Anisotropic 2-D Wavelet Packets and Rectangular Tiling: Theory and Algorithms

Transcription

1 Anisotopi -D Wavelet Pakets and Retangula Tiling: Theoy and Algoithms Dan Xu and Minh N. Do Depatment of Eletial and Compute Engineeing and Bekman Institute Univesity of Illinois at Ubana-Champaign {danxu, Web: danxu ABSTRACT We popose a new subspae deomposition sheme alled anisotopi wavelet pakets whih boadens the existing definition of -D wavelet pakets. By allowing abitay ode of ow and olumn deompositions, this sheme fully onsides the adaptivity, whih helps find the best bases to epesent an image. We also show that the numbe of andidate tee stutues in the anisotopi ase is muh lage than isotopi ase. The geedy algoithm and double-tee algoithm ae then pesented and expeimental esults ae shown. Keywods: Anisotopi wavelet pakets, etangula tiling, best bases, dynami pogamming.. INTRODUCTION The ability to epesent signals in a lage family of bases, whee a best one an be adaptively seleted fo a given signal, is a poweful onept. Wavelet pakets (WP) povide one of suh sheme. Wavelet pakets an be seen as an extension of the wavelet tansfom via geneal tee-stutued filte banks. In the one-dimensional wavelet tansfom, the building blok two-hannel filte bank is iteatively applied to the lowpass hannels. The -D wavelet paket sheme extends this by allowing the two-hannel filte bank to be applied to futhe deompose any hannel. This esults in a lage numbe of bases oesponding to diffeent binay deomposition tees. This idea an be easily extended to highe dimensions using sepaable filte banks. We stat with the building blok of the -D wavelet tansfom that onsists of a -D two-hannel filte bank applied to eah ow, followed by a -D two-hannel filte bank applied to eah olumn (see Figue ). In the -D wavelet tansfom, this equivalent fou-hannel filte bank is iteatively applied to the lowpass hannels, wheeas -D wavelet pakets allow it to be applied to any hannel. The eason fo the oupling of the ow and olumn tansfom steps is so that the esulting basis funtions have the same sale along eah dimension. 3 We thus efe to suh ommonly used wavelet pakets as isotopi wavelet pakets. Reently, seveal studies in onstuting spase expansions fo natual images with smooth ontous 4,5 have evealed that anisotopy is a desied featue fo new image expansions. Moe peisely, to aptue smooth ontous in images, the expansion should ontain basis funtions with vaiety of shapes, in patiula with diffeent aspet atios. This pompts us to onside elaxing the oupling of the ow and olumn deomposition step in the isotopi wavelet paket. The esult is a new family of bases whee the ow and olumn deomposition steps an be taken at abitay ode at eah node. We will efe to this family of bases as anisotopi wavelet pakets, whih obviously ontain isotopi wavelet pakets as a subset. It an be seen that eah esulting anisotopi wavelet paket basis oesponds to a binay tee whee eah node is assigned to eithe a ow o a olumn value. In the fequeny-domain, the isotopi wavelet paket sheme an be seen as a quad-tee patitioning of the fequeny spetum, whee eah futhe deomposition step on a subband would divide its spetum into fou squae egions. Thus the isotopi wavelet pakets have a dual family in the spae-domain as quad-tee patitioning into squae egions. Fo the anisotopi wavelet paket family, we see that its dual in the spaedomain is a patitioning sheme into etangula egions, whee eah of the egions an be divided futhe into two halves, along eithe ow o olumn. We will efe to suh a sheme as etangula tiling. The ode of ow and olumn steps an be evesed, whih would give the idential esult

2 While anisotopi WP and etangula tiling seem somewhat tivial extensions, to ou best knowledge, it has not been onsideed befoe. A key question, whih will be addessed in this pape, is whethe these extensions povide signifiant gains without signifiantly ineasing the omputational omplexity. Afte biefly eviewing the main esult of isotopi WP in Setion, we will fomally intodue and study the popeties of the anisotopi wavelet pakets in Setion 3. A main esult is the alulation of the numbe of anisotopi WP bases, whih is signifiantly lage than the numbe of isotopi WP bases. In Setion 4, we onside a estited lass of anisotopi wavelet pakets and etangula tiling whee all deomposition in one dimension ae taken befoe anothe dimension. Suh a estition might be appopiated in etain appliations. Expeiments with eal images in Setion 5 show the potential gains of anisotopi WPs in poviding bette bases than isotopi WPs.. BACKGROUND.. Two-Dimensional Sepaable Filte Banks Sepaable filte banks ae the building bloks of the -D wavelet tansfom. Figue shows a typial -D sepaable filte banks. We denote V as the th level -D appoximation spae, V and V as the th level -D ow and X[n,n] H 0 H H 0 H H 0 H LL LH HL HH Row Column Figue : -D sepaable filte banks. The H 0 and H vaiables epesent low-pass and high-pass digital filtes, espetively. The filteing is divided into two steps, fist along the ows and then along the olumns. olumn appoximation subspaes, espetively. Based on the assumption the othonomal bases an also be sepaated as V = V V () φ,(n,n ) (x,x ) = φ,n (x )φ,n (x ) = φ(x n, x n ) () Hee we denote φ,(, ) (, ) as the th level -D saling funtion, φ, as the th level -D saling funtion, x and x as vaiables in the spatial domain fo ow and olumn subspaes espetively, and n and n as shifts fo the ow and olumn subspaes espetively. Anothe esult of this assumption is the deomposition of -D appoximation spae of up to J levels V = V = (V V = (V W ) (V W ) V ) (W V ) }{{}}{{} (V W ) }{{} (W W ) }{{} V W,() W,() W,(3) (3) Hee we denote W and W,() as the th level -D ow and olumn detail subspaes, espetively, W, W,() and W,(3) as the th level -D detail subspaes of hoizontal, vetial and diagonal dietions, espetively. Equation 3 gives us the expession of othonomal bases fo V, whih ae φ,n (x )φ,n (x ),φ,n (x )ψ,n (x ),ψ,n (x )φ,n (x ),ψ,n (x )ψ,n (x ) (4)

3 .. Two-Dimensional Isotopi Wavelet Pakets The notion of wavelet pakets gives feedom to the hoie of deomposition fom both appoximation and detail subbands, instead of allowing futhe deomposition only fom the appoximation subband. We will efe to these wavelet pakets as isotopi wavelet pakets to distinguish them fom anothe lass that we will desibe in a late setion. The basi idea is the spae deomposition of W p,q, the th level detail subspae, with indies p and q, 0 p i, 0 q. That is, W p,q = W p,q + W p+,q + W p,q+ + W p+,q+ + (5) Isotopi wavelet paket uses quadtee as its tee stutue. A typial tee stutue fo -D wavelet paket is shown in Figue. W p,q W p,q + W p+,q + W p,q+ + W p+,q+ + Figue : Spae deomposition in isotopi wavelet paket tee. Afte one deomposition, the spae W p,q geneates fou subspaes, in whih level beomes +, and indexes {p, q} beome {p, q}, {p +, q}, {p, q + }, {p +, q + }. The numbe of diffeent wavelet paket bases of VJ is equal to the numbe of admissible quadtees. An admissible quadtee is a quadtee with all its nodes having 0 o fou hilden. We denote the numbe of bases in a -D isotopi WP tee of up to level J. Then it is easy to see that Q 0 = and Q J = Q 4 J +. Based on this eusive fomula, the following poposition povides an estimate of Q J. Poposition ( 3 Pop. 8.5). The numbe Q J of wavelet paket bases in a -D isotopi wavelet paket tee of up to level J satisfies 4J Q J J (6) The algoithm fo finding the best basis of -D isotopi wavelet pakets is based on the dynami pogamming. 6, 7 In a -D isotopi wavelet paket tee, eah node has an othonomal basis B p,q of W p,q. We define the ost of f with espet to basis B = {g m } 0 m<m as the patial sum C(f, B) = M m=0 Φ( f,g m f ) (7) Hee Φ( ) is any onave funtion. We all the ost the additive ost if fo any othonomal bases B 0 and B C(f, B 0 B ) = C(f, B 0 ) + C(f, B ) (8) Poposition ( 3 Pop. 9.0). Suppose that C is an additive ost funtion. If then othewise C(f, B p,q ) < C(f, O p,q + ) + C(f, Op+,q + ) + C(f, O p,q+ + ) + C(f, O p+,q+ + ) O p,q O p,q = B p,q (9) = O p,q + + Op+,q + + O p,q+ + + O p+,q+ + (0)

4 Hee f is the input -D signal. O p,q is the optimal basis of W p,q. We an see Poposition guaantees a eusive dynami pogamming algoithm. By iteating the above poess fom bottom of the tee to the top, the oveall optimal basis will be obtained. The omputational omplexity of this algoithm is O(J log J) fo the wavelet paket tees of up to level J. 6 Although -D isotopi wavelet pakets allow feedom in the seletion of a subband fo futhe deomposition, it is somewhat igid in the -onseutive-step ow-olumn deomposition. In a eal image, the dominant dietionality and textue fo eah subband dives us to think of a moe flexible way of deomposition, whih an take suitable tansfoms aoding to the atual featue of one given subband. We ty to find a moe effetive way to epesent image in wavelet domain. This usually equies lage set of andidate deompositions and fast algoithms to find the best tee. The solution is the so alled -D anisotopi wavelet pakets, whih take ow-wise and olumn-wise deomposition sepaately and adaptively. 3. ANISOTROPIC WAVELET PACKETS AND RECTANGULAR TILING The isotopi wavelet paket is a quadtee like stutue whih pefoms ow and olumn deompositions onseutively in one step. Sine ow and olumn deompositions espetively expess the vetial and hoizontal disontinuity, this stutue seems umbesome when dealing with some images whee a pevalent dietionality exists in etain subbands. By enabling an abitay ode of ow and olumn deomposition, we popose the -D anisotopi wavelet pakets hee. 3.. Anisotopi Wavelet Paket Deomposition In anisotopi wavelet pakets, eah ow deomposition is not neessaily followed by a olumn deomposition. Rathe, fo eah step, eithe a ow o a olumn deomposition will be hosen. In this way, it is possible to have the ase of seveal onseutive olumn deompositions o any ombination of ow and olumn deompositions. Denote the anisotopi wavelet paket spaes as W p,q i, (see Figue 3). Hee i and ae ow level and olumn level espetively. Vaiables p and q ae ow and olumn indexes espetively to speify a etain spae. The anisotopi wavelet paket spaes ae made up of ow spaes W p i and olumn spaes W q. W p,q i, = W p i Note that fo an image with size M N, i and satisfy W q () 0 i M 0 N () Fo a ow deomposition, the anisotopi wavelet paket spaes satisfy W p,q i, = W p,q i+, W p+,q i+, (3) Fo a olumn deomposition W p,q i, = W p,q i,+ W p,q+ i,+ (4) Two-dimensional ow (olumn) deomposition divides only the ow (olumn) spae in the tenso podut expession of -D spae. And fo seveal steps of spae deomposition, we ae able to have any ombination of ow and olumn deompositions. This atually ineases the flexibility of spae deomposition, ompaed to the -D isotopi wavelet pakets, whee a ow deomposition is always followed by a olumn deomposition. Figue 4 shows 8 vaieties of deompositions inside a squae. Note that only type () and type (8) ae possible in isotopi wavelet pakets. We make a seletion of ow o olumn deomposition in the fist step and then make anothe seletion in the seond step. Both ow and olumn deomposition in the fist step an lead to a -D isotopi wavelet paket deomposition in the seond step. Theefoe the isotopi wavelet pakets ae ust speial ases of anisotopi wavelet pakets. Note that a two-step deomposition has moe than thee vaieties whih ae listed in the figue.

5 f - W i, p,q q q+ q -( + ) p i - -i f Figue 3: Index fo W p,q i, in the fequeny domain. Both fequeny dimensions ae nomalized to the ange 0 to. The i and vaiables epesent ow level and olumn level, espetively. Axis f is divided unifomly into i segments, with eah segment i in width and f into segments, with eah segment in width. The p and q vaiables ae indies fo ow and olumn and ae suh that 0 p i, 0 q. To find the position of W p,q i,, fist we divide f and f axes into i and unifom segments. Then W p,q i, is the inteseting aea of the pth segment of f axis and the qth segment of f axis, whih is shown as the shaded aea, standing fo the tenso podut of ow and olumn spaes. Fo a futhe deomposition, a olumn deomposition of W p,q i, fo example, level beomes level + and eah segment is divided in half, suh that the width of eah segment of f axis beomes (+). The esulting lowe pat of the shaded aea is indexed by q, and the uppe pat indexed by q +. Eah step of deomposition oesponds to a -D ow(olumn) wavelet tansfom whih takes -D disete wavelet tansfom along ows(olumns). Given an input matix with size M N, the ow tansfom yields two oeffiient maties with size M N/ eah, one fo appoximation subspae and one fo detail subspae, while the olumn tansfom yields anothe two oeffiient maties with size M/ N eah, also one fo appoximation subspae and one fo detail subspae. The -D ow(olumn) deomposition is the ountepat of -D ow(olumn) wavelet tansfom in the spae deomposition sense. Hee we show the esults fo one-step ow-wise and olumnwise deomposition in Figue 5. Note that in the detail subband fo both ow and olumn tansfom, the oeffiients ae nonzeo only on two paallel lines. In this way, the oiginal image is well ompessed. The stutue fo anisotopi wavelet pakets is a binay-tee like stutue, instead of the quadtee stutue in isotopi ase. Eah node is assigned a value fom the value set {ow,ol,null}, standing fo a ow deomposition, a olumn deomposition o no deomposition at all in a etain tee banh. Note that the level of the tee annot exeed L fo an image with size L L. (Moeove, the level of ow o olumn deomposition annot exeed L eithe.) Figue 6 is a typial tee stutue fo anisotopi wavelet pakets. 3.. Retangula Tiling The sepaate and abitay deomposition of subspaes in ow and olumn is equivalent to a etangula tiling in the fequeny domain. Instead of dividing eah blok into fou equally sized sub-bloks, etangula tiling bings about feedom to both numbe and dietion of sub-bloks obtained in eah one-step deomposition. Thee ae two advantages of this. Fist, in the sense of the seletion set size, the possible ways of deomposition ae ineased lagely. Thus moe nea-to-optimal deomposition an be obtained. Seond, the adaptive seletion of ow o olumn deomposition omes moe natualy, simila to the sheme human uses when looking at a pitue.

6 () () (3) (4) (5) (6) (7) (8) Figue 4: Eight possible deompositions of a unit squae on fequeny domain fo anisotopi WP. Left side epesents the tee stutue of the deomposition. Right side epesents the oesponding tiling in fequeny domain. (a) (b) () Figue 5: -D Row and olumn tansfom. Blak aea epesents zeo oeffiients, while white epesents non-zeos. (a)oiginal image. (b)row tansfom. The left pat is the appoximation subband. The ight pat is the detail subband, in whih only one line of oeffiients ae non-zeos (shown as the gay vetial line). ()Column tansfom. The uppe pat is the appoximation subband. The lowe pat is the detail subband, in whih all the oeffieients ae zeos. A ompaison of deomposition stutue between quadtee-based isotopi wavelet pakets and etangula tiling-based anisotopi ase is shown in Figue Numbe of Bases We will show the lage numbe of possible anisotopi wavelet paket tee stutues hee. Note that the moe andidate tee stutues we have, the bette bases an be hosen fom the basis family. Poposition 3. The numbe A I,J of wavelet paket bases in an anisotopi wavelet paket tee of ow level I and olumn level J satisfies A I,J = + A I,J + A I,J A 4 I,J (5) fo I and J, and A I,0 = + A I,0, A 0,J = + A 0,J and A 0,0 =. Poof: We will pove this esult by indution. Denote A I,J as the numbe of diffeent anisotopi tee expansion with ow level I and olumn level J. In the fist level tee banh, that is the fist deomposition a tee an possibly have. It an only be a ow deomposition, o a olumn deomposition o no deomposition at all. If the fist level is a ow deomposition, the whole tee is made up of two subtees that ae set of tees of up to I ow levels and J olumn levels. similaly, if the fist level is a olumn deomposition, the whole tee is made up of two subtees that ae set of tees of up to I ow levels and J olumn levels. In the thid ase, the tee has level 0 that is edued to the oot. Note that the fist two ases have an ovelap set. In fequeny domain, this

7 ow ow ol ow ow ow ol Figue 6: Binay-tee like anisotopi wavelet deomposition Figue 7: Compaison between diffeent deomposition stutues set is all the tiling with an isotopi deomposition in the fist level. The numbe of bases in this set is A 4 I,J. Theefoe, we have the eusive fomula A I,J = + A I,J + A I,J A 4 I,J (6) Note that if I and/o J ae 0, the oesponding ow and/o olumn deomposition annot be done. Theefoe we have A I,0 = + A I,0, A 0,J = + A 0,J and A 0,0 =. Beause of symmety between ow and olumn deomposition, A I,J equals to A J,I. A ompaison among A J,J, Q J (see Equation (6)), and S J (see Equation (8)) an be found in Setion Adaptive Basis Seletion using Geedy Algoithm Beause of allowing abitay ombination of ow and olumn deompositions, the best basis seletion algoithm in isotopi WP does not wok hee in anisotopi WP. We make the adaptive basis seletion by geedy algoithm, whih is a simple but vey fast algoithm to seah fo the suboptimal tee stutue. It stats fom the oiginal image and ends at some loal optimal step. Although this ignoes a possible lage set of andidates, the esult shows that it woks vey well with espet to both ompession ate and omputational ost. We fist stat fom the oot node, whih is the oiginal image. Then we ompae the ost fo no deomposition C null, ow deomposition C ow, and olumn deomposition C ol to detemine the lowest ost C. C null = C(f, B p,q i, ) C ow = C(f, B p,q i+, C ol = C(f, B p,q i,+ C = min{c null,c ow,c ol } ) + C(f, Bp+,q i+, ) + C tee,ow ) + C(f, Bp,q+ i,+ ) + C tee,ol (7) In equation 7, C(f, B) is the ost of f with espet to basis family B and is equied to be additive. Refe to equation 7 and 8 fo moe about this. The notations C tee,ow and C tee,ol ae fo the additional ost of ow

8 and olumn deompositions, espetively. In detemining C we deide whethe deomposition is needed and in whih dietion to deompose. If futhe deomposition is needed, add the popety ow o ol to the node. Othewise, set the popety as null. Repeat the above poedue fo eah node fom top nodes to leaf nodes, until no moe nodes an be deomposed. The omputational omplexity vaies fo diffeent images. In the wose ase, the omplexity will be O(J 3 ) fo a level of J. But nomally, the omputational ost is muh less than this bound and thus usually yields a fast onvegene of this algoithm. Fo the optimal basis seletion in anisotopi WP, we an do a eusive seah fom top to the bottom of the tee DOUBLE TREE ALGORITHM In this setion, we onside some speial ases, e.g. ases whee all olumn tansfoms ae taken afte all ow tansfoms, o all ow tansfoms afte all olumn tansfoms. This subset is denoted as D. We use the double tee algoithm 9 hee, whih is fist used in time-fequeny optimality poblem. Note that the ost funtion hee, the entopy, is additive, and we want to ahieve simultaneous optimality in both ow and olumn. If we onside the subset D, we an get a suboptimal solution to ou poblem. Given the maximum level J, we fist build up a omplete binay tee with only ow deompositions. Fo eah node in this all-ow tee, we use the best tee seah algoithm in isotopi ase to find the best all-olumn tee assoiated with that node (Figue 8). Hee the best means the minimum ost of a signal epesented by Row Deomposition C o l u m n D e o m p o s i t i o n Figue 8: Double tee algoithm. Dotted lines ae deompositions along ow dimension. Solid lines ae deompositions along olumn dimension. a family of bases, with espet to a etain ost funtion, whih an be an entopy funtion, fo example. We eod the least ost in olumn dimension fo eah node in the all-ow tee. Then in the all-ow tee, we find the best tee aoding to the least ost in olumn, using the best tee seah algoithm simila to the one in isotopi ase. Speifially, fom bottom to top of the all-ow tee, we aess eah node and do a puning if neessay. The iteion fo the puning is based upon whethe the ost of the paental node is moe than the sum of osts of its hild nodes. In the all-ow tee, the osts ae the least osts alulated in the all-olumn tee. Note that the size of subset D is still muh lage than the isotopi wavelet pakets. We have the following poposition. Poposition 4. The numbe S J of elements in D fo double tees with up to level J satisfies B J S J = B MJ(k) J BJ (8) k=0 whee B J is the numbe of wavelet paket bases in a -D isotopi wavelet paket tee of up to level J and satisfies B J = B J +,B 0 = (see Mallat s book, Poposition 8.). Hee M J (k) is the numbe of leaf nodes fo the kth

9 tee stutue patten in the set of all-ow (o all-olumn) tees of up to J levels, and satisfies M J (0) = M J (k) = M J (m) + M J (n) m = k B J n = (k ) mod B J, fo k =,,...,B J. whee x is the neaest intege less than o equal to x. a mod b is the modulus by dividing a into b. Poof: The subset D is a union of D, the set of all-ow-fist deomposition and D, the set of all-olumn-fist deomposition. These two sets ovelap only when all the ow and olumn tiling sepaatos in the tansfom domain hit two sides of the bounday, as an example shown in Figue 9 (). Fo set D, the tiling stutue an only be thee types whih ae shown in Figue 9. Type (a) is in set D and type (b) is in set D. Fo (), both D and D have this stutue. This ovelap has a total of B J stutues. (9) (a) (b) () Figue 9: Thee types of tiling. (a)all the vetial sepaatos hit both sides of the bounday (b)all the hoizontal sepaatos hit both sides of the bounday ()All the hoizontal and vetial sepaatos hit both sides of the bounday Beause of symmety, the numbe of elements in D equals to that of D. Denote this numbe as T J. Then S J = T J B J (0) Now we alulate T J. Fo any tee in D, we fist hoose an abitay all-ow tee of up to depth J. Fo eah leaf node in this all-ow tee, we an futhe add an all-olumn tee ooted fom the leaf node of the all-ow tee(see Figue 0). Eah of these all-olumn tee has B J numbe of stutues. Note that fo eah tee stutue we get hee, thee is a unique tiling stutue oesponding to it. Denote the numbe of leaf nodes fo the kth patten in an all-ow tee of up to L levels as M J (k),k = 0,,,...,B J. T J = B J k=0 B MJ(k) J () Fo M J (k), k = 0 oesponds to the pue oot ase and thus M J (0) =. Fo k, the leaf nodes ae made up of leaf nodes fom left subtee and ight subtee (see Figue 0). Both of them ae of up to (J ) levels. Theefoe we have M J (k) = M J (m) + M J (n) () fo m,n = 0,,,...,B J. We an put an index to k like this o k = mb J + n + (3) { m = k B J n = (k ) mod B J (4) Now we make a ompaison among Q J, S J and A J,J in Table. We an see that both A J,J and S J ae muh geate than Q J. This will establish a lage set of andidates to ahieve bette deomposition of a -D signal. The numbe S J is smalle than A J,J, whih is ompatible with the definition of double tee.

10 ow ow ow ow Left Subtee Right Subtee Figue 0: An example of double tee. A shaded ile stands fo a leaf node with an all-olumn tee ooted fom it. J Q J S J A J,J Table : Compaison among the numbe of bases fom isotopi WP (Q J), double tee (S J) and anisotopi WP (A J,J) fo up to deomposition level J. 5. EXPERIMENTAL RESULTS We have implemented the geedy and double tee algoithms. The esults ae ompaed among best tee seah algoithm fo -D wavelets, -D isotopi wavelet pakets, geedy algoithm and double tee algoithm. Figue is the ompaison of oveall entopy. We an see fom this hat that both geedy and double tee algoithm ahieve lowe oveall entopy than the existing algoithm. Theefoe possibly highe ompession ate an be obtained by using these algoithms. Fo an 8 8 atifiial image whih is shown in Figue, the deomposition esults ae illustated by geedy tee, double tee and isotopi deomposition. Both double-tee and geedy algoithms get the same esult whee thee non-zeo oeffiients emain in oeffiient matix, while in the isotopi ase, the numbe ineases to eight. Figue also shows diffeent tiling stutues of the best isotopi wavelet paket tee, geedy tee and double tee. If we denote the set of isotopi wavelet pakets as I, the set of anisotopi ones as A, we have I A. Moeove, fom the esults we an see, the optimal stutue is usually in the set S = A I. This poves that we should onside the anisotopi ase to find the optimal deomposition stutue. We also show the esults fo a eal image hee. Figue 3 shows the best tiling stutue fo isotopi wavelet paket tee, geedy tee and double tee. 6. CONCLUSION By intoduing the abitay ode of ow and olumn deomposition, the anisotopi wavelet pakets eate a muh lage set of wavelet paket bases and thus bette deomposition stutue an be obtained. Expeiments show that the two suboptimal algoithms, geedy and double tee algoithms outpefom the existing isotopi wavelet pakets with espet to ove entopy, whih an possibly bing about bette ompession. The geedy algoithm onveges vey fast and will be of muh patial use. The double tee algoithm also uns at an aeptable time ost.

11 .0.0 Wavelet Iso WP Tee Geedy Tee Double Tee Nomalized Oveall Entopy Lena Babaa Thon Boat Cameaman Goldhill Mandill Peppes Images Figue : Compaison of oveall entopy pefomane of 4 diffeent wavelet deomposition appoahes (a) (b) () (d) Figue : Coeffiients in tansfom domain fo an 8 8 atifiial image. Dak gey stands fo oeffiient 0. Blak lines ae tiling sepaatos whih show the tiling stutue and also eflet the oesponding tee stutue. (a)oiginal image (8 8, entopy=3.78) (b)isotopi ase (8 non-zeo oeffiients, entopy=.530) ()Geedy tee (3 non-zeo oeffiients, entopy=0.75) (d)double tee (3 non-zeo oeffiients, entopy=0.75) REFERENCES. R. R. Coifman, Y. Meye, and M. V. Wikehause, Wavelet analysis and signal poessing, in Wavelets and Thei Appliations, M. B. Ruskai, G. Beylkin, R. R. Coifman, I. Daubehies, S. Mallat, Y. Meye, and L. Raphael, eds., pp , Jones and Batlett, Boston, 99.. M. Vetteli and J. Kovačević, Wavelets and Subband Coding, Pentie-Hall, S. Mallat, A Wavelet Tou of Signal Poessing, Aademi Pess, nd ed., E. J. Candès and D. L. Donoho, Cuvelets a supisingly effetive nonadaptive epesentation fo obets with edges, in Cuve and Sufae Fitting, A. Cohen, C. Rabut, and L. L. Shumake, eds., Vandebilt Univesity Pess, (Saint-Malo), M. N. Do and M. Vetteli, Contoulets, in Beyond Wavelets, G. V. Welland, ed., Aademi Pess, New Yok, 003. to appea, 6. R. R. Coifman and M. V. Wikehause, Entopy-based algoithms fo best basis seletion, 38, pp , Mah K. Ramhandan and M. Vetteli, Best wavelet paket bases in a ate-distotion sense,, pp , Apil I. Pollak, M. N. Do, and C. A. Bouman, Optimal tilings and best basis seah in lage ditionaies, in Po. of Thity-Seventh Annual Asiloma Confeene on Signals, Systems, and Computes, (Paifi Gove, CA), 003. to appea.

12 (a) (b) () (d) Figue 3: Best tiling in tansfom domain fo Lena image. (a)oiginal image. Entopy=.468 (b)isotopi WP tee. Entopy=6.985 ()Geedy tee. Entopy=6.966 (d)double tee. Entopy= C. Heley, J. Kovačević, K. Ramhandan, and M. Vetteli, Tilings of the time-fequeny plane:onstution of abitay othogonal bases and fast tiling algoithms, IEEE Tans. Signal Po. 4, pp , De. 993.