An Affine Symmetric Approach to Natural Image Compression

Transcription

1 An Affne Symmerc Approach o Naural Image Compresson Heechan Par, Abhr Bhalerao, Graham Marn and Andy C. Yu Deparmen of Compuer Scence, Unversy of Warwc, UK {heechan abhr grm andycyu}@dcs.warwc.ac.u ABSTRACT We approach mage compresson usng an affne symmerc mage represenaon ha explos roaon and scalng as well as he ranslaonal redundancy presen beween mage blocs. I resembles fracal heory n he sense ha a sngle prooypcal bloc s needed o represen oher smlar blocs. Fndng he opmal prooypes s no a rval as parcularly for a naural mage. We propose an effcen echnque ulzng ndependen componen analyss ha resuls n near-opmal prooypcal blocs. A relable affne model esmaon mehod based on Gaussan mxure models and modfed expecaon maxmzaon s presened. For compleeness, a parameer enropy codng sraegy s suggesed ha acheves as low as 0.14 bpp. Ths sudy provdes an neresng approach o mage compresson alhough he reconsrucon qualy s slghly below ha of some oher mehods. However he hgh frequency deals are well-preserved a low braes, mang he echnque poenally useful n low bandwdh moble applcaons. Keywords Affne ransform, Image codng. 1. INTRODUCTION Naural mages conan consderable redundancy and hs s exploed by sae of he ar wavele mage compresson echnques [1]. A recen endency s o decompose he mage no varous dreconal subbands []. In hs conex, explong he affne symmerc redundancy resdng beween blocs, namely ranslaon, scalng and roaon, offers a dfferen vewpon from whch o acle he compresson as. Ths approach resembles fracal heory as only a sngle prooypcal bloc s needed o replcae oher blocs of smlar exure. Hsu and Wlson developed a wo-componen analyss [3], a framewor ha combnes a srucural componen whch denfes he affne ransformaon from bloc o bloc as shown n Fg.I and a sochasc resdual componen. The echnque provdes hgh fdely n he reconsrucon of perodc exure. Recenly, he framewor was exended for affne nvaran segmenaon of naural mages by Bhalerao and Wlson [4]. Whou loss of generaly, a naural mage comprses regons of dfferen exural paern. Classfcaon of blocs no self smlar groups and selecon of he mos represenave blocs have paramoun mpac on he vsual qualy of reconsrucon. They showed ha a cluserng approach s approprae and performs well for a supervsed case[4]. We propose a praccal algorhm based on Independen Componen Analyss(ICA) for an unsupervsed case, resulng n near-opmal prooypcal blocs. We also presen an alernave srucural ransformaon esmaon echnque wh mnmal compuaonal complexy and a parameer enropy codng sraegy. In secon, we brefly revew wo-componen analyss before nroducng classfcaon of blocs n secon 3. In secon 4, we explore srucure ransform esmaon and dscuss an alernave approach. In secon 5, an effcen parameer codng sraegy s descrbed. Fnally, expermenal resuls are repored and conclusons are drawn.. TWO COMPOENENT ANALYSIS The wo-componen framewor [3] combnes a srucural elemen based on he D affne relaonshps beween exure paches and a sochasc elemen desgned o accoun for he unpredcable varaons n he exure. We focus on only he srucural componen. Gven an opmal prooypcal bloc, f and a regon of self-smlary composed of affne symmerc blocs, a arge pach, f s esmaed as f ˆ ( y) f ( T ( y)) T ( y) A y (1) Fgure 1. Affne symmerc exure synhess f f ( y) / ( y) () where T y ( p, q),0 p, q N, T s a D affne ransformaon of he coordnaes of prooype f, and α s an nensy scalng coeffcen. Fg.I llusraes he concep. The Fourer ransform s used o reduce he search space by separang he lnear and ranslaon esmaons whch relae o amplude and phase respecvely. For he lnear par, A, a wo energy cluser model s employed o represen he specra usng angular varance analyss. Ths s conduced Permsson o mae dgal or hard copes of all or par of hs wor for personal or classroom use s graned whou fee provded ha copes are no made or dsrbued for prof or commercal advanage and ha copes bear hs noce and he full caon on he frs page. To copy oherwse, or republsh, o pos on servers or o redsrbue o lss, requres pror specfc permsson and/or a fee. MobMeda 06, Second Inernaonal Moble Mulmeda Communcaons Conference, Sepember 18 0, 006, Alghero, Ialy 006 ACM /06/09 $5.00

2 Fgure. ICA overcomplee bass funcons o dscover angles, θ 1 and θ, whch mnmse he sum of he varances, σ. Readers are referred o [3] for deals. (, ) x x (3) 1 x where μ s he cenrod of a cluser and Λ s a se of coordnaes n he subdvson formed by θ 1 and θ. The correspondng affne ransform s found by algnng he cenrods of a cluser n he prooype and he arge. Translaon, s shown as a phase graden whch s found from he locaon of he pea n he cross-correlaon. 1 ~ ~ max( F f ( x) f ( x)) f T ( f ( x)) (4) The resulng blocs are sched ogeher usng a squared cosne mas, W, of sze B x B, W ( p, q) cos [ p / B]cos [ q / B] (5) 3. CLASSIFICATION AND PROTYTYPE Classfcaon and he denfcaon of approprae prooypcal blocs s an essenal sep o explong affnesymmery n our approach. The mos obvous procedure s o perform an exhausve search. Tha s, o compue he ransformaon error for all combnaons of prooype and arge and hen o choose hose ha mnmse he overall error. Ths leads o he dscovery of he opmal prooype bloc bu s clearly a compuaonally arduous as gven he number of combnaons. The compuaonal burden can be halved under he symmery assumpon: Fgure 3. Classfcaon: a coarse reconsrucon formed from a lmed number of ICA bases and a dfferen number of bases for dfferen blocs Fgure 4. Seleced prooypes and classfcaon map x, y : f ( x, y) f ( y, x) 0 (6) where f(x, y) ransforms x o f y. Noneheless, he compuaonal requremen becomes prohbve as he mage sze ncreases. 3.1 Independen Componen Analyss ICA and s varans have been proposed o provde represenaons ha ulse a se of lnear bass funcons [5], [6]. Wha he approaches have n common s ha hey ry o reduce he nformaon redundancy by capurng he sascal srucure n he mages, beyond second order nformaon. ICA fnds a lnear non-orhogonal coordnae sysem n mulvarae daa, deermned by he hgher-order sascs. ICA produces an overcomplee se of bass funcons, as shown n Fg., whch s well localsed n frequency and orenaon. Ths allows one o he fnd common srucure beween blocs as well as how smlar hey are o each oher. 3. Classfcaon Despe he fac ha ICA denfes common dreconal lnear srucures, s no rval o esmae affne nvarance beween blocs wh only he dscovered bass funcons. However, he number of he dreconal lnear bases n each bloc capured by ICA s affne nvaran feaure as shown n Fg.3. Ths maes sense as s unlely o f one o anoher f he number of dreconal lnear feaures does no mach. The dfferen number of sgnfcan bass funcons for each bloc can be found as w / w c 0 s (7) max where w s he wegh of b, c s a consan, and s s a se of bass funcon ndces sored by w / b where b s he bass funcon. max s he maxmum number of bass funcons for a bloc. Blocs are grouped based on he number of sgnfcan bass funcons n each bloc. Blocs n each group are compared, f hey are no affne nvaran, he group s sub-dvded. Ths can preven dfferen exure blocs wh he same number of dreconal feaures from beng n he same group. A classfcaon map s shown n Fg.4. For each group, a prooypcal bloc s gven by max{ ( w e )} (8) where e s he egenvalue of he nd prncpal componen.

3 Ths resuls n a bloc composed of srong dreconal bases wh hgh weghs. 4. AFFINE TRANSFORM ESTIMATION The wo-componen analyss accuraely esmaes he exure s srucure. However, s ofen he case ha a wocluser model produces an ncorrec esmae parcularly for dreconal exure. Calway addressed hs problem usng a smple merc [7]. I s assumed ha wo clusers are presen when σ 1+ / σ mn < 1 holds. However, he merc provdes a false predcon n some cases, as shown n Fg.5. Bhalerao and Wlson developed an alernave sraegy o overcome hs problem, ulsng he Gaussan mxure model (GMM)[8], [4]. The amplude specrum s modelled as a wo-dmensonal, wo componen Gaussan mxure and approxmaed usng he Levenberg Marquard (LM) algorhm. The LM s a nonlnear opmsaon and requres ha he paral dervave of each parameer s nown. Wh he esmaed Gaussan mxure model, he LM s appled agan o fnd an opmal affne ransformaon appled o he Gaussan mxure ha bes fs he amplude specrum of he arge bloc n a leas square error sense. Alhough hs sraegy overcomes he aforemenoned problem, ncreases he compuaonal requremen by nroducng anoher sep o f he model o he specrum. 4.1 Alernave Mehod 1) Expecaon Maxmsaon for GMM: Expecaon Maxmsaon (EM)[9] s an esmaon algorhm drven by each daa pons probably of membershp o a parcular class. The EM s a favoure n sascs due o s smplcy and fas convergence rae, bu s ap o fall no local opma. Esmaon s acheved by repeang wo seps, an E-sep and M-sep. An example follows wh respec o he Gaussan mxure model. The E-sep compues expeced classes for all daa pons for each class and s gven by p( x w,, c ) w w x,, c ) (9) c p( x w,, c ) w 1 where μ, c and w are he cenre, covaran marx and wegh of each Gaussan model. In he M-sep, he maxmum lelhood s esmaed gven he new membershp dsrbuon for he daa Fgure 5. Fnger prn mage es: (lef) orgnal mage, (mddle) reconsrucon by wo-componen mehod, (rgh) by modfed EM for GMM Fgure 6. Magnude modelng usng EM : (lef o rgh) mage bloc, magnude specrum, Gaussan mxure before EM eraon, and approxmaon afer eraon P w x,, c ) x (10) w x,, c ) c ( w x,, c )( x )( x w x,, c ) T ) (11) w x c,, ) w (1) # daa _ pn ) Combnaon of GMM and Two-Componen Analyss: The cluserng problem n he wo-componen analyss arose from assumng wo clusers model n a sngle cluser specrum [3], leadng o wo cenrods μ, s oo close. These cenrods resul n severe deformaon when algned o cenrods based on wo-cluser daa. A mehod o deermne he effecve number of clusers and whch does no ncrease he compuaonal requremen s a useful sep forward. Our wor adops he GMM esmaon of [4] and he algnmen of cenrods from [3]. The LM s subsued by a modfed EM, n whch he prncple componen of he Gaussan model s ulsed n favour of conrollng s orenaon and breadh. The Gaussan mxure model s used no us o model he amplude specrum bu each consuen Gaussan model represens a dfferen dreconal feaure. 3) Modfed EM for GMM: The Gaussan mxure s nalzed as havng consuen Gaussan models evenly dsrbued beween 0 and π o avod local mnma. Ths s done by assgnng a desgnaed vecor as he prncpal componen of each model. As shown n Fg.6, a sx dreconal Gaussan mxure model s used nally o approxmae he specrum. One componen s fxed as beng non-dreconal CovMarx( 0,0) CovMarx(1,1) (13) wh w = 0.4. Ths s o assure he dreconal componens focus on sgnfcan feaures whou beng dsraced by he scaered hgh frequency daa. As eraon progresses,

4 componens are fxed as addonal bacground componens when her degree of dreconaly, e /e 1 < 0.6, or her weghs are neglgble (w < 0.05), and are excluded n he M-sep. As converges, componens lef non-fxed are assumed o represen dfferen dreconal feaures. These n urn provde an affne ransformaon by algnng he prncpal componens of wo sgnfcan Gaussan models. If more han wo Gaussan models are lef non-fxed, hen he model wh he leas wegh, w, and he one wh he closes Eucldean L dsance o s prncpal vecor X are merged as her medan. Furher eraons are performed unl he number of models becomes equal o he number of componens of he mxure model of he prooypcal bloc. 5. PARAMETER ENTROPY CODING 5.1 Zero-Bass Regon Usng a sngle prooype o represen a se of self-smlar blocs sounds effcen. However, each srucural ransformaon from prooype o arge bloc enals sx affne parameers, a cosly represenaon, parcularly for a homogeneous bloc. The smpler he srucure of he bloc s conens, he less effcen s o use he srucural ransform, n erms of brae. I s requred o dsngush blocs ha are beer represened by he srucural ransform from hose ha are no. We have already solved hs problem from he ICA resul obaned n Fg.4. Blocs conanng only non-dreconal or lowpass flered le bases as gven below are replcaed by nensy scalng only, for whch Eq.() s modfed as E[ y] f b b e e hre hre (14) where E s he mean of he bloc, and α s appled on a un nensy bloc nsead of he prooype. 5. Vecor Quansaon Anoher ssue mang compresson of he ransform parameers dffcul s ha hey are floang pon numbers and sensve o loss of sgnfcan dgs when lmed precson s appled. We encode cenrod coordnaes nsead of affne parameers. Lmed precson can be flexbly appled dependng on he probably of he cenrods. For a 16 x 16 bloc, a grd s gven dependng on he dsance, d from he cenre. f ( 0 < d < 1 7 < d < 8 ) apply double-pel else f ( < d < 4 ) apply half-pel (15) else apply full-pel Each cenrod coordnae par can be convered no a sngle bye when a 16 x 16 bloc s consdered. Therefore, a ransform parameer resuls n hree byes ncludng offse coordnaes, pror o frs-order enropy codng. Frs-order enropy codng can be used as, due o he symmerc Table 1. EXPERIMENTAL RESULTS [PSNR AND BITRATE(BPP)] propery of he Fourer ransform, cenrod neghbourhood redundancy s less lely o exs. 6. EXPERIMENTAL EVALUATION The algorhm was execued on he lowes wo levels of he Laplacan pyramd. A bloc enals sx parameers for he affne ransform, an nensy scale and a prooype d. They are separaely encoded usng arhmec codng. An mage of sze W x H requres W/B x H/B blocs of sze B x B, he lowpass mage of he Laplacan pyramd of sze W/4 x H/4 and N prooypes of sze B x B. The algorhm was esed on several mages of sze 51 x 51 and a bloc sze of 16 x 16. The resuls are shown n Table.1, where each column represens he algorhm wh dfferen opons. A and B are wh prooype selecon by ICA, and C and D are wh manually seleced prooypes. A and C are wh a classfcaon map by ICA and B and D are wh applcaon of all prooypes o he whole mage. E s JPEG000. The op fve rows are wh hree prooypes and he boom fve rows are wh wo prooypes. Manually seleced prooypes may no be opmal choces, however hese were he bes combnaons of hose esed. The mages are beer reconsruced wh manually seleced prooypes han wh ICA based prooypes bu he dfference s only 0.1dB. We beleve ha he dfferences wh he opmal prooype based reconsrucon would be nsgnfcan. The more prooypes he beer he reconsrucon bu aguar and graffe n A and B show he oppose effec. Ths s due o he wea-classfcaon and sub-opmal prooypes. Comparng A wh B and C wh D demonsrae ha he Zero-Bass Regon s an effecve sraegy o reduce brae and ha classfcaon by he number of dreconal feaures

5 s effecve. In general, he rae dsoron performance does no mach ha of JPEG000, however our echnque provdes a sasfyng resul n ha hgh frequency nformaon s preserved. Ths s demonsraed by he reconsrucon of A and D shown n Fg CONCLUSION An alernave mage compresson echnque explong he affne symmerc redundancy presen n naural mages has been dscussed. The man dffculy les n he selecon of prooypcal blocs as naural mages are ofen composed of varous exures. The echnque nvolves classfcaon of blocs no self-smlar groups as well as he selecon of an opmal bloc n each group. A near-opmal soluon s suggesed ha performs ICA based classfcaon by dreconal feaures and whch selecs he mos represenave bloc as he prooype. The reconsrucon qualy dffers by only 0.1dB from hose reconsruced by manually seleced prooypes. A compuaonally effcen srucural componen esmaon echnque s presened, wh an adapve accuracy quansaon sraegy. The wor may be useful n applcaons where dealed exure s preferred o blurred mages a low braes. 8. REFERENCES [1] Taubman, S.D., and Marcelln, M.W. JPEG000: Image Compresson Fundamenals, Sandards and Pracce. Kluwer Academc Publshers,00. [] Sarc, J.L., Candes, E.J., and Donoho, D.L. The curvele ransform for mage denosng. IEEE Trans. on Image Processng, vol. 11, pp , 000. [3] Hsu, T.I. and Wlson, R. A wo-componen model of exure for analyss and synhess. IEEE Trans. on Image Processng, vol. 7, no. 10, pp , Ocober [4] Bhalerao, A., and Wlson, R. Affne nvaran mage segmenaon. BMVC., 004. [5] Hyvarnen, A., Karhunen, J., and Oa, E. Independen Componen Analyss. John Wley, 001. [6] Olshausen, B.A. Learnng sparse, overcomplee represenaons of me varyng naural mages. IEEE ICIP., 003, pp [7] Calway, A.D. Image represenaon based on he affne symmery group. IEEE ICIP., 1996, pp [8] Bhalerao, A., and Wlson, R. Warple: An mage-dependen wavele represenaon. IEEE ICIP., 005, pp [9] Bshop, C. Neural Newors for Paern Recognon. Oxford Unversy Press, 1995.

6 Fgure 7. Affne symmerc exure synhess resuls wh hree prooypes (from op o boom) frog, graffe, lena, barbara, and aguar (lef o rgh) orgnal mage, classfcaon map, reconsrucon by applyng all prooypes o he whole mage(d) and map based reconsrucon(a)