BLaC-Wavelets: A Multiresolution Analysis With Non-Nested Spaces. Georges-Pierre Bonneau Stefanie Hahmann Gregory M. Nielson z

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1 BLaC-Wavelets: A Multresoluton Analyss Wth Non-Nested Spaces Georges-Perre Bonneau Stefane Hahmann Gregory M. Nelson z CNRS - Laboratore LMC Grenoble, France Arzona State Unversty Tempe, USA ABSTRACT In the last ve years, there has been numerous applcatons of wavelets and multresoluton analyss n many elds of computer graphcs as derent as geometrc modellng, volume vsualzaton or llumnaton modellng. Classcal multresoluton analyss s based on the nowledge of a nested set of functonal spaces n whch the successve approxmatons of a gven functon converge to that functon, and can be ecently computed. Ths paper rst proposes a theoretcal framewor whch enables multresoluton analyss even f the functonal spaces are not nested, as long asthey stll have the property that the successve approxmatons converge to the gven functon. Based on ths concept we nally ntroduce a new multresoluton analyss wth exact reconstructon for large data sets dened on unform grds. We construct a one-parameter famly of multresoluton analyses whch s a blendng ofhaar and lnear multresoluton. INTRODUCTION Wth a multresoluton analyss one can represent a gven functon at multple levels of detal. The loss of detal n each level s stored nto the so-called wavelet coecents (detal coecents). Wavelets are bass functons encodng the derence between two successve levels of representaton. Multresoluton analyss conssts n a decomposton of a gven functon over a coarse subdvson of the doman together wth a sequence of wavelet coecents whch allows an exact reconstructon. Multresoluton analyss based on wavelets s a powerful tool for handlng large data sets. It allows an ecent representaton by means of compresson and fast computatons. Multresoluton analyss has ts orgns n numercal analyss and sgnal decomposton. But n the last ve years, t has become more and more popular n many elds of computer graphcs as derent as mage processng, geometrc modellng, volume vsualzaton or llumnaton modellng. LMC: Laboratore de Modelsaton et Calcul, B.P. 53, 384 Grenoble, France: bonneau@mag.fr, hahmann@mag.fr z Computer Scence Department, Arzona State Unversty, Tempe, AZ , USA: nelson@asu.edu Compresson of large data sets le mages can be done byre- movng small wavelet coecents. The number of removed coecents depends on the error bound for the resultng approxmaton. Wth amultresoluton analyss t s easy to perform progressve dsplayng of complex scenes. Whle addng progressvely the wavelet coecents the mage s dsplayed more and more n detal. One doesnotneed to load and dspay a whole large mage all at once. Multresoluton analyss s also sutable for level-of-detal control n renderng systems. One can go smoothly from a very coarse approxmaton of an obect far away from the vewer to more detaled representatons more the vewer approaches the obect. All these applcaton examples and more can be found n [2]. Classcal multresoluton analyss s based on the nowledge of a nested set of functonal spaces n whch the successve approxmatons of a gven functon converge to that functon, and can be ecently computed. Ths paper rst proposes atheoretcal framewor whch enables multresoluton analyss even f the functonal spaces are not nested, as long asthey stll have the propertythat the successve approxmatons converge to the gven functon. The purpose of ths framewor s to ntroduce a new multresoluton analyss of large data setsdened on unform grds. For these sets two well nown wavelet bases, the Haar and the lnear bass, can be used. Haar wavelets allow local analyss formulas but they are not contnuous. Lnear wavelets are contnuous but ther regularty can be a drawbac fthe analyzed data have many dscontnutes. The theoretcal framewor ntroduced n ths paper provdes a one-parameter famly of wavelet bases whch goes smoothly from the Haar to the lnear bass. The user can choose hs own compromse between regularty and localty propertes. 2 THEORETICAL BACKGROUND The purpose of ths secton s to recall brey the bascs of a classcal multresoluton analyss, omttngtechncal detals, to ntroduce our new multresoluton framewor, and to pont out lns between both theores. Multresoluton analyss based on wavelets, as rst ntroduced by Mallat n [4], leads to a herarchcal scheme for the computaton of the wavelet coecents. The use of fully orthogonal wavelet bases generally mples globally supported wavelets. But the nnte support of such

2 wavelets s usually a dsadvantage for practcal mplementatons. Ths s the reason why we focus on sem-orthogonal wavelet bases. 2. Classcal multresoluton analyss Let be some doman and L 2 () the set of all functons wth nte energy over. Classcal multresoluton analyss, s based on the nowledge of a nested set of subspaces V n of L 2 (). V n V n () It s also requred that [ n V n s dense n L 2 () (2) so that for any gven functon L 2 (), the successve approxmatons of V n wll converge. Property () ensures that s always a better approxmaton of f than. The wavelet spaces W n are ntroduced to capture the loss of detal between and.inother words there must exst somefuncton n W n so that =. Ths s the case f wechoose W n as the orthogonal complement of V n n V n. V n = V n W n (3) In order to computethe approxmatons and the detals, onehas to ntroduce bass functons for the spaces V n and W n. In ths secton, we won't state more precse any ndex set. Ths wll be done n secton 3. The bass functons (' n)ofv n are called scalng functons. Thebassfunctons ( n)ofw n are called wavelet functons. From () and (3) follows that there exst some matrces G n = ( )and H n =(h n ), so that ' n n = = (4) h n (5) (4) s nown as the renementequaton. Another consequence of (3) s thatthe functons ' n and n are mutually orthogonal, and therefore we have l h n l < l >= (6) The orthogonalty equaton (6) s used to compute the matrx H n and hence the wavelet functons ( n ) from a gven set of scalng functons (' n )and('n ). From (3) follows also that the ner scalng functons ( ) are lnear combnatons of the coarser scalng functons (' n) and the wavelet functons ( n): = 'n h n n P Gven an approxmaton = xn of V n, P P the coarser approxmaton = xn 'n andthedetals g n = are computed by yn n x n = xn (7) y n = h n xn (8) (7) and (8) are called analyss formulas. Computng the ner smoothng coecents (x n ) from the coarser one (x n)andthe detal coecents(yn ) s done nthe followng way: x n = x n Ths equaton s nown as synthess formula. h n y n (9) The complete analyss process, called lter ban, s represented by (x n ) ;! (x n ) ;! (x ) ;! (x ) & & & (y n ) (yn; ) (y ) The complete synthess process conssts of computng the - nest smoothng coecents (x n ) from the coarsest (x )and the detal coecents at each scale, by n applcatons of (9). 2.2 Multresoluton analyss wth non-nested spaces The choce of the scalng functons s the heart of the multresoluton analyss ntroduced n 2.. Ths choce determnes the approxmaton spaces V.Ifthe convergence condton (2) s qute ntutve snce we want the approxmaton to converge, t should be possble to dene amultresoluton framewor even f the condton oested spaces () s not vered. In fact ths condton, or equvalently the renementcondton (4) s very restrctve. It forces the scalng functons at all levels to have the same regularty. Ths s ncompatble wth the BLaC-scalng functons dened n secton 3ofths paper. The dea appled n secton 3 to domultresoluton analyss wthout the renement condton, s bascally to assocate to each scalng functon at the level n, a lnear combnaton of scalng functons of the ner level n, whch s not equal (snce t's not possble), but closeto t. In other words, we replace the renement condton by anapproxmated renement condton: ' n ' e' n = g' n n () The choce of the approxmaton depends on the applcaton, and therefore t wll be stated more precsely n the next secton. The new approxmaton space Vn e spanned by the functons (e' n) s close to V n,andtsalsoncluded n V n: V n ' e Vn V n : The wavelet space W n now captures the loss of detal between ev n and V n: V n = e Vn W n : ()

3 V n V n V n V n V n Fgure : The classcal and new multresoluton framewor The orthogonalty equaton (6) used to compute the wavelets s stll the same, but tmeans now that n and e' n (nstead of ' n )aremutually orthogonal. The analyss and synthess formulas (7), (8) and (9) are the same, but t corresponds now to an approxmaton n two steps: Analyss:. # e fn = P xn e' n # = P xn Fgure gves a geometrc nterpretaton of one step of analyss and synthess n the classcal and new multresoluton framewor. Ths new multresoluton framewor s appled n secton 3, where we wll need to handle scalng functons wth derent regulartes. 3 BLaC MULTIRESOLUTION 'n developng BLaC wavelets was to loo at the Haar and lnear borthogonal wavelets [], [5,6], [3]. Ths last nd of wavelets have some regularty, they are contnuous, whch s not the case of the pecewse constant Haar wavelets. But the regularty of the bass functons s a good pont onlyfthe data tobe analyzed s also contnuous. To llustrate ths fact, gure 2 shows the resultofonestep of analyss on a 6 pont data set, all but one equal to zero, by Haar and lnear wavelets. We can see that the analyss by Haar wavelets yelds only two non zero coecents, whle the analyss by lnear wavelets gves much more non zero coecents. Therefore, an exact reconstructon would requre less coecents n the the Haar than n the lnear case. The purpose of the BLaC wavelets s to nd a compromse between the localty of the analyss (whch s perfect for the Haar wavelets) and the regularty of the approxmaton (whch s much better for the lnear wavelets). Therefore BLaC stands for Blendng of Lnear and Constant. The graph on the rght of gure 2 shows the result ofoneanalyss step by afamlyof BLaC wavelets near from the Haar wavelets. In secton 3. we descrbe the BLaC scalng andwavelet functons. Secton 3.2 gves some mplementaton remars and compare the results of the Haar, BLaC and lnear multresoluton analyss on some examples. Ths secton deals wth the multresoluton analyss of data dened on unform grds. Snce we want our algorthm to be appled to very large data sets we eep the degree of the scalng andwavelet functons low. Our startng pont whle dcontnous test data Haar analyss lnear analyss BLaC analyss: =:2 Fgure 2: The test data set and comparson of Haar, lnear and BLaC analyss

4 3. BLaC scalng and wavelet functons BLaC scalng functons. The rst step n denng a multresoluton analyss as ntroduced n secton 2.2, s the choce of the scalng functons. Snce we wanted to nd a blendng between Haar and lnear wavelets, the followng fundamental scalng functon ' was chosen: ϕ Fgure 3 The scalng functon ' n s bult from' by dlaton by a factor 2 n,andtranslaton of ( 2 ;n ): ' n (x) ='; 2 n (x 2 ;n ) : The fundamental functon ' depends on the blendng factor : for =, (' n ) are the usual Haar scalng functons, and for = they are the lnear scalng functons. Fgure 4 shows the functon ' for varous values of between and. Fgure 4: The BLaC scalng functons Approxmated renement equatons. The scalng functons of the level of resoluton n have a slope of 2 n =. They are less regular than the scalng functons at one coarser level of resoluton, snce they \clmb" to more qucly. The functon ' n s a so-called L-Lpschtz functon wth regularty L =2 n =. As we ponted out n secton 2.2, the fact that the scalng functons at derent level of resoluton have derent regularty mples that norenement equaton can be found. Instead of that, an approxmaton e' n of the coarser scalng functon ' n as lnear combnaton of the ner scalng functons has to bechosen (see secton 2.2, equaton ()). For Haar scalng functons, the exact renement equaton s ' n = 'n 2; 'n,andnthe lnear case t s ' n = 2 2 'n 2;. Snce we want to blend between Haar and 2 2 'n 2 lnear, wechoose e' n tobealnear combnaton of 2;, 'n 2,. And snce we want the analyss scheme to preserve 2 constant functons we search the followng approxmaton of ' n : ' n ' e' n = 2; 'n 2 (; ) 2 (2) The last restrcton on the coecent of the lnear combnaton ensures that constant functons are ncluded n the approxmaton space e V n,asdened n 2.2, and thus that the analyss scheme wll preserve them. The constant s calculated to mnmze the L 2 dstance between ' n and e' n : ' n ; e'n L 2(IR) ;! mn The mnmum s reached f s the soluton of the followng lnear equaton: < 2; 'n 2 (;) 2 ;'n 2; ;'n 2 >= (3) From (3) t's clear that s ndependent ofthe level of resoluton n. For =, s equal to,andthe equaton (2) reduces to the Haar renementequaton. For =, ==2, and (2) becomes the lnear renement equaton. Fgure 5 llustrates the approxmated renement equaton (2), and compares t to the renement equaton for the Haar and lnear scalng functons. BLaC wavelets. Once the scalng functons and the approxmated renement equaton s xed, the wavelet spaces W n are also xed by (). The wavelet functons must be a bass of ths space. Obvously there s an nnte number of bases, and as n the Haar and lnear case we choose the functons whch have a mnmal support. Thsgoalsacheved f we mpose n to be a lnear combnaton of ve successve scalng functons: slope / 2 3 slope 2/... α.. ( α). /2.. /2. Haar renement BLaC renement lnear renement Fgure 5: The scalng functons and the renement equatons

5 n = a 2; b'n c 2 2 d'n 22 e'n 23 The coecents a b c d e are computed by the fact that n must be orthogonal to each functon of the approxmaton space Vn e. Snce there are exactly four bass functons of ths space whose support ntersects the support o,the orthogonalty condton reduces to: < n e' n >= wth = ; 2 (4) (4) s a lnear homogeneous system wth four equatons n the veunnowns a b c d e. It s unquely solved f we mpose the L 2 -norm of the wavelet to beequalto, as usual n multresoluton analyss: Z n 2 = Note that the coecents a b c d e are also ndependent of the resoluton level n. Fgure 6 shows one BLaC wavelet for varous values of =: =:7 Fgure 6: 3.2 Implementaton =: The BLaC wavelets y =: The analyss and synthess scheme (7), (8) and (9) have tobe carefully mplemented snce we want our algorthm to wor on large data sets. The results of secton 3. gve the coecents for the synthess formula (9). Normally the coecents for the analyss are computed from those of the synthess by the nverse of a (2 n ) (2 n )matrx. But we wll see that byusng orthogonalty propertes, the computaton of one analyss step can be reduced to the nverson of two smaller symmetrc postve dente banded matrces. Gven theapproxmaton of f atthe resoluton level n: 2 n = x n = y An MPEG move showngthe BLaC wavelet gong smoothly from Haar to lnear s avalable va anonymous ftp at ftp.mag.fr/pub/mga/bonneau/wav.mpg the problem s to nd the coarser approxmaton ef P 2 n = n P = xn e' n 2 and the detal = n ; = yn n. Snce s the error between and fn e,the followng equaton needs to bevered 2 n = x n = 2 n = 2 n ; x n e' n y n n (5) = To ndthe smooth coecents (x n ), we proceed by tang the scalar product of e' n wthbothsdes of (5). Snce thewavelets ( n) are orthogonal to the scalng functon e'n,we get 2 n < = x n e' n >= 2 n = x n < e' n e' n > (6) Thus weseethatthe smooth coecents (x n ) are thesoluton of a lnear system whose matrx has theelements < e' n e'n > and s nown as Gram-Schmdt matrx. Ths matrx has very good propertes: t s symmetrc, postve dente, and tsbanded. We cantherefore use the Cholesy algorthm to nverse the system (6). Ths algorthm s numercally stable for hgh dmenson matrces. Ths s crucal for the analyss of large data sets. Moreover, the Cholesy algorthm preserves the band structure of the matrx, so that the cost of the nverson of (6) s only of O(d), where d s the szeofthe data set. Analogously, the detal coecents (y n )arethe soluton of the followng lnear system: < 2 n = 2 n ; x n n >= y n < n n > : = Ths system can also be solved by a Cholesy algorthm for band matrces. Thus the cost of the computaton of the whole analyss s O(d), where d s the sze of the data set. 3.3 Examples In ths secton we show the results of partal reconstructons usng BLaC multresoluton on a 2D and a 3D data set. The examples of gure 7 show at the top the result of the reconstructon of the lena mage wth 4%of the coecents usng a BLaC multresoluton analyss for three derent valuesof. Onthe top leftwe used = (e.g. Haar analyss), n the mddle = :5 was used and tothe top rght = (e.g. lnear analyss) was employed. As a consequence of the regularty of the lnear wavelets, fuzzy areas appear where the mage has sudden dscontnutes (see for example, along the upper border of the hat). Ths s what we could expect from the ntroducton of secton 3: too much regularty mples poor reconstructon (for a xed number of coecents) where the data s dscontnuous. These fuzzy eects are less vsble for = :5 and they totally dsappear wth =.On the other hand, the three mages at the bottom, whch are the results of an edge detecton algorthm appled to the top mages, clearly show that the approxmaton becomes smoother whle ncreases. For =, the contours even follow the support of the wavelet functons wth the bggest coecents.

To conclude ths artcle, we see that BLaC multresoluton analyss enables the user to choose hs own compromse between regulartyof the bassfunctons and localty of the analyss, or what s equvalent,

6 Fgure 7: Lena Fgure 9: test data set z Fgure 9 shows an mage of a 256x256x28 volume data set generated usng the Stanford VolPac volume renderer. Fgure 8 shows the reconstructon of ths data set usng %of the wavelet coecents, wth a BLaC multresoluton for = : :43 :7 :. To conclude ths artcle, we see that BLaC multresoluton analyss enables the user to choose hs own compromse between regulartyof the bassfunctons and localty of the analyss, or what s equvalent, between the smoothness of the approxmaton, and tssharpness. =: =:43 References [] A. Cohen I. Daubeches J. Feauveau. B-orthogonal bases of compactlysupportedwavelets. Comm. Pure Appl. Math. 45, 485{56 (992) [2] M. Ec T. DeRose, et al. Multresoluton Analyss of Arbtrary Meshes. Proceedngs of SIGGRAPH'95. In Computer Graphcs, Annual conference Seres, (995) [3] A. Fnelsten D. Salesn Multresoluton Curves. Proc. of SIGGRAPH'94, In Computer Graphcs, Annual conference Seres, (994) [4] S. Mallat. A Theory for Multresoluton Sgnal Decomposton: The Wavelet Representaton. IEEE Trans. Pattern Anal. and Machne Intel., 674{693, (989) [5] E. Stollntz T. DeRose D. Salesn. Wavelets for Computer Graphcs: A Prmer, Part. IEEE Computer Graphcs & Applcatons 5, 76{84, (995) [6] E. Stollntz T. DeRose D. Salesn. Wavelets for Computer Graphcs: A Prmer, Part 2. IEEE Computer Graphcs & Applcatons 5, 75{85, (995) =:7 Fgure 8: =: Reconstructon wth %ofthe coecents z z Images generated usng the Stanford VolPac volume renderer

CCT. Abdul-Jabbar: Design And Realization Of Circular Contourlet Transform ..VLSI. z 2. z 1.

CCT. Abdul-Jabbar: Design And Realization Of Circular Contourlet Transform ..VLSI. z 2. z 1. Abdul-Jabbar: Desgn And Realzaton Of Crcular Contourlet Transform DESIGN AND REALIZATION OF CIRCULAR CONTOURLET TRANSFORM Dr Jassm M Abdul-Jabbar * Hala N Fathee ** * Ph D n Elect Eng, Dept of Computer