On uncertainty principle for quaternionic linear

canonical transform

Kit-Ian Kou

Department of Mathematics, Faculty of Science and Technology, University of Macau,Macau.

Jian-Yu Ou

Department of Mathematics, Faculty of Science and Technology, University of Macau,Macau.

Joao Morais

Center for Research and Development in Mathematics and Applications,Department of Mathematics, University of Aveiro, Portugal.

accepted to appear in ABSTRACT AND APPLIED ANALYSIS

Abstract

In this paper, we generalize the linear canonical transform (LCT) to quater-nion-valued signals, known as the quaternionic linear canonical transform(QLCT). Using the properties of the LCT we establish an uncertainty prin-ciple for the QLCT. This uncertainty principle prescribes a lower bound onthe product of the effective widths of quaternion-valued signals in the spatialand frequency domains. It is shown that only a 2D Gaussian signal minimizesthe uncertainty.

Keywords: Quaternionic analysis, quaternionic Fourier transform,quaternionic linear canonical transform, uncertainly principle,hypercomplex functions, quantum mechanics, Gaussian quaternionic signal.

1. Introduction

The classical uncertainty principle of harmonic analysis states that a non-trivial function and its Fourier transform (FT) cannot both be sharply local-ized. The uncertainty principle plays an important role in signal processing[17, 18, 31, 32, 52, 58, 59, 61, 72, 74, 83], and physics [3, 16, 39, 40, 41, 53,

Preprint submitted to Elsevier March 22, 2013

54, 66, 75, 81]. In quantum mechanics an uncertainty principle asserts thatone cannot make certain of the position and velocity of an electron (or anyparticle) at the same time. That is, increasing the knowledge of the positiondecreases the knowledge of the velocity or momentum of an electron. Inquaternionic analysis some papers combine the uncertainty relations and thequaternionic Fourier transform (QFT) [4, 36, 60].

The QFT plays a vital role in the representation of (hypercomplex) sig-nals. It transforms a real (or quaternionic) 2D signal into a quaternion-valuedfrequency domain signal. The four components of the QFT separate fourcases of symmetry into real signals instead of only two as in the complexFT. In [67] the authors used the QFT to proceed color image analysis. Thepaper [7] implemented the QFT to design a color image digital watermark-ing scheme. The authors in [6] applied the QFT to image pre-processingand neural computing techniques for speech recognition. Recently, certainasymptotic properties of the QFT were analyzed and straightforward gener-alizations of classical Bochner-Minlos theorems to the framework of quater-nionic analysis were derived [24, 25]. In this paper, we study the uncertaintyprinciple for the QLCT, the generalization of the QFT in the Hamiltonianquaternionic algebra.

The classical LCT being a generalization of the FT, was first proposedin the 1970s by Moshinsky and Collins [14, 57]. It is an effective processingtool for chirp signal analysis, such as the parameter estimation, samplingprogress for non-bandlimited signals with nonlinear Fourier atoms [51] andthe LCT filtering [80, 30, 62]. The windowed LCT [46], with a local win-dow function, can reveal the local LCT-frequency contents, and it enjoyshigh concentrations and eliminates cross terms. The analogue of the Poissonsummation formula, sampling formulas, series expansions, Paley-Wiener the-orem and uncertainly relations are studied in [46, 47]. In view of numerousapplications, one is particularly interested in higher dimensional analogues toEuclidean space. The LCT was first extended to the Clifford analysis settingin [45]. It was used to study the generalized prolate spheroidal wave functionsand the connection with energy concentration problems [45]. In the presentwork, we study the QLCT which transforms a quaternionic 2D signal intoa quaternion-valued frequency domain signal. Some important properties ofthe QLCT are analyzed. An uncertainty principle for the QLCT is estab-lished. This uncertainty principle prescribes a lower bound on the product ofthe effective widths of quaternion-valued signals in the spatial and frequencydomains. To the best of our knowledge, the study of a Heisenberg type un-

2

certainty principle for the QLCT has not been carried out yet. The resultsin this paper are new in the literature. The main motivation of the presentstudy is to develop further general numerical methods for differential equa-tions and to investigate localization theorems for summation of Fourier seriesin the quaternionic analysis setting. Further investigations and extensions ofthis topic will be reported in a forthcoming paper.

The article is organized as follows. Section 2 gives a brief introductionto some general definitions and basic properties of quaternionic analysis.The LCT of 2D quaternionic signal is introduced and studied in Section3. Some important properties such as Parseval and inversion theorems areobtained. In Section 4, we introduce and discuss the concept of QLCT,and demonstrate some important properties that are necessary to prove theuncertainty principle for the QLCT. The classical Heisenberg uncertaintyprinciple is generalized for the QLCT in Section 5. This principle prescribesa lower bound on the product of the effective widths of quaternion-valuedsignals in the spatial and frequency domains. Some conclusions are drawn inSection 6.

2. Preliminaries

The quaternionic algebra was invented by Hamilton in 1843 and is de-noted by H in his honor. It is an extension of the complex numbers to a4D algebra. Every element of H is a linear combination of a real scalar andthree orthogonal imaginary units (denoted, respectively, by i, j and k) withreal coefficients

H := {q = q0 + iq1 + jq2 + kq3| q0, q1, q2, q3 ∈ R},

where the elements i, j and k obey the Hamilton’s multiplication rules

i2 = j2 = k2 = −1; ij = −ji = k, jk = −kj = i, ki = −ik = j.

For every quaternionic number q = q0 + q, q = iq1 + jq2 + kq3, the scalarand non-scalar parts of q, are defined as Sc(q) := q0 and NSc(q) := q,respectively.

Every quaternion q = q0 +q has a quaternionic conjugate q̄ = q0−q. Thisleads to a norm of q ∈ H defined as

|q| :=√qq̄ =

√q20 + q

21 + q

22 + q

23.

3

Let |q| and θ(∈ R) be polar coordinates of the point (q0, q) ∈ H thatcorresponds to a nonzero quaternion q = q0 +q. Quaternion q can be writtenin polar form as

q = |q|(cos θ + e sin θ), (1)

where q0 = |q| cos θ, |q| = |q| sin θ, θ = arctan(|q|/q0) and e = q/|q|. If q ≡ 0,the coordinate θ is undefined; and so it is always understood that q 6= 0whenever θ = arg q is discussed.

The symbol e eθ, or exp(eθ), is defined by means of an infinite series (orEuler’s formula) as

eeθ :=∞∑n=0

(eθ)n

n!= cos θ + e sin θ,

where θ is to be measured in radians. It enables us to write the polar form(1) in exponential form more compactly as

q = |q|e eθ = |q| exp(q

|q|arctan

( |q|q0

)). (2)

Quaternions can be used for three- or four-entry vector analyses. Recently,quaternions have also been used for color image analysis. For q = q0 + iq1 +jq2 + kq3 ∈ H, we can use q1, q2 and q3 to represent, respectively, the R, G,and B values of a color image pixel, and set q0 = 0.

For p = 1 and 2, the quaternion modules Lp(R2;H) are defined as

Lp(R2;H) := {f |f : R2 → H, ‖f‖Lp(R2;H) :=∫R2|f(x1, x2)|pdx1dx2 L2(R2;H):=

∫R2f(x1, x2)g(x1, x2)dx1dx2 (3)

whose associated norm is

‖f‖L2(R2;H) :=(∫

R2|f(x1, x2)|2dx1dx2

)1/2. (4)

4

As a consequence of the inner product (4), we obtain the quaternionic Cauchy-Schwarz inequality

|Sc(< f, g >L2(R2;H))| ≤ | < f, g >L2(R2;H) | ≤ ‖f‖L2(R2;H)‖g‖L2(R2;H) (5)

for any f, g ∈ L2(R2;H).In [12, 42] a Clifford-valued generalized function theory is developed. In

the following, we adopt the definition that T is called a tempered distribution,if T is a continuous linear functional from S := S(R2) to H, where S(R2) isthe Schwarz class of rapidly decreasing functions. The set of all tempereddistributions is denoted by S′. If T ∈ S′, we denote this value for a testfunction φ by writing

T [φ] :=

∫RT (x1, x2)φ(x1, x2)dx1dx2,

using square brackets. (In literature one often sees the notation < T, φ >,but we shall avoid this, since it does not completely share the properties ofthe inner product.)

This is equivalent with the one defined in [12] using modules, and enablesus to define Fourier transforms on tempered distributions, by the formula

T̂ [φ] = T [φ̂], for all φ ∈ S,

which is just to perform Fourier transform

φ̂(ω1, ω2) =

∫R2φ(x1, x2)e

i(x1ω1+x2ω2)dx1dx2

on each of the components of the distribution. We will use the followingresults:

1̂(ω1, ω2) = (2π)2δ(ω1, ω2), (6)

̂(i−|α|Dαδ)(ω1, ω2) = ωα11 ωα22 , (7)

where α = (α1, α2), |α| = α1 + α2, Dα = ( ∂∂x1 )α1( ∂

∂x2)α2 and δ is the usual

Dirac delta function.

In the following we introduce the LCT for 2D quaternionic signals.

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3. LCTs of 2D quaternionic signals

The LCT was first introduced in the 70s and is a four-parameter classof linear integral transform, which includes among its many special cases,the FT, the fractional Fourier transform (FRFT), the Fresnel transform, theLorentz transform and scaling operations. In a way, the LCT has moredegrees of freedom and is more flexible than the FT and the FRFT, but withsimilar computation cost as the conventional FT [43]. Due to the mentionedadvantages, it is natural to generalize the classical LCT to the quaternionicalgebra.

3.1. Definition

Using the definition of the LCT [77, 80], we extend the LCT to the 2Dquaternionic signals. Let us define the left-sided and right-sided LCTs of 2Dquaternionic signals.

Definition 3.1. (Left-sided and right-sided LCTs)

Let Ai =

[ai bici di

]∈ R2×2 be a matrix parameter such that det(Ai) = 1,

for i = 1, 2. The left-sided and right-sided LCTs of 2D quaternionic signalsf ∈ L1(R2;H) are defined by

Lil(f)(u1, x2) :=

1√i2πb1∫R e

i(a12b1

x21−1b1x1u1+

d12b1

u21

)f(x1, x2)dx1, b1 6= 0;

√d1e

ic1d12u21f(d1u1, x2), b1 = 0,

and

Ljr(f)(x1, u2) :=

∫R f(x1, x2)

1√j2πb2

ej(a22b2

x22−1b2x2u2+

d22b2

u22

)dx2, b2 6= 0;

f(x1, d2u2)√d2e

jc2d22u22 , b2 = 0,

respectively.

Note that for bi = 0 (i = 1, 2), the LCT of a signal is essentially a chirpmultiplication and it is of no particular interest for our objective in this work.Hence, without loss of generality, we set bi 6= 0 in the following sections unlessstated. Therefore

Lil(f)(u1, x2) =

∫RK iA1(x1, u1)f(x1, x2)dx1, b1 6= 0 (8)

6

and

Ljr(f)(x1, u2) =

∫Rf(x1, x2)K

jA2

(x2, u2)dx2, b2 6= 0, (9)

where the kernel functions

K iA1(x1, u1) :=1√

i2πb1ei(a12b1

x21−1b1x1u1+

d12b1

u21

)(10)

and

KjA2(x2, u2) :=1√

j2πb2ei(a22b2

x22−1b2x2u2+

d22b2

u22

), (11)

respectively.

3.2. Properties

The following proposition summarizes some important properties of thekernel functions K iA1 (and K

jA2

) of the left-sided (and right-sided) LCTswhich will be useful to study the properties of LCTs, such as the PlancherelTheorem.

Proposition 3.1. Let the kernel function KA be defined by (10) or (11).Then

(i) KA(−x, ω) = KA(x,−ω);

(ii) KA(−x,−ω) = KA(x, ω);

(iii) KA(x, ω) = KB(x, ω), where B =

(a −b−c d

);

(iv)∫RKA1(x, ω)KA2(ω, y)dω = KA1A2(x, y), where A1A2 corresponds to

matrix product.

The proofs of properties (i) to (iii) follow from the definitions (10) and(11). The proof of property (iv) can be found in [62, 80].

Note that some properties of the LCT for 2D quaternionic signals followfrom the one dimensional case [62, 77].

Proposition 3.2. Let LAi(f) (i = 1, 2) be defined by Lil in (8), or L

jr in (9),

respectively. If f, g ∈ L1⋂L2(R2;H), then

7

(i) Additivity

LA2(LA1)(f) = LA2A1(f), for LAi := Lil;

LA2(LA1)(f) = LA1A2(f), for LAi := Ljr.

(ii) Reversibility

LAi−1(LAi)(f) = f, for LAi := Lil;

LAi−1(LAi)(f) = f, for LAi := Ljr.

(iii) Plancherel Theorem (right-sided LCT) If f, g ∈ S, then

〈f, g〉L2(R2;H) = 〈Ljr(f), Ljr(g)〉L2(R2;H). (12)

In particular, with f = g, we get the Parseval theorem, i.e.,

‖f‖2L2(R2;H) = ‖Ljr(f)‖2L2(R2;H).

Proof. By the Fubini’s theorem, property (iv) of Proposition 3.1 establishesthe additivity property (i) of left-sided LCTs.

LA2 (LA1 (f) (u1, x2)) (u1, x2)

=

∫RK iA2(u1, y1)LA1 (f) (u1, x2)du1

=

∫R

(∫RK iA2(u1, y1)K

iA1

(x1, u1)du1

)f(x1, x2)dx1

= LA2A1(f)(y1, x2).

The proof of the right-sided LCT LAi := Ljr is similar.

Reversibility property (ii) is an immediate consequence of additivity prop-erty (i) once we observe that A1 = Ai and A2 = A

−1i .

8

To verify property (iii), applying Fubini’s theorem, it suffices to see that

〈Ljr(f), Ljr(g)〉L2(R2;H)

=

∫R2Ljr(f)(u1, x2)L

jr(g)(u1, x2)du1dx2

=

∫R4f(x1, x2)K

jA1

(x1, u1)KjA1

(y1, u1) g(y1, x2)du1dx1dy1dx2

=1

2πb1

∫R4f(x1, x2)e

ja12b1

(x21−y21)e−j 1

b1u1(x1−y1) g(y1, x2)du1dx1dy1dx2

=

∫R3f(x1, x2)e

ja12b1

(x21−y21)δ(y1 − x1) g(y1, x2)dx1dy1dx2

= 〈f, g〉L2(R2;H),

where we have used (6).

Notice that the left-sided and right-sided LCTs of quaternionic signalsare unitary operators on L2(R2;H). In signal analysis, it is interpreted in thesense that (right-sided) LCT of quaternionic signal preserves the energy of asignal.

Remark 3.1. Note that Plancherel Theorem is not valid for the two-sidednor left-sided LCT of 2D quaternionic signal. For this reason, we study theright-sided LCT of 2D quaternionic signals in the following.

It is worth noting that when A1 = A2 =

[0 1−1 0

], the left-sided and

right-sided LCTs of f reduce to the left-sided and right-sided FTs of f . Thatis,

Lil(f)(u1, x2) =1√2πi

∫Re−ix1u1f(x1, x2)dx1 =

1√2πi

F il (f)(u1, x2)

and

Ljr(f)(x1, u2) =1√2πj

∫Rf(x1, x2)e

−jx2u2dx2 =1√2πj

F jr(f)(x1, u2),

respectively. Here

F il (f)(u1, x2) :=

∫Re−ix1u1f(x1, x2)dx1

9

and

F jr(f)(x1, u2) :=

∫Rf(x1, x2)e

−jx2u2dx2 (13)

are the left-sided FT and right-sided FT of f , respectively.

We now formulate the linear canonical integral representation of a 2Dquaternionic signal f .

Theorem 3.1. (Linear Canonical Inversion Theorem) Suppose that f ∈L1(R2;H), that f is continuous except for a finite number of finite jumps inany finite interval, and that f(s, t) = 1

2(f(s, t+) + f(s, t−)) for all t and s.

Then

f(s, t0) = limα→∞

∫ α−αLir(f)(s, ω)K

iA−1(ω, t0)dω (14)

for every t0 and s where f has (generalized) left and right partial derivatives.In particular, if f is piecewise smooth (i.e., continuous and with a piecewisecontinuous derivative), then the formula holds for all t0 and uniformly in s.

Proof. Put

I(s, t0;α) :=

∫ α−αLir(f)(s, ω)K

iA−1(ω, t0)dω

and rewrite this expression by inserting the definition of Lir(f),

I(s, t0;α) =

∫ α−α

(∫Rf(s, t)K iA(t, ω)dt

)K iA−1(ω, t0)dω

=

∫R

∫ α−αf(s, t)K iA(t, ω)K

iA−1(ω, t0)dωdt

=

∫Rf(s, t)

[1

2πbei

a2b

(t2−t20)∫ α−αei

1b(t0−t)ωdω

]dt

=1

4πe−i

a2bt20

∫Rf(s, t)

(ei

a2bt2 sin(

αb(t0 − t))t0 − t

)dt

=1

4πe−i

a2bt20

∫Rf(s, t0 − u)

(ei

a2b

(t0−u)2 sin(αbu)

u

)du.

Switching the order of integration is permitted, because the improper doubleintegral is absolutely convergent over the strip (t, ω) ∈ R × [−α, α], and in

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the last step we have put t0 − t = u. Using the formula∫ ∞0

eia2b

(t0−u)2 sin(αbu)

udu = 2πei

a2bt20 , for α, b > 0, (15)

we can write

1

2πe−i

a2bt20

∫ ∞0

f(s, t0 − u)(ei

a2b

(t0−u)2 sin(αbu)

u

)du− f(s, t0−) (16)

=1

2πe−i

a2bt20

∫ ∞0

(f(s, t0 − u)− f(s, t0−))(ei

a2b

(t0−u)2 sin(αbu)

u

)du.

Now let � > 0 be given. Since we have assumed that f ∈ L1(R2;H), thereexists a number β such that

1

2π

∫ ∞β

|f(s, t0 − u)| du < �.

Changing the variable, we find that∫ ∞β

eia2b

(t0−u)2 sin(αbu)

udu =

∫ ∞αβb

eia2b

(t0− bxα )2 sinx

xdx→ 0, as α/b→∞. (17)

The last integral in (16) can be split into three terms:

1

2πe−i

a2bt20

∫ β0

f(s, t0 − u)− f(s, t0−)u

(ei

a2b

(t0−u)2 sin(α

bu))du

+1

2πe−i

a2bt20

∫ ∞β

f(s, t0 − u)(ei

a2b

(t0−u)2 sin(αbu)

u

)du

− 12πe−i

a2bt20f(s, t0−)

∫ ∞β

(ei

a2b

(t0−u)2 sin(αbu)

u

)du = I1 + I2 − I3.

The term I3 tends to zero as b > 0 and α → ∞, because of (17). The termI2 can be estimated:

|I2| =∣∣∣∣ 12πe−i a2b t20

∫ ∞β

f(s, t0 − u)(ei

a2b

(t0−u)2 sin(αbu)

u

)du

∣∣∣∣≤ 1

2π

∫ ∞β

|f(s, t0 − u)|du ≤ �.

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In the term I1 we have the function g(s, u) = (f(s, t0 − u) − f(s, t0−))/u.This is continuous except for jumps in the interval R× (0, β), and it has thefinite limit g(s, 0+) = ∂

∂tfL(s, t0) as u ↘ 0; this means that g is bounded

uniformly in s and thus integrable on the interval. By the Riemann-Lebesguelemma, we conclude that I1 → 0 as β → ∞. All this together gives, since �can be taken as small as we wish,

1

2πe−i

a2bt20

∫ ∞0

f(s, t0−u)(ei

a2b

(t0−u)2 sin(αbu)

u

)du→ f(s, t0−), as α/b→∞.

A parallel argument implies that the corresponding integral over (−∞, 0)tends to f(s, t0+) uniformly in s. Taking the mean value of these two results,we have completed the proof of the theorem.

Remark 3.2. If Lir(f) ∈ L1(R2;H), then (14) can be written as the absolutelyconvergent integral

f(s, t0) =

∫RLir(f)(s, ω)K

iA−1(ω, t0)dω

The following lemma gives the relationship between the left(right)-sidedLCTs and Left(right)-sided FTs of f .

Lemma 3.1. Let Ai =

[ai bici di

]∈ R2×2 be a matrix parameter such that

det(Ai) = 1, for i = 1, 2. Let f ∈ L1(R2;H), we have

Lil(f)(u1, x2) = eid12b1

u21F il

(1√

i2πb1eia12b1

(·)2f(·, x2)

)(u1b1, x2

), (18)

and

Ljr(f)(x1, u2) =

(F jr

(f(x1, ·)

1√j2πb2

eja22b2

(·)2)(

x1,u2b2

))ejd22b2

u22 . (19)

Proof. By the definition of Lil(f) in (8), a direct computation shows that

Lil(f)(u1, x2) =

∫R

1√i2πb1

ei(a12b1

x21−1b1x1u1+

d12b1

u21

)f(x1, x2)dx1

= eid12b1

u21

(∫Re−ix1 u1b1

(1√

i2πb1eia12b1

x21f(x1, x2)

)dx1

)= e

id12b1

u21F il

(1√

i2πb1eia12b1

(·)2f(·, x2)

)(u1b1, x2

).

Similarly, by the definition of Ljr(f) in (9), we obtain (19).

12

The LCT can be further generalized into the offset linear canonical trans-form (offset LCT) [1, 63, 80]. It has two extra parameters which repre-sents the space and frequency offsets. The basic theories of the LCT havebeen developed including uncertainty principles [71, 75], convolution theorem[77, 78], Hilbert transform [21, 83], sampling theory [51, 77], discretization[33, 43, 70] and so on, which enrich the theoretical system of the LCT. On theother hand, since the LCT has three free parameters, it is more flexible andhas found many applications in radar system analysis, filter design, phaseretrieval, pattern recognition and many other areas [62, 77].

4. QLCTs of 2D quaternionic signals

4.1. Definition

This section leads to the quaternionic linear canonical transforms (QLCTs).Due to the non-commutative property of multiplication of quaternions, thereare many different types of QLCTs: two-sided QLCTs, left-sided QLCTs andright-sided QLCTs.

Definition 4.1. (Two-sided QLCTs)

Let Ai =

[ai bici di

]∈ R2×2 be a matrix parameter such that det(Ai) = 1,

bi 6= 0 for i = 1, 2. The two-sided QLCTs of signals f ∈ L1(R2;H) are thefunctions Li,j(f) : R2 → H given by

Li,j(f)(u1, u2) :=∫R2K iA1(x1, u1)f(x1, x2)K

jA2

(x2, u2)dx1dx2, (20)

where u1, u2 = u1e1 + u2e2, with KiA1

(x1, u1) and KjA2

(x2, u2) given by (10)and (11), respectively.

Definition 4.2. (Left-sided QLCTs)

Let Ai =

[ai bici di

]∈ R2×2 be a matrix parameter such that det(Ai) = 1,

bi 6= 0 for i = 1, 2. The left-sided QLCTs of signals f ∈ L1(R2;H) are thefunctions Li,jl (f) ∈ L(R2;H) given by

Li,jl (f)(u1, u2) :=∫R2K iA1(x1, u1)K

jA2

(x2, u2)f(x1, x2)dx1dx2,

where the kernels K iA1 and KjA2

are given by (10) and (11), respectively.

13

Due to the validity of the Plancherel Theorem, we study the right-sidedQLCTs of 2D quaternionic signals in this paper.

Definition 4.3. (Right-sided QLCTs)

Let Ai =

[ai bici di

]∈ R2×2 be a matrix parameter such that det(Ai) = 1,

bi 6= 0 for i = 1, 2. The left-sided QLCTs of signals f ∈ L1(R2;H) are thefunctions Li,jr (f) ∈ L1(R2;H) given by

Li,jr (f)(u1, u2) :=∫R2f(x1, x2)K

iA1

(x1, u1)KjA2

(x2, u2)dx1dx2, (21)

where K iA1 and KjA2

given by (10) and (11), respectively.

It is significant to note that when A1 = A2 =

[0 1−1 0

], the QLCT of f

reduces to the QFT of f . We denote it by

F i,jr (f)(u1, u2) :=∫R2f(x1, x2)e

−ix1u1e−jx2u2dx1dx2. (22)

Remark 4.1. In fact, the right-sided QLCTs defined above can be generalizedas follows:

Le1,e2r (f)(u1, u2) =∫R2f(x1, x2)K

e1A1

(x1, u1)Ke2A2

(x2, u2)dx1dx2, (23)

where e1 = e1,ii + e1,jj + e1,kk and e2 = e2,ii + e2,jj + e2,kk so that

e21,i + e21,j + e

21,k = e

22,i + e

22,j + e

22,k = 1 (i.e., e

21 = e

22 = −1)

e1,ie2,i + e1,je2,j + e1,ke2,k = 0.

Equation (21) is the special case of (23) in which e1 = i, and e2 = j.

Remark 4.2. For bi 6= 0 (i = 1, 2) and f ∈ L1(R2;R), the (right-sided)QLCT of a 2D signal f ∈ L1(R2;H) in (21) has the closed-form representa-tion:

Li,jr (f)(ω1, ω2) := Φ0(ω1, ω2) + Φ1(ω1, ω2) + Φ2(ω1, ω2) + Φ3(ω1, ω2),

14

where we put the integrals

Φ0(ω1, ω2) =

∫R2f(x1, x2)

1

2π√

kb1b2cos

(a12b1

x21 −1

b1x1ω1 +

d12b1

ω21

)cos

(a22b2

x22 −1

b2x2ω2 +

d22b2

ω22

)dx1dx2,

Φ1(ω1, ω2) =

∫R2f(x1, x2)

√i

2π√

jb1b2sin

(a12b1

x21 −1

b1x1ω1 +

d12b1

ω21

)cos

(a22b2

x22 −1

b2x2ω2 +

d22b2

ω22

)dx1dx2,

Φ2(ω1, ω2) =

∫R2f(x1, x2)

√j

2π√

ib1b2cos

(a12b1

x21 −1

b1x1ω1 +

d12b1

ω21

)sin

(a22b2

x22 −1

b2x2ω2 +

d22b2

ω22

)dx1dx2,

Φ3(ω1, ω2) =

∫R2f(x1, x2)

√k

2π√b1b2

sin

(a12b1

x21 −1

b1x1ω1 +

d12b1

ω21

)sin

(a22b2

x22 −1

b2x2ω2 +

d22b2

ω22

)dx1dx2.

These equations clearly show how the QLCT separate real signals into fourquaternion components, i.e., the even-even, odd-even, even-odd and odd-oddcomponents of f .

Let us give an example to illustrate expression (21).

Example 4.1. Consider the quaternionic distribution signal, i.e. the QLCTkernel of (21)

f(x1, x2) = K−jA2

(x2, u0)K−iA1

(x1, v0). (24)

It is easy to see that the QLCT of f is a Dirac quaternionic function, i.e.

Li,jr (f)(u1, u2) = (2π)2δ(u1, u2 − t), t = t1e1 + t2e2.

15

4.2. Properties

This subsection describes important properties of the QLCTs that willbe used to establish the uncertainty principles for the QLCTs.

We now establish a relation between the right-sided LCTs and the right-sided QLCTs of 2D quaternion-valued signals.

Lemma 4.1. Let Ai =

[ai bici di

]∈ R2×2 be a matrix parameter such that

det(Ai) = 1, bi 6= 0 for i = 1, 2. For f ∈ L1(R2;H), we have

Li,jr (f)(u1, u2) = Ljr(Lir(f)

)(u1, u2). (25)

Proof. By using the definition of right-sided QLCTs (21),

Li,jr (f)(u1, u2)

=

∫R2f(x1, x2)K

iA1

(x1, u1)KjA2

(x2, u2)dx1dx2

=

∫RLir(f)(u1, x2)K

jA2

(x2, u2)dx2

= Ljr(Lir(f)

)(u1, u2).

We then establish the Plancherel theorems, specific to the right-sidedQLCTs.

Theorem 4.1. (Plancherel theorems of QLCTs)For i = 1, 2, let fi ∈ S, the inner product (3) of two quaternionic modulefunctions and their QLCTs is related by

< f1, f2 >L2(R2;H)=< Li,jr (f1),Li,jr (f2) >L2(R2;H) . (26)

In particular, with f1 = f2 = f , we get the Parseval identity, i.e.

‖f‖2L2(R2;H) = ‖Li,jr (f)‖2L2(R2;H). (27)

16

Proof. By the inner product (3) and definition of right-sided QLCTs (21), astraightforward computation and Fubini’s theorem shows that

< Li,jr (f1),Li,jr (f2) >

=

∫R2

(∫R2f1(x1, x2)K

iA1

(x1, u1)KjA2

(x2, u2)dx1dx2

)(∫

R2f2(y1, y2)K iA1(y1, u1)K

jA2

(y2, u2)dy1dy2

)du1du2

=

∫R6f1(x1, x2)K

iA1

(x1, u1)(KjA2(x2, u2)K

jA2

(y2, u2))

f2(y1, y2)K iA1(y1, u1)dx1dx2dy1dy2du1du2

=

∫R6f1(x1, x2)K

iA1

(x1, u1)

(1

2πbeja22b2

(x22−y22)e−j 1

b2u2(x2−y2)

)f2(y1, y2)K iA1(y1, u1)du1dx1dx2dy1dy2du2

=

∫R5f1(x1, x2)K

iA1

(x1, u1)(eja22b2

(x22−y22)δ(y2 − x2))

f2(y1, y2)K iA1(y1, u1)dx1dx2dy1dy2du2

= < Lir(f1), Lir(f2) >=< f1, f2 >

where we have used the Plancherel theorem of right-sided LCTs (12) andformula (6).

Remark 4.3. Note that Plancherel Theorem is not valid for the two-sidednor left-sided QLCT of quaternionic signals. For this reason, we choose toapply the right-sided QLCT of 2D quaternionic signals in the present paper.

Theorem 4.1 shows that the total signal energy computed in the spatialdomain equals to the total signal energy in the quaternionic domain. TheParseval theorem allows the energy of a quaternion-valued signal to be con-sidered on either the spatial domain or the quaternionic domain, and thechange of domains for convenience of computation.

To proceed with, we prove the following derivative properties.

Lemma 4.2. For i = 1, 2, let Ai =

[ai bici di

]∈ R2×2 be a matrix parameter,

17

bi 6= 0 and aidi − bici = 1. If f ∈ S, then∫R2u2i∣∣Li,jr (f)(u1, u2)∣∣2 du1du2 = b2i ∫

R2

∣∣∣∣ ∂∂xif(x1, x2)∣∣∣∣2 dx1dx2 (28)

Proof. For i = 1, using (6), (7) and Fubini’s theorem, we have∫R2u21∣∣Li,jr (f)(u1, u2)∣∣2 du1du2

=

∫R2u21

(∫R2f(s1, s2)K

iA1

(s1, u1)KjA2

(s2, u2)ds1ds2

)(∫

R2f(x1, x2)K iA1(x1, u1)K

jA2

(x2, u2)dx1dx2

)du1du2

=

∫R6u21f(s1, s2)K

iA1

(s1, u1)(KjA2(s2, u2)K

jA2

(x2, u2))

K iA1(x1, u1) f(x1, x2)du2ds1ds2dx1dx2du1

=

∫R5u21f(s1, s2)K

iA1

(s1, u1)(eja22b2

(s22−x22)δ(x2 − s2))K iA1(x1, u1)

f(x1, x2)ds1ds2dx1dx2du1

=

∫R4f(s1, s2)

(u21K

iA1

(s1, u1)K iA1(x1, u1))f(x1, x2)du1ds1dx1dx2

= −b21∫R3f(s1, x2)

(eia12b1

(s21−x21) ∂2

∂x21δ(x1 − s1)

)f(x1, x2)ds1dx1dx2

= −b21∫R2f(x1, x2)

∂2

∂x21f(x1, x2)dx1dx2

= b21

∫R2

∣∣∣∣ ∂∂x1f(x1, x2)∣∣∣∣2 dx1dx2.

To prove the case i = 2, we argue in the same spirit as in the proof of the

18

case i = 1. Applying (6), (7) and Fubini’s theorem, we have∫R2u22∣∣Li,jr (f)(u1, u2)∣∣2 du1du2

=

∫R2u22

(∫R2f(s1, s2)K

iA1

(s1, u1)KjA2

(s2, u2)ds1ds2

)(∫

R2f(x1, x2)K iA1(x1, u1)K

jA2

(x2, u2)dx1dx2

)du1du2

=

∫R6f(s1, s2)K

iA1

(s1, u1)(u22K

jA2

(s2, u2)KjA2

(x2, u2))

f(x1, x2)K iA1(x1, u1)du2ds1ds2dx1dx2du1

= −b22∫R5f(s1, s2)K

iA1

(s1, u1)

(eja22b2

(s22−x22) ∂2

∂x22δ(x2 − s2)

)f(x1, x2)K iA1(x1, u1)ds1ds2dx1dx2du1

= −b22∫R5f(s1, x2)K

iA1

(s1, u1)

(∂2

∂x22f(x1, x2)

)K iA1(x1, u1)ds1dx1dx2du1

= −b22∫R4f(s1, x2)

(K iA1(s1, u1)K

iA1

(x1, u1)) ∂2∂x22

f(x1, x2)du1ds1dx1dx2

= −b22∫R3f(s1, x2)

(eia12b1

(s21−x21)δ(x1 − s1)) ∂2∂x22

f(x1, x2)ds1dx1dx2

= −b22∫R2f(x1, x2)

∂2

∂x22f(x1, x2)dx1dx2

= b22

∫R2

∣∣∣∣ ∂∂x2f(x1, x2)∣∣∣∣2 dx1dx2.

Some properties of the QLCT are summarized in Table 1. Let f1 and

f2 ∈ S, the constants α and β ∈ R, Ai =[ai bici di

]∈ R2×2, bi 6= 0, and

aidi − bici = 1.

5. Uncertainty principles for QLCTs

In signal processing much effort has been placed in the study of the classi-cal Heisenberg uncertainty principle during the last years. Shinde et al. [72]

19

Table 1: Properties of the QLCT.

Property Function QLCTReal linearity αf1(x1, x2) + βf2(x1, x2) αLi,jr (f)(u1, u2) + βLi,jr (f2)(u1, u2)

FormulaPlancherel < f1, f2 >L2(R2;H)= < Li,jr (f1),Li,jr (f2) >L2(R2;H)Parseval ‖ f ‖L2(R2;H)= ‖ Li,jr (f) ‖L2(R2;H)Derivatives

∫R2 u

2i

∣∣Li,jr (f)(u1, u2)∣∣2 du1du2 = b2i ∫R2 ∣∣∣ ∂∂xif(x1, x2)∣∣∣2 dx1dx2established an uncertainty principle for fractional Fourier transforms thatprovides a lower bound on the uncertainty product of real signal represen-tations in both time and frequency domains. Korn [44] proposed Heisen-berg type uncertainty principles for Cohen transforms which describe lowerlimits for the timefrequency concentration. In the meantime, Hitzer et al.[37, 38, 55, 56] investigated a directional uncertainty principle for the Clifford-Fourier transform, which describes how the variances (in arbitrary but fixeddirections) of a multivector-valued function and its Clifford-Fourier trans-form are related. On our knowledge, a systematic work on the investigationof uncertainty relations using the QLCT of a multivector-valued function isnot carried out.

In the following we explicitly prove and generalize the classical uncertaintyprinciple to quaternionic module functions using the QLCTs. We also give anexplicit proof for Gaussian quaternionic functions (Gabor filters) to be indeedthe only functions that minimize the uncertainty. We further emphasize thatour generalization is non-trivial because the multiplication of quaternionsand the quaternionic linear canonical kernel are both non-commutative. Forthis purpose we introduce the following definition.

Definition 5.1. For k = 1, 2, let f, xkf ∈ L2(R2;H) and Li,jr (f), ukLi,jr (f) ∈L2(R2;H). Then the effective spatial width or spatial uncertainty 4xk of fis evaluated by

4xk :=√

Vark(f),

where Vark(f) is the variance of the energy distribution of f along the xk-axis

20

defined by

Vark(f) :=‖xkf‖2L2(R2;H)‖f‖2L2(R2;H)

=

∫R2 x

2k|f(x1, x2)|2dx1dx2∫

R2 |f(x1, x2)|2dx1dx2. (29)

Similarly, in the quaternionic domain we define the effective spectral widthas

4uk :=√

Vark

(Li,jr (f)

),

where Vark(Li,jr (f)

)is the variance of the frequency spectrum of f along the

uk frequency axis given by

Vark(Li,jr (f)

)=‖ukLi,jr (f)‖2L2(R2;H)‖Li,j(f)‖2L2(R2;H)

=

∫R2 u

2k|Li,jr (f)(u1, u2)|2du1du2∫

R2 |Li,jr (f)(u1, u2)|2du1du2

. (30)

Example 5.1. Let us consider a 2D Gaussian quaternionic function (Figs.1, 2, 3 and 4) of the form

f(x1, x2) = Ce−(α1x21+α2x22),

where C = Ci0 + iCi1 + jCi2 + kCi3 ∈ H, for i = 1, 2, are quaternionicconstants and α1, α2 ∈ R are positive real constants.

Then the QLCT of f is given by

Li,j(f)(u1, u2)

= C

(∫RK iA1(x1, u1)e

−α1x21dx1

)(∫Re−α2x

22KjA2(x2, u2)dx2

)= C

√2b1π

2α1b1 − a1ie

(4α1b1−2a1i)d1+22b1(4α1b1−2a1i)

u21+(4α2b2−2a2j)d2+22b2(4α2b2−2a2j)

u22

√2b2π

2α2b2 − a2j.

This shows that the QLCT of the Gaussian quaternionic function is anotherGaussian quaternionic function.

Figures 1 and 2 visualize the quaternionic Gaussian function for α1 =α2 = 3, and α1 = 3 and α2 = 1 in the spatial domain. Figures 3 and 4visualize the quaternionic Gaussian function for α1 = 1 and α2 = 3, andα1 = α2 =

12

in the spatial domain.

21

2 1.5 1 0.5 0 0.5 1 1.5 2

2

1.5

1

0.5

0

0.5

1

1.5

22 1.5 1 0.5 0 0.5 1 1.5 2

2

1.5

1

0.5

0

0.5

1

1.5

2

Fig. 1 Fig. 2

2 1.5 1 0.5 0 0.5 1 1.5 2

2

1.5

1

0.5

0

0.5

1

1.5

22 1.5 1 0.5 0 0.5 1 1.5 2

2

1.5

1

0.5

0

0.5

1

1.5

2

Fig. 3 Fig. 4

Now let us begin the proofs of two uncertainty relations.

Theorem 5.1. For k = 1, 2, let f ∈ S. Then the following uncertaintyrelations are fulfilled:

4x14u1 ≥b12

(31)

and

4x24u2 ≥b22. (32)

22

The combination of the two spatial uncertainty principles above leads to theuncertainty principle for the 2D quaternionic signal f(x1, x2) of the form

4x14x24u14u2 ≥b1b2

4. (33)

Equality holds in (33) if and only if f is a 2D Gaussian function, i.e.

f(x1, x2) = βe−(C1x21+C2x22)/2,

where C1, C2 are positive real constants and β = ‖f‖L2(R2;H)(C1C2π2

)1/4.

Proof. Applying (28) in Lemma 4.2 and using Schwarz inequality (5), wehave (∫

R2x2k|f(x1, x2)|2dx1dx2

)(∫R2u2k∣∣Li,jr (f)(u1, u2)∣∣2 du1du2)

=

(∫R2x2k|f(x1, x2)|2dx1dx2

)(b2k

∫R2

∣∣∣∣ ∂∂xk f(x1, x2)∣∣∣∣2 dx1dx2

)

≥ b2k∣∣∣∣∫

R2xkf(x1, x2)

∂

∂xkf(x1, x2)dx1dx2

∣∣∣∣2 .Using the exponential form of a 2D quaternionic signal (2), let

f(x1, x2) = f0(x1, x2) + f(x1, x2) = |f(x1, x2)|e eα,

where e = f(x1, x2)/|f(x1, x2)| and α = arctan(|f(x1, x2)|/f0(x1, x2)), then

xkf(x1, x2)∂

∂xkf(x1, x2)

= xk|f(x1, x2)|e −eα∂

∂xk(|f(x1, x2)|e eα)

= xk|f(x1, x2)|e −eα[(

∂

∂xk|f(x1, x2)|

)e eα + |f(x1, x2)|

(∂

∂xke eα)]

= xk|f(x1, x2)|(

∂

∂xk|f(x1, x2)|

)+ xk|f(x1, x2)|2

(∂

∂xk(eα)

)=

1

2

∂

∂xk

(xk|f(x1, x2)|2

)− 1

2|f(x1, x2)|2 + xk|f(x1, x2)|2

(∂

∂xk(eα)

).

23

Therefore,

b2k

∣∣∣∣∫R2xkf(x1, x2)

∂

∂xkf(x1, x2)dx1dx2

∣∣∣∣2 (34)= b2k

∣∣∣∣∫R2

1

2

∂

∂xk

(xk|f(x1, x2)|2

)− 1

2|f(x1, x2)|2

+xk|f(x1, x2)|2(

∂

∂xk(eα)

)dx1dx2

∣∣∣∣2 .The first term is a perfect differential and integrates to zero. The secondterm gives minus one half of the energy ‖f‖2L2(R2;H).

Hence (∫R2x2k|f(x1, x2)|2dx1dx2

)(∫R2u2k|Li,jr (f)(u1, u2)|2du1du2

)≥ b2k

∣∣∣∣−12‖f‖2L2(R2;H)∣∣∣∣2 = b2k4 ‖f‖4L2(R2;H).

By definitions of 4xk, 4uk and Parseval theorem (27), we have

(4xk4uk)2 =(∫

R2 x2k|f(x1, x2)|2dx1dx2

) (∫R2 u

2k|Li,jr (f)(u1, u2)|2du1du2

)(∫R2 |f(x1, x2)|2dx1dx2

) (∫R2 |L

i,jr (f)(u1, u2)|2du1du2

)=

(∫R2 x

2k|f(x1, x2)|2dx1dx2

) (∫R2 u

2k|Li,jr (f)(u1, u2)|2du1du2

)‖f‖4L2(R2;H)

≥ b2k

4

and therefore we have the uncertainty principle as given by (31), (32) and(33).

We finally show that the equality in (31), (32) and (33) are satisfied ifand only if f is a Gaussian quaternionic function.

Since the minimum value for the uncertainty product is bk/2, we can askwhat signals have that minimum value. The Schwarz inequality (5) becomesan equality when the two functions are proportional to each other. Hence,we take g = −Cf , where C is a quaternionic constant and the −1 has beeninserted for convenience. We therefore have

∂

∂xkf(x1, x2) = −Ckxkf(x1, x2). (35)

24

This is a necessary condition for the uncertainty product to be the minimum.But it is not sufficient since we must also have the term∫

R2xk|f(x1, x2)|2

(∂

∂xk(eα)

)dx1dx2 = 0,

because by equation (34), we see that is the only way we can actually getthe value of bk/2.

Since Ck is arbitrary we can write it in terms of its scalar and non-scalarparts, Ck := Sc(Ck) +NSc(Ck). The solution of equation (35) is hence

f(x1, x2) = βe−(C1x21+C2x22)/2

= βe−(Sc(C1)x21+Sc(C2)x

22)/2e−(NSc(C1)x

21+NSc(C2)x

22)/2,

for some constant β. Since eα = −(NSc(C1)x21 + NSc(C2)x22)/2, it followsthat

∂

∂xk(eα) = −NSc(Ck)xk.

We have ∫R2xk|f(x1, x2)|2

(∂

∂xk(eα)

)dx1dx2

= −NSc(Ck)β2∫R2x2ke

−(Sc(C1)x21+Sc(C2)x22)dx1dx2.

The only way this can be zero is if NSc(Ck) = 0 and hence Ck must be areal number. We then have

f(x1, x2) = βe−(C1x21+C2x22)/2, (36)

where C1, C2 are positive real constants since f ∈ S and we have includedthe appropriate normalization β = ‖f‖L2(R2;H)

(C1C2π2

)1/4.

Since the 2D Gaussian function f(x1, x2) of (36) achieves the minimumwidth-bandwidth product, it is theoretically a very good prototype waveform. One can therefore construct a basic wave form using spatially or fre-quency scaled versions of f(x1, x2) to provide multiscale spectral resolution.Such a wavelet basis construction derived from a Gaussian quaternionic func-tion prototype waveform has been realized, for example, in the quaternionicwavelet transforms in [5]. The optimal space-frequency localization is alsoanother reason why 2D Clifford-Gabor bandpass filters were suggested in[10].

25

6. Conclusion

In this paper we developed the definition of QLCT. The various proper-ties of QLCT such as partial derivative, Plancherel and Parseval theoremsare discussed. Using the well-known properties of the classical LCT, we es-tablished an uncertainty principle for the QLCT. This uncertainty principlestates that the product of the variances of quaternion-valued signals in thespatial and frequency domains has a lower bound. It is shown that only a 2DGaussian signal minimizes the uncertainty. With the help of this principle,we hope to contribute to the theory and applications of signal processingthrough this investigation and to develop further general numerical methodsfor differential equations. The results in this article are new in the literature.Further investigations on this topic are now under investigation and will bereported in a forthcoming paper.

7. Acknowledgement

The first author acknowledges financial support from the research grantof the University of Macau No. MYRG142(Y1- L2)-FST11-KKIand the Sci-ence and Technology Development Fund FDCT/094/2011A. This work wassupported by FEDER funds through COMPETE–Operational ProgrammeFactors of Competitiveness (“Programa Operacional Factores de Competi-tividade”) and by Portuguese funds through the Center for Research andDevelopment in Mathematics and Applications (University of Aveiro) andthe Portuguese Foundation for Science and Technology (“FCT–Fundaçãopara a Ciência e a Tecnologia”), within project PEst-C/MAT/UI4106/2011with COMPETE number FCOMP-01-0124-FEDER-022690. Partial supportfrom the Foundation for Science and Technology (FCT) via the post-doctoralgrant SFRH/BPD/66342/2009 is also acknowledged by the third author.

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