑧𝑘∣2 − 2

𝑁∑𝑘,𝑙=1,𝑘<𝑙

∣𝑧𝑘∣ ∣𝑧𝑙∣⎞⎠ 𝑑𝜇(𝑥1, 𝑥2) ≥ 0.

Furthermore, it follows that

∣ℱ𝒊𝒋(𝜇)(𝜔1, 𝜔2)∣ =∣∣∫

ℝ2 𝑒−𝒊𝜔1𝑥1𝑒−𝒋𝜔2𝑥2𝑑𝜇(𝑥1, 𝑥2)

∣∣

6. Bochner–Minlos Theorem and Quaternion Fourier Transform 117

≤∫ℝ2

∣𝑒−𝒊𝜔1𝑥1𝑒−𝒋𝜔2𝑥2 ∣ 𝑑𝜇(𝑥1, 𝑥2) = 𝜇(ℝ2)< +∞. □

Likewise we can prove the following proposition.

Proposition 4.8. The functional ℱ𝒋𝒊(𝜇) is positive definite and bounded.

Below we shall assume that 𝜇 is a probably measure on 𝑠′. For every elements𝑎 ∈ 𝑠 and 𝑎′ ∈ 𝑠′ we define the functionals 𝑔𝒊𝒋 and 𝑔𝒋𝒊 on 𝑠 as follows:

𝑔𝒊𝒋 : 𝑠 −→ 𝑠, 𝑎 �−→ 𝑔𝒊𝒋(𝑎) :=

∫𝑠′𝑒𝒊⟨𝑎′,𝑎⟩𝑒𝒋⟨𝑎′,𝑎⟩𝑑𝜇(𝑎′)

and

𝑔𝒋𝒊 : 𝑠 −→ 𝑠, 𝑎 �−→ 𝑔𝒋𝒊(𝑎) :=

∫𝑠′𝑒𝒋⟨𝑎′,𝑎⟩𝑒𝒊⟨𝑎′,𝑎⟩𝑑𝜇(𝑎′).

Next we present a generalization of the classical Bochner–Minlos theorem on pos-itive definite functions to the case of quaternion functions.

Theorem 4.9 (Bochner–Minlos theorem). The functional 𝑔𝒊𝒋 satisfies the followingthree conditions:

1. Normalization: 𝑔𝒊𝒋(0) = 1;2. Positivity:

𝑛∑𝑘,𝑙=1,𝑘<𝑙

𝑧𝑘𝑧𝑙𝑔𝒊𝒋(𝑎𝑘 − 𝑎𝑙) +

𝑛∑𝑘,𝑙=1,𝑘<𝑙

𝑔ij(𝑎𝑘 − 𝑎𝑙)𝑧𝑙𝑧𝑘 +∑𝑘

∣𝑧𝑘∣2 ≥ 0;

3. Continuity: 𝑔𝒊𝒋 is continuous in the sense of Frechet topology.

Proof. We begin the proof by noting that

𝑔𝒊𝒋(0) =

∫𝑠′𝑑𝜇(𝑎′) = 𝜇(𝑠′) = 1.

For simplicity we will prove Statement 2 in the case 𝑛 = 2 only, i.e., we will provethat

𝑧1𝑧2 𝑔𝒊𝒋(𝑎1 − 𝑎2) + 𝑔ij(𝑎1 − 𝑎2)𝑧2𝑧1 + ∣𝑧1∣2 + ∣𝑧2∣2 ≥ 0.

For the sake of convenience we set 𝑧 = 𝑎1 − 𝑎2. It follows that

𝑧1𝑧2 𝑔𝒊𝒋(𝑧) + 𝑔ij(𝑧)𝑧2𝑧1 + ∣𝑧1∣2 + ∣𝑧2∣2

=

∫𝑠′

(𝑧1𝑧2𝑒

𝒊⟨𝑎′,𝑧⟩𝑒𝒋⟨𝑎′,𝑧⟩ + 𝑧1𝑧2𝑒𝒊⟨𝑎′,𝑧⟩𝑒𝒋⟨𝑎′,𝑧⟩ + ∣𝑧1∣2 + ∣𝑧2∣2

)𝑑𝜇(𝑎′)

=

∫𝑠′

[2 S(𝑧1𝑧2𝑒

𝒊⟨𝑎′,𝑧⟩𝑒𝒋⟨𝑎′,𝑧⟩ + ∣𝑧1∣2 + ∣𝑧2∣2)]

𝑑𝜇(𝑎′).(4.1)

Notice that the last equality follows from the relation

𝑧1𝑧2 𝑔𝒊𝒋(𝑧) = 𝑔𝒋𝒊(−𝑧)𝑧2𝑧1.

Now, let 𝑧𝑖 = ∣𝑧𝑖∣ 𝑒V(𝑧𝑖)

∣V(𝑧𝑖)∣ 𝜃𝑖 , with 𝜃𝑖 = arg(𝑧𝑖) (𝑖 = 1, 2). Then

S(𝑧1𝑧2 𝑔𝒊𝒋(𝑧)) ≥ − ∣𝑧1∣ ∣𝑧2∣ .

118 S. Georgiev, J. Morais, K.I. Kou and W. Sproßig

From here and (4.1) we obtain

𝑧1𝑧2 𝑔𝒊𝒋(𝑧) + 𝑔𝒋𝒊(−𝑧)𝑧2𝑧1 + ∣𝑧1∣2 + ∣𝑧2∣2

≥∫𝑠′

(∣𝑧1∣2 + ∣𝑧2∣2 − 2 ∣𝑧1∣ ∣𝑧2∣

)𝑑𝜇(𝑎′) ≥ 0.

Using induction we may conclude that Statement 2 is valid for every natural 𝑛.For the proof of the remaining statement, let lim𝑛−→∞ 𝑎𝑛 = 𝑎 be understood inthe sense of the topology of Frechet. Then lim𝑛−→∞ 𝑎𝑛 = 𝑎 holds in the usualsense. From here and from the definition of the functional 𝑔𝒊𝒋 we conclude thatlim𝑛−→∞ 𝑔𝒊𝒋(𝑎𝑛) = 𝑔𝒊𝒋(𝑎). Therefore for every 𝜖 > 0 a natural number 𝑁 =𝑁(𝜖) > 0 can be found so that for 𝑘 > 𝑁 the following holds

∥𝑔(𝑎𝑛)− 𝑔(𝑎)∥𝑘 < 𝜖.

Consequently, it follows that

𝜌 (𝑔(𝑎𝑛), 𝑔(𝑎)) =∞∑

𝑁+1

∥𝑔(𝑎𝑛)− 𝑔(𝑎)∥𝑘1 + ∥𝑔(𝑎𝑛)− 𝑔(𝑎)∥𝑘

2−𝑘 ≤ 𝜖∞∑

𝑁+1

2−𝑘. □

Proposition 4.10. For every element 𝑎 ∈ 𝑠 the functionals 𝑔𝒊𝒋 and 𝑔𝒋𝒊 satisfy theadditional properties:

1. 𝑔𝒊𝒋(−𝑎) = 𝑔𝒋𝒊(𝑎);

2. 𝑔𝒋𝒊(−𝑎) = 𝑔𝒊𝒋(𝑎).

Though the significance of our approach to concrete applications, such as thecharacterization of measurement configurations for functional spaces, was the mainreason for restricting ourselves to the quaternionic case, doubtless, the reduction incalculations for proving the results played an important role too. As was alreadymentioned, it is possible to perform an analogous study to generalized Cliffordalgebras following the same ideas. Further investigations will be presented in aforthcoming paper.

Acknowledgement

Partial support from the Foundation for Science and Technology (FCT) via thegrant DD-VU-02/90, Bulgaria is acknowledged by the first named author. The sec-ond named author acknowledges financial support from the Foundation for Scienceand Technology (FCT) via the post-doctoral grant SFRH/ BPD/66342/2009. Thiswork was supported by FEDER funds through COM PETE – Operational Pro-gramme Factors of Competitiveness (‘Programa Operacional Factores de Compet-itividade’) and by Portuguese funds through the Center for Research and Develop-ment in Mathematics and Applications (University of Aveiro) and the PortugueseFoundation for Science and Technology (‘FCT – Fundacao para a Ciencia e aTecnologia’), within project PEst-C/MAT/UI4106/2011 with COMPETE numberFCOMP-01-0124-FEDER-022690. The third named author acknowledges financialsupport from the research grant of the University of Macau No. MYRG142(Y1-L2)-FST11-KKI.

6. Bochner–Minlos Theorem and Quaternion Fourier Transform 119

References

[1] M. Bahri. Generalized Fourier transform in real clifford algebra 𝑐𝑙(0, 𝑛). Far EastJournal of Mathematical Sciences, 48(1):11–24, Jan. 2011.

[2] P. Bas, N. Le Bihan, and J.M. Chassery. Color image watermarking using quaternionFourier transform. In Proceedings of the IEEE International Conference on AcousticsSpeech and Signal Processing, ICASSP, volume 3, pages 521–524. Hong-Kong, 2003.

[3] E. Bayro-Corrochano, N. Trujillo, and M. Naranjo. Quaternion Fourier descriptorsfor preprocessing and recognition of spoken words using images of spatiotemporalrepresentations. Mathematical Imaging and Vision, 28(2):179–190, 2007.

[4] F. Brackx, N. De Schepper, and F. Sommen. The Clifford–Fourier transform. Journalof Fourier Analysis and Applications, 11(6):669–681, 2005.

[5] F. Brackx, N. De Schepper, and F. Sommen. The Fourier transform in Cliffordanalysis. Advances in Imaging and Electron Physics, 156:55–201, 2009.

[6] F. Brackx, R. Delanghe, and F. Sommen. Clifford Analysis, volume 76. Pitman,Boston, 1982.

[7] T. Bulow. Hypercomplex Spectral Signal Representations for the Processing and Anal-ysis of Images. PhD thesis, University of Kiel, Germany, Institut fur Informatik undPraktische Mathematik, Aug. 1999.

[8] T. Bulow, M. Felsberg, and G. Sommer. Non-commutative hypercomplex Fouriertransforms of multidimensional signals. In G. Sommer, editor, Geometric computingwith Clifford Algebras: Theoretical Foundations and Applications in Computer Visionand Robotics, pages 187–207, Berlin, 2001. Springer.

[9] J. Ebling and G. Scheuermann. Clifford Fourier transform on vector fields. IEEETransactions on Visualization and Computer Graphics, 11(4):469–479, July/Aug.2005.

[10] T.A. Ell. Quaternion-Fourier transforms for analysis of 2-dimensional linear time-invariant partial-differential systems. In Proceedings of the 32nd Conference on De-cision and Control, pages 1830–1841, San Antonio, Texas, USA, 15–17 December1993. IEEE Control Systems Society.

[11] T.A. Ell and S.J. Sangwine. Hypercomplex Fourier transforms of color images. IEEETransactions on Image Processing, 16(1):22–35, Jan. 2007.

[12] S. Georgiev. Bochner–Minlos theorem and quaternion Fourier transform. In Gurle-beck [14].

[13] S. Georgiev, J. Morais, and W. Sproßig. Trigonometric integrals in the framework ofquaternionic analysis. In Gurlebeck [14].

[14] K. Gurlebeck, editor. 9th International Conference on Clifford Algebras and theirApplications, Weimar, Germany, 15–20 July 2011.

[15] K. Gurlebeck and W. Sproßig. Quaternionic Analysis and Elliptic Boundary ValueProblems. Berlin: Akademie-Verlag, Berlin, 1989.

[16] K. Gurlebeck and W. Sproßig. Quaternionic and Clifford Calculus for Physicists andEngineers. Wiley, Aug. 1997.

[17] E. Hitzer. Quaternion Fourier transform on quaternion fields and generalizations.Advances in Applied Clifford Algebras, 17(3):497–517, May 2007.

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[18] E.M.S. Hitzer and B. Mawardi. Clifford Fourier transform on multivector fields anduncertainty principles for dimensions 𝑛 = 2(mod 4) and 𝑛 = 3(mod 4). Advances inApplied Clifford Algebras, 18(3-4):715–736, 2008.

[19] V.V. Kravchenko. Applied Quaternionic Analysis, volume 28 of Research and Expo-sition in Mathematics. Heldermann Verlag, Lemgo, Germany, 2003.

[20] B. Mawardi. Generalized Fourier transform in Clifford algebra 𝒞ℓ0,3. Far East Jour-nal of Mathematical Sciences, 44(2):143–154, 2010.

[21] B. Mawardi, E. Hitzer, A. Hayashi, and R. Ashino. An uncertainty principlefor quaternion Fourier transform. Computers and Mathematics with Applications,56(9):2411–2417, 2008.

[22] B. Mawardi and E.M.S. Hitzer. Clifford Fourier transformation and uncertainty prin-ciple for the Clifford algebra 𝐶ℓ3,0. Advances in Applied Clifford Algebras, 16(1):41–61, 2006.

[23] S.-C. Pei, J.-J. Ding, and J.-H. Chang. Efficient implementation of quaternionFourier transform, convolution, and correlation by 2-D complex FFT. IEEE Trans-actions on Signal Processing, 49(11):2783–2797, Nov. 2001.

[24] M. Shapiro and N.L. Vasilevski. Quaternionic 𝜓-hyperholomorphic functions, singu-lar integral operators and boundary value problems I. 𝜓-hyperholomorphic functiontheory. Complex Variables, Theory and Application, 27(1):17–46, 1995.

[25] M. Shapiro and N.L. Vasilevski. Quaternionic 𝜓-hyperholomorphic functions, singu-lar integral operators and boundary value problems II. Algebras of singular integraloperators and Riemann type boundary value problems. Complex Variables, TheoryAppl., 27(1):67–96, 1995.

S. GeorgievDepartment of Differential Equations, University of SofiaSofia, Bulgariae-mail: sgg2000bg@yahoo.com

J. MoraisCentro de Investigacao e Desenvolvimento em Matematica e Aplicacoes (CIDMA)Universidade de Aveiro, 3810-193 Aveiro, Portugale-mail: joao.pedro.morais@ua.pt

K.I. KouDepartment of Mathematics, Faculty of Science and TechnologyUniversity of Macau, Macaue-mail: kikouou@umac.mo

W. SproßigFreiberg University of Mining and TechnologyFreiberg, Germanye-mail: sproessig@math.tu-freiberg.de

Part II

Clifford Algebra

Quaternion and Clifford–Fourier Transforms and Wavelets

Trends in Mathematics, 123–153c⃝ 2013 Springer Basel

7 Square Roots of −1 in RealClifford Algebras

Eckhard Hitzer, Jacques Helmstetter and Rafal Ablamowicz

Abstract. It is well known that Clifford (geometric) algebra offers a geomet-ric interpretation for square roots of −1 in the form of blades that squareto minus 1. This extends to a geometric interpretation of quaternions as theside face bivectors of a unit cube. Systematic research has been done [33] onthe biquaternion roots of −1, abandoning the restriction to blades. Biquater-nions are isomorphic to the Clifford (geometric) algebra 𝐶ℓ3,0 of ℝ3. Furtherresearch on general algebras 𝐶ℓ𝑝,𝑞 has explicitly derived the geometric rootsof −1 for 𝑝+ 𝑞 ≤ 4 [20]. The current research abandons this dimension limitand uses the Clifford algebra to matrix algebra isomorphisms in order to al-gebraically characterize the continuous manifolds of square roots of −1 foundin the different types of Clifford algebras, depending on the type of associ-ated ring (ℝ, ℍ, ℝ2, ℍ2, or ℂ). At the end of the chapter explicit computergenerated tables of representative square roots of −1 are given for all Cliffordalgebras with 𝑛 = 5, 7, and 𝑠 = 3 (mod 4) with the associated ring ℂ. Thisincludes, e.g., 𝐶ℓ0,5 important in Clifford analysis, and 𝐶ℓ4,1 which in appli-cations is at the foundation of conformal geometric algebra. All these rootsof −1 are immediately useful in the construction of new types of geometricClifford–Fourier transformations.

Mathematics Subject Classification (2010). Primary 15A66; secondary 11E88,42A38, 30G35.

Keywords. Algebra automorphism, inner automorphism, center, centralizer,Clifford algebra, conjugacy class, determinant, primitive idempotent, trace.

1. Introduction

The young London Goldsmith professor of applied mathematics W.K. Cliffordcreated his geometric algebras1 in 1878 inspired by the works of Hamilton on

1In his original publication [11] Clifford first used the term geometric algebras. Subsequently inmathematics the new term Clifford algebras [28] has become the proper mathematical term. Foremphasizing the geometric nature of the algebra, some researchers continue [9, 16, 17] to use theoriginal term geometric algebra(s).

124 E. Hitzer, J. Helmstetter and R. Ablamowicz

quaternions and by Grassmann’s exterior algebra. Grassmann invented the anti-symmetric outer product of vectors, that regards the oriented parallelogram areaspanned by two vectors as a new type of number, commonly called bivector. Thebivector represents its own plane, because outer products with vectors in the planevanish. In three dimensions the outer product of three linearly independent vectorsdefines a so-called trivector with the magnitude of the volume of the parallelepipedspanned by the vectors. Its orientation (sign) depends on the handedness of thethree vectors.

In the Clifford algebra [16] of ℝ3 the three bivector side faces of a unit cube{𝑒1𝑒2, 𝑒2𝑒3, 𝑒3𝑒1} oriented along the three coordinate directions {𝑒1, 𝑒2, 𝑒3} cor-respond to the three quaternion units 𝒊, 𝒋, and 𝒌. Like quaternions, these threebivectors square to minus one and generate the rotations in their respective planes.

Beyond that Clifford algebra allows to extend complex numbers to higherdimensions [7, 17] and systematically generalize our knowledge of complex num-bers, holomorphic functions and quaternions into the realm of Clifford analysis.It has found rich applications in symbolic computation, physics, robotics, com-puter graphics, etc. [8, 9, 12, 14, 27]. Since bivectors and trivectors in the Cliffordalgebras of Euclidean vector spaces square to minus one, we can use them tocreate new geometric kernels for Fourier transformations. This leads to a largevariety of new Fourier transformations, which all deserve to be studied in theirown right [5, 6, 9, 13, 18, 19, 22,23,26, 29–32].

In our current research we will treat square roots of −1 in Clifford algebras𝐶ℓ𝑝,𝑞 of both Euclidean (positive definite metric) and non-Euclidean (indefinitemetric) non-degenerate vector spaces, ℝ𝑛 = ℝ𝑛,0 and ℝ𝑝,𝑞, respectively. We knowfrom Einstein’s special theory of relativity that non-Euclidean vector spaces are offundamental importance in nature [15]. They are further, e.g., used in computervision and robotics [12] and for general algebraic solutions to contact problems [27].Therefore this chapter is about characterizing square roots of −1 in all Cliffordalgebras 𝐶ℓ𝑝,𝑞, extending previous limited research on 𝐶ℓ3,0 in [33] and 𝐶ℓ𝑝,𝑞, 𝑛 =𝑝 + 𝑞 ≤ 4 in [20]. The manifolds of square roots of −1 in 𝐶ℓ𝑝,𝑞, 𝑛 = 𝑝 + 𝑞 = 2,compare Table 1 of [20], are visualized in Figure 1.

First, we introduce necessary background knowledge of Clifford algebras andmatrix ring isomorphisms and explain in more detail how we will characterize andclassify the square roots of −1 in Clifford algebras in Section 2. Next, we treatsection by section (in Sections 3 to 7) the square roots of −1 in Clifford algebraswhich are isomorphic to matrix algebras with associated rings ℝ, ℍ, ℝ2, ℍ2, andℂ, respectively. The term associated means that the isomorphic matrices will onlyhave matrix elements from the associated ring. The square roots of −1 in Section 7with associated ring ℂ are of particular interest, because of the existence of classesof exceptional square roots of −1, which all include a nontrivial term in the centralelement of the respective algebra different from the identity. Section 7 thereforeincludes a detailed discussion of all classes of square roots of −1 in the algebras𝐶ℓ4,1, the isomorphic 𝐶ℓ0,5, and in 𝐶ℓ7,0. Finally, we add Appendix A with tablesof square roots of −1 for all Clifford algebras with 𝑛 = 5, 7, and 𝑠 = 3 (mod 4).

7. Square Roots of −1 in Real Clifford Algebras 125

Figure 1. Manifolds of square roots 𝑓 of −1 in 𝐶ℓ2,0 (left), 𝐶ℓ1,1 (cen-ter), and 𝐶ℓ0,2 ∼= ℍ (right). The square roots are 𝑓 = 𝛼+ 𝑏1𝑒1 + 𝑏2𝑒2 +𝛽𝑒12, with 𝛼, 𝑏1, 𝑏2, 𝛽 ∈ ℝ, 𝛼 = 0, and 𝛽2 = 𝑏21𝑒

22 + 𝑏22𝑒

21 + 𝑒21𝑒

22.

The square roots of −1 in Section 7 and in Appendix A were all computed withthe Maple package CLIFFORD [2], as explained in Appendix B.

2. Background and Problem Formulation

Let 𝐶ℓ𝑝,𝑞 be the algebra (associative with unit 1) generated overℝ by 𝑝+𝑞 elements𝑒𝑘 (with 𝑘 = 1, 2, . . . , 𝑝 + 𝑞) with the relations 𝑒2𝑘 = 1 if 𝑘 ≤ 𝑝, 𝑒2𝑘 = −1 if 𝑘 > 𝑝and 𝑒ℎ𝑒𝑘 + 𝑒𝑘𝑒ℎ = 0 whenever ℎ ∕= 𝑘, see [28]. We set the vector space dimension𝑛 = 𝑝+ 𝑞 and the signature 𝑠 = 𝑝− 𝑞. This algebra has dimension 2𝑛, and its evensubalgebra 𝐶ℓ0(𝑝, 𝑞) has dimension 2𝑛−1 (if 𝑛 > 0). We are concerned with squareroots of −1 contained in 𝐶ℓ𝑝,𝑞 or 𝐶ℓ0(𝑝, 𝑞). If the dimension of 𝐶ℓ𝑝,𝑞 or, 𝐶ℓ0(𝑝, 𝑞)is ≤ 2, it is isomorphic to ℝ ∼= 𝐶ℓ0,0, ℝ

2 ∼= 𝐶ℓ1,0, or ℂ ∼= 𝐶ℓ0,1, and it is clear thatthere is no square root of −1 in ℝ and ℝ2 = ℝ×ℝ, and that there are two squaresroots 𝑖 and −𝑖 in ℂ. Therefore we only consider algebras of dimension ≥ 4. Squareroots of −1 have been computed explicitly in [33] for 𝐶ℓ3,0, and in [20] for algebrasof dimensions 2𝑛 ≤ 16.

An algebra 𝐶ℓ𝑝,𝑞 or 𝐶ℓ0(𝑝, 𝑞) of dimension ≥ 4 is isomorphic to one of thefive matrix algebras:ℳ(2𝑑,ℝ),ℳ(𝑑,ℍ),ℳ(2𝑑,ℝ2),ℳ(𝑑,ℍ2) orℳ(2𝑑,ℂ). Theinteger 𝑑 depends on 𝑛. According to the parity of 𝑛, it is either 2(𝑛−2)/2 or 2(𝑛−3)/2

for 𝐶ℓ𝑝,𝑞, and, either 2(𝑛−4)/2 or 2(𝑛−3)/2 for 𝐶ℓ0(𝑝, 𝑞). The associated ring (either

ℝ, ℍ, ℝ2, ℍ2, or ℂ) depends on 𝑠 in this way2:

2Compare Chapter 16 on matrix representations and periodicity of 8, as well as Table 1 on p.217 of [28].

126 E. Hitzer, J. Helmstetter and R. Ablamowicz

𝑠 mod 8 0 1 2 3 4 5 6 7

associated ring for 𝐶ℓ𝑝,𝑞 ℝ ℝ2 ℝ ℂ ℍ ℍ2 ℍ ℂ

associated ring for 𝐶ℓ0(𝑝, 𝑞) ℝ2 ℝ ℂ ℍ ℍ2 ℍ ℂ ℝ

Therefore we shall answer the following question: what can we say about the squareroots of −1 in an algebra 𝒜 that is isomorphic toℳ(2𝑑,ℝ),ℳ(𝑑,ℍ),ℳ(2𝑑,ℝ2),ℳ(𝑑,ℍ2), or, ℳ(2𝑑,ℂ)? They constitute an algebraic submanifold in 𝒜; howmany connected components3 (for the usual topology) does it contain? Which aretheir dimensions? This submanifold is invariant by the action of the group Inn(𝒜)of inner automorphisms4 of 𝒜, i.e., for every 𝑟 ∈ 𝒜, 𝑟2 = −1⇒ 𝑓(𝑟)2 = −1 ∀𝑓 ∈Inn(𝒜). The orbits of Inn(𝒜) are called conjugacy classes5; how many conjugacyclasses are there in this submanifold? If the associated ring is ℝ2 or ℍ2 or ℂ, thegroup Aut(𝒜) of all automorphisms of 𝒜 is larger than Inn(𝒜), and the action ofAut(𝒜) in this submanifold shall also be described.

We recall some properties of 𝒜 that do not depend on the associated ring.The group Inn(𝒜) contains as many connected components as the group G(𝒜) ofinvertible elements in 𝒜. We recall that this assertion is true for ℳ(2𝑑,ℝ) butnot for ℳ(2𝑑 + 1,ℝ) which is not one of the relevant matrix algebras. If 𝑓 is anelement of 𝒜, let Cent(𝑓) be the centralizer of 𝑓 , that is, the subalgebra of all𝑔 ∈ 𝒜 such that 𝑓𝑔 = 𝑔𝑓 . The conjugacy class of 𝑓 contains as many connectedcomponents6 as G(𝒜) if (and only if) Cent(𝑓)

∩G(𝒜) is contained in the neutral7

connected component of G(𝒜), and the dimension of its conjugacy class is

dim(𝒜)− dim(Cent(𝑓)). (2.1)

Note that for invertible 𝑔 ∈ Cent(𝑓) we have 𝑔−1𝑓𝑔 = 𝑓 .Besides, let Z(𝒜) be the center of 𝒜, and let [𝒜,𝒜] be the subspace spanned

by all [𝑓, 𝑔] = 𝑓𝑔 − 𝑔𝑓 . In all cases 𝒜 is the direct sum of Z(𝒜) and [𝒜,𝒜]. For

3Two points are in the same connected component of a manifold, if they can be joined by acontinuous path inside the manifold under consideration. (This applies to all topological spacessatisfying the property that each neighborhood of any point contains a neighborhood in whichevery pair of points can always be joined by a continuous path.)4An inner automorphism 𝑓 of 𝒜 is defined as 𝑓 : 𝒜 → 𝒜, 𝑓(𝑥) = 𝑎−1𝑥𝑎, ∀𝑥 ∈ 𝒜, with givenfixed 𝑎 ∈ 𝒜. The composition of two inner automorphisms 𝑔(𝑓(𝑥)) = 𝑏−1𝑎−1𝑥𝑎𝑏 = (𝑎𝑏)−1𝑥(𝑎𝑏)is again an inner automorphism. With this operation the inner automorphisms form the groupInn(𝒜), compare [35].5The conjugacy class (similarity class) of a given 𝑟 ∈ 𝒜, 𝑟2 = −1 is {𝑓(𝑟) : 𝑓 ∈ Inn(𝒜)}, compare[34]. Conjugation is transitive, because the composition of inner automorphisms is again an innerautomorphism.6According to the general theory of groups acting on sets, the conjugacy class (as a topologicalspace) of a square root 𝑓 of −1 is isomorphic to the quotient of G(𝒜) and Cent(𝑓) (the subgroupof stability of 𝑓). Quotient means here the set of left handed classes modulo the subgroup. Ifthe subgroup is contained in the neutral connected component of G(𝒜), then the number of

connected components is the same in the quotient as in G(𝒜). See also [10].7Neutral means to be connected to the identity element of 𝒜.

7. Square Roots of −1 in Real Clifford Algebras 127

example,8 Z(ℳ(2𝑑,ℝ)) = {𝑎1 ∣ 𝑎 ∈ ℝ} and Z(ℳ(2𝑑,ℂ)) = {𝑐1 ∣ 𝑐 ∈ ℂ}. If theassociated ring is ℝ or ℍ (that is for even 𝑛), then Z(𝒜) is canonically isomorphicto ℝ, and from the projection 𝒜 → Z(𝒜) we derive a linear form Scal : 𝒜 → ℝ.When the associated ring9 is ℝ2 or ℍ2 or ℂ, then Z(𝒜) is spanned by 1 (the unitmatrix10) and some element 𝜔 such that 𝜔2 = ±1. Thus, we get two linear formsScal and Spec such that Scal(𝑓)1+Spec(𝑓)𝜔 is the projection of 𝑓 in Z(𝒜) for every𝑓 ∈ 𝒜. Instead of 𝜔 we may use −𝜔 and replace Spec with −Spec. The followingassertion holds for every 𝑓 ∈ 𝒜: The trace of each multiplication11 𝑔 �→ 𝑓𝑔 or𝑔 �→ 𝑔𝑓 is equal to the product

tr(𝑓) = dim(𝒜) Scal(𝑓). (2.2)

The word “trace” (when nothing more is specified) means a matrix trace in ℝ,which is the sum of its diagonal elements. For example, the matrix 𝑀 ∈ ℳ(2𝑑,ℝ)

with elements 𝑚𝑘𝑙 ∈ ℝ, 1 ≤ 𝑘, 𝑙 ≤ 2𝑑 has the trace tr(𝑀) =∑2𝑑

𝑘=1 𝑚𝑘𝑘 [24].

We shall prove that in all cases Scal(𝑓) = 0 for every square root of −1 in𝒜. Then, we may distinguish ordinary square roots of −1, and exceptional ones.In all cases the ordinary square roots of −1 constitute a unique12 conjugacy classof dimension dim(𝒜)/2 which has as many connected components as G(𝒜), andthey satisfy the equality Spec(𝑓) = 0 if the associated ring is ℝ2 or ℍ2 or ℂ. Theexceptional square roots of −1 only exist13 if 𝒜 ∼= ℳ(2𝑑,ℂ). In ℳ(2𝑑,ℂ) thereare 2𝑑 conjugacy classes of exceptional square roots of −1, each one characterizedby an equality Spec(𝑓) = 𝑘/𝑑 with ±𝑘 ∈ {1, 2, . . . , 𝑑} [see Section 7], and theirdimensions are< dim(𝒜)/2 [see equation (7.5)]. For instance, 𝜔 (mentioned above)and −𝜔 are central square roots of −1 inℳ(2𝑑,ℂ) which constitute two conjugacyclasses of dimension 0. Obviously, Spec(𝜔) = 1.

For symbolic computer algebra systems (CAS), like MAPLE, there exist Clif-ford algebra packages, e.g., CLIFFORD [2], which can compute idempotents [3]and square roots of −1. This will be of especial interest for the exceptional squareroots of −1 in ℳ(2𝑑,ℂ).

Regarding a square root 𝑟 of −1, a Clifford algebra is the direct sum of thesubspaces Cent(𝑟) (all elements that commute with 𝑟) and the skew-centralizer

8A matrix algebra based proof is, e.g., given in [4].9This is the case for 𝑛 (and 𝑠) odd. Then the pseudoscalar 𝜔 ∈ 𝐶ℓ𝑝,𝑞 is also in Z(𝐶ℓ𝑝,𝑞).10The number 1 denotes the unit of the Clifford algebra 𝒜, whereas the bold face 1 denotes theunit of the isomorphic matrix algebra ℳ.11These multiplications are bilinear over the center of 𝒜.12Let 𝒜 be an algebra ℳ(𝑚,𝕂) where 𝕂 is a division ring. Thus two elements 𝑓 and 𝑔 of 𝒜induce 𝕂-linear endomorphisms 𝑓 ′ and 𝑔′ on 𝕂𝑚; if 𝕂 is not commutative, 𝕂 operates on 𝕂𝑚

on the right side. The matrices 𝑓 and 𝑔 are conjugate (or similar) if and only if there are two𝕂-bases 𝐵1 and 𝐵2 of 𝕂𝑚 such that 𝑓 ′ operates on 𝐵1 in the same way as 𝑔′ operates on 𝐵2.This theorem allows us to recognize that in all cases but the last one (with exceptional squareroots of −1), two square roots of −1 are always conjugate.13The pseudoscalars of Clifford algebras whose isomorphic matrix algebra has ring ℝ2 or ℍ2

square to 𝜔2 = +1.

128 E. Hitzer, J. Helmstetter and R. Ablamowicz

SCent(𝑟) (all elements that anticommute with 𝑟). Every Clifford algebra multivec-tor has a unique split by this Lemma.

Lemma 2.1. Every multivector 𝐴 ∈ 𝐶ℓ𝑝,𝑞 has, with respect to a square root 𝑟 ∈𝐶ℓ𝑝,𝑞 of −1, i.e., 𝑟−1 = −𝑟, the unique decomposition

𝐴± =1

2(𝐴± 𝑟−1𝐴𝑟), 𝐴 = 𝐴+ +𝐴−, 𝐴+𝑟 = 𝑟𝐴+, 𝐴−𝑟 = −𝑟𝐴−. (2.3)

Proof. For 𝐴 ∈ 𝐶ℓ𝑝,𝑞 and a square root 𝑟 ∈ 𝐶ℓ𝑝,𝑞 of −1, we compute

𝐴±𝑟 =1

2(𝐴± 𝑟−1𝐴𝑟)𝑟 =

1

2(𝐴𝑟 ± 𝑟−1𝐴(−1)) 𝑟−1=−𝑟

=1

2(𝑟𝑟−1𝐴𝑟 ± 𝑟𝐴)

= ±𝑟1

2(𝐴± 𝑟−1𝐴𝑟). □

For example, in Clifford algebras 𝐶ℓ𝑛,0 [23] of dimensions 𝑛 = 2 mod 4,Cent(𝑟) is the even subalgebra 𝐶ℓ0(𝑛, 0) for the unit pseudoscalar 𝑟, and thesubspace 𝐶ℓ1(𝑛, 0) spanned by all 𝑘-vectors of odd degree 𝑘, is SCent(𝑟). Themost interesting case is ℳ(2𝑑,ℂ), where a whole range of conjugacy classes be-comes available. These results will therefore be particularly relevant for construct-ing Clifford–Fourier transformations using the square roots of −1.

3. Square Roots of −1 in 퓜(2𝒅,ℝ)

Here 𝒜 = ℳ(2𝑑,ℝ), whence dim(𝒜) = (2𝑑)2 = 4𝑑2. The group G(𝒜) has twoconnected components determined by the inequalities det(𝑔) > 0 and det(𝑔) < 0.

For the case 𝑑 = 1 we have, e.g., the algebra 𝐶ℓ2,0 isomorphic to ℳ(2,ℝ).The basis {1, 𝑒1, 𝑒2, 𝑒12} of 𝐶ℓ2,0 is mapped to{(

1 00 1

),

(0 11 0

),

(1 00 −1

),

(0 −11 0

)}.

The general element 𝛼+ 𝑏1𝑒1 + 𝑏2𝑒2 + 𝛽𝑒12 ∈ 𝐶ℓ2,0 is thus mapped to(𝛼+ 𝑏2 −𝛽 + 𝑏1𝛽 + 𝑏1 𝛼− 𝑏2

)(3.1)

in ℳ(2,ℝ). Every element 𝑓 of 𝒜 =ℳ(2𝑑,ℝ) is treated as an ℝ-linear endomor-phism of 𝑉 = ℝ2𝑑. Thus, its scalar component and its trace (2.2) are related asfollows: tr(𝑓) = 2𝑑Scal(𝑓). If 𝑓 is a square root of −1, it turns 𝑉 into a vectorspace over ℂ (if the complex number 𝑖 operates like 𝑓 on 𝑉 ). If (𝑒1, 𝑒2, . . . , 𝑒𝑑) isa ℂ-basis of 𝑉 , then (𝑒1, 𝑓(𝑒1), 𝑒2, 𝑓(𝑒2), . . . , 𝑒𝑑, 𝑓(𝑒𝑑)) is an ℝ-basis of 𝑉 , and the2𝑑× 2𝑑 matrix of 𝑓 in this basis is

diag

((0 −11 0

), . . . ,

(0 −11 0

)︸ ︷︷ ︸

𝑑

)(3.2)

Consequently all square roots of −1 in 𝒜 are conjugate. The centralizer of asquare root 𝑓 of −1 is the algebra of all ℂ-linear endomorphisms 𝑔 of 𝑉 (since

7. Square Roots of −1 in Real Clifford Algebras 129

𝑖 operates like 𝑓 on 𝑉 ). Therefore, the ℂ-dimension of Cent(𝑓) is 𝑑2 and itsℝ-dimension is 2𝑑2. Finally, the dimension (2.1) of the conjugacy class of 𝑓 isdim(𝒜) − dim(Cent(𝑓)) = 4𝑑2 − 2𝑑2 = 2𝑑2 = dim(𝒜)/2. The two connected com-ponents of G(𝒜) are determined by the sign of the determinant. Because of thenext lemma, the ℝ-determinant of every element of Cent(𝑓) is ≥ 0. Therefore, theintersection Cent(𝑓)

∩G(𝒜) is contained in the neutral connected component of

G(𝒜) and, consequently, the conjugacy class of 𝑓 has two connected componentslike G(𝒜). Because of the next lemma, the ℝ-trace of 𝑓 vanishes (indeed its ℂ-traceis 𝑑𝑖, because 𝑓 is the multiplication by the scalar 𝑖: 𝑓(𝑣) = 𝑖𝑣 for all 𝑣) whenceScal(𝑓) = 0. This equality is corroborated by the matrix written above.

We conclude that the square roots of −1 constitute one conjugacy class withtwo connected components of dimension dim(𝒜)/2 contained in the hyperplanedefined by the equation

Scal(𝑓) = 0. (3.3)

Before stating the lemma that here is so helpful, we show what happens inthe easiest case 𝑑 = 1. The square roots of −1 in ℳ(2,ℝ) are the real matrices(

𝑎 𝑐𝑏 −𝑎

)with

(𝑎 𝑐𝑏 −𝑎

)(𝑎 𝑐𝑏 −𝑎

)= (𝑎2 + 𝑏𝑐)1 = −1; (3.4)

hence 𝑎2 + 𝑏𝑐 = −1, a relation between 𝑎, 𝑏, 𝑐 which is equivalent to (𝑏 − 𝑐)2 =(𝑏 + 𝑐)2 + 4𝑎2 + 4 ⇒ (𝑏 − 𝑐)2 ≥ 4 ⇒ 𝑏 − 𝑐 ≥ 2 (one component) or 𝑐 − 𝑏 ≥ 2(second component). Thus, we recognize the two connected components of squareroots of −1: The inequality 𝑏 ≥ 𝑐+ 2 holds in one connected component, and theinequality 𝑐 ≥ 𝑏+ 2 in the other one, compare Figure 2.

Figure 2. Two components of square roots of −1 in ℳ(2,ℝ).

In terms of 𝐶ℓ2,0 coefficients (3.1) with 𝑏− 𝑐 = 𝛽 + 𝑏1 − (−𝛽 + 𝑏1) = 2𝛽, weget the two component conditions simply as

𝛽 ≥ 1 (one component), 𝛽 ≤ −1 (second component). (3.5)

Rotations (det(𝑔) = 1) leave the pseudoscalar 𝛽𝑒12 invariant (and thus preserve thetwo connected components of square roots of −1), but reflections (det(𝑔′) = −1)change its sign 𝛽𝑒12 → −𝛽𝑒12 (thus interchanging the two components).

130 E. Hitzer, J. Helmstetter and R. Ablamowicz

Because of the previous argument involving a complex structure on the realspace 𝑉 , we conversely consider the complex space ℂ𝑑 with its structure of vectorspace over ℝ. If (𝑒1, 𝑒2, . . . , 𝑒𝑑) is a ℂ-basis of ℂ𝑑, then (𝑒1, 𝑖𝑒1, 𝑒2, 𝑖𝑒2, . . . , 𝑒𝑑, 𝑖𝑒𝑑)is an ℝ-basis. Let 𝑔 be a ℂ-linear endomorphism of ℂ𝑑 (i.e., a complex 𝑑 × 𝑑matrix), let trℂ(𝑔) and detℂ(𝑔) be the trace and determinant of 𝑔 in ℂ, and trℝ(𝑔)and detℝ(𝑔) its trace and determinant for the real structure of ℂ𝑑.

Example. For 𝑑 = 1 an endomorphism of ℂ1 is given by a complex number 𝑔 =𝑎+ 𝑖𝑏, 𝑎, 𝑏 ∈ ℝ. Its matrix representation is according to (3.2)(

𝑎 −𝑏𝑏 𝑎

)with

(𝑎 −𝑏𝑏 𝑎

)2

= (𝑎2 − 𝑏2)

(1 00 1

)+ 2𝑎𝑏

(0 −11 0

). (3.6)

Then we have trℂ(𝑔) = 𝑎 + 𝑖𝑏, trℝ

(𝑎 −𝑏𝑏 𝑎

)= 2𝑎 = 2ℜ(trℂ(𝑔)) and detℂ(𝑔) =

𝑎+ 𝑖𝑏, detℝ

(𝑎 −𝑏𝑏 𝑎

)= 𝑎2 + 𝑏2 = ∣ detℂ(𝑔)∣2 ≥ 0.

Lemma 3.1. For every ℂ-linear endomorphism 𝑔 we can write trℝ(𝑔) = 2ℜ(trℂ(𝑔))and detℝ(𝑔) = ∣ detℂ(𝑔)∣2 ≥ 0.

Proof. There is a ℂ-basis in which the ℂ-matrix of 𝑔 is triangular [then detℂ(𝑔)is the product of the entries of 𝑔 on the main diagonal]. We get the ℝ-matrixof 𝑔 in the derived ℝ-basis by replacing every entry 𝑎 + 𝑏𝑖 of the ℂ-matrix with

the elementary matrix

(𝑎 −𝑏𝑏 𝑎

). The conclusion soon follows. The fact that the

determinant of a block triangular matrix is the product of the determinants of theblocks on the main diagonal is used. □

4. Square Roots of −1 in 퓜(2𝒅,ℝ2)

Here 𝒜 = ℳ(2𝑑,ℝ2) = ℳ(2𝑑,ℝ) × ℳ(2𝑑,ℝ), whence dim(𝒜) = 8𝑑2. Thegroup G(𝒜) has four14 connected components. Every element (𝑓, 𝑓 ′) ∈ 𝒜 (with𝑓, 𝑓 ′ ∈ ℳ(2𝑑,ℝ)) has a determinant in ℝ2 which is obviously (det(𝑓), det(𝑓 ′)),and the four connected components of G(𝒜) are determined by the signs of thetwo components of detℝ2(𝑓, 𝑓 ′).

The lowest-dimensional example (𝑑 = 1) is 𝐶ℓ2,1 isomorphic to ℳ(2,ℝ2).Here the pseudoscalar 𝜔 = 𝑒123 has square 𝜔2 = +1. The center of the algebra is{1, 𝜔} and includes the idempotents 𝜖± = (1±𝜔)/2, 𝜖2± = 𝜖±, 𝜖+𝜖− = 𝜖−𝜖+ = 0.The basis of the algebra can thus be written as {𝜖+, 𝑒1𝜖+, 𝑒2𝜖+, 𝑒12𝜖+, 𝜖−, 𝑒1𝜖−,𝑒2𝜖−, 𝑒12𝜖−}, where the first (and the last) four elements form a basis of the

14In general, the number of connected components of G(𝒜) is two if 𝒜 = ℳ(𝑚,ℝ), and one if𝒜 = ℳ(𝑚,ℂ) or 𝒜 = ℳ(𝑚,ℍ), because in all cases every matrix can be joined by a continuouspath to a diagonal matrix with entries 1 or −1. When an algebra 𝒜 is a direct product of two

algebras ℬ and 𝒞, then G(𝒜) is the direct product of G(ℬ) and G(𝒞), and the number of connectedcomponents of G(𝒜) is the product of the numbers of connected components of G(ℬ) and G(𝒞).

7. Square Roots of −1 in Real Clifford Algebras 131

subalgebra 𝐶ℓ2,0 isomorphic toℳ(2,ℝ). In terms of matrices we have the identitymatrix (1,1) representing the scalar part, the idempotent matrices (1, 0), (0,1),and the 𝜔 matrix (1,−1), with 1 the unit matrix of ℳ(2,ℝ).

The square roots of (−1,−1) in 𝒜 are pairs of two square roots of −1 inℳ(2𝑑,ℝ). Consequently they constitute a unique conjugacy class with four con-nected components of dimension 4𝑑2 = dim(𝒜)/2. This number can be obtained intwo ways. First, since every element (𝑓, 𝑓 ′) ∈ 𝒜 (with 𝑓, 𝑓 ′ ∈ℳ(2𝑑,ℝ)) has twicethe dimension of the components 𝑓 ∈ℳ(2𝑑,ℝ) of Section 3, we get the componentdimension 2⋅2𝑑2 = 4𝑑2. Second, the centralizer Cent(𝑓, 𝑓 ′) has twice the dimensionof Cent(𝑓) ofℳ(2𝑑,ℝ), therefore dim(𝒜)−Cent(𝑓, 𝑓 ′) = 8𝑑2 − 4𝑑2 = 4𝑑2. In theabove example for 𝑑 = 1 the four components are characterized according to (3.5)by the values of the coefficients of 𝛽𝑒12𝜖+ and 𝛽′𝑒12𝜖− as

𝑐1 : 𝛽 ≥ 1, 𝛽′ ≥ 1,

𝑐2 : 𝛽 ≥ 1, 𝛽′ ≤ −1,𝑐3 : 𝛽 ≤ −1, 𝛽′ ≥ 1,

𝑐4 : 𝛽 ≤ −1, 𝛽′ ≤ −1. (4.1)

For every (𝑓, 𝑓 ′) ∈ 𝒜 we can with (2.2) write tr(𝑓) + tr(𝑓 ′) = 2𝑑Scal(𝑓, 𝑓 ′) and

tr(𝑓)− tr(𝑓 ′) = 2𝑑Spec(𝑓, 𝑓 ′) if 𝜔 = (1,−1); (4.2)

whence Scal(𝑓, 𝑓 ′) = Spec(𝑓, 𝑓 ′) = 0 if (𝑓, 𝑓 ′) is a square root of (−1,−1), compare(3.3).

The group Aut(𝒜) is larger than Inn(𝒜), because it contains the swap auto-morphism (𝑓, 𝑓 ′) �→ (𝑓 ′, 𝑓) which maps the central element 𝜔 to −𝜔, and inter-changes the two idempotents 𝜖+ and 𝜖−. The group Aut(𝒜) has eight connectedcomponents which permute the four connected components of the submanifold ofsquare roots of (−1,−1). The permutations induced by Inn(𝒜) are the permu-tations of the Klein group. For example for 𝑑 = 1 of (4.1) we get the followingInn(ℳ(2,ℝ2)) permutations

det(𝑔) > 0, det(𝑔′) > 0 : identity,

det(𝑔) > 0, det(𝑔′) < 0 : (𝑐1, 𝑐2), (𝑐3, 𝑐4),

det(𝑔) < 0, det(𝑔′) > 0 : (𝑐1, 𝑐3), (𝑐2, 𝑐4),

det(𝑔) < 0, det(𝑔′) < 0 : (𝑐1, 𝑐4), (𝑐2, 𝑐3). (4.3)

Beside the identity permutation, Inn(𝒜) gives the three permutations that permutetwo elements and also the other two ones.

The automorphisms outside Inn(𝒜) are(𝑓, 𝑓 ′) �→ (𝑔𝑓 ′𝑔−1, 𝑔′𝑓𝑔′−1) for some (𝑔, 𝑔′) ∈ G(𝒜). (4.4)

If det(𝑔) and det(𝑔′) have opposite signs, it is easy to realize that this automor-phism induces a circular permutation on the four connected components of squareroots of (−1,−1): If det(𝑔) and det(𝑔′) have the same sign, this automorphismleaves globally invariant two connected components, and permutes the other two

132 E. Hitzer, J. Helmstetter and R. Ablamowicz

ones. For example, for 𝑑 = 1 the automorphisms (4.4) outside Inn(𝒜) permute thecomponents (4.1) of square roots of (−1,−1) in ℳ(2,ℝ2) as follows

det(𝑔) > 0, det(𝑔′) > 0 : (𝑐1), (𝑐2, 𝑐3), (𝑐4),

det(𝑔) > 0, det(𝑔′) < 0 : 𝑐1 → 𝑐2 → 𝑐4 → 𝑐3 → 𝑐1,

det(𝑔) < 0, det(𝑔′) > 0 : 𝑐1 → 𝑐3 → 𝑐4 → 𝑐2 → 𝑐1,

det(𝑔) < 0, det(𝑔′) < 0 : (𝑐1, 𝑐4), (𝑐2), (𝑐3). (4.5)

Consequently, the quotient of the group Aut(𝒜) by its neutral connected compo-nent is isomorphic to the group of isometries of a square in a Euclidean plane.

5. Square Roots of −1 in 퓜(𝒅,ℍ)

Let us first consider the easiest case 𝑑 = 1, when 𝒜 = ℍ, e.g., of 𝐶ℓ0,2. The squareroots of −1 in ℍ are the quaternions 𝑎𝑖 + 𝑏𝑗 + 𝑐𝑖𝑗 with 𝑎2 + 𝑏2 + 𝑐2 = 1. Theyconstitute a compact and connected manifold of dimension 2. Every square root𝑓 of −1 is conjugate with 𝑖, i.e., there exists 𝑣 ∈ ℍ : 𝑣−1𝑓𝑣 = 𝑖 ⇔ 𝑓𝑣 = 𝑣𝑖. If weset 𝑣 = −𝑓𝑖+ 1 = 𝑎+ 𝑏𝑖𝑗 − 𝑐𝑗 + 1 we have

𝑓𝑣 = −𝑓2𝑖+ 𝑓 = 𝑓 + 𝑖 = (𝑓(−𝑖) + 1)𝑖 = 𝑣𝑖.

𝑣 is invertible, except when 𝑓 = −𝑖. But 𝑖 is conjugate with −𝑖 because 𝑖𝑗 = 𝑗(−𝑖),hence, by transitivity 𝑓 is also conjugate with −𝑖.

Here 𝒜 = ℳ(𝑑,ℍ), whence dim(𝐴) = 4𝑑2. The ring ℍ is the algebra overℝ generated by two elements 𝑖 and 𝑗 such that 𝑖2 = 𝑗2 = −1 and 𝑗𝑖 = −𝑖𝑗. Weidentify ℂ with the subalgebra generated by15 𝑖 alone.

The group G(𝒜) has only one connected component. We shall soon prove thatevery square root of −1 in 𝒜 is conjugate with 𝑖1. Therefore, the submanifold ofsquare roots of −1 is a conjugacy class, and it is connected. The centralizer of𝑖1 in 𝒜 is the subalgebra of all matrices with entries in ℂ. The ℂ-dimension ofCent(𝑖1) is 𝑑2, its ℝ-dimension is 2𝑑2, and, consequently, the dimension (2.1) ofthe submanifold of square roots of −1 is 4𝑑2 − 2𝑑2 = 2𝑑2 = dim(𝒜)/2.

Here 𝑉 = ℍ𝑑 is treated as a (unitary) module over ℍ on the right side:The product of a line vector 𝑡𝑣 = (𝑥1, 𝑥2, . . . , 𝑥𝑑) ∈ 𝑉 by 𝑦 ∈ ℍ is 𝑡𝑣 𝑦 =(𝑥1𝑦, 𝑥2𝑦, . . . , 𝑥𝑑𝑦). Thus, every 𝑓 ∈ 𝒜 determines an ℍ-linear endomorphism of𝑉 : The matrix 𝑓 multiplies the column vector 𝑣 = 𝑡(𝑥1, 𝑥2, . . . , 𝑥𝑑) on the leftside 𝑣 �→ 𝑓𝑣. Since ℂ is a subring of ℍ, 𝑉 is also a vector space of dimension 2𝑑over ℂ. The scalar 𝑖 always operates on the right side (like every scalar in ℍ). If(𝑒1, 𝑒2, . . . , 𝑒𝑑) is an ℍ-basis of 𝑉 , then (𝑒1, 𝑒1𝑗, 𝑒2, 𝑒2𝑗, . . . , 𝑒𝑑, 𝑒𝑑𝑗) is a ℂ-basis of𝑉 . Let 𝑓 be a square root of −1, then the eigenvalues of 𝑓 in ℂ are +𝑖 or −𝑖.If we treat 𝑉 as a 2𝑑 vector space over ℂ, it is the direct (ℂ-linear) sum of theeigenspaces

𝑉 + = {𝑣 ∈ 𝑉 ∣ 𝑓(𝑣) = 𝑣𝑖} and 𝑉 − = {𝑣 ∈ 𝑉 ∣ 𝑓(𝑣) = −𝑣𝑖}, (5.1)

15This choice is usual and convenient.

7. Square Roots of −1 in Real Clifford Algebras 133

representing 𝑓 as a 2𝑑× 2𝑑 ℂ-matrix w.r.t. the ℂ-basis of 𝑉 , with ℂ-scalar eigen-values (multiplied from the right): 𝜆± = ±𝑖.

Since 𝑖𝑗 = −𝑗𝑖, the multiplication 𝑣 �→ 𝑣𝑗 permutes 𝑉 + and 𝑉 −, as 𝑓(𝑣) =±𝑣𝑖 is mapped to 𝑓(𝑣)𝑗 = ±𝑣𝑖𝑗 = ∓(𝑣𝑗)𝑖. Therefore, if (𝑒1, 𝑒2, . . . , 𝑒𝑟) is a ℂ-basisof 𝑉 +, then (𝑒1𝑗, 𝑒2𝑗, . . . , 𝑒𝑟𝑗) is a ℂ-basis of 𝑉 −, consequently (𝑒1, 𝑒1𝑗, 𝑒2, 𝑒2𝑗,. . ., 𝑒𝑟, 𝑒𝑟𝑗) is a ℂ-basis of 𝑉 , and (𝑒1, 𝑒2, . . . , 𝑒𝑟=𝑑) is an ℍ-basis of 𝑉 . Since 𝑓 by𝑓(𝑒𝑘) = 𝑒𝑘𝑖 for 𝑘 = 1, 2, . . . , 𝑑 operates on the ℍ-basis (𝑒1, 𝑒2, . . . , 𝑒𝑑) in the sameway as 𝑖1 on the natural ℍ-basis of 𝑉 , we conclude that 𝑓 and 𝑖1 are conjugate.

Besides, Scal(𝑖1) = 0 because 2𝑖1 = [𝑗1, 𝑖𝑗1] ∈ [𝒜,𝒜], thus 𝑖1 /∈ Z(𝒜).Whence,16

Scal(𝑓) = 0 for every square root of − 1. (5.2)

These results are easily verified in the above example of 𝑑 = 1 when 𝒜 = ℍ.

6. Square Roots of −1 in 퓜(𝒅,ℍ2)

Here, 𝒜 = ℳ(𝑑,ℍ2) = ℳ(𝑑,ℍ) ×ℳ(𝑑,ℍ), whence dim(𝐴) = 8𝑑2. The groupG(𝒜) has only one connected component (see Footnote 14).

The square roots of (−1,−1) in 𝒜 are pairs of two square roots of −1 inℳ(𝑑,ℍ). Consequently, they constitute a unique conjugacy class which is con-nected and its dimension is 2× 2𝑑2 = 4𝑑2 = dim(𝒜)/2.

For every (𝑓, 𝑓 ′) ∈ 𝒜 we can write Scal(𝑓) + Scal(𝑓 ′) = 2 Scal(𝑓, 𝑓 ′) and,similarly to (4.2),

Scal(𝑓)− Scal(𝑓 ′) = 2 Spec(𝑓, 𝑓 ′) if 𝜔 = (1,−1); (6.1)

whence Scal(𝑓, 𝑓 ′) = Spec(𝑓, 𝑓 ′) = 0 if (𝑓, 𝑓 ′) is a square root of (−1,−1), comparewith (5.2).

The group Aut(𝒜) has two17 connected components; the neutral componentis Inn(𝒜), and the other component contains the swap automorphism (𝑓, 𝑓 ′) �→(𝑓 ′, 𝑓).

The simplest example is 𝑑 = 1, 𝒜 = ℍ2, where we have the identity pair(1, 1) representing the scalar part, the idempotents (1, 0), (0, 1), and 𝜔 as the pair(1,−1).

𝒜 = ℍ2 is isomorphic to 𝐶ℓ0,3. The pseudoscalar 𝜔 = 𝑒123 has the square𝜔2 = +1. The center of the algebra is {1, 𝜔}, and includes the idempotents 𝜖± =12 (1±𝜔), 𝜖2± = 𝜖±, 𝜖+𝜖− = 𝜖−𝜖+ = 0. The basis of the algebra can thus be writtenas {𝜖+, 𝑒1𝜖+, 𝑒2𝜖+, 𝑒12𝜖+, 𝜖−, 𝑒1𝜖−, 𝑒2𝜖−, 𝑒12𝜖−} where the first (and the last) fourelements form a basis of the subalgebra 𝐶ℓ0,2 isomorphic to ℍ.

16Compare the definition of Scal(𝑓) in Section 2, remembering that in the current section the

associated ring is ℍ.17Compare Footnote 14.

134 E. Hitzer, J. Helmstetter and R. Ablamowicz

7. Square Roots of −1 in 퓜(2𝒅,ℂ)

The lowest-dimensional example for 𝑑 = 1 is the Pauli matrix algebra𝒜 =ℳ(2,ℂ)isomorphic to the geometric algebra 𝐶ℓ3,0 of the 3D Euclidean space and 𝐶ℓ1,2.The 𝐶ℓ3,0 vectors 𝑒1, 𝑒2, 𝑒3 correspond one-to-one to the Pauli matrices

𝜎1 =

(0 11 0

), 𝜎2 =

(0 −𝑖𝑖 0

), 𝜎3 =

(1 00 −1

), (7.1)

with 𝜎1𝜎2 = 𝑖𝜎3 =

(𝑖 00 −𝑖

). The element 𝜔 = 𝜎1𝜎2𝜎3 = 𝑖1 represents the

central pseudoscalar 𝑒123 of 𝐶ℓ3,0 with square 𝜔2 = −1. The Pauli algebra has thefollowing idempotents

𝜖1 = 𝜎21 = 1, 𝜖0 = (1/2)(1+ 𝜎3), 𝜖−1 = 0 . (7.2)

The idempotents correspond via

𝑓 = 𝑖(2𝜖− 1), (7.3)

to the square roots of −1:𝑓1 = 𝑖1 =

(𝑖 00 𝑖

), 𝑓0 = 𝑖𝜎3 =

(𝑖 00 −𝑖

), 𝑓−1 = −𝑖1 =

(−𝑖 00 −𝑖

), (7.4)

where by complex conjugation 𝑓−1 = 𝑓1. Let the idempotent 𝜖′0 = 12 (1−𝜎3) corre-

spond to the matrix 𝑓 ′0 = −𝑖𝜎3. We observe that 𝑓0 is conjugate to 𝑓 ′0 = 𝜎−11 𝑓0𝜎1 =

𝜎1𝜎2 = 𝑓0 using 𝜎−11 = 𝜎1 but 𝑓1 is not conjugate to 𝑓−1. Therefore, only 𝑓1, 𝑓0, 𝑓−1

lead to three distinct conjugacy classes of square roots of −1 inℳ(2,ℂ). CompareAppendix B for the corresponding computations with CLIFFORD for Maple.

In general, if 𝒜 = ℳ(2𝑑,ℂ), then dim(𝒜) = 8𝑑2. The group G(𝒜) has oneconnected component. The square roots of −1 in 𝒜 are in bijection with theidempotents 𝜖 [3] according to (7.3). According18 to (7.3) and its inverse 𝜖 =12 (1 − 𝑖𝑓) the square root of −1 with Spec(𝑓−) = 𝑘/𝑑 = −1, i.e., 𝑘 = −𝑑 (seebelow), always corresponds to the trival idempotent 𝜖− = 0, and the square rootof −1 with Spec(𝑓+) = 𝑘/𝑑 = +1, 𝑘 = +𝑑, corresponds to the identity idempotent𝜖+ = 1.

If 𝑓 is a square root of−1, then 𝑉 = ℂ2𝑑 is the direct sum of the eigenspaces19

associated with the eigenvalues 𝑖 and −𝑖. There is an integer 𝑘 such that thedimensions of the eigenspaces are respectively 𝑑 + 𝑘 and 𝑑 − 𝑘. Moreover, −𝑑 ≤𝑘 ≤ 𝑑. Two square roots of −1 are conjugate if and only if they give the same

18On the other hand it is clear that complex conjugation always leads to 𝑓− = 𝑓+, wherethe overbar means complex conjugation in ℳ(2𝑑,ℂ) and Clifford conjugation in the isomorphicClifford algebra 𝐶ℓ𝑝,𝑞. So either the trivial idempotent 𝜖− = 0 is included in the bijection (7.3) ofidempotents and square roots of −1, or alternatively the square root of −1 with Spec(𝑓−) = −1is obtained from 𝑓− = 𝑓+.19The following theorem is sufficient for a matrix 𝑓 in ℳ(𝑚,𝕂), if 𝕂 is a (commutative) field.The matrix 𝑓 is diagonalizable if and only if 𝑃 (𝑓) = 0 for some polynomial 𝑃 that has only simpleroots, all of them in the field 𝕂. (This implies that 𝑃 is a multiple of the minimal polynomial,but we do not need to know whether 𝑃 is or is not the minimal polynomial.)

7. Square Roots of −1 in Real Clifford Algebras 135

integer 𝑘. Then, all elements of Cent(𝑓) consist of diagonal block matrices with 2square blocks of (𝑑+𝑘)×(𝑑+𝑘) matrices and (𝑑−𝑘)×(𝑑−𝑘) matrices. Therefore,the ℂ-dimension of Cent(𝑓) is (𝑑+ 𝑘)2 + (𝑑− 𝑘)2. Hence the ℝ-dimension (2.1) ofthe conjugacy class of 𝑓 :

8𝑑2 − 2(𝑑+ 𝑘)2 − 2(𝑑− 𝑘)2 = 4(𝑑2 − 𝑘2). (7.5)

Also, from the equality tr(𝑓) = (𝑑+𝑘)𝑖−(𝑑−𝑘)𝑖 = 2𝑘𝑖 we deduce that Scal(𝑓) = 0and that Spec(𝑓) = (2𝑘𝑖)/(2𝑑𝑖) = 𝑘/𝑑 if 𝜔 = 𝑖1 (whence tr(𝜔) = 2𝑑𝑖).

As announced on page 127, we consider that a square root of −1 is ordinaryif the associated integer 𝑘 vanishes, and that it is exceptional if 𝑘 ∕= 0. Thus thefollowing assertion is true in all cases: the ordinary square roots of −1 in 𝒜 con-stitute one conjugacy class of dimension dim(𝒜)/2 which has as many connectedcomponents as G(𝒜), and the equality Spec(𝑓) = 0 holds for every ordinary squareroot of −1 when the linear form Spec exists. All conjugacy classes of exceptionalsquare roots of −1 have a dimension < dim(𝒜)/2.

All square roots of −1 in ℳ(2𝑑,ℂ) constitute (2𝑑 + 1) conjugacy classes20

which are also the connected components of the submanifold of square roots of −1because of the equality Spec(𝑓) = 𝑘/𝑑, which is conjugacy class specific.

When 𝒜 = ℳ(2𝑑,ℂ), the group Aut(𝒜) is larger than Inn(𝒜) since it con-tains the complex conjugation (that maps every entry of a matrix to the conjugatecomplex number). It is clear that the class of ordinary square roots of −1 is invari-ant by complex conjugation. But the class associated with an integer 𝑘 other than0 is mapped by complex conjugation to the class associated with −𝑘. In particularthe complex conjugation maps the class {𝜔} (associated with 𝑘 = 𝑑) to the class{−𝜔} associated with 𝑘 = −𝑑.

All these observations can easily verified for the above example of 𝑑 = 1 ofthe Pauli matrix algebra 𝒜 = ℳ(2,ℂ). For 𝑑 = 2 we have the isomorphism of𝒜 = ℳ(4,ℂ) with 𝐶ℓ0,5, 𝐶ℓ2,3 and 𝐶ℓ4,1. While 𝐶ℓ0,5 is important in Cliffordanalysis, 𝐶ℓ4,1 is both the geometric algebra of the Lorentz space ℝ4,1 and theconformal geometric algebra of 3D Euclidean geometry. Its set of square roots of−1 is therefore of particular practical interest.

Example. Let 𝐶ℓ4,1 ∼= 𝒜 where 𝒜 = ℳ(4,ℂ) for 𝑑 = 2. The 𝐶ℓ4,1 1-vectors canbe represented21 by the following matrices:

𝑒1 =

⎛⎜⎜⎝1 0 0 00 −1 0 00 0 −1 00 0 0 1

⎞⎟⎟⎠ , 𝑒2 =

⎛⎜⎜⎝0 1 0 01 0 0 00 0 0 10 0 1 0

⎞⎟⎟⎠ , 𝑒3 =

⎛⎜⎜⎝0 −𝑖 0 0𝑖 0 0 00 0 0 −𝑖0 0 𝑖 0

⎞⎟⎟⎠ ,

20Two conjugate (similar) matrices have the same eigenvalues and the same trace. This sufficesto recognize that 2𝑑+ 1 conjugacy classes are obtained.21For the computations of this example in the Maple package CLIFFORD we have used theidentification 𝑖 = 𝑒23. Yet the results obtained for the square roots of −1 are independent of thissetting (we can alternatively use, e.g., 𝑖 = 𝑒12345 , or the imaginary unit 𝑖 ∈ ℂ), as can easilybe checked for 𝑓1 of (7.7), 𝑓0 of (7.8) and 𝑓−1 of (7.9) by only assuming the standard Cliffordproduct rules for 𝑒1 to 𝑒5.

136 E. Hitzer, J. Helmstetter and R. Ablamowicz

𝑒4 =

⎛⎜⎜⎝0 0 1 00 0 0 −11 0 0 00 −1 0 0

⎞⎟⎟⎠ , 𝑒5 =

⎛⎜⎜⎝0 0 −1 00 0 0 11 0 0 00 −1 0 0

⎞⎟⎟⎠ . (7.6)

We find five conjugacy classes of roots 𝑓𝑘 of −1 in 𝐶ℓ4,1 for 𝑘 ∈ {0,±1,±2}: fourexceptional and one ordinary. Since 𝑓𝑘 is a root of 𝑝(𝑡) = 𝑡2 +1 which factors overℂ into (𝑡 − 𝑖)(𝑡 + 𝑖), the minimal polynomial 𝑚𝑘(𝑡) of 𝑓𝑘 is one of the following:𝑡 − 𝑖, 𝑡+ 𝑖, or (𝑡 − 𝑖)(𝑡 + 𝑖). Respectively, there are three classes of characteristicpolynomial Δ𝑘(𝑡) of the matrix ℱ𝑘 in ℳ(4,ℂ) which corresponds to 𝑓𝑘, namely,(𝑡−𝑖)4, (𝑡+𝑖)4, and (𝑡−𝑖)𝑛1(𝑡+𝑖)𝑛2 , where 𝑛1+𝑛2 = 2𝑑 = 4 and 𝑛1 = 𝑑+𝑘 = 2+𝑘,𝑛2 = 𝑑 − 𝑘 = 2 − 𝑘. As predicted by the above discussion, the ordinary rootcorresponds to 𝑘 = 0 whereas the exceptional roots correspond to 𝑘 ∕= 0.

1. For 𝑘 = 2, we have Δ2(𝑡) = (𝑡− 𝑖)4, 𝑚2(𝑡) = 𝑡− 𝑖, and so ℱ2 = diag(𝑖, 𝑖, 𝑖, 𝑖)which in the above representation (7.6) corresponds to the non-trivial centralelement 𝑓2 = 𝜔 = 𝑒12345. Clearly, Spec(𝑓2) = 1 = 𝑘

𝑑 ; Scal(𝑓2) = 0; theℂ-dimension of the centralizer Cent(𝑓2) is 16; and the ℝ-dimension of theconjugacy class of 𝑓2 is zero as it contains only 𝑓2 since 𝑓2 ∈ Z(𝒜). Thus, theℝ-dimension of the class is again zero in agreement with (7.5).

2. For 𝑘 = −2, we have Δ−2(𝑡) = (𝑡 + 𝑖)4, 𝑚−2(𝑡) = 𝑡 + 𝑖, and ℱ−2 =diag(−𝑖,−𝑖,−𝑖,−𝑖) which corresponds to the central element 𝑓−2 = −𝜔 =−𝑒12345. Again, Spec(𝑓−2) = −1 = 𝑘

𝑑 ; Scal(𝑓−2) = 0; the ℂ-dimension ofthe centralizer Cent(𝑓−2) is 16 and the conjugacy class of 𝑓−2 contains only𝑓−2 since 𝑓−2 ∈ Z(𝒜). Thus, the ℝ-dimension of the class is again zero inagreement with (7.5).

3. For 𝑘 ∕= ±2, we consider three subcases when 𝑘 = 1, 𝑘 = 0, and 𝑘 = −1.When 𝑘 = 1, then Δ1(𝑡) = (𝑡− 𝑖)3(𝑡+ 𝑖) and 𝑚1(𝑡) = (𝑡− 𝑖)(𝑡+ 𝑖). Then theroot ℱ1 = diag(𝑖, 𝑖, 𝑖,−𝑖) corresponds to

𝑓1 =1

2(𝑒23 + 𝑒123 − 𝑒2345 + 𝑒12345). (7.7)

Note that Spec(𝑓1) =12 = 𝑘

𝑑 so 𝑓1 is an exceptional root of −1.When 𝑘 = 0, then Δ0(𝑡) = (𝑡− 𝑖)2(𝑡+ 𝑖)2 and 𝑚0(𝑡) = (𝑡− 𝑖)(𝑡+ 𝑖). Thus theroot of −1 in this case is ℱ0 = diag(𝑖, 𝑖,−𝑖,−𝑖) which corresponds to just

𝑓0 = 𝑒123. (7.8)

Note that Spec(𝑓0) = 0 thus 𝑓0 = 𝑒123 is an ordinary root of −1.When 𝑘 = −1, then Δ−1(𝑡) = (𝑡− 𝑖)(𝑡+ 𝑖)3 and 𝑚−1(𝑡) = (𝑡− 𝑖)(𝑡+ 𝑖). Then,the root of −1 in this case is ℱ−1 = diag(𝑖,−𝑖,−𝑖,−𝑖) which corresponds to

𝑓−1 =1

2(𝑒23 + 𝑒123 + 𝑒2345 − 𝑒12345). (7.9)

Since Scal(𝑓−1) = − 12 = 𝑘

𝑑 , we gather that 𝑓−1 is an exceptional root.

7. Square Roots of −1 in Real Clifford Algebras 137

As expected, we can also see that the roots 𝜔 and −𝜔 are related viathe grade involution whereas 𝑓1 = −𝑓−1 where ˜ denotes the reversion in𝐶ℓ4,1.

Example. Let 𝐶ℓ0,5 ∼= 𝒜 where 𝒜 = ℳ(4,ℂ) for 𝑑 = 2. The 𝐶ℓ0,5 1-vectors canbe represented22 by the following matrices:

𝑒1 =

⎛⎜⎜⎝0 −1 0 01 0 0 00 0 0 −10 0 1 0

⎞⎟⎟⎠ , 𝑒2 =

⎛⎜⎜⎝0 −𝑖 0 0−𝑖 0 0 00 0 0 −𝑖0 0 −𝑖 0

⎞⎟⎟⎠ , 𝑒3 =

⎛⎜⎜⎝−𝑖 0 0 00 𝑖 0 00 0 𝑖 00 0 0 −𝑖

⎞⎟⎟⎠ ,

𝑒4 =

⎛⎜⎜⎝0 0 −1 00 0 0 11 0 0 00 −1 0 0

⎞⎟⎟⎠ , 𝑒5 =

⎛⎜⎜⎝0 0 −𝑖 00 0 0 𝑖−𝑖 0 0 00 𝑖 0 0

⎞⎟⎟⎠ , (7.10)

Like for 𝐶ℓ4,1, we have five conjugacy classes of the roots 𝑓𝑘 of −1 in 𝐶ℓ0,5 for𝑘 ∈ {0,±1,±2}: four exceptional and one ordinary. Using the same notation as inExample 7, we find the following representatives of the conjugacy classes.

1. For 𝑘 = 2, we have Δ2(𝑡) = (𝑡 − 𝑖)4, 𝑚2(𝑡) = 𝑡 − 𝑖, and ℱ2 = diag(𝑖, 𝑖, 𝑖, 𝑖)which in the above representation (7.10) corresponds to the non-trivial cen-tral element 𝑓2 = 𝜔 = 𝑒12345. Then, Spec(𝑓2) = 1 = 𝑘

𝑑 ; Scal(𝑓2) = 0; theℂ-dimension of the centralizer Cent(𝑓2) is 16; and the ℝ-dimension of theconjugacy class of 𝑓2 is zero as it contains only 𝑓2 since 𝑓2 ∈ Z(𝒜). Thus, theℝ-dimension of the class is again zero in agreement with (7.5).

2. For 𝑘 = −2, we have Δ−2(𝑡) = (𝑡 + 𝑖)4, 𝑚−2(𝑡) = 𝑡 + 𝑖, and ℱ−2 =diag(−𝑖,−𝑖,−𝑖,−𝑖) which corresponds to the central element 𝑓−2= − 𝜔 =−𝑒12345. Again, Spec(𝑓−2) = −1 = 𝑘

𝑑 ; Scal(𝑓−2) = 0; the ℂ-dimension ofthe centralizer Cent(𝑓−2) is 16 and the conjugacy class of 𝑓−2 contains only𝑓−2 since 𝑓−2 ∈ Z(𝒜). Thus, the ℝ-dimension of the class is again zero inagreement with (7.5).

3. For 𝑘 ∕= ±2, we consider three subcases when 𝑘 = 1, 𝑘 = 0, and 𝑘 = −1.When 𝑘 = 1, then Δ1(𝑡) = (𝑡− 𝑖)3(𝑡+ 𝑖) and 𝑚1(𝑡) = (𝑡− 𝑖)(𝑡+ 𝑖). Then theroot ℱ1 = diag(𝑖, 𝑖, 𝑖,−𝑖) corresponds to

𝑓1 =1

2(𝑒3 + 𝑒12 + 𝑒45 + 𝑒12345). (7.11)

Since Spec(𝑓1) =12 = 𝑘

𝑑 , 𝑓1 is an exceptional root of −1.

22For the computations of this example in the Maple package CLIFFORD we have used theidentification 𝑖 = 𝑒3. Yet the results obtained for the square roots of −1 are independent of thissetting (we can alternatively use, e.g., 𝑖 = 𝑒12345, or the imaginary unit 𝑖 ∈ ℂ), as can easily be

checked for 𝑓1 of (7.11), 𝑓0 of (7.12) and 𝑓−1 of (7.13) by only assuming the standard Cliffordproduct rules for 𝑒1 to 𝑒5.

138 E. Hitzer, J. Helmstetter and R. Ablamowicz

When 𝑘 = 0, then Δ0(𝑡) = (𝑡 − 𝑖)2(𝑡 + 𝑖)2 and 𝑚0(𝑡) = (𝑡 − 𝑖)(𝑡 + 𝑖). Thusthe root of −1 is this case is ℱ0 = diag(𝑖, 𝑖,−𝑖,−𝑖) which corresponds to just

𝑓0 = 𝑒45. (7.12)

Note that Spec(𝑓0) = 0 thus 𝑓0 = 𝑒45 is an ordinary root of −1.When 𝑘 = −1, then Δ−1(𝑡) = (𝑡− 𝑖)(𝑡+ 𝑖)3 and 𝑚−1(𝑡) = (𝑡− 𝑖)(𝑡+ 𝑖). Then,the root of −1 in this case is ℱ−1 = diag(𝑖,−𝑖,−𝑖,−𝑖) which corresponds to

𝑓−1 =1

2(−𝑒3 + 𝑒12 + 𝑒45 − 𝑒12345). (7.13)

Since Scal(𝑓−1) = − 12 = 𝑘

𝑑 , we gather that 𝑓−1 is an exceptional root.Again we can see that the roots 𝑓2 and 𝑓−2 are related via the grade

involution whereas 𝑓1 = −𝑓−1 where ˜ denotes the reversion in 𝐶ℓ0,5.

Example. Let 𝐶ℓ7,0 ∼= 𝒜 where 𝒜 = ℳ(8,ℂ) for 𝑑 = 4. We have nine conjugacyclasses of roots 𝑓𝑘 of −1 for 𝑘 ∈ {0,±1,±2 ± 3 ± 4}. Since 𝑓𝑘 is a root of apolynomial 𝑝(𝑡) = 𝑡2 + 1 which factors over ℂ into (𝑡 − 𝑖)(𝑡 + 𝑖), its minimalpolynomial 𝑚(𝑡) will be one of the following: 𝑡− 𝑖, 𝑡+ 𝑖, or (𝑡− 𝑖)(𝑡+ 𝑖) = 𝑡2 + 1.

Respectively, each conjugacy class is characterized by a characteristic poly-nomial Δ𝑘(𝑡) of the matrix 𝑀𝑘 ∈ ℳ(8,ℂ) which represents 𝑓𝑘. Namely, we have

Δ𝑘(𝑡) = (𝑡− 𝑖)𝑛1(𝑡+ 𝑖)𝑛2 ,

where 𝑛1 + 𝑛2 = 2𝑑 = 8 and 𝑛1 = 𝑑 + 𝑘 = 4 + 𝑘 and 𝑛2 = 𝑑 − 𝑘 = 4 − 𝑘. Theordinary root of −1 corresponds to 𝑘 = 0 whereas the exceptional roots correspondto 𝑘 ∕= 0.

1. When 𝑘 = 4, we have Δ4(𝑡) = (𝑡− 𝑖)8, 𝑚4(𝑡) = 𝑡− 𝑖, and ℱ4 = diag(

8︷ ︸︸ ︷𝑖, . . . , 𝑖)

which in the representation used by CLIFFORD [2] corresponds to the non-trivial central element 𝑓4 = 𝜔 = 𝑒1234567. Clearly, Spec(𝑓4) = 1 = 𝑘

𝑑 ;Scal(𝑓4) = 0; the ℂ-dimension of the centralizer Cent(𝑓4) is 64; and theℝ-dimension of the conjugacy class of 𝑓4 is zero since 𝑓4 ∈ Z(𝒜). Thus, theℝ-dimension of the class is again zero in agreement with (7.5).

2. When 𝑘 = −4, we have Δ−4(𝑡) = (𝑡 + 𝑖)8, 𝑚−4(𝑡) = 𝑡 + 𝑖, and ℱ−4 =

diag(

8︷ ︸︸ ︷−𝑖, . . . ,−𝑖) which corresponds to 𝑓−4 = −𝜔 = −𝑒1234567. Again,

Spec(𝑓−4) = −1 =𝑘

𝑑; Scal(𝑓−4) = 0;

the ℂ-dimension of the centralizer Cent(𝑓) is 64 and the conjugacy class of𝑓−4 contains only 𝑓−4 since 𝑓−4 ∈ Z(𝒜). Thus, the ℝ-dimension of the classis again zero in agreement with (7.5).

3. When 𝑘 ∕= ±4, we consider seven subcases when 𝑘 = ±3, 𝑘 = ±2, 𝑘 = ±1,and 𝑘 = 0.

7. Square Roots of −1 in Real Clifford Algebras 139

When 𝑘 = 3, then Δ3(𝑡) = (𝑡− 𝑖)7(𝑡+ 𝑖) and 𝑚3(𝑡) = (𝑡− 𝑖)(𝑡+ 𝑖). Then the

root ℱ3 = diag(

7︷ ︸︸ ︷𝑖, . . . , 𝑖,−𝑖) corresponds to

𝑓3 =1

4(𝑒23 − 𝑒45 + 𝑒67 − 𝑒123 + 𝑒145 − 𝑒167 + 𝑒234567 + 3𝑒1234567). (7.14)

Since Spec(𝑓3) =34 = 𝑘

𝑑 , 𝑓3 is an exceptional root of −1.When 𝑘 = 2, then Δ2(𝑡) = (𝑡 − 𝑖)6(𝑡 + 𝑖)2 and 𝑚2(𝑡) = (𝑡 − 𝑖)(𝑡 + 𝑖). Then

the root ℱ2 = diag(

6︷ ︸︸ ︷𝑖, . . . , 𝑖,−𝑖,−𝑖) corresponds to

𝑓2 =1

2(𝑒67 − 𝑒45 − 𝑒123 + 𝑒1234567). (7.15)

Since Spec(𝑓2) =12 = 𝑘

𝑑 , 𝑓2 is also an exceptional root.

When 𝑘 = 1, then Δ1(𝑡) = (𝑡 − 𝑖)5(𝑡 + 𝑖)3 and 𝑚1(𝑡) = (𝑡 − 𝑖)(𝑡 + 𝑖). Then

the root ℱ1 = diag(

5︷ ︸︸ ︷𝑖, . . . , 𝑖,−𝑖,−𝑖,−𝑖) corresponds to

𝑓1 =1

4(𝑒23 − 𝑒45 + 3𝑒67 − 𝑒123 + 𝑒145 + 𝑒167 − 𝑒234567 + 𝑒1234567). (7.16)

Since Spec(𝑓1) =14 = 𝑘

𝑑 , 𝑓1 is another exceptional root.

When 𝑘 = 0, then Δ0(𝑡) = (𝑡 − 𝑖)4(𝑡 + 𝑖)4 and 𝑚0(𝑡) = (𝑡 − 𝑖)(𝑡 + 𝑖). Thenthe root ℱ0 = diag(𝑖, 𝑖, 𝑖, 𝑖,−𝑖,−𝑖,−𝑖,−𝑖) corresponds to

𝑓0 =1

2(𝑒23 − 𝑒45 + 𝑒67 − 𝑒234567). (7.17)

Since Spec(𝑓0) = 0 = 𝑘𝑑 , we see that 𝑓0 is an ordinary root of −1.

When 𝑘 = −1, then Δ−1(𝑡) = (𝑡 − 𝑖)3(𝑡 + 𝑖)5 and 𝑚−1(𝑡) = (𝑡 − 𝑖)(𝑡 + 𝑖).

Then the root ℱ−1 = diag(𝑖, 𝑖, 𝑖,

5︷ ︸︸ ︷−𝑖, . . . ,−𝑖) corresponds to

𝑓−1 =1

4(𝑒23 − 𝑒45 + 3𝑒67 + 𝑒123 − 𝑒145 − 𝑒167 − 𝑒234567 − 𝑒1234567). (7.18)

Thus, Spec(𝑓−1) = − 14 = 𝑘

𝑑 and so 𝑓−1 is another exceptional root.

When 𝑘 = −2, then Δ−2(𝑡) = (𝑡 − 𝑖)2(𝑡 + 𝑖)6 and 𝑚−2(𝑡) = (𝑡 − 𝑖)(𝑡 + 𝑖).

Then the root ℱ−2 = diag(𝑖, 𝑖,

6︷ ︸︸ ︷−𝑖, . . . ,−𝑖) corresponds to

𝑓−2 =1

2(𝑒67 − 𝑒45 + 𝑒123 − 𝑒1234567). (7.19)

Since Spec(𝑓−2) = − 12 = 𝑘

𝑑 , we see that 𝑓−2 is also an exceptional root.

When 𝑘 = −3, then Δ−3(𝑡) = (𝑡− 𝑖)(𝑡+ 𝑖)7 and 𝑚−3(𝑡) = (𝑡− 𝑖)(𝑡+ 𝑖). Then

the root ℱ−3 = diag(𝑖,

7︷ ︸︸ ︷−𝑖, . . . ,−𝑖) corresponds to

𝑓−3 =1

4(𝑒23 − 𝑒45 + 𝑒67 + 𝑒123 − 𝑒145 + 𝑒167 + 𝑒234567 − 3𝑒1234567). (7.20)

140 E. Hitzer, J. Helmstetter and R. Ablamowicz

Again, Spec(𝑓−3) = − 34 = 𝑘

𝑑 and so 𝑓−3 is another exceptional root of −1.As expected, we can also see that the roots 𝜔 and −𝜔 are related via

the reversion whereas 𝑓3 = −𝑓−3, 𝑓2 = −𝑓−2, 𝑓1 = −𝑓−1 where ¯ denotesthe conjugation in 𝐶ℓ7,0.

8. Conclusions

We proved that in all cases Scal(𝑓) = 0 for every square root of −1 in 𝒜 isomorphicto 𝐶ℓ𝑝,𝑞. We distinguished ordinary square roots of −1, and exceptional ones.

In all cases the ordinary square roots 𝑓 of −1 constitute a unique conjugacyclass of dimension dim(𝒜)/2 which has as many connected components as thegroup G(𝒜) of invertible elements in 𝒜. Furthermore, we have Spec(𝑓) = 0 (zeropseudoscalar part) if the associated ring is ℝ2, ℍ2, or ℂ. The exceptional squareroots of −1 only exist if 𝒜 ∼=ℳ(2𝑑,ℂ) (see Section 7).

For 𝒜 =ℳ(2𝑑,ℝ) of Section 3, the centralizer and the conjugacy class of asquare root 𝑓 of −1 both have ℝ-dimension 2𝑑2 with two connected components,pictured in Figure 2 for 𝑑 = 1.

For 𝒜 =ℳ(2𝑑,ℝ2) =ℳ(2𝑑,ℝ)×ℳ(2𝑑,ℝ) of Section 4, the square roots of(−1,−1) are pairs of two square roots of −1 inℳ(2𝑑,ℝ). They constitute a uniqueconjugacy class with four connected components, each of dimension 4𝑑2. Regardingthe four connected components, the group Inn(𝒜) induces the permutations of theKlein group whereas the quotient group Aut(𝒜)/Inn(𝒜) is isomorphic to the groupof isometries of a Euclidean square in 2D.

For 𝒜 =ℳ(𝑑,ℍ) of Section 5, the submanifold of the square roots 𝑓 of −1is a single connected conjugacy class of ℝ-dimension 2𝑑2 equal to the ℝ-dimensionof the centralizer of every 𝑓 . The easiest example is ℍ itself for 𝑑 = 1.

For 𝒜 =ℳ(𝑑,ℍ2) =ℳ(2𝑑,ℍ)×ℳ(2𝑑,ℍ) of Section 6, the square roots of(−1,−1) are pairs of two square roots (𝑓, 𝑓 ′) of −1 in ℳ(2𝑑,ℍ) and constitute aunique connected conjugacy class of ℝ-dimension 4𝑑2. The group Aut(𝒜) has twoconnected components: the neutral component Inn(𝒜) connected to the identityand the second component containing the swap automorphism (𝑓, 𝑓 ′) �→ (𝑓 ′, 𝑓).The simplest case for 𝑑 = 1 is ℍ2 isomorphic to 𝐶ℓ0,3.

For 𝒜 = ℳ(2𝑑,ℂ) of Section 7, the square roots of −1 are in bijection tothe idempotents. First, the ordinary square roots of −1 (with 𝑘 = 0) constitutea conjugacy class of ℝ-dimension 4𝑑2 of a single connected component which isinvariant under Aut(𝒜). Second, there are 2𝑑 conjugacy classes of exceptionalsquare roots of −1, each composed of a single connected component, characterizedby equality Spec(𝑓) = 𝑘/𝑑 (the pseudoscalar coefficient) with ±𝑘 ∈ {1, 2, . . . , 𝑑},and their ℝ-dimensions are 4(𝑑2 − 𝑘2). The group Aut(𝒜) includes conjugationof the pseudoscalar 𝜔 �→ −𝜔 which maps the conjugacy class associated with 𝑘to the class associated with −𝑘. The simplest case for 𝑑 = 1 is the Pauli matrixalgebra isomorphic to the geometric algebra 𝐶ℓ3,0 of 3D Euclidean space ℝ3, andto complex biquaternions [33].

7. Square Roots of −1 in Real Clifford Algebras 141

Section 7 includes explicit examples for 𝑑 = 2: 𝐶ℓ4,1 and 𝐶ℓ0,5, and for 𝑑 = 4:𝐶ℓ7,0. Appendix A summarizes the square roots of −1 in all 𝐶ℓ𝑝,𝑞 ∼= ℳ(2𝑑,ℂ)for 𝑑 = 1, 2, 4. Appendix B contains details on how square roots of −1 can becomputed using the package CLIFFORD for Maple.

Among the many possible applications of this research, the possibility of newintegral transformations in Clifford analysis is very promising. This field thus ob-tains essential algebraic information, which can, e.g., be used to create steerabletransformations, which may be steerable within a connected component of a sub-manifold of square roots of −1.

Appendix A. Summary of Roots of −1 in 𝑪ℓ𝒑,𝒒 ∼=퓜(2𝒅,ℂ)for 𝒅 = 1, 2, 4

In this appendix we summarize roots of −1 for Clifford algebras 𝐶ℓ𝑝,𝑞 ∼=ℳ(2𝑑,ℂ)for 𝑑 = 1, 2, 4. These roots have been computed with CLIFFORD [2]. Maple [25]worksheets written to derive these roots are posted at [21].

Table 1. Square roots of −1 in 𝐶ℓ3,0 ∼=ℳ(2,ℂ), 𝑑 = 1

𝑘 𝑓𝑘 Δ𝑘(𝑡)

1 𝜔 = 𝑒123 (𝑡− 𝑖)2

0 𝑒23 (𝑡− 𝑖)(𝑡+ 𝑖)

−1 −𝜔 = −𝑒123 (𝑡+ 𝑖)2

Table 2. Square roots of −1 in 𝐶ℓ4,1 ∼=ℳ(4,ℂ), 𝑑 = 2

𝑘 𝑓𝑘 Δ𝑘(𝑡)

2 𝜔 = 𝑒12345 (𝑡− 𝑖)4

1 12 (𝑒23 + 𝑒123 − 𝑒2345 + 𝑒12345) (𝑡− 𝑖)3(𝑡+ 𝑖)

0 𝑒123 (𝑡− 𝑖)2(𝑡+ 𝑖)2

−1 12 (𝑒23 + 𝑒123 + 𝑒2345 − 𝑒12345) (𝑡− 𝑖)(𝑡+ 𝑖)3

−2 −𝜔 = −𝑒12345 (𝑡+ 𝑖)4

142 E. Hitzer, J. Helmstetter and R. Ablamowicz

Table 3. Square roots of −1 in 𝐶ℓ0,5 ∼=ℳ(4,ℂ), 𝑑 = 2

𝑘 𝑓𝑘 Δ𝑘(𝑡)

2 𝜔 = 𝑒12345 (𝑡− 𝑖)4

1 12 (𝑒3 + 𝑒12 + 𝑒45 + 𝑒12345) (𝑡− 𝑖)3(𝑡+ 𝑖)

0 𝑒45 (𝑡− 𝑖)2(𝑡+ 𝑖)2

−1 12 (−𝑒3 + 𝑒12 + 𝑒45 − 𝑒12345) (𝑡− 𝑖)(𝑡+ 𝑖)3

−2 −𝜔 = −𝑒12345 (𝑡+ 𝑖)4

Table 4. Square roots of −1 in 𝐶ℓ2,3 ∼=ℳ(4,ℂ), 𝑑 = 2

𝑘 𝑓𝑘 Δ𝑘(𝑡)

2 𝜔 = 𝑒12345 (𝑡− 𝑖)4

1 12 (𝑒3 + 𝑒134 + 𝑒235 + 𝜔) (𝑡− 𝑖)3(𝑡+ 𝑖)

0 𝑒134 (𝑡− 𝑖)2(𝑡+ 𝑖)2

−1 12 (−𝑒3 + 𝑒134 + 𝑒235 − 𝜔) (𝑡− 𝑖)(𝑡+ 𝑖)3

−2 −𝜔 = −𝑒12345 (𝑡+ 𝑖)4

Table 5. Square roots of −1 in 𝐶ℓ7,0 ∼=ℳ(8,ℂ), 𝑑 = 4

𝑘 𝑓𝑘 Δ𝑘(𝑡)

4 𝜔 = 𝑒1234567 (𝑡− 𝑖)8

3 14 (𝑒23 − 𝑒45 + 𝑒67 − 𝑒123 + 𝑒145

− 𝑒167 + 𝑒234567 +3𝜔)(𝑡− 𝑖)7(𝑡+ 𝑖)

2 12 (𝑒67 − 𝑒45 − 𝑒123 + 𝜔) (𝑡− 𝑖)6(𝑡+ 𝑖)2

1 14 (𝑒23 − 𝑒45 + 3𝑒67 − 𝑒123 + 𝑒145

+ 𝑒167 − 𝑒234567 + 𝜔)(𝑡− 𝑖)5(𝑡+ 𝑖)3

0 12 (𝑒23 − 𝑒45 + 𝑒67 − 𝑒234567) (𝑡− 𝑖)4(𝑡+ 𝑖)4

−1 14 (𝑒23 − 𝑒45 + 3𝑒67 + 𝑒123 − 𝑒145

− 𝑒167 − 𝑒234567 − 𝜔)(𝑡− 𝑖)3(𝑡+ 𝑖)5

−2 12 (𝑒67 − 𝑒45 + 𝑒123 − 𝜔) (𝑡− 𝑖)2(𝑡+ 𝑖)6

−3 14 (𝑒23 − 𝑒45 + 𝑒67 + 𝑒123 − 𝑒145

+ 𝑒167 + 𝑒234567− 3𝜔)(𝑡− 𝑖)(𝑡+ 𝑖)7

−4 −𝜔 = −𝑒1234567 (𝑡+ 𝑖)8

7. Square Roots of −1 in Real Clifford Algebras 143

Table 6. Square roots of −1 in 𝐶ℓ1,6 ∼=ℳ(8,ℂ), 𝑑 = 4

𝑘 𝑓𝑘 Δ𝑘(𝑡)

4 𝜔 = 𝑒1234567 (𝑡− 𝑖)8

3 14 (𝑒4 − 𝑒23 − 𝑒56 + 𝑒1237 + 𝑒147

+ 𝑒1567− 𝑒23456 +3𝜔)(𝑡− 𝑖)7(𝑡+ 𝑖)

2 12 (−𝑒23 − 𝑒56 + 𝑒147 + 𝜔) (𝑡− 𝑖)6(𝑡+ 𝑖)2

1 14 (−𝑒4 − 𝑒23 − 3𝑒56 − 𝑒1237 + 𝑒147

+ 𝑒1567 − 𝑒23456 + 𝜔)(𝑡− 𝑖)5(𝑡+ 𝑖)3

0 12 (𝑒4 + 𝑒23 + 𝑒56 + 𝑒23456) (𝑡− 𝑖)4(𝑡+ 𝑖)4

−1 14 (−𝑒4 − 𝑒23 − 3𝑒56 + 𝑒1237 − 𝑒147

− 𝑒1567 − 𝑒23456 − 𝜔)(𝑡− 𝑖)3(𝑡+ 𝑖)5

−2 12 (−𝑒23 − 𝑒56 − 𝑒147 − 𝜔) (𝑡− 𝑖)2(𝑡+ 𝑖)6

−3 14 (𝑒4 − 𝑒23 − 𝑒56 − 𝑒1237 − 𝑒147

− 𝑒1567− 𝑒23456− 3𝜔)(𝑡− 𝑖)(𝑡+ 𝑖)7

−4 −𝜔 = −𝑒1234567 (𝑡+ 𝑖)8

Table 7. Square roots of −1 in 𝐶ℓ3,4 ∼=ℳ(8,ℂ), 𝑑 = 4

𝑘 𝑓𝑘 Δ𝑘(𝑡)

4 𝜔 = 𝑒1234567 (𝑡− 𝑖)8

3 14 (𝑒4 + 𝑒145 + 𝑒246 + 𝑒347 − 𝑒12456

−𝑒13457−𝑒23467+3𝜔)(𝑡− 𝑖)7(𝑡+ 𝑖)

2 12 (𝑒145 − 𝑒12456 − 𝑒13457 + 𝜔) (𝑡− 𝑖)6(𝑡+ 𝑖)2

1 14 (−𝑒4 + 𝑒145 + 𝑒246 − 𝑒347 − 3𝑒12456

− 𝑒13457− 𝑒23467 +𝜔)(𝑡− 𝑖)5(𝑡+ 𝑖)3

0 12 (𝑒4 + 𝑒12456 + 𝑒13457 + 𝑒23467) (𝑡− 𝑖)4(𝑡+ 𝑖)4

−1 14 (−𝑒4 − 𝑒145 − 𝑒246 + 𝑒347 − 3𝑒12456

− 𝑒13457− 𝑒23467−𝜔)(𝑡− 𝑖)3(𝑡+ 𝑖)5

−2 12 (−𝑒145 − 𝑒12456 − 𝑒13457 − 𝜔) (𝑡− 𝑖)2(𝑡+ 𝑖)6

−3 14 (𝑒4 − 𝑒145 − 𝑒246 − 𝑒347 − 𝑒12456

−𝑒13457−𝑒23467−3𝜔)(𝑡− 𝑖)(𝑡+ 𝑖)7

−4 −𝜔 = −𝑒1234567 (𝑡+ 𝑖)8

144 E. Hitzer, J. Helmstetter and R. Ablamowicz

Table 8. Square roots of −1 in 𝐶ℓ5,2 ∼=ℳ(8,ℂ), 𝑑 = 4

𝑘 𝑓𝑘 Δ𝑘(𝑡)

4 𝜔 = 𝑒1234567 (𝑡− 𝑖)8

3 14 (−𝑒23 + 𝑒123 + 𝑒2346 + 𝑒2357 − 𝑒12346

−𝑒12357+𝑒234567+3𝜔)(𝑡− 𝑖)7(𝑡+ 𝑖)

2 12 (𝑒123 − 𝑒12346 − 𝑒12357 + 𝜔) (𝑡− 𝑖)6(𝑡+ 𝑖)2

1 14 (−𝑒23 + 𝑒123 − 𝑒2346 + 𝑒2357 − 3𝑒12346

−𝑒12357−𝑒234567+𝜔)(𝑡− 𝑖)5(𝑡+ 𝑖)3

0 12 (𝑒23 + 𝑒12346 + 𝑒12357 + 𝑒234567) (𝑡− 𝑖)4(𝑡+ 𝑖)4

−1 14 (−𝑒23 − 𝑒123 + 𝑒2346 − 𝑒2357 − 3𝑒12346

−𝑒12357−𝑒234567−𝜔)(𝑡− 𝑖)3(𝑡+ 𝑖)5

−2 12 (−𝑒123 − 𝑒12346 − 𝑒12357 − 𝜔) (𝑡− 𝑖)2(𝑡+ 𝑖)6

−3 14 (−𝑒23 − 𝑒123 − 𝑒2346 − 𝑒2357 − 𝑒12346

−𝑒12357+𝑒234567−3𝜔)(𝑡− 𝑖)(𝑡+ 𝑖)7

−4 −𝜔 = −𝑒1234567 (𝑡+ 𝑖)8

Appendix B. A Sample Maple Worksheet

In this appendix we show a computation of roots of −1 in 𝐶ℓ3,0 in CLIFFORD.Although these computations certainly can be performed by hand, as shown inSection 7, they illustrate how CLIFFORD can be used instead especially whenextending these computations to higher dimensions.23 To see the actual Mapleworksheets where these computations have been performed, see [21].> restart:with(Clifford):with(linalg):with(asvd):> p,q:=3,0; ##<<-- selecting signature> B:=diag(1$p,-1$q): ##<<-- defining diagonal bilinear form> eval(makealiases(p+q)): ##<<-- defining aliases> clibas:=cbasis(p+q); ##assigning basis for Cl(3,0)

𝑝, 𝑞 := 3, 0

clibas := [Id , e1 , e2 , e3 , e12 , e13 , e23 , e123 ]

> data:=clidata(); ##<<-- displaying information about Cl(3,0)

data := [complex , 2, simple,Id

2+e1

2, [Id , e2 , e3 , e23 ], [Id , e23 ], [Id , e2 ]]

> MM:=matKrepr(); ##<<-- displaying default matrices to generators

Cliplus has been loaded. Definitions for type/climon andtype/clipolynom now include &C and &C[K]. Type ?cliprod forhelp.

23In showing Maple display we have edited Maple output to save space. Package asvd is a

supplementary package written by the third author and built into CLIFFORD. The primarypurpose of asvd is to compute Singular Value Decomposition in Clifford algebras [1].

7. Square Roots of −1 in Real Clifford Algebras 145

MM := [e1 =

⎡⎣ 1 0

0 −1

⎤⎦ , e2 =

⎡⎣ 0 1

1 0

⎤⎦ , e3 =

⎡⎣ 0 −e23e23 0

⎤⎦]

Pauli algebra representation displayed in (7.1):> sigma[1]:=evalm(rhs(MM[1]));> sigma[2]:=evalm(rhs(MM[2]));> sigma[3]:=evalm(rhs(MM[3]));

𝜎1, 𝜎2, 𝜎3 :=

⎡⎣ 0 1

1 0

⎤⎦ ,

⎡⎣ 0 −e23e23 0

⎤⎦ ,

⎡⎣ 1 0

0 −1

⎤⎦

We show how we represent the imaginary unit 𝑖 in the field ℂ and the diagonalmatrix diag(𝑖, 𝑖) :> ii:=e23; ##<<-- complex imaginary unit> II:=diag(ii,ii); ##<<-- diagonal matrix diag(i,i)

ii := e23

II :=

⎡⎣ e23 0

0 e23

⎤⎦

We compute matrices 𝑚1,𝑚2, . . . ,𝑚8 representing each basis element in 𝐶ℓ3,0isomorphic with ℂ(2). Note that in our representation element 𝑒23 in 𝐶ℓ3,0 is usedto represent the imaginary unit 𝑖.> for i from 1 to nops(clibas) do

lprint(‘The basis element‘,clibas[i],‘is represented by the followingmatrix:‘);M[i]:=subs(Id=1,matKrepr(clibas[i])) od;

‘The basis element‘, Id, ‘is represented by the following matrix:‘

𝑀1 :=

⎡⎣ 1 0

0 1

⎤⎦

‘The basis element‘, e1, ‘is represented by the following matrix:‘

𝑀2 :=

⎡⎣ 1 0

0 −1

⎤⎦

‘The basis element‘, e2, ‘is represented by the following matrix:‘

𝑀3 :=

⎡⎣ 0 1

1 0

⎤⎦

‘The basis element‘, e3, ‘is represented by the following matrix:‘

𝑀4 :=

⎡⎣ 0 −e23e23 0

⎤⎦

146 E. Hitzer, J. Helmstetter and R. Ablamowicz

‘The basis element‘, e12, ‘is represented by the following matrix:‘

𝑀5 :=

⎡⎣ 0 1

−1 0

⎤⎦

‘The basis element‘, e13, ‘is represented by the following matrix:‘

𝑀6 :=

⎡⎣ 0 −e23

−e23 0

⎤⎦

‘The basis element‘, e23, ‘is represented by the following matrix:‘

𝑀7 :=

⎡⎣ e23 0

0 −e23

⎤⎦

‘The basis element‘, e123, ‘is represented by the following matrix:‘

𝑀8 :=

⎡⎣ e23 0

0 e23

⎤⎦

We will use the procedure phi from the asvd package which gives an isomor-phism from ℂ(2) to 𝐶ℓ3,0. This way we can find the image in 𝐶ℓ3,0 of any complex2 × 2 complex matrix 𝐴. Knowing the image of each matrix 𝑚1,𝑚2, . . . ,𝑚8 interms of the Clifford polynomials in 𝐶ℓ3,0, we can easily find the image of 𝐴 inour default spinor representation of 𝐶ℓ3,0 which is built into CLIFFORD.

Procedure Centralizer computes a centralizer of 𝑓 with respect to the Clif-ford basis 𝐿:> Centralizer:=proc(f,L) local c,LL,m,vars,i,eq,sol;

m:=add(c[i]*L[i],i=1..nops(L));vars:=[seq(c[i],i=1..nops(L))];eq:=clicollect(cmul(f,m)-cmul(m,f));if eq=0 then return L end if:sol:=op(clisolve(eq,vars));m:=subs(sol,m);m:=collect(m,vars);return sort([coeffs(m,vars)],bygrade);end proc:

Procedures Scal and Spec compute the scalar and the pseudoscalar parts of 𝑓 .> Scal:=proc(f) local p: return scalarpart(f); end proc:> Spec:=proc(f) local N; global p,q;

N:=p+q:return coeff(vectorpart(f,N),op(cbasis(N,N)));end proc:

The matrix idempotents in ℂ(2) displayed in (7.2) are as follows:> d:=1:Eps[1]:=sigma[1] &cm sigma[1];

7. Square Roots of −1 in Real Clifford Algebras 147

> Eps[0]:=evalm(1/2*(1+sigma[3]));> Eps[-1]:=diag(0,0);

Eps1, Eps0, Eps−1 :=

⎡⎣ 1 0

0 1

⎤⎦ ,

⎡⎣ 1 0

0 0

⎤⎦ ,

⎡⎣ 0 0

0 0

⎤⎦

This function ff computes matrix square root of −1 corresponding to the matrixidempotent 𝑒𝑝𝑠:> ff:=eps->evalm(II &cm (2*eps-1));

ff := eps → evalm(II &cm (2 eps − 1))

We compute matrix square roots of −1 which correspond to the idempotents𝐸𝑝𝑠1, 𝐸𝑝𝑠0, 𝐸𝑝𝑠−1, and their characteristic and minimal polynomials. Note thatin Maple the default imaginary unit is denoted by 𝐼.> F[1]:=ff(Eps[1]); ##<<-- this square root of -1 corresponds to Eps[1]

Delta[1]:=charpoly(subs(e23=I,evalm(F[1])),t);Mu[1]:=minpoly(subs(e23=I,evalm(F[1])),t);

𝐹1 :=

⎡⎣ e23 0

0 e23

⎤⎦ , Δ1 := (𝑡− 𝐼)2, 𝑀1 := 𝑡− 𝐼

> F[0]:=ff(Eps[0]); ##<<-- this square root of -1 corresponds to Eps[0]Delta[0]:=charpoly(subs(e23=I,evalm(F[0])),t);Mu[0]:=minpoly(subs(e23=I,evalm(F[0])),t);

𝐹0 :=

⎡⎣ e23 0

0 −e23

⎤⎦ , Δ0 := (𝑡− 𝐼) (𝑡+ 𝐼), 𝑀0 := 1 + 𝑡2

> F[-1]:=ff(Eps[-1]); ##<<-- this square root of -1 corresponds to Eps[-1]Delta[-1]:=charpoly(subs(e23=I,evalm(F[-1])),t);Mu[-1]:=minpoly(subs(e23=I,evalm(F[-1])),t);

𝐹−1 :=

⎡⎣ −e23 0

0 −e23

⎤⎦ , Δ−1 := (𝑡+ 𝐼)2, 𝑀−1 := 𝑡+ 𝐼

Now, we can find square roots of −1 in 𝐶ℓ3,0 which correspond to the matrixsquare roots 𝐹−1, 𝐹0, 𝐹1 via the isomorphism 𝜙 : 𝐶ℓ3,0 → ℂ(2) realized with theprocedure phi.

First, we let reprI denote element in 𝐶ℓ3,0 which represents the diagonal(2𝑑) × (2𝑑) with 𝐼 = 𝑖 on the diagonal where 𝑖2 = −1. This element will replacethe imaginary unit 𝐼 in the minimal polynomials.> reprI:=phi(diag(I$(2*d)),M);

reprI := e123

148 E. Hitzer, J. Helmstetter and R. Ablamowicz

Now, we compute the corresponding square roots 𝑓1, 𝑓0, 𝑓−1 in 𝐶ℓ3,0.> f[1]:=phi(F[1],M); ##<<-- element in Cl(3,0) corresponding to F[1]

cmul(f[1],f[1]); ##<<-- checking that this element is a root of -1Mu[1]; ##<<-- recalling minpoly of matrix F[1]subs(e23=I,evalm(subs(t=evalm(F[1]),Mu[1]))); ##<<-- F[1] in Mu[1]mu[1]:=subs(I=reprI,Mu[1]); ##<<-- defining minpoly of f[1]cmul(f[1]-reprI,Id); ##<<-- verifying that f[1] satisfies mu[1]

𝑓1 := e123

−Id , 𝑡− 𝐼,

⎡⎣ 0 0

0 0

⎤⎦

𝜇1 := 𝑡− e123 , 0

> f[0]:=phi(F[0],M); ##<<-- element in Cl(3,0) corresponding to F[0]cmul(f[0],f[0]); ##<<-- checking that this element is a root of -1Mu[0]; ##<<-- recalling minpoly of matrix F[0]subs(e23=I,evalm(subs(t=evalm(F[0]),Mu[0]))); ##<<-- F[0] in Mu[0]mu[0]:=subs(I=reprI,Mu[0]); ##<<-- defining minpoly of f[0]cmul(f[0]-reprI,f[0]+reprI); ##<<-- f[0] satisfies mu[0]

𝑓0 := e23

−Id , 1 + 𝑡2,

⎡⎣ 0 0

0 0

⎤⎦

𝜇0 := 1 + 𝑡2, 0

> f[-1]:=phi(F[-1],M); ##<<-- element in Cl(3,0) corresponding to F[-1]cmul(f[-1],f[-1]); ##<<-- checking that this element is a root of -1Mu[-1]; ##<<-- recalling minpoly of matrix F[-1]subs(e23=I,evalm(subs(t=evalm(F[-1]),Mu[-1]))); ##<<-- F[-1] in Mu[-1]mu[-1]:=subs(I=reprI,Mu[-1]); ##<<-- defining minpoly of f[-1]cmul(f[-1]+reprI,Id); ##<<-- f[-1] satisfies mu[-1]

𝑓−1 := −e123

−Id , 𝑡+ 𝐼,

⎡⎣ 0 0

0 0

⎤⎦

𝜇−1 := 𝑡+ e123 , 0

Functions RdimCentralizer and RdimConjugClass of 𝑑 and 𝑘 compute thereal dimension of the centralizer Cent(𝑓) and the conjugacy class of 𝑓 (see (7.4)).> RdimCentralizer:=(d,k)->2*((d+k)ˆ2+(d-k)ˆ2); ##<<-- from the theory> RdimConjugClass:=(d,k)->4*(dˆ2-kˆ2); ##<<-- from the theory

RdimCentralizer := (𝑑, 𝑘) → 2 (𝑑+ 𝑘)2 + 2 (𝑑− 𝑘)2

RdimConjugClass := (𝑑, 𝑘) → 4 𝑑2 − 4 𝑘2

7. Square Roots of −1 in Real Clifford Algebras 149

Now, we compute the centralizers of the roots and use notation 𝑑, 𝑘, 𝑛1, 𝑛2 dis-played in Examples.Case 𝑘 = 1 :> d:=1:k:=1:n1:=d+k;n2:=d-k;

A1:=diag(I$n1,-I$n2); ##<<-- this is the first matrix root of -1

n1 := 2, n2 := 0, A1 :=

⎡⎣ 𝐼 0

0 𝐼

⎤⎦

> f[1]:=phi(A1,M); cmul(f[1],f[1]); Scal(f[1]), Spec(f[1]);

𝑓1 := e123 , −Id , 0, 1

> LL1:=Centralizer(f[1],clibas); ##<<-- centralizer of f[1]dimCentralizer:=nops(LL1); ##<<-- real dimension of centralizer of f[1]RdimCentralizer(d,k); ##<<-- dimension of centralizer of f[1] from theoryevalb(dimCentralizer=RdimCentralizer(d,k)); ##<<-- checkingequality

LL1 := [Id , e1 , e2 , e3 , e12 , e13 , e23 , e123 ]dimCentralizer := 8, 8, true

Case 𝑘 = 0 :> d:=1:k:=0:n1:=d+k;n2:=d-k;

A0:=diag(I$n1,-I$n2); ##<<-- this is the second matrix root of -1

n1 := 1, n2 := 1, A0 :=

⎡⎣ 𝐼 0

0 −𝐼

⎤⎦

> f[0]:=phi(A0,M); cmul(f[0],f[0]); Scal(f[0]), Spec(f[0]);

𝑓0 := e23 , −Id , 0, 0

> LL0:=Centralizer(f[0],clibas); ##<<-- centralizer of f[0]dimCentralizer:=nops(LL0); ##<<-- real dimension of centralizer of f[0]RdimCentralizer(d,k); ##<<-- dimension of centralizer of f[0] from theoryevalb(dimCentralizer=RdimCentralizer(d,k)); ##<<-- checking equality

LL0 := [Id , e1 , e23 , e123 ]dimCentralizer := 4, 4, true

Case 𝑘 = −1 :> d:=1:k:=-1:n1:=d+k;n2:=d-k;

Am1:=diag(I$n1,-I$n2); ##<<-- this is the third matrix root of -1

n1 := 0, n2 := 2, Am1 :=

⎡⎣ −𝐼 0

0 −𝐼

⎤⎦

> f[-1]:=phi(Am1,M); cmul(f[-1],f[-1]); Scal(f[-1]), Spec(f[-1]);

𝑓−1 := −e123 , −Id , 0, −1

150 E. Hitzer, J. Helmstetter and R. Ablamowicz

> LLm1:=Centralizer(f[-1],clibas); ##<<-- centralizer of f[-1]dimCentralizer:=nops(LLm1); ##<<-- real dimension of centralizer of f[-1]RdimCentralizer(d,k); ##<<--dimension of centralizer of f[-1] from theoryevalb(dimCentralizer=RdimCentralizer(d,k)); ##<<-- checking equality

LLm1 := [Id , e1 , e2 , e3 , e12 , e13 , e23 , e123 ]

dimCentralizer := 8, 8, true

We summarize roots of −1 in 𝐶ℓ3,0:> ’F[1]’=evalm(F[1]); ##<<-- square root of -1 in C(2)

Mu[1]; ##<<-- minpoly of matrix F[1]’f[1]’=f[1]; ##<<-- square root of -1 in Cl(3,0)mu[1]; ##<<-- minpoly of element f[1]

𝐹1 =

⎡⎣ e23 0

0 e23

⎤⎦ , 𝑡− 𝐼

𝑓1 = e123 , 𝑡− e123

> ’F[0]’=evalm(F[0]); ##<<-- square root of -1 in C(2)Mu[0]; ##<<-- minpoly of matrix F[0]’f[0]’=f[0]; ##<<-- square root of -1 in Cl(3,0)mu[0]; ##<<-- minpoly of element f[0]

𝐹0 =

⎡⎣ e23 0

0 −e23

⎤⎦ , 1 + 𝑡2

𝑓0 = e23 , 1 + 𝑡2

> ’F[-1]’=evalm(F[-1]); ##<<-- square root of -1 in C(2)Mu[-1]; ##<<-- minpoly of matrix F[-1]’f[-1]’=f[-1]; ##<<-- square root of -1 in Cl(3,0)mu[-1]; ##<<-- minpoly of element f[-1]

𝐹−1 =

⎡⎣ −e23 0

0 −e23

⎤⎦ , 𝑡+ 𝐼

𝑓−1 = −e123 , 𝑡+ e123

Finaly, we verify that roots 𝑓1 and 𝑓−1 are related via the reversion:> reversion(f[1])=f[-1]; evalb(%);

−e123 = −e123 , true

References

[1] R. Ablamowicz. Computations with Clifford and Grassmann algebras. Advances inApplied Clifford Algebras, 19(3–4):499–545, 2009.

7. Square Roots of −1 in Real Clifford Algebras 151

[2] R. Ablamowicz and B. Fauser. CLIFFORD with bigebra – a Maple package forcomputations with Clifford and Grassmann algebras. Available at http://math.

tntech.edu/rafal/, c⃝1996–2012.

[3] R. Ablamowicz, B. Fauser, K. Podlaski, and J. Rembielinski. Idempotents of Cliffordalgebras. Czechoslovak Journal of Physics, 53(11):949–954, 2003.

[4] J. Armstrong. The center of an algebra. Weblog, available athttp://unapologetic.wordpress.com/2010/10/06/the-center-of-an-algebra/,accessed 22 March 2011.

[5] M. Bahri, E. Hitzer, R. Ashino, and R. Vaillancourt. Windowed Fourier transformof two-dimensional quaternionic signals. Applied Mathematics and Computation,216(8):2366–2379, June 2010.

[6] M. Bahri, E.M.S. Hitzer, and S. Adji. Two-dimensional Clifford windowed Fouriertransform. In E.J. Bayro-Corrochano and G. Scheuermann, editors, Geometric Al-gebra Computing in Engineering and Computer Science, pages 93–106. Springer,London, 2010.

[7] F. Brackx, R. Delanghe, and F. Sommen. Clifford Analysis, volume 76. Pitman,Boston, 1982.

[8] T. Bulow. Hypercomplex Spectral Signal Representations for the Processing and Anal-ysis of Images. PhD thesis, University of Kiel, Germany, Institut fur Informatik undPraktische Mathematik, Aug. 1999.

[9] T. Bulow, M. Felsberg, and G. Sommer. Non-commutative hypercomplex Fouriertransforms of multidimensional signals. In G. Sommer, editor, Geometric computingwith Clifford Algebras: Theoretical Foundations and Applications in Computer Visionand Robotics, pages 187–207, Berlin, 2001. Springer.

[10] C. Chevalley. The Theory of Lie Groups. Princeton University Press, Princeton,1957.

[11] W.K. Clifford. Applications of Grassmann’s extensive algebra. American Journal ofMathematics, 1(4):350–358, 1878.

[12] L. Dorst and J. Lasenby, editors. Guide to Geometric Algebra in Practice. Springer,Berlin, 2011.

[13] J. Ebling and G. Scheuermann. Clifford Fourier transform on vector fields. IEEETransactions on Visualization and Computer Graphics, 11(4):469–479, July 2005.

[14] M. Felsberg. Low-Level Image Processing with the Structure Multivector. PhD thesis,Christian-Albrechts-Universitat, Institut fur Informatik und Praktische Mathematik,Kiel, 2002.

[15] D. Hestenes. Space-Time Algebra. Gordon and Breach, London, 1966.

[16] D. Hestenes. New Foundations for Classical Mechanics. Kluwer, Dordrecht, 1999.

[17] D. Hestenes and G. Sobczyk. Clifford Algebra to Geometric Calculus. D. ReidelPublishing Group, Dordrecht, Netherlands, 1984.

[18] E. Hitzer. Quaternion Fourier transform on quaternion fields and generalizations.Advances in Applied Clifford Algebras, 17(3):497–517, May 2007.

[19] E. Hitzer. OPS-QFTs: A new type of quaternion Fourier transform based on theorthogonal planes split with one or two general pure quaternions. In InternationalConference on Numerical Analysis and Applied Mathematics, volume 1389 of AIP

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Conference Proceedings, pages 280–283, Halkidiki, Greece, 19–25 September 2011.American Institute of Physics.

[20] E. Hitzer and R. Ablamowicz. Geometric roots of −1 in Clifford algebras 𝐶ℓ𝑝,𝑞 with𝑝 + 𝑞 ≤ 4. Advances in Applied Clifford Algebras, 21(1):121–144, 2010. Publishedonline 13 July 2010.

[21] E. Hitzer, J. Helmstetter, and R. Ablamowicz. Maple worksheets created withCLIFFORD for a verification of results presented in this chapter. Available at: http://math.tntech.edu/rafal/publications.html, c⃝2012.

[22] E. Hitzer and B. Mawardi. Uncertainty principle for Clifford geometric algebras𝐶ℓ𝑛,0, 𝑛 = 3(mod 4) based on Clifford Fourier transform. In T. Qian, M.I. Vai, andY. Xu, editors,Wavelet Analysis and Applications, Applied and Numerical HarmonicAnalysis, pages 47–56. Birkhauser Basel, 2007.

[23] E.M.S. Hitzer and B. Mawardi. Clifford Fourier transform on multivector fields anduncertainty principles for dimensions 𝑛 = 2(mod 4) and 𝑛 = 3(mod 4). Advances inApplied Clifford Algebras, 18(3-4):715–736, 2008.

[24] R.A. Horn and C.R. Johnson. Matrix Analysis. Cambridge University Press, Cam-bridge, 1985.

[25] W.M. Incorporated. Maple, a general purpose computer algebra system. http://www.maplesoft.com, c⃝ 2012.

[26] C. Li, A. McIntosh, and T. Qian. Clifford algebras, Fourier transform and singu-lar convolution operators on Lipschitz surfaces. Revista Matematica Iberoamericana,10(3):665–695, 1994.

[27] H. Li. Invariant Algebras and Geometric Reasoning. World Scientific, Singapore,2009.

[28] P. Lounesto. Clifford Algebras and Spinors, volume 286 of London MathematicalSociety Lecture Notes. Cambridge University Press, 1997.

[29] B. Mawardi and E.M.S. Hitzer. Clifford Fourier transformation and uncertainty prin-ciple for the Clifford algebra 𝐶ℓ3,0. Advances in Applied Clifford Algebras, 16(1):41–61, 2006.

[30] A. McIntosh. Clifford algebras, Fourier theory, singular integrals, and harmonic func-tions on Lipschitz domains. In J. Ryan, editor, Clifford Algebras in Analysis andRelated Topics, chapter 1. CRC Press, Boca Raton, 1996.

[31] T. Qian. Paley-Wiener theorems and Shannon sampling in the Clifford analysis set-ting. In R. Ablamowicz, editor, Clifford Algebras – Applications to Mathematics,Physics, and Engineering, pages 115–124. Birkauser, Basel, 2004.

[32] S. Said, N. Le Bihan, and S.J. Sangwine. Fast complexified quaternion Fourier trans-form. IEEE Transactions on Signal Processing, 56(4):1522–1531, Apr. 2008.

[33] S.J. Sangwine. Biquaternion (complexified quaternion) roots of −1. Advances inApplied Clifford Algebras, 16(1):63–68, June 2006.

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7. Square Roots of −1 in Real Clifford Algebras 153

Eckhard HitzerCollege of Liberal Arts, Department of Material ScienceInternational Christian University181-8585 Tokyo, Japane-mail: hitzer@icu.ac.jp

Jacques HelmstetterUnivesite Grenoble IInstitut Fourier (Mathematiques)B.P. 74F-38402 Saint-Martin d’Heres, Francee-mail: Jacques.Helmstetter@ujf-grenoble.fr

Rafal AblamowiczDepartment of Mathematics, Box 5054Tennessee Technological UniversityCookeville, TN 38505, USAe-mail: rablamowicz@tntech.edu

Quaternion and Clifford–Fourier Transforms and Wavelets

Trends in Mathematics, 155–176c⃝ 2013 Springer Basel

8 A General Geometric Fourier Transform

Roxana Bujack, Gerik Scheuermann and Eckhard Hitzer

Abstract. The increasing demand for Fourier transforms on geometric alge-bras has resulted in a large variety. Here we introduce one single straight-forward definition of a general geometric Fourier transform covering mostversions in the literature. We show which constraints are additionally neces-sary to obtain certain features such as linearity or a shift theorem. As a result,we provide guidelines for the target-oriented design of yet unconsidered trans-forms that fulfill requirements in a specific application context. Furthermore,the standard theorems do not need to be shown in a slightly different formevery time a new geometric Fourier transform is developed since they areproved here once and for all.

Mathematics Subject Classification (2010). Primary 15A66, 11E88; secondary42A38, 30G35.

Keywords. Fourier transform, geometric algebra, Clifford algebra, image pro-cessing, linearity, scaling, shift.

1. Introduction

The Fourier transform by Jean Baptiste Joseph Fourier is an indispensable toolin many fields of mathematics, physics, computer science and engineering, espe-cially for the analysis and solution of differential equations, or in signal and imageprocessing, fields which cannot be imagined without it. The kernel of the Fouriertransform consists of the complex exponential function. With the square root ofminus one, the imaginary unit 𝑖, as part of the argument it is periodic and thereforesuitable for the analysis of oscillating systems.

William Kingdon Clifford created the geometric algebras in 1878, [8]. Theyusually contain continuous submanifolds of geometric square roots of −1 [16, 17].Each multivector has a natural geometric interpretation so the generalization ofthe Fourier transform to multivector-valued functions in the geometric algebras isvery reasonable. It helps to interpret the transform, apply it in a target-oriented

156 R. Bujack, G. Scheuermann and E. Hitzer

way to the specific underlying problem and it allows a new point of view on fluidmechanics.

Many different application-oriented definitions of Fourier transforms in geo-metric algebras have been developed. For example the Clifford–Fourier transformintroduced by Jancewicz [19] and expanded by Ebling and Scheuermann [10]and Hitzer and Mawardi [18] or the one established by Sommen in [21] and re-established by Bulow [7]. Further we have the quaternionic Fourier transform byEll [11] and later by Bulow [7], the spacetime Fourier transform by Hitzer [15], theClifford–Fourier transform for colour images by Batard et al. [1], the CylindricalFourier transform by Brackx et al. [6], the transforms by Felsberg [13] or Ell andSangwine [20, 12]. All these transforms have different interesting properties anddeserve to be studied independently from one another. But the analysis of theirsimilarities reveals a lot about their qualities, too. We concentrate on this matterand summarize all of them in one general definition.

Recently there have been very successful approaches by De Bie, Brackx, DeSchepper and Sommen to construct Clifford–Fourier transforms from operator ex-ponentials and differential equations [3, 4, 9, 5]. The definition presented in thischapter does not cover all of them, partly because their closed integral form is notalways known or is highly complicated, and partly because they can be producedby combinations and functions of our transforms.

We focus on continuous geometric Fourier transforms over flat spaces ℝ𝑝,𝑞 intheir integral representation. That way their finite, regular discrete equivalents asused in computational signal and image processing can be intuitively constructedand direct applicability to the existing practical issues and easy numerical man-ageability are ensured.

2. Definition of the GFT

We examine geometric algebras 𝐶ℓ𝑝,𝑞, 𝑝+ 𝑞 = 𝑛 ∈ ℕ over ℝ𝑝+𝑞 [14] generated bythe associative, bilinear geometric product with neutral element 1 satisfying

𝒆𝑗𝒆𝑘 + 𝒆𝑘𝒆𝑗 = 𝜖𝑗𝛿𝑗𝑘, (2.1)

for all 𝑗, 𝑘 ∈ {1, . . . , 𝑛} with the Kronecker symbol 𝛿 and

𝜖𝑗 =

{1 ∀𝑗 = 1, . . . , 𝑝,

−1 ∀𝑗 = 𝑝+ 1, . . . , 𝑛.(2.2)

For the sake of brevity we want to refer to arbitrary multivectors

𝑨 =

𝑛∑𝑘=0

∑1≤𝑗1<⋅⋅⋅<𝑗𝑘≤𝑛

𝑎𝑗1...𝑗𝑘𝒆𝑗1 . . .𝒆𝑗𝑘 ∈ 𝐶ℓ𝑝,𝑞, (2.3)

where 𝑎𝑗1...𝑗𝑘 ∈ ℝ, as

𝑨 =∑𝒋

𝑎𝒋𝒆𝒋 . (2.4)

8. A General Geometric Fourier Transform 157

where each of the 2𝑛 multi-indices 𝒋 ⊆ {1, . . . , 𝑛} indicates a basis vector of 𝐶ℓ𝑝,𝑞by 𝒆𝒋 = 𝒆𝑗1 . . .𝒆𝑗𝑘 , 1 ≤ 𝑗1 < ⋅ ⋅ ⋅ < 𝑗𝑘 ≤ 𝑛, 𝒆∅ = 𝒆0 = 1 and its associatedcoefficient 𝑎𝒋 = 𝑎𝑗1...𝑗𝑘 ∈ ℝ.

Definition 2.1. The exponential function of a multivector 𝑨 ∈ 𝐶ℓ𝑝,𝑞 is defined bythe power series

𝑒𝑨 :=∑∞

𝑗=0

𝑨𝑗

𝑗!. (2.5)

Lemma 2.2. For two multivectors 𝑨𝑩 = 𝑩𝑨 that commute we have

𝑒𝑨+𝑩 = 𝑒𝑨𝑒𝑩. (2.6)

Proof. Analogous to the exponent rule of real matrices. □Notation 2.3. For each geometric algebra𝐶ℓ𝑝,𝑞 we will write I 𝑝,𝑞 = {𝑖 ∈ 𝐶ℓ𝑝,𝑞, 𝑖

2 ∈ℝ−} to denote the real multiples of all geometric square roots of −1 , compare [16]and [17]. We choose the symbol I to be reminiscent of the imaginary numbers.

Definition 2.4. Let 𝐶ℓ𝑝,𝑞 be a geometric algebra, 𝑨 : ℝ𝑚 → 𝐶ℓ𝑝,𝑞 be a mul-tivector field and 𝒙,𝒖 ∈ ℝ𝑚 vectors. A Geometric Fourier Transform (GFT)ℱ𝐹1,𝐹2(𝑨) is defined by two ordered finite sets 𝐹1 = {𝑓1(𝒙,𝒖), . . . , 𝑓𝜇(𝒙,𝒖)}, 𝐹2 ={𝑓𝜇+1(𝒙,𝒖), . . . , 𝑓𝜈(𝒙,𝒖)} of mappings 𝑓𝑘(𝒙,𝒖) : ℝ

𝑚×ℝ𝑚 → I 𝑝,𝑞, ∀𝑘 = 1, . . . , 𝜈and the calculation rule

ℱ𝐹1,𝐹2(𝑨)(𝒖) :=

∫ℝ𝑚

∏𝑓∈𝐹1

𝑒−𝑓(𝒙,𝒖)𝑨(𝒙)∏𝑓∈𝐹2

𝑒−𝑓(𝒙,𝒖) d𝑚𝒙. (2.7)

This definition combines many Fourier transforms into a single general one. Itenables us to prove the well-known theorems which depend only on the propertiesof the chosen mappings.

Example. Depending on the choice of 𝐹1 and 𝐹2 we obtain previously publishedtransforms.

1. In the case of 𝑨 : ℝ𝑛 → 𝒢𝑛,0, 𝑛 = 2 (mod 4) or 𝑛 = 3 (mod 4), we canreproduce the Clifford–Fourier transform introduced by Jancewicz [19] for𝑛 = 3 and expanded by Ebling and Scheuermann [10] for 𝑛 = 2 and HitzerandMawardi [18] for 𝑛 = 2 (mod 4) or 𝑛 = 3 (mod 4) using the configuration

𝐹1 = ∅,𝐹2 = {𝑓1},

𝑓1(𝒙,𝒖) = 2𝜋𝑖𝑛𝒙 ⋅ 𝒖,(2.8)

with 𝑖𝑛 being the pseudoscalar of 𝐺𝑛,0.2. Choosing multivector fields ℝ𝑛 → 𝒢0,𝑛,

𝐹1 = ∅,𝐹2 = {𝑓1, . . . , 𝑓𝑛},

𝑓𝑘(𝒙,𝒖) = 2𝜋𝒆𝑘𝑥𝑘𝑢𝑘, ∀𝑘 = 1, . . . , 𝑛

(2.9)

we have the Sommen–Bulow–Clifford–Fourier transform from [21, 7].

158 R. Bujack, G. Scheuermann and E. Hitzer

3. For𝑨 : ℝ2 → 𝒢0,2 ≈ ℍ the quaternionic Fourier transform [11, 7] is generatedby

𝐹1 = {𝑓1},𝐹2 = {𝑓2},

𝑓1(𝒙,𝒖) = 2𝜋𝑖𝑥1𝑢1,

𝑓2(𝒙,𝒖) = 2𝜋𝑗𝑥2𝑢2.

(2.10)

4. Using 𝒢3,1 we can build the spacetime, respectively the volume-time, Fouriertransform from [15]1 with the 𝒢3,1-pseudoscalar 𝑖4 as follows

𝐹1 = {𝑓1},𝐹2 = {𝑓2},

𝑓1(𝒙,𝒖) = 𝒆4𝑥4𝑢4,

𝑓2(𝒙,𝒖) = 𝜖4𝒆4𝑖4(𝑥1𝑢1 + 𝑥2𝑢2 + 𝑥3𝑢3).

(2.11)

5. The Clifford–Fourier transform for colour images by Batard, Berthier andSaint-Jean [1] for 𝑚 = 2, 𝑛 = 4,𝑨 : ℝ2 → 𝒢4,0, a fixed bivector 𝑩, and thepseudoscalar 𝑖 can intuitively be written as

𝐹1 = {𝑓1},𝐹2 = {𝑓2},

𝑓1(𝒙,𝒖) =1

2(𝑥1𝑢1 + 𝑥2𝑢2)(𝑩 + 𝑖𝑩),

𝑓2(𝒙,𝒖) = −1

2(𝑥1𝑢1 + 𝑥2𝑢2)(𝑩 + 𝑖𝑩),

(2.12)

but (𝑩+ 𝑖𝑩) does not square to a negative real number, see [16]. The specialproperty that 𝑩 and 𝑖𝑩 commute allows us to express the formula using

𝐹1 = {𝑓1, 𝑓2},𝐹2 = {𝑓3, 𝑓4},

𝑓1(𝒙,𝒖) =1

2(𝑥1𝑢1 + 𝑥2𝑢2)𝑩,

𝑓2(𝒙,𝒖) =1

2(𝑥1𝑢1 + 𝑥2𝑢2)𝑖𝑩,

𝑓3(𝒙,𝒖) = −1

2(𝑥1𝑢1 + 𝑥2𝑢2)𝑩,

𝑓4(𝒙,𝒖) = −1

2(𝑥1𝑢1 + 𝑥2𝑢2)𝑖𝑩,

(2.13)

which fulfills the conditions of Definition 2.4.

1Please note that Hitzer uses a different notation in [15]. His 𝒙 = 𝑡𝒆0 + 𝑥1𝒆1 + 𝑥2𝒆2 + 𝑥3𝒆3corresponds to our 𝒙 = 𝑥1𝒆1 + 𝑥2𝒆2 + 𝑥3𝒆3 + 𝑥4𝒆4, with 𝒆0𝒆0 = 𝜖0 = −1 being equivalent toour 𝒆4𝒆4 = 𝜖4 = −1.

8. A General Geometric Fourier Transform 159

6. Using 𝒢0,𝑛 and

𝐹1 = {𝑓1},𝐹2 = ∅,

𝑓1(𝒙,𝒖) = −𝒙 ∧ 𝒖

(2.14)

produces the cylindrical Fourier transform as introduced by Brackx, de Schep-per and Sommen in [6].

3. General Properties

First we prove general properties valid for arbitrary sets 𝐹1, 𝐹2.

Theorem 3.1 (Existence). The geometric Fourier transform exists for all integrablemultivector fields 𝑨 ∈ 𝐿1(ℝ

𝑛).

Proof. The property

𝑓2𝑘 (𝒙,𝒖) ∈ ℝ− (3.1)

of the mappings 𝑓𝑘 for 𝑘 = 1, . . . , 𝜈 leads to

𝑓2𝑘 (𝒙,𝒖)

∣𝑓2𝑘 (𝒙,𝒖)∣

= −1 (3.2)

for all 𝑓𝑘(𝒙,𝒖) ∕= 0. So using the decomposition

𝑓𝑘(𝒙,𝒖) =𝑓𝑘(𝒙,𝒖)

∣𝑓𝑘(𝒙,𝒖)∣ ∣𝑓𝑘(𝒙,𝒖)∣ (3.3)

we can write ∀𝑗 ∈ ℕ

𝑓 𝑗𝑘(𝒙,𝒖) =

⎧⎨⎩(−1)𝑙∣𝑓𝑘(𝒙,𝒖)∣𝑗 for 𝑗 = 2𝑙, 𝑙 ∈ ℕ0

(−1)𝑙 𝑓𝑘(𝒙,𝒖)∣𝑓𝑘(𝒙,𝒖)∣ ∣𝑓𝑘(𝒙,𝒖)∣

𝑗 for 𝑗 = 2𝑙 + 1, 𝑙 ∈ ℕ0(3.4)

which results in

𝑒−𝑓𝑘(𝒙,𝒖) =

∞∑𝑗=0

(−𝑓𝑘(𝒙,𝒖))𝑗

𝑗!

=

∞∑𝑗=0

(−1)𝑗∣𝑓𝑘(𝒙,𝒖)∣2𝑗(2𝑗)!

− 𝑓𝑘(𝒙,𝒖)

∣𝑓𝑘(𝒙,𝒖)∣∞∑𝑗=0

(−1)𝑗 ∣𝑓𝑘(𝒙,𝒖)∣2𝑗+1

(2𝑗 + 1)!

= cos (∣𝑓𝑘(𝒙,𝒖)∣)− 𝑓𝑘(𝒙,𝒖)

∣𝑓𝑘(𝒙,𝒖)∣ sin (∣𝑓𝑘(𝒙,𝒖)∣) .

(3.5)

160 R. Bujack, G. Scheuermann and E. Hitzer

Because of

∣𝑒−𝑓𝑘(𝒙,𝒖)∣ =∣∣∣∣cos (∣𝑓𝑘(𝒙,𝒖)∣)− 𝑓𝑘(𝒙,𝒖)

∣𝑓𝑘(𝒙,𝒖)∣ sin (∣𝑓𝑘(𝒙,𝒖)∣)∣∣∣∣

≤ ∣cos (∣𝑓𝑘(𝒙,𝒖)∣)∣+∣∣∣∣ 𝑓𝑘(𝒙,𝒖)∣𝑓𝑘(𝒙,𝒖)∣

∣∣∣∣ ∣sin (∣𝑓𝑘(𝒙,𝒖)∣)∣≤ 2

(3.6)

the magnitude of the improper integral

∣ℱ𝐹1,𝐹2(𝑨)(𝒖)∣ =∣∣∣∣∣∣∫ℝ𝑚

∏𝑓∈𝐹1

𝑒−𝑓(𝒙,𝒖)𝑨(𝒙)∏𝑓∈𝐹2

𝑒−𝑓(𝒙,𝒖) d𝑚𝒙

∣∣∣∣∣∣≤∫ℝ𝑚

∏𝑓∈𝐹1

∣∣∣𝑒−𝑓(𝒙,𝒖)∣∣∣ ∣𝑨(𝒙)∣

∏𝑓∈𝐹2

∣∣∣𝑒−𝑓(𝒙,𝒖)∣∣∣ d𝑚𝒙

≤∫ℝ𝑚

∏𝑓∈𝐹1

2 ∣𝑨(𝒙)∣∏𝑓∈𝐹2

2 d𝑚𝒙

= 2𝜈∫ℝ𝑚

∣𝑨(𝒙)∣ d𝑚𝒙

(3.7)

is finite and therefore the geometric Fourier transform exists. □

Theorem 3.2 (Scalar linearity). The geometric Fourier transform is linear with re-spect to scalar factors. Let 𝑏, 𝑐 ∈ ℝ and 𝑨,𝑩,𝑪 : ℝ𝑚 → 𝐶ℓ𝑝,𝑞 be three multivectorfields that satisfy 𝑨(𝒙) = 𝑏𝑩(𝒙) + 𝑐𝑪(𝒙), then

ℱ𝐹1,𝐹2(𝑨)(𝒖) = 𝑏ℱ𝐹1,𝐹2(𝑩)(𝒖) + 𝑐ℱ𝐹1,𝐹2(𝑪)(𝒖). (3.8)

Proof. The assertion is an easy consequence of the distributivity of the geomet-ric product over addition, the commutativity of scalars and the linearity of theintegral. □

4. Bilinearity

All geometric Fourier transforms from the introductory example can also be ex-pressed in terms of a stronger claim. The mappings 𝑓1, . . . , 𝑓𝜈, with the first 𝜇terms to the left of the argument function and the 𝜈 − 𝜇 others on the right of it,are all bilinear and therefore take the form

𝑓𝑘(𝒙,𝒖) = 𝑓𝑘

(𝑚∑𝑗=1

𝑥𝑗𝒆𝑗 ,𝑚∑𝑙=1

𝑢𝑙𝒆𝑙

)

=𝑚∑

𝑗,𝑙=1

𝑥𝑗𝑓𝑘(𝒆𝑗 , 𝒆𝑙)𝑢𝑙 = 𝒙𝑇𝑀𝑘𝒖,

(4.1)

∀𝑘 = 1, . . . , 𝜈, where 𝑀𝑘 ∈ (I 𝑝,𝑞)𝑚×𝑚, (𝑀𝑘)𝑗𝑙 = 𝑓𝑘(𝒆𝑗 , 𝒆𝑙) according to Nota-tion 2.3.

8. A General Geometric Fourier Transform 161

Example. Ordered in the same way as in the previous example, the geometricFourier transforms expressed in the way of (4.1) take the following shapes:

1. In the Clifford–Fourier transform 𝑓1 can be written with

𝑀1 = 2𝜋𝑖𝑛 Id . (4.2)

2. The 𝜈 = 𝑚 = 𝑛 mappings 𝑓𝑘, 𝑘 = 1, . . . , 𝑛 of the Bulow–Clifford–Fouriertransform can be expressed using

(𝑀𝑘)𝑙𝑗 =

{2𝜋𝒆𝑘 for 𝑘 = 𝑙 = 𝑗,

0 otherwise.(4.3)

3. Similarly the quaternionic Fourier transform is generated using

(𝑀1)𝑙𝜄 =

{2𝜋𝑖 for 𝑙 = 𝜄 = 1,

0 otherwise,

(𝑀2)𝑙𝜄 =

{2𝜋𝑗 for 𝑙 = 𝜄 = 2,

0 otherwise.

(4.4)

4. We can build the spacetime Fourier transform with

(𝑀1)𝑙𝑗 =

{𝒆4 for 𝑙 = 𝑗 = 1,

0 otherwise,

(𝑀2)𝑙𝑗 =

{𝜖4𝒆4𝑖4 for 𝑙 = 𝑗 ∈ {2, 3, 4},0 otherwise.

(4.5)

5. The Clifford–Fourier transform for colour images can be described by

𝑀1 =1

2𝑩 Id,

𝑀2 =1

2𝑖𝑩 Id,

𝑀3 = −1

2𝑩 Id,

𝑀4 = −1

2𝑖𝑩 Id .

(4.6)

6. The cylindrical Fourier transform can also be reproduced with mappingssatisfying (4.1) because we can write

𝒙 ∧ 𝒖 = 𝒆1𝒆2𝑥1𝑢2 − 𝒆1𝒆2𝑥2𝑢1

+ ⋅ ⋅ ⋅+ 𝒆𝑚−1𝒆𝑚𝑥𝑚−1𝑢𝑚 − 𝒆𝑚−1𝒆𝑚𝑥𝑚𝑢𝑚−1

(4.7)

and set

(𝑀1)𝑙𝑗 =

{0 for 𝑙 = 𝑗,

𝒆𝑙𝒆𝑗 otherwise.(4.8)

162 R. Bujack, G. Scheuermann and E. Hitzer

Theorem 4.1 (Scaling). Let 0 ∕= 𝑎 ∈ ℝ be a real number, 𝑨(𝒙) = 𝑩(𝑎𝒙) twomultivector fields and all 𝐹1, 𝐹2 be bilinear mappings then the geometric Fouriertransform satisfies

ℱ𝐹1,𝐹2(𝑨)(𝒖) = ∣𝑎∣−𝑚ℱ𝐹1,𝐹2(𝑩)(𝒖𝑎

). (4.9)

Proof. A change of coordinates together with the bilinearity proves the assertion by

ℱ𝐹1,𝐹2(𝑨)(𝒖) =

∫ℝ𝑚

∏𝑓∈𝐹

𝑒−𝑓(𝒙,𝒖)𝑩(𝑎𝒙)∏𝑓∈𝐵

𝑒−𝑓(𝒙,𝒖) d𝑚𝒙

𝑎𝒙=𝒚=

∫ℝ𝑚

∏𝑓∈𝐹

𝑒−𝑓(𝒚𝑎 ,𝒖)𝑩(𝒚)

∏𝑓∈𝐵

𝑒−𝑓(𝒚𝑎 ,𝒖)∣𝑎∣−𝑚 d𝑚𝒚

𝑓 bilin.= ∣𝑎∣−𝑚

∫ℝ𝑚

∏𝑓∈𝐹

𝑒−𝑓(𝒚,𝒖𝑎 )𝑩(𝒚)

∏𝑓∈𝐵

𝑒−𝑓(𝒚,𝒖𝑎 ) d𝑚𝒚 (4.10)

= ∣𝑎∣−𝑚ℱ𝐹1,𝐹2(𝑩)(𝒖𝑎

). □

5. Products with Invertible Factors

To obtain properties of the GFT like linearity with respect to arbitrary multi-vectors or a shift theorem we will have to change the order of multivectors andproducts of exponentials. Since the geometric product usually is neither commu-tative nor anticommutative this is not trivial. In this section we provide usefulLemmata that allow a swap if at least one of the factors is invertible. For moreinformation see [14] and [17].

Remark 5.1. Every multiple of a square root of −1, 𝑖 ∈ I 𝑝,𝑞 is invertible, sincefrom 𝑖2 = −𝑟, 𝑟 ∈ ℝ ∖ {0} follows 𝑖−1 = −𝑖/𝑟. Because of that, for all 𝒖,𝒙 ∈ ℝ𝑚 afunction 𝑓𝑘(𝒙,𝒖) : ℝ

𝑚 × ℝ𝑚 → I 𝑝,𝑞 is pointwise invertible.

Definition 5.2. For an invertible multivector 𝑩 ∈ 𝐶ℓ𝑝,𝑞 and an arbitrary multivec-tor 𝑨 ∈ 𝐶ℓ𝑝,𝑞 we define

𝑨𝒄0(𝑩) =1

2(𝑨+𝑩−1𝑨𝑩),

𝑨𝒄1(𝑩) =1

2(𝑨−𝑩−1𝑨𝑩).

(5.1)

Lemma 5.3. Let 𝑩 ∈ 𝐶ℓ𝑝,𝑞 be invertible with the unique inverse 𝑩−1 = ��/𝑩2,

𝑩2 ∈ ℝ ∖ {0}. Every multivector 𝑨 ∈ 𝐶ℓ𝑝,𝑞 can be expressed unambiguously by thesum of 𝑨𝒄0(𝑩) ∈ 𝐶ℓ𝑝,𝑞 that commutes and 𝑨𝒄1(𝑩) ∈ 𝐶ℓ𝑝,𝑞 that anticommutes withrespect to 𝑩. That means

𝑨 = 𝑨𝒄0(𝑩) +𝑨𝒄1(𝑩),

𝑨𝒄0(𝑩)𝑩 = 𝑩𝑨𝒄0(𝑩),

𝑨𝒄1(𝑩)𝑩 = −𝑩𝑨𝒄1(𝑩).

(5.2)

8. A General Geometric Fourier Transform 163

Proof. We will only prove the assertion for 𝑨𝒄0(𝑩).

Existence: With Definition 5.2 we get

𝑨𝒄0(𝑩) +𝑨𝒄1(𝑩) =1

2(𝑨+𝑩−1𝑨𝑩 +𝑨−𝑩−1𝑨𝑩)

= 𝑨(5.3)

and considering

𝑩−1𝑨𝑩 =��𝑨𝑩

𝑩2 = 𝑩𝑨𝑩−1 (5.4)

we also get

𝑨𝒄0(𝑩)𝑩 =1

2(𝑨+𝑩−1𝑨𝑩)𝑩

=1

2(𝑨+𝑩𝑨𝑩−1)𝑩

=1

2(𝑨𝑩 +𝑩𝑨)

= 𝑩1

2(𝑩−1𝑨𝑩 +𝑨)

= 𝑩𝑨𝒄0(𝑩)

(5.5)

Uniqueness: From the first claim in (5.2) we get

𝑨𝒄1(𝑩) = 𝑨−𝑨𝒄0(𝑩), (5.6)

together with the third one this leads to

(𝑨−𝑨𝒄0(𝑩))𝑩 = −𝑩(𝑨−𝑨𝒄0(𝑩))

𝑨𝑩 −𝑨𝒄0(𝑩)𝑩 = −𝑩𝑨+𝑩𝑨𝒄0(𝑩)

𝑨𝑩 +𝑩𝑨 = 𝑨𝒄0(𝑩)𝑩 +𝑩𝑨𝒄0(𝑩)

(5.7)

and from the second claim finally follows

𝑨𝑩 +𝑩𝑨 = 2𝑩𝑨𝒄0(𝑩)

1

2(𝑩−1𝑨𝑩 +𝑨) = 𝑨𝒄0(𝑩).

(5.8)

The derivation of the expression for 𝑨𝒄1(𝑩) works analogously. □Corollary 5.4 (Decomposition w.r.t. commutativity). Let 𝑩 ∈ 𝐶ℓ𝑝,𝑞 be invertible,then ∀𝑨 ∈ 𝐶ℓ𝑝,𝑞

𝑩𝑨 = (𝑨𝒄0(𝑩) −𝑨𝒄1(𝑩))𝑩. (5.9)

Definition 5.5. For 𝑑 ∈ ℕ,𝑨 ∈ 𝐶ℓ𝑝,𝑞, the ordered set 𝐵 = {𝑩1, . . . ,𝑩𝑑} of invert-ible multivectors and any multi-index 𝒋 ∈ {0, 1}𝑑 we define

𝑨𝒄𝒋(−→𝐵 )

: = ((𝑨𝒄𝑗1 (𝑩1))𝒄𝑗2 (𝑩2) . . .)𝒄𝑗𝑑 (𝑩𝑑),

𝑨𝒄𝒋(←−𝐵 )

: = ((𝑨𝒄𝑗𝑑 (𝑩𝑑))𝒄𝑗𝑑−1 (𝑩𝑑−1). . .)𝒄𝑗1 (𝑩1)

(5.10)

recursively with 𝒄0, 𝒄1 as in Definition 5.2.

164 R. Bujack, G. Scheuermann and E. Hitzer

Example. Let 𝑨 = 𝑎0 + 𝑎1𝒆1 + 𝑎2𝒆2 + 𝑎12𝒆12 ∈ 𝒢2,0 then, for example

𝑨𝒄0(𝒆1) =1

2(𝑨+ 𝒆−1

1 𝑨𝒆1)

=1

2(𝑨+ 𝑎0 + 𝑎1𝒆1 − 𝑎2𝒆2 − 𝑎12𝒆12)

= 𝑎0 + 𝑎1𝒆1

(5.11)

and further

𝑨𝒄0,0(−−−→𝒆1,𝒆2) = (𝑨𝒄0(𝒆1))𝒄0(𝒆2)

= (𝑎0 + 𝑎1𝒆1)𝒄0(𝒆2) = 𝑎0.(5.12)

The computation of the other multi-indices with 𝑑 = 2 works analogously andtherefore

𝑨 =∑

𝒋∈{0,1}𝑑𝑨𝒄𝒋(𝒆1,𝒆2)

= 𝑨𝒄00(−−−→𝒆1,𝒆2) +𝑨𝒄01(−−−→𝒆1,𝒆2) +𝑨𝒄10(−−−→𝒆1,𝒆2) +𝑨𝒄11(−−−→𝒆1,𝒆2)

= 𝑎0 + 𝑎1𝒆1 + 𝑎2𝒆2 + 𝑎12𝒆12.

(5.13)

Lemma 5.6. Let 𝑑 ∈ ℕ, 𝐵 = {𝑩1, . . . ,𝑩𝑑} be invertible multivectors and for 𝒋 ∈{0, 1}𝑑 let ∣𝒋∣ :=∑𝑑

𝑘=1 𝑗𝑘, then ∀𝑨 ∈ 𝐶ℓ𝑝,𝑞

𝑨 =∑

𝒋∈{0,1}𝑑𝑨

𝒄𝒋(−→𝐵 )

,

𝑨𝑩1 . . .𝑩𝑑 = 𝑩1 . . .𝑩𝑑

∑𝒋∈{0,1}𝑑

(−1)∣𝒋∣𝑨𝒄𝒋(−→𝐵 )

,

𝑩1 . . .𝑩𝑑𝑨 =∑

𝒋∈{0,1}𝑑(−1)∣𝒋∣𝑨

𝒄𝒋(←−𝐵 )

𝑩1 . . .𝑩𝑑.

(5.14)

Proof. Apply Lemma 5.3 repeatedly. □

Remark 5.7. The distinction of the two directions can be omitted using the equality

𝑨𝒄𝒋(−−−−−−→𝑩1,...,𝑩𝑑)

= 𝑨𝒄𝒋(←−−−−−−𝑩𝑑,...,𝑩1)

. (5.15)

We established it for the sake of notational brevity and will not formulate norprove every assertion for both directions.

Lemma 5.8. Let 𝐹 = {𝑓1(𝒙,𝒖), . . . , 𝑓𝑑(𝒙,𝒖)} be a set of pointwise invertible func-tions then the ordered product of their exponentials and an arbitrary multivector𝑨 ∈ 𝐶ℓ𝑝,𝑞 satisfies

𝑑∏𝑘=1

𝑒−𝑓𝑘(𝒙,𝒖)𝑨 =∑

𝒋∈{0,1}𝑑𝑨

𝒄𝒋(←−𝐹 )

(𝒙,𝒖)

𝑑∏𝑘=1

𝑒−(−1)𝑗𝑘𝑓𝑘(𝒙,𝒖), (5.16)

where 𝐴𝒄𝒋(←−𝐹 )

(𝒙,𝒖) := 𝐴𝒄𝒋(←−−−−𝐹 (𝒙,𝒖))

is a multivector-valued function ℝ𝑚 × ℝ𝑚 →𝐶ℓ𝑝,𝑞.

8. A General Geometric Fourier Transform 165

Proof. For all 𝒙,𝒖 ∈ ℝ𝑚 the commutation properties of 𝑓𝑘(𝒙,𝒖) dictate the prop-erties of 𝑒−𝑓𝑘(𝒙,𝒖) by

𝑒−𝑓𝑘(𝒙,𝒖)𝑨Def. 2.1=

∞∑𝑙=0

(−𝑓𝑘(𝒙,𝒖))𝑙

𝑙!𝑨

Lem. 5.3=

∞∑𝑙=0

(−𝑓𝑘(𝒙,𝒖))𝑙

𝑙!

(𝑨𝒄0(𝑓𝑘(𝒙,𝒖)) +𝑨𝒄1(𝑓𝑘(𝒙,𝒖))

).

(5.17)

The shape of this decomposition of 𝑨 may depend on 𝒙 and 𝒖. To stress this factwe will interpret 𝑨𝒄0(𝑓𝑘(𝒙,𝒖)) as a multivector function and write 𝑨𝒄0(𝑓𝑘)(𝒙,𝒖).According to Lemma 5.3 we can move 𝑨𝒄0(𝑓𝑘)(𝒙,𝒖) through all factors, because itcommutes. Analogously swapping 𝑨𝒄1(𝑓𝑘)(𝒙,𝒖) will change the sign of each factorbecause it anticommutes. Hence we get

= 𝑨𝒄0(𝑓𝑘)(𝒙,𝒖)

∞∑𝑙=0

(−𝑓𝑘(𝒙,𝒖))𝑙

𝑙!+𝑨𝒄1(𝑓𝑘)(𝒙,𝒖)

∞∑𝑙=0

(𝑓𝑘(𝒙,𝒖))𝑙

𝑙!

= 𝑨𝒄0(𝑓𝑘)(𝒙,𝒖)𝑒−𝑓𝑘(𝒙,𝒖) +𝑨𝒄1(𝑓𝑘)(𝒙,𝒖)𝑒

𝑓𝑘(𝒙,𝒖).

(5.18)

Applying this repeatedly to the product we can deduce

𝑑∏𝑘=1

𝑒−𝑓𝑘(𝒙,𝒖)𝑨 =

𝑑−1∏𝑘=1

𝑒−𝑓𝑘(𝒙,𝒖)

(𝑨𝒄0(𝑓𝑑)(𝒙,𝒖)𝑒

−𝑓𝑑(𝒙,𝒖)

+𝑨𝒄1(𝑓𝑑)(𝒙,𝒖)𝑒𝑓𝑑(𝒙,𝒖)

)

=

𝑑−2∏𝑘=1

𝑒−𝑓𝑘(𝒙,𝒖)

⎛⎜⎜⎜⎜⎜⎜⎝𝑨

𝒄0,0(←−−−−−𝑓𝑑−1,𝑓𝑑)

(𝒙,𝒖)𝑒−𝑓𝑑−1(𝒙,𝒖)𝑒−𝑓𝑑(𝒙,𝒖)

+𝑨𝒄1,0(

←−−−−−𝑓𝑑−1,𝑓𝑑)

(𝒙,𝒖)𝑒𝑓𝑑−1(𝒙,𝒖)𝑒−𝑓𝑑(𝒙,𝒖)

+𝑨𝒄0,1(

←−−−−−𝑓𝑑−1,𝑓𝑑)

(𝒙,𝒖)𝑒−𝑓𝑑−1(𝒙,𝒖)𝑒𝑓𝑑(𝒙,𝒖)

+𝑨𝒄1,1(

←−−−−−𝑓𝑑−1,𝑓𝑑)

(𝒙,𝒖)𝑒𝑓𝑑−1(𝒙,𝒖)𝑒𝑓𝑑(𝒙,𝒖)

⎞⎟⎟⎟⎟⎟⎟⎠...

...... (5.19)

=∑

𝒋∈{0,1}𝑑𝑨

𝒄𝒋(←−𝐹 )

(𝒙,𝒖)𝑑∏

𝑘=1

𝑒−(−1)𝑗𝑘𝑓𝑘(𝒙,𝒖). □

6. Separable GFT

From now on we want to restrict ourselves to an important group of geometricFourier transforms whose square roots of −1 are independent from the first argu-ment.

Definition 6.1. We call a GFT left (right) separable, if

𝑓𝑙 = ∣𝑓𝑙 (𝒙,𝒖)∣ 𝑖𝑙(𝒖), (6.1)

∀𝑙 = 1, . . . , 𝜇, (𝑙 = 𝜇+1, . . . , 𝜈), where ∣𝑓𝑙(𝒙,𝒖)∣ : ℝ𝑚×ℝ𝑚 → ℝ is a real functionand 𝑖𝑙 : ℝ

𝑚 → I 𝑝,𝑞 a function that does not depend on 𝒙.

166 R. Bujack, G. Scheuermann and E. Hitzer

Example. The first five transforms from the introductory example are separable,while the cylindrical transform (vi) can not be expressed as in (6.1) except for thetwo-dimensional case.

We have seen in the proof of Lemma 5.8 that the decomposition of a con-stant multivector 𝑨 with respect to a product of exponentials generally resultsin multivector-valued functions 𝑨𝒄𝒋(𝐹 )(𝒙,𝒖) of 𝒙 and 𝒖. Separability guaranteesindependence from 𝒙 and therefore allows separation from the integral.

Corollary 6.2 (Decomposition independent from 𝒙). Consider a set of functions𝐹 = {𝑓1(𝒙,𝒖), . . . , 𝑓𝑑(𝒙,𝒖)} satisfying condition (6.1) then the ordered product oftheir exponentials and an arbitrary multivector 𝑨 ∈ 𝐶ℓ𝑝,𝑞 satisfies

𝑑∏𝑘=1

𝑒−𝑓𝑘(𝒙,𝒖)𝑨 =∑

𝒋∈{0,1}𝑑𝑨

𝒄𝒋(←−𝐹 )

(𝒖)

𝑑∏𝑘=1

𝑒−(−1)𝑗𝑘𝑓𝑘(𝒙,𝒖). (6.2)

Remark 6.3. If a GFT can be expressed as in 6.1 but with multiples of squareroots of −1, 𝑖𝑘 ∈ I 𝑝,𝑞, which are independent from 𝒙 and 𝒖, the parts 𝑨

𝒄𝒋(←−𝐹 )

of 𝑨 will be constants. Note that the first five GFTs from the reference examplesatisfy this stronger condition, too.

Definition 6.4. For a set of functions 𝐹 = {𝑓1(𝒙,𝒖), . . . , 𝑓𝑑(𝒙,𝒖)} and a multi-index 𝒋 ∈ {0, 1}𝑑, we define the set of functions 𝐹 (𝒋) by

𝐹 (𝒋) :={(−1)𝑗1𝑓1(𝒙,𝒖), . . . , (−1)𝑗𝑑𝑓𝑑(𝒙,𝒖)

}. (6.3)

Theorem 6.5 (Left and right products). Let 𝑪 ∈ 𝐶ℓ𝑝,𝑞 and 𝑨,𝑩 : ℝ𝑚 → 𝐶ℓ𝑝,𝑞 betwo multivector fields with 𝑨(𝒙) = 𝑪𝑩(𝒙) then a left separable geometric Fouriertransform obeys

ℱ𝐹1,𝐹2(𝑨)(𝒖) =∑

𝒋∈{0,1}𝜇𝑪

𝒄𝒋(←−𝐹1)

(𝒖)ℱ𝐹1(𝒋),𝐹2(𝑩)(𝒖). (6.4)

If 𝑨(𝒙) = 𝑩(𝒙)𝑪 we analogously get

ℱ𝐹1,𝐹2(𝑨)(𝒖) =∑

𝒌∈{0,1}(𝜈−𝜇)

ℱ𝐹1,𝐹2(𝒌)(𝑩)(𝒖)𝑪𝒄𝒌(−→𝐹2)

(𝒖)(6.5)

for a right separable GFT.

Proof. We restrict ourselves to the proof of the first assertion.

ℱ𝐹1,𝐹2(𝑨)(𝒖) =

∫ℝ𝑚

∏𝑓∈𝐹1

𝑒−𝑓(𝒙,𝒖)𝑪𝑩(𝒙)∏𝑓∈𝐹2

𝑒−𝑓(𝒙,𝒖) d𝑚𝒙

Lem. 5.8=

∫ℝ𝑚

⎛⎝ ∑𝒋∈{0,1}𝜇

𝑪𝒄𝒋(←−𝐹1)

(𝒖)

𝜇∏𝑙=1

𝑒−(−1)𝑗𝑙𝑓𝑙(𝒙,𝒖)

⎞⎠𝑩(𝒙)

∏𝑓∈𝐹2

𝑒−𝑓(𝒙,𝒖) d𝑚𝒙

8. A General Geometric Fourier Transform 167

=∑

𝒋∈{0,1}𝜇𝑪

𝒄𝒋(←−𝐹1)

(𝒖)

∫ℝ𝑚

𝜇∏𝑙=1

𝑒−(−1)𝑗𝑙𝑓𝑙(𝒙,𝒖)

𝑩(𝒙)∏𝑓∈𝐹2

𝑒−𝑓(𝒙,𝒖) d𝑚𝒙

=∑

𝒋∈{0,1}𝜇𝑪

𝒄𝒋(←−𝐹1)

(𝒖)ℱ𝐹1(𝒋),𝐹2(𝑩)(𝒖).

The second one follows in the same way. □

Corollary 6.6 (Uniform constants). Let the claims from Theorem 6.5 hold. If theconstant 𝑪 satisfies 𝑪 = 𝑪

𝒄𝒋(←−𝐹1)

(𝒖) for a multi-index 𝒋 ∈ {0, 1}𝜇 then the theorem

simplifies to

ℱ𝐹1,𝐹2(𝑨)(𝒖) = 𝑪ℱ𝐹1(𝒋),𝐹2(𝑩)(𝒖) (6.6)

for 𝑨(𝒙) = 𝑪𝑩(𝒙) respectively

ℱ𝐹1,𝐹2(𝑨)(𝒖) = ℱ𝐹1,𝐹2(𝒌)(𝑩)(𝒖)𝑪 (6.7)

for 𝑨(𝒙) = 𝑩(𝒙)𝑪 and 𝑪 = 𝑪𝒄𝒌(−→𝐹2)

(𝒖) for a multi-index 𝒌 ∈ {0, 1}(𝜈−𝜇).2

Corollary 6.7 (Left and right linearity). The geometric Fourier transform is left(respectively right) linear if 𝐹1 (respectively 𝐹2) consists only of functions 𝑓𝑘 withvalues in the center of 𝐶ℓ𝑝,𝑞, that means ∀𝒙,𝒖 ∈ ℝ𝑚, ∀𝑨 ∈ 𝐶ℓ𝑝,𝑞 : 𝑨𝑓𝑘(𝒙,𝒖) =𝑓𝑘(𝒙,𝒖)𝑨.

Remark 6.8. Note that for empty sets 𝐹1 (or 𝐹2) necessarily all elements satisfycommutativity and therefore the condition in Corollary 6.7.

The different appearances of Theorem 6.5 are summarized in Table 1 andTable 2.

We have seen how to change the order of a multivector and a product ofexponentials in the previous section. To get a shift theorem we will have to separatesums appearing in the exponent and sort the resulting exponentials with respect tothe summands. Note that Corollary 6.2 can be applied in two ways here, becauseexponentials appear on both sides.

Not every factor will need to be swapped with every other. So, to keep thingsshort, we will make use of the notation 𝒄(𝐽)𝑙(𝑓1, . . . , 𝑓𝑙, 0, . . . , 0) for 𝑙 ∈ {1, . . . , 𝑑}instead of distinguishing between differently sized multi-indices for every 𝑙 thatappears. The zeros at the end substitutionally indicate real numbers. They com-mute with every multivector. That implies, that for the last 𝑑− 𝑙 factors no swapand therefore no separation needs to be made. It would also be possible to use thenotation 𝒄(𝐽)𝑙(𝑓1, . . . , 𝑓𝑙−1, 0, . . . , 0) for 𝑙 ∈ {1, . . . , 𝑑}, because every function com-mutes with itself. The choice we have made means that no exceptional treatment

2Corollary 6.6 follows directly from (𝑪𝒄𝒋(

←−𝐹1)

)𝒄𝒌(

←−𝐹1)

= 0 for all 𝒌 ∕= 𝒋 because no non-zero

component of 𝑪 can commute and anticommute with respect to a function in 𝐹1.

168 R. Bujack, G. Scheuermann and E. Hitzer

Table 1. Theorem 6.5 (Left products) applied to the GFTs of the firstexample enumerated in the same order.Notations: on the LHS ℱ𝐹1,𝐹2 = ℱ𝐹1,𝐹2(𝑨)(𝒖), on the RHS ℱ𝐹 ′

1,𝐹′2=

ℱ𝐹 ′1,𝐹

′2(𝑩)(𝒖)

GFT 𝑨(𝒙) = 𝑪𝑩(𝒙)

1. Clifford ℱ𝑓1 = 𝑪ℱ𝑓1

2. Bulow ℱ𝑓1,...,𝑓𝑛 = 𝑪ℱ𝑓1,...,𝑓𝑛

3. Quaternionic ℱ𝑓1,𝑓2 = 𝑪𝒄0(𝑖)ℱ𝑓1,𝑓2 +𝑪𝒄1(𝑖)ℱ−𝑓1,𝑓24. Spacetime ℱ𝑓1,𝑓2 = 𝑪𝒄0(𝒆4)ℱ𝑓1,𝑓2 +𝑪𝒄1(𝒆4)ℱ−𝑓1,𝑓25. Colour Image ℱ𝑓1,𝑓2,𝑓3,𝑓4 = 𝑪

𝒄00(←−−−𝑩,𝑖𝑩)

ℱ𝑓1,𝑓2,𝑓3,𝑓4

+𝑪𝒄10(

←−−−𝑩,𝑖𝑩)

ℱ−𝑓1,𝑓2,𝑓3,𝑓4+𝑪

𝒄01(←−−−𝑩,𝑖𝑩)

ℱ𝑓1,−𝑓2,𝑓3,𝑓4

+𝑪𝒄11(

←−−−𝑩,𝑖𝑩)

ℱ−𝑓1,−𝑓2,𝑓3,𝑓46. Cylindrical 𝑛 = 2 ℱ𝑓1 = 𝑪𝒄0(𝒆12)ℱ𝑓1 +𝑪𝒄1(𝒆12)ℱ−𝑓1

Cylindrical 𝑛 ∕= 2 -

of 𝑓1 is necessary. But please note that the multivectors (𝐽)𝑙 indicating the com-mutative and anticommutative parts will all have zeros from 𝑙 to 𝑑 and thereforeform a strictly triangular matrix.

Lemma 6.9. Let a set of functions 𝐹 = {𝑓1(𝒙,𝒖), . . . , 𝑓𝑑(𝒙,𝒖)} fulfil (6.1) and belinear with respect to 𝒙. Further let 𝐽 ∈ {0, 1}𝑑×𝑑 be a strictly lower triangularmatrix, that is associated column by column with a multi-index 𝒋 ∈ {0, 1}𝑑 by ∀𝑘 =

1, . . . , 𝑑 : (∑𝑑

𝑙=1 𝐽𝑙,𝑘) mod 2 = 𝑗𝑘, with (𝐽)𝑙 being its 𝑙th row, then

𝑑∏𝑙=1

𝑒−𝑓𝑙(𝒙+𝒚,𝒖) =∑

𝒋∈{0,1}𝑑

∑𝐽∈{0,1}𝑑×𝑑,∑

𝑑𝑙=1(𝐽)𝑙 mod 2=𝒋

𝑑∏𝑙=1

𝑒−𝑓𝑙(𝒙,𝒖)

𝒄(𝐽)𝑙 (←−−−−−−−−−−𝑓1,...,𝑓𝑙,0,...,0)

𝑑∏𝑙=1

𝑒−(−1)𝑗𝑙𝑓𝑙(𝒚,𝒖)

(6.8)or alternatively with strictly upper triangular matrices 𝐽 :

𝑑∏𝑙=1

𝑒−𝑓𝑙(𝒙+𝒚,𝒖) =∑

𝒋∈{0,1}𝑑

∑𝐽∈{0,1}𝑑×𝑑,∑𝑑

𝑙=1(𝐽)𝑙 mod 2=𝒋

𝑑∏𝑙=1

𝑒−(−1)𝑗𝑙𝑓𝑙(𝒙,𝒖)𝑑∏

𝑙=1

𝑒−𝑓𝑙(𝒚,𝒖)

𝒄(𝐽)𝑙 (−−−−−−−−−−→0,...,0,𝑓𝑙,...,𝑓𝑑)

.

(6.9)

We do not explicitly indicate the dependence of the partition on 𝒖 as in Corol-lary 6.2, because the functions in the exponents already contain this dependence.Please note that the decomposition is pointwise.

8. A General Geometric Fourier Transform 169

Table 2. Theorem 6.5 (Right products) applied to the GFTs of thefirst example, enumerated in the same order.Notations: on the LHS ℱ𝐹1,𝐹2 = ℱ𝐹1,𝐹2(𝑨)(𝒖), on the RHS ℱ𝐹 ′

1,𝐹′2=

ℱ𝐹 ′1,𝐹

′2(𝑩)(𝒖)

GFT 𝑨(𝒙) = 𝑩(𝒙)𝑪

1. Clif. 𝑛 = 2 (mod 4) ℱ𝑓1 = ℱ𝑓1𝑪𝒄0(𝑖) + ℱ−𝑓1𝑪𝒄1(𝑖)

Clif. 𝑛 = 3 (mod 4) ℱ𝑓1 = ℱ𝑓1𝑪

2. Bulow ℱ𝑓1,...,𝑓𝑛

=∑

𝒌∈{0,1}𝑛 ℱ(−1)𝑘1𝑓1,...,(−1)𝑘𝑛𝑓𝑛𝑪𝒄𝒌(−−−−−→𝑓1,...,𝑓𝑛)

3. Quaternionic ℱ𝑓1,𝑓2 = ℱ𝑓1,𝑓2𝑪𝒄0(𝑗) + ℱ𝑓1,−𝑓2𝑪𝒄1(𝑗)

4. Spacetime ℱ𝑓1,𝑓2 = ℱ𝑓1,𝑓2𝑪𝒄0(𝒆4𝑖4) + ℱ𝑓1,−𝑓2𝑪𝒄1(𝒆4𝑖4)

5. Colour Image ℱ𝑓1,𝑓2,𝑓3,𝑓4 = ℱ𝑓1,𝑓2,𝑓3,𝑓4𝑪𝒄00(−−−→𝑩,𝑖𝑩)

+ℱ𝑓1,𝑓2,−𝑓3,𝑓4𝑪𝒄10(−−−→𝑩,𝑖𝑩)

+ℱ𝑓1,𝑓2,𝑓3,−𝑓4𝑪𝒄01(−−−→𝑩,𝑖𝑩)

+ℱ𝑓1,𝑓2,−𝑓3,−𝑓4𝑪𝒄11(−−−→𝑩,𝑖𝑩)

6. Cylindrical ℱ𝑓1 = ℱ𝑓1𝑪

Proof. We will only prove the first assertion. The second one follows analogouslyby applying Corollary 6.2 the other way around.

𝑑∏𝑙=1

𝑒−𝑓𝑙(𝒙+𝒚,𝒖) 𝐹 lin.=

𝑑∏𝑙=1

𝑒−𝑓𝑙(𝒙,𝒖)−𝑓𝑙(𝒚,𝒖) (6.10)

Lem. 2.2=

𝑑∏𝑙=1

𝑒−𝑓𝑙(𝒙,𝒖)𝑒−𝑓𝑙(𝒚,𝒖) (6.11)

= 𝑒−𝑓1(𝒙,𝒖)𝑒−𝑓1(𝒚,𝒖)𝑑∏

𝑙=2

𝑒−𝑓𝑙(𝒙,𝒖)𝑒−𝑓𝑙(𝒚,𝒖) (6.12)

Cor. 6.2= 𝑒−𝑓1(𝒙,𝒖)(𝑒

−𝑓2(𝒙,𝒖)𝒄0(𝑓1)

𝑒−𝑓1(𝒚,𝒖)𝑒−𝑓2(𝒚,𝒖)

+ 𝑒−𝑓2(𝒙,𝒖)𝒄1(𝑓1)

𝑒𝑓1(𝒚,𝒖)𝑒−𝑓2(𝒚,𝒖))

𝑑∏𝑙=3

𝑒−𝑓𝑙(𝒙,𝒖)𝑒−𝑓𝑙(𝒚,𝒖). (6.13)

Now we use Corollary 6.2 to step by step rearrange the order of the product.

Cor. 6.2= 𝑒−𝑓1(𝒙,𝒖)

(𝑒−𝑓2(𝒙,𝒖)𝒄0(𝑓1)

𝑒−𝑓3(𝒙,𝒖)

𝒄00(←−−−𝑓1,𝑓2)

𝑒−𝑓1(𝒚,𝒖)𝑒−𝑓2(𝒚,𝒖)𝑒−𝑓3(𝒚,𝒖)

170 R. Bujack, G. Scheuermann and E. Hitzer

+ 𝑒−𝑓2(𝒙,𝒖)𝒄0(𝑓1)

𝑒−𝑓3(𝒙,𝒖)

𝒄01(←−−−𝑓1,𝑓2)

𝑒−𝑓1(𝒚,𝒖)𝑒𝑓2(𝒚,𝒖)𝑒−𝑓3(𝒚,𝒖)

+ 𝑒−𝑓2(𝒙,𝒖)𝒄0(𝑓1)

𝑒−𝑓3(𝒙,𝒖)

𝒄10(←−−−𝑓1,𝑓2)

𝑒𝑓1(𝒚,𝒖)𝑒−𝑓2(𝒚,𝒖)𝑒−𝑓3(𝒚,𝒖)

+ 𝑒−𝑓2(𝒙,𝒖)𝒄0(𝑓1)

𝑒−𝑓3(𝒙,𝒖)

𝒄11(←−−−𝑓1,𝑓2)

𝑒𝑓1(𝒚,𝒖)𝑒𝑓2(𝒚,𝒖)𝑒−𝑓3(𝒚,𝒖)

+ 𝑒−𝑓2(𝒙,𝒖)𝒄1(𝑓1)

𝑒−𝑓3(𝒙,𝒖)

𝒄00(←−−−𝑓1,𝑓2)

𝑒𝑓1(𝒚,𝒖)𝑒−𝑓2(𝒚,𝒖)𝑒−𝑓3(𝒚,𝒖)

+ 𝑒−𝑓2(𝒙,𝒖)𝒄1(𝑓1)

𝑒−𝑓3(𝒙,𝒖)

𝒄01(←−−−𝑓1,𝑓2)

𝑒𝑓1(𝒚,𝒖)𝑒𝑓2(𝒚,𝒖)𝑒−𝑓3(𝒚,𝒖)

+ 𝑒−𝑓2(𝒙,𝒖)𝒄1(𝑓1)

𝑒−𝑓3(𝒙,𝒖)

𝒄10(←−−−𝑓1,𝑓2)

𝑒−𝑓1(𝒚,𝒖)𝑒−𝑓2(𝒚,𝒖)𝑒−𝑓3(𝒚,𝒖)

+ 𝑒−𝑓2(𝒙,𝒖)𝒄1(𝑓1)

𝑒−𝑓3(𝒙,𝒖)

𝒄11(←−−−𝑓1,𝑓2)

𝑒−𝑓1(𝒚,𝒖)𝑒𝑓2(𝒚,𝒖)𝑒−𝑓3(𝒚,𝒖)

)𝑑∏

𝑙=4

𝑒−𝑓𝑙(𝒙,𝒖)𝑒−𝑓𝑙(𝒚,𝒖). (6.14)

There are only 2𝛿 ways of distributing the signs of 𝛿 exponents, so some of thesummands can be combined.

= 𝑒−𝑓1(𝒙,𝒖)

((𝑒−𝑓2(𝒙,𝒖)𝒄0(𝑓1)

𝑒−𝑓3(𝒙,𝒖)

𝒄00(←−−−𝑓1,𝑓2)

+ 𝑒−𝑓2(𝒙,𝒖)𝒄1(𝑓1)

𝑒−𝑓3(𝒙,𝒖)

𝒄10(←−−−𝑓1,𝑓2)

)𝑒−𝑓1(𝒚,𝒖)𝑒−𝑓2(𝒚,𝒖)𝑒−𝑓3(𝒚,𝒖)

+(𝑒−𝑓2(𝒙,𝒖)𝒄0(𝑓1)

𝑒−𝑓3(𝒙,𝒖)

𝒄01(←−−−𝑓1,𝑓2)

+ 𝑒−𝑓2(𝒙,𝒖)𝒄1(𝑓1)

𝑒−𝑓3(𝒙,𝒖)

𝒄11(←−−−𝑓1,𝑓2)

)𝑒−𝑓1(𝒚,𝒖)𝑒𝑓2(𝒚,𝒖)𝑒−𝑓3(𝒚,𝒖)

+(𝑒−𝑓2(𝒙,𝒖)𝒄0(𝑓1)

𝑒−𝑓3(𝒙,𝒖)

𝒄10(←−−−𝑓1,𝑓2)

+ 𝑒−𝑓2(𝒙,𝒖)𝒄1(𝑓1)

𝑒−𝑓3(𝒙,𝒖)

𝒄00(←−−−𝑓1,𝑓2)

)𝑒𝑓1(𝒚,𝒖)𝑒−𝑓2(𝒚,𝒖)𝑒−𝑓3(𝒚,𝒖)

+(𝑒−𝑓2(𝒙,𝒖)𝒄0(𝑓1)

𝑒−𝑓3(𝒙,𝒖)

𝒄11(←−−−𝑓1,𝑓2)

+ 𝑒−𝑓2(𝒙,𝒖)𝒄1(𝑓1)

𝑒−𝑓3(𝒙,𝒖)

𝒄01(←−−−𝑓1,𝑓2)

)𝑒𝑓1(𝒚,𝒖)

𝑒𝑓2(𝒚,𝒖)𝑒−𝑓3(𝒚,𝒖)

) 𝑑∏𝑙=4

𝑒−𝑓𝑙(𝒙,𝒖)𝑒−𝑓𝑙(𝒚,𝒖). (6.15)

To get a compact notation we expand all multi-indices by adding zeros untilthey have the same length. Note that the last non-zero argument in terms like

𝒄000(←−−−−𝑓1, 0, 0) always coincides with the exponent of the corresponding factor. Be-

cause of that it will always commute and could also be replaced by a zero.

= 𝑒−𝑓1(𝒙,𝒖)

𝒄000(←−−−𝑓1,0,0)((

𝑒−𝑓2(𝒙,𝒖)

𝒄000(←−−−−𝑓1,𝑓2,0)

𝑒−𝑓3(𝒙,𝒖)

𝒄000(←−−−−−𝑓1,𝑓2,𝑓3)

+ 𝑒−𝑓2(𝒙,𝒖)

𝒄100(←−−−−𝑓1,𝑓2,0)

𝑒−𝑓3(𝒙,𝒖)

𝒄100(←−−−−−𝑓1,𝑓2,𝑓3)

)𝑒−𝑓1(𝒚,𝒖)𝑒−𝑓2(𝒚,𝒖)𝑒−𝑓3(𝒚,𝒖)

8. A General Geometric Fourier Transform 171

+(𝑒−𝑓2(𝒙,𝒖)

𝒄000(←−−−−𝑓1,𝑓2,0)

𝑒−𝑓3(𝒙,𝒖)

𝒄010(←−−−−−𝑓1,𝑓2,𝑓3)

+ 𝑒−𝑓2(𝒙,𝒖)

𝒄100(←−−−−𝑓1,𝑓2,0)

𝑒−𝑓3(𝒙,𝒖)

𝒄110(←−−−−−𝑓1,𝑓2,𝑓3)

)𝑒−𝑓1(𝒚,𝒖)𝑒𝑓2(𝒚,𝒖)𝑒−𝑓3(𝒚,𝒖)

+(𝑒−𝑓2(𝒙,𝒖)

𝒄000(←−−−−𝑓1,𝑓2,0)

𝑒−𝑓3(𝒙,𝒖)

𝒄100(←−−−−−𝑓1,𝑓2,𝑓3)

+ 𝑒−𝑓2(𝒙,𝒖)

𝒄100(←−−−−𝑓1,𝑓2,0)

𝑒−𝑓3(𝒙,𝒖)

𝒄000(←−−−−−𝑓1,𝑓2,𝑓3)

)𝑒𝑓1(𝒚,𝒖)𝑒−𝑓2(𝒚,𝒖)𝑒−𝑓3(𝒚,𝒖)

+(𝑒−𝑓2(𝒙,𝒖)

𝒄000(←−−−−𝑓1,𝑓2,0)

𝑒−𝑓3(𝒙,𝒖)

𝒄110(←−−−−−𝑓1,𝑓2,𝑓3)

+ 𝑒−𝑓2(𝒙,𝒖)

𝒄100(←−−−−𝑓1,𝑓2,0)

𝑒𝑓3(𝒙,𝒖)

𝒄010(←−−−−−𝑓1,𝑓2,𝑓3)

)𝑒𝑓1(𝒚,𝒖)𝑒𝑓2(𝒚,𝒖)𝑒−𝑓3(𝒚,𝒖)

) 𝑑∏𝑙=4

𝑒−𝑓𝑙(𝒙,𝒖)𝑒−𝑓𝑙(𝒚,𝒖) (6.16)

For 𝛿 = 3 we look at all strictly lower triangular matrices 𝐽 ∈ {0, 1}𝛿×𝛿 with theproperty

∀𝑘 = 1, . . . , 𝛿 :

(𝛿∑

𝑙=1

(𝐽)𝑙,𝑘

)mod 2 = 𝑗𝑘. (6.17)

That means the 𝑙th row (𝐽)𝑙 of 𝐽 contains a multi-index (𝐽)𝑙 ∈ {0, 1}𝛿, with thelast 𝛿 − 𝑙 − 1 entries being zero and the 𝑘th column sum being even when 𝑗𝑘 = 0and odd when 𝑗𝑘 = 1. For example, the first multi-index is 𝒋 = (0, 0, 0). There areonly two different strictly lower triangular matrices that have columns summingup to even numbers:

𝐽 =

⎛⎝0 0 00 0 00 0 0

⎞⎠ and 𝐽 =

⎛⎝0 0 01 0 01 0 0

⎞⎠ . (6.18)

The first row of each contains the multi-index that belongs to 𝑒−𝑓1(𝒙,𝒖), the sec-ond one belongs to 𝑒−𝑓2(𝒙,𝒖) and so on. So the summands with exactly thesemulti-indices are the ones assigned to the product of exponentials whose signs areinvariant during the reordering. With this notation and all 𝐽 ∈ {0, 1}3×3 thatsatisfy the property (6.17) we can write

𝑑∏𝑙=1

𝑒−𝑓𝑙(𝒙+𝒚,𝒖) =∑

𝒋∈{0,1}3

∑𝐽

3∏𝑙=1

𝑒−𝑓𝑙(𝒙,𝒖)

𝒄(𝐽)𝑙 (←−−−−−−−−−−𝑓1,...,𝑓𝑙,0,...,0)

3∏𝑙=1

𝑒−(−1)𝑗𝑙𝑓𝑙(𝒚,𝒖)

𝑑∏𝑙=4

𝑒−𝑓𝑙(𝒙,𝒖)𝑒−𝑓𝑙(𝒚,𝒖).

(6.19)

Using mathematical induction with matrices 𝐽 ∈ {0, 1}𝛿×𝛿 as introduced abovefor growing 𝛿 and Corollary 6.2 repeatedly until we reach 𝛿 = 𝑑 we get

=∑

𝒋∈{0,1}𝑑

∑𝐽

𝑑∏𝑙=1

𝑒−𝑓𝑙(𝒙,𝒖)

𝒄(𝐽)𝑙 (←−−−−−−−−−−𝑓1,...,𝑓𝑙,0,...,0)

𝑑∏𝑙=1

𝑒−(−1)𝑗𝑙𝑓𝑙(𝒚,𝒖). (6.20)

□

172 R. Bujack, G. Scheuermann and E. Hitzer

Remark 6.10. The number of summands actually appearing is usually much smallerthan in Theorem 6.11. It is determined by the number of distinct strictly lower(upper) triangular matrices 𝐽 with entries being either zero or one, namely:

2𝑑(𝑑−1)

2 . (6.21)

Theorem 6.11 (Shift). Let 𝑨(𝒙) = 𝑩(𝒙 − 𝒙0) be multivector fields, 𝐹1, 𝐹2, lin-ear with respect to 𝒙, and let 𝒋 ∈ {0, 1}𝜇,𝒌 ∈ {0, 1}(𝜈−𝜇) be multi-indices, and𝐹1(𝒋), 𝐹2(𝒌) be as introduced in Definition 6.4, then a separable GFT suffices

ℱ𝐹1,𝐹2(𝑨)(𝒖) =∑𝒋,𝒌

∑𝐽,𝐾

𝜇∏𝑙=1

𝑒−𝑓𝑙(𝒙0,𝒖)

𝒄(𝐽)𝑙 (←−−−−−−−−−−𝑓1,...,𝑓𝑙,0,...,0)

ℱ𝐹1(𝒋),𝐹2(𝒌)(𝑩)(𝒖)

𝜈∏𝑙=𝜇+1

𝑒−𝑓𝑙(𝒙0,𝒖)

𝒄(𝐾)𝑙−𝜇 (−−−−−−−−−−→0,...,0,𝑓𝑙,...,𝑓𝜈)

,

(6.22)

where 𝐽 ∈ {0, 1}𝜇×𝜇 and 𝐾 ∈ {0, 1}(𝜈−𝜇)×(𝜈−𝜇) are the strictly lower, respectivelyupper, triangular matrices with rows (𝐽)𝑙, (𝐾)𝑙−𝜇 summing up to (

∑𝜇𝑙=1(𝐽)𝑙) mod

2 = 𝒋 respectively(∑𝜈

𝑙=𝜇+1(𝐾)𝑙−𝜇)mod 2 = 𝒌 as in Lemma 6.9.

Proof. First we rewrite the transformed function in terms of 𝑩(𝒚) using a changeof coordinates.

ℱ𝐹1,𝐹2(𝑨)(𝒖) =

∫ℝ𝑚

𝜇∏𝑙=1

𝑒−𝑓𝑙(𝒙,𝒖)𝑨(𝒙)𝜈∏

𝑙=𝜇+1

𝑒−𝑓𝑙(𝒙,𝒖) d𝑚𝒙

=

∫ℝ𝑚

𝜇∏𝑙=1

𝑒−𝑓𝑙(𝒙,𝒖)𝑩(𝒙− 𝒙0)

𝜈∏𝑙=𝜇+1

𝑒−𝑓𝑙(𝒙,𝒖) d𝑚𝒙

𝒚=𝒙−𝒙0=

∫ℝ𝑚

𝜇∏𝑙=1

𝑒−𝑓𝑙(𝒚+𝒙0,𝒖)𝑩(𝒚)

𝜈∏𝑙=𝜇+1

𝑒−𝑓𝑙(𝒚+𝒙0,𝒖) d𝑚𝒚

(6.23)

Now we separate and sort the factors using Lemma 6.9.

Lem. 6.9=

∫ℝ𝑚

∑𝒋∈{0,1}𝜇

∑𝐽∈{0,1}𝜇×𝜇

∑(𝐽)𝑙 mod 2=𝒋

𝜇∏𝑙=1

𝑒−𝑓𝑙(𝒙0,𝒖)

𝒄(𝐽)𝑙 (←−−−−−−−−−−𝑓1,...,𝑓𝑙,0,...,0)

𝜇∏𝑙=1

𝑒−(−1)𝑗𝑙𝑓𝑙(𝒚,𝒖)𝑩(𝒚)∑𝒌∈{0,1}(𝜈−𝜇)

∑𝐾∈{0,1}(𝜈−𝜇)×(𝜈−𝜇)∑

(𝐾)𝑙 mod 2=𝒌

𝜈∏𝑙=𝜇+1

𝑒−(−1)𝑘𝑙−𝜇𝑓𝑙(𝒚,𝒖)𝜈∏

𝑙=𝜇+1

𝑒−𝑓𝑙(𝒙0,𝒖)

𝒄(𝐾)𝑙−𝜇 (−−−−−−−−−−→0,...,0,𝑓𝑙,...,𝑓𝜈)

d𝑚𝒚

8. A General Geometric Fourier Transform 173

=∑𝒋,𝒌

∑𝐽,𝐾

𝜇∏𝑙=1

𝑒−𝑓𝑙(𝒙0,𝒖)

𝒄(𝐽)𝑙 (←−−−−−−−−−−𝑓1,...,𝑓𝑙,0,...,0)

(6.24)

ℱ𝐹1(𝒋),𝐹2(𝒌)(𝑩)(𝒖)

𝜈∏𝑙=𝜇+1

𝑒−𝑓𝑙(𝒙0,𝒖)

𝒄(𝐾)𝑙−𝜇 (−−−−−−−−−−→0,...,0,𝑓𝑙,...,𝑓𝜈)

□

Corollary 6.12 (Shift). Let 𝑨(𝒙) = 𝑩(𝒙 − 𝒙0) be multivector fields, 𝐹1 and 𝐹2

each consisting of mutually commutative functions3 being linear with respect to 𝒙,then the GFT obeys

ℱ𝐹1,𝐹2(𝑨)(𝒖) =

𝜇∏𝑙=1

𝑒−𝑓𝑙(𝒙0,𝒖)ℱ𝐹1,𝐹2(𝑩)(𝒖)

𝜈∏𝑙=𝜇+1

𝑒−𝑓𝑙(𝒙0,𝒖). (6.25)

Remark 6.13. For sets 𝐹1, 𝐹2 that each consist of less than two functions thecondition of Corollary 6.12 is necessarily satisfied, compare, e.g., the Clifford–Fourier transform, the quaternionic transform or the spacetime Fourier transformlisted in the preceeding examples.

The specific forms taken by our standard examples are summarized in Table 3.As expected they are often shorter than what could be expected from Remark 6.10.

7. Conclusions and Outlook

For multivector fields over ℝ𝑝,𝑞 with values in any geometric algebra 𝐺𝑝,𝑞 we havesuccessfully defined a general geometric Fourier transform. It covers all popularFourier transforms from current literature in the introductory example. Its exis-tence, independent of the specific choice of functions 𝐹1, 𝐹2, can be proved forall integrable multivector fields, see Theorem 3.1. Theorem 3.2 shows that ourgeometric Fourier transform is generally linear over the field of real numbers. Alltransforms from the reference example consist of bilinear 𝐹1 and 𝐹2. We provedthat this property is sufficient to ensure the scaling property of Theorem 4.1.

If a general geometric Fourier transform is separable as introduced in Defi-nition 6.1, then Theorem 6.5 (Left and right products) guarantees that constantfactors can be separated from the vector field to be transformed. As a consequencegeneral linearity is achieved by choosing 𝐹1, 𝐹2 with values in the centre of thegeometric algebra 𝐶ℓ𝑝,𝑞, compare Corollary 6.7. All examples except for the cylin-drical Fourier transform [6] satisfy this claim.

Under the condition of linearity with respect to the first argument of thefunctions of the sets 𝐹1 and 𝐹2 additionally to the separability property just men-tioned, we have also proved a shift property (Theorem 6.11).

In future publications we are going to state the necessary constraints fora generalized convolution theorem, invertibility, derivation theorem and we willexamine how simplifications can be achieved based on symmetry properties of the

3Cross commutativity between 𝐹1 and 𝐹2 is not necessary.

174 R. Bujack, G. Scheuermann and E. Hitzer

Table 3. Theorem 6.11 (Shift) applied to the GFTs of the first exam-ple, enumerated in the same order.Notations: on the LHS ℱ𝐹1,𝐹2 = ℱ𝐹1,𝐹2(𝑨)(𝒖), on the RHS ℱ𝐹 ′

1,𝐹′2=

ℱ𝐹 ′1,𝐹

′2(𝑩)(𝒖). In the second row 𝐾 represents all strictly upper

triangular matrices ∈ {0, 1}𝑛×𝑛 with rows (𝐾)𝑙−𝜇 summing up to(∑𝜈𝑙=𝜇+1(𝐾)𝑙−𝜇

)mod 2 = 𝒌. The simplified shape of the colour im-

age FT results from the commutativity of 𝑩 and 𝑖𝑩 and application ofLemma 2.2.

GFT 𝑨(𝒙) = 𝑩(𝒙− 𝒙0)

1. Clifford ℱ𝑓1 = ℱ𝑓1𝑒−2𝜋𝑖𝒙0⋅𝒖

2. Bulow ℱ𝑓1,...,𝑓𝑛 =∑

𝒌∈{0,1}𝑛∑

𝐾 ℱ(−1)𝑘1𝑓1,...,(−1)𝑘𝑛𝑓𝑛∏𝑛𝑙=1 𝑒

−2𝜋𝑥0𝑘𝑢𝑘

𝒄(𝐾)𝑙 (−−−−−−−−−−→0,...,0,𝑓𝑙,...,𝑓𝑛)

3. Quaternionic ℱ𝑓1,𝑓2 = 𝑒−2𝜋𝑖𝑥01𝑢1ℱ𝑓1,𝑓2𝑒−2𝜋𝑗𝑥02𝑢2

4. Spacetime ℱ𝑓1,𝑓2 = 𝑒−𝒆4𝑥04𝑢4ℱ𝑓1,𝑓2𝑒−𝜖4𝒆4𝑖4(𝑥1𝑢1+𝑥2𝑢2+𝑥3𝑢3)

5. Colour Image ℱ𝑓1,𝑓2,𝑓3,𝑓4 = 𝑒−12 (𝑥01𝑢1+𝑥02𝑢2)(𝑩+𝑖𝑩)ℱ𝑓1,𝑓2,𝑓3,𝑓4

𝑒12 (𝑥01𝑢1+𝑥02𝑢2)(𝑩+𝑖𝑩)

6. Cyl. 𝑛 = 2 ℱ𝑓1 = 𝑒𝒙0∧𝒖ℱ𝑓1

Cyl. 𝑛 ∕= 2 -

multivector fields to be transformed. We will also construct generalized geometricFourier transforms in a broad sense from combinations of the ones introduced inthis chapter and from decomposition into their sine and cosine parts which willalso cover the vector and bivector Fourier transforms of [9]. It would further beof interest to extend our approach to Fourier transforms defined on spheres orother non-Euclidean manifolds, to functions in the Schwartz space and to square-integrable functions.

References

[1] T. Batard, M. Berthier, and C. Saint-Jean. Clifford Fourier transform for color imageprocessing. In Bayro-Corrochano and Scheuermann [2], pages 135–162.

[2] E.J. Bayro-Corrochano and G. Scheuermann, editors. Geometric Algebra Computingin Engineering and Computer Science. Springer, London, 2010.

[3] F. Brackx, N. De Schepper, and F. Sommen. The Clifford–Fourier transform. Journalof Fourier Analysis and Applications, 11(6):669–681, 2005.

[4] F. Brackx, N. De Schepper, and F. Sommen. The two-dimensional Clifford–Fouriertransform. Journal of Mathematical Imaging and Vision, 26(1):5–18, 2006.

8. A General Geometric Fourier Transform 175

[5] F. Brackx, N. De Schepper, and F. Sommen. The Clifford–Fourier integral kernel ineven dimensional Euclidean space. Journal of Mathematical Analysis and Applica-tions, 365(2):718–728, 2010.

[6] F. Brackx, N. De Schepper, and F. Sommen. The Cylindrical Fourier Transform. InBayro-Corrochano and Scheuermann [2], pages 107–119.

[7] T. Bulow. Hypercomplex Spectral Signal Representations for the Processing and Anal-ysis of Images. PhD thesis, University of Kiel, Germany, Institut fur Informatik undPraktische Mathematik, Aug. 1999.

[8] W.K. Clifford. Applications of Grassmann’s extensive algebra. American Journal ofMathematics, 1(4):350–358, 1878.

[9] H. De Bie and F. Sommen. Vector and bivector Fourier transforms in Clifford anal-ysis. In K. Guerlebeck and C. Koenke, editors, 18th International Conference on theApplication of Computer Science and Mathematics in Architecture and Civil Engi-neering, page 11, 2009.

[10] J. Ebling. Visualization and Analysis of Flow Fields using Clifford Convolution. PhDthesis, University of Leipzig, Germany, 2006.

[11] T.A. Ell. Quaternion-Fourier transforms for analysis of 2-dimensional linear time-invariant partial-differential systems. In Proceedings of the 32nd Conference on De-cision and Control, pages 1830–1841, San Antonio, Texas, USA, 15–17 December1993. IEEE Control Systems Society.

[12] T.A. Ell and S.J. Sangwine. Hypercomplex Fourier transforms of color images. IEEETransactions on Image Processing, 16(1):22–35, Jan. 2007.

[13] M. Felsberg. Low-Level Image Processing with the Structure Multivector. PhD thesis,Christian-Albrechts-Universitat, Institut fur Informatik und Praktische Mathematik,Kiel, 2002.

[14] D. Hestenes and G. Sobczyk. Clifford Algebra to Geometric Calculus. D. ReidelPublishing Group, Dordrecht, Netherlands, 1984.

[15] E. Hitzer. Quaternion Fourier transform on quaternion fields and generalizations.Advances in Applied Clifford Algebras, 17(3):497–517, May 2007.

[16] E. Hitzer and R. Ablamowicz. Geometric roots of −1 in Clifford algebras 𝐶ℓ𝑝,𝑞 with𝑝 + 𝑞 ≤ 4. Advances in Applied Clifford Algebras, 21(1):121–144, 2010. Publishedonline 13 July 2010.

[17] E. Hitzer, J. Helmstetter, and R. Ablamowicz. Square roots of −1 in real Cliffordalgebras. In K. Gurlebeck, editor, 9th International Conference on Clifford Algebrasand their Applications, Weimar, Germany, 15–20 July 2011. 12 pp.

[18] E.M.S. Hitzer and B. Mawardi. Clifford Fourier transform on multivector fields anduncertainty principles for dimensions 𝑛 = 2(mod 4) and 𝑛 = 3(mod 4). Advances inApplied Clifford Algebras, 18(3-4):715–736, 2008.

[19] B. Jancewicz. Trivector Fourier transformation and electromagnetic field. Journalof Mathematical Physics, 31(8):1847–1852, 1990.

[20] S.J. Sangwine and T.A. Ell. The discrete Fourier transform of a colour image. In J.M.Blackledge and M.J. Turner, editors, Image Processing II Mathematical Methods, Al-gorithms and Applications, pages 430–441, Chichester, 2000. Horwood Publishing forInstitute of Mathematics and its Applications. Proceedings Second IMA Conferenceon Image Processing, De Montfort University, Leicester, UK, September 1998.

176 R. Bujack, G. Scheuermann and E. Hitzer

[21] F. Sommen. Hypercomplex Fourier and Laplace transforms I. Illinois Journal ofMathematics, 26(2):332–352, 1982.

Roxana BujackUniversitat LeipzigInstitut fur InformatikJohannisgasse 26D-04103 Leipzig, Germanye-mail: bujack@informatik.uni-leipzig.de

Gerik ScheuermannUniversitat LeipzigInstitut fur InformatikJohannisgasse 26D-04103 Leipzig, Germanye-mail: scheuermann@informatik.uni-leipzig.de

Eckhard HitzerCollege of Liberal Arts, Department of Material Science,International Christian University,181-8585 Tokyo, Japane-mail: hitzer@icu.ac.jp

Quaternion and Clifford–Fourier Transforms and Wavelets

Trends in Mathematics, 177–195c⃝ 2013 Springer Basel

9 Clifford–Fourier Transform andSpinor Representation of Images

Thomas Batard and Michel Berthier

Abstract. We propose in this chapter to introduce a spinor representation forimages based on the work of T. Friedrich. This spinor representation gener-alizes the usual Weierstrass representation of minimal surfaces (i.e., surfaceswith constant mean curvature equal to zero) to arbitrary surfaces (immersedin ℝ3). We investigate applications to image processing focusing on segmen-tation and Clifford–Fourier analysis. All these applications involve sections ofthe spinor bundle of image graphs, that is spinor fields, satisfying the so-calledDirac equation.

Mathematics Subject Classification (2010). Primary 68U10, 53C27; secondary53A05, 43A32.

Keywords. Image processing, spin geometry, Clifford–Fourier transform.

1. Introduction

The idea of this chapter is to perform grey-level image processing using the geo-metric information given by the Gauss map variations of image graphs. While itis well known that one can parameterize the Gauss map of a minimal surface bya meromorphic function (see below), it is a much more recent result (see [5]) thatsuch a parametrization can be extended to arbitrary surfaces of ℝ3 when dealingwith spin geometry.

Let us first recall that a minimal surface Σ immersed in ℝ3, that is a surfacewith constant mean curvature equal to zero, can be described with one holomor-phic function 𝜑 and one meromorphic function 𝜓 such that the product 𝜑𝜓2 isholomorphic. This is the so-called Weierstrass representation of Σ (see [6] or [8]for details). The function 𝜓 is nothing else but the composition of the Gauss mapof Σ with the stereographic projection from the unit sphere to the complex plane.

This work was partially supported by ONR Grant N00014-09-1-0493.

178 T. Batard and M. Berthier

The main result of T. Friedrich in [5] states that there is a one-to-one cor-respondance between spinor fields 𝜑∗ of constant length on a Riemannian surface(Σ, 𝑔) and satisfying

𝐷𝜑∗ = 𝐻𝜑∗ (1.1)

where 𝐷 is a Dirac operator in one hand, and isometric immersions of Σ in ℝ3

with mean curvature equal to𝐻 , on the other hand. The Weierstrass representationappears to be the particular case corresponding to 𝐻 ≡ 0.

Let us describe now the method introduced in the following. Let

𝜒 : Ω ⊂ ℝ2 −→ ℝ3

(𝑥, 𝑦) �−→ (𝑥, 𝑦, 𝐼(𝑥, 𝑦))(1.2)

be the immersion in the three-dimensional Euclidean space of a grey-level image𝐼 defined on a domain Ω of ℝ2. The first step (see § 2) consists in computing thespinor field 𝜑∗ that describes the image surface Σ. We follow here the paper of T.Friedrich [5]: 𝜑∗ is obtained from the restriction to the surface Σ of a parallel spinor𝜙 on ℝ3. The computation of 𝜑∗ requires us to deal with irreducible representationsof the complex Clifford algebra 𝐶ℓ3,0 ⊗ ℂ and with the generalized Weierstrassrepresentation of Σ based on period forms. In practice, 𝜑∗ is given by a field ofelements of ℂ2.

As said before, the spinor field 𝜑∗ characterizes the geometry of the surfaceΣ immersed in ℝ3 by the parametrization (1.2). In the same way that the normalof a minimal surface is parameterized by the meromorphic function 𝜓, the normalof the surface Σ is parameterized by the spinor field 𝜑∗. The latter explains howthe tangent plane to Σ varies in the ambient space.

There are many reasons to believe that such a generalized Weierstrass para-metrization may reveal itself to be an efficient tool in the context of image pro-cessing:

1. The field 𝜑∗ of elements of ℂ2 (see (2.26)) encodes the Riemannian structureof the surface Σ in a very tractable way (although the definition of 𝜑∗ mayappear quite complicated).

2. The geometrical methods based on the study of the so-called structure tensorinvolve only the eigenvalues of the structure tensor, that means in some sensethe values of the first fundamental form of the surface. The spinor field 𝜑∗

contains both intrinsic and extrinsic information. Studying the variations of𝜑∗ allows us to get not only information about the variations (derivative) ofthe first fundamental form, but also about the geometric embedding of thesurface Σ and in particular about the mean curvature.

3. We are dealing here with first-order instead of zero-order geometric variationsof Σ. As shown later, this appears to be more relevant by taking into accountboth edges and textures.

4. As will be detailed in the sequel, the spinor field 𝜑∗ can be decomposed as aseries of basic spinor fields using a suitable Clifford–Fourier transform. Thisseries corresponds to a harmonic decomposition of the surface Σ adapted to

9. Clifford–Fourier Transform 179

the Riemannian geometry. This is in fact the main novelty of this chaptersince the usual techniques of Fourier analysis do not involve geometric data.

5. One can envisage the possibility of performing diffusion in this context. Theusual Laplace Beltrami operator can be replaced by the squared Atiyah SingerDirac operator [7] (the Atiyah Singer Dirac operator acting as an ellipticoperator of order one on spinor fields).

To illustrate some of these ideas, we investigate rapidly in § 3 applications tosegmentation and more precisely to edge and texture detection. As stated before,the basic idea is to replace the usual order-one structure tensor by an order-twostructure tensor called the spinor tensor obtained from the derivative of the spinorfield 𝜑∗. This spinor tensor measures the variations of the unit normal of theimage surface. Experiments show that this approach is particularly well adaptedto texture detection.

We define in § 4 the Clifford–Fourier transform of a spinor field. For this,we follow the approach of [3] that relies on a spin generalization of the usualnotion of group character. We are led to compute the group morphisms fromℤ/𝑀ℤ×ℤ/𝑁ℤ to Spin(3). Since this last group acts on the sections of the spinorbundle, a Clifford–Fourier transform can be defined by averaging this action. Oneof the key ideas here is to split the spinor bundle of the surface according to theClifford multiplication by the bivector coding the tangent plane to the surface.This has two advantages: the first one is to involve the geometry in the process,the second one is to reduce the computation of the Clifford–Fourier transform totwo usual complex Fourier transforms. It is important to notice that although theFourier transform we propose is, as usual, a global transformation on the image, theway it is computed takes into account local geometric data. We finally introducethe harmonic decomposition mentioned above and show some results of filteringon standard images.

The reader will find in Appendix A the mathematical definitions and resultsused throughout the text.

2. Spinor Representation of Images

This section is devoted to the explicit computation of the spinor field 𝜑∗ of a givensurface immersed in Euclidean space. It is obtained as the restriction of a constantspinor field of ℝ3 the components of which are determined using period forms.

2.1. Spinors and Graphs

Let 𝐼 : Ω −→ ℝ be a differentiable function defined on a domain Ω of ℝ2. Weconsider the surface Σ immersed in ℝ3 by the parametrization:

𝜒(𝑥, 𝑦) = (𝑥, 𝑦, 𝐼(𝑥, 𝑦)). (2.1)

Also, let 𝑔 be the metric on Σ induced by the Euclidean metric of ℝ3. The cou-ple (Σ, 𝑔) is a Riemannian surface of global chart (Ω, 𝜒). We denote by 𝑀 theRiemannian manifold (ℝ3, ∥ ∥2) and by (𝑧1, 𝑧2, 𝜈) an orthonormal frame field of

180 T. Batard and M. Berthier

𝑀 with (𝑧1, 𝑧2) an orthonormal frame field on Σ, and by 𝜈 the global unit fieldnormal to Σ. One can choose (𝑧1, 𝑧2, 𝜈) with the following matrix representation⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

𝐼𝑥√(𝐼2

𝑥 + 𝐼2𝑦 )(𝐼

2𝑥 + 𝐼2

𝑦 + 1)

−𝐼𝑦√𝐼2𝑥 + 𝐼2

𝑦

−𝐼𝑥√𝐼2𝑥 + 𝐼2

𝑦 + 1

𝐼𝑦√(𝐼2

𝑥 + 𝐼2𝑦 )(𝐼

2𝑥 + 𝐼2

𝑦 + 1)

𝐼𝑥√𝐼2𝑥 + 𝐼2

𝑦

−𝐼𝑦√𝐼2𝑥 + 𝐼2

𝑦 + 1

𝐼2𝑥 + 𝐼2

𝑦√(𝐼2

𝑥 + 𝐼2𝑦 )(𝐼

2𝑥 + 𝐼2

𝑦 + 1)0

1√𝐼2𝑥 + 𝐼2

𝑦 + 1

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠. (2.2)

Note that 𝑧1 and 𝑧2 are not defined when 𝐼𝑥 = 𝐼𝑦 = 0. This has no consequencein the sequel since we deal only with the normal 𝜈.

Following [5] the surface Σ can be represented by a spinor field 𝜑∗ withconstant length satisfying the Dirac equation:

𝐷𝜑∗ = 𝐻𝜑∗ (2.3)

where 𝐻 denotes the mean curvature of Σ. We recall here the basic idea (seeAppendix A for notations and definitions). Let 𝜙 be a parallel spinor field of 𝑀 ,i.e., satisfying

∇𝑀𝑋 𝜙 = 0 (2.4)

for all vector fields 𝑋 on 𝑀 . Let also 𝜑 be the restriction 𝜙∣Σ of 𝜙 to Σ. The spinorfield 𝜑 decomposes into

𝜑 = 𝜑+ + 𝜑− (2.5)

with

𝜑+ =1

2(𝜑+ 𝑖𝜈 ⋅ 𝜑) 𝜑− =

1

2(𝜑− 𝑖𝜈 ⋅ 𝜑) (2.6)

and satisfies

𝐷𝜑 = −𝐻 ⋅ 𝜈 ⋅ 𝜑. (2.7)

This last equation reads

𝐷(𝜑+ + 𝜑−) = −𝐻 ⋅ 𝜈 ⋅ (𝜑+ + 𝜑−) (2.8)

and implies

𝐷𝜑+ = −𝑖𝐻𝜑− 𝐷𝜑− = 𝑖𝐻𝜑+. (2.9)

If we set 𝜑∗ = 𝜑+ − 𝑖𝜑− then 𝐷𝜑∗ = 𝐻𝜑∗ and 𝜑∗ is of constant length.

9. Clifford–Fourier Transform 181

Proposition 2.1. The spinor fields 𝜑+, 𝜑− and 𝜑∗ are given by

𝜑+ =1

2

⎛⎜⎜⎜⎜⎜⎜⎜⎝

⎛⎝1− 𝐼𝑦√1 + 𝐼2

𝑥 + 𝐼2𝑦

⎞⎠ 𝑢+

⎛⎝ 𝐼𝑥 − 𝑖√1 + 𝐼2

𝑥 + 𝐼2𝑦

⎞⎠ 𝑣

⎛⎝1 +𝐼𝑦√

1 + 𝐼2𝑥 + 𝐼2

𝑦

⎞⎠ 𝑣 +

⎛⎝ 𝐼𝑥 + 𝑖√1 + 𝐼2

𝑥 + 𝐼2𝑦

⎞⎠ 𝑢

⎞⎟⎟⎟⎟⎟⎟⎟⎠(2.10)

𝜑− =1

2

⎛⎜⎜⎜⎜⎜⎜⎜⎝

⎛⎝1 +𝐼𝑦√

1 + 𝐼2𝑥 + 𝐼2

𝑦

⎞⎠ 𝑢−⎛⎝ 𝐼𝑥 − 𝑖√

1 + 𝐼2𝑥 + 𝐼2

𝑦

⎞⎠ 𝑣

⎛⎝1− 𝐼𝑦√1 + 𝐼2

𝑥 + 𝐼2𝑦

⎞⎠ 𝑣 −⎛⎝ 𝐼𝑥 + 𝑖√

1 + 𝐼2𝑥 + 𝐼2

𝑦

⎞⎠ 𝑢

⎞⎟⎟⎟⎟⎟⎟⎟⎠(2.11)

and

𝜑∗ =1

2(1− 𝑖)

⎛⎜⎜⎜⎜⎜⎜⎜⎝

⎛⎝1− 𝑖𝐼𝑦√1 + 𝐼2

𝑥 + 𝐼2𝑦

⎞⎠ 𝑢+

⎛⎝ 1 + 𝑖𝐼𝑥√1 + 𝐼2

𝑥 + 𝐼2𝑦

⎞⎠ 𝑣

⎛⎝1 +𝑖𝐼𝑦√

1 + 𝐼2𝑥 + 𝐼2

𝑦

⎞⎠ 𝑣 +

⎛⎝ 𝑖𝐼𝑥 − 1√1 + 𝐼2

𝑥 + 𝐼2𝑦

⎞⎠ 𝑢

⎞⎟⎟⎟⎟⎟⎟⎟⎠(2.12)

where 𝑢 and 𝑣 are (constant) complex numbers.

Proof. Since 𝜙 is a parallel spinor field on 𝑀 , 𝜙 = (𝑢, 𝑣) where 𝑢 and 𝑣 are two(constant) complex numbers. Let 𝜌2 be the irreducible complex representation ofℂ𝑙(3) described in Appendix A.1. Recall that

𝜈 =1

Δ(−𝐼𝑥𝑒1 − 𝐼𝑦𝑒2 + 𝑒3) (2.13)

where Δ =√

𝐼2𝑥 + 𝐼2

𝑦 + 1, so that

𝜌2(𝜈) = −𝐼𝑥Δ

(0 𝑖𝑖 0

)− 𝐼𝑦

Δ

( −𝑖 00 𝑖

)+

1

Δ

(0 −11 0

). (2.14)

By definition:

𝜈 ⋅ 𝜑 = 𝜌2(𝜈)

(𝑢𝑣

). (2.15)

Simple computations lead now to the result. □

The next step consists in computing the components (𝑢, 𝑣) of the constant field 𝜙.This is done by considering a quaternionic structure on the spinor bundle 𝑆(Σ) ofthe surface Σ and period forms.

182 T. Batard and M. Berthier

2.2. Quaternionic Structure and Period Forms

Let 𝐼 be the complex structure on 𝑆(Σ) given by the multiplication by 𝑖. A quater-nionic structure on 𝑆(Σ) is a linear map 𝐽 that satisfies 𝐽2 = −𝐼𝑑 and 𝐼𝐽 = −𝐽𝐼.In the sequel 𝐽 is given by

𝐽

(𝜑1

𝜑2

)=

( −𝜑2

𝜑1

). (2.16)

If we write 𝜑1 = 𝛼1 + 𝑖𝛽1 and 𝜑2 = 𝛼2 + 𝑖𝛽2, the corresponding quaternion isgiven by

𝜑1 + 𝜑2𝑗 = (𝛼1 + 𝑖𝛽1) + (𝛼2 + 𝑖𝛽2)𝑗 = 𝛼1 + 𝑖𝛽1 + 𝛼2𝑗 + 𝛽2𝑘 (2.17)

and

𝑗(𝜑1 + 𝜑2𝑗) = −𝜑2 + 𝜑1𝑗, (2.18)

i.e., 𝐽 is the left multiplication by 𝑗. Since

𝑆+(Σ) =

{(𝜑1

𝜑2

), 𝜑1 =

𝐼𝑥 − 𝑖

𝐼𝑦 +Δ𝜑2

}(2.19)

and

𝑆+(Σ) =

{(𝜑1

𝜑2

), 𝜑1 =

𝐼𝑥 − 𝑖

𝐼𝑦 −Δ𝜑2

}(2.20)

then 𝐽𝑆+(Σ) ⊂ 𝑆−(Σ) and 𝐽𝑆−(Σ) ⊂ 𝑆+(Σ). We also denote by 𝐽 the quater-nionic structure (obtained in the same way) on 𝑆(𝑀).

Let us consider 𝜙 = (𝑢, 𝑣) a constant spinor field on 𝑀 and 𝜑∗ its restrictionon Σ. Let also 𝑓 : ℝ3 −→ ℝ and 𝑔 : ℝ3 −→ ℂ be the functions defined by

𝑓(𝑚) = −ℑ(𝑚 ⋅ 𝜙, 𝜙) (2.21)

and

𝑔(𝑚) = 𝑖(𝑚 ⋅ 𝜙, 𝐽(𝜙)) (2.22)

where ( , ) denotes the Hermitian product. Using the representation 𝜌2, one cancheck that

𝑚 ⋅ 𝜙 =

(−𝑖𝑚2𝑢+ (𝑖𝑚1 −𝑚3)𝑣

(𝑖𝑚1 +𝑚3)𝑢+ 𝑖𝑚2𝑣

)(2.23)

for 𝑚 = (𝑚1,𝑚2,𝑚3). The equations 𝑓(𝑚) = 𝑚1 and 𝑔(𝑚) = 𝑚2 + 𝑖𝑚3 areequivalent to:

∣𝑢∣2 = ∣𝑣∣2 , 𝑢𝑣 = −1

2(2.24)

and

𝑢𝑣 = −1

2, 𝑢2 + 𝑣2 = 1, 𝑢2 = 𝑣2. (2.25)

This implies 𝑢 = ±1/√2 and 𝑣 = −𝑢.

9. Clifford–Fourier Transform 183

Definition 2.2. The spinor representation of the image given by the parametrization(2.1) is defined by

𝜑∗ =1

2√2(1− 𝑖)

⎛⎜⎜⎜⎜⎜⎜⎜⎝

⎛⎝1− 1 + 𝑖( 𝐼𝑥 + 𝐼𝑦)√1 + 𝐼2

𝑥 + 𝐼2𝑦

⎞⎠−⎛⎝1 +

1 + 𝑖(−𝐼𝑥 + 𝐼𝑦)√1 + 𝐼2

𝑥 + 𝐼2𝑦

⎞⎠

⎞⎟⎟⎟⎟⎟⎟⎟⎠. (2.26)

This means that 𝑢 = 1/√2 and 𝑣 = −1/√2 in the expression (2.12).

The two 1-forms

𝜂𝑓 (𝑋) = 2ℜ(𝑋 ⋅ (𝜑∗)+, (𝜑∗)−) = −ℑ(𝑋 ⋅ 𝜑, 𝜑) (2.27)

𝜂𝑔(𝑋) = 𝑖(𝑋 ⋅ (𝜑∗)+, 𝐽((𝜑∗)+)) + 𝑖(𝑋 ⋅ (𝜑∗)−, 𝐽((𝜑∗)−))= 𝑖(𝑋 ⋅ 𝜑, 𝐽(𝜑)) (2.28)

are exact and verify 𝑑(𝑓∣Σ) = 𝜂𝑓 , 𝑑(𝑔∣Σ) = 𝜂𝑔. The generalized Weierstrass para-metrization is actually given by the isometric immersion:∫

(𝜂𝑓 , 𝜂𝑔) : Σ −→𝑀. (2.29)

2.3. Dirac Equation and Mean Curvature

We only mention here some results that can be used when dealing with diffusion.We do not go into further details since we will not treat this problem in the presentchapter. Let (Σ, 𝑔) be an oriented two-dimensional Riemannian manifold and 𝜑 aspinor field without zeros solution of the Dirac equation 𝐷𝜑 = 𝜆𝜑. Then 𝜑 definesan isometric immersion

(Σ, ∣𝜑∣4𝑔) −→ ℝ3 (2.30)

with mean curvature 𝐻 = 𝜆/∣𝜑∣2 (see [5]).

3. Spinors and Segmentation

The aim of this section is to introduce the spinor tensor corresponding to thevariations of the unit normal and to show its capability to detect both edges andtextures.

3.1. The Spinor Tensor

We propose here to deal with a second-order version of the classical approachof edge detection based on the so-called structure tensor (see [10]). Instead ofmeasuring edges from eigenvalues of the Riemannian metric, we focus here on the

184 T. Batard and M. Berthier

eigenvalues of the tensor obtained from the derivative of the spinor field 𝜑∗. Moreprecisely let

𝜑 =

(𝜑1

𝜑2

)(3.1)

be a section of the spinor bundle 𝑆(Σ) given in an orthonormal frame, i.e., ∣𝜑∣2 =

∣𝜑1∣2 + ∣𝜑2∣2 and let 𝑋 = (𝑋1, 𝑋2) be a section of the tangent bundle 𝑇 (Σ). Weconsider the connection ∇ on 𝑆(Σ) given by the connection 1-form 𝜔 = 0. Thus

∇𝑋𝜑 =

⎛⎜⎜⎝ 𝑋1∂𝜑1

∂𝑥+𝑋2

∂𝜑1

∂𝑦

𝑋1∂𝜑2

∂𝑥+𝑋2

∂𝜑2

∂𝑦

⎞⎟⎟⎠ (3.2)

and

∣∇𝑋𝜑∣2 = 𝑋21

∣∣∣∣∂𝜑1

∂𝑥

∣∣∣∣2 + 2𝑋1𝑋2ℜ(∂𝜑1

∂𝑥

∂𝜑1

∂𝑦

)+𝑋2

2

∣∣∣∣∂𝜑1

∂𝑦

∣∣∣∣2+𝑋2

1

∣∣∣∣∂𝜑2

∂𝑥

∣∣∣∣2 + 2𝑋1𝑋2ℜ(∂𝜑2

∂𝑥

∂𝜑2

∂𝑦

)+𝑋2

2

∣∣∣∣∂𝜑2

∂𝑦

∣∣∣∣2 . (3.3)

If we denote

𝐺𝜑 =

⎛⎜⎜⎜⎝∣∣∣∣∂𝜑1

∂𝑥

∣∣∣∣2+ ∣∣∣∣∂𝜑2

∂𝑥

∣∣∣∣2 ℜ(∂𝜑1

∂𝑥

∂𝜑1

∂𝑦+

∂𝜑2

∂𝑥

∂𝜑2

∂𝑦

)ℜ(∂𝜑1

∂𝑥

∂𝜑1

∂𝑦+

∂𝜑2

∂𝑥

∂𝜑2

∂𝑦

) ∣∣∣∣∂𝜑1

∂𝑦

∣∣∣∣2+ ∣∣∣∣∂𝜑2

∂𝑦

∣∣∣∣2⎞⎟⎟⎟⎠ (3.4)

then

(𝑋1 𝑋2)𝐺𝜑(𝑋1 𝑋2)𝑇 = ∣∇𝑋𝜑∣2 . (3.5)

𝐺𝜑 is a field of real symmetric matrices.As in the case of the usual structure tensor (i.e., Di Zenzo tensor, see [10])

the optima of ∣∇𝑋𝜑∣2 under the constraint ∥𝑋∥ = 1 (for the Euclidean norm) aregiven by the field of eigenvalues of 𝐺𝜑. Applying the above formula to the spinor𝜑∗ of Definition 2.2 leads to

𝐺𝜑∗ =1

2(1 + 𝐼2𝑥 + 𝐼2

𝑦 )2

(𝐺11

𝜑∗ 𝐺12𝜑∗

𝐺21𝜑∗ 𝐺22

𝜑∗

)(3.6)

with

𝐺11𝜑∗ = 𝐼2

𝑥𝑥 + 𝐼2𝑥𝑦 + 𝐼2

𝑥𝑥𝐼2𝑦 + 𝐼2

𝑥𝑦𝐼2𝑥 − 2𝐼𝑥𝑥𝐼𝑥𝑦𝐼𝑥𝐼𝑦

𝐺22𝜑∗ = 𝐼2

𝑦𝑦 + 𝐼2𝑥𝑦 + 𝐼2

𝑦𝑦𝐼2𝑥 + 𝐼2

𝑥𝑦𝐼2𝑦 − 2𝐼𝑦𝑦𝐼𝑥𝑦𝐼𝑥𝐼𝑦

𝐺12𝜑∗ = 𝐼𝑥𝑥𝐼𝑥𝑦 + 𝐼𝑥𝑦𝐼𝑦𝑦 + 𝐼𝑥𝑥𝐼𝑥𝑦𝐼

2𝑦 + 𝐼𝑥𝑦𝐼𝑦𝑦𝐼

2𝑥 − 𝐼2

𝑥𝑦𝐼𝑥𝐼𝑦 − 𝐼𝑥𝑥𝐼𝑦𝑦𝐼𝑥𝐼𝑦

𝐺21𝜑∗ = 𝐺12

𝜑∗ .

(3.7)

9. Clifford–Fourier Transform 185

Definition 3.1. The tensor 𝐺𝜑∗ is called the spinor tensor of the surface Σ.

Note that as already mentioned this last tensor corresponds to the tensor involvedin the measure of the variations of the unit normal 𝜈 introduced in § 2.1. Indeed,we have

(𝑋1 𝑋2)𝐺𝜑∗(𝑋1 𝑋2)𝑇 = ∥𝑑𝑋𝜈∥2. (3.8)

3.2. Experiments

We compare in Figure 1 the edge and texture detection methods based on theusual structure tensor (Figure 1(b) and 1(d)) and on the spinor tensor (Figure 1(e)and 1(f)).

The structure tensor only takes into account the first-order derivatives of thefunction 𝐼. The subsequent segmentation method detects the strongest grey-levelvariations of the image. As a consequence, this method provides thick edges, ascan be observed.

The spinor tensor takes into account the second-order derivatives of the func-tion 𝐼 too. By definition, it measures the strongest variations of the unit normalto the surface parametrized by the graph of 𝐼. We observe that this new approachprovides thinner edges than the first one. It appears also to be more relevant todetect textures.

4. Spinors and Clifford–Fourier Transform

We first define a Clifford–Fourier transform using spin characters that is groupmorphisms from ℝ2 to Spin(3). Then, we introduce a harmonic decomposition ofspinor fields and show some results of filtering applied to images.

4.1. Clifford–Fourier Transform with Spin Characters

Let us recall the idea of the construction of the Clifford–Fourier transform forcolour image processing introduced in [3]. From the mathematical viewpoint, aFourier transform is defined through group actions and more precisely throughirreducible and unitary representations of the involved group. This is closely relatedto the well-known shift theorem stating that:

ℱ𝑓𝛼(𝑢) = 𝑒𝑖𝛼𝑢ℱ𝑓(𝑢) (4.1)

where 𝑓𝛼(𝑢) = 𝑓(𝛼+ 𝑢). The group morphism

𝛼 �−→ 𝑒𝑖𝛼𝑢 (4.2)

is a so-called character of the additive group (ℝ,+), that is an irreducible unitaryrepresentation of dimension 1.

The definition proposed in [3] relies on a Clifford generalization of this notionby introducing spin characters. It can be shown that the group morphisms fromℤ/𝑀ℤ× ℤ/𝑁ℤ to Spin(3) are given by

𝜌𝑢,𝑣,𝐵 : (𝑚,𝑛) �−→ 𝑒2𝜋(𝑢𝑚/𝑀+𝑣𝑛/𝑁)𝐵 (4.3)

186 T. Batard and M. Berthier

9. Clifford–Fourier Transform 187

where

𝑒2𝜋(𝑢𝑚𝑀 + 𝑣𝑛

𝑁 )𝐵 = cos 2𝜋(𝑢𝑚𝑀 + 𝑣𝑛

𝑁

)+ sin 2𝜋

(𝑢𝑚𝑀 + 𝑣𝑛

𝑁

)𝐵 (4.4)

(𝑢, 𝑣) ∈ ℤ/𝑀ℤ× ℤ/𝑁ℤ, and

𝐵 = 𝛾1𝑒1𝑒2 + 𝛾2𝑒1𝑒3 + 𝛾3𝑒2𝑒3 (4.5)

is a unit bivector, i.e., 𝛾21 + 𝛾2

2 + 𝛾23 = 1. The map 𝜌𝑢,𝑣,𝐵 is called a spin character

of the group ℤ/𝑀ℤ × ℤ/𝑁ℤ. Recalling that Spin(3) acts on the sections of thespinor bundle, we are led to propose the following definition.

Definition 4.1. The Clifford–Fourier transform of a spinor 𝜑 of 𝑆(Σ) is given by

ℱ(𝜑)(𝑢, 𝑣) =∑

𝑛∈ℤ/𝑁ℤ

𝑚∈ℤ/𝑀ℤ

𝜌𝑢,𝑣,𝑧1∧𝑧2 (𝑚,𝑛)(−𝑚,−𝑛) ⋅ 𝜑(𝑚,𝑛) (4.6)

where (𝑧1, 𝑧2) is an orthonormal frame of 𝑇 (Σ).

Since the spinor bundle of Σ splits into

𝑆(Σ) = 𝑆+𝑧1∧𝑧2(Σ)⊕ 𝑆−𝑧1∧𝑧2(Σ) (4.7)

we have

𝜌𝑢,𝑣,𝑧1∧𝑧2(𝑚,𝑛)(−𝑚,−𝑛) ⋅ 𝜑(𝑚,𝑛) = 𝑒2𝜋𝑖(𝑢𝑚𝑀 + 𝑣𝑛

𝑁

)𝜑+(𝑚,𝑛) 𝑣−𝑖(𝑚,𝑛)

+ 𝑒−2𝜋𝑖(𝑢𝑚𝑀 + 𝑣𝑛

𝑁

)𝜑−(𝑚,𝑛) 𝑣𝑖(𝑚,𝑛) (4.8)

where 𝑣−𝑖, respectively 𝑣𝑖, is the unit eigenspinor field of eigenvalue−𝑖, respectively𝑖, relatively to the operator 𝑧1 ∧ 𝑧2 ⋅ (here ⋅ denotes the Clifford multiplication).Consequently

ℱ(𝜑)(𝑢, 𝑣) =(𝜑+

−1(𝑢, 𝑣), 𝜑−(𝑢, 𝑣)

)(4.9)

in the frame (𝑣−𝑖, 𝑣𝑖), where ˆ and ˆ −1 denote the Fourier transform on

𝐿2(ℤ/𝑀ℤ× ℤ/𝑁ℤ,ℂ),

also called discrete Fourier transform, and its inverse.

4.2. Spinor Field Decomposition

The inverse Clifford–Fourier transform of 𝜑 is

ℱ−1(𝜑)(𝑢, 𝑣) =∑

𝑛∈ℤ/𝑁ℤ

𝑚∈ℤ/𝑀ℤ

𝜌𝑢,𝑣,𝑧1∧𝑧2(𝑚,𝑛)(𝑚,𝑛) ⋅ 𝜑(𝑚,𝑛) (4.10)

This means that every spinor field 𝜑 may be written as a superposition of basicspinor fields, i.e.,

𝜑 =∑

𝜑𝑚,𝑛 (4.11)

where

𝜑𝑚,𝑛 : (𝑢, 𝑣) �−→ 𝜌𝑢,𝑣,𝑧1∧𝑧2(𝑚,𝑛)(𝑚,𝑛) ⋅ ℱ(𝜑)(𝑚,𝑛) (4.12)

188 T. Batard and M. Berthier

Following the splitting 𝑆(Σ) = 𝑆+𝑧1∧𝑧2(Σ)⊕ 𝑆−𝑧1∧𝑧2(Σ), we have

𝜑𝑚,𝑛 =(𝜑+𝑚,𝑛, 𝜑

−𝑚,𝑛

)in the frame (𝑣−𝑖, 𝑣𝑖), with

𝜑+𝑚,𝑛 : (𝑢, 𝑣) �−→ 𝑒−2𝜋𝑖(𝑢𝑚/𝑀+𝑣𝑛/𝑁)𝜑+

−1(𝑚,𝑛)

and

𝜑−𝑚,𝑛 : (𝑢, 𝑣) �−→ 𝑒2𝜋𝑖(𝑢𝑚/𝑀+𝑣𝑛/𝑁)𝜑−(𝑚,𝑛)

Moreover,

∣𝜑𝑚,𝑛∣2 = ∣𝜑+𝑚,𝑛∣2 + ∣𝜑−𝑚,𝑛∣2

since 𝑆+𝑧1∧𝑧2(Σ) and 𝑆−𝑧1∧𝑧2(Σ) are orthogonal.

4.3. Experiments

Let us now give an example of applications of the Clifford–Fourier transform onspinor fields to image processing. In order to perform filtering with the decompo-sition (4.11), we proceed as follows. Let 𝐼 be a grey-level image, and 𝜑∗ be thecorresponding spinor representation given in Definition 2.2. We apply a Gaussianmask 𝑇𝜎 of variance 𝜎 in the spectrum ℱ𝜑∗ of 𝜑∗. Then, we consider the norm ofits inverse Fourier transform, i.e., ∣ℱ−1𝑇𝜎ℱ𝜑∗∣ and the function ∣ℱ−1𝑇𝜎ℱ𝜑∗∣ 𝐼.

Figures 2 and 3 show results of this process for different values of 𝜎 (leftcolumn ∣ℱ−1𝑇𝜎ℱ𝜑∗∣ and right column ∣ℱ−1𝑇𝜎ℱ𝜑∗∣ 𝐼). It is clear that for 𝜎 suffi-ciently high, we have ∣ℱ−1𝑇𝜎ℱ𝜑∗∣ 𝐼 ≃ 𝐼 and ∣ℱ−1𝑇𝜎ℱ𝜑∗∣ ≃ 1 since ∣𝜑∗∣ = 1. Thisexplains why the two left lower images are almost white and the two right lowerimages are almost the same as the originals.

We can see in the left columns of Figures 2 and 3 that the filtering actsthrough 𝜑∗ as a smoothing of the geometry of the image. More precisely, when 𝜎is small, the modulus ∣ℱ−1𝑇𝜎ℱ𝜑∗∣ is small at points corresponding to nearly allthe geometric variations of the image. When 𝜎 increases the modulus is affectedonly at points corresponding to the strongest geometric variations, i.e., to bothedges and textures (and also where the noise is high).

The right columns of Figures 2 and 3 show that the filtering acts through∣ℱ−1𝑇𝜎ℱ𝜑∗∣ 𝐼 as a diffusion that leaves the geometric data untouched (the higherthe value of 𝜎, the more important is the diffusion). This appears clearly in Figure 4(compare the plumes of the hat) or in Figure 5 (compare the hair).

These experiments show that our approach is relevant to deal with harmonicanalysis together with Riemannian geometry.

Conclusion

Spin geometry is a powerful mathematical tool to deal with many theoreticaland applied geometric problems. In this chapter we have shown how to take ad-vantage of the generalized Weierstrass representation to perform grey-level imageprocessing, in particular edge and texture detection. Our main contribution is the

9. Clifford–Fourier Transform 189

Figure 2. Left: ∣ℱ−1(𝑇𝜎 ℱ𝜑∗)∣ for 𝜎 = 100, 1000, 10000, 100000 (fromtop to bottom). Right: ∣ℱ−1(𝑇𝜎 ℱ𝜑∗)∣𝐼

definition of a Clifford–Fourier transform for spinor fields that relies on a general-ization of the usual notion of character (the spin character). One important factis that this new transform takes into account the Riemannian geometry of the

190 T. Batard and M. Berthier

Figure 3. Left: ∣ℱ−1(𝑇𝜎 ℱ𝜑∗)∣ for 𝜎 = 100, 1000, 10000, 100000 (fromtop to bottom). Right: ∣ℱ−1(𝑇𝜎 ℱ𝜑∗)∣𝐼

image surface by involving the spinor field that parameterizes the normal and thebivector field coding the tangent plane. We have also introduced what appears to

9. Clifford–Fourier Transform 191

Figure 4. Left: original. Right: ∣ℱ−1𝑇𝜎ℱ𝜑∗∣ 𝐼 with 𝜎 = 100

Figure 5. Left: original. Right: ∣ℱ−1𝑇𝜎ℱ𝜑∗∣ 𝐼 with 𝜎 = 100

be a harmonic decomposition of the parametrization and investigated applicationsto filtering.

Note that there are only two cases where the Grassmannian 𝐺𝑛,2 of 2-planesin ℝ𝑛 admits a rational parametrization. In fact, one can show that 𝐺3,2 ≃ ℂ𝑃 1

and 𝐺4,2 ≃ ℂ𝑃 1 ×ℂ𝑃 1 (see [9]). The case treated here corresponds to 𝐺3,2. As aconsequence the generalization to colour images is not straightforward. Neverthe-less, a quite different approach is possible to tackle this problem and will be thesubject of a forthcoming paper.

Let us also mention that one may envisage performing diffusion on grey-levelimages through the heat equation given by the Dirac operator. The latter is wellknown be a square root of the Laplacian. Preliminary results are discussed in [2]that show that this diffusion better preserves edges and textures than the usualRiemannian approaches.

192 T. Batard and M. Berthier

Appendix A. Mathematical Background

We recall here some definitions and results concerning spin geometry. The readermay refer to [7] for details and conventions. We focus on the particular case of anoriented surface immersed in ℝ3.

A.1. Complex Representations of 𝑪ℓ3,0 ⊗ ℂ

Let (𝑒1, 𝑒2, 𝑒3) be an orthonormal basis of ℝ3. The Clifford algebra 𝐶ℓ3,0 is thequotient of the tensor algebra of the vectorial space ℝ3 by the ideal generatedby the elements 𝑢⊗ 𝑢+𝑄(𝑢) where 𝑄 is the Euclidean quadratic form. It can beshown that 𝐶ℓ3,0 is isomorphic to the product ℍ×ℍ of two copies of the quaternionalgebra. The complex Clifford algebra 𝐶ℓ3,0⊗ℂ is isomorphic to ℂ(2)⊕ℂ(2) whereℂ(2) denotes the algebra of 2×2-matrices with complex entries. This decompositionis given by

𝐶ℓ3,0 ⊗ ℂ ≃ (𝐶ℓ3,0 ⊗ ℂ)+ ⊕ (𝐶ℓ3,0 ⊗ ℂ)− (A.1)

where(𝐶ℓ3,0 ⊗ ℂ)± = (1± 𝜔3)𝐶ℓ3,0 ⊗ ℂ (A.2)

and 𝜔3 is the pseudoscalar 𝒆1𝒆2𝒆3. More precisely, the subalgebra (𝐶ℓ3,0 ⊗ℂ)+ isgenerated by the elements

𝛼1 =1 + 𝒆1𝒆2𝒆3

2, 𝛼2 =

𝒆2𝒆3 − 𝒆1

2, 𝛼3 =

𝒆2 + 𝒆1𝒆3

2, 𝛼4 =

𝒆3 − 𝒆1𝒆2

2(A.3)

and an isomorphism with ℂ(2) is given by sending these elements to the matrices

𝐴1 =

(1 00 1

), 𝐴2 =

(0 𝑖𝑖 0

), 𝐴3 =

(𝑖 00 −𝑖

), 𝐴4 =

(0 1

−1 0

). (A.4)

In the same way, (𝐶ℓ3,0 ⊗ ℂ)− is generated by

𝛽1 =1− 𝒆1𝒆2𝒆3

2, 𝛽2 =

𝒆2𝒆3 + 𝒆1

2, 𝛽3 =

𝒆1𝒆3 − 𝒆2

2, 𝛽4 =

−𝒆3 − 𝒆1𝒆2

2(A.5)

and an isomorphism is given by sending these elements to the above matrices 𝐴1,𝐴2, 𝐴3 and 𝐴4.

Let us denote by 𝜌 the natural representation of ℂ(2) on ℂ2. The two equiv-alent classes 𝜌1 and 𝜌2 of irreducible complex representations of 𝐶ℓ3,0 ⊗ ℂ aregiven by

𝜌1(𝜑1 + 𝜑2) = 𝜌(𝜑1) 𝜌2(𝜑1 + 𝜑2) = 𝜌(𝜑2). (A.6)

They are characterized by

𝜌1(𝜔3) = 𝐼𝑑 and 𝜌2(𝜔3) = −𝐼𝑑 (A.7)

For the sake of completeness, let us list these representations explicitly:

𝜌1(1) = 𝜌(𝛼1) = 𝐴1, 𝜌1(𝒆1) = 𝜌(−𝛼2) = −𝐴2

𝜌1(𝒆2) = 𝜌(𝛼3) = 𝐴3, 𝜌1(𝒆3) = 𝜌(𝛼4) = 𝐴4

𝜌1(𝒆1𝒆2) = 𝜌(−𝛼4) = −𝐴4, 𝜌1(𝒆1𝒆3) = 𝜌(𝛼3) = 𝐴3

𝜌1(𝒆2𝒆3) = 𝜌(𝛼2) = 𝐴2, 𝜌1(𝜔3) = 𝜌(𝛼1) = 𝐴1

(A.8)

9. Clifford–Fourier Transform 193

and

𝜌2(1) = 𝜌(𝛽1) = 𝐴1, 𝜌2(𝒆1) = 𝜌(𝛽2) = 𝐴2

𝜌2(𝒆2) = 𝜌(−𝛽3) = −𝐴3, 𝜌2(𝒆3) = 𝜌(−𝛽4) = −𝐴4

𝜌2(𝒆1𝒆2) = 𝜌(−𝛽4) = −𝐴4, 𝜌2(𝒆1𝒆3) = 𝜌(𝛽3) = 𝐴3

𝜌2(𝒆2𝒆3) = 𝜌(𝛽2) = 𝐴2, 𝜌2(𝜔3) = 𝜌(−𝛽1) = −𝐴1.

(A.9)

The complex spin representation of Spin(3) is the homomorphism

Δ3 : Spin(3) −→ ℂ(2) (A.10)

given by restricting an irreducible complex representation of 𝐶ℓ3,0⊗ℂ to the spinorgroup Spin(3) ⊂ (𝐶ℓ3,0 ⊗ ℂ)0 (see for example [4] for the definition of the Spingroup). Note that Δ3 is independant of the chosen representation.

A.2. Spin Structures and Spinor Bundles

Let us denote by 𝑀 the Riemannian manifold ℝ3 and 𝑃𝑆𝑂(𝑀) the principal𝑆𝑂(3)-bundle of oriented orthonormal frames of 𝑀 . A spin structure on 𝑀 is aprincipal Spin(3)-bundle 𝑃Spin(𝑀) together with a 2-sheeted covering

𝑃Spin(𝑀) −→ 𝑃𝑆𝑂(𝑀) (A.11)

that is compatible with 𝑆𝑂(3) and Spin(3) actions. The Spinor bundle 𝑆(𝑀)is the bundle associated to the spin structure 𝑃Spin(𝑀) and the complex spinrepresentation Δ3. More precisely, it is the quotient of the product 𝑃Spin(𝑀)×ℂ2

by the action

Spin(3)× 𝑃Spin(𝑀)× ℂ2 −→ 𝑃Spin(𝑀)× ℂ2 (A.12)

that sends (𝜏, 𝑝, 𝑧) to (𝑝𝜏−1,Δ3(𝜏)𝑧). We will write

𝑆(𝑀) = 𝑃Spin(𝑀)×Δ3 ℂ2. (A.13)

It appears that the fiber bundle 𝑆(𝑀) is a bundle of complex left modules overthe Clifford bundle 𝐶𝑙(𝑀) = 𝑃Spin(𝑀)×𝐴𝑑 ℂ𝑙(3) of 𝑀 . In the sequel

(𝑢, 𝜙) �−→ 𝑢 ⋅ 𝜙 (A.14)

denotes the corresponding multiplication for 𝑢 ∈ 𝑇 (𝑀) and 𝜙 a section of 𝑆(𝑀).We consider now an oriented surface Σ embedded in 𝑀 . Let us denote by

(𝑧1, 𝑧2) an orthonormal frame of 𝑇 (Σ) and 𝜈 the global unit field normal to Σ.Using the map

(𝑧1, 𝑧2) �−→ (𝑧1, 𝑧2, 𝜈) (A.15)

it is possible to pull back the bundle 𝑃Spin(𝑀)∣Σ to obtain a spin structure 𝑃Spin(Σ)

on Σ. Since 𝐶ℓ2,0 ⊗ ℂ is isomorphic to (𝐶ℓ3,0 ⊗ ℂ)0 under the map 𝛼 defined by

𝛼(𝜂0 + 𝜂1) = 𝜂0 + 𝜂1𝜈 (A.16)

the algebra 𝐶ℓ2,0 ⊗ℂ acts on ℂ2 via 𝜌2. This representation leads to the complexspinor representation Δ2 of Spin(2). It can be shown that the induced bundle

𝑆(Σ) = 𝑃Spin(Σ)×Δ3∘𝛼 ℂ2 (A.17)

194 T. Batard and M. Berthier

coincides with the spinor bundle of the induced spin structure on Σ. Once again𝑆(Σ) is a bundle of complex left modules over the Clifford bundle 𝐶𝑙(Σ) of Σ: theClifford multiplication is given by the map

(𝑣, 𝜑) �−→ 𝑣 ⋅ 𝜈 ⋅ 𝜑 (A.18)

for 𝑣 ∈ 𝑇 (Σ) and 𝜑 a section of 𝑇 (Σ).The Spinor bundle 𝑆(Σ) decomposes into

𝑆(Σ) = 𝑆+(Σ)⊕ 𝑆−(Σ) (A.19)

where

𝑆±(Σ) = {𝜑 ∈ 𝑆(Σ), 𝑖 ⋅ 𝑧1 ⋅ 𝑧2 ⋅ 𝜑 = ±𝜑} (A.20)

(compare [5]). Since 𝜌2(𝑧1𝑧2𝜈) is minus the identity, this is equivalent to

𝑆±(Σ) = {𝜑 ∈ 𝑆(Σ), 𝑖𝜈 ⋅ 𝜑 = ±𝜑}. (A.21)

A.3. Spinor Connections and Dirac Operators

Let ∇𝑀 and ∇Σ be the Levi–Civita connections on the tangent bundles 𝑇 (𝑀)and 𝑇 (Σ) respectively. The classical Gauss formula asserts that

∇𝑀𝑋 𝑌 = ∇Σ

𝑋𝑌 − ⟨∇𝑀𝑋 𝜈, 𝑌 ⟩𝜈 (A.22)

where 𝑋 and 𝑌 are vector fields on Σ. A similar formula exists when dealing withspinor fields. Let us first recall that one may construct on 𝑆(𝑀) and 𝑆(Σ) somespinor Levi–Civita connections compatible with the Clifford multiplication, thatis connections which we continue to denote by ∇𝑀 and ∇Σ verifying

∇𝑀𝑋 (𝑌 ⋅ 𝜑) = (∇𝑀

𝑋 𝑌 ) ⋅ 𝜑+ 𝑌 ⋅ ∇𝑀𝑋 𝜑 (A.23)

when 𝑋 and 𝑌 are vector fields on 𝑀 and 𝜑 is a section of 𝑆(𝑀) and a similarformula for ∇Σ. The analog of the Gauss formula reads

∇𝑀𝑋 𝜑 = ∇Σ

𝑋𝜑− 1

2(∇𝑀

𝑋 𝜈) ⋅ 𝜈 ⋅ 𝜑 (A.24)

for 𝜑 a section of 𝑆(Σ) and 𝑋 a vector field on Σ (see [1] for a proof). If (𝑧1, 𝑧2) isan orthonormal frame of 𝑇 (Σ), following [5], the Dirac operator on 𝑆(Σ) is definedby

𝐷 = 𝑧1 ⋅ ∇Σ𝑧1 + 𝑧2 ⋅ ∇Σ

𝑧2 (A.25)

and it can be verified that 𝐷𝑆±(Σ) ⊂ 𝑆∓(Σ).Let now 𝜙 and 𝜑 be respectively a section of 𝑆(𝑀) and the section of 𝑆(Σ)

given by the restriction 𝜙∣Σ. We obtain from the Gauss spinor formula

𝑧1 ⋅ ∇𝑀𝑧1𝜙+ 𝑧2 ⋅ ∇𝑀

𝑧2𝜙 = 𝐷𝜑− 1

2(𝑧1 ⋅ (∇𝑀

𝑧1 𝜈) ⋅ 𝜈 ⋅ 𝜑+ 𝑧2 ⋅ (∇𝑀𝑧2 𝜈) ⋅ 𝜈 ⋅ 𝜑). (A.26)

Since

𝑧1 ⋅ (∇𝑀𝑧1 𝜈) + 𝑧2 ⋅ (∇𝑀

𝑧2 𝜈) = −2𝐻 (A.27)

where 𝐻 is the mean curvature of Σ, it follows that

𝐷𝜑 = 𝑧1 ⋅ ∇𝑀𝑧1𝜙+ 𝑧2 ⋅ ∇𝑀

𝑧2𝜙−𝐻 ⋅ 𝜈 ⋅ 𝜑. (A.28)

9. Clifford–Fourier Transform 195

References

[1] C. Bar. Metrics with harmonic spinors.Geometric and Functional Analysis, 6(6):899–942, 1996.

[2] T. Batard and M. Berthier. The spinor representation of images. In K. Gurlebeck,editor, 9th International Conference on Clifford Algebras and their Applications,Weimar, Germany, 15–20 July 2011.

[3] T. Batard, M. Berthier, and C. Saint-Jean. Clifford Fourier transform for color im-age processing. In E.J. Bayro-Corrochano and G. Scheuermann, editors, GeometricAlgebra Computing in Engineering and Computer Science, pages 135–162. Springer,London, 2010.

[4] T. Batard, C.S. Jean, and M. Berthier. A metric approach to nD images edge detec-tion with Clifford algebras. Journal of Mathematical Imaging and Vision, 33:296–312,2009.

[5] T. Friedrich. On the spinor representation of surfaces in Euclidean 3-space. Journalof Geometry and Physics, 28:143–157, 1998.

[6] B. Lawson. Lectures on minimal manifolds, volume I, volume 9 of Mathematics Lec-ture Series. Publish or Perish Inc., Wilmington, Del., second edition, 1980.

[7] B. Lawson and M.-L. Michelson. Spin Geometry. Princeton University Press, Prince-ton, New Jersey, 1989.

[8] R. Osserman. A survey of minimal surfaces. Dover Publications, Inc., New York,second edition, 1986.

[9] I.A. Taimanov. Two-dimensional Dirac operator and surface theory. Russian Math-ematical Surveys, 61(1):79–159, 2006.

[10] S.D. Zenzo. A note on the gradient of a multi-image. Computer Vision, Graphics,and Image Processing, 33:116–125, 1986.

Thomas BatardDepartment of Applied MathematicsTel Aviv UniversityRamat AvivTel Aviv 69978, Israele-mail: thomas.batard@gmail.com

Michel BerthierMIA LaboratoryLa Rochelle UniversityAvenue Michel CrepeauF-17042 La Rochelle, Francee-mail: michel.berthier@univ-lr.fr

Quaternion and Clifford–Fourier Transforms and Wavelets

Trends in Mathematics, 197–219c⃝ 2013 Springer Basel

10 Analytic Video (2D + 𝒕) SignalsUsing Clifford–Fourier Transformsin Multiquaternion Grassmann–Hamilton–Clifford Algebras

P.R. Girard, R. Pujol, P. Clarysse, A. Marion,R. Goutte and P. Delachartre

Abstract. We present an algebraic framework for (2D+ 𝑡) video analytic sig-nals and a numerical implementation thereof using Clifford biquaternions andClifford–Fourier transforms. Though the basic concepts of Clifford–Fouriertransforms are well known, an implementation of analytic video sequences us-ing multiquaternion algebras does not seem to have been realized so far. Aftera short presentation of multiquaternion Clifford algebras and Clifford–Fouriertransforms, a brief pedagogical review of 1D and 2D quaternion analytic sig-nals using right quaternion Fourier transforms is given. Then, the biquater-nion algebraic framework is developed to express Clifford–Fourier transformsand (2D + 𝑡) video analytic signals in standard and polar form constitutedby a scalar, a pseudoscalar and six phases. The phase extraction procedure isfully detailed. Finally, a numerical implementation using discrete fast Fouriertransforms of an analytic multiquaternion video signal is provided.

Mathematics Subject Classification (2010). Primary 15A66; secondary 42A38.

Keywords. Multiquaternion Clifford algebras, Clifford biquaternions, Clifford–Fourier transforms, 2D + 𝑡 analytic video signals.

1. Introduction

In recent decades, new algebraic structures based on Clifford algebras have beendeveloped [19,31]. Many of these developments use a geometric approach whereaswe propose an algebraic multiquaternion approach [10–12]. This chapter focuses ona (2D+ 𝑡) Clifford biquaternion analytic signal using Clifford–Fourier transforms[1, 29]. After a short pedagogical review of 1D complex and 2D quaternion ana-lytic signals using right quaternion Fourier transforms, we shall provide a concrete

198 P.R. Girard et al.

algebraic Clifford biquaternion framework for the Clifford–Fourier transform, theanalytic Clifford–Fourier transform and the analytic signal both in standard andpolar form. Finally, we shall give a numerical implementation, using discrete fastFourier transforms, of a (2D + 𝑡) video analytic signal.

2. Multiquaternion Grassmann–Hamilton–Clifford Algebras

In 1844, in his Ausdehnungslehre, Hermann Gunther Grassmann (1809–1877) laidthe foundations of an 𝑛-dimensional associative multivector calculus [14–16,27]. Ayear before, in 1843, William Rowan Hamilton (1805–1865) discovered the quater-nions and later on, was to introduce biquaternions [6, 22–24]. In 1878, WilliamKingdon Clifford (1845–1879) was to give a concise definition of Clifford algebrasand to demonstrate a theorem relating Grassmann’s system to Hamilton’s quater-nions [4],[5, pp. 266–276]. Due to this close connection, we shall refer to Cliffordalgebras also as Grassmann–Hamilton–Clifford algebras or multiquaternion alge-bras [13, 17, 18].

Definition 2.1. A Clifford algebra 𝐶𝑛 is an algebra composed of 𝑛 generators𝒆1, 𝒆2, . . . , 𝒆𝑛 multiplying according to the rule 𝒆𝑖𝒆𝑗 = −𝒆𝑗𝒆𝑖 (𝑖 ∕= 𝑗) and suchthat 𝒆2

𝑖 = ±1. The algebra 𝐶𝑛 contains 2𝑛 elements constituted by the 𝑛 genera-tors, the various products 𝒆𝑖𝒆𝑗 , 𝒆𝑖𝒆𝑗𝒆𝑘, . . . and the unit element 1.

Theorem 2.2. If 𝑛 = 2𝑚 (𝑚 : integer), the Clifford algebra 𝐶2𝑚 is the tensorproduct of 𝑚 quaternion algebras. If 𝑛 = 2𝑚− 1, the Clifford algebra 𝐶2𝑚−1 is thetensor product of 𝑚 − 1 quaternion algebras and the algebra (1, 𝜖) where 𝜖 is theproduct of the 2𝑚 generators (𝜖 = 𝒆1𝒆2 . . .𝒆2𝑚) of the algebra 𝐶2𝑚.

Examples of Clifford algebras are the complex numbers ℂ (with 𝒆1 = 𝑖),quaternionsℍ (𝒆1 = 𝒊, 𝒆2 = 𝒋), biquaternions (𝒆1 = 𝐼𝒊, 𝒆2 = 𝐼𝒋, 𝒆3 = 𝐼𝒌, 𝐼2 = −1,𝐼 commuting with 𝒊, 𝒋,𝒌, 𝒆2

𝑖 = 1) and tetraquaternions ℍ ⊗ ℍ (𝒆0 = 𝒋, 𝒆1 =𝒌𝑰, 𝒆2 = 𝒌𝑱 , 𝒆3 = 𝒌𝑲, where the small 𝒊, 𝒋,𝒌 commute with the capital 𝑰,𝑱 ,𝑲).These examples prove the Clifford theorem up to dimension 𝑛 = 4.

3. Analytic Signal in 𝑵 Dimensions

Consider a Grassmann–Hamilton–Clifford algebra having 𝑛 generators 𝒆𝑛 (with𝒆2𝑛 = −1) and let 𝐴 be a general element of this algebra [7, 21]. Call 𝐾[𝐴] the

conjugate of 𝐴 such that

𝐾(𝐴𝐵) = 𝐾(𝐵)𝐾(𝐴),𝐾(𝒆𝑛) = −𝒆𝑛. (3.1)

Given a function 𝑓 (𝒙) having its value in the Clifford algebra with 𝒙 = (𝑥1, 𝑥2,. . ., 𝑥𝑛), let 𝐹 (𝒖) with 𝒖 = (𝑢1, 𝑢2, . . . , 𝑢𝑛) denote the Clifford–Fourier transform

𝐹 (𝒖) =

∫ℝ𝑛

𝑓(𝒙)𝑛∏

𝑘=1

𝑒−𝒆𝑘2𝜋𝑢𝑘𝑥𝑘𝑑𝑛𝒙. (3.2)

10. Analytic Video (2D + 𝑡) Signals 199

The inverse Clifford–Fourier transform is given by

𝑓(𝒙) =

∫ℝ𝑛

𝐹 (𝒖)

𝑛−1∏𝑘=0

𝑒𝒆𝑛−𝑘2𝜋𝑢𝑛−𝑘𝑥𝑛−𝑘𝑑𝑛𝒖. (3.3)

Proof. Inserting 𝐹 (𝒖) in the equation above, one obtains after integration respec-tively over 𝑑𝑛𝒖 and 𝑑𝑛𝒙′ [29]∫

ℝ2𝑛

𝑓(𝒙′)𝑛∏

𝑗=1

𝑒−𝒆𝑗2𝜋𝑢𝑗𝑥′𝑗𝑑𝑛𝒙′

𝑛−1∏𝑘=0

𝑒𝒆𝑛−𝑘2𝜋𝑢𝑛−𝑘𝑥𝑛−𝑘𝑑𝑛𝒖 (3.4)

=

∫ℝ𝑛

𝑓(𝒙′)𝛿𝑛 (𝒙− 𝒙′) 𝑑𝑛𝒙′ = 𝑓(𝒙). □

Given two functions 𝑓 (𝒙) , 𝑔 (𝒙) taking their values in the Clifford algebra,define the following product

⟨𝑓 (𝒙) , 𝑔 (𝒙)⟩ = 1

2

∫ℝ𝑛

[𝑓𝐾 (𝑔) + 𝑔𝐾 (𝑓)] 𝑑𝑛𝒙 (3.5)

and similarly ⟨𝐹 (𝒖) , 𝐺 (𝒖)⟩. These products satisfy Plancherel’s theorem

⟨𝑓 (𝒙) , 𝑔 (𝒙)⟩ = ⟨𝐹 (𝒖) , 𝐺 (𝒖)⟩ (3.6)

Proof.

𝑓(𝒙) =

∫ℝ𝑛

𝐹 (𝒖)𝑒𝒆𝑛2𝜋𝑢𝑛𝑥𝑛 . . . 𝑒𝒆12𝜋𝑢1𝑥1𝑑𝑛𝒖 (3.7)

𝐾𝑔(𝒙) =

∫ℝ𝑛

𝑒−𝒆12𝜋𝑢′1𝑥1 . . . 𝑒−𝒆𝑛2𝜋𝑢′

𝑛𝑥𝑛𝐾 [𝐺(𝒖′)] 𝑑𝑛𝒖′. (3.8)

After integration over 𝑑𝑛𝒙, one has

1

2

∫ℝ𝑛

𝑓 (𝒙)𝐾 [𝑔 (𝒙)] 𝑑𝑛𝒙 =1

2

∫ℝ2𝑛

𝐹 (𝒖) 𝛿𝑛 (𝒖− 𝒖′)𝐾 [𝐺 (𝒖′)] 𝑑𝑛𝒖𝑑𝑛𝒖′

=1

2

∫ℝ𝑛

𝐹 (𝒖)𝐾 [𝐺 (𝒖)] 𝑑𝑛𝒖; (3.9)

similarly,

1

2

∫ℝ𝑛

𝑔 (𝒙)𝐾 [𝑓 (𝒙)] 𝑑𝑛𝒙 =1

2

∫ℝ2𝑛

𝐺 (𝒖) 𝛿𝑛 (𝒖− 𝒖′)𝐾 [𝐹 (𝒖′)] 𝑑𝑛𝒖𝑑𝑛𝒖′

=1

2

∫ℝ𝑛

𝐺 (𝒖)𝐾 [𝐹 (𝒖)] 𝑑𝑛𝒖. (3.10)

Hence, adding the last two equations, one obtains equation (3.5). □

200 P.R. Girard et al.

Next consider a real scalar function 𝑓 (𝒙) and its Clifford–Fourier transform𝐹 (𝒖), the analytic Clifford–Fourier transform 𝐹𝐴(𝒖) and the analytic signal 𝑓𝐴(𝒙)are respectively defined by

𝐹𝐴(𝒖) =

𝑛∏𝑘=1

[1 + sign (𝑢𝑘)]𝐹 (𝒖) (3.11)

𝑓𝐴(𝒙) =

∫ℝ𝑛

𝐹𝐴(𝒖)

𝑛−1∏𝑘=0

𝑒𝒆𝑛−𝑘2𝜋𝑢𝑛−𝑘𝑥𝑛−𝑘𝑑𝑛𝒖 (3.12)

with sign(𝑢𝑘) yielding −1, 0, or 1 depending on whether 𝑢𝑘 is negative, zero orpositive. In particular, the analytic signal satisfies the Parseval relation

⟨𝑓𝐴 (𝒙) , 𝑓𝐴 (𝒙)⟩ = ⟨𝐹𝐴 (𝒖) , 𝐹𝐴 (𝒖)⟩ . (3.13)

4. Analytic Signals in 1 and 2 Dimensions

4.1. Complex 1D Analytic Signal

The one-dimensional analytic signal notion was introduced by D. Gabor [9] andJ. Ville [30] in 1948. Within the Clifford algebra framework it can be presentedas follows. Using the complex numbers as Clifford algebra (𝒆1 = 𝑖) for modeling1D physics, the Clifford–Fourier transform of a scalar function 𝑓(𝑥1) is simply thestandard Fourier transform

𝐹 (𝑢1) =

∫ ∞

−∞𝑓(𝑥1)𝑒

−𝑖2𝜋𝑢1𝑥1𝑑𝑥1 (4.1)

with the inverse transformation

𝑓(𝑥1) =

∫ ∞

−∞𝐹 (𝑢1)𝑒

𝑖2𝜋𝑢1𝑥1𝑑𝑢1. (4.2)

Furthermore, one defines the scalar product of two functions 𝑓(𝑥1), 𝑔(𝑥1) as

⟨𝑓(𝑥1), 𝑔(𝑥1)⟩ = 1

2

∫ ∞

−∞(𝑓𝑔∗ + 𝑔𝑓∗) 𝑑𝑥1 (4.3)

where 𝑓∗(𝑥1), 𝑔∗(𝑥1) are the complex conjugate functions. A few properties of the

Fourier transform are recalled in Table 1.Under an orthogonal symmetry transforming the basis vector 𝒆1 into −𝒆1 one

has 𝐹 (−𝑢1) = 𝐹 (𝑢1)∗ where 𝐹 (𝑢1)

∗ is the complex conjugate of 𝐹 (𝑢1). Hence,only one orthant (𝑢1 ≥ 0) of the Fourier domain is necessary to obtain the signal.The analytic Fourier transform is defined by

𝐹𝐴(𝑢1) = [1 + sign(𝑢1)]𝐹 (𝑢1) (4.4)

and the analytic signal by

𝑓𝐴(𝑥1) =

∫ ∞

−∞𝐹𝐴(𝑢1)𝑒

𝑖2𝜋𝑢1𝑥1𝑑𝑢1. (4.5)

10. Analytic Video (2D + 𝑡) Signals 201

Table 1. Properties of the complex Fourier transform (𝑎, 𝑏 ∈ ℝ)

Property Complex function Fourier transform

Linearity 𝑎𝑓 (𝑥1) + 𝑏𝑔 (𝑥1) 𝑎𝐹 (𝑢1) + 𝑏𝐺 (𝑢1)Translation 𝑓 (𝑥1 − 𝑥0) 𝑒−𝑖2𝜋𝑢1𝑥0𝐹 (𝑢1)

Scaling 𝑓 (𝑎𝑥1)1∣𝑎∣𝐹

(𝑢1𝑎)

Partial derivative∂𝑟𝑓 (𝑥1)

∂𝑥𝑟1(𝑖2𝜋𝑢1)

𝑟 𝐹 (𝑢1)

Plancherel ⟨𝑓, 𝑔⟩ = ⟨𝐹,𝐺⟩Parseval ⟨𝑓, 𝑓⟩ = ⟨𝐹, 𝐹 ⟩

The analytic signal satisfies Plancherel’s theorem∫ ∞

−∞∣𝑓𝐴(𝑥1)∣2 𝑑𝑥1 =

∫ ∞

−∞∣𝐹𝐴(𝑢1)∣2 𝑑𝑢1. (4.6)

As example, consider the function 𝑓(𝑥) = 𝑎 cos 𝜔𝑡 which yields the analytic signal𝑓𝐴 = 𝑎𝑒𝑖𝜔𝑡. The instantaneous amplitude and the instantaneous phase of a realsignal 𝑓 at a given position 𝑥 can be defined as the modulus and the angularargument of the complex-valued analytic signal. Therefore, the analytic signalplays a key role in one-dimensional signal processing. It is also important to notethat the analytic signal concept is global, i.e., 𝑓𝐴(𝑥) depends on the whole of theoriginal signal and not only on values at positions near 𝑥. This concept can besummarized into three main properties:

1. The analytic signal has a one-sided spectrum;2. The original signal can be reconstructed from its analytic signal (in particular,

the real part of the analytic signal is equal to the original signal);3. The local amplitude (envelope) and the local phase of the original signal

can be derived as the modulus and angular argument of the analytic signalrespectively.

4.2. Quaternion 2D Analytic Signal

4.2.1. Clifford Algebra. The analytic signal notion can be extended to the 2D casein several ways [2, 3, 8, 20, 25]. Taking the quaternions ℍ as a Clifford algebra, itcontains the elements

[1, 𝒆1 = 𝒊, 𝒆2 = 𝒋, 𝒆1𝒆2 = 𝒌] . (4.7)

Hence, a general element 𝑞 can be expressed as

𝑞 = 𝑞0 + 𝒊𝑞1 + 𝒋𝑞2 + 𝒌𝑞3

= [𝑞0, 𝑞1, 𝑞2, 𝑞3] (4.8)

the conjugate 𝑞𝑐 being

𝑞𝑐 = [𝑞0,−𝑞1,−𝑞2,−𝑞3] . (4.9)

202 P.R. Girard et al.

Figure 1. Orthogonal symmetry with respect to a straight line.

A 2D vector is expressed by 𝑋 = 𝒊𝑞1+𝒋𝑞2 with 𝑋𝑋𝑐 = −𝑋2 = (𝑞1)2+(𝑞2)

2,a bivector by 𝐵 = 𝒌𝑞3 and a scalar by 𝑞0.

Under an orthogonal symmetry with respect to a straight line passing throughthe origin and perpendicular to the unit vector 𝑎 = 𝒊𝑎1+𝒋𝑎2, (𝑎𝑎𝑐 = 1), a vector istransformed into the vector 𝑋 ′ = 𝑎𝑋𝑎

𝑎𝑎𝑐= 𝑎𝑋𝑎 [10, p. 76], a bivector 𝐵 = 𝑋 ∧𝑌 =

− 12 (𝑋𝑌 − 𝑌 𝑋) into 𝐵′ = −𝑎𝐵𝑎, whereas a scalar remains invariant (Figure 1).

Hence, the entire quaternion transforms as

𝑞′ = 𝑎∗𝑞𝑎∗𝑐 (4.10)

where 𝑎∗ = 𝒌𝑎 is the dual of 𝑎 (𝑎∗𝑐 = 𝑎𝑐𝒌𝑐 = 𝑎𝒌). One verifies the conservation ofthe square of the norm of the quaternion 𝑞′𝑞′𝑐 = 𝑞𝑞𝑐 under an orthogonal symme-try. Equation (4.10) allows us to deduce the involution formulas developed belowwhich are useful for the comprehension of the properties of the analytic Cliffordquaternion Fourier transform and the extraction of phases. In particular, if 𝑎 = 𝒊(orthogonal symmetry with respect to the 𝑦-axis), one has

𝑞′ = 𝐾1(𝑞) = 𝒊∗𝑞𝒊∗𝑐 = 𝒌𝒊𝑞𝒊𝒌 = −𝒋𝑞𝒋 = 𝑞0 − 𝒊𝑞1 + 𝒋𝑞2 − 𝒌𝑞3 (4.11)

where 𝐾1(𝑞) is an involution of 𝑞. Similarly, if 𝑎 = 𝒋 (orthogonal symmetry withrespect to the 𝑥-axis)

𝑞′ = 𝐾2(𝑞) = 𝒋∗𝑞𝒋∗𝑐 = 𝒌𝒋𝑞𝒋𝒌 = −𝒊𝑞𝒊 = 𝑞0 + 𝒊𝑞1 − 𝒋𝑞2 − 𝒌𝑞3. (4.12)

A combination of the two above orthogonal symmetries (𝒊 → −𝒊, 𝒋 → −𝒋) yieldsa rotation of 𝜋 around the origin

𝑞′ = 𝐾12(𝑞) = 𝒊𝒋𝑞𝒋𝒊 = −𝒌𝑞𝒌 = 𝑞0 − 𝒊𝑞1 − 𝒋𝑞2 + 𝒌𝑞3. (4.13)

10. Analytic Video (2D + 𝑡) Signals 203

Table 2. Properties of the right quaternion Fourier transform (𝑎, 𝑏 ∈ ℝ)

Property scalar functions RQFT

Linearity 𝑎𝑓 (𝑥1, 𝑥2) + 𝑏𝑔 (𝑥1, 𝑥2) 𝑎𝐹 (𝑢1, 𝑢2) + 𝑏𝐺 (𝑢1, 𝑢2)Translation 𝑓 (𝑥1 − 𝑎1, 𝑥2 − 𝑎2) 𝑒−𝒊2𝜋𝑢1𝑎1𝐹 (𝑢1, 𝑢2)𝑒

−𝒋2𝜋𝑢2𝑎2

Scaling 𝑓 (𝑎1𝑥1, 𝑎2𝑥2)1∣𝑎1∣

1∣𝑎2∣𝐹

(𝑢1𝑎1

, 𝑢2𝑎2

)Partial derivative

∂𝑟𝑓∂𝑥𝑟1

(𝑥1, 𝑥2) 𝐹 (𝑢1, 𝑢2) (𝒊2𝜋𝑢1)𝑟

(𝑞 = 4𝑝) (2𝜋𝑢2)𝑞𝐹 (𝑢1, 𝑢2)

(𝑞 = 4𝑝+ 1)∂𝑞𝑓∂𝑥𝑞2

(𝑥1, 𝑥2) (2𝜋𝑢2)𝑞𝑗𝐾1[𝐹 (𝑢1, 𝑢2)]

(𝑞 = 4𝑝+ 2) − (2𝜋𝑢2)𝑞𝐹 (𝑢1, 𝑢2)

(𝑞 = 4𝑝+ 3) − (2𝜋𝑢2)𝑞 𝑗𝐾1[𝐹 (𝑢1, 𝑢2)]

Plancherel ⟨𝑓, 𝑔⟩ = ⟨𝐹,𝐺⟩Parseval ⟨𝑓, 𝑓⟩ = ⟨𝐹, 𝐹 ⟩

4.2.2. Right Quaternion Fourier Transform (RQFT). As Clifford–Fourier trans-form, we shall take the right quaternion Fourier transform [1, 26, 28]

𝐹 (𝑢1, 𝑢2) =

∫ℝ2

𝑓(𝑥1, 𝑥2)𝑒−𝒊2𝜋𝑢1𝑥1𝑒−𝒋2𝜋𝑢2𝑥2𝑑𝑥1𝑑𝑥2 (4.14)

and its inverse transform

𝑓(𝑥1, 𝑥2) =

∫ℝ2

𝐹 (𝑢1, 𝑢2)𝑒𝒋2𝜋𝑢2𝑥2𝑒𝒊2𝜋𝑢1𝑥1𝑑𝑢1𝑑𝑢2. (4.15)

The scalar product of two quaternion functions 𝑓(𝒙), 𝑔(𝒙) with (𝒙) = (𝑥1, 𝑥2) isdefined by

⟨𝑓(𝒙), 𝑔(𝒙)⟩ = 1

2

∫ℝ2

(𝑓𝑔𝑐 + 𝑔𝑓𝑐) 𝑑𝑥1𝑑𝑥2 (4.16)

where 𝑔𝑐, 𝑓𝑐 are the quaternion conjugates. Table 2 lists a few properties of theright Fourier transform.

From the definition of the QFT and the above orthogonal symmetry proper-ties, it follows that with (𝒖) = (𝑢1, 𝑢2)

𝐹 (−𝑢1, 𝑢2) = 𝐾1 [𝐹 (𝒖)] = −𝒋𝐹 (𝒖)𝒋 (4.17)

𝐹 (𝑢1,−𝑢2) = 𝐾2 [𝐹 (𝒖)] = −𝒊𝐹 (𝒖)𝒊 (4.18)

𝐹 (−𝑢1,−𝑢2) = 𝐾12 [𝐹 (𝒖)] = −𝒌𝐹 (𝒖)𝒌 (4.19)

Hence, only one orthant of the Fourier space is necessary to represent the entireFourier space (Figure 2).

4.2.3. Analytic Signal. The analytic quaternion Fourier transform (Figure 3) isdefined as

𝐹𝐴(𝒖) = [1 + sign(𝑢1)] [1 + sign(𝑢2)]𝐹 (𝒖) (4.20)

204 P.R. Girard et al.

�𝑢1

�𝑢2

𝐹 (𝑢1, 𝑢2)𝐹 (−𝑢1, 𝑢2)= 𝐾1[𝐹 (𝑢1, 𝑢2)]= −𝒋𝐹 (𝑢1, 𝑢2)𝒋

𝐹 (−𝑢1,−𝑢2)= 𝐾12[𝐹 (𝑢1, 𝑢2)]= −𝒌𝐹 (𝑢1, 𝑢2)𝒌

𝐹 (𝑢1,−𝑢2)= 𝐾2[𝐹 (𝑢1, 𝑢2)]= −𝒊𝐹 (𝑢1, 𝑢2)𝒊

Figure 2. The quaternion Fourier spectrum of a real signal can bereconstructed from only one quadrant of the Fourier plane.

�𝑢1

�𝑢2

4𝐹 (𝑢1, 𝑢2)

Figure 3. The spectrum of the analytic quaternion Fourier transformis obtained from only one quadrant.

and the analytic signal by

𝑓𝐴(𝒙) =

∫ℝ2

𝐹𝐴(𝒖)𝑒𝒋2𝜋𝑢2𝑥2𝑒𝒊2𝜋𝑢1𝑥1𝑑𝑢1𝑑𝑢2. (4.21)

The analytic signal satisfies Plancherel’s theorem∫ℝ2

𝑓𝐴(𝒙) [𝑓𝐴(𝒙)]𝑐 𝑑𝑥1𝑑𝑥2 =

∫ℝ2

𝐹𝐴(𝒖) [𝐹𝐴(𝒖)]𝑐 𝑑𝑢1𝑑𝑢2. (4.22)

10. Analytic Video (2D + 𝑡) Signals 205

Table 3. Examples of analytic signals.

Function Analytic signal

cos𝜔1𝑥1 cos𝜔2𝑥2 𝑒𝒋𝜔2𝑥2𝑒𝒊𝜔1𝑥1

sin𝜔1𝑥1 sin𝜔2𝑥2 𝒌𝑒𝒋𝜔2𝑥2𝑒𝒊𝜔1𝑥1

sin𝜔1𝑥1 cos𝜔2𝑥2 −𝒊𝑒𝒋𝜔2𝑥2𝑒𝒊𝜔1𝑥1

cos𝜔1𝑥1 sin𝜔2𝑥2 −𝒋𝑒𝒋𝜔2𝑥2𝑒𝒊𝜔1𝑥1

cos (𝜔1𝑥1 ± 𝜔2𝑥2 + 𝜑) (1∓ 𝒌) 𝑒𝒋𝜑𝑒𝒋𝜔2𝑥2𝑒𝒊𝜔1𝑥1

Examples of analytic signals are given in Table 3. The analytic signal being aquaternion, it can be represented as

𝑞 = ∣𝑞∣ 𝑒𝒌𝜃3𝑒𝒋𝜃2𝑒𝒊𝜃1 = ∣𝑞∣ 𝑟 (4.23)

with ∣𝑞∣ = √𝑞𝑞𝑐 and where 𝑟 = 𝑒𝒌𝜃3𝑒𝒋𝜃2𝑒𝒊𝜃1 is a unit quaternion (𝑟𝑟𝑐 = 1). The

procedure of extracting the triplet of phases (𝜃1, 𝜃2, 𝜃3) is recalled below [29].Using the involutions 𝐾1,𝐾2,𝐾12 defined above one has (with ∗ representing thequaternion multiplication)

𝑑2 = 𝑟 ∗ [𝐾2 (𝑟)]𝑐

= [cos 2𝜃2 cos 2𝜃3, 0, sin 2𝜃2, cos 2𝜃2 sin 2𝜃3] (4.24)

𝑑12 = [𝐾12 (𝑟)]𝑐 ∗ 𝑟= [cos 2𝜃1 cos 2𝜃2, cos 2𝜃2 sin 2𝜃1, sin 2𝜃2, 0] . (4.25)

From 𝑑2 one obtains 𝜃2 = 12 arcsin𝑑2(3) where 𝑑2(3) means the third component

of 𝑑2. If cos 2𝜃2 ∕= 0, one has

𝜃3 =Arg [𝑑2 (1) + 𝑖𝑑2(4)]

2, 𝜃1 =

Arg [𝑑12(1) + 𝑖𝑑12(2)]

2. (4.26)

If cos 2𝜃2 = 0 (𝜃2 = ±𝜋/4), one has an indeterminacy and only (𝜃1 ± 𝜃3) can bedetermined; adopting the choice 𝜃3 = 0, one has

𝑑1 = 𝑟 ∗ [𝐾1 (𝑟)]𝑐 = [cos 2𝜃1, 0, 0,∓ sin 2𝜃1] (4.27)

and thus

𝜃1 = ∓𝐴𝑟𝑔 [𝑑1(1) + 𝑖𝑑1(4)]

2. (4.28)

Finally, having determined the phases (𝜃1, 𝜃2, 𝜃3) one computes 𝑒𝑘𝜃3𝑒𝑗𝜃2𝑒𝑖𝜃1 ; if itis equal to −𝑟 and 𝜃3 ≥ 0, one takes 𝜃3 − 𝜋; if one has −𝑟 and 𝜃3 < 0 then onetakes 𝜃3 + 𝜋. Hence, the domain of the phases is

(𝜃1, 𝜃2, 𝜃3) ∈[−𝜋

2,𝜋

2

]∪[−𝜋

2,𝜋

2

]∪ [−𝜋, 𝜋[ . (4.29)

206 P.R. Girard et al.

5. Clifford Biquaternion 2D + 𝒕 Analytic Signal: Implementation

5.1. Clifford Biquaternion Algebra

In 2D + 𝑡 dimensions, we shall take as Clifford algebra, Clifford biquaternionshaving as generators (𝒆1 = 𝜖𝑖, 𝒆2 = 𝜖𝑗, 𝒆3 = 𝜖𝑘, 𝜖 = 𝑖′𝐼, 𝜖2 = 1, 𝒆2

𝑖 = −1) with 𝑖′

designating the usual complex imaginary (𝑖′2 = −1) and where the tensor product𝐼 = 1⊗ 𝑖 (𝐼2 = −1) commutes with 𝑖 = 𝑖⊗ 1, 𝑗 = 𝑗 ⊗ 1, 𝑘 = 𝑘⊗ 1; 𝒆1 correspondsto the time axis 𝑥1, 𝒆2 and 𝒆3 correspond respectively to the 𝑥2 and 𝑥3 axes. Thefull algebra contains the elements[

1 𝑖 = 𝒆2𝒆3 𝑗 = 𝒆3𝒆1 𝑘 = 𝒆1𝒆2

𝜖 = 𝑖′𝐼 = −𝒆1𝒆2𝒆3 𝒆1 = 𝜖𝑖 𝒆2 = 𝜖𝑗 𝒆3 = 𝜖𝑘

]. (5.1)

Hence, a general element of the algebra can be expressed as a Clifford biquaternion(a clifbquat for short)

𝐴 = 𝑝+ 𝜖𝑞

= (𝑝0 + 𝑖𝑝1 + 𝑗𝑝2 + 𝑘𝑝3) + 𝜖(𝑞0 + 𝑖𝑞1 + 𝑗𝑞2 + 𝑘𝑞3)

= [𝑝0, 𝑝1, 𝑝2, 𝑝3] + 𝜖 [𝑞0, 𝑞1, 𝑞2, 𝑞3] (5.2)

where 𝑝, 𝑞 are quaternions. The product of two clifbquats 𝐴 and 𝐵 = 𝑝′ + 𝜖𝑞′ isgiven by

𝐴𝐵 = (𝑝+ 𝜖𝑞) (𝑝′ + 𝜖𝑞′)

= (𝑝𝑝′ + 𝑞𝑞′) + 𝜖 (𝑝𝑞′ + 𝑞𝑝′) . (5.3)

The conjugate of 𝐴 is

𝐴𝑐 = 𝑝𝑐 + 𝜖𝑞𝑐

= [𝑝0,−𝑝1,−𝑝2,−𝑝3] + 𝜖 [𝑞0,−𝑞1,−𝑞2,−𝑞3] (5.4)

where 𝑝𝑐, 𝑞𝑐 are respectively the quaternion conjugates of 𝑝 and 𝑞. The complexconjugate of 𝐴 is

𝐴 = 𝑝− 𝜖𝑞. (5.5)

A vector is expressed by

𝑎 = 𝜖(𝑖𝑎1 + 𝑗𝑎2 + 𝑘𝑎3) (5.6)

with 𝑎𝑎𝑐 = 𝑎21 + 𝑎2

2 + 𝑎23, a bivector by 𝐵 = 𝑎 ∧ 𝑏 = − 1

2 (𝑎𝑏− 𝑏𝑎) , a trivector by

𝑇 = 𝑐 ∧𝐵 = 12 (𝑐𝐵 +𝐵𝑐). A unit vector 𝑎 is defined by 𝑎𝑎𝑐 = 1.

Under an orthogonal symmetry with respect to a hyperplane (space of dimen-sion 𝑛 − 1) perpendicular to a unit vector 𝑎, a vector 𝑋 is transformed into thevector 𝑋 ′ = 𝑎𝑋𝑎, a bivector 𝐵 into 𝐵′ = −𝑎𝐵𝑎 and a trivector 𝑇 into 𝑇 ′ = 𝑎𝑇𝑎,whereas a scalar remains invariant. These formulas allow us to derive the trans-formation of a clifbquat under an arbitrary orthogonal symmetry (Figure 4). Inparticular, if 𝑎 = 𝜖𝑖 (orthogonal symmetry with respect to the hyperplane 𝑂𝑥2𝑥3),one has the involution

𝐴′ = 𝐾1 (𝐴) = [𝑝0, 𝑝1,−𝑝2,−𝑝3] + 𝜖 [−𝑞0,−𝑞1, 𝑞2, 𝑞3] . (5.7)

10. Analytic Video (2D + 𝑡) Signals 207

Figure 4. Orthogonal symmetry with respect to a hyperplane.

Similarly, if 𝑎 = 𝜖𝑗 (orthogonal symmetry with respect to the hyperplane 𝑂𝑥1𝑥3),one has

𝐴′ = 𝐾2(𝐴) = [𝑝0,−𝑝1, 𝑝2,−𝑝3] + 𝜖 [−𝑞0, 𝑞1,−𝑞2, 𝑞3] . (5.8)

If 𝑎 = 𝜖𝑘, one has

𝐴′ = 𝐾3(𝐴) = [𝑝0,−𝑝1,−𝑝2, 𝑝3] + 𝜖 [−𝑞0, 𝑞1, 𝑞2,−𝑞3] . (5.9)

A combination of 𝐾1 followed by 𝐾2 leads to a rotation of 𝜋 around the origin

𝐴′ = 𝐾12(𝐴) = 𝑟𝐴𝑟𝑐 = [𝑝0,−𝑝1,−𝑝2, 𝑝3] + 𝜖 [𝑞0,−𝑞1,−𝑞2, 𝑞3] (5.10)

with 𝑟 = 𝑗𝑖 = −𝑘. Similarly,

𝐴′ = 𝐾13(𝐴) = [𝑝0,−𝑝1, 𝑝2,−𝑝3] + 𝜖 [𝑞0,−𝑞1, 𝑞2,−𝑞3] (5.11)

𝐴′ = 𝐾23(𝐴) = [𝑝0, 𝑝1,−𝑝2,−𝑝3] + 𝜖 [𝑞0, 𝑞1,−𝑞2,−𝑞3] . (5.12)

A combination of three symmetries leads to

𝐴′ = 𝐾123(𝐴) = [𝑝0, 𝑝1, 𝑝2, 𝑝3] + 𝜖 [−𝑞0,−𝑞1,−𝑞2,−𝑞3] . (5.13)

5.2. Clifford–Fourier Transform

The Clifford–Fourier 2D+ 𝑡 transform and its inverse are respectively defined with𝒙 = (𝑥1, 𝑥2, 𝑥3), and 𝒖 = (𝑢1, 𝑢2, 𝑢3) by

𝐹 (𝒖) =

∫𝑅3

𝑓(𝒙)𝑒−𝜖𝑖2𝜋𝑢1𝑥1𝑒−𝜖𝑗2𝜋𝑢2𝑥2𝑒−𝜖𝑘2𝜋𝑢3𝑥3𝑑𝑥1𝑑𝑥2𝑑𝑥3 (5.14)

𝑓(𝒙) =

∫𝑅3

𝐹 (𝒖)𝑒𝜖𝑘2𝜋𝑢3𝑥3𝑒𝜖𝑗2𝜋𝑢2𝑥2𝑒𝜖2𝜋𝑢1𝑥1𝑑𝑢1𝑑𝑢2𝑑𝑢3. (5.15)

208 P.R. Girard et al.

Table 4. Computation principle of the Clifford–Fourier transform(CFT) with a clifbquat entry (for the inverse CFT, the formulas arethe same except that the FFT is replaced by an IFFT and the integra-tion order is reversed: 𝑥3, 𝑥2, 𝑥1).

CFT of 𝑓(x) = [𝑝1, 𝑝2, 𝑝3, 𝑝4] + 𝜖 [𝑝5, 𝑝6, 𝑝7, 𝑝8]

real component: 𝑝𝜇 (x) = 𝑎 (x)FFT on 𝑥1 : 𝑎1 + 𝜖𝑖𝑎2

FFT on 𝑥2 : (𝑎1𝛼 + 𝜖𝑗𝑎1𝛽) + 𝜖𝑖 (𝑎2𝛼 + 𝜖𝑗𝑎2𝛽)FFT on 𝑥3 : (𝑎1𝛼𝛾 + 𝜖𝑘𝑎1𝛼𝛿) + 𝜖𝑗 (𝑎1𝛽𝛾 + 𝜖𝑘𝑎1𝛽𝛿)

+𝜖𝑖 [(𝑎2𝛼𝛾 + 𝜖𝑘𝑎2𝛼𝛿) + 𝜖𝑗 (𝑎2𝛽𝛾 + 𝜖𝑘𝑎2𝛽𝛿)]𝐹𝜇 = 𝐶𝐹𝑇 (𝑝𝜇) = [𝑎1𝛼𝛾 , 𝑎1𝛽𝛿,−𝑎2𝛼𝛿, 𝑎2𝛽𝛾 ] +𝜖 [−𝑎2𝛽𝛿, 𝑎2𝛼𝛾 , 𝑎1𝛽𝛾 , 𝑎1𝛼𝛿]

CFT(𝑓)=(𝐹1 + 𝑖𝐹2 + 𝑗𝐹3 + 𝑘𝐹4) + 𝜖 (𝐹5 + 𝑖𝐹6 + 𝑗𝐹7 + 𝑘𝐹8)

Considering two clifbquat functions 𝑓(𝒙) and 𝑔(𝒙), one defines the product

⟨𝑓, 𝑔⟩ = 1

2

∫ℝ3

(𝑓𝑔𝑐 + 𝑔𝑓𝑐) 𝑑𝑥1𝑑𝑥2𝑑𝑥3 (5.16)

which satisfies Plancherel’s theorem

⟨𝑓 (𝒙) , 𝑔 (𝒙)⟩ = ⟨𝐹 (𝒖) , 𝐺 (𝒖)⟩ . (5.17)

To compute the direct Clifford–Fourier transform (CFT), one proceeds in cascadeintegrating first with respect to 𝑥1 using a standard FFT. A second FFT (inte-gration with respect to 𝑥2) is then applied to each real component of the previouscomplex number. Then, a third FFT (integration on 𝑥3) is applied on each of theresulting real components. Finally, all the components are properly displayed asa clifbquat. For the inverse Clifford–Fourier transform, one proceeds in the sameway on each real component of the clifbquat using an IFFT and reversing theorder of integration (see Table 4).

The above involutions lead to the following symmetries of the Clifford–Fouriertransform of a scalar function 𝑓(𝑥1, 𝑥2, 𝑥3)

𝐹 (−𝑢1, 𝑢2, 𝑢3) = 𝐾1 [𝐹 (𝒖)] , 𝐹 (𝑢1,−𝑢2, 𝑢3) = 𝐾2 [𝐹 (𝒖)] (5.18)

𝐹 (𝑢1, 𝑢2,−𝑢3) = 𝐾3 [𝐹 (𝒖)] , 𝐹 (−𝑢1,−𝑢2, 𝑢3) = 𝐾12 [𝐹 (𝒖)] (5.19)

𝐹 (−𝑢1, 𝑢2,−𝑢3) = 𝐾13 [𝐹 (𝒖)] , 𝐹 (𝑢1,−𝑢2,−𝑢3) = 𝐾23 [𝐹 (𝒖)] (5.20)

𝐹 (−𝑢1,−𝑢2,−𝑢3) = 𝐾123 [𝐹 (𝒖)] . (5.21)

Hence, only one orthant of the Fourier space is necessary to represent the en-tire Fourier space. A few properties of the Clifford–Fourier transform of a scalarfunction are given in Table 5.

5.3. Analytic Signal

The analytic Clifford–Fourier transform is defined by

𝐹𝐴(𝒖) = [1 + sign (𝑢1)] [1 + sign (𝑢2)] [1 + sign (𝑢3)]𝐹 (𝒖) (5.22)

10. Analytic Video (2D + 𝑡) Signals 209

Table 5. Properties of the Clifford–Fourier transform of a scalar func-tion (𝑎, 𝑏 ∈ ℝ).

Property Functions CFT

Linearity 𝑎𝑓 (𝒙) + 𝑏𝑔 (𝒙) 𝑎𝐹 (𝒖) + 𝑏𝐺 (𝒖)Translation 𝑓 (𝑥1 − 𝑎1, 𝑥2, 𝑥3 − 𝑎3) 𝑒−𝜖𝑖2𝜋𝑢1𝑎1𝐹 (𝒖)𝑒−𝜖𝑘2𝜋𝑢3𝑎3

Scaling 𝑓 (𝑎1𝑥1, 𝑎2𝑥2, 𝑎3𝑥3)1∣𝑎1∣

1∣𝑎2∣

1∣𝑎3∣𝐹

(𝑢1𝑎1

, 𝑢2𝑎2

, 𝑢3𝑎3

)Partial derivative

∂𝑟𝑓∂𝑥𝑟1

(𝑥1, 𝑥2, 𝑥3) 𝐹 (𝒖) (𝜖𝑖2𝜋𝑢1)𝑟

(𝑞 = 4𝑝) (2𝜋𝑢2)𝑞𝐹 (𝒖)

(𝑞 = 4𝑝+ 1)∂𝑞𝑓∂𝑥𝑞2

(𝑥1, 𝑥2, 𝑥3) (2𝜋𝑢2)𝑞𝜖𝑗𝐾1[𝐹 (𝒖)]

(𝑞 = 4𝑝+ 2) − (2𝜋𝑢2)𝑞𝐹 (𝒖)

(𝑞 = 4𝑝+ 3) − (2𝜋𝑢2)𝑞 𝜖𝑗𝐾1[𝐹 (𝒖)]

∂𝑟𝑓∂𝑥𝑟3

(𝑥1, 𝑥2, 𝑥3) 𝐹 (𝒖) (𝜖𝑘2𝜋𝑢3)𝑟

Plancherel ⟨𝑓, 𝑔⟩ = ⟨𝐹,𝐺⟩Parseval ⟨𝑓, 𝑓⟩ = ⟨𝐹, 𝐹 ⟩

Table 6. Examples of analytic signals.

Function: 𝑓(𝒙) Analytic signal: 𝑓𝐴(𝒙)

cos𝜔1𝑥1 cos𝜔2𝑥2 cos𝜔3𝑥3 𝑒𝜖𝑘𝜔3𝑥3𝑒𝜖𝑗𝜔2𝑥2𝑒𝜖𝑖𝜔1𝑥1

cos𝜔1𝑥1 sin𝜔2𝑥2 sin𝜔3𝑥3 𝑖𝑒𝜖𝑘𝜔3𝑥3𝑒𝜖𝑗𝜔2𝑥2𝑒𝜖𝑖𝜔1𝑥1

sin𝜔1𝑥1 sin𝜔2𝑥2 sin𝜔3𝑥3 𝜖𝑒𝜖𝑘𝜔3𝑥3𝑒𝜖𝑗𝜔2𝑥2𝑒𝜖𝑖𝜔1𝑥1

cos𝜔1𝑥1 cos𝜔2𝑥2 sin𝜔3𝑥3 −𝜖𝑘𝑒𝜖𝑘𝜔3𝑥3𝑒𝜖𝑗𝜔2𝑥2𝑒𝜖𝑖𝜔1𝑥1

cos (𝜔1𝑥1 + 𝜔2𝑥2 + 𝜔3𝑥3) (1− 𝑖+ 𝑗 − 𝑘) 𝑒𝜖𝑘𝜔3𝑥3𝑒𝜖𝑗𝜔2𝑥2𝑒𝜖𝑖𝜔1𝑥1

= 2𝑒𝑗𝜋/4𝑒−𝑘𝜋/4𝑒𝜖𝑘𝜔3𝑥3𝑒𝜖𝑗𝜔2𝑥2𝑒𝜖𝑖𝜔1𝑥1

and the analytic signal by

𝑓𝐴(𝒙) =

∫𝑅3

𝐹𝐴(𝒖)𝑒𝜖𝑘2𝜋𝑢3𝑥3𝑒𝜖𝑗2𝜋𝑢2𝑥2𝑒𝜖𝑖2𝜋𝑢1𝑥1𝑑𝑢1𝑑𝑢2𝑑𝑢3. (5.23)

Examples of analytic signals are given in Table 6. The analytic signal being aClifford biquaternion, it can be represented as 𝐴 = 𝜆𝑎 where 𝜆 = 𝛼+𝜖𝛽 consists ofa scalar and a pseudo-scalar and where 𝑎 is a unit Clifford biquaternion (𝑎𝑎𝑐 = 1).Writing, 𝐴𝐴𝑐 = 𝜆2 = 𝑔1 + 𝜖𝑔2, one finds

𝛼 =

√𝑔1 +

√𝑔21 − 𝑔2

2

2, 𝛽 =

√𝑔1 −

√𝑔21 − 𝑔2

2

2(5.24)

(if 𝛼 = 𝛽 = 0, one adopts the choice 𝑎 = 1, in order to have a unit Cliffordbiquaternion 𝑎 in all cases). The unit Clifford biquaternion 𝑎 is obtained via the

210 P.R. Girard et al.

relation

𝑎 = 𝜆−1𝐴 = (𝛼+ 𝜖𝛽)−1 𝐴 =

(𝛼− 𝜖𝛽

𝛼2 − 𝛽2

)𝐴, (5.25)

and is decomposed according to 𝑎 = 𝑏𝑟, with 𝑏 = 𝑏1 + 𝜖 (𝒊𝑏2 + 𝒋𝑏3 + 𝒌𝑏4) being aunit Clifford biquaternion such that 𝑏𝑐 = 𝑏 and where 𝑟 = 𝑟1+𝒊𝑟2+𝒋𝑟3+𝒌𝑟4 with𝑟𝑟𝑐 = 1. Using a procedure similar to that used in special relativity [11, p. 82], onehas

𝑏 =1 + 𝑑√

2 + 𝑑+ 𝑑𝑐(5.26)

with 𝑑 = 𝑎 (𝑎𝑐). If 𝑑 = −1, one has a particular case which has to be treated so asto yield a unit Clifford biquaternion 𝑏 in all cases. For example, if 𝑎 = ±𝜖, (with𝑑 = −1) one takes 𝑏 = 𝜖𝒊; if 𝑎 = 𝜖 (𝒊𝑎1 + 𝒋𝑎2 + 𝒌𝑎3), one adopts

𝑏 = 𝜖(𝒊𝑎1 + 𝒋𝑎2 + 𝒌𝑎3)√

𝑎21 + 𝑎2

2 + 𝑎23

. (5.27)

The unit Clifford biquaternion 𝑟 is then obtained as 𝑟 = 𝑏𝑐𝑎. Both, 𝑏 and 𝑟 can beput into a polar form, respectively

𝑏 = 𝑒𝜖𝑗𝜑2[𝑒𝜖𝑘𝜑3

(𝑒𝜖𝑖𝜑1

)𝑒𝜖𝑘𝜑3

]𝑒𝜖𝑗𝜑2 (5.28)

= cos𝜑1 cos 2𝜑2 cos 2𝜑3

+ 𝜖(𝑖 sin𝜑1 + 𝑗 cos𝜑1 cos 2𝜑3 sin 2𝜑2 + 𝑘 cos𝜑1 sin 2𝜑3) (5.29)

with 𝑏𝑐 = 𝑏 and 𝑟 = 𝑒𝑖𝜃1𝑒𝑘𝜃3𝑒𝑗𝜃2 . The phases of 𝑏 are extracted according to therules:

𝜑1 = arcsin 𝑏 (2, 2) (5.30)

where 𝑏 (2, 2) means the second component of the second quaternion of 𝑏. Ifcos𝜑1 ∕= 0,

𝜑3 =1

2arcsin

(𝑏 (2, 4)

cos𝜑1

); (5.31)

if cos𝜑1 ∕= 0 and cos 2𝜑3 ∕= 0

𝜑2 =1

2arcsin

(𝑏 (2, 3)

cos𝜑1 cos 2𝜑3

). (5.32)

The particular cases are treated as follows. If 𝜑1 = ±𝜋2 , the choice 𝜑2 = 𝜑3 = 0

is adopted; if 𝜑1 ∕= ±𝜋2 and 𝜑3 = ±𝜋

4 , one chooses 𝜑2 = 0. The sign of thereconstructed 𝑏 is then compared to that of the initial 𝑏; if it is opposed, 𝜑1

is replaced by [𝜑1 − 𝜋 sign (𝜑1)]. The phases of 𝑟 = 𝑒𝑖𝜃1𝑒𝑘𝜃3𝑒𝑗𝜃2 are extractedaccording to the procedure presented for the 2D analytic signal by Sommer [29,p. 194] (except that eventual phase shifts are reported on 𝜃3 rather than on 𝜃1).

10. Analytic Video (2D + 𝑡) Signals 211

Within the 2D + 𝑡 Clifford biquaternion framework, the procedure of extractingthe triplet of phases (𝜃1, 𝜃2, 𝜃3) is as follows. From the relations

𝑑1 = 𝑟 ∗ [𝐾2 (𝑟)]𝑐

= [cos 2𝜃1 cos 2𝜃3, sin 2𝜃1 cos 2𝜃3, 0, sin 2𝜃3] (5.33)

𝑑2 = [𝐾1 (𝑟)]𝑐 ∗ 𝑟= [cos 2𝜃2 cos 2𝜃3, 0, cos 2𝜃3 sin 2𝜃2, sin 2𝜃3] (5.34)

one obtains 𝜃3 = 12 arcsin𝑑1(4) where 𝑑1(4) means the fourth component of 𝑑1. If

cos 2𝜃3 ∕= 0, one has

𝜃1 =1

2Arg [𝑑1 (1) + 𝑖𝑑1(2)] , 𝜃2 =

1

2Arg [𝑑2(1) + 𝑖𝑑2(3)] . (5.35)

If cos 2𝜃3 = 0, one has an indeterminacy and only (𝜃1 ± 𝜃2) can be determinedfrom the relation

𝑑3 = [𝐾12 (𝑟)]𝑐 ∗ 𝑟 = [cos 2 (𝜃1 ∓ 𝜃2) , 0,∓ sin 2 (𝜃1 ∓ 𝜃2) , 0] ; (5.36)

adopting the choice 𝜃1 = 0, one has

𝜃2 =1

2Arg [𝑑3(1) + 𝑖𝑑3(3)] . (5.37)

Finally, having determined the phases (𝜃1, 𝜃2, 𝜃3) one computes 𝑒𝑖𝜃1𝑒𝑘𝜃3𝑒𝑗𝜃2 ; if itis equal to −𝑟 and 𝜃3 ≥ 0, one takes 𝜃3 − 𝜋; if one has −𝑟 and 𝜃3 < 0 then onetakes 𝜃3 + 𝜋. Hence, the phase domains for 𝑟 and for 𝑏 are respectively

(𝜃1, 𝜃2, 𝜃3) ∈[−𝜋

2,𝜋

2

]∪[−𝜋

2,𝜋

2

]∪ [−𝜋, 𝜋[ (5.38)

(𝜑1, 𝜑2,𝜑3) ∈ [−𝜋, 𝜋[ ∪[−𝜋

2,𝜋

2

]∪[−𝜋

2,𝜋

2

]. (5.39)

Finally, the analytic signal is characterized by a scalar, a pseudo-scalar and thesix phases above.

6. Results

6.1. Introductory Example

One considers the function

𝑓(𝑡, 𝑥, 𝑦) = cos(𝜔1𝑡+ 𝑘1𝑥+ 𝑝1𝑦) + cos(𝜔2𝑡+ 𝑘2𝑥+ 𝑝2𝑦) (6.1)

with 𝜔𝑖 = Ω± 𝜔2 , 𝑘𝑖 = 𝐾 ± 𝑘

2 , 𝑃𝑖 = 𝑃 ± 𝑝2 and 𝜔2 > 𝜔1, 𝑘2 > 𝑘1, 𝑝2 > 𝑝1.

212 P.R. Girard et al.

The analytic signal is given by

𝑓𝐴(𝑡, 𝑥, 𝑦) =

⎡⎢⎢⎣2 cos(𝜔𝑡+ 𝑘𝑥+ 𝑝𝑦) cos(Ω𝑡+𝐾𝑥+ 𝑃𝑦),

−2 cos(𝜔𝑡− 𝑘𝑥+ 𝑝𝑦) cos(Ω𝑡−𝐾𝑥+ 𝑃𝑦),2 cos(𝜔𝑡+ 𝑘𝑥− 𝑝𝑦) cos(Ω𝑡−𝐾𝑥− 𝑃𝑦),

−2 cos(𝜔𝑡− 𝑘𝑥− 𝑝𝑦) cos(Ω𝑡−𝐾𝑥− 𝑃𝑦)

⎤⎥⎥⎦

+𝜖

⎡⎢⎢⎣2 cos(𝜔𝑡− 𝑘𝑥+ 𝑝𝑦) sin(Ω𝑡−𝐾𝑥+ 𝑃𝑦),2 cos(𝜔𝑡+ 𝑘𝑥+ 𝑝𝑦) sin(Ω𝑡+𝐾𝑥+ 𝑃𝑦),

−2 cos(𝜔𝑡− 𝑘𝑥− 𝑝𝑦) sin(Ω𝑡−𝐾𝑥− 𝑃𝑦),−2 cos(𝜔𝑡+ 𝑘𝑥− 𝑝𝑦) sin(Ω𝑡+𝐾𝑥− 𝑃𝑦)

⎤⎥⎥⎦ . (6.2)

One then computes

𝑑4 = [𝑓𝐴(1)]2+ [𝑓𝐴(6)]

2= 4 cos2(𝜔𝑡+ 𝑘𝑥+ 𝑝𝑦) (6.3)

𝑑5 = 𝑓𝐴𝑓𝐴𝑐 =[8(1 + cos 2𝜔𝑡 cos 2𝑘𝑥 cos 2𝑝𝑦), 0, 0, 0]+𝜖 [8 sin 2𝜔𝑡 sin 2𝐾𝑥 cos 2𝑝𝑦, 0, 0, 0]

. (6.4)

The phase and group velocities are obtained as follows. Write

Φ1 = Ω𝑡+𝐾𝑥+ 𝑃𝑦,Φ2 = 𝜔𝑡+ 𝑘𝑥+ 𝑝𝑦; (6.5)

one has

Φ1 = arctan𝑓𝐴(6)

𝑓𝐴(1)(6.6)

Φ2 =1

2arccos

(𝑑4

2− 1

). (6.7)

Hence the phase and group velocities are

𝑣𝜑𝑥 =

∂Φ1∂𝑡∂Φ1∂𝑥

, 𝑣𝜑𝑦 =

∂Φ1∂𝑡∂Φ1∂𝑦

(6.8)

𝑣𝑔𝑥 =

∂Φ2∂𝑡∂Φ2∂𝑥

, 𝑣𝑔𝑦 =

∂Φ2∂𝑡∂Φ2∂𝑦

. (6.9)

Finally, one might notice that the scalar part of 𝑑5 contains only the modulationof the signal.

6.2. Analytic Video Signal 2D + 𝒕 Implementation

The implementation considers the scalar function

𝑓(𝑥1, 𝑥2, 𝑥3) = cos𝑥1 cos𝑥2 cos𝑥3 (6.10)

10. Analytic Video (2D + 𝑡) Signals 213

(𝑥1 being the time and 𝑥2, 𝑥3 respectively 𝑥, 𝑦) where (𝑥1, 𝑥2, 𝑥3) vary between[0, 2𝜋]. The explicit form of the analytic video signal is

𝐴 =

⎡⎢⎢⎣(

cos𝑥1 cos𝑥2 cos𝑥3,− cos𝑥1 sin𝑥2 sin𝑥3,sin𝑥1 cos𝑥2 sin𝑥3,− sin𝑥1 sin𝑥2 cos𝑥3

)+𝜖

(sin𝑥1 sin𝑥2 sin𝑥3, sin𝑥1 cos𝑥2 cos𝑥3,cos𝑥1 sin𝑥2 cos𝑥3, cos𝑥1 cos𝑥2 sin𝑥3

)⎤⎥⎥⎦ (6.11)

with 𝐴𝐴𝑐 = 1, hence 𝜆 = 1. The eight components of 𝐴 are represented at thetime 𝑥1 = 2𝜋

8 s in Figure 5

𝐴 = (𝑎, 𝑏, 𝑐, 𝑑) + 𝜖 (𝑒, 𝑓, 𝑔, ℎ) . (6.12)

The eight components in polar form (𝛼, 𝜃1, 𝜃2, 𝜃3, 𝛽, 𝜑1, 𝜑2, 𝜑3) are represented inFigure 6.

7. Conclusion

This chapter has presented a concrete algebraic framework, i.e., Clifford’s biquater-nions for the expression of Clifford–Fourier transforms and 2D+ 𝑡 analytic signals.Then, we have shown how to put the analytic signal into a polar form constitutedby a scalar, a pseudoscalar and six phases. Finally, using discrete fast Fouriertransforms we have implemented numerically the Clifford biquaternion Fouriertransform, the analytic Fourier transform and the analytic signal both in standardand polar form. Our next objective will be to extract physical information fromthe phases in medical images.

Appendix A. Examples of (2D + 𝒕) Clifford–Fourier Transforms

A.1. Example 1

Signal

𝑓(𝑥1, 𝑥2, 𝑥3) = cos 2𝜋𝑓1𝑥1 cos 2𝜋𝑓2𝑥2 cos 2𝜋𝑓3𝑥3.

Clifford–Fourier transform

𝐹 (𝑢1, 𝑢2, 𝑢3) =1

8[𝛿 (𝑓1 − 𝑢1) + 𝛿 (𝑓1 + 𝑢1)] [𝛿 (𝑓2 − 𝑢2) + 𝛿 (𝑓2 + 𝑢2)]

× [𝛿 (𝑓3 − 𝑢3) + 𝛿 (𝑓3 + 𝑢3)] .

Analytic Clifford–Fourier transform

𝐹𝐴(𝑢1, 𝑢2, 𝑢3) = 𝛿 (𝑓1 − 𝑢1) 𝛿 (𝑓2 − 𝑢2) 𝛿 (𝑓3 − 𝑢3) .

Analytic signal

𝑓𝐴(𝑥1, 𝑥2, 𝑥3) = 𝑒𝜖𝑘2𝜋𝑓3𝑥3𝑒𝜖𝑗2𝜋𝑓2𝑥2𝑒𝜖𝑖2𝜋𝑓1𝑥1 .

214 P.R. Girard et al.

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Figure 5. The eight components of the analytic video signal 𝐴 =(𝑎, 𝑏, 𝑐, 𝑑) + 𝜖(𝑒, 𝑓, 𝑔, ℎ) at the time 𝑥1 = 2𝜋

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10. Analytic Video (2D + 𝑡) Signals 215

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Figure 6. Polar form of the analytic video signal at the time 𝑥1 = 2𝜋8

s, (𝑎) is the scalar, (𝑏, 𝑐, 𝑑) are the three phases (𝜃1, 𝜃2, 𝜃3); (e) is thepseudo-scalar and (𝑓, 𝑔, ℎ) are the three phases (𝜑1, 𝜑2, 𝜑3).

216 P.R. Girard et al.

A.2. Example 2

Signal

𝑓(𝑥1, 𝑥2, 𝑥3) = cos (2𝜋𝑓1𝑥1 + 2𝜋𝑓2𝑥2 + 2𝜋𝑓3𝑥3) = 𝑓1 + 𝑓2 + 𝑓3 + 𝑓4

with

𝑓1 = cos 2𝜋𝑓1𝑥1 cos 2𝜋𝑓2𝜋𝑥2 cos 2𝜋𝑓3𝑥3

𝑓2 = − sin 2𝜋𝑓1𝑥1 sin 2𝜋𝑓2𝜋𝑥2 cos 2𝜋𝑓3𝑥3

𝑓3 = − sin 2𝜋𝑓1𝑥1 cos 2𝜋𝑓2𝜋𝑥2 sin 2𝜋𝑓3𝑥3

𝑓4 = − cos 2𝜋𝑓1𝑥1 sin 2𝜋𝑓2𝜋𝑥2 sin 2𝜋𝑓3𝑥3.

Clifford–Fourier transform

𝐹 (𝑢1, 𝑢2, 𝑢3) = 𝐹1 + 𝐹2 + 𝐹3 + 𝐹4, 𝐹𝑖 = 𝐶𝐹𝑇 (𝑓𝑖)

where

𝐹1 =1

8[𝛿 (𝑓1 − 𝑢1) + 𝛿 (𝑓1 + 𝑢1)] [𝛿 (𝑓2 − 𝑢2) + 𝛿 (𝑓2 + 𝑢2)]

× [𝛿 (𝑓3 − 𝑢3) + 𝛿 (𝑓3 + 𝑢3)]

and

𝐹2 =−18

[−𝜖𝑖𝛿 (𝑓1 − 𝑢1) + 𝜖𝑖𝛿 (𝑓1 + 𝑢1)] [−𝜖𝑗𝛿 (𝑓2 − 𝑢2) + 𝜖𝑗𝛿 (𝑓2 + 𝑢2)]

× [−𝛿 (𝑓3 − 𝑢3) + 𝛿 (𝑓3 + 𝑢3)]

=−𝑘

8[−𝛿 (𝑓1 − 𝑢1) + 𝛿 (𝑓1 + 𝑢1)] [−𝛿 (𝑓2 − 𝑢2) + 𝛿 (𝑓2 + 𝑢2)]

× [−𝛿 (𝑓3 − 𝑢3) + 𝛿 (𝑓3 + 𝑢3)]

and

𝐹3 =−18

[−𝜖𝑖𝛿 (𝑓1 − 𝑢1) + 𝜖𝑖𝛿 (𝑓1 + 𝑢1)] [𝛿 (𝑓2 − 𝑢2) + 𝛿 (𝑓2 + 𝑢2)]

× [−𝜖𝑘𝛿 (𝑓3 − 𝑢3) + 𝜖𝑘𝛿 (𝑓3 + 𝑢3)]

=𝑗

8[−𝛿 (𝑓1 − 𝑢1) + 𝛿 (𝑓1 + 𝑢1)] [𝛿 (𝑓2 − 𝑢2) + 𝛿 (𝑓2 + 𝑢2)]

× [−𝛿 (𝑓3 − 𝑢3) + 𝛿 (𝑓3 + 𝑢3)]

and

𝐹4 =−18

[𝛿 (𝑓1 − 𝑢1) + 𝛿 (𝑓1 + 𝑢1)] [−𝜖𝑗𝛿 (𝑓2 − 𝑢2) + 𝜖𝑗𝛿 (𝑓2 + 𝑢2)]

× [−𝜖𝑘𝛿 (𝑓3 − 𝑢3) + 𝜖𝑘𝛿 (𝑓3 + 𝑢3)]

=−𝑖

8[𝛿 (𝑓1 − 𝑢1) + 𝛿 (𝑓1 + 𝑢1)] [−𝛿 (𝑓2 − 𝑢2) + 𝛿 (𝑓2 + 𝑢2)]

× [−𝛿 (𝑓3 − 𝑢3) + 𝛿 (𝑓3 + 𝑢3)] .

10. Analytic Video (2D + 𝑡) Signals 217

Analytic Clifford–Fourier transform

𝐹𝐴(𝑢1, 𝑢2, 𝑢3) = (1− 𝑖+ 𝑗 − 𝑘) 𝛿 (𝑓1 − 𝑢1) 𝛿 (𝑓2 − 𝑢2) 𝛿 (𝑓3 − 𝑢3)

Analytic signal

𝑓𝐴(𝑥1, 𝑥2, 𝑥3) = (1− 𝑖+ 𝑗 − 𝑘)𝑒𝜖𝑘2𝜋𝑓3𝑥3𝑒𝜖𝑗2𝜋𝑓2𝑥2𝑒𝜖𝑖2𝜋𝑓1𝑥1 = 𝑝+ 𝜖𝑞

where

𝑝 =

[cos (2𝜋𝑓1𝑥1 + 2𝜋𝑓2𝑥2 + 2𝜋𝑓3𝑥3) ,− cos (2𝜋𝑓1𝑥1 − 2𝜋𝑓2𝑥2 + 2𝜋𝑓3𝑥3) ,cos (2𝜋𝑓1𝑥1 + 2𝜋𝑓2𝑥2 − 2𝜋𝑓3𝑥3) ,− cos (2𝜋𝑓1𝑥1 − 2𝜋𝑓2𝑥2 − 2𝜋𝑓3𝑥3)

]𝑞 =

[sin (2𝜋𝑓1𝑥1 − 2𝜋𝑓2𝑥2 + 2𝜋𝑓3𝑥3) , sin (2𝜋𝑓1𝑥1 + 2𝜋𝑓2𝑥2 + 2𝜋𝑓3𝑥3) ,

− sin (2𝜋𝑓1𝑥1 − 2𝜋𝑓2𝑥2 − 2𝜋𝑓3𝑥3) ,− sin (2𝜋𝑓1𝑥1 + 2𝜋𝑓2𝑥2 − 2𝜋𝑓3𝑥3)

]

References

[1] F. Brackx, N. De Schepper, and F. Sommen. The two-dimensional Clifford–Fouriertransform. Journal of Mathematical Imaging and Vision, 26(1):5–18, 2006.

[2] T. Bulow. Hypercomplex Spectral Signal Representations for the Processing and Anal-ysis of Images. PhD thesis, University of Kiel, Germany, Institut fur Informatik undPraktische Mathematik, Aug. 1999.

[3] T. Bulow and G. Sommer. A novel approach to the 2D analytic signal. In F. Solinaand A. Leonardis, editors, Computer Analysis of Images and Patterns, volume 1689of Lecture Notes in Computer Science, pages 25–32. Springer, Berlin/Heidelberg,1999.

[4] W.K. Clifford. Applications of Grassmann’s extensive algebra. American Journal ofMathematics, 1(4):350–358, 1878.

[5] W.K. Clifford. Mathematical Papers. Chelsea Publishing Company, New York, 1968.First published 1882, edited by R. Tucker.

[6] M.J. Crowe. A History of Vector analysis: The Evolution of the Idea of a VectorialSystem. University of Notre Dame, Notre Dame, London, 1967.

[7] P. Delachartre, P. Clarysse, R. Goutte, and P.R. Girard. Mise en oeuvre du signalanalytique dans les algebres de Clifford. In GRETSI, page 4, Bordeaux, France, 2011.

[8] T.A. Ell. Hypercomplex Spectral Transformations. PhD thesis, University of Min-nesota, June 1992.

[9] D. Gabor. Theory of communication. Journal of the Institution of Electrical Engi-neers, 93(26):429–457, 1946. Part III.

[10] P.R. Girard. Quaternions, Algebre de Clifford et Physique Relativiste. PPUR, Lau-sanne, 2004.

[11] P.R. Girard. Quaternions, Clifford Algebras and Relativistic Physics. Birkhauser,Basel, 2007. Translation of [10].

[12] P.R. Girard. Quaternion Grassmann-Hamilton-Clifford-algebras: new mathematicaltools for classical and relativistic modeling. In O. Dossel and W.C. Schlegel, editors,World Congress on Medical Physics and Biomedical Engineering, September 7–12,2009, volume 25/IV of IFMBE Proceedings, pages 65–68. Springer, 2010.

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[13] P.R. Girard. Multiquaternion Grassmann-Hamilton-Clifford algebras in physics andengineering: a short historical perspective. In K. Gurlebeck, editor, 9th InternationalConference on Clifford Algebras and their Applications, page 9. Weimar, Germany,15–20 July 2011.

[14] H. Grassmann. Die lineale Ausdehungslehre: ein neuer Zweig der Mathematik,dargestellt und durch Anwendungen auf die ubrigen Zweige der Mathematik, wie auchdie Statik, Mechanik, die Lehre von Magnetismus und der Krystallonomie erlautert.Wigand, Leipzig, second, 1878 edition, 1844.

[15] H. Grassmann. Der Ort der Hamilton’schen Quaternionen in der Ausdehnungslehre.Mathematische Annalen, 12:375–386, 1877.

[16] H. Grassmann. Gesammelte mathematische und physikalische Werke. B.G. Teubner,Leipzig, 1894–1911. 3 volumes in 6 parts.

[17] A. Gsponer and J.P. Hurni. Quaternions in mathematical physics (1): Alphabeti-cal bibliography. Preprint, available at: http://www.arxiv.org/abs/arXiv:mathph/0510059, 2008. 1430 references.

[18] A. Gsponer and J.P. Hurni. Quaternions in mathematical physics (2): Analyticalbibliography. Preprint, available at: http://www.arxiv.org/abs/arXiv:mathphy/

0511092, 2008. 1100 references.

[19] K. Gurlebeck and W. Sproßig. Quaternionic and Clifford Calculus for Physicists andEngineers. Wiley, Aug. 1997.

[20] S.L. Hahn. Multidimensional complex signals with single-orthant spectra. Proceed-ings of the IEEE, 80(8):1287–1300, Aug. 1992.

[21] S.L. Hahn and K.M. Snopek. The unified theory of 𝑛-dimensional complex andhypercomplex analytic signals. Bulletin of the Polish Academy of Sciences TechnicalSciences, 59(2):167–181, 2011.

[22] H. Halberstam and R.E. Ingram, editors. The Mathematical Papers of Sir WilliamRowan Hamilton, volume III Algebra. Cambridge University Press, Cambridge, 1967.

[23] W.R. Hamilton. Elements of Quaternions. Chelsea Publishing Company, New York,reprinted 1969 edition, 1969. 2 volumes (1899–1901).

[24] T.L. Hankins. Sir William Rowan Hamilton. Johns Hopkins University Press, Bal-timore, London, 1980.

[25] J.P. Havlicek, J.W. Havlicek, and A.C. Bovik. The analytic image. In Proceedings1997 International Conference on Image Processing (ICIP ′97), volume 2, pages446–449, Washington, DC, USA, October 26–29 1997.

[26] E. Hitzer. Quaternion Fourier transform on quaternion fields and generalizations.Advances in Applied Clifford Algebras, 17(3):497–517, May 2007.

[27] H.-J. Petsche. Grassmann. Birkhauser, Basel, 2006.

[28] S.J. Sangwine and T.A. Ell. The discrete Fourier transform of a colour image. In J.M.Blackledge and M.J. Turner, editors, Image Processing II Mathematical Methods, Al-gorithms and Applications, pages 430–441, Chichester, 2000. Horwood Publishing forInstitute of Mathematics and its Applications. Proceedings Second IMA Conferenceon Image Processing, De Montfort University, Leicester, UK, September 1998.

[29] G. Sommer, editor. Geometric computing with Clifford Algebras: Theoretical Foun-dations and Applications in Computer Vision and Robotics. Springer, Berlin, 2001.

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[30] J. Ville. Theorie et applications de la notion de signal analytique. Cables et Trans-mission, 2A:61–74, 1948.

[31] J. Vince. Geometric Algebra for Computer Graphics. Springer, London, 2008.

P.R. Girard, P. Clarysse, A. Marion, R. Goutte and P. DelachartreUniversite de Lyon, CREATIS; CNRS UMR 5220Inserm U1044; INSA-Lyon;Universite Lyon 1, FranceBat. Blaise Pascal7 avenue Jean CapelleF-69621 Villeurbanne, France

e-mail: patrick.girard@creatis.insa-lyon.frpatrick.clarysse@creatis.insa-lyon.fr

adrien.marion@creatis.insa-lyon.fr

robert.goutte@creatis.insa-lyon.fr

philippe.delachartre@creatis.insa-lyon.fr

R. PujolUniversite de LyonPole de Mathematiques, INSA-LyonBat. Leonard de Vinci21 avenue Jean CapelleF-69621 Villeurbanne, France

e-mail: romaric.pujol@insa-lyon.fr

Quaternion and Clifford–Fourier Transforms and Wavelets

Trends in Mathematics, 221–246c⃝ 2013 Springer Basel

11 Generalized Analytic Signalsin Image Processing: Comparison,Theory and Applications

Swanhild Bernstein, Jean-Luc Bouchot, Martin Reinhardtand Bettina Heise

Abstract. This article is intended as a mathematical overview of the general-izations of analytic signals to higher-dimensional problems, as well as of theirapplications to and of their comparison on artificial and real-world imagesamples.

We first start by reviewing the basic concepts behind analytic signaltheory and derive its mathematical background based on boundary valueproblems of one-dimensional analytic functions. Following that, two gener-alizations are motivated by means of higher-dimensional complex analysis orClifford analysis. Both approaches are proven to be valid generalizations ofthe known analytic signal concept.

In the last part we experimentally motivate the choice of such higher-dimensional analytic or monogenic signal representations in the context ofimage analysis. We see how one can take advantage of one or the other rep-resentation depending on the application.

Mathematics Subject Classification (2010). Primary 94A12; secondary 44A12,30G35.

Keywords. Monogenic signal, monogenic functional theory, image processing,texture, Riesz transform.

1. Introduction

In the past years and since the pioneer work of Gabor [11], the analytic signalhas attracted much interest in signal processing and information theory. Due toan orthogonal decomposition of oscillating signals into envelope and instantaneousphase or respectively into energetic and structural components, this concept hasbecome very suitable for analyzing signals. In this context such a property is calleda split of identity and allows to separate the different characteristics of a signalinto useful components.

222 S. Bernstein, J.-L. Bouchot, M. Reinhardt and B. Heise

While this approach has given rise to many one-dimensional signal process-ing methods, other developments have been directed towards higher-dimensionalgeneralizations. Of particular interest is the two-dimensional case, i.e., how to dealwith images in an analytic way. As it will be demonstrated in our chapter, twomain directions have been taken, one based on multidimensional complex analysisand another one based on Clifford analysis.

This article is intended as an overview of the mathematical concepts behindanalytic signals based on the Hilbert transform (Section 2). Then, the mathemat-ical generalizations are detailed in Section 3. The end of that section is dedicatedto illustrative examples of the detailed differences between the two generaliza-tions. Section 4 describes the use of spinors for image analysis tasks. The lastsection of this article (Section 5) illustrates their applications like demodulationof two-dimensional AM-FM signals as provided, e.g., in interferometry and someapplications to the processing of natural images.

2. Analytic Signal Theory and Signal Decomposition

Analytic signals were introduced for signal processing in the context of communi-cation theory in the late 40𝑠 [11]. Since then, there has been a growing interestin the analytic signal as a useful tool for representing real-valued signals [25]. Westart here by first reviewing the basics of analytic signal theory and the Hilberttransform and see why the so-called split of identity is an interesting property. Inthe last part we review the mathematical basics and see how we can derive theanalytic signal from a boundary value problem in complex analysis.

2.1. Basic Analytic Signal Theory and the Hilbert Transform

Definition 2.1 (One-dimensional Fourier Transform). In the following, we use asFourier transform ℱ :

ℱ(𝑓)(𝑢) = 𝑓(𝑢) =1√2𝜋

∫ℝ

𝑓(𝑡)𝑒−𝑖𝑡𝑢d𝑡 (2.1)

for 𝑡 ∈ ℝ, 𝑢 ∈ ℝ and 𝑓 ∈ 𝐿2(ℝ)

Definition 2.2 (Hilbert Transform). The Hilbert transform of a signal 𝑓 ∈ 𝐿2(ℝ)(or more generally 𝑓 ∈ 𝐿𝑝(ℝ), 1 < 𝑝 <∞) is defined, either in the spatial domainas a convolution with the Hilbert kernel (2.2), or as a Fourier multiplier (2.3):

ℋ𝑓 = ℎ ∗ 𝑓 (2.2)

ℱ(ℋ𝑓)(𝑢) = −𝑖 sign(𝑢)ℱ(𝑓)(𝑢) (2.3)

where we have made use of two functions:

∙ The Hilbert kernel ℎ(𝑡) = 1𝜋𝑡 .

∙ The operator sign(𝑢) =

⎧⎨⎩1 𝑢 > 0

0 𝑢 = 0

−1 𝑢 < 0.

11. Generalized Analytic Signals in Image Processing 223

Following its definition, we notice that the Hilbert transform acts as an asym-metric phase shift: if we write ±𝑖 = 𝑒±𝑖𝜋/2, the phase of the Fourier spectrum ofthe Hilbert transform is obtained by a rotation of ±90∘.Proposition 2.3 (Properties of the Hilbert Transform). Given a signal 𝑓 the fol-lowing hold true:

∙ ∀𝑢 ∕= 0, ∣ℋ𝑓(𝑢)∣ = ∣ℱ(𝑓)(𝑢)∣,∙ ℋℋ𝑓 = −𝑓 ⇒ ℋ−1 = −ℋ.

Note that a constant function being not in 𝐿2 can not be reconstructed in this way.

The analytic signal is computed as a complex combination of both the originalsignal and its Hilbert transform:

Definition 2.4 (Analytic Signal).

𝑓𝐴 = 𝑓 + 𝑖ℋ𝑓. (2.4)

Due to its definition, an analytic signal has a one-sided Fourier spectrum.Moreover, its values are doubled on the positive side. We also remark that it ispossible to recover the original signal from its analytic description by taking thereal part.

The following proposition holds:

Proposition 2.5.

⟨𝑓,ℋ𝑓⟩𝐿2= 0 Orthogonality, (2.5)

∥𝑓∥22 = ∥ℋ𝑓∥22 Energy conservation. (2.6)

The energy equality is valid only if the DC, or zero-frequency component ofthe signal is neglected [10].

Note that it is possible to write the complex analytic signal in polar coor-dinates. In this case we have: ∀𝑡 ∈ ℝ, 𝑓𝐴(𝑡) = 𝐴(𝑡)𝑒𝑖𝜙(𝑡). 𝐴 is called the localamplitude and 𝜙 is called the local phase. These local features are defined asfollows [11]:

Definition 2.6 (Local features).

𝐴(𝑡) =√

𝑓(𝑡)2 +ℋ𝑓(𝑡)2 (2.7)

𝜙(𝑡) = arctan

(ℋ𝑓(𝑡)

𝑓(𝑡)

)= arctan

(ℑ (𝑓𝐴(𝑡))

ℜ (𝑓𝐴(𝑡))

). (2.8)

Proposition 2.7 (Invariance–Equivariance, Split of identity [10]). The local phase,together with the local amplitude, fulfil the properties of invariance and equivari-ance:

∙ The local phase depends only on the local structure.∙ The local amplitude depends only on the local energy.

If moreover these features constitute a complete description of the signal, they aresaid to perform a split of identity.

224 S. Bernstein, J.-L. Bouchot, M. Reinhardt and B. Heise

However as stated in [10], a split of identity is strictly valid only for band-limited signals with a local zero mean property.

If these conditions are fulfilled the analytic signal representation relies on anorthogonal decomposition of the structural information (the local phase), and theenergetic information (the local amplitude).

This split of identity is illustrated in Figure 1. The first plot represents threesignals. They are sine waves generated from a mother sine wave (the red one). Theblue curve corresponds to a modification in terms of amplitude of the red one,while the green curve has half the frequency of the red one. Figures 1(b) and 1(c)are respectively the local amplitudes and phases of these three signals. Note thata small phase shift has been added to the blue curve for better readability. We canclearly see that due to the split of identity, modifying one local characteristic ofthe signal does not affect the other one and vice versa.

Figure 1. Illustration of the split of identity. (Explanation in the text.)

2.2. From Analytic Function to Analytic Signal

While the analytic signal is a very common concept in the field of signal theory, itsbasic mathematics can be derived from the theory of analytic functions. The closeconnection can be understood when considering the following Riemann–Hilbert

11. Generalized Analytic Signals in Image Processing 225

problem with respect to the complex parameter 𝑧 = 𝑥+ 𝑖𝑦:

∂𝐹

∂𝑧= 0, 𝑧 ∈ ℂ, 𝑦 ≥ 0, (2.9)

ℜ (𝐹 (𝑥)) = 𝑓(𝑥), 𝑥 ∈ ℝ. (2.10)

One solution of this problem is given by the Cauchy integral

𝐹 (𝑧) = 𝐹Γ𝑓(𝑧) :=1

2𝜋𝑖

∫ℝ

1

𝜏 − 𝑧𝑓(𝜏)d𝜏. (2.11)

Of course this solution is unique only up to a constant. Normally, this constantwill be fixed by the condition ℑ (𝐹 (𝑧0)) = 𝑐, i.e., the imaginary part of 𝐹 given atan interior point.

When we now consider the trace of 𝐹Γ, i.e., the boundary value, we arriveat the so-called Plemelj–Sokhotzki formula:

tr𝐹Γ𝑓 =1

2(𝐼 + 𝑖ℋ)𝑓 =

1

2𝑓 +

1

2𝑖ℋ𝑓 =: 𝑃Γ𝑓. (2.12)

Up to the factor 1/2 this corresponds to our above definition of an analytic signal.

In this way an analytic signal represents the boundary values of an ana-lytic function in the upper half-plane (or for periodic functions in the unit disc).Starting from this concept we are now going to take a look at higher-dimensionalgeneralizations.

3. Higher-dimensional Generalizations

Different approaches have been studied in past years to extend the definition of ananalytic signal to higher-dimensional spaces. Two of them have gained the great-est interest based respectively on multidimensional complex analysis and Cliffordanalysis.

3.1. Using Multiple Complex Variables

3.1.1. Mathematics. In 1998 Bulow proposed a definition of a hypercomplex signalbased on the so-called partial and total Hilbert transform [6]. To adapt this to ourpoint of view, that analytic signals are functions in a Hardy space, we consider thefollowing Riemann–Hilbert problem in ℂ2:

∂𝐹

∂ 𝑧1= 0, (𝑧1, 𝑧2) ∈ ℂ2, 𝑦1, 𝑦2 ≥ 0, (3.1)

∂𝐹

∂ 𝑧2= 0, (𝑧1, 𝑧2) ∈ ℂ2, 𝑦1, 𝑦2 ≥ 0, (3.2)

ℜ (𝐹 (𝑥1, 𝑥2)) = 𝑓(𝑥1, 𝑥2), 𝑥1, 𝑥2 ∈ ℝ2. (3.3)

For the solution, (see, e.g., [8] or [21]), we need to point out that the domain isa poly-domain in the sense of ℂ𝑛, so that we can give it in the form of a Cauchy

226 S. Bernstein, J.-L. Bouchot, M. Reinhardt and B. Heise

integral:

𝐹 (𝑧1, 𝑧2) =1

4𝜋2

∫ℝ2

1

(𝝃1 − 𝑧1)(𝝃2 − 𝑧2)𝑓(𝝃1, 𝝃2)d𝝃1d𝝃2. (3.4)

Now again looking at the corresponding Plemelj–Sokhotzki formula we get

tr𝐹 (𝑥1, 𝑥2) =1

4𝑓(𝑥1, 𝑥2)− 1

4

∫ℝ2

1

(𝝃1 − 𝑥1)(𝝃2 − 𝑥2)𝑓(𝝃1, 𝝃2)d𝝃1d𝝃2

+ 𝑖1

4

(∫ℝ

1

𝝃1 − 𝑥1𝑓(𝝃1, 𝑥2)d𝝃1 +

∫ℝ

1

𝝃2 − 𝑥2𝑓(𝑥1, 𝝃2)d𝝃2

), (3.5)

which up to the factor 1/4 corresponds to the definition of an analytic signal givenby Hahn [13]. Here

ℋ1𝑓(𝑥1, 𝑥2) =

∫ℝ

1

𝝃1 − 𝑥1𝑓(𝝃1, 𝑥2)d𝝃1 (3.6)

ℋ2𝑓(𝑥1, 𝑥2) =

∫ℝ

1

𝝃2 − 𝑥2𝑓(𝑥1, 𝝃2)d𝝃2 (3.7)

is called a partial Hilbert transform, and

ℋ𝑇 𝑓 =1

4

∫ℝ2

1

(𝝃1 − 𝑥1)(𝝃2 − 𝑥2)𝑓(𝝃1, 𝝃2)d𝝃1d𝝃2 (3.8)

a total Hilbert transform. On the level of Fourier symbols we get

ℱ(tr𝐹 )(𝑢1, 𝑢2) = (1 + sign𝑢1)(1 + sign𝑢2)ℱ𝑓(𝑢1, 𝑢2). (3.9)

Let us now take a look at the definition of Bulow. To this end we consider𝐹 to be a function of two variables 𝑧1 and 𝔷2 with two different imaginary units𝒊 and 𝒋 (with 𝒊2 = 𝒋2 = −1), i.e., 𝑧1 = 𝑥1 + 𝒊𝑦1 and 𝔷2 = 𝑥2 + 𝒋𝑦2. We remarkthat both imaginary units can be understood as elements of the quaternionic basiswith multiplication rules 𝒊𝒋 = −𝒋𝒊 = 𝒌. In this way the above Riemann–Hilbertproblem can be rewritten as

∂

∂𝑧1𝐹 = 0, (𝑧1, 𝔷2) ∈ ℂ2, 𝑦1, 𝑦2 ≥ 0, (3.10)

𝐹∂

∂𝔷2= 0, (𝑧1, 𝔷2) ∈ ℂ2, 𝑦1, 𝑦2 ≥ 0, (3.11)

ℜ (𝐹 (𝑥1, 𝑥2)) = 𝑓(𝑥1, 𝑥2), 𝑥1, 𝑥2 ∈ ℝ2, (3.12)

where the second equation should be understood as ∂ 𝔷2 being applied from theright due to the non-commutativity of the complex units 𝒊 and 𝒋.

The solution is given by

𝐹 (𝑧1, 𝔷2) =1

4𝜋2

∫ℝ2

1

(𝝃1 − 𝑧1)(𝝃2 − 𝔷2)𝑓(𝝃1, 𝝃2)d𝝃1d𝝃2, (3.13)

11. Generalized Analytic Signals in Image Processing 227

so that we get from the Plemelj–Sokhotzki formulae

tr𝐹 (𝑥1, 𝑥2) =1

4(𝐼 + 𝒊𝐻1)(𝐼 + 𝒋𝐻2)𝑓(𝑥1, 𝑥2) (3.14)

=1

4(𝑓 + 𝒊ℋ1𝑓 + 𝒋ℋ2𝑓 + 𝒌ℋ𝑇 𝑓)(𝑥1, 𝑥2). (3.15)

While (3.15) is a quaternionic-valued function, it still corresponds to a boundaryvalue of a function holomorphic in two variables. For the representation in theFourier domain one has to keep in mind that one has to apply one Fourier transformwith respect to the complex plane in 𝒊, and one Fourier transform with respect tothe complex plane generated by 𝒋. Taking into account that 𝒊𝒋 = −𝒋𝒊 one arrivesat the so-called quaternionic Fourier transform [16, 6]:

𝒬ℱ𝑓 =

∫ℝ2

𝑒𝒊𝑥1𝝃1𝑓(𝑥1, 𝑥2)𝑒𝒋𝑥2𝝃2d𝑥1d𝑥2, (3.16)

and the following representation in Fourier symbols

𝒬ℱ(tr𝐹 )(𝑢1, 𝑢2) = (1 + sign𝑢1)(1 + sign𝑢2)𝒬ℱ𝑓(𝑢1, 𝑢2). (3.17)

3.1.2. Image Analysis. In image analysis problems, we can introduce the followingfeatures according to [13]

Amplitude. The local amplitude of a multidimensional analytic signal is definedin a similar way as in the one-dimensional case:

𝐴𝐴(𝑥, 𝑦) =√∣𝑓(𝑥, 𝑦)∣2 + ∣ℋ1𝑓(𝑥, 𝑦)∣2 + ∣ℋ2𝑓(𝑥, 𝑦)∣2 + ∣ℋ𝑇 𝑓(𝑥, 𝑦)∣2 (3.18)

This is also called energetic information.

Phase. The phase is a feature describing how much a vector or quaternion numberdiverge from the real axis. It is defined in a manner similar to the classical complexplane.

𝜙𝐴 = arctan

(√ℋ1𝑓2 +ℋ2𝑓2 +ℋ𝑇 𝑓2

𝑓

). (3.19)

This angle 𝜙𝐴 is what is called phase or structural information.

Orientation. Because we are currently considering 2D signals (that is, images), wecan also describe orientation information, as the principal direction carrying thephase information. The imaginary plane, spanned by {𝒊, 𝒋}, is two dimensionaland therefore we can also define an angle 𝜃𝐴 in this plane:

𝜃𝐴 = arctan

(ℋ2𝑓

ℋ1𝑓

). (3.20)

This new angle is called the orientation of the signal or geometric information.

228 S. Bernstein, J.-L. Bouchot, M. Reinhardt and B. Heise

3.2. Using Clifford Analysis

Another approach to higher dimensions is Clifford analysis.

3.2.1. Mathematics. Here we use Clifford algebra 𝐶ℓ0,𝑛 [4]. This is the free algebraconstructed over ℝ𝑛 generated modulo the relation

𝑥2 = −∣𝑥∣2𝒆0, 𝑥 ∈ ℝ𝑛 (3.21)

where 𝒆0 is the identity of 𝐶ℓ0,𝑛.For the algebra 𝐶ℓ0,𝑛 we have the anti-commutation relationship

𝒆𝑖𝒆𝑗 + 𝒆𝑗𝒆𝑖 = −2𝛿𝑖𝑗𝒆0, (3.22)

where 𝛿𝑖𝑗 is the Kronecker symbol. Each element 𝑥 of ℝ𝑛 may be represented by

𝑥 =

𝑛∑𝑖=1

𝑥𝑖𝒆𝑖. (3.23)

A first-order differential operator which factorizes the Laplacian is given bythe Dirac operator

𝐷𝑓(𝑥) =𝑛∑

𝑗=1

∂𝑓

∂𝑥𝑗. (3.24)

The Riemann–Hilbert problem for the Dirac operator in ℝ3 can be stated inthe form

𝐷𝐹 (𝑥) = 0, 𝑥 ∈ ℝ3, 𝑥3 > 0, (3.25)

ℜ (𝐹 (𝑥1, 𝑥2)) = 𝑓(𝑥1, 𝑥2), 𝑥1, 𝑥2 ∈ ℝ2. (3.26)

To solve this problem we follow the same idea as above.

𝐹Γ𝑓 =

∫ℝ2

𝑥− 𝑦

∣𝑥− 𝑦∣2 𝒆3𝑓(𝑥1, 𝑥2)d𝑥1d𝑥2 (3.27)

tr𝐹Γ𝑓 =1

2(𝐼 + 𝑆Γ)𝑓

=1

2𝑓(𝑦1, 𝑦2) +

1

2

∫ℝ2

𝒆1(𝑥1 − 𝑦1) + 𝒆2(𝑥2 − 𝑦2)

∣𝑥− 𝑦∣2 𝒆3𝑓(𝑥1, 𝑥2)d𝑥1d𝑥2. (3.28)

Because quaternions ℍ are isomorphic to the even subalgebra 𝐶ℓ+0,3, i.e., allelements of the form

𝑐0 + 𝑐1𝒆1𝒆2 + 𝑐2𝒆1𝒆3 + 𝑐3𝒆2𝒆3, 𝑐0, 𝑐1, 𝑐2, 𝑐3 ∈ ℝ (3.29)

we can set 𝒊 = 𝒆1𝒆2 and 𝒋 = 𝒆2𝒆3 so that

tr𝐹Γ𝑓 =1

2(𝐼 + 𝑆Γ)𝑓 (3.30)

=1

2𝑓(𝑦1, 𝑦2) +

1

2

∫ℝ2

𝒊(𝑥1 − 𝑦1) + 𝒋(𝑥2 − 𝑦2)

∣𝑥− 𝑦∣2 𝑓(𝑥1, 𝑥2)d𝑥1d𝑥2. (3.31)

Up to the factor 1/2 this is the monogenic signal 𝑓𝑀 = 𝑓 + 𝒊ℛ1𝑓 + 𝒋ℛ2𝑓 :=𝑓+(𝒊, 𝒋)ℛ𝑓 of Sommer and Felsberg [10]. Here ℛ1, ℛ2 and ℛ denote respectively

11. Generalized Analytic Signals in Image Processing 229

the first and second component of the Riesz transform, and the Riesz transformitself [23]. Defined as Fourier multipliers, it holds:

ℛ𝑓(𝑢1, 𝑢2) =𝑖(𝑢1, 𝑢2)

∥(𝑢1, 𝑢2)∥2 𝑓(𝑢1, 𝑢2), (3.32)

ℛ1𝑓(𝑢1, 𝑢2) =𝑖𝑢1

∥(𝑢1, 𝑢2)∥2 𝑓(𝑢1, 𝑢2), (3.33)

ℛ2𝑓(𝑢1, 𝑢2) =𝑖𝑢2

∥(𝑢1, 𝑢2)∥2 𝑓(𝑢1, 𝑢2), (3.34)

where ∥(𝑢1, 𝑢2)∥2 =√

𝑢21 + 𝑢2

2. An equivalent definition in the spatial domaincan be obtained by convolution with the two-dimensional Riesz kernel, i.e., for𝑚 = 1, 2

ℛ𝑖𝑓 = 𝑐𝑥𝑖∥𝑥∥32

∗ 𝑓, (3.35)

with 𝑐 being a constant.

3.2.2. Image Analysis. Following [10], three features can be computed, and willalso be denoted as energetic, structural and geometrical information, as alreadyintroduced for the multidimensional analytic signal.

Amplitude. The local amplitude of a monogenic signal is defined in a similarmanner as for the analytic signal:

𝐴𝑀 (𝑥, 𝑦) =√∣𝑓(𝑥, 𝑦)∣2 + ∣ℛ𝑓(𝑥, 𝑦)∣2 =

√𝑓𝑀 (𝑥, 𝑦)𝑓𝑀 (𝑥, 𝑦), (3.36)

where the overbar denotes the conjugation of a quaternion.

Phase.

𝜙𝑀 (𝑥, 𝑦) = arctan∣ℛ𝑓(𝑥, 𝑦)∣𝑓(𝑥, 𝑦)

, (3.37)

and we also have that 𝜙𝑀 denotes the angle between 𝐴(𝑥, 𝑦) and 𝑓𝑀 (in the planespanned by the two complex vectors). This yields values 𝜙𝑀 ∈ [−𝜋/2;𝜋/2].

An alternative but equivalent definition is using the arccosine:

𝜙𝑀 = arccos𝑓

∣𝑓𝑀 ∣ . (3.38)

In (3.38), we have 𝜙𝑀 ∈ [0;𝜋].

Orientation. Once again, we can derive an orientation 𝜃𝑀 ∈ [−𝜋, 𝜋] based on themonogenic signal which represents the direction of the phase information.

𝜃𝑀 = arctanℛ2𝑓

ℛ1𝑓. (3.39)

We note that this definition actually only provides an orientation modulo𝜋. To determine the orientation respectively the direction modulo 2𝜋, a furtherorientation unwrapping step or sign estimation is needed [18, 5].

230 S. Bernstein, J.-L. Bouchot, M. Reinhardt and B. Heise

Representation. In the rest of this chapter the different features of images arerepresented by using different colourmaps: a grey colour map is used for the am-plitude representation, which is normalized between [0, 1], a jet colormap for thephase (running between blue and red and mapping the interval [0, 𝜋)) [24], and aHSV colormap for the orientation (running between red, blue and red in a peri-odic way and mapping the interval [−𝜋/2, 𝜋/2)), as depicted in Figure 2 (left, resp.from top to bottom) [24]. As a last representation, we use a colormap accordingto the Middlebury representation [1] where the colour encodes the orientation andthe intensity is computed according to the phase (or structure) information. Thisrepresentation can be seen on the right-hand side of Figure 2.

Figure 2. Scales used for the different colour coding. (See explanationin text.)

3.3. Illustrations

We want here to illustrate the differences between the generalizations proposed.We will visually assess the characteristics of both approaches first applied to aSiemens star1 then to a checkerboard image. Both examples are interesting fortheir regularity (point symmetry for the star and many horizontal and verticalline symmetries for the checkerboard).

An example of such star is depicted in Figure 3(a). The two other imagesin the first row of Figure 3 illustrate the two components of the Riesz transform.As we can see, and we will come back to this property later, the partial Riesztransforms show in some directions behaviour similar to steered derivatives. The

1The Siemens star is a test image used to characterize the resolution of different optical and

graphical devices such as printers or computer projectors. The image is interesting as it showsmuch regularity, as well as many intrinsic one-dimensional and two-dimensional parts.

11. Generalized Analytic Signals in Image Processing 231

Figure 3. The Siemens star together with the different Riesz andHilbert transforms presented in this section. (Additional explanationin the text.)

first component tends to emphasize horizontal edges while the second one tendsto respond more to vertical ones.

The second row shows the results of applying the different Hilbert transformsto the Siemens star. The two first images represent the results of the two partialHilbert transforms and the last one depicts the result of the total Hilbert transform.We can notice the high anisotropy of these transforms at, for instance, the strongvertical respectively horizontal delineation through the centres of the images. Wecan also notice the patchy response of the total Hilbert transform.

As the Riesz kernel in polar coordinate [𝑟, 𝛼] of the spatial domain reads

𝑅(𝑟, 𝛼) ∼ 1

𝑟2𝑒𝑖𝛼, (3.40)

it exhibits an isotropic behavior with respect to its magnitude. In comparison,the partial and the total Hilbert transforms both induce a strict relationship tothe orthogonal coordinate system, and therefore also the two-dimensional analyticsignal inherits this characteristic.

Next we consider local features computed according to the formulas intro-duced above. The results are depicted in Figure 4. The first row corresponds to

232 S. Bernstein, J.-L. Bouchot, M. Reinhardt and B. Heise

Figure 4. Local features computed with the monogenic signal repre-sentation (first row), and the multidimensional analytic signal (secondrow). The images are depicted in a pseudo-colour representation withamplitude: grey, phase: jet, orientation: HSV-Middlebury.

monogenic features, while the second one corresponds to analytic features. Thephase is displayed in a jet colormap, the orientation in a hue-saturation-valueHSV colormap. The last column shows the orientations with intensities weightedproportionally to the cosine of the phase. It is shown according to the Middleburyrepresentation2: strength (cosine of the phase) is encoded as an intensity value ofthe colour and the colour itself corresponds to the orientation. The main differ-ences between these two sets of features lie in the shape and in the boundaries.While monogenic features yield rather smooth boundaries, the analytic represen-tation creates abrupt changes due to its anisotropy. We remark that the phasegives reasonable insights into the structure in the images.

In comparison to the Siemens star, the checkerboard example (see Figure 5(a))shows many orthogonal features. In this case, we see that the partial Hilberttransforms give some good insights into the closeness of an edge and preservethe checkerboard structure (Figure 5(d) and 5(e)), while the Riesz transform gives

2The Middlebury benchmark for optical flow is a web resource for comparing results on optical

flow computations. The colour error representation is well suited for encoding our orientation.More information can be found in [1].

11. Generalized Analytic Signals in Image Processing 233

Figure 5. The checkerboard together with the different Riesz andHilbert transforms presented in this section.

more local responses. The total Hilbert transform acts as an accurate corner de-tector, as can be seen from its response in Figure 5(f).

When discussing the analytic and monogenic features (Figure 6) we remarkthat this effect is preserved. The Riesz transform, being well localized at the edges,does not yield many differences inside any of the squares and seems to jumpfrom one extreme to another across the edges. See in particular Figure 6(b) foran illustrative example of the phase. On the other hand, the Hilbert transformcontains more neighbourhood information and yields a smoother transition in thephase from one square to the next. These features have to be considered carefullybased on the application problem one wishes to solve.

4. The Geometric Approach

For a better understanding of signals a geometric interpretation can help. The fol-lowing considerations about complex numbers, quaternions, rotations, the unitarygroup, special unitary and special orthogonal groups, as well as the spin group, arewell known and can be found in numerous papers. We would like to suggest thebook by Lounesto [19], which provides a comprehensive knowledge of these topics.

234 S. Bernstein, J.-L. Bouchot, M. Reinhardt and B. Heise

Figure 6. Local features computed with the monogenic signal repre-sentation (first row), and the multidimensional analytic signal (secondrow). The images are depicted in a pseudo-colour representation withamplitude: grey, phase: jet, orientation: HSV-Middlebury.

The analytic signal 𝑓𝐴(𝑡) = 𝐴(𝑡)𝑒𝑖𝜙(𝑡) consists of boundary values of an analyticfunction, but the analytic signal can also be seen as a complex-valued function,where 𝑒𝑖𝜙(𝑡) = cos𝜙(𝑡) + 𝑖 sin𝜙(𝑡) has unit modulus and hence can be identifiedwith the unit circle 𝑆1. But there is even more. The set of unit complex numbers isa group with the complex multiplication as group operation, which is the unitarygroup 𝑈(1) = {𝑧 ∈ ℂ : 𝑧𝑧 = 1}. On the other hand a unit complex number canalso be seen as a rotation in ℝ2 if we identify the unit complex number with thematrix

𝑅𝜙 =

(cos𝜙 − sin𝜙sin𝜙 cos𝜙

)∈ 𝑆𝑂(2), (4.1)

i.e., the group of all counter-clockwise rotations in ℝ2. Now all this can also bedescribed inside Clifford algebras. Let us consider the Clifford algebra 𝐶ℓ0,2 withgenerators 𝑒1, 𝑒2. The complex numbers can be identified with all elements 𝑥 +𝑦𝑒12, 𝑥, 𝑦 ∈ ℝ, i.e., the even subalgebra 𝐶ℓ+0,2 of the Clifford algebra 𝐶ℓ0,2. The

rotation (4.1) can also be described by a Clifford multiplication. To see that, weidentify (𝑥, 𝑦) ∈ ℝ2 with 𝑥𝑒1 + 𝑦𝑒2 ∈ 𝐶ℓ0,2, and set

𝑅𝜙(𝑥, 𝑦)𝑇 = (cos 𝜙

2 + 𝑒12 sin𝜙2 )−1(𝑥𝑒1 + 𝑦𝑒2)(cos

𝜙2 + 𝑒12 sin

𝜙2 ), (4.2)

11. Generalized Analytic Signals in Image Processing 235

where cos 𝜙2 + 𝑒12 sin

𝜙2 ∈ Spin(2) = {𝑠 ∈ 𝐶ℓ+0,2 : 𝑠𝑠 = 1}, the spin group of even

products of Clifford vectors. It is easily seen that 𝑠 and −𝑠 in Spin(2) represent thesame rotation, which means that Spin(2) is a two-fold cover of 𝑆𝑂(2). The basis forall these interpretations is the description of complex numbers in a trigonometricway, which is possible by using a logarithm function, which is well known forcomplex numbers. All of that can be generalized into higher dimensions and hasbeen used for monogenic signals. We will start with quaternions because they arethe even subalgebra of the Clifford algebra 𝐶ℓ0,3.

4.1. Quaternions and Rotations

A quaternion 𝑞 ∈ ℍ can be written as

𝑞 = 𝑞0 + 𝑞 = S(𝑞) +V(𝑞) = ∣𝑞∣ 𝑞

∣𝑞∣ , (4.3)

where ∣𝑞∣ is the absolute value or norm of 𝑞 in ℝ4 and𝑞∣𝑞∣ ∈ ℍ1 is a unit quaternion.

Because of ∣∣ 𝑞∣𝑞∣∣∣2 =

3∑𝑖=0

𝑞2𝑖

∣𝑞∣2 = 1, (4.4)

the set of unit quaternions ℍ1 can be identified with 𝑆3, the three-dimensionalsphere in ℝ4.

On the other hand the Clifford algebra 𝐶ℓ0,3 is generated by the elements𝒆1, 𝒆2 and 𝒆3 with 𝒆2

1 = 𝒆22 = 𝒆2

3 = −1 and 𝒆𝑖𝒆𝑗+𝒆𝑗𝒆𝑖 = −2𝛿𝑖,𝑗. Its even subalgebra

𝐶ℓ+0,3, as defined in (3.29), can be identified with quaternions by 𝒆1𝒆2 ∼ 𝒊, 𝒆1𝒆3 ∼ 𝒋and 𝒆2𝒆3 ∼ 𝒌.

Furthermore,

Spin(3) = {𝑢 ∈ 𝐶ℓ+0,3 : 𝑢�� = 1} = ℍ1. (4.5)

That means a unit quaternion can be considered as a spinor. Because Spin(3)is a double cover of the group 𝑆𝑂(3), rotations can be described by unit quater-nions. The monogenic signal is interpreted as a spinor in [26] and lately in [2].

4.2. Quaternions in Trigonometric Form

In this section we represent the monogenic signal in a similar manner to the ana-lytic signal. The analytic signal is a holomorphic and analytic function and there-fore connected to complex numbers. Complex numbers can be written in algebraicor trigonometric form as:

𝑧 = 𝑥+ 𝑖𝑦 = 𝑟𝑒𝑖𝜙.

The analytic signal is given by

𝐴(𝑡)𝑒𝑖𝜙(𝑡),

with amplitude 𝐴(𝑡) and (local) phase 𝜙(𝑡). We want to obtain a similar represen-tation of the monogenic signal using quaternions. A simple computation leads to

𝑞 = ∣𝑞∣(

𝑞0∣𝑞∣ +

𝑞

∣𝑞∣∣𝑞∣∣𝑞∣)

= ∣𝑞∣ (cos𝜙+ 𝑢 sin𝜙),

236 S. Bernstein, J.-L. Bouchot, M. Reinhardt and B. Heise

where 𝜙 = arccos𝑞0∣𝑞∣ and 𝑢 =

𝑞

∣𝑞∣ ∈ 𝑆2. (Alternatively, the argument 𝜙 can be

defined by the arctan.)We have to mention that this representation is different from all previous ones,specifically the vector 𝑢 is a unit vector in ℝ3 and not a unit quaternion.We can represent the quaternion 𝑞 by its amplitude ∣𝑞∣, the phase 𝜙 and theorientation 𝑢. Moreover,

𝑞 = ∣𝑞∣ 𝑒𝑢𝜙,where 𝑒 is the usual exponential function.

With the aid of an appropriate logarithm we can compute 𝑢𝜙 from𝑞∣𝑞∣ = 𝑒𝑢𝜙.

Next, we want to explain the orientation 𝑢. We have already obtained that

𝑞 = ∣𝑞∣ (cos𝜙+ 𝑢 sin𝜙),

where 𝑢 =𝑞

∣𝑞∣ ∈ 𝑆2 and 𝑢2 = −1, i.e., 𝑢 behaves like a complex unit. But because

of 𝑢 ∈ 𝑆2 we can express 𝑢 in spherical coordinates. We have

𝑢 =𝑞1𝒊+ 𝑞2𝒋 + 𝑞3𝒌

∣𝑞1𝒊+ 𝑞2𝒋 + 𝑞3𝒌∣ = 𝒊

(𝑞1∣𝑞∣ +

(𝑞2(−𝒊𝒋) + 𝑞3𝒋)

∣𝑞∣), (4.6)

and if we set cos 𝜃 =𝑞1∣𝑞∣ we get

𝑢 = 𝒊(cos 𝜃 + 𝑢 sin 𝜃), 𝑢 =𝑞

∣𝑞∣ and 𝑞 = 𝒋𝑞3 − 𝒊𝒋𝑞2. (4.7)

Because of

𝑞 = 𝒋𝑞3 − 𝒊𝒋𝑞2 = 𝒋(𝑞3 + 𝒊𝑞2) (4.8)

and with cos 𝜏 =𝑞3∣𝑞∣ we get that

𝑢 = 𝒋(cos 𝜏 + 𝒊 sin 𝜏). (4.9)

Finally, we put everything together we obtain

𝑞 = 𝑞0 + 𝑞1𝒊+ 𝑞2𝒋 + 𝑞3𝒌 (4.10)

= ∣𝑞∣ (cos𝜙+ 𝑢 sin𝜙) = ∣𝑞∣ (cos𝜙+ 𝒊(cos 𝜃 + 𝑢 sin 𝜃

)sin𝜙

)(4.11)

= ∣𝑞∣ (cos𝜙+ 𝒊 (cos 𝜃 + 𝒋 (cos 𝜏 + 𝒊 sin 𝜏) sin 𝜃) sin𝜙) (4.12)

= ∣𝑞∣ (cos𝜙+ 𝒊 sin𝜙 cos 𝜃 + 𝒋 sin𝜙 sin 𝜃 sin 𝜏 + 𝒌 sin𝜙 sin 𝜃 cos 𝜏) , (4.13)

where 𝜙, 𝜃 ∈ [0, 𝜋] and 𝜏 ∈ [0, 2𝜋]. In case of a reduced quaternion, i.e., 𝑞3 = 0, asimilar computation leads to

𝑞 = 𝑞0 + 𝑞1𝒊+ 𝑞2𝒋 (4.14)

= ∣𝑞∣ (cos𝜙+ 𝑢 sin𝜙) = ∣𝑞∣ (cos𝜙+ 𝒊 (cos 𝜃 − 𝒌 sin 𝜃) sin𝜙) (4.15)

= ∣𝑞∣ (cos𝜙+ 𝒊 sin𝜙 cos 𝜃 + 𝒋 sin𝜙 sin 𝜃) , (4.16)

where 𝜙 ∈ [0, 𝜋] and 𝜃 ∈ [0, 2𝜋].

11. Generalized Analytic Signals in Image Processing 237

It is easily seen that 𝜃 can be computed by

tan 𝜃 =𝑞2𝑞1

⇐⇒ 𝜃 = arctan𝑞2𝑞1

.

If we compare that with the monogenic signal

𝑓𝑀 (𝑥, 𝑦) = 𝑓(𝑥, 𝑦) + 𝒊(ℛ1𝑓)(𝑥, 𝑦) + 𝒋(ℛ2𝑓)(𝑥, 𝑦)

we see that (compare with (3.39))

𝜃 = arctan(ℛ2𝑓)(𝑥, 𝑦)

(ℛ1𝑓)(𝑥, 𝑦)= 𝜃𝑀 (𝑥, 𝑦). (4.17)

Therefore the vector 𝑢 = 𝒊 cos 𝜃+ 𝒋 sin 𝜃 can also be considered as the orientation.

4.3. Exponential Function and Logarithm for Quaternionic Arguments

The exponential function for quaternions and para-vectors in a Clifford algebra isdefined in [12] and many other papers.

Definition 4.1. For 𝑞 ∈ ℍ the exponential function is defined as

𝑒𝑞 :=

∞∑𝑘=0

𝑞𝑘

𝑘!. (4.18)

Lemma 4.2. With 𝑢 =𝑞

∣𝑞∣ the exponential function can be written as

𝑒𝑞 = 𝑒𝑞0 (cos ∣𝑞∣+ 𝑢 sin ∣𝑞∣) = 𝑒𝑞0𝑒𝑢∣𝑞∣. (4.19)

Remark 4.3. The formula

𝑒𝑢∣𝑞∣ = cos ∣𝑞∣+ 𝑢 sin ∣𝑞∣ (4.20)

can be considered as a generalized Euler formula.

It is always a challenge to define a logarithm. We will use the followingdefinition.

Definition 4.4. Let 𝑢 =𝑞

∣𝑞∣ . Then the logarithm is defined as

ln 𝑞 :=

{ln ∣𝑞∣+ 𝑢 arccos

𝑞0∣𝑞∣ , ∣𝑞∣ ∕= 0, or ∣𝑞∣ = 0 and 𝑞0 > 0,

undefined, ∣𝑞∣ = 0 and 𝑞0 ≤ 0.(4.21)

Remark 4.5. A logarithm cannot be uniquely defined for −1 because

𝑒𝑢𝜋 = cos𝜋 + 𝑢 sin𝜋 = −1, (4.22)

for all 𝑢 ∈ 𝑆2.

Remark 4.6. More precisely, we can define the 𝑘th branch, 𝑘 ∈ ℤ, of the logarithmbecause cos 𝑡 is a 2𝜋 periodic function.

238 S. Bernstein, J.-L. Bouchot, M. Reinhardt and B. Heise

Theorem 4.7 (Exponential and logarithm function).

1. For ∣𝑞∣ ∕= 0 or ∣𝑞∣ = 0 and 𝑞0 > 0,

𝑒ln 𝑞 = 𝑞. (4.23)

2. For ∣𝑞∣ ∕= 𝑘𝜋, 𝑘 ∈ ℤ∖{0} the following holds true

ln 𝑒𝑞 = 𝑞. (4.24)

Lemma 4.8. For 𝑞 ∈ ℍ1 and 𝑞 ∕= −1 both relations are true:

𝑒ln 𝑞 = ln 𝑒𝑞 = 𝑞. (4.25)

5. Applications to Image Analysis

5.1. Motivations

In several imaging applications only intensity-based images (encoded mostly ingray-scale representation) are provided. Apart from monochromatic camera im-ages, we can cite, e.g., computerized tomography images which encodes local ab-sorption inside a body, or optical coherence tomography images which representsthe back-scattering at an interface. These kinds of images directly describe naturalscenes or physical quantities. In other types of images information is encoded in-directly, e.g., in varying amplitude or frequency of fringe patterns. They are calledamplitude modulated (AM) or frequency modulated (FM) signals. Textures canbe interpreted as a trade-off between both ideas: they depict natural scenes andcan be described as generalized AM-FM signals.

To enrich the information content of a pure intensity image (i.e., imagesencoded with a single value at each pixel), we test the concept of analytic signalsin image processing.

5.2. Application to AM-FM Image Demodulation

Here we study the applicability of the monogenic signal representation to AM-FM signal demodulation, as needed for instance in interferometric imaging [18].A certain given two-dimensional signal (= an image, Figure 7(a)) exhibits bothamplitude modulations (Figure 7(b)) and frequency modulations (Figure 7(c)).The aim is to separate each component of the signal by means of monogenic signalanalysis.

The three features described in the previous section are computed and theirresults are depicted as local orientation in Figure 7(d), local amplitude in Fig-ure 7(e) and local phase in Figure 7(f).

It appears that for such AM-FM signals, the orientation is able to describethe direction of the phase modulation, while the local amplitude gives a goodapproximation of the amplitude modulation (corresponding to the energy of thetwo-dimensional signal) and the phase encodes information about the frequencymodulation (understood as the structural information).

11. Generalized Analytic Signals in Image Processing 239

Figure 7. Example of a two-dimensional AM-FM signal. The first rowshows the input ground truth image together with its amplitude and fre-quency modulations. The second row depicts the recovered orientation,amplitude and phases. The images are displayed using the conventionaljet colormap.

The next example, in Figure 8(a), shows a fringe pattern as an example of areal-world interferometric AM-FM image. The following images show the mono-genic analysis of this image. The local amplitude is depicted beneath the fringepattern (Figure 8(d)). This image gives us a coarse idea of how much structure is tobe found within a given neighbourhood. The second column illustrates the phasecalculation either on the whole image (Figure 8(b)) or only where the local ampli-tude is above a given threshold (Figure 8(e)). The two images in the last columnrepresent the monogenic orientation encoded in HSV without or with the previousmask. As we would expect, illumination changes are appearing in the amplitude,while local structures are contained in both phase and orientation features.

5.3. Application to Texture Analysis

A task of particular interest in artificial vision, is the characterization or descrip-tion of textures. The problem here is to find interesting features to describe agiven texture the best we can in order to classify it for instance [14]. The use of

240 S. Bernstein, J.-L. Bouchot, M. Reinhardt and B. Heise

Figure 8. Example of a fringe pattern and its monogenic decomposi-tion. Phase (second column) is encoded as a jet colormap and orientationas HSV. The two last images show phase and orientation masked witha binary filter set to one when the local amplitude gets over a certainthreshold.

steerable filters could both optimize feature computations and affect the classifi-cation. In other words, if we can compute good descriptive features, we can bettercharacterize a texture.

Considering textures from a more general viewpoint as approximate AM-FM signals, we examine here the use of monogenic representation for the localcharacterization of a textured object as depicted in Figure 9(a).

When looking at the local monogenic signal description (amplitude in Fig-ure 9(b), phase in Figure 9(c) and orientation in Figure 9(d)), we indeed see theserepetitive features along the textured object. Moreover, these estimated valuesseem to be robust against small imperfections in the periodicity.

5.4. Applications to Natural Image Scenes

In this part of our chapter, we want to highlight the interest of the monogenicsignal for natural images. Such images have completely different characteristicsfrom those introduced above. For instance, natural images are often embeddedin a fully cluttered background, encoded with several colour channels, and have

11. Generalized Analytic Signals in Image Processing 241

Figure 9. Example of a textured image superposed with a reliabil-ity mask together with its monogenic analysis. Regions with too littleamplitude are masked out as unreliable.

information at many different scales, etc. In practical applications one needs toapply band-pass filters before analyzing such images [10]. Note that this workconsiders only grey-scale images, but there is literature dealing with multichannelimages [3].

In the following we will describe two tasks useful for image processing. Thefirst part deals with edge detection. We will see how the Riesz transform can beused as an edge detector in images. Then we will see how the orientation estimationis useful, for instance, in computer vision tasks, and how monogenic signal analysiscan help with this, as has already been done for structure interpretation [22, 15].

5.4.1. Edge Detection. The Riesz transform acts as an edge detector for severalreasons. This becomes clear when one has a closer look at its definition as a Fouriermultiplier. Indeed, let us recall the 𝑗𝑡ℎ Riesz multiplier (𝑗 = 1, 2, see (3.32)):

ℛ𝑗𝑓 = 𝑖𝑢𝑗∣𝑢∣𝑓, (5.1)

and we have

ℛ𝑗𝑓 = 𝑖1

∣𝑢∣ ∂𝑗𝑓, (5.2)

so that the Riesz transform acts as a normalized derivative operator.Another (eventually better) way to see this derivative effect is to consider the

Fourier multipliers in polar coordinates [17], as given in (3.40). Figure 10 illustratesthis behavior. The first column shows examples of natural grey-level images. Thesecond and third columns show the first and second Riesz components respectively.It appears that they act as edge detectors steered in the 𝑥 and 𝑦 directions. If wecompare the two Riesz components, we can see the response to different kinds ofedges.

5.4.2. Orientation Estimation of Edges. An important task in image processingand higher level computer vision is to estimate the orientation of edges. As thisis often the first step towards feature description and image interpretation (we

242 S. Bernstein, J.-L. Bouchot, M. Reinhardt and B. Heise

Figure 10. First and second components of the Riesz transform onsome natural images. Notice for instance the table leg appearing inFigure 10(f) but not in Figure 10(e), showing the directions of the com-ponents.

refer the reader to [7, 20] for some non-exhaustive surveys), one wants to have anorientation estimator which is as reliable as possible.

As stated in earlier sections, an orientation can be computed from an analyticor a monogenic signal analysis. For simplicity, let us consider the case of images,where the input function is defined on a set 𝐷 ⊂ ℝ2. Using polar coordinates inthe Fourier domain (𝜌, 𝛽), it holds that

ℛ𝑓 = 𝑖(cos𝛽, sin𝛽)𝑇 𝑓, (5.3)

but on the other hand, we also have

∇𝑓 = 𝑖𝜌(cos𝛽, sin𝛽)𝑇 𝑓, (5.4)

so that both the gradient and the Riesz operator have similar effects on the anglesin the Fourier domain.

It has been shown in [9] that using monogenic orientation estimation in-creases the robustness compared to the traditional Sobel operator. Moreover in

11. Generalized Analytic Signals in Image Processing 243

Figure 11. Local features computed by means of monogenic signal analysis.

their work Felsberg and Sommer introduced an improved version based on localneighbourhood considerations and by using the phase as a confidence value.

Figure 11 illustrates the monogenic analysis of our two test images. The firstcolumn represents the local amplitude of the image whereas the second columnshows the local monogenic phases. The last two columns illustrate the computationof the monogenic orientation. The colours are encoded on a linear periodic basisaccording to the Middlebury colour coding. The last column shows exactly thesame orientation but with the phase in the important role of intensity information.The basic idea is to keep relevant orientation only where the structural information(i.e., the phase) is high.

Note that we do not discuss here the local-zero mean property in naturalimage scenes. So, for example, background and illumination effects may influencethe procedure and will be discussed elsewhere.

6. Conclusion

In this chapter the specificity and analysis of two generalizations of the analyticsignal to higher dimensions have been detailed mathematically, based respectivelyon multiple complex analysis and Clifford analysis. It is shown that they are bothvalid extensions of the one-dimensional concept of the analytic signal. The maindifference between the two approaches is with regard to rotation invariance due

244 S. Bernstein, J.-L. Bouchot, M. Reinhardt and B. Heise

to the point symmetric definition of the sign function in the case of the mono-genic approach compared to the single orthant definition of the multidimensionalanalytic signal.

In a second part we have illustrated the analytic or monogenic analysis of im-ages on both artificial samples and real-world examples in terms of fringe analysisand texture analysis. In the context of AM-FM signal demodulation the monogenicsignal analysis yields a robust decomposition into energectic, structural and geo-metric information. Finally some ideas for the use of generalized analytic signalsin higher-level image processing and computer vision tasks were given showing thehigh potential for further research.

Acknowledgment

We thank the ERASMUS program, and gratefully acknowledge financial supportfrom the Federal Ministry of Economy, Family and Youth, and from the NationalFoundation for Research, Technology and Development. This work was furthersupported in part by the Austrian Science Fund under grant number P21496 N23.

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[26] D. Zang and G. Sommer. Signal modeling for two-dimensional image structures.Journal of Visual Communication and Image Representation, 18(1):81–99, 2007.

246 S. Bernstein, J.-L. Bouchot, M. Reinhardt and B. Heise

Swanhild Bernstein and Martin ReinhardtTechnische Universitat Bergakademie FreibergFakultat fur Mathematik und InformatikInstitut fur Angewandte AnalysisD-09596 Freiberg, Germany

e-mail: swanhild.bernstein@math.tu-freiberg.demartin.reinhardt.87@googlemail.com

Jean-Luc BouchotJohannes Kepler UniversityDepartment of Knowledge-Based Mathematical Systems, FLLLAltenbergerstr., 69A-4040 Linz, Austria

e-mail: jean-luc.bouchot@jku.at

Bettina HeiseJohannes Kepler UniversityDepartment of Knowledge-Based Mathematical Systems, FLLL

and

Christian Doppler Laboratory MS-MACHCenter for Surface- and Nanoanalytics, ZONAAltenbergerstr., 69A-4040 Linz, Austria

e-mail: bettina.heise@jku.at

Quaternion and Clifford–Fourier Transforms and Wavelets

Trends in Mathematics, 247–268c⃝ 2013 Springer Basel

12 Colour Extension of MonogenicWavelets with Geometric Algebra:Application to Color Image Denoising

Raphael Soulard and Philippe Carre

Abstract. We define a colour monogenic wavelet transform. This is basedon recent greyscale monogenic wavelet transforms and an extension to coloursignals aimed at defining non-marginal tools. Wavelet based colour image pro-cessing schemes have mostly been made by separately using a greyscale toolon every colour channel. This may have some unexpected effects on coloursbecause those marginal schemes are not necessarily justified. Here we proposea definition that considers a colour (vector) image right at the beginning ofthe mathematical definition so that we can expect to create an actual colourwavelet transform – which has not been done so far to our knowledge. Thisprovides a promising multiresolution colour geometric analysis of images. Weshow an application of this transform through the definition of a full denoisingscheme based on statistical modelling of coefficients.

Mathematics Subject Classification (2010). Primary 68U10; secondary 15A66,42C40.

Keywords. Colour wavelets, analytic, monogenic, wavelet transforms, imageanalysis, denoising.

1. Introduction

Wavelets have been widely used for handling images for more than 20 years. Itseems that the human visual system sees images through different channels relatedto particular frequency bands and directions; and wavelets provide such decom-positions. Since 2001, the analytic signal and its 2D generalizations have broughta great improvement to wavelets [1, 8, 9] by a natural embedding of an AM/FManalysis in the subband coding framework. This yields an efficient representation of

This work is part of the French ANR project VERSO-CAIMAN.

248 R. Soulard and P. Carre

geometric structures in greyscale images thanks to a local phase carrying geomet-ric information complementary to an amplitude envelope having good invarianceproperties. So it codes the signal in a more coherent way than standard wavelets.The last and seemingly most appropriate proposition [9] of analytic wavelets forimage analysis is based on the monogenic signal [5] defined with geometric algebra.

In parallel a colour monogenic signal was proposed [3] as a mathematicalextension of the monogenic signal; paving the way to non-marginal colour toolsespecially by using geometric algebra and above all by considering a colour signalright at the foundation of the mathematical construction.

We define here a colour monogenic wavelet transform that extends the mono-genic wavelets of [9] to colour. These analytic wavelets are defined for colour 2Dsignals (images) and avoid the classical pitfall of marginal processing (greyscaletools used separately on colour channels) by relying on a sound mathematicaldefinition. We can therefore expect to handle coherent information of multireso-lution colour geometric structure, which would ease wavelet based colour imageprocessing. To our knowledge colour wavelets have not been proposed so far.

We first give a technical study of analytic signals and wavelets with the intentto popularize them since they rely on non-trivial concepts of geometric algebra,complex and harmonic analysis, as well as non-separable wavelet frames. Thenwe describe our colour monogenic wavelet transform, and finally an application tocolour denoising will be presented.

Notation:2D vector coordinates: 𝒙=(𝑥, 𝑦) , 𝝎=(𝜔1, 𝜔2) ∈ ℝ2; 𝒌 ∈ ℤ2

Euclidean norm: ∥𝒙∥ =√𝑥2 + 𝑦2

Complex imaginary number: 𝒋 ∈ ℂ

Argument of a complex number: argConvolution symbol: ∗Fourier transform: ℱ

2. Analytic Signal and 2D Generalization

2.1. Analytic Signal (1D)

An analytic signal 𝑠𝐴 is a multi-component signal associated to a real signal 𝑠 tobe analyzed. The definition is well known in the 1D case where 𝑠𝐴(𝑡) = 𝑠(𝑡)+𝒋 (ℎ∗𝑠)(𝑡) is the complex signal made of 𝑠 and its Hilbert transform (with ℎ(𝑡) = 1/𝜋𝑡).

The polar form of the 1D analytic signal provides an AM/FM representationof 𝑠 with ∣𝑠𝐴∣ being the amplitude envelope and 𝜑 = arg (𝑠𝐴) the instantaneousphase. This classical tool can be found in many signal processing books and is forexample used in communications.

Interestingly, we can also interpret the phase in terms of the signal shape,i.e., there is a direct link between the angle 𝜑 and the local structure of 𝑠. Sucha link between a 2D phase and the local geometric structures of an image isvery attractive in image processing. That is why there were several attempts to

12. Colour Monogenic Wavelets 249

generalize it to 2D signals; and among them the monogenic signal introduced byFelsberg [5] seems the most advanced since it is rotation invariant.

2.2. Monogenic Signal (2D)

Without going beyond the strictly necessary details, we review here the key pointsof the fundamental construction of the monogenic signal, which will be necessaryto understand the colour extension.

The definition of the 1D case given above can be interpreted in terms ofsignal processing: the Hilbert transform makes a ‘pure 𝜋

2 -phase-shift’. But such aphase-shift is not straightforward to define in 2D (similar to many other 1D signaltools) so let us look at the equivalent complex analysis definition of the 1D analyticsignal. It says that 𝑠𝐴 is the holomorphic extension of 𝑠 restricted to the real line.But complex algebra is limited regarding generalizations to higher dimensions. Tobypass this limitation we can see a holomorphic function as a 2D harmonic fieldthat is an equivalent harmonic analysis concept involving the 2D Laplace equationΔ𝑓=0. It then can be generalized within the framework of 3D harmonic fields by

using the 3D Laplace operator Δ3=(

𝛿2

𝛿𝑥2 + 𝛿2

𝛿𝑦2 + 𝛿2

𝛿𝑧2

). The whole generalization

relies on this natural choice and the remaining points are analogous to the 1Dcase (see [5] for more details). Note that in Felsberg’s thesis [5] this construction isexpressed in terms of geometric algebra but here we have avoided this for the sakeof simplicity. Finally the 2D monogenic signal 𝑠𝐴 associated to 𝑠 is the 3-vector-valued signal:

𝑠𝐴(𝒙) =

⎡⎢⎢⎢⎢⎣𝑠(𝒙)

𝑠𝑟1(𝒙) =𝑥

2𝜋 ∥𝒙∥3 ∗ 𝑠(𝒙)

𝑠𝑟2(𝒙) =𝑦

2𝜋 ∥𝒙∥3 ∗ 𝑠(𝒙)

⎤⎥⎥⎥⎥⎦ . (2.1)

Where 𝑠𝑟1 and 𝑠𝑟2 are analogous to the imaginary part of the complex 1D ana-lytic signal. Interestingly, this construction reveals the two components of a Riesztransform:

ℛ{𝑠} = (𝑠𝑟1(𝒙), 𝑠𝑟2(𝒙)) =

(𝑥

2𝜋 ∥𝒙∥3 ∗ 𝑠(𝒙),𝑦

2𝜋 ∥𝒙∥3 ∗ 𝑠(𝒙))

, (2.2)

in the same way that the 1D case exhibits a Hilbert transform. Note that we getback to a signal processing interpretation since the Riesz transform can also beviewed as a pure 2D phase-shift. In the end, by focusing on the complex analysisdefinition of the analytic signal we end up with a convincing generalization of theHilbert transform.

Now recall that the motivation to build 2D analytic signals arises from thestrong link existing between the phase and the geometric structure. To define the2D phase related to the Riesz transform the actual monogenic signal must beexpressed in spherical coordinates, which yields the following amplitude envelope

250 R. Soulard and P. Carre

𝑠 𝐴 𝜑 𝜃(𝜋)

−1 0 1 0 max 0 𝜋 −𝜋2 0 𝜋

2

Figure 1. Felsberg’s monogenic signal associated to a narrow-bandsignal 𝑠. The orientation 𝜃 is shown modulo 𝜋 for visual convenience.Phase values of small coefficients have no meaning so they are replacedby black pixels.

and 2-angle phase:

Amplitude: 𝐴 =√

𝑠2 + 𝑠2𝑟1 + 𝑠2𝑟2 , 𝑠 = 𝐴 cos𝜑 ,Orientation: 𝜃 = arg (𝑠𝑟1 + 𝒋𝑠𝑟2) , 𝑠𝑟1 = 𝐴 sin𝜑 cos 𝜃 ,1D Phase: 𝜑 = arccos (𝑠/𝐴) , 𝑠𝑟2 = 𝐴 sin𝜑 sin 𝜃 .

(2.3)

A monogenic signal analysis is illustrated in Figure 1.

2.3. Physical Interpretation

Felsberg shows a direct link between the angles 𝜃 and 𝜑 and the geometric localstructure of 𝑠. The signal is thus expressed as an ‘𝐴-strong’ (𝐴 = amplitude) 1Dstructure with orientation 𝜃. 𝜑 is analogous to the 1D local phase and indicateswhether the structure is a line or an edge. A direct drawback is that intrinsically2D structures are not handled. Yet this tool has found many applications in im-age analysis, from contour detection to motion estimation (see [9] and referencestherein p. 1).

From a signal processing viewpoint the AM/FM representation provided byan analytic signal is well suited for narrowband signals. That is why it seems natu-ral to embed it within a wavelet transform that performs subband decomposition.We now present the monogenic wavelet analysis proposed by Unser in [9].

3. Monogenic Wavelets

So far there is one proposition of computable monogenic wavelets in the literature[9]. It provides 3D vector-valued monogenic subbands consisting of a rotation-covariant magnitude and a new 2D phase. This representation – specially definedfor 2D signals – is a great theoretical improvement over complex and quaternionwavelets [8, 1], similar to the way that the monogenic signal itself is an improve-ment over its complex and quaternion counterparts.

12. Colour Monogenic Wavelets 251

The proposition of [9] consists of one real-valued ‘primary’ wavelet transformin parallel with an associated complex-valued wavelet transform. Both transformsare linked to each other by a Riesz transform so they carry out a monogenic mul-tiresolution analysis. We end up with three vector coefficients forming subbandsthat are monogenic.

3.1. Primary Transform

The primary transform is real-valued and relies on a dyadic pyramid decompo-sition tied to a wavelet frame. Only one 2D wavelet is needed and the dyadicdownsampling is done only on the low frequency branch; leading to a redundancyof 4 : 3. The scaling function 𝜑𝛾 and the mother wavelet 𝜓 are defined in theFourier domain as

𝜑𝛾ℱ←→(4(sin2 1

2𝜔1 + sin2 12𝜔2

)− 83 sin

2 12𝜔1 sin

2 12𝜔2

) 𝛾2

∥𝝎∥𝛾 , (3.1)

𝜓(𝒙) = (−Δ)𝛾2 𝜑2𝛾(2𝒙) . (3.2)

Note that 𝜑𝛾 is a cardinal polyharmonic spline of order 𝛾 and spans the space ofthose splines with its integer shifts. It also generates – as a scaling function – avalid multiresolution analysis.

This particular construction is made by an extension of a wavelet basis (non-redundant) related to a critically-sampled filterbank. This extension to a waveletframe (redundant) adds some degrees of freedom used by the authors to tune theinvolved functions. In addition a specific subband regression algorithm is used onthe synthesis side. The construction is fully described in [10].

3.2. The Monogenic Transform

The second ‘Riesz part’ transform is a complex-valued extension of the primaryone. We define the associated complex-valued wavelet by including the Riesz com-ponents

𝜓′ = −(

𝑥

2𝜋 ∥𝒙∥3 ∗ 𝜓(𝒙))

+ 𝒋

(𝑦

2𝜋 ∥𝒙∥3 ∗ 𝜓(𝒙))

. (3.3)

It can be shown that this generates a valid wavelet basis and that it can be extendedto the pyramid described above. The joint consideration of both transforms leadsto monogenic subbands from which the amplitude and the phase can be extractedwith an overall redundancy of 4 : 1. The monogenic wavelet transform by Unseret al. is illustrated in Figure 2.

So far no applications of the monogenic wavelets have been proposed. In [9]a demonstration of AM/FM analysis is done with fine orientation estimation andgives very good results in terms of coherency and accuracy. Accordingly this toolmay be used for analysis tasks rather than for processing.

Motivated by the powerful analysis provided by the monogenic wavelet trans-form we propose now to extend it to colour images.

252 R. Soulard and P. Carre

𝑠 𝐴 𝜑 𝜃(𝜋)

0 1 0 max 0 𝜋 −𝜋2 0 𝜋

2

Figure 2. Unser’s MWT of the image ‘face’. Same graphic chart asFigure 1. We used 𝛾 = 3 and the scales are 𝑖 ∈ {−1,−2,−3}.

4. Colour Monogenic Wavelets

We define here our proposition that combines a fundamental generalization ofthe monogenic signal for colour signals with the monogenic wavelets describedabove. The challenge is to avoid the classical marginal definition that would applya greyscale monogenic transform to each of the three colour channels of a colourimage. We believe that the monogenic signal has a favorable theoretical frameworkfor a colour extension and this is why we propose to start from this particularwavelet transform rather than from a more classical one.

The colour generalization of the monogenic signal can be expressed withinthe geometric algebra framework. This algebra is very general and embeds thecomplex numbers and quaternions as subalgebras. Its elements are ‘multivectors’,naturally linked with various geometric entities. The use of this fundamental toolis gaining popularity in the literature because it allows rewriting sophisticatedconcepts with simpler algebraic expressions and so paves the way to innovativeideas and generalizations in many fields.

For simplicity’s sake and since we would not have enough space to present thefundamentals of geometric algebra, we express the construction here in classicalterms, as we did above in Section 2.2. We may sometimes point out some necessaryspecific mechanisms but we refer the reader to [3, 5] for further details.

4.1. The Colour Monogenic Signal

Starting from Felsberg’s approach that is originally expressed in the geometricalgebra of ℝ3; the extension proposed in [3] is written in the geometric algebra ofℝ5 for 3-vector-valued 2D signals of the form (𝑠𝑅, 𝑠𝐺, 𝑠𝐵). By simply increasingthe dimensions we can embed each colour channel along a different axis and the

12. Colour Monogenic Wavelets 253

original equation from Felsberg involving a 3D Laplace operator can be generalizedin 5D with

Δ5 =

(𝛿2

𝛿𝑥21

+𝛿2

𝛿𝑥22

+𝛿2

𝛿𝑥23

+𝛿2

𝛿𝑥24

+𝛿2

𝛿𝑥25

).

Then the system can be simplified by splitting it into three systems each witha 3D Laplace equation, reduced to be able to apply Felsberg’s condition to eachcolour channel. At this stage the importance of geometric algebra appears since analgebraic simplification between vectors leads to a 5-vector colour monogenic signalthat is non-marginal. Instead of naively applying the Riesz transform to each colourchannel, this fundamental generalization leads to the following colour monogenicsignal: 𝑠𝐴=(𝑠𝑅, 𝑠𝐺, 𝑠𝐵, 𝑠𝑟1, 𝑠𝑟2) where 𝑠𝑟1 and 𝑠𝑟2 are the Riesz transform appliedto 𝑠𝑅+𝑠𝐺+𝑠𝐵.

Now that the colour extension of Felsberg’s monogenic signal has been de-fined, let us construct the colour extension of the monogenic wavelets.

4.2. The Colour Monogenic Wavelet Transform

We can now define a wavelet transform whose subbands are colour monogenicsignals. The goal is to obtain vector coefficients of the form

(𝑐𝑅, 𝑐𝐺, 𝑐𝐵, 𝑐𝑟1, 𝑐𝑟2) (4.1)

such that

𝑐𝑟1 =𝑥

2𝜋 ∥𝒙∥3 ∗ (𝑐𝑅 + 𝑐𝐺 + 𝑐𝐵),

𝑐𝑟2 =𝑦

2𝜋 ∥𝒙∥3 ∗ (𝑐𝑅 + 𝑐𝐺 + 𝑐𝐵).

It turns out that we can very simply use the transforms presented above by ap-plying the primary one on each colour channel and the Riesz part on the sum ofthe three. The five related colour wavelets illustrated in Figure 3 and forming onecolour monogenic wavelet 𝜓𝐴 are:

𝜓𝑅 =

⎛⎝𝜓00

⎞⎠ , 𝜓𝐺 =

⎛⎝0𝜓0

⎞⎠ , 𝜓𝐵 =

⎛⎝00𝜓

⎞⎠ (4.2)

𝜓𝑟1 =

⎛⎜⎝𝑥

2𝜋∥𝒙∥3 ∗ 𝜓𝑥

2𝜋∥𝒙∥3 ∗ 𝜓𝑥

2𝜋∥𝒙∥3 ∗ 𝜓

⎞⎟⎠ 𝜓𝑟2 =

⎛⎜⎝𝑦

2𝜋∥𝒙∥3 ∗ 𝜓𝑦

2𝜋∥𝒙∥3 ∗ 𝜓𝑦

2𝜋∥𝒙∥3 ∗ 𝜓

⎞⎟⎠ (4.3)

𝜓𝐴 =(𝜓𝑅, 𝜓𝐺, 𝜓𝐵, 𝜓𝑟1, 𝜓𝑟2

)(4.4)

We then get five vector coefficients verifying our conditions, and forming a colourmonogenic wavelet transform. The associated decomposition is described by thediagram of Figure 4. This provides a multiresolution colour monogenic analysismade of a 5-vector-valued pyramid transform. The five decompositions of two im-ages are shown in Figure 5 from left to right. Each one consists of four juxtaposed

254 R. Soulard and P. Carre

𝜓𝑅 𝜓𝐺 𝜓𝐵 𝜓𝑟1 𝜓𝑟2

Figure 3. Space representation of the 5 colour wavelets.

Figure 4. Colour MWT scheme. Each colour channel is analyzed withthe primary wavelet transform symbolized by a 𝜓 block and the sum‘𝑅 + 𝐺 + 𝐵’ is analyzed with the ‘Riesz part’ wavelet transform (𝜓𝑟1

and 𝜓𝑟2 blocks).

image-like subbands resulting from a 3-level decomposition. We fixed 𝛾 = 3, be-cause it gave good experimental results.

4.3. Interpretation

Let us look at the first three greymaps. These are the three primary transforms𝑐𝑅, 𝑐𝐺 and 𝑐𝐵 where white (respectively black) pixels are high positive (respec-tively negative) values. Note that our transform is non-separable and so providesat each scale only one subband related to all orientations. We are not subjectto the arbitrarily separated horizontal, vertical and diagonal analyses usual withwavelets. This advantage is even greater in colour. Whereas marginal separabletransforms show three arbitrary orientations within each colour channel – which isnot easily interpretable – the colour monogenic wavelet transform provides a morecompact energy representation of the colour image content regardless of the localorientation. The colour information is well separated through 𝑐𝑅, 𝑐𝐺 and 𝑐𝐵: see,e.g., that the blue contours of the first image are present only in 𝑐𝐵 , and in each ofthe three decompositions it is clear that every orientation is equally represented allalong the round contours. This is different from separable transforms that preferparticular directions. The multiresolution framework causes the horizontal blue

12. Colour Monogenic Wavelets 255

Images 𝑐𝑅 𝑐𝐺 𝑐𝐵 Riesz part

Figure 5. Colour MWT of images. The two components of the Rieszpart are displayed in the same figure part with the magnitude of 𝑐𝑟1+𝒋𝑐𝑟2encoded in the intensity, and its argument (local orientation) encodedin the hue.

low-frequency structure of the second image to be coded mainly in the third scaleof 𝑐𝐵.

But the directional analysis is not lost thanks to the Riesz part that completesthis representation. Now look at the ‘2-in-1’ last decomposition forming the Riesz

part. It is displayed in one colour map where the geometric energy√

𝑐2𝑟1 + 𝑐2𝑟2 isencoded into the intensity (with respect to the well-known HSV colour space) andthe orientation arg(𝑐𝑟1 + 𝒋𝑐𝑟2)(𝜋) is encoded in the hue (e.g., red is for {0, 𝜋} andcyan is for ±𝜋

2 ). This way of displaying the Riesz part reveals well the providedgeometric analysis of the image.

The Riesz part gives a precise analysis that is local both in space and scale.If there is a local colour geometric structure in the image at a certain scale theRiesz part exhibits a high intensity in the corresponding position and subband.This is completed with an orientation analysis (hue) of the underlying structure.For instance a horizontal (respectively vertical) structure in the image will becoded by an intense cyan (respectively red) point in the corresponding subband.The orientation analysis is strikingly coherent and accurate. See, for example, thatcolour structures with constant orientation (second image) exhibit a constant huein the Riesz part over the whole structure.

256 R. Soulard and P. Carre

Note that low intensity corresponds to ‘no structure’, i.e., where the imagehas no geometric information. It is sensible not to display the orientation (lowintensity makes the hue invisible) for these coefficients since the values have nomeaning in these cases.

In short, the colour and geometric information of the image are well separatedfrom each other, and the orientation analysis is very accurate. In addition theinvariance properties of the primary and Riesz wavelet transforms are kept in thecolour extension for a slight overall redundancy of 20 : 9 ≈ 2.2. This transform isnon-marginal because the RGB components are considered as well as the intensity(𝑅+𝐺+𝐵) – which involves two different colour spaces.

4.4. Reconstruction Issue

Image processing tasks such as denoising need the synthesis part of filterbanks. Inthe case of redundant representations, there are often several ways to perfectly re-construct the transformed image. However, when wavelet coefficients are processed,the reconstruction method affects the way in which the wavelet domain processingwill modify the image. In other words, we have to combine the redundant data sothat the retrieved graphical elements are consistent with the modifications thathave been done to the wavelet coefficients.

This issue occurs in the scalar case, since the pyramids we use have a redun-dancy of 4 : 3. The associated reconstruction algorithm has been well defined bythe authors in [10] and consists of using the spatial redundancy of each subbandat the synthesis stage, by using the so-called subband regression algorithm.

In our case, we have to face another kind of redundancy, which stems fromthe monogenic model. Apart from the wavelet decomposition, the monogenic rep-resentation (as well as the analytic representation) is already basically redundantsince additional signals are processed (the Riesz part). In our case, the followingdistinct reconstructions are possible:

∙ We can reconstruct the whole colour image (𝑅,𝐺,𝐵) solely from the primarypart (𝑐𝑅, 𝑐𝐺, 𝑐𝐵).

∙ The Riesz part 𝑐𝑟1 + 𝒋𝑐𝑟2 can be used to reconstruct 𝑅+𝐺+𝐵– which is only a partial reconstruction.

∙ One can also combine both reconstructions with a specific application drivenmethod.

In every case, the reconstruction is perfect. What is unknown is the meaning ofwavelet domain processing with respect to the chosen reconstruction method.

Let us now study the new colour wavelet transform from an experimentalpoint of view.

12. Colour Monogenic Wavelets 257

5. Wavelet Coefficients Study for Denoising

We propose in this section a data restoration algorithm based on the colour mono-genic wavelet transform defined above. Classical denoising methods consist in per-forming non-linear processing in the wavelet domain by thresholding the coef-ficients. However in our case, information carried by the wavelet coefficients isricher than in the usual orthogonal case. The colour monogenic wavelet decompo-sition is not orthogonal, so we have to further study modelling of this new kind ofcoefficient.

In addition, we saw that the colour monogenic decomposition is composed oftwo kinds of data, i.e.:

∙ The ‘primary part’: A set of coefficients associated to colorimetric informa-tion, forming three pyramids linked to the colour channels.

∙ The ‘Riesz part’: A geometric measure composed of a norm and an angle ateach point, processed by the Riesz transform and giving some informationabout shape and structure.

In order to carry out an image restoration algorithm, a coefficient selection processhas to be defined.

We first experimentally characterize noise-related coefficients in the differentsubbands in order to identify their distribution and correlation. Then both thecolorimetric information and the geometric information are merged into a singlethresholding to perform a colour monogenic wavelet based denoising.

5.1. Modeling of Noise-related Coefficients

Classical wavelet based denoising techniques rely on the assumption that the distri-bution of noise-related wavelet coefficients can be modelled efficiently by a centredGaussian law. This usually implies a constant threshold over the whole transform(see [4]). Here we have a non-orthogonal transform so we need to observe the be-haviour of the noise term of a noisy image through the colour monogenic waveletanalysis. Despite the singularity of our transform, we retain the classical Gaussianmodel, a method which will be experimentally validated.

5.1.1. Primary Coefficients. As shown in Figure 6 we observe that the decompo-sition of the centred Gaussian noise with variance 𝜎2 = 1 remains centered andGaussian even after the decomposition, but with different variances.

Experimental values for the standard deviations are given in Table 1. Thecontinuous theoretical distributions have been plotted above the histograms (leftside of Figure 6) so as to confirm the Gaussian model:

𝑓 =1

𝜎𝑠√2𝜋

exp

(−𝑥2

2𝜎2𝑠

)(5.1)

where 𝜎𝑠 is the estimated variance of the coefficients. The decomposition is per-formed through a set of band-pass filters, with very limited correlation between thebasis functions, which implies that we retain Gaussian signals with some degreeof independence.

258 R. Soulard and P. Carre

0.00.10.20.30.4

-5 -4 -3 -2 -1 0 1 2 3 4 5

0.0000.1170.2330.350

-5 -4 -3 -2 -1 0 1 2 3 4 5 0.0000.2330.4670.700

-5 -4 -3 -2 -1 0 1 2 3 4 5

0.00.10.20.3

-5 -4 -3 -2 -1 0 1 2 3 4 5 0.00.20.40.6

-5 -4 -3 -2 -1 0 1 2 3 4 5

0.0000.1170.2330.350

-5 -4 -3 -2 -1 0 1 2 3 4 5 0.00.20.40.6

-5 -4 -3 -2 -1 0 1 2 3 4 5

0.0000.1170.2330.350

-5 -4 -3 -2 -1 0 1 2 3 4 5 0.00.20.40.6

-5 -4 -3 -2 -1 0 1 2 3 4 5

Original noise

Primary part 1st scale

Primary part 2nd scale

Primary part 3rd scale

Primary part 4th scale

Riesz part modulus 1st scale

Riesz part modulus 2nd scale

Riesz part modulus 3rd scale

Riesz part modulus 4th scale)

Figure 6. Histograms (bars) of primary subbands and modulus ofRiesz subbands, and Probability Density Function models (line).

Table 1. Standard deviations 𝜎𝑠 of subbands after decomposition ofGaussian noise with variance 1 (with 𝛾 = 3).

Scale 𝑠 1(High freq.) 2 3 4Standard deviation 𝜎𝑠 1.339 1.502 1.512 1.560

The values of 𝜎𝑠 can also be derived analytically from the definitions of thefilters. Recall that linear filtering of a stationary random signal 𝑥 by a filter 𝐻with output 𝑦 can be studied with power spectral densities Ψ𝑥 = ∣ℱ [𝑥]∣2 (PSD)

and autocorrelations 𝑅𝑥(𝜏) = ℱ−1Ψ𝑥. In particular we have Ψ𝑦 = Ψ𝑥 ∣𝐻∣2. Theoutput variance 𝜎2

𝑦 is equal to 𝑅𝑦(0) which reduces to

𝜎2𝑦 =

𝜎2𝑥

4𝜋2

(2𝜋,2𝜋)∫∫(0,0)

∣𝐻(𝝎)∣2 𝑑𝝎.

For example, the first scale output of our filterbank is directly linked to thefirst stage high-pass filter:

𝜎21 =

𝜎2

4𝜋2

(2𝜋,2𝜋)∫∫(0,0)

∣∣∣∣∣(4(sin2 1

2𝜔1 + sin2 12𝜔2)− 8

3 sin2 1

2𝜔1 sin2 1

2𝜔2

)𝛾2 ∥𝝎∥𝛾

∣∣∣∣∣2

𝑑𝝎 (5.2)

The remaining coefficients are tied to equivalent filters of each filterbank output.To have more realistic data, we introduce the image peppers altered by

additive Gaussian white noise (SNR = 83dB) in Figure 7.

12. Colour Monogenic Wavelets 259

Figure 7. Image peppers altered by additive whiteGaussian noise.

We illustrate in Figure 8 a histogram of 𝑐𝑅 (primary part, red channel) on thefirst scale. As in classical denoising, we assume that the first scale mainly containsnoise related to coefficients since natural images should not have substantial highfrequency content. Again, this histogram visually suggests that the Gaussian modelis justified.

To finally confirm the Gaussian model, we computed a Kolmogorov-Smirnovtest comparing experimental coefficients with a true Gaussian distribution. Thetest was positive for the image peppers with a 𝑝 -value of 0.9. Let us now studythe modelling of the Riesz part coefficients.

5.1.2. Riesz Part Coefficients. Recall that structure information is both carriedby an angle and a modulus. Handling and thresholding of circular data such as anangle is a difficult issue. In this exploratory work, and as a natural first step, we willconcentrate on the modulus. Moreover, the modulus is tied to certain amplitudeinformation related to geometrical structures, and for which thresholding will berelevant. Thresholding an angle is less intuitive.

The Riesz part coefficients can also be viewed as outputs of filtering pro-cesses, so we again propose to make a Gaussian assumption for their distribution.Real and imaginary parts of subbands follow centered Gaussian distributions withvariance 𝜎2

𝑠 .

Studying the modulus requires us to find the distribution law of the random

variable 𝑍 = (𝑋2 + 𝑌 2)12 where (𝑋,𝑌 ) are two independent Gaussian random

variables with zero-mean and standard variation 𝜎𝑠. In this case, it is well known

260 R. Soulard and P. Carre

Figure 8. Distribution of first scale coefficients of red primary decom-position 𝑐𝑅.

that 𝑍 follows a Rayleigh distribution

𝑓𝑍(𝑚) =𝑚

𝜎2𝑠

exp(−𝑚2/2𝜎2

𝑠

)1𝑚>0

with 1𝑚>0 being the Heaviside step. The moments of 𝑍 are then 𝐸(𝑍) = 𝜎𝑠√

𝜋/2and 𝑉 (𝑍) = 𝜎2

𝑠 (2−𝜋/2). The sequence of operations in the experiment of Figure 6were:

∙ Generate a test colour image made of Gaussian noise only,∙ analyze it through the colour monogenic wavelet transform,∙ process histograms of the modulus of the primary subbands and the Rieszsubbands,

∙ compute standard deviations of the primary subbands to get experimentalvalues for 𝜎𝑠,

∙ compute theoretical distributions 𝑓 and 𝑓𝑍 with the measured 𝜎𝑠,∙ plot histograms and theoretical curves in the same diagram for each scale.

We can see that the modeled Riesz-part curves – on the right side of Figure 6 –correspond well to the measured histograms. Histograms of higher scales are lessregular because of the small number of coefficients in those subbands.

Note that the Rayleigh distribution would theoretically be fully justified ifstatistical independence were guaranteed between 𝑋 and 𝑌 . In our case, the goal

12. Colour Monogenic Wavelets 261

Figure 9. Experimental distribution of the Riesz-part modulus on thefirst scale.

of the modelling is simply to define a threshold so that we can keep the Rayleighmodel for the modulus of the Riesz part.

In the case of the noisy natural image peppers introduced above, we observethe first scale histogram of the Riesz part modulus in Figure 9. According to thehypothesis that the first scale contains mainly noise, the related histogram is stillRayleigh-shaped.

To further illustrate this choice, Figure 10 shows the Riesz part modulushistograms for several scales of the decomposed noisy image peppers. We cansee that the dispersion increases when useful information becomes – with scale– more important than noise-related coefficients. Therefore, it is a good idea toestimate the threshold on the first scale, which will discard all noise coefficientswhile preserving structural information within other scales.

Finally, a Gaussian model according to the primary subbands and a Rayleighmodel for the modulus of Riesz subbands will be chosen. We can now apply auto-matic thresholding to perform colour denoising.

5.2. Thresholding

We propose to first focus on thresholding the primary part – which is the mostintuitive. Thresholding will be done in the following way:

∙ Estimation of noise level from the first scale (by assumption it is the solescale containing only noise).

262 R. Soulard and P. Carre

Figure 10. Experimental distribution of the Riesz-part modulus on thefirst three scales.

∙ Soft thresholding of the different scales with the threshold equal to threetimes the estimated noise level.

Using the first scale to estimate the threshold is classical (see [7]). By assumingthat the first scale contains mainly noise, one can use the experimental median ofthe coefficients to estimate the standard deviation of the Gaussian noise distribu-tion well. So we have the estimate �� = median(∣𝑊1∣)/0.6745 with 𝑊1 being thecoefficients of the first scale.

Recall that the soft thresholding is defined as follows:

thresholding(𝑧, 𝑔) = sign(𝑧)⌊ ∣𝑧∣ − 𝑔

⌋+, (5.3)

where 𝑧 is the coefficient, 𝑔 the threshold and ⌊.⌋+ is max(., 0).

As already discussed in Section 4.4, about the synthesis part of the colourmonogenic filterbank, there are several ways to reconstruct the denoised image.Within our work we must ensure that the Riesz part controls the denoising process.Otherwise, this would be equivalent to marginal wavelet denoising without anyuse of the monogenic analysis. That means, it would consider the colorimetricinformation (primary part) without taking into account the geometric information(Riesz part). But how to combine both the primary and the Riesz decomposition ina unified reconstruction is an open issue. In this work we propose to use the Rieszpart jointly with the primary part in the thresholding process rather than in thefilterbank synthesis. So the filterbank synthesis will be done for the thresholdedprimary part only. Finally, the denoising scheme actually fully uses the colourmonogenic analysis.

12. Colour Monogenic Wavelets 263

Thresholding from the modulus of the Riesz coefficients is not straightforwardbecause of the Rayleigh distribution for which classical schemes do not hold. Asimilar issue is dealt with in [6] with the modulus of Gabor wavelets, where settingthe following threshold is suggested:

𝐸(𝑍) + 𝑘 ∗√

𝑉 (𝑍), (5.4)

where 𝐸(𝑍) and 𝑉 (𝑍) are the moments of the Rayleigh distribution already givenabove, and estimated from the modulus of the first scale Riesz part. We proposeto use 𝑘 = 3 which gave satisfactory experimental results. In the context of thehistograms plotted in Figure 10, this corresponds to a threshold of 193. We cansee that it actually discards most of the noise from the first scale, while preservinghigher coefficients on the following scales.

The combination of both thresholding processes is done through a simple log-ical AND operator, i.e., at a given position, we keep the primary coefficient only ifboth the primary part and the Riesz part lie above their respective thresholds. Thisstep may be easily improved through a wider study about how to combine colori-metric information with geometric information for this kind of colour monogenicanalysis.

5.3. Experimental Results

We finally show different results of colour image restoration in Figures 11, 12,13 and 14. Note that our test images are strongly altered so as to easily revealthe particularities of different denoising methods. With less substantial levels ofnoise, the visual comparison is too difficult. We compare our approach to the twoclassical techniques that have become the reference in wavelet based denoising [2]:

∙ Soft thresholding of the decimated orthogonal wavelet transform(filter Daubechies 4),

∙ Hard thresholding of the undecimated wavelet transform (Daubechies 4).

Thresholds are established by the estimation of the noise variance as describedabove, whereas the low frequency subband is not modified.

We can see that the colour monogenic wavelet transform performance is in-termediate between those of decimated wavelets and of undecimated wavelets. Thelatter has a large redundancy (𝐿 scales on an image of 𝑁 pixels will give (3𝐿+1)𝑁coefficients) while preserving orthogonality properties between interlaced decom-positions, which makes it one of the most efficient denoising tools at the moment.On the other hand, decimated wavelets are known to produce unwanted oscilla-tions – the so-called pseudo-Gibbs phenomenon – as the price to pay for their veryfast implementation.

We observe that the colour monogenic approach allows us to preserve themain colour information of a colour image with limited redundancy. However,these experimental results hint that this is not fully satisfactory, since unwantedoscillation artifacts appear due to the thresholding process. Contrary to classicalwavelets, the artifacts are smoother and more rounded, which may be thought of

264 R. Soulard and P. Carre

Figure 11. Restoration of image house. Upper row: Noisy image anddecimated wavelet based denoising. Lower row: Undecimated waveletbased denoising and proposed approach.

as visually less annoying. The ‘artifact shape’ is due to the reconstruction func-tions, which is independent of the monogenic analysis. All the information is wellselected and preserved. First we can see that image discontinuities are retainedwell. Textures are substantially smoothed – see, e.g., the mandrill image shownin Figure 14 – which is usual whatever the denoising method. Note that no falsecolour is introduced, contrary to the decimated wavelet method. See for examplethe parrot image shown in Figure 13, where the decimated wavelet method in-troduces many coloured artifacts: green artifacts around the black area of the redparrot’s beak, yellow artifacts on the right of the beak, green artifacts on the largeyellow area on the right side of the parrot.

Given the results, it is clear that one of the possible future directions ofthis work is about the numerical scheme used to perform the monogenic wavelet

12. Colour Monogenic Wavelets 265

Figure 12. Restoration of image peppers. Upper row: Noisy image anddecimated wavelet based denoising. Lower row: Undecimated waveletbased denoising and proposed approach.

analysis, since the current version is very sensitive to coefficient modification. Theshape of the basis functions would be explored as well.

We think that the main way to improve this work is to focus on the physicalinterpretation when designing the key monogenic colour concept. According to us,this transform suffers from a lack of unity around its different components, as wellas a poor link between them and the visual features. Although the generalizationis not strictly marginal, it has a marginal style since it reduces to apply the Riesztransform to the intensity of the image. We are currently working on a new def-inition of colour monogenic wavelets, where the physical interpretation is takenmore into account, and the local geometry is studied more deeply from a vectordifferential geometry viewpoint.

266 R. Soulard and P. Carre

Figure 13. Restoration of image parrot. Upper row: Noisy image anddecimated wavelet based denoising. Lower row: Undecimated waveletbased denoising and proposed approach.

6. Conclusion

We have defined a colour extension of the recent monogenic wavelet transformproposed in [9]. This extension is non-marginal since we have taken care to con-sider a vector signal at the very beginning of the fundamental construction and itleads to a definition fundamentally different from the marginal approach. The useof non-separable wavelets together with the monogenic framework permits a goodorientation analysis, well separated from the colour information. This colour trans-form can be a great colour image analysis tool thanks to its good separation ofinformation into various data. A statistical modelling of the coefficients for thresh-olding as well as a full denoising scheme is given and compared to state-of-the artwavelet based denoising methods.

12. Colour Monogenic Wavelets 267

Figure 14. Restoration of image mandrill. Upper row: Noisy im-age and decimated wavelet based denoising. Lower row: Undecimatedwavelet based denoising and proposed approach.

Although it is not marginal the colour generalization has a marginal style,since it reduces to applying the Riesz transform to the intensity of the image. Sothe geometric analysis is done without considering the colour information and itwould be much more attractive to have a complete representation of the colourmonogenic signal in terms of magnitude and phase(s) with both colour and geo-metric interpretation. The numerical scheme used is fragile with respect to thevisual impact of modifying wavelet coefficients. Our future work includes a newdefinition of monogenic filterbanks by focusing on reconstruction.

268 R. Soulard and P. Carre

References

[1] W.L. Chan, H.H. Choi, and R.G. Baraniuk. Coherent multiscale image process-ing using dual-tree quaternion wavelets. IEEE Transactions on Image Processing,17(7):1069–1082, July 2008.

[2] I. Daubechies. Ten Lectures on Wavelets. Society for Industrial and Applied Math-ematics, 1992.

[3] G. Demarcq, L. Mascarilla, and P. Courtellemont. The color monogenic signal: Anew framework for color image processing. application to color optical flow. In 16thIEEE International Conference on Image Processing (ICIP), pages 481–484, 2009.

[4] D.L. Donoho. De-noising by soft-thresholding. IEEE Transactions on InformationTheory, 41(3):613–627, 1995.

[5] M. Felsberg. Low-Level Image Processing with the Structure Multivector. PhD thesis,Christian-Albrechts-Universitat, Institut fur Informatik und Praktische Mathematik,Kiel, 2002.

[6] P. Kovesi. Image features from phase congruency. VIDERE: Journal of ComputerVision Research, 1(3):2–26, 1999.

[7] S. Mallat. A Wavelet Tour of Signal Processing. Academic Press, third edition, 2008.First edition published 1998.

[8] I.W. Selesnick, R.G. Baraniuk, and N.G. Kingsbury. The dual-tree complex wavelettransform. IEEE Signal Processing Magazine, 22(6):123–151, Nov. 2005.

[9] M. Unser, D. Sage, and D. Van De Ville. Multiresolution monogenic signal analysisusing the Riesz-Laplace wavelet transform. IEEE Transactions on Image Processing,18(11):2402–2418, Nov. 2009.

[10] M. Unser and D. Van De Ville. The pairing of a wavelet basis with a mildly re-dundant analysis via subband regression. IEEE Transactions on Image Processing,17(11):2040–2052, Nov. 2008.

Raphael Soulard and Philippe CarreXlim-SIC laboratoryUniversity of Poitiers, Francee-mail: raphael.soulard@univ-poitiers.fr

carre@sic.univ-poitiers.fr

Quaternion and Clifford–Fourier Transforms and Wavelets

Trends in Mathematics, 269–284c⃝ 2013 Springer Basel

13 Seeing the Invisible andMaxwell’s Equations

Swanhild Bernstein

Abstract. In this chapter we study inverse scattering for Dirac operators withscalar, vector and quaternionic potentials. For that we consider factorizationsof the Helmholtz equation and related fundamental solutions; the standardGreen’s function and Faddeev’s Green function. This chapter is motivated byoptical coherence tomography.

Mathematics Subject Classification (2010). Primary 30G35; secondary 45B05.

Keywords. Optical coherence tomography, Dirac operator, inverse scattering,Faddeev’s Green function.

1. Preliminaries

Let ℍ be the algebra of real quaternions and 𝔹 the complex quaternions or bi-quaternions. The vectors 𝒆1, 𝒆2, 𝒆3 are the generating vectors with

𝒆𝑗𝒆𝑘 + 𝒆𝑘𝒆𝑗 = −2𝛿𝑗𝑘and 𝒆0 the unit element. An arbitrary element 𝑎 ∈ ℍ is given by

𝑎 =3∑

𝑗=0

𝑎𝑗𝒆𝑗 , 𝑎𝑗 ∈ ℝ

and an arbitrary element 𝑏 ∈ 𝔹 by

𝑏 =3∑

𝑗=0

𝑏𝑗𝒆𝑗 , 𝑏𝑗 ∈ ℂ.

We denote by S(𝑞) the scalar part 𝑞0, by V(𝑞) = q =∑3

𝑗=1 𝑞𝑗𝒆𝑗 the vector partand define the conjugated quaternion by

𝑞 = S(𝑞)−V(𝑞)

270 S. Bernstein

for a quaternion 𝑞 ∈ ℍ or a biquaternion 𝑞 ∈ 𝔹. The algebra of quaternions is freeof zero divisors, i.e., if 𝑎1𝑎2 = 0 for 𝑎1, 𝑎2 ∈ ℍ then 𝑎1 = 0 or 𝑎2 = 0. This is nottrue for biquaternions, for example

(1 + 𝑖𝒆3)(1− 𝑖𝒆3) = 1− 𝑖2𝒆23 = 1− 1 = 0 and

(𝒆1 + 𝑖𝒆3)(𝒆1 + 𝑖𝒆3) = 𝒆21 + 𝑖(𝒆1𝒆3 + 𝒆3𝒆1) + 𝑖2𝒆2

3 = −1 + 1 = 0.

A zero divisor 𝑎 ∈ 𝔹 is an element 𝑎 ∕= 0 such that there exists a 𝑏 ∈ 𝔹 with 𝑏 ∕= 0and 𝑎𝑏 = 0.

Let 𝐺 ∈ ℝ3 be a domain and 𝐺𝑐 = {𝑥 ∈ ℝ3 : 𝑥 ∕∈ 𝐺} = ℝ3∖𝐺. A(bi)quaternion-valued function 𝑢 belongs to 𝐶(𝐺), 𝐿𝑝(𝐺), 𝐻1(𝐺) etc. when eachreal respectively complex-valued component 𝑢𝑗 belongs to that function space. For𝑠 ∈ ℝ, 𝐿2,𝑠 denotes the set of (scalar-valued) functions 𝑢 such that

∥𝑢∥𝑠 =∥∥∥(1 + ∣𝑥∣2)𝑠/2𝑢∥∥∥

𝐿2(ℝ3)<∞.

By 𝐻𝛼, 𝛼 ≥ 0, we denote the Sobolev space of (scalar-valued) functions 𝑢 suchthat

∥𝑢∥𝐻𝛼 =∥∥∥(1 + ∣𝜉∣𝛼/2)��∥∥∥

𝐿2(ℝ3)<∞,

and the weighted Sobolev spaces 𝐻𝛼,𝑠, 𝑠 ∈ ℝ∥∥∥(1 + ∣𝑥∣𝑠/2 𝑢)∥∥∥𝐻𝛼

<∞.

Furthermore, we identify a vector k ∈ ℂ3 with the bi-quaternion

𝑘1𝒆1 + 𝑘2𝒆2 + 𝑘3𝒆3.

We will denote the vector and the element in 𝔹 or ℍ with k. By ⋅ we denote theinner product in ℝ3

k ⋅ k =

3∑𝑗=1

𝑘2𝑗 ,

whereas

kk = k2 = −3∑

𝑗=1

𝑘2𝑗

is the product of the quaternion k with itself. By 𝐷 we denote the Dirac operator

𝐷 =

3∑𝑗=1

𝒆𝑗∂

∂𝑥𝑗=

3∑𝑗=1

𝒆𝑗∂𝑗 .

In particular, we have for a vector field u = (𝑢1, 𝑢2, 𝑢3) ∼ 𝑢1𝒆1 + 𝑢2𝒆2 + 𝑢3𝒆3:

𝐷u =

(− divucurlu

),

13. Seeing the Invisible 271

where

divu = ∇ ⋅ u =

3∑𝑗=1

∂𝑗𝑢𝑗 and curlu = ∇× u =

⎛⎝ ∂2𝑢3 − ∂3𝑢2

∂3𝑢1 − ∂1𝑢3

∂1𝑢2 − ∂2𝑢1

⎞⎠ .

Because the multiplication in the algebra of quaternions is not commutative weintroduce two different multiplication operators with vectors k ∈ ℍ or k ∈ 𝔹:

k𝑀𝑞 = k 𝑞 but 𝑀k𝑞 = 𝑞 k, 𝑞 ∈ 𝔹.

2. Motivation

2.1. Optical Coherence Tomography

This chapter is motivated by optical coherence tomography (OCT). Methods intomography usually use diffraction of beams to reconstruct an image. OCT isdifferent because it uses the interference of light waves. Therefore the mathematicalmodel is given by scattering of waves, see for example [7] and [8].

In this section we present the mathematical treatment of single OCT fol-lowing [7]. Unscattered photons like x-rays and 𝛾-rays have been used to obtaintomographic ray projections for a long time. The mathematical problem of recon-structing a function from its straight ray projections has already been presentedby Radon in 1917 [31]. Its solution, the Fourier slice theorem, shows that someof the three-dimensional Fourier data of the object can be obtained from two-dimensional Fourier transforms of its projections.Optical tomography techniques, in particular, OCT, deviate in several respectsfrom the better known coherence tomography (CT) concept.

∙ Diffraction optical tomography (DOT) uses highly diffracted and scatteredradiation, straight ray propagation can only be assumed for a fraction of thephotons.

∙ OCT images are synthesized from a series of adjacent interferometric depth-scans performed by a straight propagating low-coherence probing beam. Thisleads to an advantageous decoupling of transversal resolution from depthresolution.

∙ OCT uses backscattering, i.e., light propagates twice through the same ob-ject.

Let us consider a weakly inhomogeneous sample illuminated by the Rayleigh lengtharound the beam waist where we can assume plane-wave illumination with incidentwaves:

𝑉 (𝑖)(r, k, 𝑡) = 𝐴(𝑖)𝒆𝑖k(𝑖)⋅r−𝑖𝜔𝑡,

k(𝑖) is the wave vector of the illumination wave, ∣k(𝑖)∣ = 𝑘 = 2𝜋/𝜆 the wavenumber. Using the outgoing free-space Green’s function

𝐹𝑘(r, r′) =

𝒆𝑖𝑘(r−r′)

4𝜋 ∣r− r′∣

272 S. Bernstein

of the Helmholtz operator, the first-order Born approximation yields the scatteredwave as an approximate solution of the Helmholtz equation

𝑉𝑠(r,k(𝑠), 𝑡) = 𝑉 (𝑖)(r,k(𝑖), 𝑡) +

∫𝑉 (′)

𝑉 (𝑖)(r,k(𝑖), 𝑡) ⋅ 𝐹𝑠(r′, 𝑘) ⋅ 𝐹𝑘(r, r

′) 𝑑r′.

k(𝑠) is the wave vector of the scattered wave, ∣k(𝑠)∣ = 𝑘. This integral is extendedover wavelets originating from the illuminated sample volume 𝑉 (r(′)). The relativeamplitudes of these wavelets are determined by the scattering potential of thesample

𝐹𝑠(r, 𝑘) = 𝑘2[𝑚2(r, 𝑘)− 1],

where 𝑚 is the complex refractive index distribution of the sample structure:

𝑚(r) = 𝑛(r)[1 + 𝑖𝜅(r)],

with 𝑛(r) being the phase refractive index, and 𝜅(r) the attenuation index. In OCT,backscattered light originating from the coherently illuminated sample volume isdetected at a distance 𝑑 much larger than the linear dimensions of that volume.Therefore

𝑉𝑠(r,k, 𝑡) =𝐴(𝑖)

4𝜋 𝑑𝑒𝑖k

(𝑠)⋅r−𝑖𝜔𝑡∫𝑉 (′)

𝐹𝑠(r′)𝑒−k⋅r′ 𝑑r′ = 𝐴𝑠(r,k

(𝑠), 𝑡)𝑒𝑖k(𝑠)⋅r−𝑖𝜔𝑡,

where the amplitude 𝐴(𝑖) of the illuminating wave has been assumed constantwithin the coherent probe volume.

3. Helmholtz Equation and Maxwell’s Equation

The propagation of waves can be described by Maxwell’s equations which consti-tute a first-order differential system. That system can be reduced to second-orderdifferential equations. Due to their importance, Maxwell’s equations under dif-ferent boundary conditions, radiation conditions and other properties are widelystudied.

In this chapter we want to discuss the scattering problem for Maxwell’s equa-tions in terms of the Dirac operator. The description of Maxwell’s equations withthe Dirac operator and Clifford algebras as well as quaternions has been studiedin several papers ([12, 16, 15, 23, 3, 11, 10, 27]). Usually, time harmonic equationsin homogeneous media were investigated. Inhomogeneous media and in particularchiral media are considered in [15] and [19].

The Helmholtz equation and its application in acoustics and electromagneticshave been widely investigated (see for example [30, 32, 25]). Also some kind ofDirac equation and scattering has been considered [13].

We will study some scattering problems in connection with the Dirac-typeoperator 𝐷 +𝑀 𝑖k, where k is a vector, described with the aid of quaternions.

This problem is also motivated by so-called Lax pairs and the Abiowitz, Kaup,Newell and Segur (AKNS) method (see [1]). In very simple terms this methodmeans that a nonlinear partial differential equation can be written as a pair of

13. Seeing the Invisible 273

linear equations where one operator describes the spectral problem and the otheroperator is the operator governing the associated time evolution. Most knownmethods use the complex ∂-operator and can be reduced to a scalar or matrixRiemann–Hilbert problem [9]. But it can also be done with a Dirac operator. Thefactorization of the Helmholtz equation is also an important tool, because we haveto consider the operator 𝐷 +𝑀 𝑖k and the scattering data.

4. Maxwell’s Equations

The well-known Maxwell’s equations are usually used to describe electromagneticphenomena. In optics the interaction of light with a medium is characterized bythese equations. Maxwell’s equations in optics can be found in [28]. Under theassumption of an isotropic material that obeys Ohm’s law for electric conductionand can act in a para- or diamagnetic manner, Maxwell’s equationsare as follows:

curlE = −∂B

∂𝑡, divD = 𝜌,

curlH = 𝜎E+ 𝜀∂E

∂𝑡, divB = 0,

where E is the electric field, H the magnetic field, D the electric induction, B themagnetic induction, 𝜌 the charge density not due to polarization of the medium, µthe permeability, 𝜎 describes the electric conductivity and 𝜀 the permittivity of themedium. The permittivity, permeability and electric conductivity are parametersthat are related to material properties of the medium and do not change withtime. In optics µ ≈ 1 can be used. If we consider a perfectly transparent insulatorthen its electric conductivity and its external charge density can be taken to bezero.

Maxwell’s equations are considered together with so-called constitutive rela-tions, which describe the relation between induction vectors and field vectors:

D = 𝜀E and B = µH.

Interference occurs when radiation follows more than one path from its source tothe point of detection. The simplest example of interference is that between planewaves. We restrict our consideration to plane waves, which means that we canconsider time-harmonic Maxwell’s equations, i.e.,

E(𝑥, 𝑡) = Re (E(𝑥)𝑒−𝑖𝜔𝑡), H(𝑥, 𝑡) = Re (H(𝑥)𝑒−𝑖𝜔𝑡).

If we take everything into account we obtain the system

curlE = 𝑖𝜔µH, div(𝜀E) = 0

curlH = −𝑖𝜔𝜀E, div(µH) = 0.

274 S. Bernstein

5. Maxwell’s Equations Written as Dirac Equations

5.1. Quaternionic Formulation

Let 𝜀0 be the free space permittivity and µ0 be the free space permeability, then1√µ0𝜀0

= 𝑐, the speed of light, and 𝑘 = 𝜔√µ𝜀 is the wave number. We further

introduce the complex refractive index 𝑁 by

𝑁2 = µ𝑟𝜀𝑟 =

(µ

µ0

)(𝜀

𝜀0

),

where µ𝑟 and 𝜀𝑟 are called relative permittivity and permeability.

𝑁 = 𝑛+ 𝑖𝜅,

where the real part 𝑛 is the conventional refractive index, whereas 𝜅 is called theextinction coefficient, which describes the attenuation of the electric field in themedium. Maxwell’s equations, medium properties in optics, and their mathemat-ical description are studied in [28].

Remark 5.1. Force-free magnetic fields are an important special solution of non-linear equations of magnetohydrodynamics. They are characterized by

divB = 0 and curlB+ 𝛼(𝑥)B = 0,

where 𝛼(𝑥) is a scalar-valued function. This system is equivalent to

𝐷𝛼𝐵 = 0. (5.1)

This relation between force-free fields and (5.1) can be found in [17].

Remark 5.2. In case of static electric and magnetic fields in an inhomogeneousmedium we obtain the decoupled system

𝐷𝜀E = − 𝜌√𝜀, 𝐷µH =

õj.

A very elegant quaternionic reformulation of the Maxwell’s equations (see[14]) in inhomogeneous media can now be obtained by the substitution

ℰ :=√𝜀E and ℋ :=

õH,

which leads to the system {𝐷𝜀𝜀𝜀ℰ = −𝑖𝑘ℋ− 𝜌√

𝜀 ,

𝐷𝜇ℋ = 𝑖𝑘ℰ +√µ𝑗,

where 𝜀𝜀𝜀 =grad

√𝜀√

𝜀 (𝜇 =grad

√µ√

µ) and 𝐷𝜀 = 𝐷 + 𝜀 (𝐷µ = 𝐷 + µ). This system

can also be written using matrices:(𝐷𝜀𝜀𝜀 −𝑖𝑘𝑖𝑘 𝐷𝜇

)( ℰℋ)

=

[(𝐷 −𝑖𝑘𝑖𝑘 𝐷

)+

(𝜀𝜀𝜀 00 𝜇

)]( ℰℋ)

=

(− 𝜌√

𝜀√µj

).

13. Seeing the Invisible 275

6. Factorization of the Wave Equation

Using vector and scalar potentials, Maxwell’s equations can be rewritten as second-order differential equations, that, in turn can be factorized with the help of Diracoperators:

1

𝑐2∂2

∂𝑡2−Δ =

(1

𝑐

∂

∂𝑡+ 𝑖𝐷

)(1

𝑐

∂

∂𝑡+ 𝑖𝐷

),

where 𝑖 ∈ ℂ is the complex unit. If we just take the time-harmonic case, i.e., thetime dependence is given by the expontial 𝑒𝑖𝜔𝑡, differentiation by 𝑡 is simply givenby multiplication with 𝑖𝜔 and we can consider the operator

−𝜔2

𝑐2−Δ =

(𝐷 +

𝜔

𝑐

)(𝐷 − 𝜔

𝑐

).

These factorizations show the deep connection between the Helmholtz equationand Maxwell’s equations. But there is yet another factorization. Let k be a vector,then

(𝐷 +𝑀k)(𝐷 −𝑀k) = (𝐷 −𝑀k)(𝐷 +𝑀k) = −Δ− k2 = −Δ+ ∣k∣2 ,and a factorization of the Helmholtz equation arises from

(𝐷 +𝑀 𝑖k)(𝐷 −𝑀 𝑖k) = −Δ− 𝑖2k2 = −Δ− ∣k∣2 .

7. Scattering

We will need some properties of the fundamental solution of the Helmholtz equa-tion.

Proposition 7.1 ([5]). Let

𝐹𝑘(𝑥− 𝑦) := − 𝑒𝑖𝑘∣𝑥− 𝑦∣

4𝜋 ∣𝑥− 𝑦∣ , 𝑥, 𝑦 ∈ ℝ3, 𝑥 ∕= 𝑦.

𝐹𝑘(⋅, 𝑦) solves the Helmholtz equation Δ𝑢 + 𝑘2𝑢 = 0 in ℝ3∖{𝑦}, and 𝑦 ∈ 𝐺 forevery bounded subset 𝐺 ⊂ ℝ3.

𝐹𝑘(𝑥 − 𝑦) = − 𝑒𝑖𝑘∣𝑥∣

4𝜋 ∣𝑥∣𝑒−𝑖𝑘��⋅𝑦 +O(∣𝑥∣−2),

𝐷𝑥𝐹𝑘(𝑥 − 𝑦) = (𝐷𝑥𝐹𝑘(𝑥))𝑒−𝑖𝑘��⋅𝑦 +O(∣𝑥∣−2),

uniform in �� = 𝑥∣𝑥∣ ∈ 𝑆2 and 𝑦 ∈ 𝐺 for every bounded subset 𝐺 ⊂ ℝ3.

Let 𝑘 = const and ℑ (𝑘) ≥ 0. Then application of −𝐷−𝑘 to the fundamentalsolution of the Helmholtz operator gives

−𝐷−𝑘𝐹𝑘(𝑥) = 𝐶𝑘(𝑥) =

(𝑘 +

𝑥

∣𝑥∣2 − 𝑖𝑘𝑥

∣𝑥∣

)𝑒𝑖𝑘∣𝑥∣

4𝜋.

This fundamental solution was obtained in [20].

276 S. Bernstein

Proposition 7.2 ([4]). Let 𝐺 be a bounded Lipschitz domain with boundary ∂𝐺 andoutward pointing unit normal n. Let u ∈ 𝐻1(𝐺). If u ∈ ker𝐷𝑘(𝐺), ℑ (𝑘) ≥ 0, then

u(𝑥) = 𝒞𝑘[u](𝑥) = −∫∂𝐺

𝐶𝑘(𝑥 − 𝑦)n(𝑦)u(𝑦)𝑑𝑦.

If u ∈ ker𝐷k = ker(𝐷 +𝑀k), then

u(𝑥) = 𝒞k[u](𝑥) = −∫∂𝐺

(−𝐷𝑥𝐹𝑘)(𝑥− 𝑦)n(𝑦)u(𝑦) + n(𝑦)u(𝑦)𝐹𝑘(𝑥− 𝑦)k 𝑑𝑦,

= −∫∂𝐺

(𝑥− 𝑦

∣𝑥− 𝑦∣2 − 𝑖𝑘(𝑥− 𝑦)

∣𝑥− 𝑦∣

)𝑒𝑖𝑘∣𝑥− 𝑦∣

4𝜋 ∣𝑥− 𝑦∣n(𝑦)u(𝑦)

− 𝑒𝑖𝑘∣𝑥− 𝑦∣

4𝜋 ∣𝑥− 𝑦∣n(𝑦)u(𝑦)k 𝑑𝑦,

where 𝑘 =√k2 ∈ ℂ is chosen such that ℑ (𝑘) ≥ 0.

Proposition 7.3 (Radiation condition [18]). Let 𝑓 ∈ 𝐻1loc(𝐺

𝑐), 𝑓 ∈ ker𝐷𝑘(𝐺𝑐) and

𝑓 satisfy the radiation condition(𝑘 − 𝑥

∣𝑥∣2 + 𝑖𝑘𝑥

∣𝑥∣

)𝑓(𝑥) = o(∣𝑥∣−1) as ∣𝑥∣ → ∞,

then

𝑓(𝑥) = −𝒞𝑘[𝑓 ](𝑥) ∀𝑥 ∈ 𝐺𝑐.

If 𝑓 satisfies the radiation condition

𝑘𝑓(𝑥) +𝑖𝑥

∣𝑥∣𝑓(𝑥)k = o(∣𝑥∣−1) as ∣𝑥∣ → ∞,

where 𝑘 =√k2 and ℑ (𝑘) ≥ 0, then

𝑓(𝑥) = −𝒞k[𝑓 ](𝑥), for all 𝑥 ∈ 𝐺𝑐.

If k is a zero divisor we suppose additionally 𝑓(𝑥) = o(∣𝑥∣−1).

Remark 7.4. It looks as if the condition for a scalar 𝑘 can be written without theterm 𝑥/ ∣𝑥∣2 which apparently gives a faster decay. But this is not true because(1 + 𝑖𝑥/ ∣𝑥∣) is a zero divisor. See also [18].

Remark 7.5. This proposition is true for a quaternion-valued function, but wewill restrict ourselves to vector fields u, i.e., quaternion-valued functions with zeroscalar part.

8. The Scattering Problem

In this section we want to consider the scattering problem for the operator𝐷+𝑀k.The case 𝐷 + 𝑘 can be treated similarly.

13. Seeing the Invisible 277

Statement of the problem: Let m(𝑥) be a quaternion-valued potential with com-pact support = 𝐺. Let k be a vector which will be identified with 𝑘1𝑒1 + 𝑘2𝑒2 +

𝑘3𝑒3 ∈ ℍ and 𝑘 =√k2 with ℑ (𝑘) ≥ 0.

Further, let u𝑖(𝑥) be a solution of

𝐷u𝑖(𝑥) + u𝑖(𝑥)k = 0 in ℝ3. (8.1)

The scattering problem then consists in determining a scattering solution u𝑠(𝑥)such that

𝐷u+ uk = um(𝑥)k in ℝ3, (8.2)

u = u𝑖 + u𝑠, (8.3)

and u𝑠 fullfils the radiation condition

𝑘u𝑠(𝑥) +𝑖𝑥

∣𝑥∣u𝑠(𝑥)k = o(∣𝑥∣−1

), as ∣𝑥∣ → ∞. (8.4)

If k is a zero divisor we additionally assume u𝑠(𝑥) = o(1).

Remark 8.1. The unknown vector field m with compact support could also bequaternion-valued, i.e., it could have a non-zero scalar part. We writem to indicatethat we consider a scalar- or vector- or quaternion-valued function and not only ascalar-valued function.

Theorem 8.2. If u is a solution of the scattering problem (8.1)–(8.4) then u ∣𝐺solves the Lippmann–Schwinger integral equation

u(𝑥)=u𝑖(𝑥)−∫𝐺

(−𝐷𝑥𝐹𝑘(𝑥− 𝑦)u(𝑦))m(𝑦)k+u(𝑦)m(𝑦)k𝐹𝑘(𝑥− 𝑦)k 𝑑𝑦. (8.5)

Conversely, if u is a solution of the Lippmann–Schwinger equation then u is asolution of the scattering problem.

Proof. Let u be a solution of the scattering problem and v the integral on theright-hand side of (8.5). Because

𝐷v − vk = um(𝑥)k = 𝐷u+ u(𝑥)k,

w = u− v satisfies𝐷w +wk = 0 in ℝ3.

Furthermore,

w(𝑥) = (u𝑖(𝑥)− u(𝑥))

−∫𝐺

(−𝐷𝑥𝐹𝑘(𝑥− 𝑦))u(𝑦)m(𝑦)k+ u(𝑦)m(𝑦)k𝐹𝑘(𝑥− 𝑦)k 𝑑𝑦,

𝑥 ∈ ℝ3, and it satisfies the radiation condition which will be seen as follows:u𝑖(𝑥) − u(𝑥) = u𝑠(𝑥), which satisfies the radiation condition by assumption. Letus now consider the integral. First we look at the kernels

𝐹𝑘(𝑥− 𝑦) = 𝐹𝑘(𝑥)𝑒−𝑖𝑘��⋅𝑦 +O(∣𝑥∣−2) and

(𝐷𝑥𝐹𝑘)(𝑥 − 𝑦) = (𝐷𝑥𝐹𝑘)(𝑥)𝑒−𝑖𝑘��⋅𝑦 +O(∣𝑥∣−2

).

278 S. Bernstein

We conclude that

(−𝐷𝑥𝐹𝑘(𝑥− 𝑦)u(𝑦))m(𝑦)k+ u(𝑦)m(𝑦)k𝐹𝑘(𝑥− 𝑦)k

=((−𝐷𝑥𝐹𝑘(𝑥)u(𝑦))m(𝑦)k + 𝑘2u(𝑦)m(𝑦)𝐹𝑘(𝑥− 𝑦)

)𝑒−𝑖𝑘��⋅𝑦 +O(∣𝑥∣−2

).

Therefore it is enough to consider

(−𝐷𝑥𝐹𝑘(𝑥)u(𝑦))m(𝑦)k + 𝑘2u(𝑦)m(𝑦)𝐹𝑘(𝑥− 𝑦).

We obtain

𝑘((−𝐷𝑥𝐹𝑘(𝑥)u(𝑦))m(𝑦)k + 𝑘2u(𝑦)m(𝑦)𝐹𝑘(𝑥− 𝑦)

)+

𝑖𝑥

∣𝑥∣((−𝐷𝑥𝐹𝑘(𝑥)u(𝑦))m(𝑦)k + 𝑘2u(𝑦)m(𝑦)𝐹𝑘(𝑥− 𝑦)

)k

+ 𝑘((−𝐷𝑥𝐹𝑘(𝑥)u(𝑦))m(𝑦)k + 𝑘2u(𝑦)m(𝑦)𝐹𝑘(𝑥− 𝑦)

)+

𝑖𝑥

∣𝑥∣((−𝐷𝑥𝐹𝑘(𝑥)u(𝑦))m(𝑦)k2 +

𝑖𝑥

∣𝑥∣𝑘2u(𝑦)m(𝑦)𝐹𝑘(𝑥− 𝑦)k

)=

(𝑘(−𝐷𝑥𝐹𝑘)(𝑥) +

𝑖𝑘2𝑥

∣𝑥∣ 𝐹𝑘(𝑥)

)u(𝑦)m(𝑦)k

+ 𝑘2

(𝑖𝑥

∣𝑥∣ (−𝐷𝑥𝐹𝑘)(𝑥) + 𝑘𝐹𝑘(𝑥)

)u(𝑦)m(𝑦)

= −(

𝑘𝑥

∣𝑥∣3 − 𝑖𝑘2 𝑥

∣𝑥∣2 + 𝑖𝑘2 𝑥

∣𝑥∣2)

𝑒𝑖𝑘∣𝑥∣

4𝜋u(𝑦)m(𝑦)k

− 𝑘2

(𝑖𝑥2

∣𝑥∣4 + 𝑘𝑥2

∣𝑥∣3 + 𝑘1

∣𝑥∣

)𝑒𝑖𝑘∣𝑥∣

4𝜋u(𝑦)m(𝑦)

= −𝑘𝑥𝑒𝑖𝑘∣𝑥∣

4𝜋 ∣𝑥∣3 u(𝑦)m(𝑦)k− 𝑘2 𝑖𝑥2𝑒𝑖𝑘∣𝑥∣

4𝜋 ∣𝑥∣4 u(𝑦)m(𝑦)

= O(∣𝑥∣−2). □

Remark 8.3. We can replace the region of integration by any domain 𝐺 such thatthe support of m is contained in 𝐺

Using the unique continuation principle (8.5) we will prove that the homo-geneous equation has only the trivial solution and by the Fredholm theory theexistence of a solution for functions in 𝐶(𝐺) or 𝐿𝑝(𝐺), 1 < 𝑝 < ∞. Then thesolution of the Lippmann–Schwinger equation can be computed and we obtain thesolution u from

u(𝑥) = u𝑖(𝑥) −∫𝐺

(−𝐷𝑥𝐹𝑘(𝑥 − 𝑦))u(𝑦)m(𝑦)k+ u(𝑦)m(𝑦)k𝐹𝑘(𝑥− 𝑦)k 𝑑𝑦.

Lemma 8.4. We have

∣𝐹𝑘(𝑥− 𝑦)∣ ≤ 𝑐

∣𝑥− 𝑦∣ and ∣𝐷𝐹𝑘(𝑥− 𝑦)∣ ≤ 𝑐1

∣𝑥− 𝑦∣2 +𝑐2

∣𝑥− 𝑦∣ .

13. Seeing the Invisible 279

Because the kernels of the Lippmann–Schwinger equation are weakly singular,the integral operators are compact as mappings in spaces of continuous functionsas well as 𝐿𝑝, 1 < 𝑝 <∞, but only for bounded domains.

To get information about the solution of the Lippmann–Schwinger equationwe would like to apply the Fredholm theory for compact operators. So far we knowthat the integral operator is a compact operator. Now we need to prove that thehomogeneous equation has only the trivial solution. For that we need the uniquecontinuation principle. The proof of this principle goes back to [24]. For a proofwe refer to [5, Lemma 8.5], which also uses ideas from [29] and [21].

Lemma 8.5 (Unique continuation principle, [5]). Let 𝐺 be a domain in ℝ3 and let𝑢1, . . . , 𝑢𝑝 ∈ 𝐶2(𝐺), be real-valued functions satisfying

∣Δ𝑢𝑝∣ ≤ 𝑐

𝑃∑𝑞=1

{∣𝑢𝑞∣+ ∣grad𝑢𝑞∣} in 𝐺,

for 𝑝 = 1, . . . , 𝑃 and some constant 𝑐. Assume that 𝑢𝑝 vanishes in a neighborhoodof some 𝑥0 ∈ 𝐺 for 𝑝 = 1, . . . , 𝑃. Then 𝑢𝑝 is identically zero in 𝐺 for 𝑝 = 1, . . . , 𝑃.

To be able to apply the unique continuation principle to

(𝐷 +𝑀k)u−m(𝑥)uk = 0

we apply the operator 𝐷 again to obtain

𝐷((𝐷 +𝑀k)u−m(𝑥)uk) = −Δu+ (𝐷u)k−𝐷(m(𝑥)uk).

and we can use the equation

Δu = (𝐷u)k−𝐷(m(𝑥)uk).

The last equation equation can be rewritten as the following system

Δu = −(divu)k+ (curlu)× k+ (divm)(u× k)

−(curlm)(u ⋅ k)− 2

(3∑

𝑗=1

𝑚𝑗∂𝑗u

)k.

If there are constants 𝐶1, 𝐶2 > 0 such that

max𝑥∈𝐺

∣𝑚𝑗(𝑥)∣ ≤ 𝐶1 and max𝑥∈𝐺

∣grad𝑚𝑗(𝑥)∣ ≤ 𝐶2,

then we can apply the unique continuation principle to conclude that the homoge-neous Lippmann–Schwinger equation over a bounded domain has only the trivialsolution.

Theorem 8.6. Let k ∈ ℂ3∖{0}. There exists a unique solution to the inverse scat-tering problem (8.1)–(8.4) and the solution u depends continuously, with respectto the maximum norm, on the incident field u𝑖.

Proof. Due to the Fredholm theory it is enough to prove that the homogeneousequation has only the trivial solution. If that is proven, the integral equationis a bounded invertible operator in 𝐶(��𝑅). From this it follows that u dependscontinuously on the incident field u𝑖 with respect to the maximum norm. Let 𝐵𝑅 :=

280 S. Bernstein

{𝑥 ∈ ℝ3 : ∣𝑥∣ ≤ 𝑅}. Due to Proposition 7.2 the function u can be represented asan integral over ∂𝐵𝑅, and due to the properties of 𝐹𝑘(𝑥− 𝑦) we have u(𝑥) = o(1)as ∣𝑥∣ → ∞. Now,

u(𝑥) = 𝒞k[𝑓 ](𝑥) =∫∂𝐵𝑅

(−𝐷𝑥𝐹𝑘)(𝑥− 𝑦)n(𝑦)u(𝑦) + n(𝑦)u(𝑦)𝐹𝑘(𝑥− 𝑦)k 𝑑𝑦

∼∫∂𝐵𝑅

(𝐷𝑦𝐹𝑘)(𝑥− 𝑦)𝑦

∣𝑦∣u(𝑦) +𝑦

∣𝑦∣u(𝑦)𝐹𝑘(𝑥− 𝑦)k 𝑑𝑦

∼∫∂𝐵𝑅

𝐹𝑘(𝑦)

{(1

∣𝑦∣ − 𝑖𝑘

)u(𝑦) +

𝑦

∣𝑦∣u(𝑦)k}

𝑑𝑦

∼∫∂𝐵𝑅

𝐹𝑘(𝑦)

{1

∣𝑦∣u(𝑦)− 𝑖

(𝑘𝑓(𝑦) + 𝑖

𝑦

∣𝑦∣u(𝑦)k)}

𝑑𝑦

∼∫∂𝐵𝑅

𝐹𝑘(𝑦)

{1

∣𝑦∣u(𝑦) + o(∣𝑦∣−1)

}𝑑𝑦 → 0 as 𝑦 →∞,

because u fulfils the radiation condition and u(𝑦) = o(1). We have proven thatu = 0 for ∣𝑥∣ ≥ 𝑅. The unique continuation principle (Lemma 8.5) implies thatu = 0 in ℝ3. □

Up to now we have used the usual Green’s function for the Helmholtz equation𝐹𝑘(𝑥) and the Green’s functions 𝐶𝑘(𝑥) and 𝐶k(𝑥) for the Dirac operators 𝐷 + 𝑘and 𝐷 + 𝑀k. Another type of Green’s functions are the exponentially growingGreen’s functions. These Green’s functions were introduced by Faddeev [6] andlater on used by Nachman and Ablowitz [25], Beals and Coifman [2], Sylvesterand Uhlmann [32], Paivarinta [30] and Isozaki [13] in inverse scattering. Someinverse scattering problems for the Dirac operator are also discussed in [26] and[22].

The main idea is to consider plane waves

𝑒𝑖k⋅𝑥, k ∈ ℂ3, 𝑥 ∈ ℝ3, k ⋅ k = 𝑘2.

Therefore we analyze how the operators change when u is replaced by 𝑒𝑖k⋅𝑥v. Weobtain

(Δ + 𝑘2)u = (Δ + 𝑘2)(𝑒𝑖k⋅𝑥v) = 𝑒𝑖k⋅𝑥(Δ + 2𝑖k ⋅ ∇)v,

(𝐷 − 𝑖k)u = (𝐷 − 𝑖k)(𝑒𝑖k⋅𝑥v) = 𝑒𝑖k⋅𝑥 (𝑖kv +𝐷v − 𝑖kv) = 𝑒𝑖k⋅𝑥𝐷v

(𝐷 +𝑀 𝑖k)u = (𝐷 +𝑀 𝑖k)(𝑒𝑖k⋅𝑥v) = 𝑒𝑖k⋅𝑥 (𝑖kv +𝐷v + v(𝑖k)) .

In the last equation it does matter whether v is truly quaternion-valued or just avector. In the latter case we see that kv+vk = −k⋅v+k×v−v⋅k+v×k = −2k⋅v.Hence

(𝐷 +𝑀 𝑖k)u = (𝐷 +𝑀 𝑖k)(𝑒𝑖k⋅𝑥v) = 𝑒𝑖k⋅𝑥(𝐷v − 2𝑖k ⋅ v).We will relate these new operators to the operator

−Δ− 2𝑖𝛾𝛾𝛾 ⋅ ∇ − 𝜆2,

and Faddeev’s Green function.

13. Seeing the Invisible 281

8.1. Faddeev’s Green Function

The idea of Faddeev to obtain a nice Green function starts with decomposing thevector k = 𝜂𝜂𝜂+ 𝑡𝛾𝛾𝛾, where 𝛾𝛾𝛾 ∈ 𝑆2 is an arbitrary direction and 𝜂𝜂𝜂 ⋅𝛾𝛾𝛾 = 0. If we applyΔ + 𝑘2 to 𝑒𝑖𝑡𝛾𝛾𝛾⋅𝑥w(𝑥) we obtain

(Δ + 𝑘2)(𝑒𝑖𝑡𝛾𝛾𝛾⋅𝑥w(𝑥)) = 𝑒𝑖𝑡𝛾𝛾𝛾⋅𝑥(−𝑡2 + 𝑘2 + 2𝑖𝑡𝛾𝛾𝛾 ⋅ ∇+Δ)w(𝑥).

Let 𝜆2 = 𝑘2 − 𝑡2, take the Fourier transform of the differential operator −Δ −2𝑖𝑡𝛾𝛾𝛾 ⋅ ∇ − 𝜆2. We then get Faddeev’s Green operator defined as

(𝑔𝛾𝛾𝛾(𝜆, 𝑧)𝑓)(𝑥) =1

(2𝜋)3

∫𝑒𝑖𝑥⋅𝜉

𝜉2 + 2𝑧𝛾𝛾𝛾 ⋅ 𝜉 − 𝜆2𝑓(𝜉) 𝑑𝜉,

where 𝛾𝛾𝛾 ∈ 𝑆2, 𝜆 ≥ 0, and 𝑧 ∈ ℂ+ = {𝑧 ∈ ℂ : ℑ (𝑧) > 0}. If ℑ (𝑧) ∕= 0, (𝜉2 + 2𝑧𝛾𝛾𝛾 ⋅𝜉 − 𝜆2)−1 ∈ 𝐿1

loc(ℝ3). Therefore the integral is absolutely convergent for 𝑓 ∈ 𝒮.

For 𝑡 ∈ ℝ, 𝑔𝛾𝛾𝛾(𝜆, 𝑡) is defined as the boundary value 𝑔𝛾𝛾𝛾(𝜆, 𝑡+ 𝑖0).

Proposition 8.7 ([13]). Let 𝑠 > 12 . Then

1. 𝑔𝛾𝛾𝛾(𝜆, 𝑧) is continuous with respect to 𝜆 ≥ 0, 𝛾𝛾𝛾 ∈ 𝑆2, 𝑧 ∈ ℂ+ except for (𝜆, 𝑧) =(0, 0).

2. 𝑔𝛾𝛾𝛾(𝜆, 𝑧) is analytic in 𝑧 ∈ ℂ+.3. For any 𝛿0 > 0, there exists a constant 𝐶 > 0 such that

∥𝑔𝛾𝛾𝛾(𝜆, 𝑧)∥(𝐿2,𝑠,𝐻𝛼,𝑠) ≤𝐶

(𝜆 + ∣𝑧∣)1−𝛼 ,

with 𝜆+ ∣𝑧∣ ≥ 𝛿0, and 0 ≤ 𝛼 ≤ 2.

We would like to have a similar Faddeev’s Green function for the operators𝐷+ 𝑖k = 𝐷+ 𝑖k𝑀 and 𝐷+𝑀 𝑖k. There are differences between both cases due tothe fact that

(𝐷 + 𝑖k)(𝐷 + 𝑖k)u = −Δu+ 𝑖k𝐷u+ 𝑖𝐷(ku)− k2u

= −Δu+ 𝑖k𝐷u− 𝑖k𝐷u− 2𝑖3∑

𝑗=1

𝑘𝑗∂𝑗u+ k ⋅ ku

= (−Δ− 2𝑖k ⋅ ∇+ 𝑘2)u.

With 𝑖k = 𝑖𝑧𝛾𝛾𝛾 we have

(𝐷 + 𝑖𝑧𝛾𝛾𝛾)(𝐷 + 𝑖𝑧𝛾𝛾𝛾)u = (−Δ− 2𝑖𝑧𝛾𝛾𝛾 ⋅ ∇+ 𝑧2)u,

and hence

(𝐷 + 𝑖𝑧𝛾𝛾𝛾 − 𝑘)(𝐷 + 𝑖𝑧𝛾𝛾𝛾 + 𝑘)u

= (𝐷 + 𝑖𝑧𝛾𝛾𝛾)(𝐷 + 𝑖𝑧𝛾𝛾𝛾)u+ (𝐷 + 𝑖𝑧𝛾𝛾𝛾)(𝑘u) + 𝑘(𝐷 + 𝑖𝑧𝛾𝛾𝛾)u− 𝑘2u

= (−Δ− 2𝑖𝑧𝛾𝛾𝛾 ⋅ ∇ − 𝑘2 + 𝑧2)u

= (−Δ− 2𝑖𝑧𝛾𝛾𝛾 ⋅ ∇ − 𝜆2)u.

282 S. Bernstein

This is the way that Isozaki got a Faddeev’s Green function. We obtain

𝐺𝛾𝛾𝛾(𝜆, 𝑧) = (𝐷 + 𝑖𝑧𝛾𝛾𝛾 − 𝑘)𝑔(𝜆, 𝑧).

The multiplication from the other side leads to a different Faddeev’s Greenfunction. In this case we have

(𝐷 +𝑖𝑧𝛾𝛾𝛾 𝑀 +𝑀 𝑖(𝑧𝛾𝛾𝛾+𝜂𝜂𝜂))(𝐷 +𝑖𝑧𝛾𝛾𝛾 𝑀 −𝑀 𝑖(𝑧𝛾𝛾𝛾+𝜂𝜂𝜂))u = (−Δ− 2𝑖𝑧𝛾𝛾𝛾 ⋅ ∇ − 𝜆2)u,

and thus with 𝑖k = 𝑖𝑧𝛾𝛾𝛾 + 𝑖𝜂𝜂𝜂 we obtain

𝐺𝛾𝛾𝛾(𝜆, 𝑧) = (𝐷 +𝑖𝑧𝛾𝛾𝛾 𝑀 −𝑀 𝑖k)𝑔(𝜆, 𝑧).

This easily shows that for both operators 𝐺𝛾𝛾𝛾(𝜆, 𝑧) the following is true.

Theorem 8.8. Let 𝑠 > 12 . Then

1. 𝑔𝛾𝛾𝛾(𝜆, 𝑧) is continuous with respect to 𝜆 ≥ 0, 𝛾𝛾𝛾 ∈ 𝑆2, 𝑧 ∈ ℂ+ except for (𝜆, 𝑧) =(0, 0).

2. 𝑔𝛾𝛾𝛾(𝜆, 𝑧) is analytic in 𝑧 ∈ ℂ+.3. For any 𝛿0 > 0, and 0 ≤ 𝛼 ≤ 1, there exists a constant 𝐶 > 0 such that

∥𝑔𝛾𝛾𝛾(𝜆, 𝑧)∥(𝐿2,𝑠,𝐻𝛼,𝑠) ≤ 𝐶(𝜆 + ∣𝑧∣)𝛼,with 𝜆+ ∣𝑧∣ ≥ 𝛿0.

We conclude that we can also use Faddeev’s Green function to solve theinverse scattering problem and we can assume the solution u to have the structureu = 𝑒𝑖k⋅𝑥v, which is the appropriate setting for optical coherence tomography.

Acknowledgment

I would like to thank Dr. B. Heise and the Christian Doppler Laboratory for Micro-scopic and Spectroscopic Material Characterization, Johannes Kepler UniversityLinz, Austria, for making me realize the importance of Maxwell’s equations andinverse scattering for OCT and for the opportunity to learn the physics behindthe mathematical formulae.

References

[1] M.J. Ablowitz and A.P. Clarkson. Solitons, Nonlinear Evolution Equations and In-verse Scattering, volume 149 of London Mathematical Society Lecture Note Series.Cambridge University Press, 1991.

[2] R. Beals and R.R. Coifman. Multidimensional inverse scattering and nonlinear par-tial differential equations. In F. Treves, editor, Pseudodifferential Operators and Ap-plications, volume 43 of Proceedings of Symposia in Pure Mathematics, pages 45–70.American Mathematical Society, 1984.

[3] S. Bernstein. Factorization of the Schrodinger operator. In W. Sproßig, editor, Pro-ceedings of the Symposium Analytical and Numerical Methods in Quaternionic andClifford analysis, pages 1–6, 1996.

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papers/46/M_46.pdf, 2003.

13. Seeing the Invisible 283

[5] D. Colton and R. Kress. Inverse Acoustic and Electromagnetic Scattering Theory,volume 93 of Applied Mathematical Sciences. Springer-Verlag, Berlin, Heidelberg,New York, 1992.

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[8] A.F. Fercher, W. Drexler, C.K. Hitzenberger, and T. Lasser. Optical coherence to-mography – principles and applications. Reports on Progress in Physics, 66:239–303,2003.

[9] A.S. Fokas. A unified transform method for solving linear and certain nonlinearPDEs. Proceedings of the Royal Society A, Mathematical, Physical & EngineeringSciences, 453:1411–1443, 1997.

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[16] V.V. Kravchenko. Applied Quaternionic Analysis, volume 28 of Research and Expo-sition in Mathematics. Heldermann Verlag, 2003.

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[18] V.V. Kravchenko and R.P. Castillo. An analogue of the Sommerfeld radiation condi-tion for the Dirac operator. Mathematical Methods in the Applied Sciences, 25:1383–1394, 2002.

[19] V.V. Kravchenko and M.P. Ramirez. New exact solutions of the massive Dirac equa-tion with electric or scalar potential. Mathematical Methods in the Applied Sciences,23:769–776, 2000.

[20] V.V. Kravchenko and M.V. Shapiro. On a generalized system of Cauchy-Riemannequations with a quaternionic parameter. Russian Academy of Sciences, Doklady.,47:315–319, 1993.

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[24] C. Muller. On the behavior of solutions of the differential equation 𝛿𝑢 = 𝑓(𝑥, 𝑢)in a neighborhood of a point. Communications on Pure and Applied Mathematics,7:505–515, 1954.

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Swanhild BernsteinTU Bergakademie FreibergInstitute of Applied AnalysisPruferstr. 9D-09596 Freiberg, Germanye-mail: swanhild.bernstein@math.tu-freiberg.de

Quaternion and Clifford–Fourier Transforms and Wavelets

Trends in Mathematics, 285–298c⃝ 2013 Springer Basel

14A Generalized Windowed Fourier Trans-form in Real Clifford Algebra 𝑪ℓ0,𝒏

Mawardi Bahri

Abstract. The Clifford–Fourier transform in 𝐶ℓ0,𝑛 (CFT) can be regarded as ageneralization of the two-dimensional quaternionic Fourier transform (QFT),which was first introduced from the mathematical aspect by Brackx. In thischapter, we propose the Clifford windowed Fourier transform using the kernelof the CFT. Some important properties of the transform are investigated.

Mathematics Subject Classification (2010). Primary 15A66; secondary 42B10.

Keywords. Multivector-valued function, Clifford algebra, Clifford–Fouriertransform.

1. Introduction

Recently researchers have paid much attention to the generalization of the classi-cal windowed Fourier transform (WFT) using the quaternion algebra and Cliffordalgebra. The first attempt to extend the WFT to the quaternion algebra was byBulow and Sommer [6, 7]. They introduced a special case of the quaternionicwindowed Fourier transform (QWFT) known as quaternionic Gabor filters. Theyapplied these filters to obtain a local two-dimensional quaternionic phase. Theirgeneralization was obtained using the inverse (two-sided) quaternion Fourier ker-nel. A further extension of the quaternionic Gabor filter to quaternionic Gaborwavelets was introduced by Bayro-Corrochano [3] and Xi et al.. [20]. Hahn [12]constructed a Fourier-Wigner distribution of 2D quaternionic signals, which is infact closely related to the QWFT.

The WFT has been also studied in the quaternion algebra framework. Thegeneralization uses the kernel of the (right-sided) quaternion Fourier transform[18, 13] which was introduced in [2] and the kernel of the (two-sided) quaternionFourier transform recently proposed by Fu et al. in [9]. In [15], we applied the 𝐶ℓ𝑛,0Clifford windowed Fourier transform to linear time-varying systems. In this chap-ter, we continue the generalization of the quaternionic windowed Fourier transform

This work was partially supported by Bantuan Seminar Luar Negeri oleh DP2M DIKTI 2011,Indonesia.

286 M. Bahri

to the real Clifford algebra 𝐶ℓ0,𝑛 called the 𝐶ℓ0,𝑛 Clifford windowed Fourier trans-form (CWFT) which differs appreciably1 from the 𝐶ℓ𝑛,0 Clifford windowed Fouriertransform (see [17, 16]).

This chapter is organized as follows. A brief review of real Clifford algebrais given in § 2. § 3 introduces the Clifford–Fourier transform in 𝐶ℓ0,𝑛 (CFT) andderives its important properties. § 4 discusses the basic ideas for constructing theClifford windowed Fourier transform in 𝐶ℓ0,𝑛 (CWFT) using the kernel of theClifford–Fourier transform in 𝐶ℓ0,𝑛. We show that some properties of the two-dimensional quaternionic windowed Fourier transform (see [2]) are not valid in theCWFT such as the shift property and the Heisenberg uncertainty principle.

2. Preliminaries

We will be working with real Clifford algebras. Let {𝒆1, 𝒆2, 𝒆3, . . . , 𝒆𝑛} be an or-thonormal vector basis of the real 𝑛-dimensional Euclidean vector space ℝ𝑛. Thereal Clifford algebra over ℝ𝑛 denoted by 𝐶ℓ0,𝑛 then has the graded 2𝑛-dimensionalbasis

{1, 𝒆1, 𝒆2, . . . , 𝒆𝑛, 𝒆12, 𝒆31, 𝒆23, . . . , 𝑖𝑛 = 𝒆1𝒆2 ⋅ ⋅ ⋅ 𝒆𝑛}. (2.1)

Obviously, for 𝑛 = 2 (mod 4) the pseudoscalar 𝑖𝑛 = 𝒆1𝒆2 ⋅ ⋅ ⋅ 𝒆𝑛 anti-commuteswith each basis of the Clifford algebra while 𝑖2𝑛 = −1. The associative geometricmultiplication of the basis vectors is governed by the rules:

𝒆𝑘 𝒆𝑙 = −𝒆𝑙 𝒆𝑘 for 𝑘 ∕= 𝑙, 1 ≤ 𝑘, 𝑙 ≤ 𝑛,

𝒆2𝑘 = 1 for 1 ≤ 𝑘 ≤ 𝑛. (2.2)

An element of a Clifford algebra is called a multivector and has the following form

𝑓 =∑𝐴

𝒆𝐴𝑓𝐴, (2.3)

where 𝑓𝐴 ∈ ℝ, 𝒆𝐴 = 𝒆𝛼1𝛼2⋅⋅⋅𝛼𝑘= 𝒆𝛼1𝒆𝛼2 ⋅ ⋅ ⋅𝒆𝛼𝑘

, and 1 ≤ 𝛼1 ≤ 𝛼2 ≤ ⋅ ⋅ ⋅ ≤ 𝛼𝑘 ≤ 𝑛with 𝛼𝑗 ∈ {1, 2, . . . 𝑛}. For convenience, we introduce ⟨𝑓⟩𝑘 =

∑∣𝐴∣=𝑘 𝑓𝐴𝒆𝐴 to

denote the 𝑘-vector part of 𝑓 (𝑘 = 0, 1, 2, . . . , 𝑛), then

𝑓 =

𝑘=𝑛∑𝑘=0

⟨𝑓⟩𝑘 = ⟨𝑓⟩+ ⟨𝑓⟩1 + ⟨𝑓⟩2 + ⋅ ⋅ ⋅+ ⟨𝑓⟩𝑛, (2.4)

where ⟨. . .⟩0 = ⟨. . .⟩.The Clifford conjugate 𝑓 of a multivector 𝑓 is defined as the anti-auto-

morphism for which

𝑓 =∑

𝐴𝒆𝐴𝑓𝐴, 𝒆𝐴 = (−1)𝑘(𝑘+1)/2, 𝒆𝑘 = −𝒆𝑘, 𝑘 = 1, 2, 3, . . . , 𝑛. (2.5)

1The CWFT presented here is constructed using the kernel of the 𝐶ℓ0,𝑛 Clifford–Fourier trans-form whose properties are quite different from the kernel of the 𝐶ℓ𝑛,0 Clifford–Fourier transform

(see [1, 14]).

14. Clifford Windowed Fourier Transform 287

and hence

𝑓𝑔 = 𝑔𝑓 for arbitrary 𝑓, 𝑔 ∈ 𝐶ℓ0,𝑛. (2.6)

The scalar product of multivectors 𝑓 and 𝑔 and its associated norm is definedby, respectively,

⟨𝑓 𝑔⟩ = 𝑓 ∗ 𝑔 =∑

𝐴𝑓𝐴𝑔𝐵, and ∣𝑓 ∣2 =

∑𝐴𝑓2𝐴. (2.7)

The product of two Clifford multivectors 𝒙 =∑𝑛

𝑖=1 𝑥𝑖𝒆𝑖 and 𝒚 =∑𝑛

𝑗=1 𝑦𝑗𝒆𝑗 splitsinto a scalar part and a 2-vector or so-called bivector part

𝒙𝒚 = 𝒙 ⋅ 𝒚 + 𝒙 ∧ 𝒚 (2.8)

where

𝒙 ⋅ 𝒚 =

𝑛∑𝑖=1

𝑥𝑖𝑦𝑖, and 𝒙 ∧ 𝒚 =

𝑛∑𝑖=1

𝑛∑𝑗=𝑖+1

𝒆𝑖𝒆𝑗(𝑥𝑖𝑦𝑗 − 𝑥𝑗𝑦𝑖). (2.9)

In the following we give the definition of a 𝐶ℓ0,𝑛-valued function.

Definition 2.1. Let Ω ⊂ ℝ𝑛 be an open connected set. Functions 𝑓 defined in Ωwith values in 𝐶ℓ0,𝑛 can be expressed as:

𝑓 : Ω −→ 𝐶ℓ0,𝑛.

They are of the form:

𝑓(𝒙) =∑

𝐴𝒆𝐴𝑓𝐴(𝒙), (2.10)

where 𝑓𝐴 are real-valued functions in Ω.

More specifically, we let 𝐿𝑝(Ω;𝐶ℓ0,𝑛), 1 ≤ 𝑝 < ∞ and 𝐿∞(Ω;𝐶ℓ0,𝑛) denotethe usual Lebesgue space of integrable or essentially bound 𝐶ℓ0,𝑛-valued functionon Ω. Notice that 𝐿𝑝(Ω;𝐶ℓ0,𝑛) is a 𝐶ℓ0,𝑛-bimodule. Moreover, it can be provedto be a Banach module. The norms on the space are denoted ∥⋅∥𝐿𝑝(ℝ𝑛;𝐶ℓ0,𝑛) and

∥⋅∥𝐿∞𝑝(ℝ𝑛;𝐶ℓ0,𝑛), respectively. The set of 𝐶𝑘-functions in Ω with values in 𝐶ℓ0,𝑛 is

denoted by

𝐶𝑘(Ω;𝐶ℓ0,𝑛) = {𝑓 ∣𝑓 : Ω −→ 𝐶ℓ0,𝑛, 𝑓(𝒙) =∑𝐴

𝒆𝐴𝑓𝐴(𝒙)} (2.11)

Notice that if 𝑓𝐴 ∈ 𝐶𝑘(Ω) then we say 𝑓 ∈ 𝐶𝑘(Ω;𝐶ℓ0,𝑛).Let us consider 𝐿2(ℝ𝑛;𝐶ℓ0,𝑛) as a left module. For two multivector functions

𝑓, 𝑔 ∈ 𝐿2(ℝ𝑛;𝐶ℓ0,𝑛), an inner product is defined by

(𝑓, 𝑔)𝐿2(ℝ𝑛;𝐶ℓ0,𝑛) =

∫ℝ𝑛

𝑓(𝒙)𝑔(𝒙) 𝑑𝑛𝒙 =∑

𝐴,𝐵𝒆𝐴𝒆𝐵

∫ℝ𝑛

𝑓𝐴(𝒙)𝑔𝐵(𝒙) 𝑑𝑛𝒙 (2.12)

In particular, if 𝑓 = 𝑔, then the scalar part of the above inner product gives the𝐿2-norm

∥𝑓∥2𝐿2(ℝ𝑛;𝐶ℓ0,𝑛) =

∫ℝ𝑛

∑𝐴𝑓2𝐴(𝒙) 𝑑

𝑛𝒙 (2.13)

288 M. Bahri

3. Clifford–Fourier Transform (CFT)

In the following, we introduce the Clifford–Fourier transform (CFT) (see [8, 5, 19]).We can regard the CFT as an alternative representation of the classical tensorialFourier transform, i.e., we apply a one-dimensional Fourier transform 𝑛 times(each time with a different imaginary unit).

3.1. Definition of CFT

Definition 3.1. The CFT of a multivector function 𝑓 ∈ 𝐿1(ℝ𝑛;𝐶ℓ0,𝑛) is the func-tion ℱ{𝑓}: ℝ𝑛 → 𝐶ℓ0,𝑛 given by

ℱ{𝑓}(𝝎) =∫ℝ𝑛

𝑓(𝒙)

𝑛∏𝑘=1

𝑒−𝒆𝑘𝜔𝑘𝑥𝑘𝑑𝑛𝒙, (3.1)

with 𝝎,𝒙 ∈ ℝ𝑛.

Note that

𝑑𝑛𝒙 =𝑑𝒙1 ∧ 𝑑𝒙2 ∧ ⋅ ⋅ ⋅ ∧ 𝑑𝒙𝑛

𝑖𝑛(3.2)

and is scalar valued (𝑑𝒙𝑘 = 𝑑𝑥𝑘𝒆𝑘, 𝑘 = 1, 2, 3, . . . , 𝑛, no summation). Notice alsothat the Clifford–Fourier kernel

∏𝑛𝑘=1 𝑒

−𝒆𝑘𝜔𝑘𝑥𝑘 in general does not commute withelements of 𝐶ℓ0,𝑛. Furthermore, the product has to be performed in a fixed order.

The existence of the inverse CFT is given by the following theorem. For moredetail and for proofs see [5, 7].

Theorem 3.2. Suppose that 𝑓 ∈ 𝐿1(ℝ𝑛;𝐶ℓ0,𝑛) and ℱ{𝑓} ∈ 𝐿1(ℝ𝑛;𝐶ℓ0,𝑛). Thenthe CFT is invertible and its inverse is calculated by

𝑓(𝒙) =1

(2𝜋)𝑛

∫ℝ𝑛

ℱ{𝑓}(𝝎)𝑛−1∏𝑘=0

𝑒𝒆𝑛−𝑘𝜔𝑛−𝑘𝑥𝑛−𝑘 𝑑𝑛𝝎. (3.3)

The CFT is a generalization of the quaternionic Fourier transform (QFT),so most of the properties of the QFT such as the shift property, convolution, andPlancherel’s theorem, have their corresponding CFT generalizations. Observe thatthis Fourier transform can be extended to the whole of 𝐿2(ℝ𝑛;𝐶ℓ0,𝑛) in the usualway by considering it in the weak or distributional sense.

The following subsection investigates some important properties of the CFT,which will be necessary to establish the Clifford windowed Fourier transform in𝐶ℓ0,𝑛 (CWFT).

3.2. Main Properties of CFT

We first establish a scalar Plancherel theorem. It states that for every 𝑓 ∈𝐿2(ℝ𝑛;𝐶ℓ0,𝑛)

∥𝑓∥2𝐿2(ℝ𝑛;𝐶ℓ0,𝑛) =1

(2𝜋)𝑛∥ℱ{𝑓}∥2𝐿2(ℝ𝑛;𝐶ℓ0,𝑛) . (3.4)

This shows that the total signal energy computed in the spatial domain is equal tothe total signal energy computed in the Clifford domain. The Plancherel theorem

14. Clifford Windowed Fourier Transform 289

allows the energy of a Clifford-valued signal to be considered in either the spatialdomain or the Clifford domain and the change of domains for convenience ofcomputation.

Let us now formulate the Clifford–Parseval theorem, which is needed to provethe orthogonality relation of the CWFT.

Theorem 3.3 (CFT Parseval). The inner product (2.12) of two Clifford functions𝑓, 𝑔 ∈ 𝐿2(ℝ𝑛;𝐶ℓ0,𝑛) and their CFTs are related by

(𝑓, 𝑔)𝐿2(ℝ𝑛;𝐶ℓ0,𝑛) =1

(2𝜋)𝑛(ℱ{𝑓},ℱ{𝑔})𝐿2(ℝ𝑛;𝐶ℓ0,𝑛). (3.5)

Proof. We have

(𝑓, 𝑔)𝐿2(ℝ𝑛;𝐶ℓ0,𝑛)

=

∫ℝ𝑛

𝑓(𝒙)𝑔(𝒙) 𝑑𝑛𝒙

(3.3)=

1

(2𝜋)𝑛

∫ℝ𝑛

[∫ℝ𝑛

ℱ{𝑓}(𝝎)𝑛−1∏𝑘=0

𝑒𝒆𝑛−𝑘𝜔𝑛−𝑘𝑥𝑛−𝑘 𝑑𝑛𝝎

]𝑔(𝒙)𝑑𝑛𝒙

(2.6)=

1

(2𝜋)𝑛

∫ℝ𝑛

ℱ{𝑓}(𝝎)[∫

ℝ𝑛

𝑔(𝒙)

𝑛∏𝑘=1

𝑒−𝒆𝑘𝜔𝑘𝑥𝑘 𝑑𝑛𝒙

]𝑑𝑛𝝎

=1

(2𝜋)𝑛

∫ℝ𝑛

ℱ{𝑓}(𝝎)ℱ{𝑔}(𝝎) 𝑑𝑛𝝎

=1

(2𝜋)𝑛(ℱ{𝑓},ℱ{𝑔})𝐿2(ℝ𝑛;𝐶ℓ0,𝑛).

This proves the theorem. □

Notice that equation (3.5) is multivector valued. In particular, with 𝑓 = 𝑔,we get the multivector version of the Plancherel theorem, i.e.,

(𝑓, 𝑓)𝐿2(ℝ𝑛;𝐶ℓ0,𝑛) =1

(2𝜋)𝑛(ℱ{𝑓},ℱ{𝑓})𝐿2(ℝ𝑛;𝐶ℓ0,𝑛). (3.6)

Due to the non-commutativity of the Clifford exponential product factors we havea left linearity property for general linear combinations with Clifford constantsand a scaling property.

Theorem 3.4 (Left linearity property). The CFT of two functions in the Cliffordmodule 𝑓, 𝑔 ∈ 𝐿1(ℝ𝑛;𝐶ℓ0,𝑛) is a left linear operator 2, i.e.,

ℱ{𝛼𝑓 + 𝛽𝑔}(𝝎) = 𝛼ℱ{𝑓}(𝝎) + 𝛽ℱ{𝑔}(𝝎), (3.7)

where 𝛼 and 𝛽 ∈ 𝐶ℓ0,𝑛 are Clifford constants.

2The CFT is also right linear for real constants 𝜇, 𝜆 ∈ ℝ.

290 M. Bahri

Theorem 3.5 (Scaling property). Suppose 𝑓 ∈ 𝐿1(ℝ𝑛;𝐶ℓ0,𝑛). Let 𝑎 be a positivereal constant. Then the CFT of the function 𝑓𝑎(𝒙) = 𝑓(𝑎𝒙) is

ℱ{𝑓𝑎}(𝝎) = 1

𝑎𝑛ℱ{𝑓}

(𝝎𝑎

). (3.8)

Remark 3.6. The usual form of the shift and modulation properties of the complexFT does not hold for the CFT because of the non-commutativity of the Clifford–Fourier kernel

𝑛∏𝑘=1

𝑒𝒆𝑘𝜔𝑘𝑥𝑘

𝑛−1∏𝑘−0

𝑒𝒆𝑛−𝑘𝜔𝑛−𝑘𝑥𝑛−𝑘 ∕=𝑛−1∏𝑘=0

𝑒𝒆𝑛−𝑘𝜔𝑛−𝑘𝑥𝑛−𝑘

𝑛∏𝑘=1

𝑒𝒆𝑘𝜔𝑘𝑥𝑘 . (3.9)

The following properties are extensions of the QFT, which are very usefulin solving partial differential equations in Clifford algebra. First, let us give anexplicit proof of the derivative properties stated in Table 1.

Table 1. Properties of the CFT of Clifford functions𝑓, 𝑔 ∈ 𝐿2(ℝ𝑛;𝐶ℓ0,𝑛).The constants are 𝛼, 𝛽 ∈ 𝐶ℓ0,𝑛, 𝑎 ∈ ℝ ∖ {0}, and 𝑛 ∈ ℕ.

Property Clifford Function Clifford–Fourier Transform

Left linearity 𝛼𝑓(𝒙)+𝛽𝑔(𝒙) 𝛼ℱ{𝑓}(𝝎)+ 𝛽ℱ{𝑔} (𝝎)

Scaling 𝑓(𝑎𝒙) 1

𝑎𝑛ℱ{𝑓}(𝝎𝑎 )

Partial derivative ∂𝑛

∂𝑥𝑛1𝑓(𝒙) 𝒆−𝑛1 𝜔𝑛

1ℱ{𝑓}(𝝎)∂𝑛

∂𝑥𝑛1𝑓(𝒙) (𝒆1𝜔1)

𝑛ℱ{𝑓}(𝝎),𝑓 = 𝑓0 + 𝒆1𝑓1 + 𝑖𝑛𝑓123⋅⋅⋅𝑛if 𝑛 = 3 (mod 4)

∂𝑚𝑓∂𝑥𝑚

𝑟(𝜔𝑟𝒆𝑟)

𝑚ℱ{𝑓}(𝝎),𝑟 = 2, . . . , 𝑛− 1,𝑚 = 2𝑠, 𝑠 ∈ ℕ

∂𝑚𝑓∂𝑥𝑚

𝑛ℱ{𝑓}(𝝎)(𝒆𝑛𝜔𝑛)

𝑚

Power 𝑓(𝒙)(𝑥1𝒆1)𝑚 (−1)𝑚 ∂𝑚

∂𝜔𝑚1ℱ{𝑓}(𝝎)

𝑓(𝒙)(𝑥𝑟𝒆𝑟)𝑚 ∂𝑚

∂𝜔𝑚𝑟ℱ{𝑓}(𝝎),

𝑟 = 2, . . . , 𝑛− 1,𝑚 = 2𝑠, 𝑠 ∈ ℕ

𝑓(𝒙)𝑥𝑚𝑛∂𝑚

∂𝜔𝑚𝑛ℱ{𝑓}(𝝎)𝒆𝑚𝑛

Plancherel (𝑓, 𝑔)𝐿2(ℝ𝑛;𝐶ℓ0,𝑛) =1

(2𝜋)𝑛 (ℱ{𝑓},ℱ{𝑓})𝐿2(ℝ𝑛;𝐶ℓ0,𝑛)

Scalar Parseval 1(2𝜋)𝑛 ∥𝑓∥2𝐿2(ℝ𝑛;𝐶ℓ0,𝑛) =

1(2𝜋)𝑛 ∥ℱ{𝑓}∥2𝐿2(ℝ𝑛;𝐶ℓ0,𝑛)

14. Clifford Windowed Fourier Transform 291

Theorem 3.7. Suppose 𝑓 ∈ 𝐿1(ℝ𝑛;𝐶ℓ0,𝑛). Then the CFT of the 𝑛th partial deriv-ative of 𝑥1𝑓 ∈ 𝐿1(ℝ𝑛;𝐶ℓ0,𝑛) with respect to the variable 𝑥1 is given by

ℱ{∂𝑛𝑓

∂𝑥𝑛1𝒆−𝑛1

}(𝝎) = 𝜔𝑛

1ℱ{𝑓}(𝝎), ∀𝑛 ∈ ℕ (3.10)

and if 𝑥𝑟𝑓 ∈ 𝐿1(ℝ𝑛;𝐶ℓ0,𝑛), then for 𝑟 = 2, 3, . . . ,ℕ− 1 we have

ℱ{∂𝑚𝑓

∂𝑥𝑚𝑟

}(𝝎) =

∫ℝ𝑛

𝑓(𝒙) 𝑒−𝒆1𝜔1𝑥1𝑒−𝒆2𝜔2𝑥2 ⋅ ⋅ ⋅ (𝜔𝑟 𝒆𝑟)𝑚

×𝑛∏

𝑘=𝑟

𝑒−𝒆𝑘𝜔𝑘𝑥𝑘 𝑑𝑛𝒙, 𝑚 = 2𝑠+ 1, 𝑠 ∈ ℕ, (3.11)

and

ℱ{∂𝑚𝑓

∂𝑥𝑚𝑟

}(𝝎) = (𝜔𝑟𝒆𝑟)

𝑚ℱ{𝑓}(𝝎), 𝑚 = 2𝑠, 𝑠 ∈ ℕ. (3.12)

Proof. We only prove (3.10) of Theorem 3.7, the others being similar. In this proof,we first prove the theorem for 𝑛 = 1. Applying integration by parts and using thefact that 𝑓 tends to zero for 𝑥1 →∞ we immediately obtain

ℱ{

∂

∂𝑥1𝑓 𝒆−1

1

}(𝝎)

=

∫ℝ𝑛

(∂

∂𝑥1𝑓(𝒙) 𝒆−1

1

) 𝑛∏𝑘=1

𝑒−𝒆𝑘𝜔𝑘𝑥𝑘 𝑑𝑛𝒙

=

∫ℝ𝑛−1

[∫ℝ

(∂

∂𝑥1𝑓(𝒙) 𝒆−1

1

)𝑒−𝒆1𝜔1𝑥1 𝑑𝑥1

] 𝑛∏𝑘=2

𝑒−𝒆𝑘𝜔𝑘𝑥𝑘 𝑑𝑛−1𝒙

=

∫ℝ𝑛−1

[𝑓(𝒙) 𝒆−1

1 𝑒−𝒆1𝜔1𝑥1 ∣𝑥1=∞𝑥1=−∞

−∫ℝ

𝑓(𝒙) 𝒆−11

∂

∂𝑥1𝑒−𝒆1𝜔1𝑥1𝑑𝑥1

] 𝑛∏𝑘=2

𝑒−𝒆𝑘𝜔𝑘𝑥𝑘 𝑑𝑛−1𝒙

=

∫ℝ𝑛

𝑓(𝒙)𝜔1

𝑛∏𝑘=1

𝑒−𝒆𝑘𝜔𝑘𝑥𝑘 𝑑𝑛𝒙

= 𝜔1ℱ{𝑓}(𝝎). (3.13)

In a similar way we get

ℱ{

∂

∂𝑥1𝑓

}(𝝎) = 𝜔1ℱ{𝑓𝒆1}(𝝎). (3.14)

292 M. Bahri

For 𝑛 = 2 we obtain

ℱ{

∂2

∂𝑥21

𝑓 𝒆−21

}(𝝎)

=

∫ℝ𝑛

(∂

∂𝑥1(

∂

∂𝑥1𝑓(𝒙)) 𝒆−2

1

) 𝑛∏𝑘=1

𝑒−𝒆𝑘𝜔𝑘𝑥𝑘 𝑑𝑛𝒙

=

∫ℝ𝑛−1

[∫ℝ

(∂

∂𝑥1(

∂

∂𝑥1𝑓(𝒙)) 𝒆−2

1

)𝑒−𝒆1𝜔1𝑥1 𝑑𝑥1

] 𝑛∏𝑘=2

𝑒−𝒆𝑘𝜔𝑘𝑥𝑘 𝑑𝑛−1𝒙

=

∫ℝ𝑛−1

[(

∂

∂𝑥1𝑓(𝒙)) 𝒆−2

1 𝑒−𝒆1𝜔1𝑥1 ∣𝑥1=∞𝑥1=−∞

−𝜔1

∫ℝ

∂

∂𝑥1𝑓(𝒙) 𝒆−1

1 𝑒−𝒆1𝜔1𝑥1𝑑𝑥1

] 𝑛∏𝑘=2

𝑒−𝒆𝑘𝜔𝑘𝑥𝑘 𝑑𝑛−1𝒙

= 𝜔21

∫ℝ𝑛

𝑓(𝒙)𝑛∏

𝑘=1

𝑒−𝒆𝑘𝜔𝑘𝑥𝑘 𝑑𝑛𝒙

= 𝜔21ℱ{𝑓}(𝝎). (3.15)

By repeating this process 𝑛−2 additional times we finish the proof of Theorem 3.7.□

Remark 3.8. Observe that when we assume that 𝑓 = 𝑓0 + 𝒆1𝑓1 + 𝑖𝑛𝑓123⋅⋅⋅𝑛, 𝑛 = 3(mod 4), then equation (3.14) takes the form

ℱ{∂𝑛𝑓

∂𝑥𝑛1

}(𝝎) = (𝒆1𝜔1)

𝑛ℱ{𝑓}(𝝎), 𝑛 ∈ ℕ. (3.16)

Theorem 3.9. Let 𝑥𝑛𝑓 ∈ 𝐿1(ℝ𝑛;𝐶ℓ0,𝑛). Then the CFT of the 𝑚th partial derivativeof a Clifford-valued function 𝑓 ∈ 𝐿1(ℝ𝑛;𝐶ℓ0,𝑛) with respect to the variable 𝑥𝑛 isgiven by

ℱ{∂𝑚𝑓

∂𝑥𝑚𝑛

}(𝝎) = ℱ{𝑓}(𝝎)(𝒆𝑛𝜔𝑛)

𝑚, 𝑚 ∈ ℕ. (3.17)

Proof. For 𝑚 = 1 direct calculation gives

∂𝑓(𝒙)

∂𝑥𝑛=

∂

∂𝑥𝑛

1

(2𝜋)𝑛

∫ℝ𝑛

ℱ{𝑓}(𝝎)𝑛−1∏𝑘=0

𝑒𝒆𝑛−𝑘𝜔𝑛−𝑘𝑥𝑛−𝑘 𝑑𝑛𝝎

=1

(2𝜋)𝑛

∫ℝ𝑛

ℱ{𝑓}(𝝎)(

∂

∂𝑥𝑛𝑒𝒆𝑛𝜔𝑛𝑥𝑛

) 𝑛−1∏𝑘=1

𝑒𝒆𝑛−𝑘𝜔𝑛−𝑘𝑥𝑛−𝑘 𝑑𝑛𝝎

=1

(2𝜋)𝑛

∫ℝ𝑛

[ℱ{𝑓}(𝝎) 𝒆𝑛𝜔𝑛]𝑛−1∏𝑘=0

𝑒𝒆𝑛−𝑘𝜔𝑛−𝑘𝑥𝑛−𝑘 𝑑𝑛𝝎

= ℱ−1 [ℱ{𝑓}(𝝎) 𝒆𝑛𝜔𝑛] . (3.18)

14. Clifford Windowed Fourier Transform 293

We therefore get

ℱ{

∂𝑓

∂𝑥𝑛

}(𝝎) = ℱ{𝑓}(𝝎)𝒆𝑛𝜔𝑛. (3.19)

By successive differentiation with respect to the variable 𝑥𝑛 and by induction weeasily obtain

ℱ{∂𝑚𝑓

∂𝑥𝑚𝑛

}(𝝎) = ℱ{𝑓}(𝝎)(𝒆𝑛𝜔𝑛)

𝑚, ∀𝑚 ∈ ℕ. (3.20)

This ends the proof of (3.17). □

Next we derive the power properties of the CFT stated in Table 1.

Theorem 3.10. If we assume that 𝑥1𝑓 ∈ 𝐿1(ℝ𝑛;𝐶ℓ0,𝑛). Then the CFT of the 𝑛thpartial derivative of 𝑓 ∈ 𝐿1(ℝ𝑛;𝐶ℓ0,𝑛) with respect to the variable 𝑥1 is given by

ℱ{𝑓(𝒙)(𝑥1𝒆1)𝑛}(𝝎) = (−1)𝑛 ∂𝑛

∂𝜔𝑛1

ℱ{𝑓}(𝝎), ∀𝑛 ∈ ℕ. (3.21)

If 𝑥𝑚𝑟 𝑓 ∈ 𝐿1(ℝ𝑛;𝐶ℓ0,𝑛), then for 𝑟 = 2, 3, . . . , 𝑛− 1 we have

∂𝑚

∂𝜔𝑚𝑟

ℱ{𝑓}(𝝎) =∫ℝ𝑛

𝑓(𝒙) 𝑒−𝒆1𝜔1𝑥1𝑒−𝒆2𝜔2𝑥2 ⋅ ⋅ ⋅ (−𝒆𝑟𝑥𝑟)𝑚

×𝑛∏

𝑘=𝑟

𝑒−𝒆𝑘𝜔𝑘𝑥𝑘 𝑑𝑛𝒙,𝑚 = 2𝑠+ 1, 𝑠 ∈ ℕ, (3.22)

and for 𝑚 = 2𝑠, 𝑠 ∈ ℕ

ℱ{(𝒆𝑟𝑥𝑟)𝑚𝑓}(𝝎) = ∂𝑚

∂𝜔𝑚𝑟

ℱ{𝑓}(𝝎). (3.23)

If 𝑥𝑛𝑓 ∈ 𝐿1(ℝ𝑛;𝐶ℓ0,𝑛), then

ℱ{𝑥𝑚𝑛 𝑓}(𝝎) = ∂𝑚

∂𝜔𝑚𝑛

ℱ{𝑓}(𝝎) 𝒆𝑚𝑛 , 𝑚 = 1, 2, 3, . . . , 𝑛. (3.24)

Proof. We only prove (3.21) of Theorem 3.10. It is not difficult to check that

𝑓(𝒙)(𝑥1𝒆1)𝑛

𝑛∏𝑘=1

𝑒−𝒆𝑘𝜔𝑘𝑥𝑘 = (−1)𝑛 ∂𝑛

∂𝜔𝑛1

𝑓(𝒙)𝑛∏

𝑘=1

𝑒−𝒆𝑘𝜔𝑘𝑥𝑘 . (3.25)

We immediately obtain∫ℝ𝑛

𝑓(𝒙)(𝑥1𝒆1)𝑛

𝑛∏𝑘=1

𝑒−𝒆𝑘𝜔𝑘𝑥𝑘 𝑑𝑛𝒙 = (−1)𝑛 ∂𝑛

∂𝜔𝑛1

∫ℝ𝑛

𝑓(𝒙)

𝑛∏𝑘=1

𝑒−𝒆𝑘𝜔𝑘𝑥𝑘 𝑑𝑛𝒙,

(3.26)which gives the desired result. □

294 M. Bahri

4. Clifford Windowed Fourier Transform (CWFT)

In this section, we introduce the Clifford windowed Fourier transform as a gener-alization of the two-dimensional quaternionic Fourier transform to higher dimen-sions. For this let us define the Clifford–Gabor filter, which is a special case of theClifford atom operator [9].

4.1. Two-dimensional Clifford–Gabor Filters

The two-dimensional Clifford–Gabor filter3 is the extension of the complex Gaborfilter to the two-dimensional Clifford algebra. It takes the form

𝐺𝑐(𝒙, 𝜎1, 𝜎2) = 𝑒𝒆2𝑢0𝑥1𝑒𝒆1𝑣0𝑥2𝑔(𝒙,𝝈)

= 𝑒𝒆2𝑢0𝑥1𝑒𝒆1𝑣0𝑥2𝑒−[(𝑥1/𝜎1)2+(𝑥2/𝜎2)

2]/2. (4.1)

Equation (4.1) is often called thequaternionic Gabor filter. Bulow and Sommer[6, 7] have applied it to get the local quaternionic phase of a two-dimensional realsignal. From this, we get the following facts:

∙ It is generated using the kernel of the 𝐶𝑙(0, 2) CFT.∙ If the Gaussian function 𝑔(𝒙,𝝈) is replaced by the Clifford window function

𝜙(𝒙− 𝒃), then it becomes the Clifford atom operator, i.e.,

𝜙𝝎,𝒃(𝒙) = 𝑒𝒆2𝑣0𝑥2𝑒𝒆1𝑢0𝑥1𝜙(𝒙− 𝒃), 𝒙, 𝒃 ∈ ℝ2. (4.2)

∙ Since the modulation property does not hold for the CFT, (4.2) can not beexpressed in terms of the CFT.

4.2. Definition of CWFT

Definition 4.1. The CWFT of a multivector function 𝑓 ∈ 𝐿2(ℝ𝑛;𝐶ℓ0,𝑛) withrespect to the non-zero Clifford window function 𝜙 ∈ 𝐿2(ℝ𝑛;𝐶ℓ0,𝑛) such that

∣𝒙∣1/2 𝜙(𝒙) ∈ 𝐿2(ℝ𝑛;𝐶ℓ0,𝑛) is given by

𝐺𝜙𝑓(𝝎, 𝒃) = (𝑓, 𝜙𝝎,𝒃)𝐿2(ℝ𝑛;𝐶ℓ0,𝑛)

=

∫ℝ𝑛

𝑓(𝒙)𝜙𝝎,𝒃(𝒙) 𝑑𝑛𝒙

=

∫ℝ𝑛

𝑓(𝒙) 𝜙(𝒙− 𝒃)

𝑛∏𝑘=1

𝑒−𝒆𝑘𝜔𝑘𝑥𝑘 𝑑𝑛𝒙. (4.3)

We then call

𝜙𝝎,𝒃(𝒙) =

𝑛−1∏𝑘=0

𝑒𝒆𝑛−𝑘𝜔𝑛−𝑘𝑥𝑛−𝑘𝜙(𝒙− 𝒃), (4.4)

the atom operator as the kernel of the CWFT in (4.3). Notice that for 𝑛 = 2 theCWFT above is identical to the two-dimensional quaternionic windowed Fourier

3Here we start with the Clifford–Gabor filter to obtain the Clifford atom operator, which isneeded to construct the Clifford windowed Fourier transform.

14. Clifford Windowed Fourier Transform 295

transform (see [2]) and for 𝑛 = 1 is the classical windowed Fourier transform (see[10, 11]).

Example. Consider the Clifford Gaussian window 𝜙 ∈ 𝐿2(ℝ2;𝐶ℓ0,2) given by:

𝜙(𝒙) = (2 + 𝒆1 + 𝒆2 − 𝒆12)𝑒−(𝑥2

1+𝑥22). (4.5)

Thus we obtain the Clifford window daughter functions (4.4) of the form

𝜙𝝎,𝒃(𝒙) = {𝑒𝒆2𝜔2𝑥2𝑒𝒆1𝜔1𝑥1(2 + 𝒆1 + 𝒆2 − 𝒆12)} 𝑒−((𝑥1−𝑏1)2+(𝑥2−𝑏2)2)

= {(2𝑒𝒆2𝜔2𝑥2𝑒𝒆1𝜔1𝑥1 + 𝒆1𝑒−𝒆2𝜔2𝑥2𝑒𝒆1𝜔1𝑥1 + 𝒆2𝑒

𝒆2𝜔2𝑥2𝑒−𝒆1𝜔1𝑥1

− 𝒆12𝑒−𝒆2𝜔2𝑥2𝑒−𝒆1𝜔1𝑥1)}𝑒−((𝑥1−𝑏1)2+(𝑥2−𝑏2)2). (4.6)

We first notice that, for fixed 𝒃,

𝐺𝜙𝑓(𝝎, 𝒃) = ℱ{𝑓 𝜙(⋅ − 𝒃)}(𝝎) = ℱ{𝑓 𝑇𝒃𝜙}(𝝎), (4.7)

where 𝑇𝒃 is the translation operator defined by 𝑇𝒃𝑓 = 𝑓(𝒙−𝒃). It thus means thatthe CWFT can be regarded as the CFT of the product of a multivector-valuedfunction 𝑓 and a shifted and Clifford reversion of the Clifford atom operator (4.4).

4.3. Properties of the CWFT

The following proposition describes the elementary properties of the CWFT. Itsproof can be easily obtained.

Proposition 4.2. Let 𝜙 ∈ 𝐿2(ℝ𝑛;𝐶ℓ0,𝑛) be a Clifford window function.

Left linearity:

[𝐺𝜙(𝜆𝑓 + 𝜇𝑔)](𝝎, 𝒃) = 𝜆𝐺𝜙𝑓(𝝎, 𝒃) + 𝜇𝐺𝜙𝑔(𝝎, 𝒃), (4.8)

for arbitrary Clifford constants 𝜆, 𝜇 ∈ 𝐶ℓ0,𝑛.

Parity:𝐺𝑃𝜙(𝑃𝑓)(𝝎, 𝒃) = 𝐺𝜙𝑓(𝝎,−𝒃), (4.9)

where 𝑃 is the parity operator defined by 𝑃𝑓(𝑥) = 𝑓(−𝑥).

Theorem 4.3 (Orthogonality relation). Let 𝜙, 𝜓 be Clifford window functions and𝑓, 𝑔 ∈ 𝐿2(ℝ𝑛;𝐶ℓ0,𝑛) be arbitrary. Then we have∫

ℝ𝑛

∫ℝ𝑛

𝐺𝜙𝑓(𝝎, 𝒃)𝐺𝜓𝑔(𝝎, 𝒃) 𝑑𝑛𝝎 𝑑𝑛𝒃

= (2𝜋)𝑛(𝑓(𝜙, 𝜓)𝐿2(ℝ𝑛;𝐶ℓ0,𝑛), 𝑔)𝐿2(ℝ𝑛;𝐶ℓ0,𝑛). (4.10)

Proof. Applying (4.7) we have∫ℝ𝑛

𝐺𝜙𝑓(𝝎, 𝒃)𝐺𝜓𝑔(𝝎, 𝒃) 𝑑𝑛𝝎 =

∫ℝ𝑛

ℱ{𝑓𝑇𝒃𝜙}ℱ{𝑔𝑇𝒃𝜓} 𝑑𝑛𝝎

=(ℱ{𝑓𝑇𝒃𝜙},ℱ{𝑔𝑇𝒃𝜓}

)𝐿2(ℝ𝑛;𝐶ℓ0,𝑛)

. (4.11)

We assume that Clifford windows 𝜙, 𝑔 ∈ 𝐿1(ℝ𝑛;𝐶ℓ0,𝑛)∩

𝐿∞(ℝ𝑛;𝐶ℓ0,𝑛) so that

𝑓𝑇𝒃𝜙, 𝑔𝑇𝒃𝜓 ∈ 𝐿2(ℝ𝑛;𝐶ℓ0,𝑛). We know [6, 13] that Parseval’s theorem is valid for

296 M. Bahri

the CFT. So, applying it to the right-hand side of (4.11) we easily get (compareto Grochenig [10])∫

ℝ𝑛

𝐺𝜙𝑓(𝝎, 𝒃)𝐺𝜓𝑔(𝝎, 𝒃) 𝑑𝑛𝝎 = (ℱ{𝑓𝑇𝒃𝜙},ℱ{𝑔𝑇𝒃𝜓})𝐿2(ℝ𝑛;𝐶ℓ0,𝑛)

= (2𝜋)𝑛(𝑓𝑇𝒃𝜙, 𝑔𝑇𝒃𝜓

)𝐿2(ℝ𝑛;𝐶ℓ0,𝑛)

= (2𝜋)𝑛∫ℝ𝑛

𝑓(𝒙)𝜙(𝒙 − 𝒃)𝜓(𝒙− 𝒃)𝑔(𝒙) 𝑑𝑛𝒙. (4.12)

Observe that 𝑓𝜙 and 𝜓𝑔 are in 𝐿1(ℝ𝑛;𝐶ℓ0,𝑛). Then integrating (4.12) with respectto 𝑑𝑛𝒃 we immediately get∫

ℝ𝑛

∫ℝ𝑛

𝐺𝜙𝑓(𝝎, 𝒃)𝐺𝜓𝑔(𝝎, 𝒃) 𝑑𝑛𝝎 𝑑𝑛𝒃

= (2𝜋)𝑛∫ℝ𝑛

∫ℝ𝑛

𝑓(𝒙)𝜙(𝒙− 𝒃)𝜓(𝒙− 𝒃)𝑔(𝒙) 𝑑𝑛𝒙 𝑑𝑛𝒃

= (2𝜋)𝑛∫ℝ𝑛

∫ℝ𝑛

𝑓(𝒙)𝜙(𝒙− 𝒃)𝜓(𝒙− 𝒃) 𝑑𝑛𝒃 𝑔(𝒙) 𝑑𝑛𝒙

= (2𝜋)𝑛∫ℝ𝑛

𝑓(𝒙)

∫ℝ𝑛

𝜙(𝒙− 𝒃)𝜓(𝒙− 𝒃) 𝑑𝑛𝒃 𝑔(𝒙) 𝑑𝑛𝒙, (4.13)

where from the second to the third line of (4.13) we applied Fubini’s theorem tointerchange the order of integration. Using a standard density argument we cannow extend the result to the 𝐿2(ℝ𝑛;𝐶ℓ0,𝑛)-case. This proves the theorem. □

From the above theorem, we obtain the following consequences.

(i) If 𝜙 = 𝜓 and 𝐶𝜙 = (𝜙, 𝜙)𝐿2(ℝ𝑛;𝐶ℓ0,𝑛) is a multivector constant, then∫ℝ𝑛

∫ℝ𝑛

𝐺𝜙𝑓(𝝎, 𝒃)𝐺𝜙𝑔(𝝎, 𝒃) 𝑑𝑛𝒃 𝑑𝑛𝝎 = (2𝜋)𝑛⟨𝐶𝜙⟩(𝑓, 𝑔)𝐿2(ℝ𝑛;𝐶ℓ0,𝑛)

+ (2𝜋)𝑛(𝑓⟨𝐶𝜙⟩1, 𝑔)𝐿2(ℝ𝑛;𝐶ℓ0,𝑛) + ⋅ ⋅ ⋅+ (2𝜋)𝑛(𝑓⟨𝐶𝜙⟩𝑛, 𝑔)𝐿2(ℝ𝑛;𝐶ℓ0,𝑛). (4.14)

(ii) If 𝑓 = 𝑔 is a paravector, then (4.14) reduces to∫ℝ𝑛

∫ℝ𝑛

𝐺𝜙𝑓(𝝎, 𝒃)𝐺𝜙𝑓(𝝎, 𝒃) 𝑑𝑛𝒃 𝑑𝑛𝝎 = (2𝜋)𝑛⟨𝐶𝜙⟩ ∥𝑓∥2𝐿2(ℝ𝑛;𝐶ℓ0,𝑛)

+ (2𝜋)𝑛(𝑓⟨𝐶𝜙⟩1, 𝑓)𝐿2(ℝ𝑛;𝐶ℓ0,𝑛) + ⋅ ⋅ ⋅+ (2𝜋)𝑛(𝑓⟨𝐶𝜙⟩𝑛, 𝑓)𝐿2(ℝ𝑛;𝐶ℓ0,𝑛). (4.15)

Theorem 4.4 (Reconstruction formula). Let 𝜙, 𝜓 ∈ 𝐿2(ℝ𝑛;𝐶ℓ0,𝑛) be two Clifford

window functions with (𝜙, 𝜓)𝐿2(ℝ𝑛;𝐶ℓ0,𝑛) ∕= 0. Then every 𝑛-D Clifford signal 𝑓 ∈𝐿2(ℝ𝑛;𝐶ℓ0,𝑛) can be fully reconstructed by

𝑓(𝒙) =1

(2𝜋)𝑛

∫ℝ𝑛

∫ℝ𝑛

𝐺𝜙𝑓(𝝎, 𝒃)𝜓𝝎,𝒃(𝒙) (𝜙, 𝜓)−1𝐿2(ℝ𝑛;𝐶ℓ0,𝑛

)𝑑𝑛𝒃 𝑑𝑛𝝎. (4.16)

14. Clifford Windowed Fourier Transform 297

Proof. By direct calculation, we obtain for every 𝑔 ∈ 𝐿2(ℝ𝑛;𝐶ℓ0,𝑛)∫ℝ𝑛

∫ℝ𝑛

𝐺𝜙𝑓(𝝎, 𝒃)𝐺𝜓𝑔(𝝎, 𝒃) 𝑑𝑛𝝎 𝑑𝑛𝒃

=

∫ℝ𝑛

∫ℝ𝑛

∫ℝ𝑛

𝐺𝜙𝑓(𝝎, 𝒃)𝜓𝝎,𝒃(𝒙)𝑔(𝒙) 𝑑𝑛𝝎 𝑑𝑛𝒃 𝑑𝑛𝒙

=

(∫ℝ𝑛

∫ℝ𝑛

𝐺𝜙𝑓(𝝎, 𝒃)𝜓𝝎,𝒃 𝑑𝑛𝝎 𝑑𝑛𝒃, 𝑔

)𝐿2(ℝ𝑛;𝐶ℓ0,𝑛)

. (4.17)

Applying equation (4.10) of Theorem 4.3 to the left-hand side of (4.17) gives forevery 𝑔 ∈ 𝐿2(ℝ𝑛;𝐶ℓ0,𝑛)

(2𝜋)𝑛(𝑓(𝜙, 𝜓)𝐿2(ℝ𝑛;𝐶ℓ0,𝑛), 𝑔)𝐿2(ℝ𝑛;𝐶ℓ0,𝑛)

=

(∫ℝ𝑛

∫ℝ𝑛

𝐺𝜙𝑓(𝝎, 𝒃)𝜓𝝎,𝒃 𝑑𝑛𝝎 𝑑𝑛𝒃, 𝑔

)𝐿2(ℝ𝑛;𝐶ℓ0,𝑛)

. (4.18)

Because the inner product identity (4.18) holds for every 𝑔 ∈ 𝐿2(ℝ𝑛;𝐶ℓ0,𝑛) weconclude that

(2𝜋)𝑛𝑓(𝜙, 𝜓)𝐿2(ℝ𝑛;𝐶ℓ0,𝑛) =

∫ℝ𝑛

∫ℝ𝑛

𝐺𝜙𝑓(𝝎, 𝒃)𝜓𝝎,𝒃 𝑑𝑛𝝎 𝑑𝑛𝒃. (4.19)

If it is assumed that the inner product (𝜙, 𝜓)𝐿2(ℝ𝑛;𝐶ℓ0,𝑛) is invertible. Then multi-

plying both sides of (4.19) from the right side by (𝜙, 𝜓)−1𝐿2(ℝ𝑛;𝐶ℓ0,𝑛) we immediately

obtain

(2𝜋)𝑛𝑓 =

∫ℝ𝑛

∫ℝ𝑛

𝐺𝜙𝑓(𝝎, 𝒃)𝜓𝝎,𝒃 (𝜙, 𝜓)−1𝐿2(ℝ𝑛;𝐶ℓ0,𝑛) 𝑑

𝑛𝝎 𝑑𝑛𝒃, (4.20)

which was to be proved. □

Acknowledgment

The author would like to thank the reviewer whose deep and extensive commentsgreatly contributed to improve this chapter. He thanks Ass. Prof. Eckhard Hitzerfor his helpful guidance. He also wants to thank ICCA9 organizer Professor KlausGurlebeck.

References

[1] M. Bahri. Generalized Fourier transform in real clifford algebra 𝑐𝑙(0, 𝑛). Far EastJournal of Mathematical Sciences, 48(1):11–24, Jan. 2011.

[2] M. Bahri, E. Hitzer, R. Ashino, and R. Vaillancourt. Windowed Fourier transformof two-dimensional quaternionic signals. Applied Mathematics and Computation,216(8):2366–2379, June 2010.

[3] E. Bayro-Corrochano. The theory and use of the quaternion wavelet transform. Jour-nal of Mathematical Imaging and Vision, 24(1):19–36, 2006.

[4] E. Bayro-Corrochano and G. Scheuermann, editors. Applied Geometric Algebras inComputer Science and Engineering. Springer, London, 2010.

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[5] F. Brackx, R. Delanghe, and F. Sommen. Clifford Analysis, volume 76. Pitman,Boston, 1982.

[6] T. Bulow. Hypercomplex Spectral Signal Representations for the Processing and Anal-ysis of Images. PhD thesis, University of Kiel, Germany, Institut fur Informatik undPraktische Mathematik, Aug. 1999.

[7] T. Bulow, M. Felsberg, and G. Sommer. Non-commutative hypercomplex Fouriertransforms of multidimensional signals. In G. Sommer, editor, Geometric computingwith Clifford Algebras: Theoretical Foundations and Applications in Computer Visionand Robotics, pages 187–207, Berlin, 2001. Springer.

[8] T. Bulow and G. Sommer. Hypercomplex signals – a novel extension of the ana-lytic signal to the multidimensional case. IEEE Transactions on Signal Processing,49(11):2844–2852, Nov. 2001.

[9] Y. Fu, U. Kahler, and P. Cerejeiras. The Balian–Low theorem for the windowedquaternionic Fourier transform. Advances in Applied Clifford Algebras, page 16, 2012.published online 3 February 2012.

[10] K. Grochenig. Foundations of Time-Frequency Analysis. Applied and Numerical Har-monic Analysis. Birkhauser, Boston, 2001.

[11] K. Grochenig and G. Zimmermann. Hardy’s theorem and the short-time Fouriertransform of Schwartz functions. Journal of the London Mathematical Society, 2:205–214, 2001.

[12] S.L. Hahn. Wigner distributions and ambiguity functions of 2-D quaternionic andmonogenic signals. IEEE Transactions on Signal Processing, 53:3111–3128, 2005.

[13] E. Hitzer. Quaternion Fourier transform on quaternion fields and generalizations.Advances in Applied Clifford Algebras, 17(3):497–517, May 2007.

[14] E.M.S. Hitzer and B. Mawardi. Clifford Fourier transform on multivector fields anduncertainty principles for dimensions 𝑛 = 2(mod 4) and 𝑛 = 3(mod 4). Advances inApplied Clifford Algebras, 18(3-4):715–736, 2008.

[15] B. Mawardi. Clifford windowed Fourier transform applied to linear time-varyingsystems. Applied Mathematical Sciences, 6:2857–2864, 2012.

[16] B. Mawardi, S. Adji, and J. Zhao. Real Clifford windowed Fourier transform. ActaMathematica Sinica, 27:505–518, 2011.

[17] B. Mawardi, E. Hitzer, and S. Adji. Two-dimensional Clifford windowed Fouriertransform. In Bayro-Corrochano and Scheuermann [4], pages 93–106.

[18] B. Mawardi, E. Hitzer, A. Hayashi, and R. Ashino. An uncertainty principlefor quaternion Fourier transform. Computers and Mathematics with Applications,56(9):2411–2417, 2008.

[19] F. Sommen. A product and an exponential function in hypercomplex function theory.Applicable Analysis, 12:13–26, 1981.

[20] Y. Xi, X. Yang, L. Song, L. Traversoni, and W. Lu. QWT: Retrospective and newapplication. In Bayro-Corrochano and Scheuermann [4], pages 249–273.

Mawardi BahriDepartment of MathematicsUniversitas HasanuddinTamalanrea Makassar 90245, Indonesiae-mail: mawardibahri@gmail.com

Quaternion and Clifford–Fourier Transforms and Wavelets

Trends in Mathematics, 299–319c⃝ 2013 Springer Basel

15 The Balian–Low Theorem for theWindowed Clifford–Fourier Transform

Yingxiong Fu, Uwe Kahler and Paula Cerejeiras

Abstract. In this chapter, we provide the definition of the Clifford–Zak trans-form associated with the discrete version of the kernel of a windowed Clifford–Fourier transform. We proceed with deriving several important properties ofsuch a transform. Finally, we establish the Balian–Low theorem for a Cliffordframe under certain natural assumptions on the window function.

Mathematics Subject Classification (2010). 15A66; 30G35.

Keywords. Clifford–Zak transform, Clifford frame, Balian–Low theorem, Clif-ford–Fourier transform, windowed Clifford–Fourier transform.

1. Introduction

The last decade has seen a growing interest in generalizations of the Fourier trans-form, motivated by applications to higher-dimensional signal processing. Firststeps in that direction where already made in Brackx et al. [10], where a Fouriertransform for multivector-valued distributions in 𝐶ℓ0,𝑛 with compact support waspresented. Also worth mention is the alternative definition of the Fourier trans-form of Sommen [27], based on a generalization of the exponential function toℝ𝑛×ℝ𝑛+1. A quaternionic Fourier transform was given in Bulow et al. [12] in thecontext of quaternionic-valued two-dimensional signals (the so-called hypercom-plex signals). Shortly after, motivated by spectral analysis of colour images Sang-wine et al. proposed in [26] a quaternionic Fourier transform in which the imaginaryunit 𝑖 was replaced by a unit quaternion. Almost in parallel, Felsberg defined in[17] his Clifford–Fourier transform (CFT) for the low-dimensional Clifford alge-bras 𝐶ℓ2,0 and 𝐶ℓ3,0, using the pseudoscalar 𝑖𝑛 as imaginary unit. Following thisapproach, several authors have extended this transform to a three-dimensional set-ting and successfully detected vector-valued patterns in the frequency domain (cf.[15, 16, 24]). However, a major problem did remain: as the classical Fourier trans-form, the Clifford–Fourier transform is ineffective for representing and computing

300 Y. Fu, U. Kahler and P. Cerejeiras

local information of signals. In fact, the harmonic analysis version of Heisenberg’suncertainty principle states that it is impossible to localize simultaneously a func-tion and its Fourier transform.

One way to overcome this difficulty is by means of Clifford–Gabor filters.They were initially proposed in [11], and later on also in [8], which extended theapplications of the complex Gabor filters. In general, they correspond to modula-tions of Gaussians. A good account on this subject can be found in [9].

A more general approach to this problem is by means of the windowed Fouriertransform (WFT), also called continuous Gabor transform or short-time Fouriertransform. Given 𝑓 ∈ 𝐿2(ℝ) and a fixed non-zero window function 𝑔 ∈ 𝐿2(ℝ), onedefines the WFT 𝑉𝑔{𝑓} ∈ 𝐿2(ℝ2) as

𝑉𝑔{𝑓}(𝑡, 𝜔) =∫ℝ

𝑓(𝑥)𝑔(𝑥− 𝑡)𝑒−2𝜋𝑖𝑥𝜔𝑑𝑥. (1.1)

Mawardi et al. extended the theory of WFT to the Clifford case [1–4, 23]. In [3]the definition of a windowed Clifford–Fourier transform (WCFT) was establishedand several important properties were obtained such as shift, modulation, recon-struction formulae, etc. Furthermore, a Heisenberg type uncertainty principle forthe WCFT was derived.

Practical applications require a discrete version of this continuous transform.One can establish a discrete form of the WFT 𝑉𝑔 linked to a given countable setΛ ⊂ ℝ2 = ℝ × ℝ as the transformation that assigns to every function 𝑓 ∈ 𝐿2(ℝ)the number sequence

𝑓 �→ { ⟨𝑓, 𝑔𝑚,𝑛⟩ : (𝑚,𝑛) ∈ Λ}, (1.2)

where ⟨⋅, ⋅⟩ denotes the ordinary inner product in 𝐿2(ℝ) and the functions 𝑔𝑚,𝑛

are shifts and modulations of the window 𝑔 given by

𝑔𝑚,𝑛(𝑥) = 𝑒2𝜋𝑖𝑚𝑥𝑔(𝑥− 𝑛) (1.3)

for all (𝑚,𝑛) ∈ Λ ⊂ ℝ2. Such a collection {𝑔𝑚,𝑛 : (𝑚,𝑛) ∈ Λ} is called Gaborsystem, or Weyl-Heisenberg system, generated by 𝑔 and Λ. To recover the originalfunction 𝑓 from the number sequence {⟨𝑓, 𝑔𝑚,𝑛⟩ : (𝑚,𝑛) ∈ Λ} it is necessary thatthe system forms an orthonormal basis or at least a frame.

Gabor systems are related to the classical uncertainty principle by the Balian–Low theorem (a stronger version of the said principle), as it expresses the fact thattime-frequency concentration and non-redundancy are incompatible properties ofa Gabor system if such a system is a frame for 𝐿2(ℝ) [7,13,14,18,21]. Specifically,if the window 𝑔 is such that the Gabor system {𝑔𝑚,𝑛 : 𝑚,𝑛 ∈ ℤ} constitutes anexact frame for 𝐿2(ℝ), i.e., if there exist constants 0 < 𝐵 ≤ 𝐶 <∞ such that

𝐵∥𝑓∥2 ≤∑

𝑚,𝑛∈ℤ∣⟨𝑓, 𝑔𝑚,𝑛⟩∣2 ≤ 𝐶 ∥𝑓∥2 , ∀𝑓 ∈ 𝐿2(ℝ), (1.4)

15. The Balian–Low Theorem for the WCFT 301

and the system ceases to be a frame when any of its elements is removed then, itholds (∫ +∞

−∞𝑡2∣𝑔(𝑡)∣2𝑑𝑡

)(∫ +∞

−∞𝜉2∣𝑔(𝜉)2𝑑𝜉

)=∞. (1.5)

In other words, the window function 𝑔 maximizes the uncertainty principle insome sense. This result has been extended to higher dimensions and to a moregeneral set of time-frequency shifts in the standard coordinate system [6, 19]. Itwas also proved to be valid in a multi-window setting [25, 28] and in the case ofsuperframes [5].

Regarding the important question of discretizing the WCFT it is naturalto ask if the Balian–Low theorem holds for Gabor systems generated by certainClifford-valued window functions 𝑔 and countable sets Λ ⊂ ℝ2𝑛. Our goal in thischapter is to obtain it for Gabor systems which form a Clifford frame arising fromthe discrete version of the kernel of the WCFT. To this end it is necessary tointroduce a Clifford–Zak transform and study some of its properties.

This chapter is organized as follows: Section 2 is devoted to the review ofthe necessary results on Clifford algebra and the definitions of both the CFT andthe WCFT. In Section 3 we provide the definition of a Clifford frame associatedwith the discrete version of the kernel of the WCFT, establish the definition of aClifford–Zak transform and derive some properties of it, which will play a key rolein the proof of the Balian–Low theorem. In Section 4 we demonstrate the Balian–Low theorem for Gabor systems which form a Clifford frame. Some conclusionsare drawn in Section 5.

2. Preliminaries

Let 𝐶ℓ𝑛,0 be the 2𝑛-dimensional universal real Clifford algebra over ℝ𝑛 constructed

from the basis {𝒆1, 𝒆2, . . . , 𝒆𝑛} under the usual relations

𝒆𝑘𝒆𝑙 + 𝒆𝑙𝒆𝑘 = 2𝛿𝑘𝑙, 1 ≤ 𝑘, 𝑙 ≤ 𝑛, (2.1)

where 𝛿𝑘𝑙 is the Kronecker delta function. An element 𝑓 ∈ 𝐶ℓ𝑛,0 can be representedas 𝑓 =

∑𝐴 𝑓𝐴𝑒𝐴, 𝑓𝐴 ∈ ℝ, where 𝒆𝐴 = 𝒆𝑗1𝑗2⋅⋅⋅𝑗𝑘 = 𝒆𝑗1𝒆𝑗2 ⋅ ⋅ ⋅𝒆𝑗𝑘 , 𝐴 = {𝑗1, 𝑗2, . . . 𝑗𝑘}

with 1 ≤ 𝑗1 ≤ 𝑗2 ≤ ⋅ ⋅ ⋅ ≤ 𝑗𝑘 ≤ 𝑛, and 𝒆0 = 𝒆∅ = 1 is the identity element of𝐶ℓ𝑛,0. The elements of the algebra 𝐶ℓ𝑛,0 for which ∣𝐴∣ = 𝑘 are called k-vectors.We denote the space of all k-vectors by

𝐶ℓ𝑘𝑛,0 := spanℝ{𝒆𝐴 : ∣𝐴∣ = 𝑘}. (2.2)

It is clear that the spaces ℝ and ℝ𝑛 can be identified with 𝐶ℓ0𝑛,0 and 𝐶ℓ1𝑛,0, re-spectively.

Of interest for this work is the (unit oriented) pseudoscalar element 𝑖𝑛 =𝒆1𝒆2 ⋅ ⋅ ⋅ 𝒆𝑛. Observe that 𝑖2𝑛 = −1 for 𝑛 = 2, 3(mod 4). For the sake of simplicity,if not otherwise stated, 𝑛 is always assumed to be 𝑛 = 2, 3(mod 4) for the remainingof this chapter.

302 Y. Fu, U. Kahler and P. Cerejeiras

We define the anti-automorphism reversion˜ : 𝐶ℓ𝑛,0 → 𝐶ℓ𝑛,0 by its action

on the basis elements 𝑒𝐴 = (−1) 𝑘(𝑘−1)2 𝑒𝐴, for ∣𝐴∣ = 𝑘, and its reversion property

𝑓𝑔 = 𝑔𝑓 for every 𝑓, 𝑔 ∈ 𝐶ℓ𝑛,0. In particular, we remark that 𝑖𝑛 = −𝑖𝑛.In what follows, we will require two types of scalar products. First, we in-

troduce the (real-valued) scalar product of 𝑓, 𝑔 ∈ 𝐶ℓ𝑛,0 as the scalar part of theirgeometric product

𝑓 ⋅ 𝑔 := [𝑓 𝑔 ]0 =∑𝐴

𝑓𝐴𝑔𝐴. (2.3)

As usual, when we set 𝑓 = 𝑔 we obtain the square of the modulus (or magnitude)of the multivector 𝑓 ∈ 𝐶ℓ𝑛,0,

∣𝑓 ∣2 =[𝑓𝑓]0=∑𝐴

𝑓2𝐴. (2.4)

Also, an additional useful property of the scalar part [ ]0 is the cyclic sym-metric product

[𝑝𝑞𝑟]0 = [𝑞𝑟𝑝]0, ∀𝑝, 𝑞, 𝑟 ∈ 𝐶ℓ𝑛,0. (2.5)

Second, we require an inner product in the function space under considera-tion. We denote by 𝐿𝑝(ℝ𝑛;𝐶ℓ𝑛,0) the left module of all Clifford-valued functions𝑓 : ℝ𝑛 → 𝐶ℓ𝑛,0 with finite norm

∥𝑓∥𝑝 =

{ (∫ℝ𝑛 ∣𝑓(x)∣𝑝𝑑𝑛x

) 1𝑝 , 1 ≤ 𝑝 <∞

ess supx∈ℝ𝑛 ∣𝑓(x)∣, 𝑝 =∞ , (2.6)

where 𝑑𝑛x = 𝑑𝑥1𝑑𝑥2 ⋅ ⋅ ⋅ 𝑑𝑥𝑛 represents the usual Lebesgue measure in ℝ𝑛. In theparticular case of 𝑝 = 2, we shall denote this norm by ∥𝑓∥.

Given two functions 𝑓, 𝑔 ∈ 𝐿2(ℝ𝑛;𝐶ℓ𝑛,0), we define a Clifford-valued bilinearform

𝑓, 𝑔 → (𝑓, 𝑔) :=

∫ℝ𝑛

𝑓(x)𝑔(x)𝑑𝑛x, (2.7)

from which we construct the scalar inner product

⟨𝑓, 𝑔⟩ : = [(𝑓, 𝑔)]0 =

∫ℝ𝑛

[𝑓(x)𝑔(x)

]0𝑑𝑛x. (2.8)

We remark that (2.8) satisfies the (Clifford) Cauchy–Schwarz inequality

∣⟨𝑓, 𝑔⟩∣ ≤ ∥𝑓∥ ∥𝑔∥ , ∀𝑓, 𝑔 ∈ 𝐿2(ℝ𝑛;𝐶ℓ𝑛,0). (2.9)

In the following, we recall the CFT, originally introduced by M. Felsberg(see [17]).

Definition 2.1. Let 𝑓 ∈ 𝐿1(ℝ𝑛;𝐶ℓ𝑛,0). The CFT of 𝑓 at the point 𝜔 ∈ ℝ𝑛 is definedas the 𝐶ℓ𝑛,0-valued (Lebesgue) integral

ℱ{𝑓}(𝜔) =∫ℝ𝑛

𝑓(x)𝑒−2𝜋𝑖𝑛𝜔⋅x𝑑𝑛x. (2.10)

The function 𝜔 → ℱ{𝑓}(𝜔) s called the CFT of 𝑓.

15. The Balian–Low Theorem for the WCFT 303

Lemma 2.2 (Parseval’s equality for CFT). If 𝑓 ∈ 𝐿1(ℝ𝑛;𝐶ℓ𝑛,0) ∩ 𝐿2 (ℝ𝑛;𝐶ℓ𝑛,0),then

∥𝑓∥ = ∥ℱ{𝑓}∥. (2.11)

Lemma 2.2 asserts that the CFT is a bounded linear operator on 𝐿1(ℝ𝑛;𝐶ℓ𝑛,0) ∩ 𝐿2(ℝ𝑛;𝐶ℓ𝑛,0). Hence, standard density arguments allow us to extendthe CFT in an unique way to the whole of 𝐿2(ℝ𝑛;𝐶ℓ𝑛,0). In what follows wealways consider the properties of the CFT as an operator from 𝐿2(ℝ𝑛;𝐶ℓ𝑛,0) into𝐿2(ℝ𝑛;𝐶ℓ𝑛,0).

Definition 2.3. Let 𝑔 ∈ 𝐿2(ℝ𝑛;𝐶ℓ𝑛,0) be a non-zero window function such that

∣x∣1/2𝑔(x) is in 𝐿2(ℝ𝑛;𝐶ℓ𝑛,0). Then, the WCFT of 𝑓 ∈ 𝐿2(ℝ𝑛;𝐶ℓ𝑛,0) with respectto 𝑔 is defined by

𝑄𝑔𝑓(𝜔,b) : =

∫ℝ𝑛

𝑓(x) ˜𝑔(x− b)𝑒−2𝜋𝑖𝑛𝜔⋅x𝑑𝑛x

= (𝑓, 𝑔𝜔,b), (2.12)

where 𝑔𝜔,b(x) := 𝑒2𝜋𝑖𝑛𝜔⋅x𝑔(x− b) denotes the kernel of the WCFT.

3. Clifford Frame and Clifford–Zak Transform

The Balian–Low theorem is regarded as a strong version of the uncertainty princi-ple for the Gabor system associated with the discrete version of the kernel of theclassical WFT. To establish the Balian–Low theorem for a Gabor system in theClifford algebra module 𝐿2(ℝ𝑛;𝐶ℓ𝑛,0) we consider the following discrete versionof the kernel of the WCFT

𝑔m,n(x) := 𝑒2𝜋𝑖𝑛m⋅x𝑔(x− n), x ∈ ℝ𝑛, m,n ∈ ℤ𝑛. (3.1)

A frame for a vector space equipped with an inner product allows each el-ement in the space to be written as a linear combination of the elements in theframe. In general frame elements are neither orthogonal to each other nor linearlyindependent. We now introduce the definition and properties of a Clifford framein 𝐿2(ℝ𝑛;𝐶ℓ𝑛,0) as follows.

Definition 3.1. {𝑔m,n : m,n ∈ ℤ𝑛} is a Clifford frame for 𝐿2(ℝ𝑛;𝐶ℓ𝑛,0) if thereexist real constants 0 < 𝐵 ≤ 𝐶 <∞ such that

𝐵 ∥𝑓∥2𝐿2(ℝ𝑛;𝐶ℓ𝑛,0)≤

∑m,n∈ℤ𝑛

∣⟨𝑓, 𝑔m,n⟩∣2 ≤ 𝐶 ∥𝑓∥2𝐿2(ℝ𝑛;𝐶ℓ𝑛,0),

∀𝑓 ∈ 𝐿2(ℝ𝑛;𝐶ℓ𝑛,0), (3.2)

where 𝑔m,n is defined by (3.1) and the scalar inner product ⟨⋅, ⋅⟩ is defined by (2.8).

Any two constants 𝐵,𝐶 satisfying condition (3.1) are called frame bounds.If 𝐵 = 𝐶, then {𝑔m,n : m,n ∈ ℤ𝑛} is called a tight frame.

To understand frames and reconstruction methods better, we study someimportant associated operators.

304 Y. Fu, U. Kahler and P. Cerejeiras

Definition 3.2. For any subset {𝑔m,n : m,n ∈ ℤ𝑛} ⊆ 𝐿2(ℝ𝑛;𝐶ℓ𝑛,0), the coefficientoperator 𝐹 is defined by

𝐹𝑓 = {⟨𝑓, 𝑔m,n⟩ : m,n ∈ ℤ𝑛}. (3.3)

The reconstruction operator 𝑅 for a sequence 𝑐 = (𝑐m,n)m,n∈ℤ𝑛 is given by

𝑅𝑐 =∑

m,n∈ℤ𝑛

𝑐m,n𝑔m,n ∈ 𝐿2(ℝ𝑛;𝐶ℓ𝑛,0). (3.4)

Finally, the frame operator 𝑆 in 𝐿2(ℝ𝑛;𝐶ℓ𝑛,0) is defined by

𝑆𝑓 =∑

m,n∈ℤ𝑛

⟨𝑓, 𝑔m,n⟩ 𝑔m,n. (3.5)

Based on the classic frame theory [18], under the assumption that {𝑔m,n :m,n ∈ ℤ𝑛} is a frame defined by (3.1) for 𝐿2(ℝ𝑛;𝐶ℓ𝑛,0), we know that the frameoperator 𝑆 maps 𝐿2(ℝ𝑛;𝐶ℓ𝑛,0) to 𝐿2(ℝ𝑛;𝐶ℓ𝑛,0) and it is a self-adjoint, positiveand invertible operator satisfying

𝐵𝐼 ≤ 𝑆 ≤ 𝐶𝐼, 𝐶−1𝐼 ≤ 𝑆−1 ≤ 𝐵−1𝐼 (3.6)

with 𝐼 being the identity operator. Moreover, observe that

⟨𝑆𝑓, 𝑓⟩ =∑

m,n∈ℤ𝑛

∣⟨𝑓, 𝑔m,n⟩∣2 (3.7)

and ∑m,n∈ℤ𝑛

∣∣⟨𝑓, 𝑆−1𝑔m,n

⟩∣∣2 =∑

m,n∈ℤ𝑛

∣∣⟨𝑆−1𝑓, 𝑔m,n

⟩∣∣2=⟨𝑆(𝑆−1𝑓), 𝑆−1𝑓

⟩=⟨𝑆−1𝑓, 𝑓

⟩. (3.8)

Therefore, we have

𝐶−1 ∥𝑓∥2 ≤ ⟨𝑆−1𝑓, 𝑓⟩=

∑m,n∈ℤ𝑛

∣∣⟨𝑓, 𝑆−1𝑔m,n

⟩∣∣2 ≤ 𝐵−1 ∥𝑓∥2 . (3.9)

Thus the collection {𝑆−1𝑔m,n : m,n ∈ ℤ𝑛} is a so-called dual frame with framebounds 𝐶−1 and 𝐵−1. Using the factorizations 𝐼 = 𝑆−1𝑆 = 𝑆𝑆−1, we obtain theseries expansions

𝑓 = 𝑆(𝑆−1𝑓) =∑

m,n∈ℤ𝑛

⟨𝑆−1𝑓, 𝑔m,n

⟩𝑔m,n

=∑

m,n∈ℤ𝑛

⟨𝑓, 𝑆−1𝑔m,n

⟩𝑔m,n (3.10)

and

𝑓 = 𝑆−1𝑆𝑓 =∑

m,n∈ℤ𝑛

⟨𝑓, 𝑔m,n⟩ 𝑆−1𝑔m,n. (3.11)

15. The Balian–Low Theorem for the WCFT 305

Furthermore, it is well known that the classic Zak transform is a very usefultool to analyze Gabor systems

{𝑒2𝜋𝑖𝑚𝑡𝑔(𝑡− 𝑛) : 𝑚,𝑛 ∈ ℤ

}in 𝐿2(ℝ). The classic

Zak transform, 𝑍 : 𝐿2(ℝ)→ 𝐿2([0, 1)2), is defined by

𝑓 = 𝑓(𝑥), 𝑥 ∈ ℝ �→ 𝑍𝑓 = 𝑍𝑓(𝑡, 𝜔)

=∑𝑘∈ℤ

𝑒2𝜋𝑖 𝑘𝜔𝑓(𝑡− 𝑘), (𝑡, 𝜔) ∈ ℝ2. (3.12)

Moreover, the Zak transform is a unitary transformation from 𝐿2(ℝ) to 𝐿2([0, 1)2).Interest in this transform has been revived in recent years due to its relationshipto different types of coherent states, one of which is the affine coherent state,commonly known as wavelet [14, 20].

To analyse Clifford–Gabor systems {𝑒2𝜋𝑖𝑛m⋅x𝑔(x − n) : m,n ∈ ℤ𝑛} in𝐿2(ℝ𝑛;𝐶ℓ𝑛,0), we need to establish a definition of the Clifford–Zak transform andto show some of its properties.

First, we define the Clifford–Zak transform pointwisely. The Clifford–Zaktransform of a Clifford-valued function 𝑓 ∈ 𝐿2(ℝ𝑛;𝐶ℓ𝑛,0), with t �→ 𝑓(t), at thepoint (x, 𝜔) ∈ ℝ𝑛 × ℝ𝑛, is given as

𝑍𝑐𝑓(x, 𝜔) :=∑k∈ℤ𝑛

𝑒2𝜋𝑖𝑛k⋅𝜔𝑓(x− k). (3.13)

As in the one-dimensional case, periodicity properties allow us to consider asmaller domain for the variables (x, 𝜔) in this transform. In fact, the Clifford–Zaktransform 𝑍𝑐 satisfies the following relations

𝑍𝑐𝑓(x, 𝜔 + n) = 𝑍𝑐𝑓(x, 𝜔), n ∈ ℤ𝑛, (3.14)

𝑍𝑐𝑓(x+ n, 𝜔) = 𝑒2𝜋𝑖𝑛n⋅𝜔𝑍𝑐𝑓(x, 𝜔), n ∈ ℤ𝑛. (3.15)

Thus, 𝑍𝑐𝑓 is uniquely determined by its values on [0, 1)𝑛 × [0, 1)𝑛 := 𝑄2𝑛 ⊂ℝ𝑛×ℝ𝑛. Henceforward, we consider 𝑍𝑐𝑓 as a function on 𝑄2𝑛, where its extensionto ℝ𝑛 × ℝ𝑛 is trivially obtained by the quasiperiodic properties (3.14) and (3.15)of the transform.

Finally, we prove that the series defining 𝑍𝑐𝑓 converges in 𝐿2(𝑄2𝑛;𝐶ℓ𝑛,0).This will be achieved by showing that 𝑍𝑐 is a unitary map from 𝐿2(ℝ𝑛;𝐶ℓ𝑛,0)onto 𝐿2(𝑄2𝑛;𝐶ℓ𝑛,0).

Theorem 3.3. The Clifford–Zak transform 𝑍𝑐 is a unitary map of 𝐿2(ℝ𝑛;𝐶ℓ𝑛,0)onto 𝐿2(𝑄2𝑛;𝐶ℓ𝑛,0).

Proof. Let 𝑓, 𝑔 ∈ 𝐿2(ℝ𝑛;𝐶ℓ𝑛,0). In order to show that 𝑍𝑐𝑓 is a unitary mappingwe consider the auxiliary functions in 𝐿2(𝑄2𝑛;𝐶ℓ𝑛,0)

(x, 𝜔) �→ 𝐹k(x, 𝜔) := 𝑒2𝜋𝑖𝑛k⋅𝜔𝑓(x− k), (3.16)

and

(x, 𝜔) �→ 𝐺k(x, 𝜔) := 𝑒2𝜋𝑖𝑛k⋅𝜔𝑔(x− k), (3.17)

306 Y. Fu, U. Kahler and P. Cerejeiras

each obtained from the original 𝑓, 𝑔 ∈ 𝐿2(ℝ𝑛;𝐶ℓ𝑛,0) by a specific modulation, anda translation, dependent on the parameter k ∈ ℤ𝑛. Note that we have

𝑍𝑐𝑓(x, 𝜔) =∑k∈ℤ𝑛

𝐹k(x, 𝜔)

𝑍𝑐𝑔(x, 𝜔) =∑k∈ℤ𝑛

𝐺k(x, 𝜔),(3.18)

for all (x, 𝜔) ∈ [0, 1)𝑛 × [0, 1)𝑛. It is easy to see that 𝐹k, 𝐺k ∈ 𝐿2(𝑄2𝑛;𝐶ℓ𝑛,0), forall k ∈ ℤ𝑛.

We have

⟨𝐹k, 𝐺m⟩𝐿2(𝑄2𝑛;𝐶ℓ𝑛,0)

=

[∫𝑄2𝑛

𝑒2𝜋𝑖𝑛k⋅𝜔𝑓(x− k) ˜𝑔(x−m)𝑒−2𝜋𝑖𝑛m⋅𝜔𝑑𝑛x𝑑𝑛𝜔]0

=

[∫[0,1)𝑛×[0,1)𝑛

𝑓(x− k) ˜𝑔(x−m)𝑒−2𝜋𝑖𝑛(m−k)⋅𝜔𝑑𝑛x𝑑𝑛𝜔

]0

=

[(∫[0,1)𝑛

𝑓(x− k) ˜𝑔(x−m)𝑑𝑛x

)(∫[0,1)𝑛

𝑒−2𝜋𝑖𝑛(m−k)⋅𝜔𝑑𝑛𝜔

)]0

, (3.19)

due to the cyclic property (2.5) and the relation 𝑖𝑛 = −𝑖𝑛. Hence

⟨𝐹k, 𝐺m⟩𝐿2(𝑄2𝑛;𝐶ℓ𝑛,0)=

{⟨𝑓(⋅ − k), 𝑔(⋅ − k)⟩𝐿2([0,1)𝑛;𝐶ℓ𝑛,0)

, m = k

0 , m ∕= k.(3.20)

Hence,

⟨𝑍𝑐𝑓, 𝑍𝑐𝑔⟩𝐿2(𝑄2𝑛;𝐶ℓ𝑛,0)=

∑k,m∈ℤ𝑛

⟨𝐹k, 𝐺m⟩𝐿2(𝑄2𝑛;𝐶ℓ𝑛,0)

=∑k∈ℤ𝑛

⟨𝐹k, 𝐺k⟩𝐿2(𝑄2𝑛;𝐶ℓ𝑛,0)

=∑k∈ℤ𝑛

⟨𝑓(⋅ − k), 𝑔(⋅ − k)⟩𝐿2([0,1)𝑛;𝐶ℓ𝑛,0)

=∑k∈ℤ𝑛

[∫[0,1)𝑛

𝑓(x− k) ˜𝑔(x − k)𝑑𝑛x

]0

=

∫ℝ𝑛

𝑓(y)𝑔(y)𝑑𝑛y = ⟨𝑓, 𝑔⟩𝐿2(ℝ𝑛;𝐶ℓ𝑛,0), (3.21)

therefore, completing the proof of the unitary property of the Clifford–Zak trans-form. □

Consequently, based on the above theorem, we get the following corollary.

15. The Balian–Low Theorem for the WCFT 307

Corollary 3.4. In particular, it holds

∥𝑍𝑐𝑓∥2𝐿2(𝑄2𝑛;𝐶ℓ𝑛,0)= ∥𝑓∥2𝐿2(ℝ𝑛;𝐶ℓ𝑛,0)

, (3.22)

where ∥𝑍𝑐𝑓∥2𝐿2(𝑄2𝑛;𝐶ℓ𝑛,0)=∫𝑄2𝑛 ∣𝑍𝑐𝑓(x, 𝜔)∣2 𝑑𝑛x𝑑𝑛𝜔.

The unitary nature of the Clifford–Zak transform allows us to translate con-ditions on Clifford frames for 𝐿2(ℝ𝑛;𝐶ℓ𝑛,0) into those for 𝐿2(𝑄2𝑛;𝐶ℓ𝑛,0), wherethings are frequently easier to deal with.

Let us now define the space 𝒵 as the set of all 𝐹 : ℝ𝑛 → 𝐶ℓ𝑛,0 such that

𝐹 (x+ n, 𝜔) = 𝑒2𝜋𝑖𝑛n⋅𝜔𝐹 (x, 𝜔),

𝐹 (x, 𝜔 + n) = 𝐹 (x, 𝜔),

∥𝐹∥2𝐿2(𝑄2𝑛;𝐶ℓ𝑛,0)=

∫𝑄2𝑛

∣𝐹 (x, 𝜔)∣2 𝑑𝑛x𝑑𝑛𝜔 <∞. (3.23)

In consequence, as 𝒵 is a subset of 𝐿2(𝑄2𝑛;𝐶ℓ𝑛,0) the Clifford–Zak transform 𝑍𝑐

is a unitary mapping between 𝐿2(ℝ𝑛;𝐶ℓ𝑛,0) and 𝒵.The following theorem provides some inversion formulas.

Theorem 3.5. If 𝑓 ∈ 𝐿2(ℝ𝑛;𝐶ℓ𝑛,0) ∩ 𝐿1(ℝ𝑛;𝐶ℓ𝑛,0), then the following relations,

𝑓(x) =

∫[0,1)𝑛

𝑍𝑐𝑓(x, 𝜔)𝑑𝑛𝜔, x ∈ ℝ𝑛, (3.24)

and

ℱ{𝑓 }(−𝜔) =

∫[0,1)𝑛

˜𝑍𝑐𝑓(x, 𝜔)𝑒2𝜋𝑖𝑛𝜔⋅x𝑑𝑛x, 𝜔 ∈ ℝ𝑛, (3.25)

hold true, where ℱ denotes the CFT operator given by (2.10).

Before proceeding with the proof, we remark that the Zak transform can beextended in both arguments to the whole of ℝ𝑛 by relations (3.14) and (3.15). In

consequence, the above identities state that both the signal 𝑓, and the CFT ℱ{𝑓 },can be reconstructed on the whole of ℝ𝑛 via the Clifford–Zak transform. Standarddensity arguments allow us to extend this result in an unique way to the whole of𝐿2(ℝ𝑛;𝐶ℓ𝑛,0).

Proof. By definition, there exist unique y ∈ [0, 1)𝑛 and n ∈ ℤ𝑛 such that x = y+n,and∫

[0,1)𝑛𝑍𝑐𝑓(x, 𝜔)𝑑

𝑛𝜔 =

∫[0,1)𝑛

𝑍𝑐𝑓(y + n, 𝜔)𝑑𝑛𝜔

=

∫[0,1)𝑛

𝑒2𝜋𝑖𝑛n⋅𝜔𝑍𝑐𝑓(y, 𝜔)𝑑𝑛𝜔

=

∫[0,1)𝑛

∑k∈ℤ𝑛

𝑒2𝜋𝑖𝑛(k+n)⋅𝜔𝑓(y − k)𝑑𝑛𝜔

308 Y. Fu, U. Kahler and P. Cerejeiras

=

∫[0,1)𝑛

∑m∈ℤ𝑛

𝑒2𝜋𝑖𝑛m⋅𝜔𝑓(y + n−m)𝑑𝑛𝜔

=

∫[0,1)𝑛

∑m∈ℤ𝑛

𝑒2𝜋𝑖𝑛m⋅𝜔𝑓(x−m)𝑑𝑛𝜔

=

∫[0,1)𝑛

𝑓(x)𝑑𝑛𝜔 +

∫[0,1)𝑛

∑m∕=0

𝑒2𝜋𝑖𝑛m⋅𝜔𝑓(x−m)𝑑𝑛𝜔

= 𝑓(x) +

∫[0,1)𝑛

∑m∕=0

𝑒2𝜋𝑖𝑛m⋅𝜔𝑓(x−m)𝑑𝑛𝜔. (3.26)

To calculate the remaining integral, we use Fubini’s Theorem to validate the in-terchange between integration and summation so that∫

[0,1)𝑛

∑m∕=0

𝑒2𝜋𝑖𝑛m⋅𝜔𝑓(x−m)𝑑𝑛𝜔

=∑m∕=0

(∫[0,1)𝑛

𝑒2𝜋𝑖𝑛m⋅𝜔𝑑𝑛𝜔

)𝑓(x−m) = 0. (3.27)

This completes the proof of the first identity. For the second, a direct calculationleads to∫

[0,1)𝑛

˜𝑍𝑐𝑓(x, 𝜔)𝑒2𝜋𝑖𝑛𝜔⋅x𝑑𝑛x =

∫[0,1)𝑛

∑k∈ℤ𝑛

˜𝑓(x− k)𝑒−2𝜋𝑖𝑛k⋅𝜔𝑒2𝜋𝑖𝑛𝜔⋅x𝑑𝑛x

=∑k∈ℤ𝑛

∫[0,1)𝑛

˜𝑓(x− k)𝑒2𝜋𝑖𝑛(x−k)⋅𝜔𝑑𝑛x (3.28)

=

∫ℝ𝑛

𝑓(y)𝑒2𝜋𝑖𝑛y⋅𝜔𝑑𝑛y = ℱ{𝑓}(−𝜔). □

In particular, we obtain the following reconstruction formula: for any 𝐹 ∈ 𝒵,we have (

𝑍−1𝑐 𝐹

)(x) =

∫[0,1)𝑛

𝐹 (x, 𝜔)𝑑𝑛𝜔, x ∈ ℝ𝑛. (3.29)

Lemma 3.6. If 𝑔m,n is defined by (3.1), then

𝑍𝑐𝑔m,n(x, 𝜔) = 𝑒−2𝜋𝑖𝑛n⋅𝜔𝑒2𝜋𝑖𝑛m⋅x𝑍𝑐𝑔(x, 𝜔), (x, 𝜔) ∈ 𝑄2𝑛. (3.30)

Proof. From the definitions of the Clifford–Zak transform and of 𝑔m,n we obtain

𝑍𝑐𝑔m,n(x, 𝜔) =∑k∈ℤ𝑛

𝑒2𝜋𝑖𝑛k⋅𝜔𝑒2𝜋𝑖𝑛m⋅(x−k)𝑔(x− k− n)

=∑k∈ℤ𝑛

𝑒2𝜋𝑖𝑛k⋅𝜔𝑒2𝜋𝑖𝑛m⋅x𝑔(x− k− n)

= 𝑒2𝜋𝑖𝑛m⋅x∑k∈ℤ𝑛

𝑒2𝜋𝑖𝑛k⋅𝜔𝑔(x− k− n)

15. The Balian–Low Theorem for the WCFT 309

= 𝑒2𝜋𝑖𝑛m⋅x∑k∈ℤ𝑛

𝑒2𝜋𝑖𝑛(k−n)⋅𝜔𝑔(x− k)

= 𝑒−2𝜋𝑖𝑛n⋅𝜔𝑒2𝜋𝑖𝑛m⋅x∑k∈ℤ𝑛

𝑒2𝜋𝑖𝑛k⋅𝜔𝑔(x− k) (3.31)

= 𝑒−2𝜋𝑖𝑛n⋅𝜔𝑒2𝜋𝑖𝑛m⋅x𝑍𝑐𝑔(x, 𝜔). □

Theorem 3.7. If 𝑓, 𝑔 ∈ 𝐿2(ℝ𝑛;ℝ ⊕ 𝑖𝑛ℝ) and 𝑔m,n is defined as in (3.1) then wehave ∑

m,n∈ℤ𝑛

∣∣⟨𝑓, 𝑔m,n⟩𝐿2(ℝ𝑛;𝐶ℓ𝑛,0)

∣∣2 =∥∥∥𝑍𝑐𝑓𝑍𝑐𝑔

∥∥∥2𝐿2(𝑄2𝑛;𝐶ℓ𝑛,0)

. (3.32)

Proof. We remark that 𝑓 ∈ 𝐿2(ℝ𝑛;ℝ ⊕ 𝑖𝑛ℝ) ⊂ 𝐿2(ℝ𝑛;𝐶ℓ𝑛,0) so that we have∥𝑓∥𝐿2(ℝ𝑛;ℝ⊕𝑖𝑛ℝ) = ∥𝑓∥𝐿2(ℝ𝑛;𝐶ℓ𝑛,0)

. Then, based on Theorem 3.3 and Lemma 3.6,

we obtain that∑m,n∈ℤ𝑛

∣∣⟨𝑓, 𝑔m,n⟩𝐿2(ℝ𝑛;𝐶ℓ𝑛,0)

∣∣2=

∑m,n∈ℤ𝑛

∣∣⟨𝑍𝑐𝑓, 𝑍𝑐𝑔m,n⟩𝐿2(𝑄2𝑛;𝐶ℓ𝑛,0)

∣∣2=

∑m,n∈ℤ𝑛

∣∣∣[∫𝑄2𝑛 𝑍𝑐𝑓𝑍𝑐𝑔m,n𝑑𝑛x𝑑𝑛𝜔

]0

∣∣∣2=

∑m,n∈ℤ𝑛

∣∣∣[∫𝑄2𝑛 𝑍𝑐𝑓𝑍𝑐𝑔𝑒2𝜋𝑖𝑛n⋅𝜔𝑒−2𝜋𝑖𝑛m⋅x𝑑𝑛x𝑑𝑛𝜔

]0

∣∣∣2 (3.33)

=∑

m,n∈ℤ𝑛

∣∣∣∣⟨𝑍𝑐𝑓𝑍𝑐𝑔, 𝐸m,n

⟩𝐿2(𝑄2𝑛;𝐶ℓ𝑛,0)

∣∣∣∣2=∥∥∥𝑍𝑐𝑓𝑍𝑐𝑔

∥∥∥2𝐿2(𝑄2𝑛;𝐶ℓ𝑛,0)

,

where the set of all 𝐸m,n := 𝑒−2𝜋𝑖𝑛n⋅𝜔𝑒2𝜋𝑖𝑛m⋅x constitutes an orthonormal basisfor 𝐿2(𝑄2𝑛;ℝ⊕ 𝑖𝑛ℝ). □

4. Balian–Low Theorem for WCFT

Before proceeding with the Balian–Low theorem for a Clifford–Gabor frame, letus recall some basic facts on the modulus of a Clifford number. In general, themodulus of arbitrary Clifford numbers is not multiplicative. In fact, for any twoelements 𝑎, 𝑏 ∈ 𝐶ℓ𝑛,0 we have ∣𝑎𝑏∣ ≤ 2

𝑛2 ∣𝑎∣ ∣𝑏∣. However, in some special cases, the

multiplicative property does hold. An easy calculation leading to such a case isdescribed in the following lemma.

Lemma 4.1. Let 𝑏 ∈ 𝐶ℓ𝑛,0 be such that 𝑏�� = ∣𝑏∣2. Then∣𝑎𝑏∣ = ∣𝑎∣ ∣𝑏∣ , ∀𝑎 ∈ 𝐶ℓ𝑛,0. (4.1)

310 Y. Fu, U. Kahler and P. Cerejeiras

In this section, and since we need the Clifford–Zak transform 𝑍𝑐 of the windowfunction 𝑔 to satisfy ∣𝑓𝑍𝑐𝑔∣ = ∣𝑓 ∣ ∣𝑍𝑐𝑔∣ for any multivector 𝑓 ∈ 𝐶ℓ𝑛,0, it is necessaryto impose some restrictions on the non-zero window function 𝑔. In what followswe will require

𝑔 ∈ 𝐿2(ℝ𝑛;ℝ⊕ 𝑖𝑛ℝ). (4.2)

Some simple examples of possible real-valued window functions are the two-dimen-sional Gaussian function and the two-dimensional first-order B-spline [23], whichcan be generalized to 𝑛 dimensions and to cases of linear combinations of real-valued window functions and pseudoscalars.

Lemma 4.2. Suppose that the non-zero window function 𝑔 satisfies (4.2). Then wehave

𝑍𝑐𝑔𝑍𝑐𝑔 = ∣𝑍𝑐𝑔∣2 , (4.3)

and for any multivector 𝑎 ∈ 𝐶ℓ𝑛,0

∣𝑎𝑍𝑐𝑔∣ = ∣𝑎∣ ∣𝑍𝑐𝑔∣ . (4.4)

Proof. Since 𝑔 ∈ 𝐿2(ℝ𝑛;ℝ ⊕ 𝑖𝑛ℝ), an easy computation shows that there exist𝐺,𝐻 ∈ 𝐿2(𝑄2𝑛;ℝ) such that

𝑍𝑐𝑔 = 𝐺+ 𝑖𝑛𝐻. (4.5)

Thus, based on the fact that 𝑖𝑛 = −𝑖𝑛 and �� = 𝐺, �� = 𝐻 for 𝐺,𝐻 ∈ 𝐿2(𝑄2𝑛;ℝ),it follows that

𝑍𝑐𝑔𝑍𝑐𝑔 = (𝐺+ 𝑖𝑛𝐻)(��+ ��𝑖𝑛) = (𝐺+ 𝑖𝑛𝐻)(𝐺−𝐻𝑖𝑛)

= 𝐺𝐺−𝐺𝐻𝑖𝑛 + 𝑖𝑛𝐻𝐺− 𝑖𝑛𝐻𝐻𝑖𝑛

= 𝐺2 −𝐻𝐺𝑖𝑛 + 𝑖𝑛𝐻𝐺− 𝑖𝑛𝐻2𝑖𝑛. (4.6)

Note that the pseudoscalar 𝑖𝑛 commutes with the scalar elements of the algebra.Thus,

𝐻𝐺𝑖𝑛 = 𝑖𝑛𝐻𝐺, 𝐺2 = ∣𝐺∣2 , 𝐻2 = ∣𝐻 ∣2 . (4.7)

Substituting (4.7) into (4.6) leads to

𝑍𝑐𝑔 𝑍𝑐𝑔 = ∣𝐺∣2 + ∣𝐻 ∣2 = ∣𝑍𝑐𝑔∣2 . (4.8)

Moreover, by Lemma 4.1 we see that for any 𝑎 ∈ 𝐶ℓ𝑛,0

∣𝑎𝑍𝑐𝑔∣ = ∣𝑎∣ ∣𝑍𝑐𝑔∣ , (4.9)

which completes the proof. □

Based on the properties of the Clifford–Zak transform discussed in the pre-vious section, we are going to study the time-frequency localization property ofa Clifford–Gabor system {𝑔m,n : m,n ∈ ℤ𝑛} for 𝐿2(ℝ𝑛;ℝ ⊕ 𝑖𝑛ℝ), which is thecontent of the following theorem. This theorem will enable us later on to derivethe Clifford version of the Balian–Low theorem.

15. The Balian–Low Theorem for the WCFT 311

Theorem 4.3. Suppose that {𝑔m,n(x) : m,n ∈ ℤ𝑛} constitutes a frame for𝐿2(ℝ𝑛;ℝ ⊕ 𝑖𝑛ℝ) with a non-zero window function 𝑔 satisfying (4.2). Then wehave

△𝑥𝑘△𝜔𝑘 =∞, 𝑘 = 1, 2, . . . , 𝑛, (4.10)

where

△𝑥𝑘 =

∫ℝ𝑛

𝑥2𝑘 ∣𝑔(x)∣2 𝑑𝑛x, △𝜔𝑘 =

∫ℝ𝑛

𝜔2𝑘 ∣ℱ𝑔(𝜔)∣2 𝑑𝑛𝜔 (4.11)

and the CFT of 𝑔, ℱ𝑔(𝜔), is defined by (2.10).

We will divide the proof by demonstrating a sequence of lemmas. Denote by𝐷𝑘 and 𝑀𝑘 the following partial derivative and multiplication operators

𝐷𝑘𝑓(x) := ∂𝑥𝑘𝑓(x)

1

2𝜋𝑖𝑛, 𝑀𝑘𝑓(x) := 𝑥𝑘𝑓(x), 𝑘 = 1, 2, . . . , 𝑛, (4.12)

where ∂𝑥𝑘:= ∂

∂𝑥𝑘. One can easily see that the product of these operators depends

on their order. In fact, they satisfy the commutation relation

[𝑀𝑘, 𝐷𝑘]𝑓(x) := (𝑀𝑘𝐷𝑘 −𝐷𝑘𝑀𝑘) 𝑓(x) = −𝑓(x)1

2𝜋𝑖𝑛. (4.13)

This is traditionally expressed by saying that the time and frequency variables arecanonically conjugate. In the first place, we review the following lemma which wasshown in [22, 24].

Lemma 4.4 (CFT partial derivative). The CFT of ∂𝑥𝑘𝑓(x) ∈ 𝐿2(ℝ𝑛;𝐶ℓ𝑛,0) is

given by

ℱ{∂𝑥𝑘𝑓(x)}(𝜔) = 2𝜋𝜔𝑘ℱ{𝑓}(𝜔)𝑖𝑛, (4.14)

that is,

ℱ{𝐷𝑘𝑓(x)}(𝜔) = 𝑀𝑘ℱ{𝑓}(𝜔), 𝑘 = 1, 2, . . . , 𝑛. (4.15)

Lemma 4.5. Let 𝑔, 𝜇 ∈ 𝐿2(ℝ𝑛;ℝ⊕ 𝑖𝑛ℝ). If we have

𝐷𝑘𝑔,𝐷𝑘𝜇,𝑀𝑘𝑔,𝑀𝑘𝜇 ∈ 𝐿2(ℝ𝑛;ℝ⊕ 𝑖𝑛ℝ), 𝑘 = 1, 2, . . . , 𝑛, (4.16)

then the following relation holds

⟨𝑀𝑘𝑔,𝐷𝑘𝜇⟩ − ⟨𝐷𝑘𝑔,𝑀𝑘𝜇⟩ = ± 1

2𝜋𝑖𝑛⟨𝑔, 𝜇⟩ , (4.17)

where the scalar inner product ⟨⋅, ⋅⟩ is given by (2.8).

Proof. Let us choose 𝜑𝑗 , 𝜓𝑗 ∈ 𝒮(ℝ𝑛;ℝ⊕ 𝑖𝑛ℝ), where 𝒮(ℝ𝑛;ℝ⊕ 𝑖𝑛ℝ) denotes theSchwartz class. We recall that 𝒮(ℝ𝑛;ℝ⊕𝑖𝑛ℝ) is a dense subspace in 𝐿2(ℝ𝑛;ℝ⊕𝑖𝑛ℝ)and it is defined as the set of all smooth functions from ℝ𝑛 to ℝ⊕ 𝑖𝑛ℝ such thatall of its partial derivatives are rapidly decreasing. Therefore, we have that if𝜑𝑗 → 𝑔, 𝜓𝑗 → 𝜇, then 𝐷𝑘𝜑𝑗 → 𝐷𝑘𝑔,𝑀𝑘𝜑𝑗 → 𝑀𝑘𝑔 and 𝐷𝑘𝜓𝑗 → 𝐷𝑘𝜇,𝑀𝑘𝜓𝑗 →𝑀𝑘𝜇, 𝑘 = 1, 2, . . . , 𝑛. Moreover, the convergence is in the 𝐿2-sense. Since it is easyto check that for fixed 𝑘 the operators 𝐷𝑘 and 𝑀𝑘 are self-adjoint, we obtain

⟨𝑀𝑘𝜑𝑗 , 𝐷𝑘𝜓𝑗⟩ − ⟨𝐷𝑘𝜑𝑗 ,𝑀𝑘𝜓𝑗⟩ = ⟨𝐷𝑘𝑀𝑘𝜑𝑗 , 𝜓𝑗⟩ − ⟨𝑀𝑘𝐷𝑘𝜑𝑗 , 𝜓𝑗⟩

312 Y. Fu, U. Kahler and P. Cerejeiras

= −⟨[𝑀𝑘, 𝐷𝑘]𝜑𝑗 , 𝜓𝑗⟩ =⟨𝜑𝑗

1

2𝜋𝑖𝑛, 𝜓𝑗

⟩= ± 1

2𝜋𝑖𝑛⟨𝜑𝑗 , 𝜓𝑗⟩ , (4.18)

where we use the fact that the pseudoscalar 𝑖𝑛 commutes with the scalar elementsof the algebra. Moreover, since the scalar inner product is continuous, the desiredresult holds in the limit. □

The utility of the Clifford–Zak transform 𝑍𝑐 for constructing Gabor basesstems from the following result.

Lemma 4.6. Suppose that the non-zero window function 𝑔 satisfies (4.2). Then wehave

1. {𝑔m,n : m,n ∈ ℤ𝑛} is a frame for 𝐿2(ℝ𝑛;ℝ⊕ 𝑖𝑛ℝ) if and only if

0 < 𝐵 ≤ ∣𝑍𝑐𝑔(x, 𝜔)∣2 ≤ 𝐶 <∞ 𝑎.𝑒. (x, 𝜔) ∈ 𝑄2𝑛. (4.19)

2. {𝑔m,n : m,n ∈ ℤ𝑛} is an orthonormal basis for 𝐿2(ℝ𝑛;ℝ⊕ 𝑖𝑛ℝ) if and only

if ∣𝑍𝑐𝑔(x, 𝜔)∣2 = 1 for almost all (x, 𝜔) ∈ 𝑄2𝑛.

Proof. Let us first prove statement 1. Assume that {𝑔m,n : m,n ∈ ℤ𝑛} is a framefor 𝐿2(ℝ𝑛;ℝ⊕ 𝑖𝑛ℝ). According to the definition, there exist 0 < 𝐵 ≤ 𝐶 <∞ suchthat for all 𝑓 ∈ 𝐿2(ℝ𝑛;ℝ⊕ 𝑖𝑛ℝ), it holds

𝐵 ∥𝑓∥2𝐿2(ℝ𝑛;ℝ⊕𝑖𝑛ℝ) ≤∑

m,n∈ℤ𝑛

∣⟨𝑓, 𝑔m,n⟩∣2 ≤ 𝐶 ∥𝑓∥2𝐿2(ℝ𝑛;ℝ⊕𝑖𝑛ℝ) . (4.20)

Then 𝐹 = 𝑍𝑐𝑓 ∈ 𝐿2(𝑄2𝑛;ℝ⊕ 𝑖𝑛ℝ) and by Corollary 3.4 and Theorem 3.7 we get

𝐵 ∥𝐹∥2𝐿2(𝑄2𝑛;ℝ⊕𝑖𝑛ℝ) ≤∥∥∥𝐹𝑍𝑐𝑔

∥∥∥2𝐿2(𝑄2𝑛;ℝ⊕𝑖𝑛ℝ)

≤ 𝐶 ∥𝐹∥2𝐿2(𝑄2𝑛;ℝ⊕𝑖𝑛ℝ) , (4.21)

which implies that 𝐵 ≤ ∣𝑍𝑐𝑔∣2 ≤ 𝐶 𝑎.𝑒. due to Lemma 4.2. Conversely, if 𝐵 ≤∣𝑍𝑐𝑔∣2 ≤ 𝐶 𝑎.𝑒., then by Theorem 3.7 and Corollary 3.4, we conclude that for all𝑓 ∈ 𝐿2(ℝ𝑛;ℝ⊕ 𝑖𝑛ℝ)

𝐵 ∥𝑓∥2𝐿2(ℝ𝑛;ℝ⊕𝑖𝑛ℝ) ≤∑

m,n∈ℤ𝑛

∣⟨𝑓, 𝑔m,n⟩∣2 =∥∥∥𝑍𝑐𝑓𝑍𝑐𝑔

∥∥∥2𝐿2(𝑄2𝑛;ℝ⊕𝑖𝑛ℝ)

≤ 𝐶 ∥𝑓∥2𝐿2(ℝ𝑛;ℝ⊕𝑖𝑛ℝ) . (4.22)

Now, let us take a look at statement 2. If {𝑔m,n : m,n ∈ ℤ𝑛} is an orthonor-mal basis for 𝐿2(ℝ𝑛;ℝ⊕ 𝑖𝑛ℝ) then, for all 𝑓 ∈ 𝐿2(ℝ𝑛;ℝ⊕ 𝑖𝑛ℝ), it holds∑

m,n∈ℤ𝑛

∣⟨𝑓, 𝑔m,n⟩∣2 = ∥𝑓∥2𝐿2(ℝ𝑛;ℝ⊕𝑖𝑛ℝ) = ∥𝑓∥2𝐿2(ℝ𝑛;𝐶ℓ𝑛,0)

= ∥𝑍𝑐𝑓∥2𝐿2(𝑄2𝑛;𝐶ℓ𝑛,0). (4.23)

Thus, by Theorem 3.7 we get∥∥∥𝑍𝑐𝑓𝑍𝑐𝑔∥∥∥2𝐿2(𝑄2𝑛;𝐶ℓ𝑛,0)

= ∥𝑍𝑐𝑓∥2𝐿2(𝑄2𝑛;𝐶ℓ𝑛,0), (4.24)

15. The Balian–Low Theorem for the WCFT 313

which implies that ∣𝑍𝑐𝑔(x, 𝜔)∣2 = 1 for almost all (x, 𝜔) ∈ 𝑄2𝑛 due to Lemma 4.2.

Conversely, if ∣𝑍𝑐𝑔(x, 𝜔)∣2 = 1 for almost all (x, 𝜔) ∈ 𝑄2𝑛, then by Theorem 3.7,{𝑔m,n : m,n ∈ ℤ𝑛} is a tight frame for 𝐿2(ℝ𝑛;ℝ⊕ 𝑖𝑛ℝ). Moreover, Corollary 3.4

yields ∥𝑔∥2𝐿2(ℝ𝑛;𝐶ℓ𝑛,0)=∥𝑍𝑐𝑔∥2𝐿2(𝑄2𝑛;𝐶ℓ𝑛,0)

=1, which leads to ∥𝑔m,n∥2𝐿2(ℝ𝑛;𝐶ℓ𝑛,0)=1.

Consequently, {𝑔m,n : m,n ∈ ℤ𝑛} is an orthonormal basis for 𝐿2(ℝ𝑛;ℝ⊕𝑖𝑛ℝ). □

Combining Corollary 3.4, Theorem 3.7 and (3.7), we can assert that

⟨𝑍𝑐𝑆𝑓, 𝑍𝑐𝑓⟩ =∥∥∥𝑍𝑐𝑓𝑍𝑐𝑔

∥∥∥2𝐿2(𝑄2𝑛;𝐶ℓ𝑛,0)

, (4.25)

where 𝑆 is the frame operator defined by (3.2). Thus, we get that 𝑍𝑐𝑆𝑍−1𝑐 corre-

sponds to a multiplication by ∣𝑍𝑐𝑔∣2 on the space 𝒵 defined by (3.23).The following lemma shows that ∂𝑥1𝑍𝑐𝑔 ∈ 𝐿2(𝑄2𝑛;ℝ ⊕ 𝑖𝑛ℝ) implies that

∂𝑥1 ∣𝑍𝑐𝑔∣ ∈ 𝐿2(𝑄2𝑛;ℝ) under a certain condition. In a similar way, we can get thecorresponding results for ∂𝑥2𝑍𝑐𝑔, ∂𝜔1𝑍𝑐𝑔 and ∂𝜔2𝑍𝑐𝑔 in 𝐿2(𝑄2𝑛;ℝ⊕ 𝑖𝑛ℝ).

Lemma 4.7. Under the hypotheses of Theorem 4.3 we see that if ∂𝑥1𝑍𝑐𝑔 ∈𝐿2(𝑄2𝑛;ℝ⊕ 𝑖𝑛ℝ) then ∂𝑥1 ∣𝑍𝑐𝑔∣ ∈ 𝐿2(𝑄2𝑛;ℝ) ⊂ 𝐿2(𝑄2𝑛;ℝ⊕ 𝑖𝑛ℝ).

Proof. Since 𝑔 ∈ 𝐿2(ℝ𝑛;ℝ ⊕ 𝑖𝑛ℝ), an easy computation shows that there exist𝐺,𝐻 ∈ 𝐿2(𝑄2𝑛;ℝ⊕ 𝑖𝑛ℝ) such that

𝑍𝑐𝑔 = 𝐺+ 𝑖𝑛𝐻, 𝐺,𝐻 ∈ ℝ. (4.26)

Thus ∂𝑥1𝑍𝑐𝑔 = ∂𝑥1𝐺+ ∂𝑥1𝐻𝑖𝑛 and

∥∂𝑥1𝑍𝑐𝑔∥2 =

∫𝑄2𝑛

∣∂𝑥1𝑍𝑐𝑔∣2 𝑑𝑛x𝑑𝑛𝜔

=

∫𝑄2𝑛

((∂𝑥1𝐺)2 + (∂𝑥1𝐻)2

)𝑑𝑛x𝑑𝑛𝜔 <∞, (4.27)

which means that

∥∂𝑥1𝐺∥ <∞, ∥∂𝑥1𝐻∥ <∞. (4.28)

Moreover, since {𝑔m,n : m,n ∈ ℤ𝑛} is a frame for 𝐿2(ℝ𝑛;ℝ⊕𝑖𝑛ℝ) by Lemma4.6 we have

0 < 𝐵 ≤ ∣𝑍𝑐𝑔(x, 𝜔)∣2 ≤ 𝐶 <∞, 𝑎.𝑒. (x, 𝜔) ∈ 𝑄2𝑛, (4.29)

which tells us that

𝐺2 ≤ 𝐶 <∞, 𝐻2 ≤ 𝐶 <∞. (4.30)

A simple calculation leads to

∂𝑥1 ∣𝑍𝑐𝑔∣ = ∣𝑍𝑐𝑔∣−1(𝐺∂𝑥1𝐺+𝐻∂𝑥1𝐻). (4.31)

Now, based on (4.28), (4.30), and the Minkowski inequality in 𝐿2(𝑄2𝑛;ℝ), it followsthat

∥∂𝑥1 ∣𝑍𝑐𝑔∣∥2 =

∫𝑄2𝑛

∣𝑍𝑐𝑔∣−2 ∣(𝐺∂𝑥1𝐺+𝐻∂𝑥1𝐻)∣2 𝑑𝑛x𝑑𝑛𝜔

314 Y. Fu, U. Kahler and P. Cerejeiras

≤ 𝐵−1

∫𝑄2𝑛

∣(𝐺∂𝑥1𝐺+𝐻∂𝑥1𝐻)∣2 𝑑𝑛x𝑑𝑛𝜔

≤ 𝐵−1(∥𝐺∂𝑥1𝐺∥+ ∥𝐻∂𝑥1𝐻∥)2 (4.32)

≤ 𝐵−1𝐶(∥∂𝑥1𝐺∥+ ∥∂𝑥1𝐻∥)2 <∞. □

Now, let us consider the dual frame 𝑔m,n := 𝑆−1𝑔m,n given by (3.9). Since𝑍𝑐𝑆𝑍

−1𝑐 corresponds to a multiplication by ∣ 𝑍𝑐𝑔 ∣2 on the space 𝒵, by Lemma 4.6

it follows that

𝑍𝑐𝑔m,n = 𝑍𝑐𝑆−1𝑍−1

𝑐 𝑍𝑐𝑔m,n = ∣𝑍𝑐𝑔∣−2𝑍𝑐𝑔m,n (4.33)

or

𝑍𝑐𝑔m,n = ∣𝑍𝑐𝑔∣−2𝑒−2𝜋𝑖𝑛n⋅𝜔𝑒2𝜋𝑖𝑛m⋅x𝑍𝑐𝑔

= 𝑒−2𝜋𝑖𝑛n⋅𝜔𝑒2𝜋𝑖𝑛m⋅x𝑍𝑐𝑔, (4.34)

which belongs to the space 𝒵 with 𝑍𝑐𝑔 = ∣𝑍𝑐𝑔∣−2 𝑍𝑐𝑔. In particular, (4.34) impliesthat

𝑔m,n(x) := 𝑒2𝜋𝑖𝑛m⋅x𝑔(x− n), (4.35)

which proves the following lemma associated with Lemma 4.2.

Lemma 4.8. Under the hypotheses of Theorem 4.3 the dual window function 𝑔

satisfies 𝑍𝑐𝑔 = ∣𝑍𝑐𝑔∣−2𝑍𝑐𝑔 and 𝑍𝑐𝑔 𝑍𝑐𝑔 = 1.

Analogous to the Minkowski inequality in 𝐿2(ℝ𝑛;ℝ), by the definition ofthe norm (2.6) and the Clifford–Cauchy–Schwarz inequality (2.9), the followingClifford–Minkowski inequality holds true in 𝐿2(𝑄2𝑛;𝐶ℓ𝑛,0). This will be necessaryin the proof of Theorem 4.3.

Lemma 4.9. For 𝜑, 𝜓 ∈ 𝐿2(𝑄2𝑛;𝐶ℓ𝑛,0), we have 𝜑+ 𝜓 ∈ 𝐿2(𝑄2𝑛;𝐶ℓ𝑛,0) and

∥𝜑+ 𝜓∥ ≤ ∥𝜑∥ + ∥𝜓∥ . (4.36)

Now, we are in position to demonstrate Theorem 4.3.

Proof. We prove the theorem by contradiction. Suppose that {𝑔m,n : m,n ∈ ℤ𝑛}is a frame for 𝐿2(ℝ𝑛;ℝ⊕ 𝑖𝑛ℝ) with a non-zero window function 𝑔 satisfying (4.2),and, furthermore, suppose that

△𝑥𝑘 =

∫ℝ𝑛

𝑥2𝑘 ∣𝑔(x)∣2 𝑑𝑛x <∞, (4.37)

and

△𝜔𝑘 =

∫ℝ𝑛

𝜔2𝑘 ∣ℱ𝑔(𝜔)∣2 𝑑𝑛𝜔 <∞, 𝑘 = 1, 2, . . . , 𝑛. (4.38)

Now, based on Lemmas 2.2 and 4.4, we get 𝑀𝑘𝑔,𝐷𝑘𝑔 ∈ 𝐿2(ℝ𝑛;ℝ ⊕ 𝑖𝑛ℝ), wherethe multiplication and partial derivative operators 𝑀𝑘, 𝐷𝑘 are defined by (4.12).

One has that

[𝑍𝑐(𝑀𝑘𝑔)] (x, 𝜔) = 𝑥𝑘𝑍𝑐𝑔 − 1

2𝜋𝑖𝑛∂𝜔𝑘

𝑍𝑐𝑔, (4.39)

15. The Balian–Low Theorem for the WCFT 315

which means that 𝑀𝑘𝑔 ∈ 𝐿2(ℝ𝑛;ℝ⊕𝑖𝑛ℝ) if and only if ∂𝜔𝑘𝑍𝑐𝑔 ∈ 𝐿2(𝑄2𝑛;ℝ⊕𝑖𝑛ℝ)

by Theorem 3.4 and Lemma 4.9. In a similar way, we find that 𝐷𝑘𝑔 ∈ 𝐿2(ℝ𝑛;ℝ⊕𝑖𝑛ℝ) if and only if ∂𝑥𝑘

𝑍𝑐𝑔 ∈ 𝐿2(𝑄2𝑛;ℝ⊕ 𝑖𝑛ℝ). By Lemma 4.8 we know that the

dual window function 𝑔 satisfies 𝑍𝑐𝑔 = ∣𝑍𝑐𝑔∣−2 𝑍𝑐𝑔. Consequently, we see that

∂𝑥𝑘(𝑍𝑐𝑔) = ∂𝑥𝑘

(∣𝑍𝑐𝑔∣−2)𝑍𝑐𝑔 + ∣𝑍𝑐𝑔∣−2

∂𝑥𝑘𝑍𝑐𝑔

= −2 ∣𝑍𝑐𝑔∣−3𝑍𝑐𝑔 ∂𝑥𝑘

∣𝑍𝑐𝑔∣+ ∣𝑍𝑐𝑔∣−2∂𝑥𝑘

𝑍𝑐𝑔 (4.40)

and∂𝜔𝑘

(𝑍𝑐𝑔) = −2 ∣𝑍𝑐𝑔∣−3𝑍𝑐𝑔 ∂𝜔𝑘

∣𝑍𝑐𝑔∣+ ∣𝑍𝑐𝑔∣−2∂𝜔𝑘

𝑍𝑐𝑔 (4.41)

are in 𝐿2(𝑄2𝑛;ℝ ⊕ 𝑖𝑛ℝ) by Lemmas 4.9, 4.6, and 4.7. Hence, we conclude that𝑀𝑘𝑔,𝐷𝑘𝑔 ∈ 𝐿2(ℝ𝑛;ℝ ⊕ 𝑖𝑛ℝ). For the functions 𝑔 and 𝑔, we shall next prove thefact

⟨𝑀𝑘𝑔,𝐷𝑘𝑔⟩ = ⟨𝐷𝑘𝑔,𝑀𝑘𝑔⟩ , (4.42)

from which we derive the contradiction. In fact, in the first place, by Corollary 3.4and Lemma 4.8 we find that

⟨𝑔, 𝑔m,n⟩ = ⟨𝑍𝑐𝑔, 𝑍𝑐𝑔m,n⟩ =[∫

𝑄2𝑛

𝑍𝑐𝑔𝑍𝑐𝑔𝑒2𝜋𝑖𝑛n⋅𝜔𝑒−2𝜋𝑖𝑛m⋅x𝑑𝑛x𝑑𝑛𝜔

]0

=

[∫𝑄2𝑛

𝑒2𝜋𝑖𝑛n⋅𝜔𝑒−2𝜋𝑖𝑛m⋅x𝑑𝑛x𝑑𝑛𝜔]0

= 𝛿m,0𝛿n,0, (4.43)

where 𝛿m,k denotes the Kronecker delta function. Similarly, we can check that

⟨𝑔, 𝑔m,n⟩ = 𝛿m,0𝛿n,0. (4.44)

Now, since 𝑀𝑘𝑔,𝐷𝑘𝑔 ∈ 𝐿2(ℝ𝑛;ℝ⊕ 𝑖𝑛ℝ) and {𝑔m,n}, {𝑔m,n} constitute dualframes for 𝐿2(ℝ𝑛;ℝ⊕ 𝑖𝑛ℝ), we obtain

⟨𝑀𝑘𝑔,𝐷𝑘𝑔⟩ =∑

m,n∈ℤ𝑛

⟨𝑀𝑘𝑔, 𝑔m,n⟩ ⟨𝑔m,n, 𝐷𝑘𝑔⟩ . (4.45)

Based on (4.35), (4.44), and (2.5), a simple calculation leads to

⟨𝑔−m,−n,𝑀𝑘𝑔⟩ =[∫

ℝ𝑛

𝑒−2𝜋𝑖𝑛m⋅x𝑔(x+ n)𝑥𝑘𝑔(x)𝑑𝑛x

]0

=

[∫ℝ𝑛

𝑒−2𝜋𝑖𝑛m⋅x𝑔(x+ n)𝑥𝑘 ˜𝑔(x)𝑑𝑛x]0

=

[∫ℝ𝑛

𝑥𝑘𝑔(x+ n) ˜𝑔(x)𝑒−2𝜋𝑖𝑛m⋅x𝑑𝑛x]0

=

[∫ℝ𝑛

(𝑥𝑘 − 𝑛𝑘)𝑔(x) ˜𝑔(x− n)𝑒−2𝜋𝑖𝑛m⋅(x−n)𝑑𝑛x

]0

= ⟨𝑀𝑘𝑔, 𝑔m,n⟩ − 𝑛𝑘 ⟨𝑔, 𝑔m,n⟩ = ⟨𝑀𝑘𝑔, 𝑔m,n⟩ . (4.46)

On the other hand, by (4.43) and (2.5) we obtain

⟨𝐷𝑘𝑔, 𝑔−m,−n⟩ =[∫

ℝ𝑛

∂𝑥𝑘𝑔(x)

1

2𝜋𝑖𝑛˜𝑔(x+ n)𝑒2𝜋𝑖𝑛m⋅x𝑑𝑛x

]0

316 Y. Fu, U. Kahler and P. Cerejeiras

= 0−[∫

ℝ𝑛

𝑔(x)1

2𝜋𝑖𝑛∂𝑥𝑘

(˜𝑔(x+ n)𝑒2𝜋𝑖𝑛m⋅x

)𝑑𝑛x

]0

= −

⎡⎢⎢⎣∫ℝ𝑛

𝑔(x)1

2𝜋𝑖𝑛

⎛⎜⎜⎝∂𝑥𝑘

˜𝑔(x+ n)𝑒2𝜋𝑖𝑛m⋅x

+

˜𝑔(x+ n)𝑚𝑘2𝜋𝑖𝑛𝑒2𝜋𝑖𝑛m⋅x

⎞⎟⎟⎠ 𝑑𝑛x

⎤⎥⎥⎦0

=

[∫ℝ𝑛

𝑒2𝜋𝑖𝑛m⋅x𝑔(x− n)𝐷𝑘𝑔(x)𝑑𝑛x

]0

−𝑚𝑘

[∫ℝ𝑛

𝑔(x)1

2𝜋𝑖𝑛˜𝑔(x+ n)2𝜋𝑖𝑛𝑒

2𝜋𝑖𝑛m⋅x𝑑𝑛x]0

= ⟨𝑔m,n, 𝐷𝑘𝑔⟩ ±𝑚𝑘 ⟨𝑔m,n, 𝑔⟩ = ⟨𝑔m,n, 𝐷𝑘𝑔⟩ . (4.47)

Consequently, substituting (4.46) and (4.47) into (4.45), we get

⟨𝑀𝑘𝑔,𝐷𝑘𝑔⟩ =∑

m,n∈ℤ𝑛

⟨𝑔−m,−n,𝑀𝑘𝑔⟩ ⟨𝐷𝑘𝑔, 𝑔−m,−n⟩

=∑

m,n∈ℤ𝑛

⟨𝐷𝑘𝑔, 𝑔−m,−n⟩ ⟨𝑔−m,−n,𝑀𝑘𝑔⟩ = ⟨𝐷𝑘𝑔,𝑀𝑘𝑔⟩ , (4.48)

which means that

⟨𝑔, 𝑔⟩ = 0 (4.49)

by Lemma 4.5. However, by (4.44), taking m = n = 0, we have

⟨𝑔, 𝑔⟩ = 1, (4.50)

which leads to the contradiction. Thus, we get

△𝑥𝑘△𝜔𝑘 =∞, 𝑘 = 1, 2, . . . , 𝑛. (4.51)

□

We are now in a position to extend Theorem 4.3 to the case of an arbitraryClifford frame.

Theorem 4.10 (Balian–Low theorem). Suppose that {𝑔m,n(x) : m,n ∈ ℤ𝑛} con-stitutes a frame for 𝐿2(ℝ𝑛;𝐶ℓ𝑛,0) associated to a non-zero window function 𝑔 in𝐿2(ℝ𝑛;𝐶ℓ𝑛,0). Then we have

△𝑥𝑘△𝜔𝑘 =∞, 𝑘 = 1, 2, . . . , 𝑛, (4.52)

where

△𝑥𝑘 =

∫ℝ𝑛

𝑥2𝑘 ∣𝑔(x)∣2 𝑑𝑛x, △𝜔𝑘 =

∫ℝ𝑛

𝜔2𝑘 ∣ℱ𝑔(𝜔)∣2 𝑑𝑛𝜔 (4.53)

and the CFT of 𝑔, ℱ𝑔(𝜔), is defined by (2.10).

Proof. If {𝑔m,n : m,n ∈ ℤ𝑛} constitutes a frame for 𝐿2(ℝ𝑛;𝐶ℓ𝑛,0) then the system{ℎm,n = [𝑔m,n]0 + 𝑖𝑛[𝑔m,n]𝑛 : m,n ∈ ℤ𝑛} is a frame for 𝐿2(ℝ𝑛;ℝ ⊕ 𝑖𝑛ℝ). ByTheorem 4.3 the result follows. □

15. The Balian–Low Theorem for the WCFT 317

5. Conclusions

The classical Balian–Low theorem is a strong form of the uncertainty principle forGabor systems which can be obtained by discretization of the kernel of the WFT.The WCFT is a generalization of the WFT in the framework of Clifford analysis.In this chapter, associated with the discretization of the kernel of the WCFT, weestablished a new kind of Gabor system

𝑔m,n(x) := 𝑒2𝜋𝑖𝑛m⋅x𝑔(x− n), m,n ∈ ℤ𝑛, (5.1)

which constitutes a Clifford frame satisfying the frame condition

𝐵 ∥𝑓∥2 ≤∑

m,n∈ℤ𝑛

∣⟨𝑓, 𝑔m,n⟩∣2 ≤ 𝐶 ∥𝑓∥2 , 𝐵, 𝐶 > 0, ∀𝑓 ∈ 𝐿2(ℝ𝑛;𝐶ℓ𝑛,0).

(5.2)To analyse Gabor systems {𝑔m,n(x) : m,n ∈ ℤ𝑛} in 𝐿2(ℝ𝑛, 𝐶ℓ𝑛,0), we establishedthe definition of the Clifford–Zak transform for 𝑓 ∈ 𝐿2(ℝ𝑛;𝐶ℓ𝑛,0) by

𝑍𝑐𝑓(x, 𝜔) :=∑k∈ℤ𝑛

𝑒2𝜋𝑖𝑛k⋅𝜔𝑓(x− k), (x, 𝜔) ∈ 𝑄2𝑛, (5.3)

and showed some properties of it. Furthermore, we proved the correspondingBalian–Low theorem for such Gabor systems which form a Clifford frame.

Acknowledgment

The first author is the recipient of a postdoctoral grant from Fundacao para aCiencia e a Tecnologia, ref. SFRH/BPD/46250/2008. This work was supportedby FEDER funds through COMPETE – Operational Programme Factors of Com-petitiveness (‘Programa Operacional Factores de Competitividade’) and by Por-tuguese funds through the Center for Research and Development in Mathematicsand Applications (University of Aveiro) and the Portuguese Foundation for Scienceand Technology (‘FCT – Fundacao para a Ciencia e a Tecnologia’), within projectPEst-C/MAT/UI4106/2011 with COMPETE number FCOMP-01-0124-FEDER-022690. The work was also supported by the Foundation of Hubei EducationalCommittee (No. Q20091004) and the NSFC (No. 11026056).

References

[1] M. Bahri. Generalized Fourier transform in Clifford algebra 𝐶𝑙(0, 3). Far East Jour-nal of Mathematical Science, 44(2):143–154, Sept. 2010.

[2] M. Bahri. Generalized Fourier transform in real clifford algebra 𝑐𝑙(0, 𝑛). Far EastJournal of Mathematical Sciences, 48(1):11–24, Jan. 2011.

[3] M. Bahri, S. Adji, and J.M. Zhao. Real Clifford windowed Fourier transform. ActaMathematica Sinica, 27(3):505–518, 2011.

[4] M. Bahri, E. Hitzer, R. Ashino, and R. Vaillancourt. Windowed Fourier transformof two-dimensional quaternionic signals. Applied Mathematics and Computation,216(8):2366–2379, June 2010.

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[5] R. Balan. Extensions of no-go theorems to many signal systems. In A. Aldroubi andE.B. Lin, editors, Wavelets, Multiwavelets, and Their Applications, volume 216 ofContemporary Mathematics, pages 3–14. American Mathematical Society, 1998.

[6] I.J. Benedetto, W. Czaja, and A.Y. Maltsev. The Balian–Low theorem for the sym-plectic form on ℝ2𝑑. Journal of Mathematical Physics, 44(4):1735–1750, 2003.

[7] J.J. Benedetto, C. Heil, and D.F. Walnut. Differentiation and the Balian–Low the-orem. Journal of Fourier Analysis and Applications, 1(4):355–402, 1994.

[8] F. Brackx, N. De Schepper, and F. Sommen. The two-dimensional Clifford–Fouriertransform. Journal of Mathematical Imaging and Vision, 26(1):5–18, 2006.

[9] F. Brackx, N. De Schepper, and F. Sommen. The Fourier transform in Cliffordanalysis. Advances in Imaging and Electron Physics, 156:55–201, 2009.

[10] F. Brackx, R. Delanghe, and F. Sommen. Clifford Analysis, volume 76. Pitman,Boston, 1982.

[11] T. Bulow, M. Felsberg, and G. Sommer. Non-commutative hypercomplex Fouriertransforms of multidimensional signals. In G. Sommer, editor, Geometric computingwith Clifford Algebras: Theoretical Foundations and Applications in Computer Visionand Robotics, pages 187–207, Berlin, 2001. Springer.

[12] T. Bulow and G. Sommer. Quaternionic Gabor filters for local structure classifi-cation. In Proceedings Fourteenth International Conference on Pattern Recognition,volume 1, pages 808–810, 16–20 Aug 1998.

[13] W. Czaja and A.M. Powell. Recent developments in the Balian–Low theorem. InC. Heil, editor, Harmonic Analysis and Applications, pages 79–100. Birkhauser,Boston, 2006.

[14] I. Daubechies. Ten Lectures on Wavelets. Society for Industrial and Applied Math-ematics, 1992.

[15] J. Ebling and G. Scheuermann. Clifford Fourier transform on vector fields. IEEETransactions on Visualization and Computer Graphics, 11(4):469–479, JulyAug.2005.

[16] J. Ebling and J. Scheuermann. Clifford convolution and pattern matching on vectorfields. In Proceedings IEEE Visualization, volume 3, pages 193–200, Los Alamitos,CA, 2003. IEEE Computer Society.

[17] M. Felsberg. Low-Level Image Processing with the Structure Multivector. PhD thesis,Christian-Albrechts-Universitat, Institut fur Informatik und Praktische Mathematik,Kiel, 2002.

[18] K. Grochenig. Foundations of Time-Frequency Analysis. Applied and Numerical Har-monic Analysis. Birkhauser, Boston, 2001.

[19] K. Grochenig, D. Han, and G. Kutyniok. The Balian–Low theorem for the symplecticlattice in higher dimensions. Applied and Computational Harmonic Analysis, 13:169–176, 2002.

[20] C. Heil and D.F. Walnut. Continuous and discrete wavelet transforms. SIAM Review,31:628–666, 1989.

[21] S. Held, M. Storath, P. Massopust, and B. Forster. Steerable wavelet frames based onthe Riesz transform. IEEE Transactions on Image Processing, 19(3):653–667, 2010.

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[22] E.M.S. Hitzer and B. Mawardi. Clifford Fourier transform on multivector fields anduncertainty principles for dimensions 𝑛 = 2(mod 4) and 𝑛 = 3(mod 4). Advances inApplied Clifford Algebras, 18(3-4):715–736, 2008.

[23] B. Mawardi, E. Hitzer, and S. Adji. Two-dimensional Clifford windowed Fouriertransform. In E. Bayro-Corrochano and G. Scheuermann, editors, Applied GeometricAlgebras in Computer Science and Engineering, pages 93–106. Springer, London,2010.

[24] B. Mawardi and E.M.S. Hitzer. Clifford Fourier transformation and uncertainty prin-ciple for the Clifford algebra 𝐶ℓ3,0. Advances in Applied Clifford Algebras, 16(1):41–61, 2006.

[25] M. Porat, Y.Y. Zeevi, and M. Zibulski. Multi-window Gabor schemes in signal andimage representations. In H.G. Feichtinger and T. Strohmer, editors, Gabor Analysisand Algorithms, pages 381–408. Birkhauser, 1998.

[26] S.J. Sangwine and T.A. Ell. Hypercomplex Fourier transforms of color images. InIEEE International Conference on Image Processing (ICIP 2001), volume I, pages137–140, Thessaloniki, Greece, 7–10 Oct. 2001. IEEE.

[27] F. Sommen. Hypercomplex Fourier and Laplace transforms I. Illinois Journal ofMathematics, 26(2):332–352, 1982.

[28] Y.Y. Zeevi and M. Zibulski. Analysis of multiwindow Gabor-type schemes by framemethods. Applied and Computational Harmonic Analysis, 4:188–221, 1997.

Yingxiong FuHubei Key Laboratory of Applied MathematicsHubei University

and

Faculty of Mathematics and Computer ScienceHubei University, Hubei, China

and

Department of MathematicsUniversity of Aveiro, Portugal

e-mail: fyx@ua.pt

Uwe Kahler and Paula CerejeirasDepartment of MathematicsUniversity of Aveiro, Portugal

e-mail: ukaehler@ua.ptpceres@ua.pt

Quaternion and Clifford–Fourier Transforms and Wavelets

Trends in Mathematics, 321–332c⃝ 2013 Springer Basel

16 Sparse Representation of Signalsin Hardy Space

Shuang Li and Tao Qian

Abstract. Mathematically, signals can be seen as functions in certain spaces.And processing is more efficient in a sparse representation where few coeffi-cients reveal the information. Such representations are constructed by decom-posing signals into elementary waveforms. A set of all elementary waveformsis called a dictionary. In this chapter, we introduce a new kind of sparse rep-resentation of signals in Hardy space 𝐻2(𝔻) via the compressed sensing (CS)technique with the dictionary

D = {𝑒𝑎 : 𝑒𝑎(𝑧) =

√1− ∣𝑎∣21− 𝑎𝑧

, 𝑎 ∈ 𝔻}.where 𝔻 denotes the unit disk. In addition, we give examples exhibiting thealgorithm.

Mathematics Subject Classification (2010). 30H05, 42A50, 42A38.

Keywords. Hardy space, compressed sensing, analytic signals, reproducingkernels, sparse representation, redundant dictionary, 𝑙1 minimization.

1. Introduction

A basis gives unique representations for signals in some certain space. However,it does not always give sparse expressions. One of the problems of approximationtheory is to approximate functions with elements from a large candidate set calleda dictionary. Let 𝐻 be a Hilbert space. Using terminology introduced by Mal-lat and Zhang [18], a dictionary is defined as a family of parameterized vectors

D = {𝑔𝛾}𝛾∈Γ in 𝐻 such that ∥𝑔𝛾∥ = 1 and span(𝑔𝛾) = 𝐻 . The 𝑔𝛾 are usuallycalled atoms. For the discrete-time situation, the approximation problem can bewritten as

𝑠 = D𝑥 (1.1)

where 𝑠 is the discrete signal, matrix D represents the dictionary with atoms ascolumns and 𝑥 is the vector of coefficients. Notice that we adopt the vector inner

322 S. Li and T. Qian

product instead of the Hilbert inner product. In general, D has more columnsthan rows because of redundancy. A natural question is: can we find the best𝑀 -term approximation in a redundant dictionary for a given signal? That is anoptimization problem:

min ∥𝑠−D𝑥∥𝑙2 subject to ∥𝑥∥𝑙0 ≤𝑀 (1.2)

where ∥𝑥∥𝑙0 is the number of nonzero coefficients of 𝑥. Unfortunately, findingan optimal 𝑀 -term approximation in redundant dictionaries is computationallyintractable because it is NP-hard [8, 9, 17]. Thus, it is necessary to rely on goodbut not optimal approximations with computational algorithms. Until now, threemain strategies have been investigated, they are matching pursuit , basis pursuitand compressed sensing.

1.1. Matching Pursuit and Basis Pursuit

Matching pursuit (MP) introduced by Mallat and Zhang computes signal approx-imations from a redundant dictionary by iteratively selecting one vector at a time.It is an example of a greedy algorithm. For a detailed description, please referto [17, 18]. A recent development is the Adaptive Fourier Decomposition (AFD),which is a variation of the greedy algorithm in the particular context of Hardyspaces [20]. An intrinsic feature of the algorithm is that when stopped after afew steps, it yields an approximation using only a few atoms. If the dictionary isorthogonal, the method works perfectly. If the dictionary is not orthogonal, thesituation is less clear [6]. The MP algorithm often yields locally optimal solutionsdepending on initial values. In contrast, basis pursuit (BP) performs a more globalsearch. It finds signal representations by solving the following problem

min ∥𝑥∥𝑙1 subject to 𝑠 = D𝑥. (1.3)

Given 𝑠 and D , we find 𝑥 with minimal 𝑙1 norm. Notice that (1.3) is a convexoptimization program which is not NP-hard. Actually, the use of 𝑙1 minimizationto promote sparsity has a long history, dating back at least to the work of Beurling[1] on Fourier transform extrapolation from partial observations. Basis pursuit isan optimization principle, not an algorithm. Empirical evidence suggests that BPis more powerful than MP [6]. And the stability of BP has been proved in thepresence of noise for sufficiently sparse representations [11]. BP is closely connectedwith convex programming. The interior-point method and the homotopy methodcan be applied to BP in nearly linear time [6, 13].

1.2. Compressed Sensing

If the original signal is sparse in some sense, compressed sensing (CS) gives anexcellent recovery of the signal. CS is a new concept in signal processing. The ideashave their origins in certain abstract results by Kashin [7, 15] but were broughtinto the forefront by the work of Candes, Romberg and Tao [2–5] and Donoho [10].The core idea behind CS is that a signal or image, unknown but supposed to becompressible by a known transform, can be subjected to fewer measurements than

16. Sparse Representation of Signals in Hardy Space 323

the nominal number of pixels, and yet be accurately reconstructed [12]. Basically,CS relies on random projection and BP. Suppose we have

𝑦 = Φ𝑥, (1.4)

where 𝑥 is a finite vector, Φ is observation matrix and 𝑦 is the vector of availablemeasurements. Then the BP solution 𝑥∗ of

min ∥𝑥∥𝑙1 subject to 𝑦 = Φ𝑥 (1.5)

recovers 𝑥 exactly provided that 𝑥 is sufficiently sparse and the matrix obeys theRestricted Isometry Property (RIP) [3,4]. However the RIP of a fixed matrix is veryhard to check, thus in practice we use random matrices instead. A Gaussian matrixΦ ∈ 𝑅𝑚×𝑁 whose entries Φ𝑖,𝑗 are independent and follow a normal distributionwith expectation 0 and variance 1/𝑚 is often adopted because we have [21] that:

Theorem 1.1. Let Φ ∈ ℝ𝑚×𝑁 be a Gaussian random matrix. Let 𝜀, 𝛿 ∈ (0, 1) andassume

𝑚 ≥ 𝐶𝛿−2(𝑠 log(𝑁/𝑠) + log 𝜀−1

)for a universal constant 𝐶 > 0. Then with probability at least 1 − 𝜀 the restrictedisometry constant of Φ satisfies 𝛿𝑠 ≤ 𝛿.

The theorem tells us that we can raise the probability by increasing thenumber of rows of the matrix Φ. We will not discuss details of CS, for more aboutthis technique, please see [2–5, 10, 12, 13, 21].

1.3. Hardy Space 𝑯2(𝔻)

Hardy space is an important class of spaces connected to analytic signals. Controltheory and rational approximation theory are also bound with the research in thisfield. In this chapter, we discuss signal decomposition in 𝐻2(𝔻). 𝐻2(𝔻) containsanalytic signals with finite energy. Denote 𝔻 = {𝑧 ∈ ℂ : ∣𝑧∣ < 1}, and let Hol(𝔻)be the space of analytic functions on 𝔻. For 𝑝 > 0, Hardy space 𝐻𝑝(𝔻) is definedas follows:

𝐻𝑝(𝔻) =

⎧⎨⎩𝑓 ∈ Hol(𝔻) : ∥𝑓∥𝑝𝐻𝑝 = sup0≤𝑟<1

2𝜋∫0

∣𝑓(𝑟𝑒𝑖𝑥)∣𝑝 d𝑥/2𝜋 <∞⎫⎬⎭ , (1.6)

and

𝐻∞(𝔻) =

{𝑓 ∈ Hol(𝔻) : ∥𝑓∥𝐻∞ = sup

𝑧∈𝔻∣𝑓(𝑧)∣ <∞

}. (1.7)

Indeed, 𝐻2(𝔻) is a Hilbert space consisting of all functions 𝑓(𝑧) =∑∞

𝑛=0 𝑎𝑛𝑧𝑛

analytic in the unit disc 𝔻 such that ∥𝑓∥2 =∑∞

𝑛=0 ∣𝑎𝑛∣2 <∞. It has reproducingkernels 𝑘𝑎(𝑧) =

11−𝑎𝑧, 𝑎 ∈ 𝔻. Besides the Fourier basis {1, 𝑧, 𝑧2, . . . , 𝑧𝑛, . . .}, 𝐻2(𝔻)

324 S. Li and T. Qian

has another orthonormal basis {𝑔𝑛}∞𝑛=1 named after Takenaka and Malmquist(TM):

𝑔1(𝑧) =

√1− ∣𝑧1∣21− 𝑧1𝑧

(1.8)

and

𝑔𝑛(𝑧) =

√1− ∣𝑧𝑛∣21− 𝑧𝑛𝑧

𝑛−1∏𝑘=1

𝑧 − 𝑧𝑘1− 𝑧𝑘𝑧

(1.9)

for 𝑛 ≥ 2, where {𝑧𝑛}∞𝑛=1 must satisfy

∞∑𝑛=1

1− ∣𝑧𝑛∣ =∞. (1.10)

It is clear that the TM basis can be obtained by applying the Gram–Schmidtprocedure to the reproducing kernels 𝑘𝑧𝑛 = 1/ (1− 𝑧𝑛𝑧) under condition (1.10).TM systems have long been associated with fruitful results in applied mathematicssuch as control theory, signal processing and system identification. Qian et al.recently proposed an adaptive Fourier decomposition algorithm (AFD) that resultsin a TM system (not necessarily a basis) with selected poles according to thegiven signal [20]. However AFD is valid only for one-dimensional signals whichnaturally raises the question: what happens in higher dimensions? Inspired by theaforementioned works, the adaptive decomposition of any function of two or threevariables is obtained in [19] by means of quaternionic analysis. Unfortunately, theresult can not be directly generalized into (𝑚+1)-dimensional space in the Cliffordsetting because a Clifford number is not invertible in general.

It is clear that the normalized reproducing kernel

D = {𝑒𝑎 : 𝑒𝑎(𝑧) =√1− ∣𝑎∣21− 𝑎𝑧

, 𝑎 ∈ 𝔻} (1.11)

forms a dictionary of 𝐻2(𝔻), and this redundant dictionary does give a sparse rep-resentation of signals in Hardy space. In [19] Qian et al. construct a dictionary fordecomposition of quaternionic-valued signals of finite energy. It is a generalizationof (1.11). This chapter will focus on the method based on CS to decompose signalsin 𝐻2(𝔻). Later we will generalize it into quaternionic Hardy space.

The chapter is concisely arranged as follows. We discuss the main results inSection 2. In Section 3, examples are given to illustrate our algorithm.

2. Main Results

The singular value decomposition (SVD) is an effective tool for the analysis oflinear operators. It is also useful in this chapter. We recall that:

16. Sparse Representation of Signals in Hardy Space 325

Theorem 2.1 (Singular Value Decomposition [16]). Let 𝐴 denote an arbitrary ma-trix with elements in ℂ𝑚×𝑛 and let {𝑠𝑖}𝑟𝑖=1 be the nonzero singular values of 𝐴.Then 𝐴 can be represented in the form

𝐴 = 𝑈𝐷𝑉 ∗, (2.1)

where 𝑈 ∈ ℂ𝑚×𝑚, 𝑉 ∈ ℂ𝑛×𝑛 are unitary and the 𝑚× 𝑛 matrix 𝐷 has elements 𝑠𝑖in the 𝑖, 𝑖 position (1 ≤ 𝑖 ≤ 𝑟) and zeros elsewhere. The 𝑠𝑖 are singular values of

𝐴, 𝑠𝑖 =√

𝜆𝑖(𝐴∗𝐴). In fact, the diagonal matrix 𝐷 can be written as(Σ𝑟 00 0

)(2.2)

where Σ𝑟 = diag(𝑠1, 𝑠2, . . . , 𝑠𝑟) with 𝑠1 ≥ 𝑠2 ≥ ⋅ ⋅ ⋅ , 𝑠𝑟.In the model (1.1), (1.11), we select 𝑁 points {𝑎𝑘}𝑁𝑘=1 in 𝔻, and sample 𝑀

points for each 𝑒𝑎𝑘equally spaced in [0, 2𝜋]. So we get a matrix D ∈ ℂ𝑀×𝑁 . For

a signal in 𝐻2(𝔻), we also sample 𝑀 points on its boundary value equally spacedto form a vector 𝑠 ∈ ℂ𝑀 . Thus, the representation problem is

𝑠 = D𝑥. (2.3)

Suppose that D = 𝑈𝐷𝑉 ∗, then 𝑈∗𝑠 = 𝐷𝑉 ∗𝑥. Note that 𝐷 is a diagonal matrix

and ∥𝑈∗𝑠∥2 = ∥𝑠∥2 , ∥𝑉 ∗𝑥∥2 = ∥𝑥∥2. If the singular values of D decay fast, we canexpect a relatively sparse representation 𝑥 with a small energy error. The mostimportant thing is that the assertion should be established in the sense of 𝑁 →∞because one must explain what the situation would be when the number of atomsis large.

First, we give two lemmas, they are also the simple cases of our theorem.

Lemma 2.2. Suppose 𝑁 points {𝑎𝑘}𝑁−1𝑘=0 are selected equally spaced on the circle

of radius 𝑟. Let 𝜆1 ≥ 𝜆2 ≥ ⋅ ⋅ ⋅ ≥ 𝜆𝑁 be eigenvalues of the matrix D∗D . Then wehave

lim𝑁→∞

1

𝑁

𝑚∑𝑗=1

𝜆𝑗 ≥ 1− 𝑟2𝑚. (2.4)

Proof. (Sketch of proof.) Let

D =(𝑒𝑎0 𝑒𝑎1 . . . 𝑒𝑎𝑁−1

)(2.5)

then the Hermitian matrix 𝐻 can be written as

𝐻 = D∗D =

⎛⎜⎜⎜⎝𝑒𝑎0

𝑒𝑎1

...𝑒𝑎𝑁−1

⎞⎟⎟⎟⎠(𝑒𝑎0 𝑒𝑎1 . . . 𝑒𝑎𝑁−1

). (2.6)

with elements 𝐻𝑖𝑗 =⟨𝑒𝑎𝑖 , 𝑒𝑎𝑗

⟩=⟨𝑒𝑎0 , 𝑒𝑎𝑗−𝑖

⟩= 𝑏𝑗−𝑖. Notice that

𝑏𝑘 = ⟨𝑒𝑎0 , 𝑒𝑎𝑘⟩ = 1− 𝑟2

1− 𝑟2𝑒𝑖𝑘𝜃=

1− 𝑟2

1− 𝑟2𝑒𝑖𝜃𝑘, (𝑘 = 0, 1, . . . , 𝑁 − 1) (2.7)

326 S. Li and T. Qian

0 10 20 30 40 500

2

4

6

8

10

12

14

16

18

20

0 10 20 30 40 500

1

2

3

4

5

6

7

8

9

10

Figure 1. Selecting 𝑁 points on the circle of radius 𝑟 (2.9) gives that𝜆𝑗 ≈ (1 − 𝑟2)𝑟2(𝑗−1)𝑁 when 𝑁 is large. The small red circles are thecorresponding estimation whereas the blue points represent the realeigenvalues given by numerical calculation. They fit amazingly well eventhough 𝑁 is not very large.

where 𝜃𝑘 = 𝑘𝜃 is the argument of 𝑎𝑘. Recall Ky Fan’s maximum principle: let 𝐴be any Hermitian operator, then for 𝑘 = 1, 2, . . . , 𝑛, we have

𝑘∑𝑗=1

𝜆𝑗(𝐴) = max

𝑘∑𝑗=1

⟨𝐴𝑥𝑗 , 𝑥𝑗⟩ (2.8)

where the eigenvalues 𝜆1(𝐴) ≥ 𝜆2(𝐴) ≥ ⋅ ⋅ ⋅ ≥ 𝜆𝑛(𝐴), and the maximum are takenover all orthonormal 𝑘-tuples {𝑥1, . . . , 𝑥𝑘}.

Sampling 𝑁 points equally spaced on the orthonormal functions{𝑒−𝑖𝑗𝑡}𝑚−1

𝑗=0 , we prove that

lim𝑁→∞

𝜆𝑗+1

𝑁= (1− 𝑟2)𝑟2𝑗 (2.9)

where 𝜆1 ≥ 𝜆2 ≥ ⋅ ⋅ ⋅ ≥ 𝜆𝑚. Hence

lim𝑁→∞

1

𝑁

𝑚∑𝑘=1

𝜆𝑘 ≥𝑚∑𝑘=1

(1 − 𝑟2)𝑟2(𝑘−1) = 1− 𝑟2𝑚. (2.10)

□

Remark 2.3. Note that trace(𝐻) = 𝑁 , thus (2.9) shows that the eigenvalues decayas a geometric series with common ratio 𝑟2 as 𝑁 →∞. See Figure 1.

Lemma 2.4. Suppose 𝑁 points {𝑎𝑘}𝑁𝑘=1 are selected equally spaced on the segment

[0, 1), and 𝜆1 is the largest eigenvalue of the matrix D∗D . Then we have

lim𝑁→∞

𝜆1

𝑁≥

1∫0

1∫0

√1− 𝑠2

√1− 𝑟2

1− 𝑠𝑟d𝑟 d𝑠 ≈ 0.815784. (2.11)

16. Sparse Representation of Signals in Hardy Space 327

0 10 20 30 40 50

40

45

35

30

25

20

15

10

5

0

−50 10 20 30 40 50 60

−10

0

10

20

30

40

50

Figure 2. Eigenvalue distribution

Sketch of Proof. The proof is similar to the previous lemma. In this situation, theentries of the Hermitian matrix 𝐻 in (2.6) are

𝐻𝑖𝑗 =

√1− 𝑟2𝑖

√1− 𝑟2𝑗

1− 𝑟𝑖𝑟𝑗, (2.12)

where 𝑟𝑖 is the 𝑖th point on [0, 1). Let 𝑥 =(

1√𝑁, 1√

𝑁, . . . , 1√

𝑁

)∈ ℂ𝑁 with ∥𝑥∥2 =

1. We use Ky Fan’s principle to estimate the maximum eigenvalue. We find that⟨𝐻𝑥, 𝑥⟩ is a double Riemann sum of

1∫0

1∫0

√1− 𝑠2

√1− 𝑟2

1− 𝑠𝑟d𝑟 d𝑠. (2.13)

Hence,𝜆1(𝐻)

𝑁≥ ⟨𝐻𝑥, 𝑥⟩

𝑁≈ 0.815784. (2.14)

□Remark 2.5. Select 𝑁 points on the interval [0, 1). The maximum eigenvalues are41.1816 and 49.5133 respectively. 𝜆1(𝐻)/𝑁 satisfy (2.11). See Figure 2.

Generally, we select {𝑎𝑘} in the whole disc as follows. Divide [0, 2𝜋] and [0, 1)into 𝑁1 and 𝑁2 parts respectively. Hence, the number of 𝑎𝑘s is 𝒪(𝑁1𝑁2). Weobtain the main theorem as follows.

Theorem 2.6. Let 𝜆1 ≥ 𝜆2 ≥ 𝜆3 ≥ ⋅ ⋅ ⋅ be eigenvalues of 𝐻 = D∗D . Then we have

lim𝑁1→∞𝑁2→∞

𝑛∑𝑘=1

𝜆𝑘

𝑁1𝑁2≥ 1− 1

2𝑛+ 1. (2.15)

328 S. Li and T. Qian

Sketch of Proof. 𝐻 is actually a block matrix in this situation

𝐻 =

⎛⎜⎜⎜⎜⎜⎝𝐵∗1𝐵1 𝐵∗1𝐵2 𝐵∗1𝐵3 . . . 𝐵∗1𝐵𝑁2

𝐵∗2𝐵1 𝐵∗2𝐵2 𝐵∗2𝐵3 . . . 𝐵∗2𝐵𝑁2

𝐵∗3𝐵1 𝐵∗3𝐵2 𝐵∗3𝐵3 . . . 𝐵∗3𝐵𝑁2

......

.... . .

...𝐵∗𝑁2

𝐵1 𝐵∗𝑁2𝐵2 𝐵∗𝑁2

𝐵3 . . . 𝐵∗𝑁2𝐵𝑁2

⎞⎟⎟⎟⎟⎟⎠ (2.16)

with each block 𝐵∗𝑖 𝐵𝑗 ∈ ℂ𝑁1×𝑁1 . Let

𝑓𝑛(𝑟) = 𝑟𝑛√1− 𝑟2 (2.17)

and

𝑔𝑛(𝑟) =𝑓𝑛(𝑟)

∥𝑓𝑛(𝑟)∥𝐿2(0,1)

(2.18)

where 𝑟 ∈ (0, 1), 𝑛 ≥ 0. Denote 𝑒𝑛(𝜃) = 𝑒−𝑖𝑛𝜃, 𝑛 ≥ 0.

Sample 𝑁2 points on 𝑔𝑛(𝑟) and get the vector 𝐺𝑛 ∈ ℂ𝑁2 . Sample 𝑁1 pointson 𝑒𝑛 to get 𝐸𝑛 ∈ ℂ𝑁1 . Consider

⟨𝐻(𝐺𝑛

⊗𝐸𝑛), 𝐺𝑛

⊗𝐸𝑛⟩

𝑁1𝑁2(2.19)

where the tensor product 𝐺𝑛

⊗𝐸𝑛 is a vector in ℂ𝑁1𝑁2 . We prove that

lim𝑁1→∞𝑁2→∞

⟨𝐻(𝐺𝑛

⊗𝐸𝑛), 𝐺𝑛

⊗𝐸𝑛⟩

𝑁1𝑁2=

1

2𝑛+ 1− 1

2𝑛+ 3, 𝑛 ≥ 0. (2.20)

Hence, Ky Fan’s maximum principle gives

lim𝑁1→∞𝑁2→∞

𝑛∑𝑘=1

𝜆𝑘

𝑁1𝑁2≥

𝑛∑𝑘=1

(1

2𝑘 − 1− 1

2𝑘 + 1

)= 1− 1

2𝑛+ 1. (2.21)

□

Remark 2.7. Note that trace(𝐻) = 𝑁1𝑁2, the theorem shows that the eigenvaluesdecay rapidly. Figure 3 shows the numerical calculation when 𝑁1 = 𝑁2 = 40. Wehave that 𝜆1/(𝑁1𝑁2) = 0.6544178 ≈ 2/(1× 3), 𝜆2/(𝑁1𝑁2) = 0.1312327 ≈ 2/(3×5), 𝜆3/(𝑁1𝑁2) = 0.057843 ≈ 2/(5×7), 𝜆4/(𝑁1𝑁2) = 0.03319499≈ 2/(7×9), andso on. The several largest eigenvalues contribute a large part of the sum of all eigen-values. Since singular value 𝑠𝑖 =

√𝜆𝑖(𝐻), we assert that 𝑠𝑖 tends to zero rapidly.

3. Numerical Examples

We give two numerical examples. In our examples, D ∈ ℂ900×3000. This means thedictionary has 3000 atoms and each atom vector is of size 900. We embed D into

16. Sparse Representation of Signals in Hardy Space 329

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

1 2 3 4 5 6 7 8 9 10

Figure 3. Decay of eigenvalues (see Remark 2.7).

ℝ1800×6000 for programming convenience by

D =

(ℜ (D) −ℑ (D)ℜ (D) ℜ (D)

). (3.1)

Therefore,

𝑠 = D𝑥⇐⇒ 𝑠 =

(ℜ (𝑠)ℑ (𝑠)

)= D

(ℜ (𝑥)ℑ (𝑥)

)= D 𝑥. (3.2)

where ℜ (𝑠) and ℑ (𝑠) are the real and imaginary parts respectively.

Consider the minimization problem

min ∥𝑥∥1 subject to 𝑦 = ΦD 𝑥 (3.3)

where Φ ∈ ℝ𝑛×𝑁 is a Gaussian random matrix satisfying Φ𝑖𝑗 ∼ 𝒩 (0, 1/𝑛). We

recover the signal by D𝑥∗ when the solution 𝑥∗ of (3.3) is obtained by convexprogramming. As we mentioned above, 𝑠 belongs to the vector space of 900× 2 =1800 dimensions with a 900-dimensional real part and 900-dimensional imaginary

part. D has 3000× 2 = 6000 columns and 𝑥 is a 6000-element coefficient vector.

3.1. Example 1

𝑠(𝑧) =0.247𝑧4 + 0.0355𝑧3

(1− 0.9048𝑧)(1− 0.3679𝑧)(3.4)

Set the Gaussian randommatrix Φ with 160 rows and its entries Φ𝑖𝑗 ∼ 𝒩 (0, 1/160).Runtime is 16.437 seconds and relative error is 3.7188×10−4. See Figures 4 and 5.

3.2. Example 2

𝑠(𝑧) = 𝑧20 + 𝑧10 + 𝑧5. (3.5)

Choose the Gaussian randommatrix Φ with 160 rows and entries Φ𝑖,𝑗∼𝒩 (0,1/160).Runtime is 34.891 seconds and relative error is 1.7× 10−3. See Figures 6 and 7.

330 S. Li and T. Qian

0 200 400 600 800 1000 1200 1400 1600 1800 0 200 400 600 800 1000 1200 1400 1600 1800−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

1.2

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

1.2

recovered signaloriginal signal

Figure 4. Example 1. The recovered signal almost coincides with theoriginal signal. See Section 3.1.

0 200 400 600 800 1000 1200 1400 1600 1800−2.5

−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

2.5

0 1000 2000 3000 4000 5000 6000−0.05

0

0.05

0.1

0.2

0.3

0.15

0.25

Figure 5. Example 1. See Section 3.1.

References

[1] A. Beurling. Sur les integrales de Fourier absolument convergentes et leur applica-tion a une transformation functionelle. In Proceedings Scandinavian MathematicalCongress, Helsinki, Finland, 1938.

[2] E. Candes, J. Romberg, and T. Tao. Robust uncertainty principles: Exact signalreconstruction from highly incomplete frequency information. IEEE Transactionson Information Theory, 52(2):489–509, 2006.

[3] E. Candes, J. Romberg, and T. Tao. Stable signal recovery from incomplete and in-accurate measurements. Communications on Pure Applied Mathematics, 59(8):1207–1223, 2006.

[4] E. Candes and T. Tao. Decoding by linear programming. IEEE Transactions onInformation Theory, 51(12):4203–4215, 2005.

16. Sparse Representation of Signals in Hardy Space 331

0 200 400 600 800 1000 1200 1400 1600 1800 0 200 400 600 800 1000 1200 1400 1600 1800−3

−2

−1

0

1

2

3recovered signaloriginal signal

−3

−2

−1

0

1

2

3

Figure 6. Example 2. The recovered signal almost coincides with theoriginal signal. See Section 3.2.

0 200 400 600 800 1000 1200 1400 1600 1800 0 1000 2000 3000 4000 5000 6000−40

−30

−20

−10

0

10

20

30

−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

0.4

0.5

Figure 7. Example 2. See Section 3.2.

[5] E. Candes and T. Tao. Near optimal signal recovery from random projections: Uni-versal encoding stategies? IEEE Transactions on Information Theory, 52(12):5406–5425, 2006.

[6] S. Chen, D. Donoho, and M. Saunders. Atomic decomposition by basis pursuit. SIAMJournal on Scientific Computing, 20(1):33–61, 1999.

[7] A. Cohen, W. Dahmen, and R. DeVore. Compressed sensing and best 𝑘-term ap-proximation. Journal of the American Mathematical Society, 22:211–231, 2009.

[8] G. Davis. Adaptive Nonlinear Approximations. PhD thesis, New York University,Courant Institute, 1994.

[9] G. Davis and S. Mallat. Adaptive greedy approximations. Constructive Approxima-tion, 13(1):57–98, 1997.

332 S. Li and T. Qian

[10] D. Donoho. Compressed sensing. IEEE Transactions on Information Theory, 52(4):1289–1306, 2006.

[11] D. Donoho and M. Elad. On the stability of the basis pursuit in the presence ofnoise. Signal Processing, 86(3):511–532, 2006.

[12] D. Donoho and Y. Tsaig. Extensions of compressed sensing. Signal Processing,86(3):533–548, 2006.

[13] M. Fornasier. Numerical methods for sparse recovery. In Theoretical Foundationsand Numerical Methods for Sparse Recovery [14], pages 93–200.

[14] M. Fornasier, editor. Theoretical Foundations and Numerical Methods for SparseRecovery, volume 9 of Radon Series on Computational and Applied Mathematics.De Gruyter, Germany, 2010.

[15] B. Kashin. The widths of certain finite dimensional sets and classes of smooth func-tions. Izvestia, 41:334–351, 1977.

[16] P. Lancaster and M. Tismenetsky. The Theory of Matrices with Applications. Aca-demic Press, second edition, 1985.

[17] S. Mallat. A Wavelet Tour of Signal Processing. Academic Press, third edition, 2008.First edition published 1998.

[18] S. Mallat and Z. Zhang. Matching pursuit with time-frequency dictionaries. IEEETransactions on Signal Processing, 41(12):3397–3415, Dec. 1993.

[19] T. Qian, W. Sproßig, and J. Wang. Adaptive Fourier decomposition of functions inquaternionic Hardy spaces. Mathematical Methods in the Applied Sciences, 35(1):43–64, 2012.

[20] T. Qian and Y.-B. Wang. Adaptive Fourier series – a variation of greedy algorithm.Advances in Computational Mathematics, 34(3):279–293, 2011.

[21] H. Rauhut. Compressive sensing and structured random matrices. In Fornasier [14],pages 1–92.

Shuang Li and Tao QianDepartment of MathematicsUniversity of Macau, Macaue-mail: ya97418@umac.mo

fsttq@umac.mo

Quaternion and Clifford–Fourier Transforms and Wavelets

Trends in Mathematics, 333–338c⃝ 2013 Springer Basel

Index

analytic signal, 42, 67, 222, 223, 247–250

– Clifford, 208

– biquaternion, 197

– complex, 197

– examples, 209

– hypercomplex, 48

– local amplitude, 223

– local phase, 223

– 𝑛-dimensionnal, 198

– one-dimensional, 200

– phases, 210

– properties, 201

– quaternion, 197

– quaternion 2D, 201

– quaternionic, 68

– two-dimensional, 200, 226

– local amplitude, 227

– local orientation, 227

– local phase, 227

– video, 198, 212

angular velocity, 50

anticommutative part, 162

atomic function, 58, 64

– 2D, 60

– quaternionic, 65

– up(𝑥), 58

Balian–Low theorem, 303, 316, 317

Banach module, 287

basis pursuit, 322

bivector, geometric interpretation, 124

Bochner theorem, 85

Bochner–Minlos theorem, 85, 113, 117

Cauchy–Riemann equations, 63

Cayley–Dickson form, 43

– polar, 44

checkerboard, 233

chrominance, 22

– plane, 22

clifbquat, see Clifford biquaternion

Clifford algebra, 198, 286

– 𝐶ℓ0,3, basis, 133

– 𝐶ℓ1,2, 134

– 𝐶ℓ3,0, 134

– automorphism group, 126

– basis element matrices, 145

– center, 126

– central pseudoscalar, 134

– characteristic polynomial, 136

– complex, 192

– complex conjugation, 135

– conformal geometric algebra, 135

– connected components

– Klein group, 131

– square isometries, 132

– even subalgebra, 125

– Fourier transform

– steerable, 141

– geometric algebras, 123

– grade involution, 137

– group of invertible elements, 126

– idempotents, 133, 134

– isomorphism

– CLIFFORD package, 146

– Lorentz space, 135

– matrix idempotents, 147

– matrix ring isomorphisms, 124, 125

– dimension index 𝑑, 125

– signature, 125

334 Index

– minimal polynomial, 134

– Pauli matrix algebra, 134

– pseudoscalar, 127

– relation to quaternions, 124

– reversion, 137

– scalar part, 127

– signature, 125

– software package, 127

– spinor representation, 146

– swap automorphism, 131

– symbolic computer algebra, 127

– trace, 127

– wavelet transform, steerable, 141

Clifford analysis, applications, 124

Clifford biquaternion, 206

Clifford conjugate, 286

Clifford functions, 289

Clifford module, 289

Clifford polynomials, 146

Clifford, William Kingdon, 123, 198

Clifford–Fourier transform, see Fouriertransform

colour

– denoising, 257, 259, 261–264

– image

– Fourier transform, 158

– grey line, 22

– processing, 15, 22

commutative part, 162

complex

– Clifford algebra, 192

– degenerate, 10

– envelope, 50

– representation, 181, 192

compressed sensing, 322

convolution, 12

correlation, 11

cylindrical Fourier transform, 159

decomposition

– luminance and chrominance, 22

– singular value, 324

demodulation, 238

Dirac equation, 180, 183

Dirac operator, 194, 228

dup(𝑥), 61

dyadic shifts, 63

edge detection, 183

eigen-angle, 7

eigen-axis, 7

envelope, 42

– complex, 50

Euler formula for quaternions, 237

even-odd form, 6

exponential function, 157

Faddeev’s Green function, 281

– Dirac operators, 281

filter design

– steerable, 39

filter, steerable quaternionic, 67

filtering, 188

force-free fields, 274

Fourier transform, 222

– Clifford–, 157, 185, 197, 200, 207,288, 299, 302

– examples, 213

– properties (scalar function), 209

– windowed, 294, 300, 303, 317

– cylindrical, 159

– geometric, see geometric Fouriertransform

– properties, 201

– quaternion, 43, 44

– properties, 203

– quaternionic, see quaternionicFourier transform

– Sommen–, 157

Fourier–Stieltjes transform, 85

– quaternionic, 88

frame

– bound, 303

– Clifford, 303, 307, 317

– dual, 304, 314

– tight, 303

frequency modulation, 11

Fubini’s theorem, 296

Gabor filter

– complex, 294

– quaternionic, 294

Index 335

Gabor system, 300

– Clifford–, 303, 305, 310

Gabor, D., 200

Gauss spinor formula, 194

generalized Weierstrass parametrization,183

geometric Fourier transform

– definition, 157

– existence, 159

– linearity, 160

– product theorem, 166

– scaling theorem, 162

– shift theorem, 172

geometric product, 156

GFT, 157

Gram–Schmidt procedure, 324

Grassmann, Hermann Gunther, 198

group velocity, 212

Hamilton, William Rowan, 43, 198

Hardy space, 323

Hausdorff space, 110

Helmholtz equation, 275

– factorization, 275

Hilbert space, 321, 323

Hilbert transform, 42, 47, 63, 67, 70, 222

– partial, 66, 226

– quaternion Fourier transform of, 47

– total, 66, 226

image processing

– grey line, 22

impulse response, 10

instantaneous amplitude, 42, 201

– geometric, 49

instantaneous phase, 42, 201

– geometric, 49

interferometry, 238

kernel

– Gauss, 64

– Poisson, 64

linearity, 10

Lippmann–Schwinger integral equation,277

local phase, 67

luminance, 22

matching pursuit, 322

Maxwell’s equations, 273

– in inhomogeneous media, 274

mean curvature, 183

monogenic

– coefficients, modeling of, 257

– colour signal, 248, 252, 253

– colour wavelet, 252

– colour wavelet transform, 248, 253,256

– signal, 63, 228, 248–250

– local amplitude, 229

– local orientation, 229

– local phase, 229

– wavelet transform, 248, 250, 251

multivector, 286

– scalar product, 287

operator

– coefficient, 304

– frame, 304

– reconstruction, 304

optical coherence tomography (OCT),271

optimization, 322

orthogonal 2D planes split, 15

– determination from given planes, 26,27

– exponential factor

– identities, 21, 22

– general, 18

– geometric interpretation, 26

– rotation, 26

– orthogonality of OPS planes, 19

– single pure unit quaternion, 21

– subspace bases, 22, 25

parallel spinor field, 180

paravector, 296

Parseval relation, 200

Parseval theorem

– Clifford–, 289

Parseval’s equality, 303

period form, 181, 183

336 Index

phase, 248, 250, 251, 267

– information, 67

– quaternionic, 69, 294

– velocity, 212

Plancherel’s theorem, 204, 288

Plemelj–Sokhotzki formula, 225

probability measure, 108, 113

pseudoscalar, 286

pursuit

– basis, 322

– matching, 322

quaternion

– algebra, 16, 65

– definition, 5

– over ℝ, 132

– algebraic identites, 20

– chrominance, 22

– plane, 22

– colour image

– processing, 15, 22

– complex sub-field, 6

– conjugate, 5

– decomposition, luminance andchrominance, 22

– Euler formula, 237

– exponential, 237

– exponential factors

– identity, 25, 36

– grey line, 22

– half-turn, 22, 25

– Coxeter, 20

– rotation, 18, 20

– involution, 18

– quaternion conjugation, 32

– line reflection

– pointwise invariant, 33

– real line, 32

– logarithm, 237

– luminance, 22

– modulus identity, 17

– norm, 5

– orientation, 236

– orthogonal 2D planes split

– general, 25

– real coefficients, 20

– orthogonal basis, 18, 19

– orthogonality, 17

– orthonormal basis, 22, 25

– perplex part, 6

– plane subspace bases, 18

– polar Cayley–Dickson form, 44

– polar form, 43

– properties, 43

– pure, 5

– quaternion maps, 38

– reflection

– hyperplane, 32

– invariant hyperplane, 32

– rotary axis, 33

– rotary invariant line, 33

– relation to Clifford algebra, 124

– rotary reflection, 33

– rotation angle, 35

– rotation plane basis, 34

– rotation, 33

– double, 33

– four-dimensional, 33

– reflection, 33

– rotary, 33

– scalar part

– symmetries, 17

– scalar-part, 5

– simplex and perplex parts, 24

– simplex and perplex split, 21

– simplex part, 6

– split

– orthogonality, 18

– steerable, 18

– subspace bases, 19

– vector-part, 5

quaternionic

– analytic function, 66

– analytic signal, 68

– filter, steerable, 67

– phases, 69, 294

– structure, 182

– wavelet multiresolution, 66

quaternionic Fourier transform, 105,158, 202, 227, 299

– analysis planes, 31

Index 337

– asymptotic behaviour, 111

– discrete, 30

– dual-axis form, 8

– factored form, 8

– fast, 30

– filter design

– steerable, 39

– forward transform, 8

– generalization, 28

– geometric interpretation, 31

– geometric understanding

– local, 38

– inverse transform, 9

– local phase rotations, 31

– new forms, 28

– new types, 39

– of the Hilbert transform, 47

– operator pairs, 8

– phase angle, 31

– phase angle transformation, 31

– discrete, 32

– fast, 32

– split parts, 32

– steerable, 32

– phase rotation planes, 31

– quasi-complex, 30

– quaternion conjugation, 32, 36

– local geometric interpretation, 37

– local invariant line, 37

– local phase rotation, 38

– local rotation angle, 37

– local rotation axis, 37

– phase angle transformation, 38

– phase angle transformation,discrete, 38

– phase angle transformation, fast,38

– phase angle transformation,interpretation, 38

– phase angle transformation,quasi-complex, 38

– phase angle transformation, splitparts, 38

– phase angle transformation,steerable, 38

– quasi-complex, 36

– split parts, 36

– reverse transform, 9

– sandwich form, 8

– single-axis definition, 9

– split parts transformation, 30

– split theorems, 39

– steerable, 15

– two pure unit quaternions, 28

– windowed, 286

radiation condition, 276

Riemann–Hilbert problem, 225

Riesz transform, 64, 229, 249, 251, 253,255, 257, 259, 262

– wavelet, 256

scattering problem, 276

Schwartz space, 114

separability, 165

Siemens star, 231

singular value decomposition, 324

Sobel operator, 242

Sommen–Fourier transform, 157

spacetime Fourier transform, 158

spin

– character, 185

– group, 235

– structure, 193

spinor, 235

– bundle, 187, 193

– connection, 194

– field, 179

– field, parallel, 180

– formula, Gauss, 194

– representation, 146, 183

– tensor, 185

split

– form, 7

– of identity, 222, 223

– w.r.t. commutativity, 162

square roots of −1, 157

– 𝐶ℓ0,2, 132

– 𝐶ℓ0,3, 133

– 𝐶ℓ0,5, 135, 137

– 𝐶ℓ1,2, 134

– 𝐶ℓ2,0, 128

338 Index

– 𝐶ℓ2,1, 130

– 𝐶ℓ2,3, 135

– 𝐶ℓ3,0, 134

– 𝐶ℓ4,1, 135

– 𝐶ℓ7,0, 138

– ℂ, 125

– ℝ, ℝ2, 125

– ℍ2, 133

– ℳ(2,ℂ), 135

– ℳ(2𝑑,ℂ), 134

– ℳ(2𝑑,ℝ), 128

– ℳ(2𝑑,ℝ2), 130

– ℳ(𝑑,ℍ2), 133

– ℳ(𝑑,ℍ), 132

– 𝑛 ≤ 4, 125

– algebraic submanifold, 126

– inner automorphism, 126

– bijection with idempotents, 134

– biquaternions, 𝐶ℓ3,0, 125

– central pseudoscalar, 134

– centralizer, 126

– centralizer computation, 149

– compact manifold, 132

– computation

– CLIFFORD package, 141

– conjugacy class, 126

– dimension, 126

– conjugate square root of −1, 132

– connected component, 126

– dimension, 126

– exceptional, 124, 127, 135

– Fourier transformations, 124

– idempotents, 134

– Klein group, 131

– Maple worksheets, 141, 144

– matrix square root, 147

– multivector split, 128

– ordinary, 127, 135

– Pauli matrix algebra, 134

– quaternions, 132

– scalar part zero, 127

– skew-centralizer, 128

– square isometries, 132

– stability subgroup, 126

– table ℳ(2𝑑,ℂ), 𝑑 = 1, 2, 4, 141

– visualization, 124

swap rule, 6

symmetry, 70

symplectic form, 6

texture, 238

– detection, 185

tomography, optical coherence, 271

trivector, geometric interpretation, 124

uncertainty principle, 300, 303, 317

unique continuation principle, 279

up(𝑥), 58

vector space, non-Euclidean, 124

Ville, J., 200

wavelet

– quaternionic multiresolution, 66

– transform, analytic, 248

Weierstrass

– generalized parametrization, 183

– representation, 177

Zak transform, 305

– Clifford–, 305, 307, 308, 312, 317