PARAMETER OPTIMIZATION FOR ACTIVE SHAPE MODELS. Contact:

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1 PARAMEER OPIMIZAION FOR ACIVE SHAPE MODELS Chu Che * Mg Zhao Sa Z.L Jaju Bu School of Compuer Scece ad echology, Zhejag Uvery, Hagzhou, Cha Mcroof Reearch Cha, Bejg Sgma Ceer, Bejg, Cha Coac: chec@zju.edu.c ABSRAC Acve Shape Model () a powerful acal ool for eracg objec, e.g. face, from mage. I compoed of wo par: model ad earch. I, hee wo par are reaed eparaely. Fr, model raed. he, earch performed ug h model. However, we fd ha hee wo par are cloely errelaed. he performace of deped o boh of hem. Improveme o oe of hem doe o coequeally mprove he overall performace, for may wore he oher. I h paper, we fd he key parameer ha relae hee wo par: ubpace eplaao proporo. By opmzg ubpace eplaao proporo, he overall performace of ca mprove by a perceage of abou our eperme. Furhermore, h paper propoe o decompoe he overall error o model ubpace oruco error ad earch error, provg ha he quare of he ubpace oruco error learly relaed wh he ubpace eplaao proporo ad fdg ha he quare of he earch error a pecewe fuco of he eplaao proporo. h decompoo a ew mehod for furher aaly ad poble mproveme. Baed o h decompoo, we propoe a mehod o emae he opmal eplaao proporo. Eperme how ha he emao afacory.. INRODUCION Accurae algme of face very mpora for eraco of good facal feaure for ucce of applcao uch a face ogo, epreo aaly ad face amao. Eeve reearch ha bee coduced he pa year. Ka e al [] roduced Acve Coour Model, a eergy mmzao approach for hape algme. Krby ad Srovch [] decrbed acal modelg of grey-level appearace bu dd o addre face varably. Wko e al [3] ued Gabor Wavele o geerae a daa rucure amed Elac Buch Graph o locae facal feaure. Acve Shape Model () ad Acve Appearace Model (AAM), propoed by Cooe e al [4][5], are wo popular hape ad appearace model for objec localzao. hey have bee developed ad mproved for year. I [4], he local appearace model, whch repree he local ac aroud each ladmark, effcely fd he be caddae po for each ladmark earchg he mage. he oluo pace coraed by he properly raed global hape model. Baed o he accurae modelg of he local feaure, oba ce reul hape localzao. AAM [5] combe cora o boh hape ad eure characerzao of face appearace. he hape eraced by mmzg he eure oruco error. Accordg o he dffere opmzao crera, perform more accuraely hape localzao whle AAM gve a beer mach o mage eure. I h paper, we wll cocerae o. compoed of wo par: model ad earch, whch are reaed eparaely. Fr, model raed. he, earch performed ug h model. However, we fd ha hee wo par are cloely errelaed. he performace of deped o boh of hem. Improveme o oe of hem doe o coequeally mprove he overall performace, for may wore he oher. Uforuaely, h relaohp ofe egleced by prevou work. Some work aemped o mprove he model [6]; oher aemped o mprove he earch procedure [7][8][9][]. I h paper, model ad earch are codered ogeher. We fr fd he key parameer ha relae hee wo par: ubpace eplaao proporo, a proporo whch he ubpace ca epla of he varace ehbed he rag daa. he performace of boh model ad earch affeced by he ubpace eplaao proporo. he, we decompoe he overall error * h paper uppored by Naoal Naural Scece Foudao of Cha (633).

2 o model ubpace oruco error ad earch error, provg ha he oruco error learly relaed wh he ubpace eplaao proporo ad fdg ha he earch error a pecewe fuco of he eplaao proporo. Wh h decompoo, he overall error become a fuco of he eplaao proporo. So, o mmze he overall error o opmze a parameer, he eplaao proporo. Fally, we propoe a parameer opmzao mehod o fd he opmal eplaao proporo. he re of he paper arraged a follow. he aaly of algorhm decrbed Seco. I Seco 3, we pree he decompoo of overall error. Ad eco 4, emao for opmal eplaao proporo dcued. Epermeal reul are preeed Seco 5 before cocluo are draw Seco 6.. ANALYSIS OF ALGORIHM compoed of wo par: model ad earch. model a acal hape model. I alo called po drbuo model (PDM). model o buld a PCA ubpace o appromae he objec hape pace. earch o ue model o locae he arge objec... Model - Sacal Shape Model echque rele upo each objec or mage rucure beg repreeed by a e of po. he po ca repree he boudary, eral feaure, or eve eeral oe, uch a he ceer of a cocave eco of boudary. Po are placed he ame way o each eample of he rag e of eample of he objec. h doe maually. Oe eample for face how fgure. By eamg he ac of he poo of he labeled po, a Po Drbuo Model derved [4]. he model gve he average poo of he po, ad ha a umber of parameer ha corol he ma model of varao foud he rag e. Fgure. Labeled mage wh 87 ladmark for face. he po from each mage are repreeed a a vecor ad alged o a commo co-ordae frame. Prcple Compoe Aaly [] appled o he alged hape vecor o geerae he model. hree ep are eeded for h ak. ) Compue he mea of he alged hape, where he umber of hape. ) Compue he covarace of he daa S ( )( ) 3) Compue he egevecor, φ ad correpodg egevalue λ λ ). λ of S (ored o ha + Fally, he model ca be wre a: + Φb () where he mea hape vecor, Φ { φ φ φ } coa he egevecor correpodg o he large egevalue, ad b a vecor of hape parameer. For a gve hape, hape parameer b gve by b Φ ( ) () he vecor b defe a e of parameer of a deformable model. By varyg he eleme of b we ca vary he hape, ug he equao (). By applyg lm of he parameer b we eure ha he hape geeraed mlar o hoe he orgal rag daa. he acal hape model a PCA ubpace of he objec hape pace. I h paper, we fd ha he ze of h PCA ubpace model crcal. I ca be repreeed by he umber of egevecor or he ubpace eplaao proporo. he umber of mode (egevecor),, o rea ca be choe everal way: he uual way o chooe o a o epla a gve proporo (e.g. 98%) of he varace ehbed he rag daa. We call h proporo a (ubpace) eplaao proporo. Le λ be he egevalue of he covarace mar of he rag daa. Each egevalue gve he varace of he daa abou he mea he do of he correpodg egevecor. he oal varace he rag daa he um of all he egevaluev λ. We ca he chooe he large egevalue uch ha λ αv, whereα defe he eplaao

3 proporo of he oal varao (for ace,.98 for 98%). Aoher way o chooe o ha he redual erm ca be codered a oe. Ad a alerave approach o chooe eough mode ha he model ca appromae ay rag eample o wh a gve accuracy. No maer whch approach ued, we ca ue he eplaao proporo α λ / V a he characerc of he PCA ubpace model. he hgher he eplaao proporo, he maller he ubpace oruco error. So, he uual way, α choe a hgh a 95%~98%. hu he model ca appromae he objec hape accuraely. he uderlyg aumpo ha f he oruco error maller, he overall error wll be maller oo. Bu reul are o olely decded by model. hey are decded by earch oo. Wha more, earch alo affeced by he eplaao proporo. I eco., we wll ee ha earch embed hape model earchg procedure. So he eplaao proporo α fluece boh model ad earch. Now clear ha model ad earch erac wh each oher by he eplaao proporo. hg become more comple. he uderlyg aumpo for choog α o rgh. he overall error deped o model ad earch. I eco 3, we wll decompoe error o oruco error ad earch error. Uforuaely, h ofe egleced by prevou work. Some work aemped o mprove he model; oher aemped o mprove he earch. I h paper, model ad earch are codered ogeher. We fd ha for he be performace of, he eplaao proporoα ca o be a hgh a 95%~98%. I much lower. I our eperme, he opmal eplaao proporo 7%~75%... Search he earch procedure a erao procedure of wo ep: local appearace machg ad emag of hape parameer. Each me ue local appearace model o fd a ew hape. he updae he hape parameer o be f he ew earch hape [4]. he local appearace model, whch decrbe local mage feaure aroud each ladmark, are modeled a he fr dervave of he ample profle perpedcular o he ladmark coour o reduce he effec of global ey chage. I aumed ha he local model are drbued a a mulvarae Gaua drbuo. Whe earchg ew hape, every hape po ue local appearace model o fd a be machg po eghborhood. All he be machg po form a ew hape. Whe he ew hape foud, probably o a plauble objec hape. So he model ued o raform he ew hape o a plauble objec hape. o do h, equao () ued o projec he ew hape o he ubpace model o ge he hape parameer. hough earch a very comple procedure, we ca hardly fd he mahemac decrpo for. I h paper, we ue he earch error fuco o decrbe performace. 3. DECOMPOSIION OF ERROR A aed eco., overall error affeced by model ad earch. I h eco, we aemp o decompoo overall error o oruco error ad earch error. he oruco error roduced by he model, ad he earch error roduced by earch. We ue E for overall error, E (α ) for oruco error, ad E ( α, γ) for earch error, where α he eplaao Recoruco Error Model Subpace proporo, ad γ repree he earch procedure. A how fgure, he relaohp bewee hem : E E ( α ) + E ( α, γ ) So, o mmze E, we oly eed o mmze ( α ) E ( α, γ ) E +. If we kow he formulao of (α ) E ad E ( α, γ), he work doe. I h paper, we prove ha he quare of oruco error learly relaed wh he eplaao proporo,.e. E ( α ) ( α )V. A how fgure Error Search Error Fgure. he relaohp bewee error, oruco error ad earch error

4 , we ca have where b ad b b he projeco coordae o he hegevecor model ubpace, he umber of egevecor reaed, ad he oal egevecor of he objec hape pace. he we ca ge quare of he oruco error for po E ( ) (α ) b + So he quare of he oal oruco error E ( α ) E( ) ( α ) M M b + + M + M + ( λ ) ( λ ) λ λ ( α ) V b ( b ) ( b ) where M he umber of ample. Fgure 3. he relaohp bewee he quare of error, oruco error, earch error ad he eplaao proporo. Uforuaely, we ca o fd he mahemac formulao bewee he earch error ad eplaao proporo, for he earch a dyamc procedure ad ca hardly be mahemacally decrbed. So we ca fd her relaohp by eperme. We ra he model o mage ad e o oher mage. he we ca ge he quare of overall error ad oruco error. he quare of earch error he dfferece bewee hem. Fgure 3 how he relaohp bewee he hree error ad he eplaao proporo. We ca fd ha he quare of he earch error a pecewe fuco of he eplaao proporo wh wo pece. he fr pece a lear fuco ad he ecod pece a quadrac fuco. 4. ESIMAION FOR OPIMAL EXPLANAION PROPORION A foud eco 3, he quare of he earch error a pecewe fuco of he eplaao proporo wh wo pece. he fr pece a lear fuco ad he ecod pece a quadrac fuco. o emae he opmal eplaao, we eed o emae h pecewe fuco. Bu emao of he quadrac fuco eher robu or effce. So we ry o decreae he order of h fuco. We fr coder he hree error, o he quare of hem. Bu we ca o fd lear relaohp eher. So we eed o fd oher relaohp. We defe a ew earch error where β E ( β, γ ) E E ( β ) λ / λ, he umber of egevecor reaed, ad he oal egevecor of he objec hape pace. he relaohp bewee hee hree error ad β how fgure 4. We ca ee ha alhough oe of hem a eac lear fuco, hey are early lear or pecewe lear. h propery very ueful for he emao of he opmal eplaao proporo. Now he emao mehod farly mple. Fr, we calculae wo or hree value for he hree error ear boh. Secod, we learly f he β ad β oruco error ad pecewe learly f he earch error o ge he oruco error fuco ad earch error fuco. Fally, we mmze he um of hee wo fuco o ge he opmal β. Wh h opmal β, he umber of egevecor he model ca be calculaed ad hu he eplaao proporo here. 5. EXPERIMENS Our daabae coa 4 face mage he FERE daabae, he AR daabae ad oher colleco. 87 ladmark are labeled o each face. We radomly elec mage a he rag ad he oher mage a he eg mage. Mul-reoluo earch ued, ug 4 level wh reoluo of /8, /4, /, of he orgal mage each dmeo. A mo 5 erao are ru a each level. he ue profle model of 9 pel log (4 po o eher de) ad earche pel eher de.

5 Po-Po Error(Pel) 6 4 Recoruco Error New Search Error.5.5 Bea Overall Error Fgure 4. he relaohp bewee error, oruco error, ew earch error ad bea. 5.. Po Locao Accuracy Comparo O each e mage, we alze he arg mea hape wh dplaceme from he rue poo by± pel boh ad y, 9 alzao oal. Wh hee dplaceme, mo of he earch wll coverge o he arge hape. he earchg reul are compared wh he labeled hape. We ue po-o-po error ad po-oboudary error (he dace from he foud po o he aocaed boudary o he labeled hape) a he comparo meaure. he comparo reul are how Fgure 5. We ca ee ha he opmal eplaao proporo.7~.75 for boh meaure. he po-o-po error 4.56 for eplaao proporo 98%, 4.7 for eplaao proporo 95%, 3.46 for eplaao proporo 7.6%, for eplaao proporo 75.8%. he po-o-po accuracy mproveme.8~. pel, wh mproveme rae 8~4%. he po-o-boudary error.6 for eplaao proporo 98%,.47 for eplaao proporo 95%,.38 for eplaao proporo 7.6%,.35 for eplaao proporo 75.8%. he po-o-boudary accuracy mproveme.3~.5 pel, wh mproveme rae 4~9%. Fgure 6 how he perceage of he locaed hape whoe po-o-po error are le ha a gve hrehold wh dffere egevecor (eplaao proporo). Fve kd of egevecor/eplaao proporo are ploed. 5.. Capure Rage Comparo O each e mage, we alze he arg mea hape wh dplaceme from he rue poo by up o ± 3 pel. he a earch performed o aemp o locae he arge hape. Fgure 7 how he po-opo error wh dffere al dplaceme ad dffere egevecor (eplaao proporo). We ca ee ha egevecor 6 (eplaao proporo 7%) he be wh dplaceme of ± pel. Dace(Pel) 6 4 Po-o-Po Error Po-o-Boudary Error Eplaao Proporo(Alpha) Fgure 5. error wh dffere eplaao proporo Frequecy EgeVecor3,Alpha 54%. EgeVecor6,Alpha7% EgeVecor5,Alpha9%. EgeVecor4,Alpha95% EgeVecor65,Alpha98% 5 5 Po-o-Po Error(Pel) Fgure 6. he perceage of locaed hape whoe poo-po error are le ha a hrehold wh dffere egevecor (eplaao proporo). Po-Po Error(Pel) EgeVecor3 EgeVecor5 EgeVecor Ial Dplaceme(Pel) EgeVecor6 EgeVecor4 Fgure 7. he po-o-po error wh dffere al dplaceme ad dffere egevecor Emao for Opmal Eplaao Proporo

6 We ue he propoed mehod eco 4 o emae he opmal eplaao proporo. hree value are calculaed for he hree error ear boh. he value are led able ad β ad β alo refer o fgure 4. If we oly ue β,.8,.98,, he pecewe leary fuco of he ew earch error :.β +.59 ( β ) y 5.7β.98 ( β ) he emaed opmal β value.387 correpodg o he umber of egevecor, he eplaao proporo.8. If we ue all hee value, he pecewe fuco leary of he ew earch error :.5β +.59 ( β ) y 4.9β.7 ( β ) he emaed opmal β value.8 correpodg o he umber of egevecor 5, he eplaao proporo.69. he rue opmal umber of egevecor 6, correpodg o he eplaao.7~.75. So he emaed reul are afacory. Bea Egevecor Recoruco New Search Error Error able. he oruco error ad ew earch error wh dffere egevecor. 6. CONCLUSIONS compoed of wo par: model ad earch, whch are reaed eparaely. Some of he mproveme o are for model, oher are for earch. However, he relaohp bewee hem egleced. h paper fd he key parameer ha relae hee wo par: ubpace eplaao proporo. By opmzg ubpace eplaao proporo, he overall performace of ca mprove by a perceage of abou our eperme. Furhermore, h paper propoe o decompoe he overall error o ubpace oruco error ad earch error, provg ha he quare of he ubpace oruco error learly relaed wh he ubpace eplaao proporo ad fdg ha he quare of he earch error a pecewe fuco of he eplaao proporo. h decompoo a ew mehod for furher aaly ad poble mproveme. Baed o h decompoo, we propoe a mehod o emae he opmal eplaao proporo. Eperme how ha he emao afacory. 7. REFERENCES [] M. Ka, A. Wk, ad D. erzopoulo. Same, Acve coour model. Ieraoal Coferece o Coferece o Compuer Vo, Lodo, Jue 987, pp [] M. Krby ad L. Srovch, Applcao of he karhue-loeve procedure for he characerzao of huma face. IEEE raaco o Paer Aaly ad Mache Iellgece, vol., o., Ja 99, pp. 3-8 [3] Laurez Wko, jea-marc Fellou, Norber Kruger, ad Chroph vo der Malburg, Face Recogo by Elac Graph Machg, Iellge Bomerc echque Fgerpr ad Face Recogo, ed. L.C. Ja e al., publ. CRC Pre, ISBN , Chaper, 999, pp [4].F. Cooe, C.J. aylor, D.H. Cooper, ad J. Graham, Acve Shape Model - her rag ad applcao. Compuer Vo ad Image Uderadg, vol.6, o., Ja 995, pp [5].F. Cooe, G.J. Edward ad C.J. aylor, Acve Appearace Model, IEEE raaco o Paer Aaly ad Mache Iellgece, vol.3, o.6,, pp [6] Shucheg Ya, Ce Lu, Sa Z. L, Hogjag Zhag, Heug-Yeug Shum, Qaheg Cheg, Face algme ug eure-coraed acve hape model, Image ad Vo Compug (3): [7] Ce Lu, Heug-Yeug Shum, Chaghu Zhag, Herarchcal Shape Modelg for Auomac Face Localzao. ECCV () : [8] M. Roger ad J. Graham "Robu Acve Shape Model Search," ECCV (4) : [9] Z. Xue, S.Z. L, E.K. eoh. "AI-EgeSame: A Affe-vara Deformable Coour Model for Objec Machg," Image ad Vo Compug. Vol., No., pp 77-84, Feb. [] Y Zhou, Le Gu, Hog-Jag Zhag, Bayea age Shape Model: Emag Shape ad Poe Parameer va Bayea Iferece, CVPR3.