A Wreath Product Group Approach to Signal and Image Processing ...

A Wreath Product Group Approach to Signal and Image Processing:Part II | Convolution, Correlation, and ApplicationsG. Mirchandani R. Foote yD. Rockmore zD. Healy xT. Olson {September 23, 1999AbstractThis paper continues the investigation of the use of spectral analysis on certain noncommutative nitegroups|wreath product groups|in digital signal processing. We describe here the generalization of discretecyclic convolution to convolution over these groups and show how it reduces to multiplication in the spectraldomain. Finite group-based convolution is dened in both the spatial and spectral domains and its propertiesestablished. We pay particular attention to wreath product cyclic groups and further describe convolutionproperties from a geometric view point, in terms of operations with specic signals and lters. Group-basedcorrelation is dened in a natural way and its properties follow from those of convolution. We nally consideran application of convolution: the detection of similarity of perceptually similar signals, and an applicationof correlation: the detection of similarity of group transformed signals. Several examples using images areincluded to demonstrate the ideas pictorially.EDICS numbers: SP 2-HIER and SP 4-MTHEDepartment of Electrical and Computer Engineering, University ofVermont, Burlington VT 05405. Supported by NASA/JOVEyDepartment of Mathematics and Statistics, University ofVermont, Burlington VT 05405. Supported by NASA VT SpaceGrantzDepartment of Mathematics, Dartmouth College, Hanover, NH 03755. Supported by NSF, ONR, ARPA, National Center ofAtmospheric ResearchxDepartment of Mathematics, Dartmouth College, Hanover NH 03755. Supported by ARPA{Department of Mathematics, Univ. of Florida, Gainesville, FL. Supported by ONR1

1 IntroductionThis paper continues the theoretical development of [6] and extends wreath product group-based signal processingto apply to two major properties: group-based convolution and correlation. We outline the generalizationof discrete cyclic convolution to group-based convolution for an arbitrary nite group. This generalized convolutionenjoys many ofthe properties of cyclic convolution, including transformation laws with respect to thegroup action. We consider group-based convolution and describe its properties. We also consider its use as anon-commutative lter in the detection of perceptually similar images. Group-based correlation is also denedand we consider its application in (peak) detection of group-transformed images.Cyclic (circular) convolution f?hof discrete signals f and h in a spatial (or time) domain plays a fundamentalrole in signal processing, particularly in the context of ltering. Each xed h gives a lter on the spatialdomain by f 7! f?h. Dierent choices of h extract dierent types of information from the inputs f. The classicalDFT turns convolution in the spatial domain into multiplication in the spectral (frequency) domain. Weshall see that for arbitrary nite groups the generalized Fourier transform also turns convolution in the spatialdomain into matrix multiplication in the spectral domain. Moreover, when the representation of the group onthe spatial domain is multiplicity-free|as is the case with the WPC groups|this matrix multiplication reducesto the multiplication of matrices by scalars. Thus by passing to the spectral domain, group-based convolutionmay be eectively computed. In particular, we sawin[6] that the analysis stage of an M-channel DFT lterbank outputs the spectrum of a discrete signal when the underlying group is a WPC group. Recalling that theWPC groups here are non-abelian, the associated convolution becomes non-commutative leading us to a wholenew class of DFT-based non-commutative lters. This is one of the signicant new additions of this work. Bothconvolution and correlation are illustrated with a number of examples to images.1.1 Main ResultsAlthough the theory is developed in full generality, for applications we conne ourselves primarily to the wreathproduct cyclic (WPC) group generated from the nine level tree with four branches descending from each node.Signals dened on the leaves of the tree are typically obtained here from a quadtree scan of 2 9 2 9 images. Themain results are as follows:(1) Group-based convolution for nite arbitrary groups and WPC groups is dened in both the spatial andspectral domains and its properties are established. We show howitmay be eectively computed via thegeneralized Fourier transform and we explicitly consider the special case of WPC groups.(2) A general formula for WPC convolution in terms of -functions is established. WPC convolution is alsointerpreted in terms of some basic images and scale-selective lters.(3) For images that are perceptually similar, we see that transforming them rst through WPC group-basedconvolution generates a higher correlation coecient than that obtained without transformation.(4) Group-based correlation is dened naturally from group-based convolution and we see that it extends thepeak matching property of correlation to signals (with a specic support) that are group transformedversion of each other.2

1.2 Organization and NotationIn order that the paper may beread smoothly without constant reference to [6], we recall the notation usedearlier. Finite group-based convolution, its properties, and applications of WPC group-based convolution aredescribed in Section 2. In a similar fashion, we address nite group-based correlation in Section 3. Section 4summarizes our current ndings, and we conclude with some directions for further work. In the Appendix wegive a proof for the general formula for convolution for WPC groups acting on a quadtree.Note that while we invoke two-dimensional images as examples to work with, and while this permitsillustration of the WPC group spectrum in a standard two-dimensional \wavelet-type" decomposition, we stillhave a one-dimensional transform. Thus one-dimensional signals could also have been used for applications.For convenience we summarize the notation from [6]; the numbers in parentheses indicate the section in [6]where each term is introduced.X = fx 0 ;x 1 ;:::;x N,1 g (3:1):G = any nite group acting on X as permutations (3.1).L(X) = ff j f : X ! C g, the \spatial domain" of discrete signals of length N (3.1).L(G) = ff j f : G ! C g, the special case when X = G (also the complex group ring of G) (3.1).Tm = the spherically homogeneous rooted tree, where m =(m 0 ;:::;m n,1 ) (3.2.1).Xm = the set of leaves of the tree Tm (3.2.1).Zm = the WPC group that acts on Tm (3.2.1).Q(k; n) = T (4 k ;4 k ;:::;4 k ), the quadtree with n nonzero levels (3.2.2).Z(k; n) = the WPC group that acts on Q(k; n) (3.2.2).L(Xm) = ff j f : Xm ! C g, the spatial domain indexed by the leaves of the tree (4.2).L(k; n) = the spatial domain indexed by the leaves of the quadtree Q(k; n) (5).Q(f) = the quadtree spectrum of f where f 2 L(k; n) (5.1).2 Finite Group-Based ConvolutionWe describe here nite group-based convolution and determine its properties. Since implementation in thespatial domain, which depends on the group order, can be computationally prohibitive in L(Xm) even forsmall trees, as 3.18 of [6] indicates, we consider its implementation in the spectral domain. Here convolutioncorresponds to a matrix multiplication of the spectra; for WPC groups in particular we see that it assumes anespecially simple form in that it corresponds to multiplication of a matrix by a scalar.Before considering group-based convolution further, we note other work in obtaining Fourier-like convolutiontheorems for multiresolution representations. While the spectral multiplication property does not hold for theserepresentations, there has however been eort towards deriving linear convolution from the multiresolutionspectra. A method for computing convolution with orthogonal and biorthogonal lter banks that reducesthe determination of convolution to computing convolutions of the respective subbands and adding has beendescribed by Vaidyanathan, [19]. A dierent subband convolution theorem, [20], for reducing convolutioncomplexity, computes convolution using multirate lter banks and short-time Fourier transforms. Anothertechnique for implementing convolution is given in [14]. None of these, however, provide the analogue to theconvolution theorem wherein convolution in one domain becomes multiplication in the other. In that context,group-based convolution provides the exact analog. Group operations (corresponding to the shifting process in3

cyclic convolution) result in a multiplicative property in the spatial domain, thereby entailing a product of thespectra.For arbitrary nite groups a general formula for convolution with delta functions is established which allowscomputation of convolution for arbitrary signals f and h. The general formula is made more explicit in thespecial case of WPC groups acting on the leaves of a quadtree Q(k; n) by dening convolution and its propertiesin terms of specic signals f and a class of scale-selective lters h. This will permit easier visualization of theconvolution process. We observe for the tree Q(1;n) that WPC convolution of signals f and h, may be seen asthe superposition of certain cyclic convolutions and averaging of subsequences of f, weighted by elements of h.A signicant consequence of this is that WPC convolution generates sequences constant over successive lengthsof size 4 k , for k =0; 1;:::;n, 1. From a ltering perspective, elements of h provide a scale-selective lteringoperation on f. This leads to a new class of linear, noncommutative lters that are both scale-selective aswellas capable of convolving cyclically and smoothing.We investigate the use of WPC convolution in determining signal similarity. Perceptually similar signals(images) f i and f are compared using standard linear correlation, and then compared again after transformationby WPC convolution. For the examples tested, the WPC convolution transformed (lowpass) signals exhibitsignicantly higher correlation coecients. Highpass signals, after specic WPC convolutions exhibit similargains. This points to the possibility of using WPC convolution for problems in pattern recognition.2.1 A Generalization of Cyclic ConvolutionLet G be a nite group acting on a set X = fx 0 ;x 1 ;:::;x N,1 g. Recall that for discrete signals f and h oflength N (i.e., f;g 2 L(X)), their cyclic convolution is(f ?h)(x n )=N,1Xm=0f(x m )h(x n,m )=N,1Xm=0f( m x 0 )h( ,m x n ); (2.1)where is the N{cycle that cyclically permutes x 0 ;x 1 ;:::;x N,1 , and the underlying group G is the cyclic groupof order N generated by . We dene the formula for group-based convolution over an arbitrary nite groupG acting on X which is the direct generalization. The group-based convolution of signals f and h in L(X),written henceforth as f?h, is dened as(f ?h)(x n )= jXjjGjX2Gf(x 0 )h( ,1 x n ); for all x n 2 X: (2.2)As with continuous and discrete cyclic convolutions, group-based convolution enjoys many properties (linearity,associative, etc.) which are most easily understood by passing to the spectral domain.Following the development scheme in Section 3.1 of [6], we rst describe convolution in L(G) and thendescribe how it restricts to the subspace L(X) of L(G).When G is a cyclic group acting faithfully andtransitively on X we showed that L(X) may be identied with L(G), so the discussion of convolution on L(G)includes cyclic convolution as a special case.The vector space L(G) has a natural multiplicative structure coming directly from the group operation onG. For any 2 G, let f denote the delta function in L(G) supported on . Thenf ?f = f ; (2.3)4

where the product is taken in the group G. This multiplication dened on the basis of delta functionsextends uniquely by linearity and through the distributive lawtoamultiplication on all of L(G). In this wayL(G) admits a multiplication that gives it the same structure as the group algebra of G over C . Moreover, itis easily seen that when X = G, multiplication dened this way onL(G) is the same as convolution dened in(2.2)|the two formulas agree on the basis of delta functions since both products are bilinear (linear in eachvariable), where we choose an enumeration of the group elements such that x 0 is the identity element.More generally, recall (cf. Section 3.1 of [6]) that when X is an arbitrary set acted upon transitively by G,then L(X) is identied with the subspace L(G=G 0 )ofL(G) consisting of the functions on G that are constanton the left cosets of G 0 , where G 0 is the subgroup of G xing x 0 . If f and h are constant on each left cosetof any subgroup H, then so is their convolution in L(G). Under the identication of X with the coset spaceG=G 0 , the convolution formula (2.2) is the same as convolution multiplication in L(G) restricted to L(G=G 0 ).The following properties are easily checked by the reader.Theorem 2.1 Let the nite group G act transitively on a set X and let f?gdenote the group-based convolutionof f and g in L(X) (which depends on both G and its action on X).(1) Convolution in L(X) is bilinear, and is associative.(2) Convolution is compatible with the (left) actions of G on L(G) and L(X): for the basis element f 2 L(G)and any in Gf = f = f ?f : (2.4)(3) The left action by on L(G) is linear, and so is left convolution by f . In particular, f = f ?f for allf 2 L(G), and hence inL(G=G 0 )=L(X) we havef = f ?f; for all f 2 L(X) and all 2 G: (2.5)(4) Associativity, (2.4) and (2.5) together imply that convolution transforms under the group action by(f ?h)=(f) ?h; and f?(h) =(f?f ) ?h for all f;h 2 L(X) and all 2 G: (2.6)Note that when G is not commutative, convolution on L(G) is not commutative and convolution on itssubspace L(G=G 0 )may not be commutative.Next we describe convolution viewed in the spectral domain; we rst describe this in L(G), and then restrictto the subspace L(G=G 0 ) = L(X). By Wedderburn's Theorem (Theorem 4.1 of [6]), L(G) decomposes as avector space into a direct sum of isotypic components, M k , of dimensions d 2 k. These components are also closedunder convolution multiplication, and their structure is again described by Wedderburn's Theorem:Theorem 2.2 Let M 0 ;M 1 ;:::;M d,1 be the distinct isotypic components of L(G) described in Theorem 4.1 of[6]. Then M k is the algebra, M dk d k(C ), of all d k d k matrices with complex entries. Furthermore, L(G) isthe direct product of these matrix algebras:L(G) = M d0d 0(C ) M d1d 1(C ) M dr,1d r,1(C ); (2.7)where under this isomorphism the convolution of two functions in L(G) corresponds to the matrix product oftheir components on the righthand side (i.e., it is an algebra homomorphism as well).5

This isomorphism implies that with respect to some basis of L(G) chosen from the isotypic componentsM k , the generalized Fourier transform of each f 2 L(G) may be written as a matrix in block diagonal form:bf =0B@ 0 (f) 1 (f). ... .. r,1 (f)1CA; (2.8)where each k (f) is now a d k d k matrix representing the projection (4.4 of [6]) of f onto the k th isotypiccomponent M k . Since this result holds for all discrete signals, in particular if f and h lie in the subspace L(X)of L(G) wehave:Corollary 2.3 Convolution of functions in the spatial domain L(X) corresponds to the (block diagonal) matrixmultiplication of their spectra.Thus the convolution in the spatial domain of two functions may be computed eciently by passing to thespectral domain via a generalized Fourier transform, performing a matrix multiplication there, and passing backto the spatial domain by the inverse Fourier transform.In the special case when the representation of G on L(X) ismultiplicity-free, bases for the isotypic componentsmay bechosen so that the block matrix description of the spectrum of each f in L(X) has a particularlysimple form. (Recall that \multiplicity free" means that each isotypic component, k (L(X)), of the spatialdomain is either 0 or irreducible.) These observations apply, in particular, to WPC groups acting on the leavesof a tree as in Section 3.2 of [6] (cf. Corollary 4.10 of [6]). In this case a basis of L(G) maybechosen so thatthe elements of L(X) are represented by block diagonal matrices whose blocks are zero in all but their rstcolumns. We summarize this as follows:Lemma 2.4 If G acts on X in such a way that L(X) is multiplicity-free, then there is a basis of L(G) suchthat for every k with k (L(X)) 6= 0the matrices in (2.8) are of the form01a k 1;1(f) 0 0 k a(f) = Bk 2;1(f) 0 0.@C. 0 0A : (2.9)a k (f) 0 0 d k ;1With respect to this basis the Fourier transform of the convolution f?his the matrix product, b f b h, which iseasily computed for block matrices described in (2.8) and (2.9): for each k multiply every entry in the matrixblock k (f) by the upper lefthand entry of the corresponding block ofb h, namely by ak1;1 (h).In the case of multiplicity-free representations, each xed h 2 L(X) acts as a linear lter on L(X) byf 7! f?h. In the spectral domain right-multiplication by b h then rescales each component of b f by a constant(which depends on the component). Dierent signals h may act as the same lter, since for i 2 the entriesa k i;1 (h) of b h in (2.9) have no eect on the product b f b h.Some specic choices of lters h are of special interest.6

Example 1. Let x0 be the unit impulse delta function in L(X) supported on x 0 . Then x0 is the delta functionin L(G) supported at the identity element of G projected onto the subspace L(X). Thus f? x0= f for allf 2 L(X) i.e., x0 is a right identity of L(X). The Fourier transform of x0 has a one in the upper lefthandentry of each block k ( x0 ) and zeros in all other entries.Example 2. Let h k 2 L(X) be the signal whose Fourier transform b hkis the delta function in the spectraldomain with a1inthe upper left entry of block k (h k ) and zeros in all other entries of all blocks. In otherwords h k is a bandpass lter: the Fourier transform of f?h k is zero in all but the k th component, and f?h khas the same k th component asf. Thus the lter h k is the k th component of x0 , and so may be computed byprojecting x0 onto this component, as described by Theorem 4.2 of [6]:h k (x j )= k ( x0 )= d kj G jX2G 0 k ( j ); for all x j 2 X; (2.10)where j is any element ofG that sends x 0 into x j , and d k is the dimension of the k th isotypic component ofL(X).2.2 Convolution Over WPC GroupsIn this section let G be the WPC group Zm acting on the leaves of the tree Xm. In this case the convolution f?hdened by (2.2) is called WPC convolution. Since WPC convolution may require as many asj G j multiplicationsto compute each value, as noted earlier this method can be computationally prohibitive even for small trees. It iscomputationally more eective to pass to the spectral domain; this is especially ecient since the representationof Zm on L(Xm) is multiplicity-free. The bases in Section 4.2 of [6] giving the decomposition of L(Xm)into irreducible components are also precisely the ones that give the isomorphism of multiplicative structuresdescribed in (2.7), (2.8), and (2.9). In particular, the generalized Fourier coecients are precisely the matrixentries in (2.9).We may easily describe convolution in the case when G = Z(k; n) and signals f and h in L(k; n) havea quadtree spectrum (cf. Section 5.1 in [6]). Both spectrums Q(f) and Q(h) are formed from submatricescontaining the coecients of the representations of Z(k; n) on the irreducible subspaces, with bases compatiblewith convolution computations. The upper lefthand entry of each of these submatrices is the coecient of thecorresponding bandpass signal, h k . Consequently, wehave the following result.Theorem 2.5 Convolution of signals f and h in L(k; n) may be computed in the spectral domain by multiplyingeach entry in each nested grid irreducible submatrix of Q(f) by the upper lefthand entry of the correspondingblock matrix of Q(h).Example. Following the notation of Example 1 in Section 5.1 in [6], if f and h are in L(1; 2), denote Q(f) andQ(h) by the 4 4 matrices (a i;j ) and (b i;j ) respectively. Then01a 1;1 b 1;1 a 1;2 b 1;2 a 1;3 b 1;3 a 1;4 b 1;3Q(f) Q(h) = Ba 2;1 b 2;1 a 2;2 b 2;2 a 2;3 b 1;3 a 2;4 b 1;3@Ca 3;1 b 3;1 a 3;2 b 3;1 a 3;3 b 3;3 a 3;4 b 3;3A :a 4;1 b 3;1 a 4;2 b 3;1 a 4;3 b 3;3 a 4;4 b 3;37

We now describe the spectral multiplication for the quadtree spectra of 2 kn 2 kn images f and h in L(k; n)(although, as is our style, we only detail the process for k = 1). WPC convolution of signals in general is likewiseeasily eected (cf. Theorem 2.2). If Q(f) =(a p;q ) and Q(h) =(b p;q ) are the 2 n 2 n quadtree spectral matricesof 2 n 2 n images f and h respectively, their product in the spectral domain is computed as follows: workingin stages from j =0ton, divide the upper lefthand 2 n,j 2 n,j submatrix of Q(f) into a grid of four blocks:U0 U 1U 3 U 2, where each U p isa2 n,j,1 square submatrix. At the j th stage multiply every entry of the block U pby the upper lefthand entry of the submatrix of Q(h) in the same location as U p , for p =1; 2; 3. In other words:step 1: multiply each entry of U 1 by b 1;2 n,j +1,step 2: multiply each entry of U 2 by b 2 n,j +1;2 n,j +1, andstep 3: multiply each entry of U 3 by b 2 n,j +1;1.The submatrix U 0 is passed unaltered to the next stage where j ! j +1 and the process is repeated in thesmaller matrix U 0 . At the last (j = n) stage multiply a 1;1 by b 1;1 .This shows that WPC convolution is not generally commutative, nor is it unique in the right variable h.Dierent lters h giving the same product f?hare obtained by simply altering the spectrum of h in any positionexcept the upper lefthand entries of the submatrix blocks. Convolution is invertible when all these entries ofthe spectrum of h are nonzero.2.3 WPC Convolution via Scale-Selective FiltersIn this section we consider WPC convolution from the geometric point of view. In particular, we analyze indetail the eect of convolution on images, which are interpreted as functions on the leaves of regularly branchingtrees (cf. Section 3.2.2 of [6]). For simplicity wework with trees of the form Q(1;n); corresponding results forQ(k; n), k>1, follow easily.Theorem 2.6 Let fx 0 ;x 1 ;:::;x p ;:::;x 4 n,1g be the set of leaves of the tree Q(1;n) acted upon by the WPCgroup G = Z(1;n). Let G 0 be the subgroup of G xing leaf x 0 . For any index i let h i 2 L(1;n) be the unitimpulse delta function supported atx i , 0 i 4 n , 1. Denote h 0 by f 0 . Then the following hold:(1) For any index i>0 the convolution f 0 ?h i is the function dened by(0 if x i and x p do not lie in a common subtree not containing x 0(f 0 ?h i )(x p )=4 s,n if the largest subtree containing x i and x p but not x 0 has root on level s,(2.11)where f 0 ?h 0 = f 0 .(2) If for each j we let j be an element of G sending x 0 to x j , then(f ?h i )(x p )=4Xn ,1j=0f(x j ) j (f 0 ?h i )(x p )=4Xn ,1j=0f(x j )(f 0 ?h i )( ,ijx p ): (2.12)(3) The linearity of convolution in the second variable reduces the computation of f?hfor an arbitrary h tolinear combinations of convolutions of the types in parts (1) and (2).8

Proof: See Appendix A.For 2 9 2 9 images scanned in the Q(1; 9) quadtree fashion the convolution formula in part (1) of thetheorem may be visualized as follows. The signal f 0 is represented by the image with a 1 in position 1,1 andzeros elsewhere. Let h i;j be the image with 1 in position i; j and zeros elsewhere. Let Q s dbe the unique largest2 9,s 2 9,s square among the nested grid subquadrants depicted in Figure 8 of [6] that contains the i; j pixelbut not the 1,1 pixel, where we choose Q s d = Q9 0 when i; j =1; 1. Theorem 2.6 asserts that the convolutionimage f 0 ?h i;j is the array with value 4 s,9 in each pixel of the square Q s dand zeros in all other positions. (Notethat the quadrants Q r d with r =1; 2;:::;8 and d =1; 2; 3 together with the 1,1 entry Q9 0 are the complete set oforbits of the stabilizer subgroup G 0 acting on the quadtree array, and G 0 acts independently as the full WPCgroup Z(1; 9 , r) on each Q r; this gives an explicit visualization of the decomposition of G d 0 in the displayedline (A2) of the proof.2.3.1 A Geometric ViewBecause cyclic subgroups comprise the wreath product factors of the WPC group Z(k; n), cyclic convolutionis inherent in WPC convolution. However, overall eects are quite dierent from those obtained with eithercyclic or standard convolution. As observed in Theorem 2.6, a WPC group induces eects in the convolutionprocess that are dependent on the location of the support of both f and h on the leaves of the tree Q(k; n).This induces a scale-based ltering of f. Accordingly, to understand the process of convolving signal f withlter h, we again dene signals and lters with specic support.Dene f C to be any signal with support concentrated on the rst 4 k leaves of the tree Q(k; n) (i.e., thosedescending from the 0 th node at level n , 1). In our quadtree correspondence, interpreting f C as an image, it isnonzero only in the upper left 2 k 2 k block Q n,10. (Recall that the superscript accompanying Q signies thelevel). By Theorem 2.1, convolution of any arbitrary f in Q(k; n) is determined by the knowledge of convolutionof f Cwith the lters h i . We illustrate with the special case k = 1 and n =9,and refer to Sections 3.3 and5.2 of [6]. As indicated there, through the quadtree scan the i th level of the WPC spectrum is seen to lie insub-quadrants 1, 2 and 3 (where quadrant 0 is the upper left one). Since signal f C dened on the 22 quadrantQ 8 0is subject to averaging operations in the convolution process, we dene the following: let s be the sum ofthe values of f C , and for each i =0; 1; 2;:::;8 dene f Ci byf Ci is the image with value s=4 9,i at each pixel in quadrant Q i 0, and 0 elsewhere, (2.13)as depicted in Figure 1. Thus f Ci is constant on the upper left quadrant at level i, is zero outside this quadrant,and has the same DC term (sum of all its values) as f C .For lter h we consider the following set of lters for the tree Q(1; 9). Changing the earlier labeling slightlyto accommodate two-dimensional images, we now let((1 in position 1 ; 11 in position 1 ; 1+2 i,1h 0 =h i =0 elsewhere0 elsewhere1 i 9; (2.14)as depicted in Figure 2. For 1

supported in the upper lefthand 22 block extensions of lters h 0 or h 1 (so h 1 and h 0 are considered extensionsof each other). The generators for the group Z(1; 9) described in Section 3.3 of [6] permute the lters h i in thesense that h i = (9,i)0(h 0 ), 1 i 9, where i 0 denotes the 0 th generator from among these families actingat level i, permuting the four trees descending from node 0 at level i. Moreover, all generators acting at levelsbelow i simply permute the extensions of h i among themselves (with many such group elements actually xingh i ).By Theorem 2.1, for lter h 0 wehavef C ?h 0 = f C g 1f C ? ( (8)0 )k (h 0 ) =( (8)0 )k (f C ) =( (8)0 )k (g 1 ) k =0; 1; 2; 3;where ( (8)0 )k 2 Z(1; 9) is the group operation at level 8, acting on the rst four leaves. Note that extensions ofh 0 , that is, lters supported on Q 8 0 (the rst 4 leaves of the tree), convolved with signal f C (also dened on Q 8 0)results in the standard cyclic convolution on the set Q 8 0. Hence in two dimensions, Theorem 2.1 implies that animage f convolved with lter h 0 and its extensions results in an in-place circular convolution of all 2 2 blocksof the image with the lter. We proceed in this fashion, examining the eect of group operations on lter h ionly at level 9 , i for i =2; 3;:::;9. Group operations at higher levels are seen to have no eect, while those atlower levels are formulated in terms of the corresponding h i .For lter h 2 ,by again using Theorem 2.1 we havef C ?h 2 = f C ? (7)(h 0 0) = (7)( 0f C8 ) g 2f C ? ( (8)1 )k (h 2 ) = f C ? (7)0 ((8) 0 )k (h 0 ) = (7)( 0f C8 )f C ? ( (7)0 )k (h 2 ) =( (7)0 )k (g 2 ) k =0; 1; 2; 3:Thus f C convolved with h 2 and its extensions generates output g 2 whichis f C8 , but with support on subquadrantQ 8 1. Equivalently, g 2 is f C8 operated on by group element (7)0. Local group operations ((8)1 )k on h 2 do notaect the output g 2 . However, group operations ( (7)0 )k on h 2 and its extensions that move the impulse inquadrant Q 7 0 result in corresponding permutations on g 2.Extending our analysis this way, we see that convolving signal f C with the other unit impulse lters h i fori =3; 4;:::;9 and their extensions results in an output g i consisting of averages f C10,iQ 10,i1, the same as that for lters h i . Accordingly, for lter h 9 wehavef C ?h 9 = f C ? (0)(h 0 0)= (0)( 0f C1 ) g 9f C ? ( (0)0 )k (h 9 ) =( (0)0 )k (g 9 ) k =0; 1; 2; 3:located in quadrantsFinally, by the bilinearity of WPC convolution and Theorem 2.1, we see that f?hconsists of a superpositionof cyclic convolutions and averaging operations. All 4-element sequences of f are cyclically convolved in-placewith the 4-element sequence of h dened on quadrant Q 8 0 . A lter h supported on quadrants Qk jand convolvedwith f C generates weighted averages of f C on the same quadrants Q k j , for k =8; 7;:::;1 and j =1; 2; 3. Theremaining elements of f|which are of the form (f C ) for some 2 G| when convolved with h appear as rotated(by ) versions of the operations on f C . Hence, we observe that the convolved output g = f?happears as apattern in blocks of size 4 k , for k =0; 1;:::;8; the cyclic convolution eects appear in all 4-element sequences10

(k = 0), while the weighted averages appear in sequences of increasing size 4 k , for k =1; 2;:::;8. In summary,we conclude:Theorem 2.7 (1) Filter h 0 and its extensions cyclically convolve in-place blocks of f of size 2 2.(2) For i =2; 3;:::;9, lters h i and their extensions act as scale-selective lters, and they average f C over ablock of size 2 i,1 2 i,1 . The averaging results g i are supported insubquadrants Q 10,i1, the same as thelter, thereby reecting a local rotation of the image. Group operations on h i acting at level 9 , i resultin the corresponding group operation on the outputs g i = f C ?h i , for i =2; 3;:::;9.(3) WPC convolution f?hof arbitrary images is then obtained using transformation of convolution undergroup operations on f C and h i and bilinearity.A correcting operation whereby scale-selective ltering of f appears coincident with f can be obtainedby simply convolving g i , as dened above, with appropriately rotated versions of h i .That is, g i ? ( 2 i (h i))where i = (9,i)0, for i =2; 3;:::;9 respectively, generates output averages f C10,i. Accordingly, for image f C ,scale-ltered images f CLPi and the corresponding scale-selective lowpass (LP) lters h LPi are obtained asf CLP i= g i+1 ? ( 2 (h i+1 )); i =1; 2;:::;8= f C ?h i+1 ? ( 2 (h i+1 ))= f C ?h LP i;where, by the associative property of convolution, h LPi= h i+1 ? ( 2 (h i+1 )) is the average of h i+1 uniformlydistributed on the rst 4 i leaves of the tree, or equivalently, with support on quadrant Q 9,i0. Hence, givenan image f, its lowpass scale-selective ltered image f LP , dened as f ?h LP , may be obtained using thegroup transformation and bilinearity properties of convolution. Dening h LP9as the average of h 0 uniformlydistributed on quadrant Q 0 (the whole image), it is easily seen that convolving f with h LP9 generates the averagevalue of f over quadrant Q 0 . Figure 3 shows the structure of the lowpass lters and their associated spectra.Hence f convolved with, for example, h LP8 generates the average of f in quadrants Q 1 i , for i =0; 1; 2; 3, placingthe average in the same quadrants. Recall that multiresolution subspaces V i and W i of L(1; 9) were denedin Section 5.2 of [6] as being associated to level i, for i = 0; 1;:::;9, where V 9corresponded to the highestdetail and V 0 to the coarsest. We now associate V i and W i as being at scales 9 , i, for i =0; 1;:::;9, wheresmall scale corresponds to high resolution. Hence, identifying lters h LPisee that convolving f with h LP ias scale-selective lowpass lters, wegenerates scale-selective reconstructions of f at the nine scales or levels. Fori =1; 2;:::;9, h LPi lters spectra in subspaces V 9,i , and rejects spectra in subspaces W 9,i and all lower scales.In terms of ltering we thus have:Theorem 2.8 WPC convolving f with h LPiltering out details at scales i, for i =1; 2;:::;9 and those below it.results in a lowpass zonal ltering of the spectrum, successivelyWhile h i , for i =2; 3;:::;9, and its extensions act as scale-selectivelowpass lters, we have seen that h 0 andits 2 2 extensions act as local cyclic convolvers with 2 2 blocks of the image f. However, cyclic convolutioncan also generate scale-selective lters through an appropriate choice of lter coecients. For example, consider11

a scaled extension of h 0 dened by h HP1 = [h1 h2 h3 h4] =[:75 , :25 , :25 , :25]. It is easily seen thatits spectrum, H HP1 , consists of ones located in the 1,1 term of the quadtree spectral subspaces W 8;1 , W 8;2 ,W 8;3 and zeros elsewhere. Hence convolving f with h HP1 results in a scale 1 highpass ltering of f. Thus thespectrum of f is altered such that coecients at all scales except the highest resolution subspaces W 8;1 , W 8;2 ,W 8;3 are set to zero. This is equivalent to highpass ltering at scale 1, giving the detail at scale 1.In a similar fashion we can generate other scale-selective lters h HPi , for i =2; 3;:::;9, that act as bandpasslters, providing the detail at those levels. That is, these lters will lter only the spectrum at scales 2 through9, corresponding respectively to subspaces W i;j , for i =7; 6;:::;1; 0 and j =1; 2; 3. Such bandpass lters maybe obtained by extending the support of h HP1from a 2 2 grid to a 2 i 2 i grid, wherein each element ofh HP1 is averaged over a 2 i,1 2 i,1 region to form h HPi , for i =2; 3;:::;9. The spectrum, H HPi ,ofh HPi isnonzero in subbands such that ltering f with h HPi , for i =1; 2;:::;9, lters subbands W 9,i to give scale 1to 9 reconstruction of the detail images of f. Figure 4 shows the construction of the bandpass lters. Thus wehave:Theorem 2.9 Filters h LPi and h HPi form scale-selective lowpass and bandpass lters. Since WPC convolutionis bilinear, a variety of scale-selective lters may be constructed through a combination of these lowpass andbandpass lters.While an averaging interpretation has been provided for lters h i , for i =2; 3;:::;9, we can also interprettheir action in terms of cyclic convolution, as we did for h 0 . Since convolving f C with h 2 generates an averagef C8 , this is easily seen to be equivalent toconvolving f C with h 2 , where h 2 is the average of h 2 over Q 8 1. Nowf C ? h 2 gives the cyclic convolution of the two 4-point sequences, which isf C8 (in Q 8 1). Similar interpretationsfollow for convolving f C with h i for i =3; 4 :::;9, and with their extensions. Convolution of a 256 256 imagef with three lters h LP1 , h LP2 , and h LP3 is shown in Figure 5.From the convolution properties described we reinforce the previously observed fact that dierent lters hmay produce the same outputs f?hfor a given signal f. An arbitrary lter h may, for example, be replaced bya h EQ , where h EQ is constructed from h by replacing every element ofh in quadrants Q i jby its average valuein the respective quadrants of h EQ , for i =1; 2;:::;8 and j =1; 2; 3. Equivalently, h may be replaced by thesum of h in, say, the 1,1 terms of the respective quadrants, or by other descriptions as long as the correspondingsums in Q i j in both h and h EQ are preserved. Regardless of the equivalent h utilized, the process of convolvingf with a given lter h (on the right) is always achieved by convolving f with the same canonic lter h CN whosespectrum is multiplied by that of f. The canonic lter is dened as that whose spectrum in all the irreduciblesubspaces (i.e., subquadrants) is identical to the 1,1 term in the corresponding subspaces (i.e., in the upperlefthand entry). Hence convolving f with h always amounts to convolving f with h CN . For a convolution ofa 256 256 image f with itself, i.e., with the lter h = f, we see in Figures 6(a), (b), (c) and (d): the imagef and lter h, an equivalent lter h EQ , the canonic lter h CN , and the resulting convolved image g. With Hrepresenting the spectral matrix of h, h EQ here corresponds to a lter whose spectral matrix H EQ retains onlythe 1,1 terms of H in the four subquadrants of Q 7 0 and Q i j , for i =6; 5;:::;1 and j =1; 2; 3, and sets all otherterms to zero. Note that this h EQ is also h 0 ?h, which we dene as the impulse response of lter h. Accordingly,the non-uniqueness of h is illustrated by h EQ and h CN . We also note that for the tree Q(k; n), only when theimage has support on the rst 4 k locations of the tree does the impulse response become identical to the lter.12

2.4 WPC Convolution and Scale SimilarityHaving formulated the notion of convolution for WPC groups we nowinvestigate some potential ltering applications.In particular, we apply it to the similarity determination of two signals. Our method is to rsttransform a signal through WPC convolution and then measure distance (\closeness") in that space. Twoexamples, one synthetically generated and the other consisting of natural images, are shown to illustrate theapplication.2.4.1 Distance, Similarity, and CorrelationThe notion of distance of two signals takes many forms in the literature. In signal processing, minimizing variousmetrics leads to the design of optimal lters. For example, the L 1 -norm leads to the Chebyschev lter [5], whilemaximally at approximations in the Taylor series sense generate the classic Butterworth and Bessel lters [9](pp.245{250), and [13]. For signal-noise problems, using the L 2 -norm leads to the matched and Wiener lters[17]. In pattern recognition, distance is often measured by the nearest neighbor classier which assigns a testsignal to a class whose template vector is closest or most \similar" to it. While many dierent functions havebeen described in the literature to measure similarity, as cited by Hadar et al., [8], correlation (i.e., the standardl 2 -correlation) or template matching has found wide use in the image processing literature. Its use in measuringsimilarity dates back to the very earliest eorts [4] in pattern recognition. Previous work of Anuta [2] and Ballard[3] describes how the correlation coecient serves as a good measure for a variety of tasks including registrationand steropsis of images. Use of the correlation coecient tends to normalize out intensity dierences that mightadversely aect the match in a spurious way. Many other applications of correlation in image analysis aredescribed in [8]. Calculating correlation through multiresolution representations has recently been investigatedby Stone [15] using the earlier work of Vaidyanathan [19]. Correlations coecients are calculated by correlationoperations that are computed using Fourier and wavelet-packet transforms. Application to hierarchical imageanalysis, where the goal is to correlate the template and test signal at various scales, is given in [15]. In thatapproach matching is performed sequentially from coarse to ne scale, proceeding from one level to the nextafter the corresponding cost function becomes minimal.The specic application we consider here falls generally within the above framework. We have seen that thecorrelation coecient serves as a good indicator for measuring similarity when the signals are not necessarilyidentical but are linearly correlated. Normalization of correlation makes the metric less dependent on localproperties, such as image intensity dierences, that can adversely aect the ability to match images. However,such an adjustment does not allow for important factors such as images at dierent scales, or other factors suchas imaging distortions, rotations, and texture dierences. In such cases we can have perceptually similar, that isqualitatively similar images but yet have low correlation values that may imply a mismatch and thus vitiate theadvantages of normalized correlation. To address the problem of low correlation for images at dierent scaleswe investigate the eects of measuring the correlation of images transformed to a WPC representation, whereperceptually similar images can become measurably closer. The technique of applying local linear transformationshas been adopted by Picard [12] and others [18], [1], all utilizing some application of the Karhunen-Loevetransform. Picard, for example, in a texture matching problem, uses subsets of features obtained from theKarhunen-Loeve transform, and compares these through a mean-square error criterion. The transform itselfis generated from the covariance matrix of a small random subset of DFT magnitudes of the data set. Inthe approach adopted here, we attempt to match images, especially those generated at various scales, by rst13

transforming them to a multiresolution representation and then using normalized correlation. For determiningsimilarity of a scaled image f i to a prototype image f we compare f i ?f and f?fusing standard correlationand compare that to the correlation of f i and f. We carry out this analysis by utilizing the WPC multiresolutionrepresentation: for the tree Q(1; 9), using the decomposition f = P 9j=1 f j , where f j is f dened overquadrants Q 8 0 and Q i 1 + Q i 2 + Q i 3, for i =8; 7;:::;1, we have seen that WPC convolution of signal g with f is thesuperposition of signals constant over successive lengths of size 4 k , for k =0; 1;:::;8. Specically, f i ?f gives aweighted decomposition of f i in all 9 subspaces, consisting of a cyclic convolution and the sum of weighted byf j averages of f i that are projections on subspaces W 8 ;W 7 ;:::;W 0 and V 0 (discussed prior to Theorem 2.8).We use this representation for comparison.Example 1. As a rst example we see in Figures 7(a) and (b) two 256 256 test images f 1 and f 2 obtainedfrom the WPC convolution of the prototype image f of Figure 6(a) with lters h LP 1and h HP 2. Accordingly, f 1and f 2 being respectively, projections of f at scale 1 lowpass and of f at scale 2 highpass, have spectra that liein certain xed bands. We attempt to measure similarity of both to f. Note that for the synthetic image casehere, a simple dierence of the WPC spectra of f i and f would yield the fact that both have a scale similarityto f. However, for the general case where similarity may not be exact or where the two images may stem fromdierent multiresolution bases, we try the more general approach of WPC convolution indicated above.Consider rst the lowpass image f 1 . By construction f 1 is dissimilar to f at scale 1 highpass where it iszero, but similar at all other scales. Accordingly, the spectrum of f 1 ?f is similar to that of f?fat all scalesexcept scale 1 highpass, where the former is zero. Consequently, all successive four-point sequences in f 1 ?f willbe constant. Since their sum is kept the same as in the corresponding part of signal f?f, the spectrum of thetwo convolutions is the same at all other scales. The resulting eect is that the two convolution patterns willbe similar at all scales except scale 1 highpass, where f 1 ?f will be constant. We observe the general similarityin Figure 7(c). The dissimilarity at scale 1 is seen by observing the 4-point sequences in Figure 7(e). Herewe observe that f 1 ?f is constant in all successive 4-point sequences while f?f is not. Similarity at scalesgreater than 1 is manifested by the fact that f 1 ?f has the same frequency characteristics as f?f for sequencesconsidered in successive blocks of length 4 k , for k =1; 2;:::;7.We devise a bandpass image f 2 similar to f at scale 2 highpass but dissimilar at others, where it has zerospectrum. Spectra of the resulting convolutions f 2 ?f and f?fcarry those same characteristics. Figure 7(d)shows the two convolutions, which clearly appear dissimilar. Examining convolutions at the scale of interest,namely scale 2, we see two typical 16-point sequences in Figure 7(f). We observe that both convolutions havethe same frequency characteristics at that scale, that is, for successive segments of length 4 2 , taken in blocks ofsize 4. Dissimilarity at scale 1 is discernible. Dissimilarity at other scales exists due to f 2 ?f having a zero sumover successive sequences of length 4 k , for k =2; 3;:::;7.We now correlate to measure signal similarity. Before transformation the correlation coecient, , off i andf is 0.9421 and 0.3817 for i =1; 2 respectively. After WPC convolution, as described above, it is 1 and 0.0019.A corresponding eect holds for f i similar to f at the other lowpass and highpass scales. Hence for this exampleand for the lowpass case, transformation using WPC convolution provides greater normalized correlation thanthat obtained without convolution.A correlation coecient of 0.3817 is not high, and would typically not imply a good match. For the WPCcase however, since the low value of 0.0019 is due to matching at only 1 (low-energy) scale and a mismatch at14

all others, it would appear logical to compare convolutions at each scale rather than simultaneously across allscales. Accordingly, we compare f 2 ?h HP iand f?h HP ifor i =1; 2;:::;8, where h HP iare the highpass ltersof Figure 4. Using covariances, since the sum of the values of the projections of f 2 are zero for i =1; 2;:::;8, weget, as expected, covariance of value 0 at all scales except scale 2 where is 1. Hence we are able to establishhigh correlation for this highpass signal using projections at each scale. Accordingly, convolving to obtain amultiresolution representation for the lowpass case, and at each scale as in the highpass example, results in acharacterization that generates higher correlation values than those obtained directly.2.4.2 Relationship between correlation coecientsBefore proceeding with a more general example we rst establish relationships between the l 2 -norms of signalsbefore and after WPC convolution. We consider again the tree Q(1; 9). Recall that for f;g 2 L(1; 9), theconvolution g?f achieved via spectral multiplication uses WPC spectra without normalization. Since only4-point DFTs are involved, l 2 -norms are preserved if scaling factors are introduced. Hence it is easily seen thatfor a signal g and spectrum ^g we havekgk 2 =4 9 Xn=1j g(n) j 2 =3X Xj=1W 8;jj ^g i2 j 2+ +3X Xj=1W 0;jj ^g i2 9 j 2+ X V 0j ^g i2 9 j 2(2.15)= k^g W8;1 k 2 + k^g W8;2 k 2 + k^g W8;3 k 2 + + k^g W0;1 k 2 + k^g W0;2 k 2 + k^g W0;3 k 2 + k^g V0 k 2 ;where ^g Xi refers to the normalized spectrum of g in the irreducible subspaces W i;j of Section 5.2 of [6]. ForWPC convolution g?f we then havekg ?fk 2 = k^g W8;1 k 2 j f 11W 8;1j 2 + + k^g V0 k 2 j f 11V 0j 2 ; (2.16)where f 11X idenotes the 1,1 term of the spectrum of f in the corresponding subscripted subspace, divided by theappropriate scaling factor 2 k , for k =1; 2;:::;9. We express the above sum simply askg ?fk 2 = kS 11 k 2 j K 11 j 2 + + kS 94 k 2 j K 94 j 2 ; (2.17)with the natural identication of terms in (2.16). Referring to the examples of the previous subsection, thecorrelation coecients (scaled by 2)off i and f, and also of f i ?f and f?f, assuming zero mean real sequences,can be written as2 kf i kkfk= kf ikkfk + kfkkf i k , k(f i , f)k 2kf i kkfk(2.18)and2 kf i ?fkkf?fkThe normalized mean-square error in terms of the spectrum is= kf i ?fk kf ?fk+kf ?fk kf i ?fk , k(f i , f) ?fk 2kf i ?fkkf?fk : (2.19)kf i , fk 2kf i kkfk = kS 11k 2 + + kS 94 k 2; (2.20)kf i kkfk15

andk(f i , f) ?fk 2kf i , fkkfk = kS 11k 2 j K 11 j 2 + + kS 94 k 2 j K 94 j 2kf i , fkkfk(2.21)where S ij refers to the spectrum of f i , f and K ij to that of f. Equations (2.18) to (2.21) describe therelationships between the correlation coecients before and after WPC convolution.Example 2. We proceed now with a somewhat more general example. Figure 8(a) to (d) shows 16 imageresolutions f i that are the lowpass (f 1 to f 8 ) and bandpass (f 9 to f 16 ) approximations to the Mandrill imagef at scales k = 1; 2;:::;8 obtained using the Daubechies wavelets, db8. The bandpass consists of all threebands at each scale. We investigate the similarity off i to f. Using our earlier highpass example as a guide,we could compare the two signals in a variety ofmultiresolution bases. We instead utilize the WPC basis: Itprovides a simple, piecewise constant approximation at dierent scales, it is local, and projections at dierentscales are easily obtainable. As in the previous synthetic example, we use the same process for comparison.Lowpass images are convolved and correlated; highpass images are correlated after projection at each scale.These correlation coecients|referred to as multiresolution correlation, MR |are compared to the correlationcoecients obtained directly.Each subgure in Figures 8(e) and (f) shows one f i ?f, for i =1; 2;:::;16 and f?f. For easier comparison,f i ?f is scaled so that its rst term is the same as f?f. We obtain the multiresolution correlation coecients MRand correlation coecient , and for convenience plot them as errors: the normalized multiresolutionerror (MRE), 1{abs( MR ), and the normalized mean square error (MSE), 1{abs(). As expected, and as seenin Figures 9(a) and (b), both the MRE and MSE in the lowpass case (scaled images i = 1 to 8) increasewith increasing scale while remaining high for the bandpass signals. To obtain a measure of discrimination fordissimilar images, we repeat the process, correlating nine other natural images 1 gi k , for k =1; 2;:::;9tof usingthe same procedure at scales i =1; 2;:::;16. Results are shown in Figures 9(a) and (b). We obtain a correlationcoecient 0:95 abs( MR ) 1 for images 1 to 6, whereas only images 1 and 2 have 0:85 abs() 0:95. Weconclude from MR that images f i , for i =1; 2;::: ;6, have a good match tof at some scale.Images f 9 to f 16 in Figures 8(c) and (d) are 8 bandpass images at scales 1 to 8. For each image weadopt the same procedure as for the highpass image of Figure 7(b). That is, for each image f i we correlateprojections f i ?h HP jto f?h HP jfor j =1; 2;:::;8. For each i, correlation coecients are calculated for all8 projections. While a \best match" correlation coecient could be selected in a number of ways, we selectthe one corresponding to the known scale of the test image, which also tends to be the highest. As before, werepeat the process for the other nine images. Results are shown in Figure 10. Here too we have a strongerindication of matches than with regular correlation, with abs( MR ) greater than abs() for all 8 images, andtypically, substantially greater. We note also the stronger discrimination capability with WPC convolution:For scaled image 1 to 6 the ratio of the lowest MRE for all the dissimilar images to the MRE for perceptuallysimilar images ranged from 15,000 to 12; for the bandpass images 9 to 14 it ranged from 6.4 to 1.4. Withoutconvolution, the corresponding mean-square errors were 15.4 to 1.3 and 1.4 to 1.3.It should be noted that in the process of comparing f i and f by convolving rst with lter f, the ltermerely acts as a weighting function to generate signals that are a sum of projections on subspaces. It is theprocess of projections, rather than the specic weights of lter f that helps establish similarity. Hence, many1 These images were obtained from the proposed collection of standard images at the Waterloo BragZone.16

lters f that generate a full decomposition and which are also invertible can be used. Furthermore, since f i (,x)and f(,x) are equally valid for determination of similarity, WPC correlation, as dened in the next section,may also have been utilized instead of WPC convolution.3 WPC Group-based CorrelationClassically, correlation plays an important role in comparing signals. Our goal here is to give a formulation ofgroup-based correlation which appears to achieve some of the same recognition features as classical correlation.Mimicking the formulation of discrete cyclic correlation, for an arbitrary nite group we dene the groupcorrelation of two functions byf h =(e f) ? h; for all f;h 2 L(X); (3.1)where f e is the signal f in reverse order, h is the complex conjugate of h, and ? is group-based convolution.When the underlying group is a WPC group we shall denote the WPC correlation of f and h by f has well (where WPC correlation may be computed eciently via the convolution algorithm presented earlier).When the WPC group is just a cyclic group, WPC correlation reduces to standard cyclic correlation. Whenf;h 2 L(k; n), f e is equivalently obtained from f by reversing it about the independent variable and shifting soas to make a causal sequence on Q(k; n). When f and h are images scanned in the Q(1;n) quadtree manner, f eis seen to be the image f reected about the horizontal line bisecting it. In order to make the WPC correlatedoutput consistent with the location of the image f we modify our WPC correlation denition slightly: Forp = f h we let g = ep and dene g = f h, giving us WPC correlation as g(x) =(f(,x) ? h(x))(,x).We shall see that WPC correlation, which is a natural outgrowth of WPC convolution, extends that notionof similarity, where similarity is measured by a peak matching operation. We see here that data that has forexample been cyclically shifted can be correlated to the unshifted data through WPC correlation and determinedas \similar." Moreover, for certain signals, peak values obtained through WPC correlation can be the sameas those obtained through standard correlation. As before, we use for our examples images obtained as onedimensionalsequences through quadtree scans. We note that the scale-selective property of WPC convolutionextends also to WPC correlation. Hence, as noted earlier, examination of signals for perceptual similarity asconsidered in Examples 1 and 2 may also be eected through WPC correlation.In the following subsection we give properties of WPC correlation analogous to Theorem 2.1. WPC correlationis characterized by its group and scale-invariances as also by peak values achieved. The group invariantproperty of WPC correlation is dened and illustrated by examples, and results are compared with those of linearcorrelation. The unique capability of WPC correlation in identifying perceptually similar, group-transformedpatterns is seen. We also see that WPC correlation can achieve values similar to standard correlation.3.1 Properties of WPC CorrelationTheorem 3.1 Let the nite group G act transitively on the set X.(1) Group-based correlation in L(X) is bilinear, but need not be associative.17

(2) Correlation is compatible with the left action of G on L(X):(f h) =(f) h; for all f;h 2 L(X) and all 2 G: (3.2)(3) The maximum modulus of f h (i.e., the largest absolute value of (f ?h)(x) over all x 2 X) isequal tothe maximum modulus of (f) h for all 2 G.(4) The maximum modulus of the WPC correlation (f) f is equal to the maximum modulus of the linearcorrelation of f and f when both signals have support on the rst 4 k leaves of Q(k; n), and 2 G = Z(k; n).Proof: Properties (1), (2) and (3) follow directly from those of convolution, so we verify (4). When f hassupport on the rst 4 k leaves of Q(k; n), then for determining the maximum modulus of WPC correlation f fwe need only compare circular correlation of f and f to their linear correlation. As is the case for circularconvolution ([10], pp.548{553), it follows directly that the circular correlation of two nite length sequencesis equivalent to the linear correlation of the two sequences followed by aliasing. In the case here, since boththe length and period of f is the same (4 k ), it is easily seen that in periodic correlation, all points but one arecorrupted by aliasing. The uncorrupted point is at 0, which contains the maximum modulus of linear correlation.Furthermore, invoking property (3), we also see that the maximum modulus of a group transformed version off WPC correlated with f is the same as that of the linear correlation of f and f. This completes the proof.For the remainder of this subsection we assume G = Z(1; 9) acting on the tree Q(1; 9), and signals f and hlie in L(1; 9). For a geometrical visualization of WPC correlation we address, as before, correlation with f C andthe delta functions h i , for i =0; 1;:::;9. Based on the denition of correlation involving a time-reversal of bothf and f?h, the correlation properties evolve in a natural way from the convolution properties of Theorem 2.7.Hence, for lter h 0 wehaveFor lter h 2 wehavef C h 0 = f C g 1f C ( (8)0 )k (h 0 ) =( (8)0 ),k f C = ( (8)0 ),k g 1 ; for k =0; 1; 2; 3:f C h 2 = f C (7)(h 0 0) =( (7)0 ),1 ( f C8 ) g 2f C ( (8)1 )k (h 2 ) = f C ? (7)0 ((8) 0 )k (h 0 ) =( (7)0 ),1 ( f C8 )f C ( (7)0 )k (h 2 ) =( (7)0 ),k (g 2 ); for k =0; 1; 2; 3:Similarly, we see that correlating signal f C with the other unit impulse lters h i and their extensions, fori =3; 4;:::;9, results in an output g i consisting of averages f C10,ilocated in quadrants Q 10,i3. As before, theonly group transformations on h i that we need consider are operations at levels 9 , i. Accordingly, for lter h 9we havef C h 9 = f C ? (0)(h 0 0) =( (0)0 ),1 ( f C1 ) g 9f C1 ( (0)0 )k (h 9 ) =( (0)0 ),k (g 9 ); for k =0; 1; 2; 3:18

3.2 WPC Correlation and Group Transformed ImagesWe now look to its physical interpretation of WPC correlation and its relationship to linear correlation in thecontext of similarity of signals. Linear correlation of two signals f and g essentially performs a text-stringmatching operation, the matched lter exemplifying this process in the classic problem of detection of a knownsignal in noise. Correlation then is measured by its maximum value, which occurs when two sequences orpatterns match exactly; however, linear correlation perforce is unable to classify as similar, sequences thatmay have, for example, elements interchanged, or may be circularly rotated (or group transformed) versions ofeach other. As a simple example, the 64-point sequence f =[7; 2; 6; 5; 0; 0;:::;0] when linearly correlated withitself gives a maximum value of 114; whereas f 1 =[0; 0; 0; 0; 5; 7; 2; 6; 0; 0;:::;0]|a slightly modied version off|when correlated with f, gives a maximum value of 91. With the Q(1; 3) tree, WPC correlations f f andf 1 f both show a maximum of 114, the same as that obtained with the linear correlation of f and f. (Notethat WPC convolution does not achieve that maximum, nor does cyclic convolution.)Sequence f 1 has the form (f), i.e., is a Z(1; 3) group transformed version of f. By property (4) ofTheorem 3.1, such transformations preserve the maximum achieved in linear correlation of f 1 and f whenthe two signals have the appropriate support. Hence we observe (as follows from the properties) that WPCcorrelation of f 1 and f achieves the maximum value obtained with linear correlation when f has support onthe rst 4 k leaves of the Q(k; n) tree and when f 1 is a group transformed version of f. When WPC correlationof two signals f h is employed for matching using peak detection, it should be noted (keeping in mind thenon-commutativity of correlation) that it is h that forms the lter or template and f the noisy or transformedsignal that we attempt to match. Since segments of the convolving lter h lter dierent scales of the inputsignal, it is appropriate to use the transformed or noisy signal as the input signal f and the template as thefeature selecting lter h.We generalize to a slightly more complex example wherein we desire to match a specic 4 4 templateimage h to group transformations f of that image. All patterns are located in a 256 256 array. Accordingly,we correlate the transformed images f with the template h. Here we assume that the WPC group is Z(2; 4)acting on the tree Q(2; 4), and that the template image is supported on the rst 4 2 leaves of the tree, i.e., theleftmost subtree in Q(2; 4). At each level of the tree the image is subdivided into a succession of 44 grids, witha clockwise spiral ordering of the blocks. Note that with this ordering, the eect of a cyclic shift is to rotate theblocks and spiral them towards the center, with the \central" block moved to the upper left corner (so this isnot a planar symmetry of the image). We consider sixteen 44 images embedded in 256256 arrays. For easiervisualization Figure 11 shows all 16 binary images in their actual location in one array. The upper lefthandcorner contains the template \T" image which represents the lter h and which also represents the rst of theimages to be detected (f 1 ). The other 15 images, f 2 through f 16 , constitute the remainder of the transformedimage set. Images f 2 , f 3 and f 4 , which are three rotated (in multiples of 90 )versions of h, are shown belowthe lter image. The second column contains images f 5 , f 6 , f 7 and f 8 , which are rotated versions of h at anglesof 45 , 135 , 225 and 315 respectively. The third and fourth columns contain eight group transformations ofh. That is, f 9 ;:::;f 16 are k 0(h), where k =1; 2; 3; 4; 5; 9; 11; 13, and where 0 2 S 16 is the group element oforder 16 cycling the rst 16 leaves of Q(2; 4). Note that other than the lter image h, which must exist in thetop lefthand corner, all other rotated and group transformed images are translated to lie in any 4 4 quadrantnaturally created by the Q(2; 4) subdivision.19

We correlate f i with h for i =1; 2;:::;16 using both WPC correlation and linear correlation. Since thetemplate image \T" contained in h is entirely dened on the leftmost subtree of Q(2; 4), we know that WPCcorrelation of f 1 and h will achieve the maximum value of 12, which islikewise reached with both linear 1-Dcorrelation and also 2-D correlation. As expected, 2-D correlation is unable to identify any of the rotated orgroup transformed images, in that the peak value of 12 is reached only for the case of f 1 correlated with h. On theother hand, WPC correlation, as expected, also achieves the peak value 12 for i =9; 10;:::;16, correspondingto the cases of group transformed images. For comparison purposes we also WPC correlate using the Q(1; 8)decomposition. For the Q(1; 8) tree, WPC correlation identies as similar (same peak values), rotated imagesf 2 , f 3 and f 4 since they correspond to group transformations of Q(1; 8); f 9 through f 16 are not so identied. Asexpected, the maximum value of 12 is not reached since the h is dened on more than the leaves of the leftmostsubtree of Q(1; 8). Numerical results are given in Table I. Note that gure \T" supported on the rst 4 2 leavesof Q(2; 4) and scanned in a clockwise spiral ordering generates a symmetric signal. Hence results for WPCconvolution and correlation based on the Q(2; 4) tree, involving f 1 and f 9 to f 16 , yield the same maximum of12. Other similar values are coincidental, and stem from the utilization of binary images.4 ConclusionThe algebraic structure of a group theoretic approach gives the advantage of a natural collection of (noncommutative)convolution operators. When this is applied to the quadtree indexing of a digitized image, theseconvolution operators have potential for application in signal processing. The main contribution of this paperand its predecessor [6] has been to establish some of the basic properties of nite group-based signal processing|in particular, WPC convolution and correlation|and illustrate their application through some examples. Mostof the exposition here has been conned to the transform generated from the WPC group Z(k; n) acting onthe quadtree Q(k; n). This group action provides a natural multiresolution analysis of 2 kn 2 kn images whichgeneralizes the DFT and circular convolution.An explicit formula for WPC convolution has been derived. Properties of WPC convolution and correlationhave been developed to aid their use for this new class of linear, non-commutative lters. The basic utility of thelters in frequency as well as in a scale-selectivemodewere given. Wehave seen that the process of WPC lteringestablishes a multiresolution decomposition of a signal where the signal is represented as a sum of projectionson all the subspaces. This corresponds to a transformation of the signal, where it is represented as a sum ofsignals that are piecewise constant over intervals of length 4 k , for k =0; 1;:::;n, 1. This representation wasutilized in assessing the capability of WPC ltering in determining similarity of scaled images. Signals were rsttransformed by WPC convolution and then the standard correlation measure applied to establish similarity. Itwas seen that lowpass perceptually similar test images had markedly higher correlations after WPC convolutionthan that before the transformation. Bandpass images also showed higher correlations when both the templateand test images were correlated at individual scales. In addition, the discrimination capability for dissimilarimages was considerably greater than that obtained with standard correlation. This was illustrated through anexample using 16 scaled versions of one image and comparing them to the unscaled (original) image as well asnine other (dissimilar) images.WPC correlation was used to identify images that were group transformed versions of a prototype image.For these images, and with the appropriate selection of quadtree decomposition, WPC correlation was seen to20

achieve the same peak value as standard correlation obtains for two identical images.4.1 Open ResearchThis paper has explored some of the signal processing implications of spectral analysis of the WPC groupassociated with the Q(k; n) quadtree. Applications were conned to two-dimensional images reduced to onedimensionalsignals through a quadtree scan. At this early stage of development and application of WPC grouptheory, a host of open problems exist. A few of them are as follows:1. Further investigations of WPC convolution and correlation: We have seen that transforming signals usingWPC convolution can be useful in establishing similarity of certain types of images, while still maintaininga discrimination capability. A quantitative treatment of similarity determination, classication accuracy,and scalability using WPC convolution is needed. Common image distortions such as rotation, scale, andtranslation need to be considered. WPC correlation should be investigated to determine if both its peakpreserving and pattern preserving properties can aid similarity determination.2. Applications of WPC group spectral properties to multiscale analysis: General multiscale analysis oftenuses the spectrum at various scales in applications such as pattern recognition, noise removal, imageregistration, etc. The complex WPC group spectrum makes available amplitude and, more importantlyphase, at all scales, as also a group invariance and a localized transform property. Consideration of theseproperties to applications should provide a rich area for investigation. For instance, like the unnormalizedHaar transform [16], the unnormalized WPC transform can also be looked upon as a multiscale analysistool for Poisson processes. How might the additional group invariant structure of the WPC transform andWPC convolution and correlation contribute to the modeling and estimation of Poisson processes?3. Noise ltering: What process does the WPC transform diagonalize and what is its practical signicance?For signal and noise ltering or prediction problems, what is the MSE in terms of the WPC convolution,and what conditions on the convolution ensure that it is the smallest?4. Extensions to 2-D: Given the one-dimensional unitary WPC transform, a two-dimensional separable unitarytransform can be dened; however, its group theoretic implications are not presently clear. Does thereexist such a theory? If so, what are its implications, especially those analogous to the one-dimensionalcase?21

5 Appendix A | Proof of Theorem 2.6Parts (2) and (3) of the theorem are direct consequences of parts (1) and (4) in Theorem 2.1, so we concentrateon verifying (1). From formula (2.2) and the denitions of f 0 and h i wehave(f 0 ?h i )(x p )= j X jj G j= j X jj G jX2Gf 0 (x 0 )h i ( ,1 x p )= j X jj G jX 2 G 0x i = x pX2G 0h i ( ,1 x p )1= j X jj G j jf 2 G 0 j x i = x p gj:(A1)Now G 0 consists of those symmetries of the tree Q(1;n) (as depicted in Figure 7 of [6] for n = 9) that xthe leftmost path from node 0 down to the zeroth leaf, x 0 . The subgroup G 0 therefore acts as symmetries ofthe three trees of type Q(1;n, 1) descending from nodes 1, 2 and 3 at level 1. It also acts as symmetries ofthe three trees of type Q(1;n, 2) descending from nodes 1, 2 and 3 at level 2, and so on. The symmetries ofeach of these subtrees may be specied independently, so that G 0 is the direct product of the full WPC groupsof symmetries of all these subtrees:G 0 = Z(1;n, 1) 3 Z(1;n, 2) 3 Z(1; 1) 3 ;(A2)where Z(1;m) 3 denotes the direct product Z(1;m) Z(1;m) Z(1;m). From this perspective the number ofelements in G 0 such that x i = x p can easily be computed as follows. Let s be the smallest integer such thatboth x i and x p descend from a common node at level s other than the zeroth node at level s; if no such commonancestor exists, then (f 0 ?h i )(x p ) = 0 because there is no element in G 0 with x i = x p (i.e., x i and x p lie indierent orbits of G 0 acting on the leaves). In other words, we consider the unique largest subtree containingboth x i and x p whose root node is the d th node, s d,atlevel s, for some d 1, if such exists. Then the symmetrygroup Z(1;n, s) of the subtree descending from node s d is one of the direct factors of G 0 described in (A2).By the remark following (3.6) in [6], in the group Z(1;n, s) the number of elements that send one leaf ofQ(1;n, s) toany other is the same as the number of elements that x a leaf, namely j Z(1;n, s) j=4 n,s . Sinceall the remaining direct factors of G 0 permute sets of leaves that are disjoint from the block of leaves descendingfrom s d ,any element ofZ(1;n, s) that maps x i to x p may bemultiplied by any elements from the remainingdirect factors to result in another element ofG 0 that still sends x i to x p . It therefore follows that the numberof elements in the set jf 2 G 0 j x i = x p gj is j G 0 j=4 n,s (again assuming x i and x p lie in a common subtreenot rooted at a zeroth node). Since j X j = j G j=j G 0 j, the formula in part (1) of the theorem is seen to hold.This completes the proof.22

References[1] S. Akamatsu, T. Sasaki, H. Fukamachi and Y. Suenaga, Arobust face identication scheme | KL expansionof an invariant feature space, Proc. SPIE Conf. on Intell. Robots and Computer Vision, vol. 1607, Nov.1991,71{84.[2] P. Anuta, Spatial registration of multispectral and multitemporal imager using fast Fourier transform techniques,IEEE Trans. Geosci. Electron., GE{8, Oct. 1970, 353{368.[3] D.H. Ballard and C.M. Brown, Computer Vision, Prentice{Hall, (1982), 65{70.[4] J.S. Bomba, Alpha-numeric character recognition using local operations, 1959 Fall Joint Comput.Conf.,218{224.[5] P.J. Davis, Interpolation and Approximation, Blaisdell Publishing Co., (1963).[6] R. Foote, G. Mirchandani, D. Rockmore, D. Healy, and T. Olson, Awreath product group approach to signaland image processing: part I|multiresolution analysis, to appear in IEEE Trans. on Signal Processing,2000.[7] J. Granata, M. Conner and R. Tolimieri, The tensor product: a mathematical programming language forFFTs and other fast DSP operations, IEEE Signal Processing Magazine, 9(1) Jan. 1992, 40{48.[8] I.A. Hadar, J.A. Van Mieghem, and L. Rub, Multiple subclass pattern recognition: a maximin correlationapproach, IEEE Trans. Pattern Analysis and Machine Intelligence, 17(4), April 1995, 418{431.[9] C.S. Lindquist, Active Network Design, Steward & Sons, (1977).[10] A.V. Oppenheim and R.W. Schafer, Discrete-Time Signal Processing, Prentice{Hall, (1989).[11] E.J. Pauwels, L.J. Van Gool, P. Fiddelaers, and T. Moons, An extended class of scale-invariant and recursivescale space lters, IEEE Trans. Pattern Analysis and Machine Intelligence, 17(7), July 1995, 691{700.[12] R.W. Picard and T. Kabir, Finding similar patterns in large image databases, Proceedings of the 1993ICASSP, Vol.V, April 1993, 161{164.[13] J.R. Rice, The Approximation of Functions, Addison-Wesley, (1964).[14] A. Steen, Digital Pulse Compression Using Multirate Filter Banks, Hartung{Gore Verlag, (1991).[15] H.S. Stone, Progressive wavelet correlation using Fourier methods, IEEE Trans. on Signal Processing, 17(1),Jan. 1999, 97{107.[16] K.E. Timmermann and R.D. Nowak, Multiscale modeling and estimation of Poisson processes with applicationto photon-limited imaging, IEEE Trans. on Information Theory, 45(3) April 1999, 846{862.[17] G.L. Turin, An introduction to matched lters, IRE Trans. on Information Theory, June 1960, 349{360.[18] M. Unser, Local linear transforms for texture measurements, Signal Processing, 11, (1986), 61{78.[19] P.P. Vaidyanathan, Multirate Systems and Filter Banks, Prentice{Hall, (1993).[20] M. Vetterli, Running FIR and IIR ltering using multirate lter banks, IEEE Trans. ASSP, Jan. 1988,730{738.23

Q802x2ƒ cQ09Q80_ _ _ƒc ƒc ƒc2x28 7 6Q704 x 4Q608 x 8_ƒ c0Q0512 x 512Figure 1: Average of f C over nine quadrants3Q 8 01Q 8 0hh1 21 1Q18h31Q 71h 91h 01Q1Figure 2: Unit impulse lters and extensions24

hLP12 1 21/2HLP 1 1 1 11 11112 80111 1H LP 2 80h LP2LP 1 1 11/2 4 1 12 2 1 11H LP201112 700127010h LP 912H LP91 00 02 10 01/21892Figure 3: Scale selective lowpass lters and spectrahHP1h1 h2h4 h32 1 2 2 0 1 1HHP102 8 2 912 81192hHP2H HP262 2 7 h1/2 2 h2/2 210h4/2 2 h3/2 22 6 1 12 7 0 0hHP9HHP12 9 9 22 80 11 1 1216 16h1/2 h2/202 816 16h4/2 h3/22 900 0Figure 4: Scale selective bandpass lters and spectra25

50 100 150 200 250505010010015015020020025025050 100 150 200 250(a) Image f(b) f ? hLP1505010010015015020020025050 100 150 200 25025050 100 150 200 250(c) f ? hLP2(d) f ? hLP3Figure 5: WPC convolution of image f with lters h505010010015015020020025050 100 150 200 25025050 100 150 200 250(a) Image f and lter h(b) Equivalent lter hEQ505010010015015020020025050 100 150 200 25025050 100 150 200 250(c) Canonic lter hCN(d) Convolution g = f ? fFigure 6: Canonic and equivalent lters26

505010010015015020020025025050 100 150 200 25050 100 150 200 250(a) Lowpass image f1 at scale 1(b) Highpass image f2 at scale 29850985098009800975097009650975096009550970095001 2 3 4 5 6x 10 49650196000.509550−0.595001 2 3 4 5 6x 10 4−11 2 3 4 5 6x 10 4(c) f ? fand f1 ?f(d) f ? fand f2 ?f9699.19695.29699.05969996959698.959698.99694.89698.859694.69698.81 4 8 12 169694.49694.296940.150.10.050−0.05−0.19693.81 4 8 12 16 20 24 28 32 36 40 44 48 52 56 60 641 4 8 12 16(e) f ? fand f1 ?f - zoomed(f) f ? fand f2 ?f - zoomedFigure 7: WPC convolution using WP ltered images27

5050505010010010010015015015015020020020020025050 100 150 200 250scale 125050 100 150 200 250scale 225050 100 150 200 250scale 525050 100 150 200 250scale 65050505010010010010015015015015020020020020025050 100 150 200 250scale 325050 100 150 200 250scale 425050 100 150 200 250scale 725050 100 150 200 250scale 8(a) Lowpass images f1 ! f4 at scales 1-4(b) Lowpass images f5 ! f8 at scales 5-85050505010010010010015015015015020020020020025050 100 150 200 250scale 125050 100 150 200 250scale 225050 100 150 200 250scale 525050 100 150 200 250scale 65050505010010010010015015015015020020020020025050 100 150 200 250scale 325050 100 150 200 250scale 425050 100 150 200 250scale 725050 100 150 200 250scale 8(c) Bandpass images f9 ! f12 at scales 1-4(d) Bandpass images f13 ! f16 at scales 5-81.12 x 1091.1scale 11.12 x 1091.1scale 21.151.11.151.11.080 2 4 6 8x 10 41.080 2 4 6 8x 10 41.2 x 109 scale 1 0 2 4 6 81.050 2 4 6 8x 10 41.2 x 109 scale 21.05x 10 41.12 x 1091.1scale 31.12 x 1091.1scale 41.151.11.11.2 x 109 scale 41.080 2 4 6 8x 10 41.080 2 4 6 8x 10 41.2 x 109 scale 3 0 2 4 6 81.050 2 4 6 8x 10 41x 10 41.12 x 1091.1scale 51.12 x 1091.1scale 61.111.11.051.15 x 109 scale 61.080 2 4 6 8x 10 41.080 2 4 6 8x 10 41.2 x 109 scale 5 0 2 4 6 80.90 2 4 6 8x 10 41x 10 41.12 x 1091.1scale 71.121.11.2 x 1091.1scale 71.11.051.15 x 109 scale 81.080 2 4 6 8x 10 41.14 x 109 scale 81.080 2 4 6 8x 10 410 2 4 6 8x 10 410 2 4 6 8x 10 4(e) fi ?f (lowpass) and f ? f(f) fi ?f (bandpass) and f ? hFigure 8: WPC convolution using db8 ltered images28

110.9g ik and f0.90.80.70.80.7g ik and f0.60.61−abs(ρ MR)0.51−abs(ρ)0.50.40.4← f iand f0.30.2← f iand f0.30.20.10.101 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16Scaled images i=1 to 161 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16Scaled images i=1 to 16(a) Comparing fi to f and gi k to f after WPC convolution (b) Comparing fi to f and gik to f before WPC convolution0Figure 9: Comparison of errors with and without WPC convolution10.9g ik and f0.80.70.61−abs(ρ MR)0.50.40.3← f iand f0.20.109 10 11 12 13 14 15 16Scaled images i=9 to 16Figure 10: Correlation coecient error after WPC convolution (bandpass images)29

10203040506010 20 30 40 50 60Figure 11: Image \T" and 15 rotated and group transformed imagesf and h1f and h2f and h3f and h4f and h5f and h6f and h7f and h8f and h9f and h10f and h11f and h12f and h13f and h14f and h15f and h16X(2,4)Corr.1212111099991212121212121212X(2,4)Conv.1212111099991212121212121212X(1,8)Corr.101010107.57.57.57.59.59.59.59.59.59.59.510X(1,8)Conv.101010107.57.57.57.59.59.59.59.59.59.59.5102–DCorr.1298989981089109999Table 1: Comparison of WPC correlation and convolution with linear correlation30