Learning in Gibbsian Fields: How Accurate and How Fast Can It Be?

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1 IEEE TRANSACTIONS ON PATTERN ANAYSIS AND MACHINE INTEIGENCE, VO. 24, NO. 7, JUY earg Gbbsa Fels: Hw Accurate a Hw Fast Ca It Be? Sg Chu Zhu a Xuwe u AbstractÐGbbsa fels r Markv ram fels are wely use Bayesa mage aalyss, but learg Gbbs mels s cmputatally expesve. The cmputatal cmplexty s pruce by the recet mmax etrpy (FRAME) mels whch use large eghbrhs a hures f parameters [22]. I ths paper, we preset a cmm framewrk fr learg Gbbs mels. We etfy tw key factrs that eterme the accuracy a spee f learg Gbbs mels: The effcecy f lkelh fucts a the varace apprxmatg partt fucts usg Mte Carl tegrat. We prpse three ew algrthms. I partcular, we are tereste a maxmum satellte lkelh estmatr, whch makes use f a set f precmpute Gbbs mels calle ªsatelltesº t apprxmate lkelh fucts. Ths algrthm ca apprxmately estmate the mmax etrpy mel fr textures secs a HP wrkstat. The perfrmaces f varus learg algrthms are cmpare ur expermets. Iex TermsÐMarkv ram fels, mmax etrpy learg, texture melg, Markv cha Mte Carl, maxmum-lkelh estmate, mprtace samplg. INTRODUCTION æ GIBBSIAN fels r Markv ram fels (MRF) are wely use Bayesa mage aalyss fr characterzg stchastc patters r prr kwlege, but learg accurate Gbbs mels s cmputatally expesve. I the lterature, may learg algrthms are fcuse aut-mels, such as aut-bmal mels [7] a Gaussa Markv ram fels [5] whse parameters ca be cmpute aalytcally. Fr MRF mels bey the aut-mel famles, there are three algrthms. The frst s a maxmum pseulkelh estmatr (MPE) [4]. The sec s a stchastc graet algrthm [8], [2], [22]. The thr s Markv cha Mte Carl maxmum-lkelh estmatr (MCMCME) [2], [3], [9]. The sec a thr meths apprxmate partt fucts by mprtace samplg techquesða tpc stue extesvely the lterature [5], [6], [7], [8]. There are als meths [0], [] that estmate Gbbs parameters wth precmpute ervatves f lg-partt fucts. These algrthms were use prmarly fr learg MRF mels wth par clques, such as Isg mels a Ptts mels. The cmputatal cmplexty s pruce by the recet mmax etrpy mels (FRAME) fr texture [22] whch vlve large eghbrhs (e.g., flters wth ww szes f pxels) a large umber f parameters (e.g., ). Recetly, fur algrthms were prpse t speeup mmax etrpy learg wth mxe success:. A mutema mmax algrthm [6], 2. A varatal meth [2], [], 3. A meth f learg by ffus [9], a 4. A geeralze MPE (Prvate cmmucat wth Y.N. Wu). Beses the ew algrthms, a esemble equvalece therem eables the separat f the mel select prceure frm the parameter learg step [20].. S.C. Zhu s wth the Departmets f Statstcs a Cmputer Scece, Uversty f Calfra at s Ageles, s Ageles, CA E-mal: sczhu@stat.ucla.eu.. X.W. u s wth the Departmet f Cmputer Scece, Flra State Uversty, Tallahassee, F E-mal: lux@cs.fsu.eu. Mauscrpt receve Feb. 2000; revse 26 Feb. 200; accepte 29 Nv Recmmee fr acceptace by C. Buma. Fr frmat btag reprts f ths artcle, please se e-mal t: tpam@cmputer.rg, a referece IEEECS g Number 60. I ths paper, we stuy a cmm statstcal framewrk fr learg the parameters f Gbbs mels wth a emphass cmputatal effcecy. There are tw factrs that eterme the accuracy a spee f these learg algrthms. The frst s the effcecy f the frmulate lkelh fucts measure by the Fsher's frmat. The sec s the apprxmat f partt fucts by mprtace samplg. Ths aalyss leas t three ew learg algrthms:. A maxmum partal lkelh estmatr, 2. A maxmum patch lkelh estmatr, a 3. A maxmum satellte lkelh estmatr. Our emphass wll be the thr algrthm whch apprxmates partt fucts by a set f precmpute referece Gbbs mels a smlar sprt t [0], []. Ths algrthm ca apprxmately cmpute Gbbs mels abut 0 secs a HP wrkstat. The paper s rgaze as fllws: Sect 2 presets a cmm framewrk fr Gbbs learg. Sect 3 presets three ew algrthms. Sect 4 emstrates expermets texture sythess. We cclue wth a scuss Sect 5. 2 EARNING IN GIBBSIAN FIEDSÐA COMMON FRAMEWORK et I be a mage efe a lattce a I ts buary cts. s the eghbrh f. et h I be the feature statstcs f I uer buary cts Fr example, h s a vectr fr the hstgrams f fltere mages [23]. Wthut lss f geeralty, a Gbbs mel s f the fllwg frm (see [22]), p I ; ˆ Z ; exp f <; h I g: I (), s a vectr value parameter crrespg t the Julesz esemble fte mages [20] a t s varat t the szes a shapes f. S, ca be estmate a arbtrary. I learg Gbbs mels, we are gve a bserve mage I bs, where may have may scecte cmpets t accut fr multple bservats. Equally, we may break t smaller patches I bs ;ˆ ; 2;...;M lattces f arbtrary shapes a szes. These patches may verlap wth each ther. The, s leare by maxmzg a lg-lkelh, ˆ arg max G ; wth G ˆXM ˆ lg p I bs ; : 2 We shw that exstg Gbbs learg algrthms are ufe as M-estmatrs a they ffer the fllwg tw chces. Chce. The umber, szes, a shapes f the fregru patches ;ˆ ;...;M. Fg. splays fur typcal chces fr. The brght pxels are the fregru ;ˆ ; 2;...;M, whch are surrue by ark pxels the backgru ;ˆ ; 2;...;M. I the frst three cases, are square patches wth m m pxels. I e extreme, Fg. a chses e largest patch ete by,.e., M ˆ a m ˆ N 2w wth w beg the wth f the buary. G s calle the lg-lkelh, a t s apte by the stchastc graet [2], [22] a MCMCME [2], [3], [9] meths. I the ther extreme, Fg. c chses the mmum patch sze m ˆ a G s calle the lgpseulkelh, use the maxmum pseulkelh estmat (MPE) [4]. Fg. b s a example betwee the tw extremes a G s calle the lg-patch-lkelh. I the furth case, Fg. chses ly e (M ˆ ) rregular-shape patch, ete by, where s a set f ramly selecte pxels wth the rest f the pxels beg the a G s calle the lg-partal-lkelh. I Fgs. b a c, a fregru pxel may serve as backgru /02/$7.00 ß 2002 IEEE

2 002 IEEE TRANSACTIONS ON PATTERN ANAYSIS AND MACHINE INTEIGENCE, VO. 24, NO. 7, JUY 2002 Fg.. Varus chces f, ˆ ; 2;...;M. The brght pxels are fregru whch are surrue by ark backgru pxels. (a) kelh, (b) patch lkelh (r satellte lkelh), (c) pseulkelh, a () partal lkelh. fferet patches. It s straghtfrwar t prve that maxmzg G leas t a csstet estmatr fr all fur chces [4]. The flexblty f lkelh fuct stgushes Gbbs learg frm the prblem f estmatg partt fucts [5], [6], [7]. The latter cmputes the ªpressureº a large lattce rer t vercme buary effects. Chce 2. The referece mels use fr estmatg the partt fucts. Fr a chse fregru a lg-lkelh fuct, the sec step s t apprxmate the partt fucts Z I bs ; fr each ;ˆ ;...;M by Mte Carl tegrat usg a referece mel at. Z I bs ; Z ˆ exp <; h I I ; Z I sy j ˆ Z ; Z I bs ; X jˆ exp < ; h I exp < ; h I sy j jibs p I ; : ;jˆ ; 2;...; are typcal samples frm the referece mel p I ; fr each patch ˆ ; 2;...;M. Sce Z ; ;ˆ ; 2;...M are epeet f, we ca maxmze the estmate lg-lkelh G by graet escet. Ths leas t t ˆ XM ˆ jˆ 3 ( X )! j h I sy j jibs h I bs : 4! j s the weght fr sample I sy j, exp < ; h I sy j jibs! j ˆ P : j 0ˆ exp < ; h I sy j 0 jibs The select f the referece mels p I ; epes the szes f the patches ;ˆ ;...;M. I geeral, mprtace samplg s ly val whe the tw strbuts p I ; a p I ; heavly verlap. I e extreme case m ˆ, the MPE meth [4] selects ˆ 0 a p I ; a ufrm strbut. I ths case, Z I bs ; ca be cmpute exactly. I the ther extreme case fr a large fregru m ˆ N 2w, the stchastc graet a the MCMCME meths have t chse ˆ rer t bta sesble apprxmats. Thus, bth meths must sample p I; teratvely startg frm 0 ˆ 0. Ths s the algrthm apte learg the FRAME mels [22]. T summarze, Fg. 2 llustrates tw factrs that eterme the accuracy a spee f learg. These curves are verfe thrugh expermets Sect 4 (see Fg. 7). The hrztal axs s the sze f a vual fregru lattce j j.. The varaces f ME r verse Fsher frmat. et ^ I bs be the estmatr maxmzg G a let be the ptmal slut. The ashe curve Fg. 2 llustrates the varace ^ I bs 2 ; E f where f I s a uerlyg strbut represetg the Julesz esemble. Fr chces shw Fg., f we fx the ttal umber f fregru pxels P M ˆ j j, the the varace (r estmat errr) ecreases as the patch sze (ameter f the hle) creases. 2. The varace f estmatg Z by Mte Carl tegrat E p ^Z Z 2 Š. Fr a gve referece mel ˆ ;ˆ ; 2;...;k (see sl curves Fg. 2), ths estmat errr creases wth the lattce szes. Therefre, fr very large patches, such as m ˆ 200, we must cstruct a sequece f referece mels t apprach, 0 ˆ 0!! 2!...;! k! : Ths s the majr reas why the stchastc graet algrthm was s slw FRAME [22]. 3 THREE NEW AGORITHMS The aalyss the prevus sect leas t three ew algrthms by selectg lkelhs that trae-ff betwee the tw factrs a the thr algrthm mprves accuracy by precmpute referece mels. Algrthm : Maxmzg partal lkelh. We chse a lattce shw Fg. by chsg at ram a certa Fg. 2. Estmat varaces fr varus selects f patch szes m m a referece mels. The ashe curve shws the verse Fsher's frmat whch ecreases as m m creases. The sl curves shw the varaces the mprtace samplg fr a sequece f mels apprachg.

3 IEEE TRANSACTIONS ON PATTERN ANAYSIS AND MACHINE INTEIGENCE, VO. 24, NO. 7, JUY Fg. 3. The shaw areas aru llustrate the varace f the estmate r effcecy f the lg-lkelh fucts. (a) Stchastc graet a Algrthms a 2 geerate a sequece f satelltes le t apprach clsely, m ca be small r large. (b) The maxmum satellte lkelh estmatr uses a geeral set f satelltes cmpute ffle a ca be upate cremetally. Ths ca be use fr small sze m. (c) MPE uses a sgle satellte: ˆ 0. percetage (say, 30 percet) f pxels as fregru a the rest are treate as backgru. We efe a lg-partal-lkelh G ˆlg p I bs ; : Maxmzg G by graet escet, we upate teratvely. t ˆ E p I ; h h I h I bs : 5 Ths algrthm fllws the same prceure as the rgal meth FRAME [22]. It traes ff betwee accuracy a spee a better way tha the rgal algrthm FRAME [22]. The lgpartal-lkelh has lwer Fsher frmat tha the lg-lkelh; hwever, ur expermets emstrate that t s abut 25 tmes faster tha the rgal mmax learg meth wthut lsg much accuracy. We bserve that the reas fr ths speeup s that the rgal samplg meth [22] spes a majr prt f ts tme sytheszg I sy uer ªtypcalº buary cts startg wth whte se mages. I ctrast, the ew algrthm wrks typcal buary ct I bs where the prbablty mass f the Gbbs mel p I; s fcuse. The spee appears t be ece by the ameter f the fregru lattce measure by the maxmum crcle that ca ft the fregru lattce. Algrthm 2. Maxmzg patch lkelh. Algrthm 2 chses a set f M verlappg patches frm I bs a ªgsº a hle each patch, as Fg. b shws. Thus, we efe a patch lg-lkelh G 2 ˆXM ˆ lg p I bs ; : Maxmzg G 2 by graet escet, we upate teratvely as Algrthm es. t ˆ XM ˆ XM h I sy = ˆ h I bs = : 6 I cmpars wth Algrthm, the ameters f the lattces are evely ctrlle. Algrthm has smlar perfrmace as Algrthm. Algrthm 3. Maxmzg satellte lkelh. Bth Algrthms a 2 stll ee t sythesze mages le, whch s a cmputatally tesve task. Nw, we prpse a thr algrthm whch may apprxmately cmpute the spee f a few secs wthut sytheszg mages le. We select a set f referece mels the expetal famly t whch the Gbbs mel p I; belgs, Rˆ p I; j : j 2 ;jˆ ; 2;...;s: We sample (r sythesze) e large typcal mage I sy j p I; j fr each referece mel ffle. These referece mels estmate frm fferet ªvewg agles.º By aalgy t the glbal pstg system, we call the referece mels the ªsatelltes.º The lg-satellte-lkelh s efe as G 3 ˆG 3 ; G 2 3 ; 2 G s 3 ; s ; 7 where each satellte ctrbutes e lg-lkelh apprxmat, G j 3 ; j ˆXM ˆ lg ^Z j exp <; h I bs : 8 Fllwg the mprtace samplg meth (3), we estmate Z I bs ; by Mte Carl tegrat. ^Z j ˆ Z Ibs ; j X `ˆ exp < j ; h I sy j` jibs : 9 Ntce that, fr every hle a fr every referece mel p I; j, we have a set f sythesze patches I sy j` t fll the hle: H sy j ˆ I sy ; ` ˆ ; 2;...;;8; j : There are tw ways fr geeratg H sy j.. Samplg I sy j` j` p I ; j Ðusg the ctal strbut. Ths s expesve a has t be cmpute le. 2. Samplg I sy j` p I ; j Ðusg the margal strbut. I practce, ths s just t fll the hles wth ramly selecte patches frm the sythesze mage I sy j cmpute ffle. I ur expermets, we tre bth cases a we fu that ffereces are very lttle fr mle szes m m lattces, say 4 m 3. Maxmzg G 3 by graet ascet, we have, t ˆ Xs jˆ ( X M X )! j h I sy j` jibs h I bs ˆ `ˆ! j s the weght fr sample I sy j`, exp < j ; h I sy j` jibs! j` ˆ P : `0ˆ exp < j ; h I sy j`0 jibs 0 Equat (0) cverges the spee f secs fr a average texture mel. Hwever, we shul be aware f the rsk that the lg-satelltelkelh G 3 may t be upper bue. It s almst surely t upper bue fr the MCMCME meth. Ths case ccurs whe h I bs cat be escrbe by a lear cmbat f the statstcs f the sample patches P `ˆ! jh I sy j` jibs. Whe ths ccurs, es t cverge. We hale ths prblem by clug the bserve patch I bs H sy j ; therefre, the satellte lkelh s always upper bue. Itutvely, let I sy j ˆ Ibs, s leare s that the ctal prbabltes! j! a! j`! 0; 8` 6ˆ.

4 004 Fg. 4. Sythesze texture mages usg IEEE TRANSACTIONS ON PATTERN ANAYSIS AND MACHINE INTEIGENCE, VO. 24, NO. 7, JUY 2002 leare frm varus algrthms. Fr each clum frm left t rght: : stchastc graet algrthm as the gru truth, 2: pseulkelh, 3: satellte lkelh, 4: patch lkelh, 5: partal lkelh. Sce ` s fte very large, say ` 20, ag e extra sample wll t srt the sample set. T summarze, we cmpare exstg algrthms a the ewly prpse algrthms frm the perspectve f estmatg, a ve them t three grups. Fg. 3 llustrates the cmpars where the ellpse stas fr the space a each Gbbs mel s represete by a sgle pt. Grup. As Fg. 3a llustrates, the maxmum lkelh estmatrs (clug stchastc graet a MCMCME) a the maxmum partal/patch lkelh estmatrs geerate a sample a sequece f ªsatelltesº 0 ; ;... ; k le. These satelltes get clser a clser t (suppse truth). The shaw area aru represets the ucertaty cmputg, whse sze ca be measure by the Fsher frmat. Grup 2. As Fg. 3c shws, the maxmum pseulkelh estmatr uses a ufrm mel 0 as a ªsatellteº t estmate ay mel a, thus, has large varace. Grup 3. The maxmum satellte lkelh estmatrs Fg. 3b use a geeral set f satelltes whch are precmpute a sample ffle. T save tme, e may select a small subset f satelltes fr cmputg a gve mel. Oe ca chse satelltes base the bs. The smaller the ffereces are, the ffereces h Isy j a h I clser the satellte s t the estmate mel a, thus, better apprxmat. Ather crter s that these satellte shul be strbute evely aru t bta g estmat. 4 EXPERIMENTS I ths sect, we evaluate the perfrmace f varus algrthms the ctext f learg Gbbs mels fr textures. We use 2 flters clug a testy flter, tw graet flters, three aplaca f Gaussa flters, a sx Gabr flters at a fxe scale a fferet retats. Thus, h I clues 2 hstgrams f flter respses a each hstgram has 2 bs. S, has 2 free parameters. We chse 5 atural texture mages. Fr each texture, we use the stchastc graet algrthm [22] t lear whch s treate as gru truth fr cmpars. I ths way, we als btae 5 satelltes wth 5 sythesze mages Isy cmpute ffle. j

5 IEEE TRANSACTIONS ON PATTERN ANAYSIS AND MACHINE INTEIGENCE, VO. 24, NO. 7, JUY Fg. 5. Perfrmace evaluat f the satellte algrthm. (a) Observe texture mage. (b) Sythesze mage usg leare wthut buary cts. (c) Sythesze mage usg leare wth buary cts. () Sythesze mage usg leare by stchastc graet. Expermet. Cmpars f fve algrthms. I the frst expermet, we cmpare the perfrmace f fve algrthms texture sythess. Fg. 4 emstrates 6 texture patters f pxels. Fr each rw, the frst clum s the sythesze mage (gru truth) usg a stchastc graet meth use the FRAME mel [22], the ther fur mages are, respectvely, sythesze mages usg maxmum pseulkelh, maxmum satellte lkelh, maxmum patch lkelh, a maxmum partal lkelh. Fr the last three algrthms, we fxe the ttal umber f fregru pxels t 5; 000. The patch sze s fxe t 5 5 pxels fr patch lkelhs a satellte lkelhs. We select 5 satelltes ut f the rest f the 4 precmpute mels fr each texture. Sce fr fferet textures the mel p I; may be mre sestve t sme elemets f (such as tal bs) tha t the rest f the parameters a the vectrs are hghly crrelate betwee ts cmpets, t s t very meagful t cmpare the accuracy f the leare usg a errr measure j j. Istea, we sample each leare mel I sy p I; a cmpare the hstgram errrs f the sythesze mage agast the bserve,.e., jh I sy h I bs j, summe ver 2 pars f hstgrams each beg rmalze t. The table belw shws the errrs fr each algrthms fr the sythesze mages Fg. 4. The umbers are subject t sme cmputatal fluctuats clug the samplg prcess. The expermetal results shw that the fur algrthms wrk reasably well. I cmpars, the satellte meth s fte clse t the patch a partal lkelh meths. Thugh t smetmes may yel slghtly better results tha ther meths epeg the smlarty betwee the referece mels a the mel t be leare. The pseulkelh meth ca als capture sme large mage features. I partcular, t wrks well fr textures f stchastc ature. Fr example, the three textures Fgs. 4, 4e, a 4f. I terms f cmputatal cmplexty, the satellte algrthm s the fastest, a t cmputes the rer f 0 secs a HP-wrkstat. The sec fastest s the pseulkelh. It takes a few mutes. Hwever, the pseulkelh meth csumes a large amut f memry, as t ees t remember all the k hstgrams fr the g gray levels N N pxels. The space cmplexty s O N 2 g k B wth B beg the umber f bs. It fte ees mre tha e Ggabyte f memry. The partal lkelh a patch lkelh are very smlar t the stchastc graet algrthm [22]. Sce the tal buary ct s typcal, these tw estmatrs take ly, geeral, /0th f the umber f sweeps t cvergece. I at, ly a prt f pxels ee t be sythesze, whch ca save further cmputat. The cmputat tme s ly abut /20th f the stchastc graet algrthm. Expermet 2. Aalyss f the maxmum satellte lkelh estmatr. I the sec expermet, we stuy hw the perfrmace f the satellte algrthm s fluece by ) the buary ct, a 2) the sze f patch m m.. Ifluece f buary cts. Fg. 5a splays a texture mage as I bs. We ru three algrthms fr cmpars. Fg. 5 s a result frm the FRAME (stchastc graet meth). Fgs. 5b a 5c are results usg the satellte algrthms. The fferece s that Fg. 5c uses bserve buary ct fr each patch a es le samplg, whle Fg. 5b gres the buary ct. Fr all the fllwg results f satellte lkelh meth (Algrthm 3), H sy j are geerate frm the margal prbabltes wthut le samplg. 2. Iflueces f the hle sze m m. We fx the ttal umber f fregru pxels P j j a stuy the perfrmace f satellte algrthm wth fferece hle szes m. Fgs. 6a, 6b, a 6c shw three sythesze mages usg leare by satellte algrthm wth fferet hle szes m ˆ 2; 6; 9, respectvely. It s clear frm Fgs. 6a, 6b, a 6c that the hle sze wth 6 6 pxels gves better result. T expla why the hle sze f m ˆ 6 gves better satellte apprxmat, we cmpute the tw key factrs that eterme perfrmace. Fg. 7a shws the umerc results crrespece t the theretcal aalyss splaye Fg. 2. Whe the hle sze s small, the partt fuct ca be estmate accurately as shw by the small E p ^Z Z 2 Š sl, ash-tte, a ashe curves Fg. 7. Hwever, the varace E f ^ 2 Š s large fr small hles, whch s shw by the tte curve Fg. 7a. The ptmal chce f the hle sze thus s apprxmately the tersect f the tw curves. Sce the referece mels that we use are clse t the ash-tte le shw Fg. 7a, we prect ptmal hle sze s betwee 5 5 a 6 6. Fg. 7b shws the average errr betwee the statstcs f sythesze mage I sy p I; a the bserve statstcs err ˆ 2 jh Ibs h I sy j, where s leare usg the satellte meth fr m ˆ ; 2;...; 9. Here, the hle sze f 6 6 pxels gves better result. 5 CONCUSION T cclue ur stuy, we qualtatvely cmpare 0 Gbbs learg algrthms Fg. 8 alg three factrs (r mess): Fg. 6. Sythesze mages usg leare by the satellte meth wth fferet hle szes. (a) m ˆ 2. (b)m ˆ 6. (c) m ˆ 9.

6 006 IEEE TRANSACTIONS ON PATTERN ANAYSIS AND MACHINE INTEIGENCE, VO. 24, NO. 7, JUY 2002 Fg. 8. A cmm framewrk fr learg Gbbs mels. The hrztal axs s the sze f fregru patches whch s prprtal t Fsher's frmat. The vertcal axs s the accuracy estmatg lg Z. The brghtess f the ellpses mples the learg spee, a arker s slwer. Ths fgure s tee ly fr a qualtatve cmpars. Fg. 7. The x-axes are the hle sze m 2. (a) Dtte curve s E f ^ 2 Š pltte agast the hle sze m 2. The sl, ash-tte, a ashe curves are E p ^Z Z 2 Š fr three fferet referece mels. (b) Average sythess errr per flter wth respect t the hle sze m 2.. Accuracy apprxmatg lg Z. 2. The ameter f fregru lattces a, thus, effcecy f the lkelh. 3. Cmputatal cmplexty. The 0 algrthms are:. Stchastc graet ME [22], 2. Maxmum pseulkelh (MPE) [3], [4], 3. MCMCME [2], [3], [9], [6], 4. Maxmum patch lkelh, 5. Maxmum partal lkelh, 6. Maxmum satellte lkelh, 7. Mutema mmax [6], 8. Varatal meth [2], [], 9. earg by ffus [2], 0. Geeralze maxmum pseulkelh (Y.N. Wu, prvate cmmucat). ACKNOWEDGMENTS The prject was supprte by a US Natal Scece Fuat grat NSF a a US Natal Scece Fuat Career awar IIS S.C. Zhu wul lke t thak Yga Wu fr valuable scusss. A earler vers f ths paper appear the Prceegs f Cmputer Vs a Patter Recgt, REFERENCES [] M.P. Almea a B. Gas, ªA Varatal Meth fr Estmatg the Parameters f MRF frm Cmplete a Icmplete Data,º The Aals f Apple Statstcs, vl. 3, pp , 993. [2] C.H. Aers a W.D. ager, ªStatstcal Mels f Image Texture,º Upublshe preprt, Washgt Uv., St us, M., 996. [3] J. Besag, ªSpatal Iteract a the Statstcal Aalyss f attce Systems (wth scuss),º J. Ryal Statstcal Sc. B, vl. 36, pp , 973. [4] J. Besag, ªEffcecy f Pseu-kelh Estmat fr Smple Gaussa Fels, º Bmetrka, vl. 64, pp , 977. [5] R. Chellappa a A.K. Ja, Markv Ram Fels: Thery a Applcats. Acaemc Press, 993. [6] J. Cughla a A.. Yulle, ªMutemax: A Fast Apprxmat fr Mmax earg,º Prc. Neural Ifrmat Prcessg Systems, 998. [7] G.R. Crss a A.K. Ja, ªMarkv Ram Fel Texture Mels,º IEEE Tras. Patter Aalyss a Mache Itellgece, vl. 5, pp , 983. [8] H. Der a H. Elltt, ªMelg a Segmetat f Nsy a Texture Images Usg Gbbs Ram Fels,º IEEE Tras. Patter Aalyss a Mache Itellgece, vl. 9,., pp , 987. [9] X. Descmbes, R. Mrrs, J. Zeruba, a M. Berth, ªMaxmum kelh Estmat f Markv Ram Fel Parameters Usg Markv Cha Mte Carl Algrthms,º Prc. It'l Cf. Eergy Mmzat Meths Cmputer Vs a Patter Recgt, May 997. [0] S. Gema a D. McClure, ºBayesa Images Aalyss: A Applcat t Sgle Pht Emss Tmgraphy,º Prc. Statstcal Cmputer Sect Am. Statstcal Assc., pp. 2-8, 985. [] S. Gema a D. McClure, ºStatstcal Meths fr Tmgraphc Image Recstruct,º Bull. It'l Statstcal Ist., vl. II-4, pp. 5-2, 987. [2] C.J. Geyer a E.A. Thmps, ªCstrae Mte Carl Maxmum kelh fr Depeet Data,º J. Ryal Statstcal Sc. B, vl. 54, pp , 992. [3] C.J. Geyer, ªO the Cvergece f Mte Carl Maxmum kelh Calculats,º J. Ryal Statstcal Sc. B, vl. 56, pp , 994. [4] B. Gas, ªCsstecy f Maxmum kelh a Pseu-kelh Estmatrs fr Gbbs Dstrbuts,º Stchastc Dfferetal Systems, Stchastc Ctrl Thery a Applcats, W. Flemg a P.. s, es., New Yrk: Sprger, 988. [5] M. Jerrum a A. Sclar, ªPlymal-Tme Apprxmat Algrthms fr the Isg Mel,º SIAM J. Cmputg, vl. 22, pp , 993. [6] G.G. Ptamas a J.K. Gutsas, ªPartt Fuct Estmat f Gbbs Ram Fel Images Usg Mte Carl Smulats,º IEEE Tras. Ifrmat Thery, vl. 39, pp , 993. [7] G.G. Ptamas a J. Gutsas, ªStchastc Apprxmat Algrthms fr Partt Fuct Estmat f Gbbs Ram Fels,º IEEE Tras. Ifrmat Thery, vl. 43, pp , 997. [8] H. Rbbs a S. Mr, ªA Stchastc Apprxmat Meth,º Aals Math. Statstcs, vl. 22, pp , 95. [9] J. Shah, ªMmax Etrpy a earg by Dffus,º Prc. Cmputer Vs a Patter Recgt, 998. [20] Y.N. Wu, S.C. Zhu, a X.W. u, ªEquvalece f Gbbs a Julesz Esembles,º Prc. It'l Cf. Cmputer Vs, 999. [2]. Yues, ªEstmat a Aealg fr Gbbsa Fels,º Aales e l'isttut Her Pcare, Sect B, Calcul es Prbabltes et Statstque, vl. 24, pp , 988. [22] S.C. Zhu, Y.N. Wu, a D.B. Mumfr, ªMmax Etrpy Prcple a Its Applcat t Texture Melg,º Neural Cmputat, vl. 9,. 8, Nv [23] S.C. Zhu, X.W. u, a Y.N. Wu, ªExplrg Julesz Esembles by Effcet Markv Cha Mte Carl,º IEEE Tras. Patter Aalyss a Mache Itellgece, vl. 22,. 6, pp , Jue 2000.