scale-space image analysis based on hermite polynomials...

International Journal of Computer Vision 64(2/3), 125–141, 2005c© 2005 Springer Science + Business Media, Inc.. Manufactured in The Netherlands.

Scale-Space Image Analysis Based on Hermite Polynomials Theory

SHERIF MAKRAM-EBEID* AND BENOIT MORYPhilips Medical Systems Research in Paris, 51 rue Carnot, B.P. 301, F-92156, SURESNES Cedex, France

[email protected]

[email protected]

Received October 8, 2003; Revised November 11, 2004; Accepted November 11, 2004

First online version published in June, 2005

Abstract The Hermite transform allows to locally approximate an image by a linear combination of polynomials.For a given scale σ and position ξ , the polynomial coefficients are closely related to the differential jet (set ofpartial derivatives of the blurred image) for the same scale and position. By making use of a classical formuladue to Mehler (late 19th century), we establish a linear relationship linking the differential jets at two differentscales σ and positions ξ involving Hermite polynomials. For multi-dimensional images, anisotropic excursions inscale-space can be handled in this way. Pattern registration and matching applications are suggested.

We introduce a Gaussian windowed correlation function K (v) for locally matching two images. When taking themutual translation parameter v as an independent variable, we express the Hermite coefficients of K (v) in termsof the Hermite coefficients of the two images being matched. This new result bears similarity with the Wiener-Khinchin theorem which links the Fourier transform of the conventional (flat-windowed) correlation function withthe Fourier spectra of the images being correlated. Compared to the conventional correlation function, ours is moresuited for matching localized image features.

Numerical simulations using 2D test images illustrate the potentials of our proposals for signal and imagematching in terms of accuracy and algorithmic complexity.

Keywords: Hermite transform, anisotropic scale-space, pattern matching, affine registration, Gaussian windowedcorrelation, local correlation

1. Introduction

One of the key issues in signal and image process-ing is to identify appropriate data representations tosuit a given range of applications. The widest spreadsuch representation is the Fourier transform whichis quite suited, for example, to capture the oscilla-tory behaviour of signals and noise (Bracewell, 1999;Schroder and Blume, 2000). When local features orpatterns have to be analysed or processed, Gaussianwindowed Fourier or Gabor transforms are preferred(Kruger and Sommer, 2002; Lee, 1996). A large fam-

*To whom correspondence should be addressed.

ily of wavelet transforms aim at fulfilling similar needs(Daubechies, 1992; Mallat, 1998).

In the present article, we study orthogonal polyno-mial representations that are suited to represent signaland image data within a Gaussian window. Such a win-dow is unique in the sense that it is optimal with respectto scale-space axioms and is dimensionally separable(Duits et al., 2002; Florack et al., 1996). Furthermore,as engineers and physicists know, it offers the best si-multaneous frequency-space localization compromise(Arfken, 1985).

For a fixed Gaussian window, the correspondingHermite polynomial representation is strongly relatedto the biologically motivated Gabor representation

126 Makram-Ebeid and Mory

(Lee, 1996). It has advantages over Gabor waveletsbecause it is more convenient for handling rotations,translations and scale changes as will be shown in thenext sections. As with Gabor or with other wavelettransforms, systematic analysis or processing of a com-plete signal or image requires an array of overlap-ping windows (Kruger and Sommer, 2002; Lee, 1996;Martens, 1997). In general, this should be preferablydone for a range of different scales (i.e. different sizesof Gaussian windows in our case) within a multi-resolution pyramid. A very powerful tool for dealingwith such window arrays is the wavelet frames theory(Daubechies, 1992; Lee, 1996; Mallat, 1998; Martens,1997). In this article, we concentrate on the localaspects that constitute essential building blocks ofa multiple window multi-resolution pyramid that wewish to develop in the future.

1.1. Orthogonal Polynomials for Signaland Image Processing

We consider signals and images as scalar valuedfunctions of one or several variables. We start hereby dealing with one dimensional signals or, equiv-alently, functions of one real variable. Orthogonalpolynomials provide a convenient tool for locally ap-proximating such functions (Dunkl and Xu, 2001;Szego, 1967). Thus, one approximates a function f (x)as a linear combination of polynomials pi (x) havingdegrees i going from 0 to infinity. The approximationis performed by a least mean square technique thatconsists of minimizing the weighted quadratic error Qdefined by

Q =∫ +∞

−∞w(x)

(f (x) −

∞∑i=0

ai pi (x)

)2

dx (1)

where ai are the polynomial coefficients and wherew(x) is a non-negative weight function. Regions ofx for which w(x) is large are those for which onewishes the approximation to be accurate even whenkeeping a small number of terms. The coefficientsai are those which minimize the quadratic error Q. Fora polynomial basis which is orthogonal for the weightfunction w(x), they are given by

ai = fi/ci where fi =∫ +∞

−∞w(x) f (x)pi (x) dx

and

ci =∫ +∞

−∞w(x)(pi (x))2dx . (2)

The weight function w(x) defines an L2 metric in thevector-space of functions f (x) for which any distinctpair of polynomials in the basis are orthogonal. A directconsequence of this is that one can define a scalar prod-uct between any two functions f (x) and g(x) throughParseval’s theorem (Arfken, 1985)

〈 f, g〉w ≡∫ +∞

−∞w(x) f (x)g(x) dx =

∞∑i=0

fi gi

ci

where

gi =∫ +∞

−∞w(x)g(x)pi (x) dx . (3)

The weighted scalar product 〈 f, g〉w can therefore beexpressed either as an integral in x-space or else interms of polynomial transform coefficients fi and gi .The discrete sum in the rightmost member of Eq. (3) de-fines an equivalent transform-space (index i-labelled)scalar product; its corresponding metric is referred to,in this article, as the l2-metric. The inverse of the nor-malization constants ci introduce weights in the l2-metric playing a role similar to that of w(x) in thex-space L2-metric corresponding to the scalar product〈 f, g〉w.

The above results allow the representation of a func-tion of a real variable by an infinite polynomial se-ries. In practice only a finite number of terms may beused. Skipping the high rank terms gives rise to the socalled Gibbs oscillations phenomenon (Mallat, 1998).For the particular case of Hermite polynomials dealtwith in this article, large polynomial order asymptoticexpansions can be found in Szego (1967). Those canbe used to show that missing high rank terms have anoscillatory behaviour with amplitudes behaving like1/

√w(x). In other words, the square of the approx-

imation error is inversely proportional to the weightfunction w(x). This is what one would intuitively ex-pect from Eq. (1); i.e. the square of the approximationerror is large when its constraining weight factor w(x)is small.

Scale-Space Image Analysis based on Hermite Polynomials Theory 127

1.2. Hermite Transform and Related Scale-SpaceDifferential Structure

As mentioned earlier, the Gaussian window is a naturalchoice for image processing. For one dimension, weuse for w(x) a normalized window of arbitrary size σ

and centroid ξ

w(σ, ξ ; x) = 1√2πσ 2

exp

(− (x − ξ )2

2σ 2

). (4)

The corresponding orthogonal basis is that of the Her-mite polynomials (Szego, 1967) Hi ((x − ξ )/σ

√2) for

which the normalization constants ci defined in Eq.(2) is given by ci = 2i i!. The Hermite transform ofan arbitrary function f (x) is given by the coefficientsfi defined in Eq. (2) with the above special weightfunction and with pi (x) replaced by the above Her-mite polynomials. Hermite coefficients fi can also bemathematically written in terms of derivatives of thefunction F = w ⊗ f obtained by convolving func-tion f (x) with a normalized Gaussian kernel of sizeσ . By using the basic properties of Hermite polynomi-als (Szego, 1967), it is a simple exercise to show that(Koenderink and Van Doorn, 1992)

fi (σ, ξ ) = 2i/2σ i di F

dxi

∣∣∣∣x=ξ

. (5)

A particular class of functions f is that of the poly-nomials. In this case, the blurred function F is also apolynomial and the resulting Hermite expansion willinclude a finite number of non-vanishing terms. The re-sults obtained in this article can then be considered asexact tools to manipulate polynomial approximations.A similar remark extends to the multi-dimensional case(see Section 3 below).

If the independent variable x is expressed in units ofσ√

2 , the pre-factor 2i/2σ i in Eq. (5) can be omitted.Equation (5) will be used later to derive other results.It can be used, in practice, to obtain low-order Hermitecoefficients fi with i < 4 by numerically deriving ablurred version F(x) of f (x). However, this numeri-cal method cannot be extended to compute higher or-der coefficients because of the occurrence of severerounding errors. Computation of the set of coefficientsfi , from Eq. (2), requires the knowledge of the non-blurred function f (x) for x-values at which the win-dow function w(x) is not negligibly small. With thisreservation in mind, Eq. (5) shows that the Hermite

transform is directly related to the derivatives of F(x)which define the differential structure of function f (x)at scale σ .

2. Basic Properties of Hermite Transform

In this main section, we study the basic properties ofthe Hermite transform that can be useful for signal andimage processing. We make use of the theory of orthog-onal polynomials which is well established (Szego,1967) and which has been extended to any number ofdimensions (Dunkl and Xu, 2001).

2.1. Dimensional Separabilityand its Computational Advantages

A d-dimensional image is defined here as a real-valuedfunction f (x) of the d-dimensional position vectorx = (x1, x2, . . . , xd )T . As for the 1D case, one canlocally analyse f (x) within an isotropic Gaussian win-dow of size σ (representing scale) around any win-dow centre ξ = (ξ1, ξ2, . . . , ξd )T . The isotropic d-dimensional window can be considered as a productof one-dimensional windows. Likewise, the orthogonalpolynomial basis is made up of products of 1D Hermitepolynomials, which are shown to form a complete set(Dunkl and Xu, 2001). Each one of those polynomialsis identified by a multi-index or d-dimensional vec-tor index I = (i1, i2, . . . , id ) where all component in-dices ik are non-negative integers. The correspondingd-dimensional Hermite transform coefficient f I cannow be written as

f I (σ, ξ ) = 1

(2πσ 2)d/2∫x∈Rd

exp

(−‖x − ξ‖2

2σ 2

)PI

(x)

f (x) dx

where

PI (x) =d∏

k=1

Hik

(xk − ξk√

2σ 2

)(6)

and the corresponding polynomial approximation forf (x) in the window is

f (x) ∼=∑

I

f I (σ, ξ )PI (x)/cI (7)


where cI is the normalization constant which for amulti-index I = (i1, i2, . . . , id ) is expressed as a prod-uct of 1D normalization factors. The summation in Eq.(7) extends over all multi-indices I one wishes to usein the approximation, all integer component indices ik

are constrained to be non-negative.Equations (6) and (7) define the d-dimensional trans-

form pair for going from direct x-space representationto transform I-space representation and vice-versa. Nu-merical computation algorithms can take good advan-tage of the dimensional factorisation that is inherent inthese equations. To avoid burdening the notation, welimit ourselves here to the 2D case but our observationsare readily generalized to any number of dimensionsd. For the 2D case, the multi-indices I are written aspairs of non-negative integers I = (i, j). A change ofvariable is used so as to bring the centre of the Gaus-sian window to the origin ξ = (0, 0)T , the coordinateunits are selected so that σ

√2 = 1 and the independent

space positional variable is written as (x, y)T . Equation(6) can now be reformulated as

fi, j = 1

π

∫x∈R2

e−x2−y2Hi (x)Hj (y)

× f (x, y) dx dy

= 1√π

∫ +∞

y=−∞e−y2

Hj (y) fi (y) dy

where

fi (y) = 1√π

∫ +∞

x=−∞e−x2

Hi (x) f (x, y) dx . (8)

In other words, one may first take the 1D Hermitetransform fi (y) along the x-axis for each of the y-coordinates and then take the 1D Hermite transformalong the y-axis for each value of i to get the set of co-efficients fi, j . A symmetric procedure allows gettingthe inverse transform in a similar manner. To estimatethe algorithmic complexity of those procedures, as-sume that numerical integration along the x- or y-axesmake use of L discretization points and that the i and jindices take values in the interval [0, M −1]. The num-ber of operations needed for each of the transform andits inverse is of the order of L2 M+L M2. So, if N standsfor the largest of L and M, the computation load is ofthe order of 2N 3 and for any number of dimensions d itcan be easily seen to be of the order of d × N d+1. If theorthogonal basis was not dimensionally separable, thecorresponding computational load would have been of

the order of N 2d which is much larger. The larger theimage dimension, the more significant is the saving.

Another useful consequence of Eq. (8) is that f0, j

can be interpreted as the 1D Hermite transform ofthe function f0(y) obtained by Gaussian smoothingf (x, y) in the x-direction around point (0, y)T .

2.2. Dealing with Rotation in 2D

In two dimensions, the Hermite transform is partic-ularly suited for dealing with rotations (Koenderinkand Van Doorn, 1992; Martens, 1997). This is be-cause the Hermite coefficients fi, j can be expressedas multiple partial derivatives of an invariant scalarfunction (see Eq. (29) in Section 3). Tensor theory al-lows us to deduce that the set of fi, j coefficient withrank p ≡ i + j should behave like symmetric rank pcovariant tensors (Spain, 1960) under isometric trans-formations (pure rotations and/or mirror symmetriesaround the window centre). Another familiar instanceof rank p tensors is that of degree p monomials xi y j

(with p = i+ j). Such monomials behave like symmet-ric contravariant tensors under linear transformationsof the coordinates. Both covariant and contravarianttensors lead to identical linear transformation laws forthe pure rotation transformations that are dealt with inthis section (Spain, 1960). We now determine the ma-trix A(p) which allows the conversion of the monomialsxi y j under rotation, which is the same as that neededto convert the Hermite coefficients fi, j . Since thereare p + 1 elements of rank p, A(p) is a (p + 1)2 squarematrix. As in the previous section, we bring the centreof the Gaussian window to the origin ξ = (0, 0)T . Ifthe coordinate axes are rotated by an angle θ so thatany (x, y)T vector is changed to (x ′, y′)T according tothe law:

(x ′

y′

)=

(cos θ sin θ

− sin θ cos θ

) (xy

), (9)

then, in the new referential, any monomial (x ′)i (y′)p−i

(with 0 < i < p) can be expressed as

(x ′)i (y′)p−i

= (x . cos θ + y. sin θ )i (−x . sin θ + y. cos θ )p−i

≡j=p∑j=0

A(p)i, j x j y p− j . (10)


The coefficients of matrix A(p) are evaluated by ex-panding the middle expression of Eq. (10) in terms ofthe monomials (x) j (y)p− j . As already stated above,we can deduce how Hermite coefficients are convertedunder rotation namely

f ′i,p−i =

j=p∑j=0

A(p)i, j f j,p− j . (11)

The matrix coefficients take a particularly simple formwhen i = 0 and when i = p (top and bottom lines ofA(p)) since they can then be expressed (using Eq. (10))from the binomial theorem

A(p)0, j = (−1) j

(pj

)sin j θ. cosp− j θ

and

A(p)p, j =

(pj

)cos j θ. sinp− j θ (12)

where (pj

)= p!

(p − j)! j!=

(p

p − j

)(13)

are binomial coefficients.More generally, the coefficients of matrices A(p) can

be recursively calculated starting with total order p =0. Since the zero order term is rotation invariant, thecorresponding 1 × 1 matrix has its single entry equalto unity i.e. A(0)

0,0 = 1. Using Eqs. (9) and (10), we getthe following recursive relations

A(p+1)i, j = cos θ A(p)

i, j − sin θ A(p)i, j−1 for i ≤ p

and

A(p+1)p+1, j= cos θ A(p)

p, j−1 + sin θ A(p)p, j (14)

with the convention that the matrix entries A(p)i, j with

j > p and A(p)i, j−1 with j = 0 are replaced by zero.

For image dimensions d > 2 any rotation can bedecomposed into a succession of in-plane rotations in-volving two coordinates at a time (Gallier, 2000). Astraightforward extension of the above approach cantherefore deal with multi-dimensional rotations in Her-mite domain.

If only rotations have to be dealt with, the polarGauss-Laguerre transform (Jacovitti and Neri, 2000;Koenderink and Van Doorn, 1992; Martens, 1997) maybe more efficient than the Cartesian Hermite transform.The two transform types are strongly related so that go-ing from one to the other is a simple matter (Martens,1997). However, it is always recommendable to startfrom the computationally efficient (dimensionally sep-arable) Cartesian form and then deduce the polar formwhen needed.

2.3. Mehler’s Formula and its Applicationsto Scale-Space

To deal with Hermite domain translation, scale changeand blurring, we suggest using a formula due to F.G.Mehler (late 19th Century, see Szego (1967), p. 380).This formula is very valuable when studying the con-vergence of Hermite polynomial expansions. Mehler’sformula reads

(1 − t2)−1/2 exp

(2t xy − t2(x2 + y2)

(1 − t2)

)

=∞∑

i=0

t i

2i i !Hi (x)Hi (y) (15)

for any value of t in the range −1 < t < 1. The proofof this key formula is straightforward using 2D Fouriertransform (Watson, 1933). By multiplying both sidesof Eq. (15) by e−x2

f (ξ + σ√

2x)/√

π where f is anarbitrary 1D function and integrating both sides from−∞ to +∞, one gets

1√π (1 − t2)

∫ +∞

x=−∞e− (t y−x)2

1−t2 f (ξ + σ√

2x) dx

=∞∑

i=0

fi (σ ; ξ )

2i i !t i Hi (y) (16)

where fi (σ ; ξ ) = 1√2πσ 2

∫ +∞−∞ e− (x−ξ )2

2σ2 Hi (x−ξ

σ√

2)

f (x) dx is the Hermite coefficient of f for a Gaus-sian window centred at x = ξ and of size σ . Withchange of variables, this result can be rewritten as

F(σ√

1 − t2; ξ + tv)

=∞∑

i=0

(σ t)i

2i/2i !Hi

(v

σ√

2

)F (i)(σ ; ξ )

=∞∑

i=0

t i fi (σ ; ξ )

2i i !Hi

(v

σ√

2

)(17)


where F(γ ; η) stands for the signal f Gaussiansmoothed with kernel σ = γ and evaluated at x = η

and F (i)(γ ; η) is the corresponding ith derivative. Bytaking t close to unity, one reconstructs the signal with-out blurring.

Mehler’s Formula Related Signal ProcessingLEMMA. Equation 17 implies that, given a factort with |t | < 1, if the Hermite coefficients are modifiedby multiplying order i coefficient by t i ; then the result-ing Hermite series expansion yields a version of signalf blurred with a Gaussian kernel of size σ

√1 − t2 and

then zoomed by a factor 1/t about the window centre.An equivalent formulation was derived by Martens

(1992) for deblurring isotropic Gaussian blur in im-ages.

New Analytical Expression for 1D Scale-SpaceTransformations. Florack et al. (1996) have anal-ysed the influence of simultaneous translation and scalereduction excursions on scale-space local jets. Theyhave studied the constraints on such excursions that en-sure convergence of their Taylor expansions. Mehler’sformula allows us to more specifically put the corre-sponding results in an explicit and compact analyticalform provided appropriate pairs of “conjugate zoom-ing and blurring operators” are applied. To do so, werecall that Eq. (5) allows to mathematically expressHermite coefficients in terms of derivatives of a Gaus-sian blurred function. We apply Eq. (17) with signal freplaced by its jth derivative and the values of F andof its derivatives expressed in terms of Hermite coeffi-cients to get

f j (σα ; ξ + βv)

= α j

( ∞∑i=0

β i

2i i!Hi

(v

σ√

2

)f j+i (σ ; ξ )

)(18)

where α = √1 − t2 and β = t so that α2 + β2 = 1.

This allows to express Hermite coefficients for a Gaus-sian window translated to x = ξ +βv and referred to awindow of size σα in terms of the Hermite coefficientsfor an original window centred at x = ξ and of sizeσ . It is useful to note that the Gaussian kernel sizes σ

and σα are scales (i.e. stick lengths) in units of whichthe independent variable x may be measured. Sinceα ≤ 1, the Hermite expansion for the Gaussian kernelσα may be considered as a zoomed version of that ofthe Gaussian kernel σ .

If one is interested in polynomial approximationsnot exceeding a maximum degree N, all coefficientsof order larger than N can be replaced by zeros andthe above relations are expressed as a linear relationbetween the (N + 1)-dimensional Hermite coefficientvectors for the two windows. The matrix incurred inthis transformation is an upper triangular one whereany row j is obtained from the first one ( j = 0) by ashift and a scalar multiplication by α j . In the specialcase where f (x) is a polynomial of degree N , theresulting formulae are exact.

When the function f (x) is not a pure polynomial,one still wishes to use Eq. (18) with all terms of or-der larger than N neglected. This is justified as long asthe approximation errors due to higher order terms arewithin acceptable limits. Figure 1 illustrates the valid-ity condition. The coefficients f j+i (σ ; ξ ) correspond tothe Gaussian window W1(x) with kernel size σ and cen-tre at x = ξ whereas coefficients f j (σα ; ξ+βv) corre-spond to the Gaussian window W2(x) with kernel sizeσα and centre at x = ξ +βv. As discussed at the end ofSection 1.1, approximation errors within window W1

can be expected to be small for the range of x values forwhich W1(x) is larger than a positive fraction ε.W1(ξ )of its peak value W1(ξ ) (with 0 < ε < 1, e.g. ε = 0.1).Likewise, the validity range within window W2 canbe defined by the condition W2(x) > ε.W2(ξ + βv).Since Eq. (18) expresses Hermite coefficients for W2

Figure 1. Illustrating the validity condition for the 1D scale-space transformation deduced from Mehler’s formula. Giventhe Hermite coefficients fi (σ ; ξ ) corresponding to the Gaussianwindow W1(x), Eq. (18) allows the computation of the Hermitecoefficients f j (σα ; ξ + βv) corresponding to the Gaussian win-dow W2(x). For W2(x), the kernel size σα must be smaller thanthat the kernel size σ of W1(x) (i.e. α < 1). The relative trans-lation βv as well as the relative scale factor α must be such that{x | W1(x) > ε.W1(ξ )} ⊂ {x | W2(x) > ε.W2(ξ + βv)} where ε isa small positive fraction of unity (ε = 0.1 say). In other words, theacceptable Hermite expansion accuracy range for W2(x) should beincluded in that of W1(x)


in terms of those of W1, its validity requires that the set{x | W2(x) > ε.W2(ξ + βv)} be included within theset {x | W1(x) > ε.W1(ξ )} . In other words, the goodaccuracy range of W2 should be included in that of W1.It is clear, in particular, that window W2 could not bea pure translated version of W2 as this would alwaysviolate the above accuracy range inclusion rule for anychoice of ε > 0. This is why translation must alwaysbe accompanied by zooming (i.e. reducing the kernelsize). The same remarks also explain why one cannotdeal with signal shrinkage (i.e. taking α > 1).

A useful special case is that of 1D homethety(Gallier, 2000) which is an affine transformation forwhich the translation parameter v is zero and for whichthe centre of the Gaussian window is a fixed point. Allodd order Hermite polynomials in Eq. (18) vanish and,making use of the identity β2 = 1−α2, one is left withthe simpler expression

f j (σα ; ξ ) = α j

( ∞∑i=0

(α2 − 1)i

22i i!f j+2i (σ ; ξ )

). (19)

New 1D l2-Matching based on a Further Use ofMehler’s Formula. Take two signals f and g forwhich Hermite coefficients are known in an originalwindow centred at x = ξ and of size σ . Equation (18)can be used to compare scaled and translated versionsof the two signals in Hermite coefficient domain. Todo so, we operate a scale and translation change on fand g with parameters α = αi , β = βi and v = vi

with i = 1 for f and i = 2 for g and with constraintsthat α2

i + β2i = 1. One can, furthermore, smooth and

zoom the two functions before comparing them. Thecorresponding Hermite approximations to order n are

F(ξ + β1v1 + α1tu) ∼=N∑

i=0

t i fi (σα1 ; ξ + β1v1)

2i i!

× Hi

(u

σ√

2

)(20)

and

G(ξ + β2v2 + α2tu)

∼=N∑

i=0

t i gi (σα2 ; ξ + β2v2)

2i i!Hi

(u

σ√

2

)(21)

where F and G are obtained from the functions f andg by convolving with Gaussian low-pass kernels of

size σ α1

√1 − t2 and σ α2

√1 − t2 respectively. Par-

seval theorem allows writing the L2 (x-space) dif-ference norm of the difference of blurred functionsin terms of a modified index-labelled l2 differencenorm:

1√π

∫ +∞

u=−∞

(e− u2

2 F(ξ + β1v1 + α1tu)

− e− u2

2 G(ξ + β2v2 + α2tu))2

du

∼=N∑

i=0

t2i

2i i!

(fi (σα1 ; ξ + β1v1)

−gi (σα2 ; ξ + β2v2))2

. (22)

2.4. New Hermite-Domain 1D WindowedCross-Correlation

In the previous section, we have proposed a methodto evaluate the mean square difference of two signalsfor matching purposes. We now investigate the corre-lation function for two signals versus relative trans-lation and scale parameters. We attempt to do this insuch a way that the two signals being correlated playsymmetrical roles. Furthermore, we build the correla-tion function in a hierarchical fashion (coarse to fine)using the smallest possible number of Hermite coeffi-cients of each signal. To simplify notation, we assumethat the Gaussian window is centred at the ordinatesorigin and that the ordinates unit is chosen so that√

2σ 2 = 1.Take two 1D functions f (x) and g(x) with Hermite

transforms fi and gi . Rather than predicting how eachof these two coefficient sets are modified with transla-tion or scale change, we study instead the behaviour ofthe external product h(x, y) = f (x).g(y) of these twofunctions. We thus generate a 2D function out of thetwo 1D functions. The 2D Hermite coefficients of thisnew 2D function are given by hi, j = fi g j . Let us nowperform a (−θ ) rotation from the (x, y) referential tothe rotated (u, v) so that

(u

v

)=

(cos θ sin θ

− sin θ cos θ

) (x

y

)and hence

(x

y

)

=(

cos θ − sin θ

sin θ cos θ

) (u

v

). (23)


Equation (12) tells us how to use the above hi, j co-efficients in the (x, y) referential in order to com-pute the coefficients h′

0,p in the (u, v) referential withthe first index equal to zero (i = 0) and the sec-ond index equal to an arbitrary non-negative integer( j = p). From the remark in the last paragraph ofSection 2.1, one sees that these h′

0,p coefficients arejust the Hermite coefficients of the 1D function K (v)obtained by Gaussian smoothing in the u direction,i.e.

K (v) = 1√π

∫ +∞

−∞e−u2

f (α.u − β.v)g(β.u + α.v) du

(24)

where α = cos θ and β = sin θ . In other words, theHermite coefficients K p of K (v) are given by h′

0,pwhich from Eq. (12) gives

K p = h′0,p =

j=p∑j=0

((−1) j

(pj

)sin j θ cosp− j θ

)

· f j .gp− j . (25)

The fact that K (v) is a windowed cross-correlation ofthe two functions f and g is particularly clear if onechooses θ = π/4 which, with a change of independentvariable in the integrand of Eq. (25), yields

K

(τ√2

)=

√2

π

∫ +∞

−∞e−2u2

f(

u − τ

2

)× g

(u + τ

2

)du. (26)

The special case for θ = π/4 and with identical fand g functions defines a “windowed auto-correlation”.More generally, one may remark that the indepen-dent variable u in the integral of Eq. (25) is multi-plied by a factor cos θ in the argument of f and bysin θ in the argument of g. The independent variable vparameterises a translation of the argument of f rela-tive to that of g. For a fixed translation parameter, bysetting

cos θ = α = 1/√

1 + ζ 2 and

sin θ = β = ζ/√

1 + ζ 2 (27)

the above rotation matrix equations relating (x, y) to(u, v) can be reinterpreted by eliminating the parameter

u; they are equivalent to an affine transformation fromargument x of f to argument y of g which can be writtenas

y = ζ x + α−1v, with inverse x = ζ−1 y − β−1v

(28)

where ζ is a relative scaling parameter and α−1v (or−β−1v) as relative translations.

Advantages of our Hermite-Domain Windowed Cor-relation. Equations (25)–(28) provide new originalfeatures relative to earlier proposed correlation func-tions. They allow to deal with translation for anygiven scale change when computing the Gaussian win-dowed cross-correlation (Eq. (24)). Furthermore, theHermite transform of the cross-correlation functioncan be deduced explicitly from the Hermite coeffi-cients of the two functions f and g. When search-ing for a best match, our computational savings canbe achieved by working in transform space and con-centrating on noise-robust low-order coefficients. DenBrinker (1993) have earlier reported related Laguerretransform domain correlation results. However, theRef. (Den Brinker, 1993) Laguerre counterpart of Eq.(25) is a less convenient infinite series (in place of thefinite sum in Eq. (25)). In addition to its more com-pact form, our Hermite domain correlation can readilyextend to several dimensions by virtue of dimensionalseparability.

3. Dealing with any Number of ImageDimensions

The dimensional separability of the Hermite trans-form greatly facilitates dealing with any number ofdimensions. A particularly useful feature is the rela-tion with the differential structure of an image. Theresult expressed by Eq. (5) is readily generalized for ad-dimensional image with an isotropic Gaussian win-dow of size σ around a centre ξ . The Hermite coef-ficient for a multi-index I = (i1, i2, . . . , id ) of rankn = ∑d

k=1 ik is given by:

f I (σ, ξ ) = 2n/2σ n ∂n F

∂xi11 ∂xi2

2 . . . ∂xidd

∣∣∣∣∣x=ξ

. (29)


3.1. Reducing and Extending the Number ofTransform Dimensions

In Section 2.4, we have generated a 2D functionby taking the external product of two 1D functions.This proved useful for computing the localized cross-correlation and its Hermite transform. This procedurecan be extended to any number of dimensions. Take afunction f (u) defined for m-dimensional vector argu-ments u and a function g(v) defined for n-dimensionalpositional vectors v, their product f (u).g(v) can betreated as a function of the m + n-dimensional vec-tor x = (u, v) = (u1, . . . , um, v1, . . . , vn). It is nothard to see that the Hermite coefficient of this prod-uct function for the m + n-dimensional multi-indexI = (i1, i2, . . . , im+n) is equal to f(i1,...,im )g(im+1,...,im+n ) .

The inverse situation is also of interest. In Section2.1, for example, it was shown that f0, j are the 1DHermite coefficients of f0(y) obtained by Gaussiansmoothing a 2D function f (x, y) along the x-axis. Thiswas instrumental in deriving the new Hermite domaincorrelation result of Eq. (25).

3.2. Locally Affine Warping is Readily Handled inHermite Transform Domain Using our Results

The results obtained in Sections 2.3 to 2.4 readily ex-tend to any number of dimensions through the dimen-sional separability properties of the Hermite transform.

For any one of the image dimensions (labelled k),we apply Mehler’s formula (Eq. (17)) to compute eachone of the multi-dimensional Hermite coefficients (Eq.(29)) considered as functions of xk . This allows us tolinearly express the differential jet for given translationand scale-reduction along xk in terms of the originaldifferential jet. This operation can be repeated for allimage dimensions thus providing a generalization ofthe results of Section 2.3. It is noteworthy that dif-ferent scale reductions (inverse of zooming factors)can be applied to the different dimensions thus allow-ing anisotropic excursions in scale-space. As for the1D case, neither pure translation nor size shrinkage(equivalent to scale increase) can be directly dealt within Hermite domain as this would violate the accuracyrange inclusion principle explained in Section 2.3 andFig. 1. In order to match images for more general typesof mutual warping laws using Hermite domain pro-cessing, different methods are possible. One methodconsists in applying different zoom and translation toeach of the d-dimensional images f (x) and g(y) before

comparing them together. To do so, we linearly expressthe pair of d-dimensional vectors (x, y)T in terms ofa new pair of d-dimensional vectors (u, v)T . In thisnew representation, v is a fixed mutual translation vec-tor parameter whereas u is the new independent spacevariable. In order to apply the results of Sections 2.3and 2.4 to the kth dimension, the coordinates xk and yk

of the positional vectors x and y are transformed intouk and vk through a unitary transformation defined by

(xk

yk

)=

(αk −βk

βk αk

) (uk

vk

)where α2

k + β2k = 1

(30)

If this is done for all image dimensions k, we geta partition-matrix relation between the two (2 × d)-dimensional vectors (x, y)T and (u, v)T in the form

(x

y

)=

(A −B

B A

) (u

v

)(31)

where A and B are d diagonal matrices with diagonal el-ements given by αk and βk respectively. Recalling thatv is the translation parameter and eliminating the com-mon independent space vector u by using Eq. (31), thecorresponding image-warping transformation (from xto y) is expressed by

y = Z x + A−1v, with inverse x = Z−1 y − B−1v

(32)

where Z = B A−1 = A−1 B is a diagonal d × d matrixwith diagonal elements given by ζk = βk/αk and ma-trix Z−1 is its inverse with diagonal elements given by1/ζk = αk/βk .

This, however, is not the most general image-warping class that can be conveniently dealt with inHermite transform domain. As shown earlier, pure ro-tations are easily handled in view of the tensor proper-ties of the Hermite coefficients. One can apply rotationoperators R1 and R2 in x and in y respectively aroundtheir window centres so that the generic form of trans-formation that can conveniently be handled is of theform:

y = RT2 Z R1x + RT

2 A−1v, with inverse

x = RT1 Z−1 R2 y − RT

1 B−1v. (33)


If all axial scaling factors ζk are non-zero, the abovewarping laws are invertible affine transformations(Gallier, 2000). Conversely, any invertible affine trans-formation can be put in such form by virtue of the Sin-gular Value Decomposition Theorem (Gallier, 2000).

3.3. Extending our l2 Norm and Correlation Resultsto Several Dimensions

Generalizing Eq. (17) to several variables is straightfor-ward. It is easily shown, for example, that multiplyingeach Hermite coefficient by a factor tn where |t | < 1and where n is the coefficient’s rank (sum of the in-dices ik), the resulting Hermite expansion is a versionof f blurred with an isotropic Gaussian kernel of sizeσ√

1 − t2 and zoomed about the window centre by afactor of 1/t . The other results of Sections 2.3 and 2.4extend to several dimensions by making use of the di-mensional separability of the Hermite transform. Thegeneric class of referential changes of Section 3.2 canbe easily dealt with in this manner.

4. Numerical Study for Hermite DomainRegistration and Matching

We have performed numerical simulations to illustratethe results reported in this article using 2D images. InFig. 2(a), a 128 × 128 butterfly test-image is approx-imated by Hermite polynomial expansions up to rankN = 64. The Hermite transform implementation usedfor this purpose is of the continuous wavelet-transformtype (Daubechies, 1992; Mallat and Wavelet, 1998;Martens, 1997), and is based on the numerical evalu-ation of the integrals involved in Eq. (8). This imple-mentation allows us to extract a bivariate polynomialapproximation of the image valid for a region of thereal x, y plane within a Gaussian analysis window (i.e.not only for discrete-valued x, y coordinates). The re-sults of Sections 2 and 3 show how one can handlesuch polynomial approximants in an exact analyticalmanner. In Figs. 2(a) and (e), the same butterfly im-ages, having grey values in the closed interval [0,255],is analysed using two different Gaussian analysis win-dows schematized by dotted circles having their centresat the Gaussian window centre and radii equal to 2σ

Figure 2. The same butterfly 128 × 128 test image is analysed within two different Gaussian analysis windows (a) and (e) schematized by thedotted circles. Both circle radii are equal to 2σ with σ = 16. The corresponding Hermite expansions to maximum total rank N = 64 are shownwith different blur and zoom parameter t in (b), (c) and (d) starting from the window in (a) and in (f), (g) and (h) starting from the window in(e). The absolute grey-level difference: between (b) and (f) is shown in (i), between (c) and (g) is shown in (j) and between (d) and (h) is shownin (k).


with σ = 16. Figure 2(b) and (f) depict the resultingHermite polynomial approximations for the two Gaus-sian windows. The grey-levels are clipped when above255 or below 0. It is clear that the approximation er-rors become excessive beyond a distance of 4σ fromthe window centre and yield the large amplitude Gibbsoscillation rings visible in the figures. Such errors aredue to missing high rank terms as alluded to in Section1.1. Those oscillations are expected to have an ampli-tude roughly inversely proportional to the square rootof the Gaussian window and should thus behave likee

x2+y2

4σ2 where the x, y coordinates are measured fromthe window centre. Our simulations show that essentialmorphological features are captured even when limit-ing the expansion to low rank terms but, in order to getsharp looking approximations, extremely high ranksare required. As noted by Martens (1997), this is bestunderstood by reasoning in Fourier domain. Hermitecoefficients of total rank n are seen to represent imageinformation within a ring of the 2D-spatial frequency

plane about an average radius of | �ω| ∼=√

n+ 12

σ. For

large Gaussian window sizes (σ � 1 in pixel units),unreasonably high ranks would therefore be neededto cover frequencies up to Shannon limit (equal to π

in pixel units). For this reason, one has to use an ar-ray of Gaussian windows (with small window sizes σ )and use the theory of frames in applications for whichthe approximation error is required to be negligible(Martens, 1997). This is an essential requirement, forexample, in image restoration or enhancement.

For matching or registration of local features in twoimages, the approximation error does not need to benegligible. There is even an important noise-robustnessadvantage to exclude the high rank noise-prone termsfrom the matching procedure. When a local patternundergoes an affine warping, one needs to examinewhether its low rank polynomial approximant followsthe same warping law. Figure 2 essentially shows howthe Hermite approximation behaves following a trans-lation of the analysis window. The polynomial ap-proximants resulting from the two analysis windowsare shown in Fig. 2(b) and (f). The map of the abso-lute difference of their grey-values is shown in Figs.2(i). Within the elliptical dark region at the centre ofthis map, the grey-values are less than 5. Therefore,if one sets a Gaussian matching window well withinthis black-region, the resulting Gaussian weighted L2

matching error can be expected to be very small com-pared to signal energy. Consequently, the equivalentHermite domain l2 matching error is also expected to

be very small in this case. In Fig. 2(c), (g), the Her-mite coefficients of total rank n are attenuated by afactor tn with t = 0.95. It can be seen that the Gibbsripples are attenuated and appear much further awayfrom the analysis window centre. The correspondingabsolute difference image in Fig. 2(j) is nearer to zerothan in Fig. 2(i). Figure 2(d), (h), (k) have been sim-ilarly computed but with t = 0.90. As expected fromMehler’s formula Eq. (17), for decreasing values of t (t = 1, 0.95, 0.90), the Hermite approximations appearprogressively more blurred and zoomed. For isolatedpatterns, the resulting L2 or l2 matching accuracy andrandom-noise robustness can be expected to improvewith decreasing t values (i.e. when giving more weightto low rank terms). In the presence of background im-age features, however, pattern blurring would not al-low a good discrimination between them and the localpattern of interest so that higher order ranks may berequired.

4.1. Handling Affine Warping in Hermite Domain

Figure 3 illustrates how an affine warping may be dealtwith in Hermite domain. It is factored into elementarytransformations that can be handled in Hermite domain(c.f. Sections 2.3, 2.4 and 3.2). The top row shows abutterfly image A of size 128 × 128 subjected to a ro-tation by π/4 to generate image B. Then B is axiallyzoomed in the x direction (x-Zoom) by a factor 1/ζ

(with ζ = 0.7) to produce image C. Image C is thenrotated back by an angle φ = −tan−1(ζ ) to produceimage D. In all these transformations, the image centreis fixed. The overall warping that produces image Dis a shear plus an isotropic zoom. In the second row,image E is obtained from image A by applying the Her-mite transform followed by an inverse transform up toa total rank N = 16. We refer to this image E as theHermite approximation of A up to a total rank N = 16.Images F, G and H are obtained from E by Her-mite domain warping (i.e. by applying the methods ofSections 2 and 3 for converting Hermite coefficients).The geometric warping operations applied in Hermitedomain in the second row are the same as those appliedin the first row in space domain. In the bottom row, theimages I, J and K correspond to the Hermite approx-imations of the spatially warped images B, C and D.A circle is shown in the bottom left corner of radius2σ = 32 in pixel units to visually provide the currentscale for all images (or reference window size) usedin the Hermite analysis. Even though the maximum


Figure 3. Handling affine warping in the Hermite domain and comparing with the effect of spatial warping. The top row shows a butterflyimage A of size 128 × 128, subjected to successive warping operations to produce spatially warped images (B, C and D); the abbreviation SDstands for space-domain warping and ζ = 0.7. In the second row, Image E is the Hermite approximation of A up to rank N = 16. Images (F, Gand H) are obtained by applying the same warping operations of the first row in Hermite domain; the abreviation HD stands for Hermite domainwarping. In the bottom row, the images (I, J and K) correspond to Hermite approximations (up to N = 16) of the spatially warped images ofthe first row. The reference Gauss window size is shown in the left bottom corner (σ = 16, at image centre).

reconstruction rank is small, one can still observe Gibbsoscillations. The only appreciable differences betweenthe Hermite domain warped images (F, G and H) andthose warped in space domain (second and third row)are observed to occur in regions where the Gauss win-dow is small (about 3σ from the window centre). Thuswithin the reference Gauss window, Hermite domainwarping operations correctly predict the effect of spa-tial warping operations. The simulations of the twonext sections will shed more light on the numerical ac-curacy achievable with such Hermite domain isotropicand anisotropic warping operations.

4.2. Hermite-Domain Rotation and RelativeScale Estimation

Figure 4 illustrates a rotation estimation in the Hermitedomain. The two patterns to be matched are shown inFig. 4(a) and (b). They consist of two identical butterflyimages which have been rotated by an angle θ0 = 5π/6about the Gaussian analysis window centre. The same

fixed (not rotated) background scene is present in bothimages (mountain and cloudy sky). The Hermite co-efficients of Fig. 4(a) are “rotated” for a range of ro-tation angles θ using the procedure explained in Sec-tion 2.2. In Fig. 4(c), the resulting l2 matching errorin the Hermite domain is computed versus the rota-tion angle θ . Minimum matching error is obtained foran angle θ very near to the true value θ0 = 5π/6 inspite of the disturbing effect of the fixed background.Good accuracy is obtained even with low total rankN .

Figures 5 to 8 illustrate Hermite domain relativescale estimations for a maximum Hermite coefficientsrank N = 16. In Fig. 5 and 6, a reference 128×128 but-terfly image is scaled isotropically and anisotropicallyto produce target test images. The Hermite domain l2

matching errors are computed as function of the scal-ing parameters (ζx , ζy). In Figs. 7 and 8 the same isrepeated but with a disturbing background consistingof mountains and a cloudy sky. All Hermite domainl2 matching error plots show a single minimum. Thepositions of the minima are found with high accuracy


Figure 4. A butterfly image on a mountain and cloudy sky background is shown in (a). In image (b), the butterfly is rotated with an angleθ0 = 5π/6 around the Gaussian analysis window centre. The Gauss analysis windows are represented schematically by dotted circles in bothtest-images (σ = 16). The first pattern (a) is rotated in the Hermite domain and the Hermite domain l2 matching error is shown in (c) versus therotation angle θ for different maximum ranks N of the Hermite expansions. The l2 matching error goes through a minimum for an angle θ closeto θ0 even for small N values.

Figure 5. Relative scale estimation for an isotropically zoomed test image. The original 128 × 128 butterfly image is shown in (a). Thetarget test image is defined in (b) by isotropically zooming (a) by a factor 1/ζ0 with ζ0 = 0.7 about the Gaussian analysis window centre(Gauss window size: σ = 16). The Hermite domain quadratic matching error is computed with t = 0.75 and maximum Hermite rank N = 16for independent x and y scale-reduction parameters (ζx , ζy ) applied to the original image (a). In (c) is a contour plot of the Hermite domainquadratic error (the spacing between the contours is logarithmic with a factor 1.15 between successive contours). The ideal minimum errorlocation is marked by the intersection of a horizontal with a vertical dotted line. The minimum error location coincides almost exactly with itsideal (ground truth) position.


Figure 6. Relative scale estimation for an anisotropically zoomed test image. The original 128 × 128 butterfly image is shown in (a). Thetarget test image is defined in (b) by shrinking (a) by a factor 1/ζx0 in the x direction with ζx0 = 1.8 and zooming it by a factor 1/ζy0 withζy0 = 0.8 in the y direction about the Gaussian analysis window centre (Gauss window size: σ = 16). The Hermite domain quadratic matchingerror is computed with t = 0.75 and maximum Hermite rank N = 16 versus the independent x and y scaling parameters (ζx , ζy ). Since scalereductions are not possible in Hermite domain, the relative Hermite domain warping is achieved by applying a variable x zooms ζx > 1 to theHermite coefficients of the target test image (b) and variable y zoom 1/ζy > 1 to the Hermite coefficients of the original test image (a). In (c)is a contour plot of the Hermite domain quadratic error (the spacing between the contours is logarithmic with a factor 1.15 between successivecontours). The ground truth minimum error location is marked by the intersection of a horizontal with a vertical dotted line. The minimum errorlocation coincides almost exactly with the ground truth.

Figure 7. Relative isotropic scale estimation in presence of a disturbing background. The numerical study is identical to that of Fig. 5 exceptfor the test patterns in which a disturbing background is added. Target image (b) is obtained from image (a) by scaling the foreground (butterfly)pattern only. In the contour plot of the Hermite domain quadratic error (c), the spacing between the contours is logarithmic with a factor 1.05between successive contours. The minimum error is close to the ideal position (crossing of dotted lines).


Figure 8 Relative anisotropic scale estimation in presence of a disturbing background. The numerical study is identical to that of Fig. 6 exceptfor the test patterns in which a disturbing background is added. Target image (b) is obtained from image (a) by scaling the foreground (butterfly)pattern only. In the contour plot of the Hermite domain quadratic error (c), the spacing between the contours is logarithmic with a factor 1.05between successive contours. The minimum error is close to the ideal position (crossing of dotted lines).

in Figs.(5 and 6) and with reasonably good accuracy inFigs. 7 and 8 in spite of the disturbing background.

4.3. Hermite-Domain Correlation and TranslationEstimation

Figure 9 illustrates an application of the Hermite do-main correlation proposed in this article. The two im-ages in Figs. 9(a) and (b) are identical except thatthe butterfly in the second image is translated relativeto that of the first one. The Hermite domain correla-tion has been computed and then, the inverse Hermitetransform is computed to show the results in spatial-translation domain. In Fig. 9(c), the computed cor-relation function is shown as contour plots for max-imum rank N = 8 and in Fig. 9(d) for N = 32.In both cases, the peak observed within a 2σ radiusfrom the window centre leads to a good translationestimation.

4.4. Need for Further Work on Hermite-DomainRegistration and Matching

The numerical studies presented in this article weremainly meant to illustrate the potential advantagesof working in Hermite domain in a visually con-

vincing manner. Many possibilities still need to bestudied. In particular, one observes that, for low or-der approximations, the matching error is a smoothfunction of the individual warping parameters (rota-tion angle, translation and mutual scaling). Computingthe gradient of such errors is possible because theyare explicit analytical functions of those parameters.The extra computation load is expected to be moder-ate. Therefore making use of gradient-based optimi-sation techniques may appreciably accelerate Hermitedomain registration and matching. The full computa-tional value of such techniques should appear if oneapplies them within a hierarchical coarse to fine strat-egy. Systematic benchmarking has still to be done forcomparisons with other approaches in terms of compu-tational burden and matching accuracy. In particular,comparisons with hierarchical techniques making useof multi-resolution Gaussian pyramids would be veryinteresting.

Our current work shows that the local charac-ter of our Hermite domain auto-correlation generatesseveral interesting properties. In particular, it pro-vides information on whether or not a local imagefeature can be used for translation estimation. An-other possible application is the automatic selectionof suitable scales for reliable local registration andmatching.


Figure 9 Illustrations are shown for the use of Hermite domain correlation for translation estimation. A butterfly image on a mountain andcloudy sky background is shown in (a). In image (b), the butterfly is translated by �x0 = σ and �y0 = σ relative to the Gaussian analysiswindow centre. The Gauss analysis windows are represented schematically by dotted circles in both test-images (σ = 16). Gaussian windowedcorrelation maps are computed from the Hermite domain correlation of image (a) with image (b) computed for N = 8 in (c) and for N = 32 in(d) and are shown as contour plots within a translation vector magnitude of 3σ . Images and contour plots are all displayed at the same scale. Thelevel difference between two adjacent iso-contours is constant (linear scaling). The peak in correlation within a distance 2σ from the origin oftranslation vectors (�x, �y) is near the ground-truth pattern translation (�x0, �y0, marked by crossing of dotted lines). The behaviour of thewindowed correlation map for translation vector magnitudes larger than 2σ cannot be used for local translation estimation because it capturesbackground features correlation and is too sensitive to the Hermite approximation errors.

5. Summary and Conclusions

In this article, we suggest new links between scale-space and Hermite transform theory. A formula dueto Mehler is found to have interesting implicationsfor Hermite transform pattern manipulations. In scale-space language, this formula tells us that attenuat-ing rank n coefficients of a local jet by a factor oftn (with |t | < 1) results in a Hermite series expan-sion of a blurred and zoomed version of the image(around a fixed window centre). Based on this for-mula, we derive a new result allowing the analyti-cal expression of the effect of scaling and transla-tion in the Hermite domain. The differential jet (orequivalently Hermite coefficients vector) can be usedfor pattern matching. The corresponding l2 metric isshown to be related to Gaussian weighted L2 met-ric in the spatial domain. This allows the comparison,in the Hermite domain, of mutually warped patterns.The warping laws that can be handled in this way in-clude rotations, translations as well as isotropic and

anisotropic scale changes. Comparing blurred versionsof the patterns is shown to be equivalent to a simplemodification of the l2 metric. A new weighted cross-correlation method is also proposed. It plays a rolesimilar to that of the Wiener-Khinchin theorem thatdeals with flat-windowed correlation in the Fourier do-main. Our result allows the evaluation of Gaussian win-dowed correlation in the Hermite domain. By using ourtechnique, advantages can be expected in computationand robustness. Several applications may be envisagedfor registration, motion estimation and pattern match-ing. The windowed correlation result may, further-more, be applied to code and characterize textures inimages.

Acknowledgments

The write-up of this article has greatly benefited fromconstructive comments of Luc Florack, Jean-BernardMartens, Raoul Florent and two of the IJCV reviewers.


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