CRISP-contourlets: a critically sampled directional multiresolution image representation

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1 CRISP-ontourlets: ritilly smple iretionl multiresolution imge representtion Yue Lu n Minh N. Do Deprtment of Eletril n Computer Engineering University of Illinois t Urn-Chmpign, Urn IL 68, US STRCT Diretionl multiresolution imge representtions hve ltely ttrte muh ttention. numer of new systems, suh s the urvelet trnsform n the more reent ontourlet trnsform, hve een propose. ommon issue of these trnsforms is the reunny in representtion, n unesirle feture for ertin pplitions (e.g. ompression. Though some ritilly smple trnsforms hve lso een propose in the pst, they n only provie limite iretionlity or limite flexiility in the frequeny eomposition. In this pper, we propose filter nk struture hieving nonreunnt multiresolution n multiiretionl expnsion of imges. It n e seen s ritilly smple version of the originl ontourlet trnsform (hene the nme CRISP-ontourets in the sense tht the orresponing frequeny eomposition is similr to tht of ontourlets, whih ivies the whole spetrum oth ngulrly n rilly. However, inste of performing the multisle n iretionl eomposition steps seprtely s is one in ontourlets, the key ie here is to use omine iterte nonseprle filter nk for oth steps. sie from ritil smpling, the propose trnsform possesses other useful properties inluing perfet reonstrution, flexile onfigurtion of the numer of iretions t eh sle, n n effiient tree-struture implementtion. Keywors: Multiresolution, multiiretionl, filter nks, imge representtion, mximl eimtion. INTRODUCTION If ske to ientify wish list for new effiient imge representtion, we woul put multiresolution, loliztion, iretionlity, ritil smpling, nisotropy, neffiient implementtion on top of the list. The first three requirements were suggeste y stuies relte to the humn visul system, n nturl imge sttistis. 4 In prtiulr, multiresolution sks for the imge e suessively pproximte, from orse version to finer etils; loliztion mens the sis elements in the representtion shoul e lolize in oth the sptil n frequeny omins; iretionlity requires the representtion hve high ngulr resolution, ontining elements oriente t lrge numer of ifferent iretions. For some pplitions (e.g. ompression, nonreunnt representtion is essentil, thus we lso put ritil smpling in the list. In this se, the representtion will spn sis of the imge spe. To pture smooth ontours in imges, the support size of the representtion elements shoul lso oey the nisotropy sling lw 5 for urves: with length. Lst ut not lest, n effiient implementtion is essentil for ny prtil systems. numer of imge representtions hve een propose in the literture, eh stisfying severl, ut not ll, of the ove ojetives. Cnès n Donoho 5, 6 pioneere new multisle n iretionl expnsion, lle urvelets, tht is shown to hieve optiml pproximtion ehvior in ertin sense for -D pieewise smooth funtions. One of the key fetures of urvelets is their support sizes oey the nisotropy sling lw mentione ove. Originlly evelope in the ontinuous omin, the urvelet onstrutions require rottion trnsform, whih mkes the implementtion of the trnsform on isrete imges very hllenging. Menwhile, urvelet trnsform is reunnt representtion for imges. Severl other well-known systems inlue the iretionl wvelets 7, 8 n the omplex wvelets, 9 to nme few. nie thing out these systems is tht they n e implemente rther effiiently using tree-struture Further uthor informtion: Yue Lu: yuelu@uiu.eu; Minh N. Do: minho@uiu.eu

2 (, multisle e. iretionl e. Figure. The ontourlet trnsform. lok igrm. It uses n iterte omintion of the Lplin pyrmi n the iretionl filter nk. Resulting frequeny ivision, where the whole spetrum is ivie oth ngulrly n rilly n the numer of iretions is inrese with frequeny. onstrution. However, they n only provie limite iretionlity n o not llow for ifferent numer of iretions t eh sle. More reently, Do n Vetterli, propose omputtionl frmework, lle ontourlets, for the iretionl multiresolution representtion of isrete imges. s shown in Fig., the ontourlet trnsform employs n effiient tree-struture implementtion, whih is n iterte omintion of the Lplin pyrmi n the iretionl filter nk. Menwhile, it hs esirle frequeny eomposition, where the whole spetrum is ivie oth ngulrly n rilly. It n e shown tht the ontourlet trnsform hieve most of the ojetives in our wish list. tully the only thing left is ritil smpling, whih is ue to the use of the Lplin pyrmi. reunny rte of up to 4/ is rought into the system. The purpose of this pper is to introue new iretionl multiresolution imge representtion tht n hieve similr frequeny eomposition to tht of ontourlets, while with nonreunnt expnsion. In priniple, this new system n e seen s ritilly smple version of the originl ontourlets trnsform, n hene is nme CRISP-ontourlets. Inste of performing the multisle n iretionl eomposition steps inepenently s in ontourlets, the key ie here is to use omine iterte nonseprle filter nk for oth steps. Fig. shows typil frequeny eomposition of CRISP-ontourlets. Here we hve four lowpss regions, enote s,,, in the figure. Strting from the seon level, the numer of iretions is oule with the inrese of frequeny. In prtiulr, eh npss region n hve n (n =,,... iretions. The similrity in frequeny ivision llows for the new system to inherit the merits of the ontourlet trnsform. While the e feture of nonreunnt representtion mkes it potentilly more promising system for pplitions like ompression. In similr effort, Hong n Smith propose n otve-n fmily of nonreunnt iretionl filter nks (ODF, whih lso hieves ertin ril n iretionl eomposition of the spetrum. sie from the ifferene in the shpes of iretionl npss regions, our system llows more flexiility in the numer of iretions t eh n. s speil se (not shown in Fig., CRISP-ontourlets n lso oule the numers of iretions t every other multisle level, whih is similr to the ontourlets. The outline of the pper is s follows. Setion introues severl results in -D multirte systems tht serves s kgroun n provies intuitions for the esign of the new system. Setion onentrtes on the etils of the CRISP-ontourlet trnsform, inluing the speifi filter nks use n the iterte expnsion rule. In Setion 4 we esign the FIR filters use in the system.

3 Figure. typil frequeny eomposition of the CRISP-ontourlet trnsform. For the ske of lrity, we only show the etils of frequeny ivision in the gry re. The ivision in other res n e got vi symmetry. In CRISP-ontourlets, we hve four lowpss regions, enote s,,,. The numer of iretions is inrese with frequeny. In prtiulr, eh npss region n hve n (n =,,...iretions.. MXIMLLY DECIMTED FREQUENCY SUPPORT The theory of multiimensionl multirte systems hs een extensively stuie in the pst. Exellent tutorils on this topi n e foun in referenes. 5 In this setion, we will review severl importnt results, whih not only ly the groun for the following isussion, ut lso revel the importnt intuitions in the CRISP-ontourlets system. Let x(n e -D isrete signl n y(n e the eimte version of x(n through nonsingulr mtrix M, i.e. y(n =x(mn. The reltionship etween the two signls in the Fourier omin n e expresse s: 4 Y (ω = µ X(M T (ω πk l, ( µ l= where X(ω, Y (ω re the Fourier trnsform of the orresponing signls, µ = et(m, n{k l } µ l= re the oset vetors of M T. We will first restrit our ttention to the se of iel signls, where X(ω = in its pssn n elsewhere. The issue of esigning FIR filters pproximting the iel supports will e isusse in Setion 4. For now, the signls re solely speifie y the shpes of their frequeny supports n it is onvenient to introue the following notion. For ny iel spetrum X(ω, we use the point set X {ω : ω support of X(ω} to represent its support region. Note tht X is perioi with n intervl π π. Let F {ω : ω [ π, π] } enote the funmentl perio in the frequeny omin. Now the support of X(ω in one perio n e efine s X F X F. In ition to the ommon set opertions suh s union n intersetion, we efine two more opertions on the point sets here, nmely the shift X + {ω + : ω X}n the liner wrp MX {Mω : ω X}, where is n ritrry vetor n M is nonsingulr integer mtrix.

4 oring to ( n with the notions ove, the frequeny support of Y (ω n e expresse s Y = M T ( = k N ( X F +πm T k ( k N (M T X F +πk ( In (, we first otin the union of ll the shifte versions of X F long the gri πm T k ( k N, n then pply the liner wrp M T to get the support Y. While in (, X F is wrpe first n then shifte long the retngulr gri πk ( k N. Though ( n ( re equivlent in esriing the sme eimtion proess, it might e more onvenient to use one interprettion thn the other, epening on the prolem. Definition.. We sy support X F llows lisfree M-fol eimtion, if there is no overlp etween ll shifte versions of the support, i.e. (X F +πm T k (X F +πm T n=, k n, (4 or equivlently, (M T X F +πk (M T X F +πn =, k n. (5 Definition.. Furthermore, we sy frequeny support X F n e mximlly eimte y M, if sie from lisfree eimtion, the support of the eimte signl lso overs the whole spetrum (i.e. no hole left. In this se, we nee R = ( X F +πm T k (6 k N = (M T X F +πk (7 k N Strting from the efinition, we n get severl useful properties ut mximl eimtion. Proposition. Suppose X F n e mximlly eimte y M. For ny vetor, the shifte support X F + n lso e mximlly eimte y M. Proposition. For spetrum support X F to e mximlly eimte y M, neessry onition is the re 4π of X F must equl etm, whih is n integer frtion of 4π. Proof. Suppose X F n e mximlly eimte y M. oring to Definition., the π perioilly shifte versions of M T X F o not overlp with eh other n their union ompletely overs the whole spetrum. This implies re(m T X F = re(f et(m re(x F =4π (8 re(x F = 4π et(m Sine M is n integer mtrix, the re of X F must e n integer frtion of 4π. s n importnt orollry of the ove proposition, we now know it is impossile to fin ritilly smple filter nk system hieving the frequeny eomposition of ontourlets shown in Fig.. tully it is esy to verify tht the re of ny iretionl npss region in the figure is π /(4 m n, where m =,,... orrespons to the multiresolution level, while n =,,... orrespons to the numer of iretions in tht level. We n see no mtter wht vlues of m, n re we hoosing, the ftor in the nomintor n never e nelle out n the re is not n integer frtion of 4π. Due to Proposition, these shpes nnot e mximlly eimte y ny integer mtrix. (9

5 ω ( π 8, π 8 ( π 4, π 8 ( 5π 4, π 8 Figure. The pitoril explntion of Proposition. The gry region is the originl prllelogrm support X F, whih n e mximlly eimte y mtrix M. X F is split into two piees. One piee is move upwrs y shift of ( π, π 8 8 T, n the other is move ownwrs y shift of ( π, π 8 8 T. Sine the shift stisfies the onition in Proposition, the new split-n-shifte support n still e mximlly eimte y M. This prompts us to onsier the frequeny ivision for CRISP-ontourlets s shown in Fig.. Here, eh iretionl frequeny region of the ontourlet trnsform is further prtitione into smller ones, eh with n re s the integer frtion of 4π. Sine Proposition is just neessry onition, we still nee to fin out whether the frequeny shpes in the CRISP-ontourlets n relly e mximlly eimte. We will show this y iretly fining the suitle mximl eimtion mtrix M. The following proposition will gretly filitte the serh for M. Proposition. Suppose frequeny shpe X F n e mximlly eimte y M. If we ritrrily ivie X F into n piees n shift eh piee vi ifferent vetor +πm T k i,where{k i N } n i= n is onstnt vetor, then the new split n shifte support is lso mximlly eimte y M. We will explin this proposition pitorilly. In Fig., the gry support region is prllelogrm efine s X F = {( / /8 /8 πx, x [, }. Clerly X F n e mximlly eimte y igonl mtrix M = ig(8, 8. In Fig, we ivie X F into two piees n shift them long the gri πm T k (shown s she lines plus onstnt vetor =( π/8,π/8. From Proposition, the vlue of oes not relly mtter n we n just ssume = for simpliity. The ft tht X F n e mximlly eimte y M mens the shifts of X F long the gri o not overlp with eh other n their union overs the whole plne. Now we just put the new split-n-shifte support uner the sme opertion n lerly the shifts of it still o not overlp n their union still overs the whole plne (R. The reer might notie tht the new support is tully one of the iretionl npss regions of CRISPontourlets in Fig., n now we know it n e mximlly eimte y igonl mtrix ig(8, 8. Similrly, we n show the following proposition. Proposition 4. ll the iretionl npss supports in CRISP-ontourlets n e mximlly eimte y

6 ω Q Q Q Q Q DF Q Figure 4. The first two levels of CRISP-ontourlets. lok igrm. They re two se quinunx filter nks. The equivlent frequeny eomposition. The whole spetrum is ivie into 4 iretionl regions. igonl mtries with the following generl forms. ( M (m,n m+n+ v = m+ n ( M (m,n m+ h = m+n+, ( where M (m,n v n M (m,n h orrespon to the priniplly vertil n horizontl regions respetively, while m =,,... orrespons to the multiresolution level, n n =,,... orrespons to the numer of iretions on tht level.. THE CRISP-CONTOURLETS SYSTEM With the ssurne tht the frequeny ivision of CRISP-ontourlets is fesile, the next question is just how to fin iterte filter nks to hieve tht. In this setion, we will go through the etils of the system. First, the eimtion mtries use in the system re s follows. Q = (, Q = (, D = (, D = (, P = (, where Q n Q re the quinunx mtries, while D, D n P re mtries with retngulr smpling ltties. In ition, we will lso use the following two resmpling mtries ( ( R =, n H =. The first two levels of CRISP-ontourlets, lle DF, re the sme s those of the iretionl filter nk in, ontourlets, using sing of quinunx filter nks. Fig. 4 shows the lok igrm. In the frequeny omin, eh output orrespons to one of the four iretionl regions in Fig. 4. Thnks to the symmetry mong the outputs, we n fous our following isussion on just one of them (n. The gry region in Fig. 5 orrespons to the iretionl n in the support of the input signl. fter going through DF, it is mppe to the spetrum of output in the wy s shown in Fig. 5. We n see the support is tully oule without hnging of shpe, sine the equivlent eimtion mtrix M = Q Q = ig(,. On the thir level, we use new filter nk lle the shere-hekeror (SC s shown in Fig. 6 n Fig. 6. We n see from Fig. 6( n Fig. 6( tht the input spetrum is effetively ivie into two

7 Figure 5. The frequeny mpping of DF. The gry region shows one of the iretionl ns in the spetrum of the input signl. fter DF, the frequeny is mppe to the spetrum of output. The support is simply oule without hnging of shpe. ω ω L SC D D π/ ω 4π/ ( ( Figure 6. The shere-hekeror (SC filter nk. lok igrm. The spetrum support of the iel filter. ( The spetrum mpping t output. ( The spetrum mpping t output. groups, with the lowpss n (region n one iretionl highpss (region ppering t output rnh, n two other iretionl highpss (regions ( n ( t output rnh. Now the next step is just to further seprte region from t rnh n seprte region ( from ( t rnh. For the former one, it is ler tht we n use the prllelogrm filter nk struture s shown in Fig. 7 n Fig. 7. The two resmpling mtries R n H re introue to get the esirle frequeny mpping for possile sing. Note tht they will not ffet perfet reonstrution, sine their effet n e nelle out t the reonstrution filter nks y using their inverse mtries, whih is lso unimoulr. The support region omes out t the lowpss rnh, while region omes out t the highpss rnh, with the etile mpping shown in Fig. 7( n Fig. 7( respetively. Compring Fig. 5 with Fig. 7(, we n see tht region is enlrge y times ut still remins the originl shpe. tully this is the key to multiresolution, sine now we n hieve extly the sme iretionl frequeny eomposition on the lower ns, y tthing the SC filter nk fter output L n repeting ll the proesses ove. To seprte region ( from (, we n use the shere-prllelogrm (SPR filter nk (Fig. 8 n Fig. 8. gin, the frequeny mppings in oth rnhes re shown in Fig. 8( n Fig. 8(. From the figure, we fin tht region ( n e further ivie (long the she line y using the SPR filter nk, while region ( n e further ivie (long the she line y using the PR filter nk. tully it n e shown vi inution tht ritrry multiresolution n multiiretionl ( n frequeny eomposition n e hieve y n itertive sing of the ove filter nks, oring to the following expnsion rule:

8 ω (pi,pi] ω PR P P H R L ( pi, pi] ( ( Figure 7. The prllelogrm (PR filter nk. lok igrm. The spetrum support of the iel filter. ( The spetrum mpping t the lowpss rnh. ( The spetrum mpping t the highpss rnh. SPR D D π π ω ω ( ( Figure 8. The shere-prllelogrm (SPR filter nk. lok igrm. The spetrum support of the iel filter. ( The spetrum mpping t the lowpss rnh. ( The spetrum mpping t the highpss rnh.. For multiresolution, the type L output is followe y the SC filter nk, whih in turn genertes one type n one type output.. For multiiretion, the type output is followe y the PR filter nk, whih in turn genertes two type outputs.. lso for multiiretion, the type output is followe y the SPR filter nk, whih in turn genertes one type n one type output. s n exmple to show how these rules n e pplie, we present in Fig. 9 the lok igrm of the system tht hieves the frequeny eomposition in Fig.. gin, for simpliity we only give the etils fter one of the 4 iretionl rnhes. ll other rnhes n e simply otine vi symmetry. 4. IMPLEMENTTION ISSUES In this setion we riefly isuss the esign of the FIR filters whose spetr pproximte the iel support shpes introue ove. Totlly we nee to esign 4 prototype filters, nmely the fn, the prllelogrm (PR, the shere-prllelogrm (SPR, n the shere-hekeror (SC. In prtie, we will esign the moulte (frequeny shifte versions of them, s shown in Fig.. The first two filters, i.e. fn n prllelogrm, re ommonly use in multiimensionl systems n their esign hs lrey een stuie y severl, 6, 7 uthors. Here, we will just introue the esign of the ltter two filters, i.e. shere-prllelogrm n shere-hekeror, y using the metho propose y Ty n Kingsury. 6 Tht metho is se on trnsformtion of vrile tehnique n n esign liner phse perfet reonstrution FIR filters. The key to

9 Itertion on Lowpss ns R R L SC Filter nk PR Filter nk 4 6 L SC Filter nk PR Filter nk SPR Filter nk SPR Filter nk DF SPR Filter nk SPR Filter nk 9 PR Filter nk Figure 9. The lok igrm of the CRISP-ontourlets system hieving the frequeny eomposition shown in Fig.. ( ( Figure. The moulte (frequeny shifte versions of the 4 prototype filters. The fn (imon filter. The prllelogrm (PR filter. ( The shere-prllelogrm (SPR filter. ( The shere-hekeror (SC filter. esign filters pproximting the esire support shpe is to get the impulse response of the iel filter. In our se, we hve h SPR (n,n =sin( n π sin(n π n π, ( 6 n h SC (n,n =sin( n ( π ( n sin( n π os((n + n π+ sin((n n π, ( where h SPR (n,n nh SC (n,n re the impulse responses of the iel SPR n SC filters respetively. Finlly, we show in Fig. the -D mgnitue response of the FIR filters we esigne. 5. CONCLUSION We hve presente new ritilly smple iretionl multiresolution imge representtion, lle CRISPontourlets. Using n effiient iterte filter nk struture, it hieves frequeny ivision similr to tht

10 .5.5 Mgnitue.5 Mgnitue F y.5.5 F x F y F x Figure. -D mgnitue response of the esigne FIR filters. The shere-prllelogrm (SPR filter. The shere-hekeror (SC filter. of the originl ontourlet trnsform. While the e feture of nonreunnt representtion mkes it more ttrtive to pplitions like ompression. Inste of performing the multisle n iretionl eomposition steps seprtely s is one in ontourlets, the key ie here is to use omine iterte nonseprle filter nk for oth steps. s the future reserh iretion, we pln to investigte the regulrity issue n the nonliner pproximtion (NL ehvior of the propose system. REFERENCES. M. N. Do, Diretionl Multiresolution Imge Representtions. PhD thesis, Swiss Feerl Institute of Tehnology, Lusnne, Switzerln, Deemer, D. H. Huel n T. N. Wiesel, Reeptive fiels, inoulr intertion n funtionl rhiteture in the t s visul ortex, Journl of Physiology (6, pp. 6 54, Wtson, The ortex trnsform: Rpi omputtion of simulte neurl imges, Computer Vision, Grphis, n Imge Proessing 9(, pp. 7, Olshusen n D. J. Fiel, Emergene of simple-ell reeptive fiel properties y lerning sprse oe for nturl imges, Nture (8, pp , E. J. Cnès n D. L. Donoho, Curvelets suprisingly effetive nonptive representtion for ojets with eges, in Curve n Surfe Fitting,. Cohen, C. Rut, n L. L. Shumker, es., Vnerilt University Press, (Sint-Mlo, E. J. Cnès n D. L. Donoho, Curvelets, multiresolution representtion, n sling lws, in SPIE Wvelet pplitions in Signl n Imge Proessing VIII,. lroui,. F. Line, n M.. Unser, es., 49,. 7. J. P. ntoine, R. Murenzi, n P. Vnergheynst, Diretionl wvelets revisite: Cuhy wvelets n symmetry etetion in ptterns, pplie n Computtionl Hrmoni nlysis 6, pp. 4 45, F. C.. Fernnes, R. L. vn Spenonk, n C. S. urrus, iretionl, shift-insensitive, lowreunny, wvelet trnsform, in Pro. IEEE Int. Conf. on Imge Pro., (Thessloniki, Greee, Ot. 9. N. Kingsury, Complex wvelets for shift invrint nlysis n filtering of signls, Journl of ppl. n Comput. Hrmoni nlysis, pp. 4 5,.. M. N. Do n M. Vetterli, Contourlets: omputtionl frmework for iretionl multiresolution imge representtion, IEEE Trns. Imge Pro.,. sumitte,

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