Bayesian Tangent Shape Model: Estimating Shape and Pose Parameters via Bayesian Inference *

Transcription

1 Bayesan angent Shape Model: Estmatng Shape and Pose Paametes va Bayesan Infeence * Y Zhou ** Mcosoft Reseach Asa SMS, Pekng Unv. yzhou@mschna.eseach.mc -osoft.com Le Gu CSD, Canege Mellon Unv. gu+@cs.cmu.edu Hong-Jang Zhang Mcosoft Reseach Asa hjzhang@mcosoft.com Abstact In ths pape we study the poblem of shape analyss and ts applcaton n locatng facal featue ponts on fontal faces. We popose a Bayesan nfeence soluton based on tangent shape appomaton called Bayesan angent Shape Model (BSM). Smlaty tansfom coeffcents and the shape paametes n BSM ae detemned though MAP estmaton. angent shape vecto s teated as the hdden state of the model, and accodngly, an EM based seachng algothm s poposed to mplement the MAP pocedue. he majo esults of ou algothm ae: ) tangent shape s updated by a weghted aveage of two shape vectos, the pojecton of the obseved shape onto tangent space, and the econstucton of shape paametes. ) Shape paametes ae egulazed by multplyng a ato of the nose vaatons, whch s a contnuous functon nstead of a tuncated functon. We dscussed the advantages conveyed by these esults, and demonstate the accuacy and the stablty of the algothm by etensve epements.. Intoducton he geometcal descpton of an object can be decomposed nto two pats: the geometcal tansfom and the shape. A common vson task s to ecove both pose paametes and low-dmensonal epesentatons of the undelyng shape fom obseved mages. hs pocedue s usually efeed as shape analyss o shape egstaton. Shape analyss has been advanced n both the lteatue of statstcs and vson. he statstcal theoy of geneal shape space began wth the wok of Kendall [6] n 977. Kendall descbed shape dstbuton n a Remann manfold whch s hghly cuved and nonlnea. Statstcal * he wok pesented n ths pape s pefomed n Mcosoft Reseach Asa. ** hs autho s patally suppoted by SFC gant # technques wee fst ntoduced to analyze the pobablstc dstbuton of shape n ths manfold. Subsequent developments [7][5][] have led to seveal pactcal statstcal appoaches to analyzng objects usng pobablty dstbutons of shape and lkelhood based nfeence. A compehensve suvey can be found n Small [5]. Geneal shape space has been poved to be hghly nonlnea. Howeve, as fo a set of concentated data, tangent space povdes a good lnea appomaton to geneal shape space. Moe mpotantly, modelng shape n tangent space can convet statstcal shape analyss to standad multvaate analyss [7]. In mage analyss lteatues, pactcal paametc defomable models [3][0][] have been developed to deal wth the poblems lke segmentaton o featue ponts localzaton. hese models ae geneally capable of ncopoatng po knowledge wth obsevatons dectly deved fom mage data. In patcula, Actve Shape Model [3] poposed by Cootes et.al. n 99 attacts a wde ange of attenton. ASM conssts of a pont dstbuton model captung shape vaatons of vald object nstances, and a set of gey gadent dstbuton models, whch descbe local tetue of each landmak pont. Cootes developed an teatve seachng algothm to actvely update the model paametes accodng to the obseved mage. he majo advantage of ASM s that the model can only defom n the ways leant fom the tanng set. hat s, t can accommodate consdeable vaablty and t s stll specfc to the class of object t ntends to epesent. Specfcally, n ASM the pncple component analyss (PCA) technque s used to model both D shape vaatons and local gey level stuctues. In ths pape, we addess the poblem of shape analyss fom two aspects. Fst, shape analyss poblem s fomulated n Bayesan famewok. Specfcally, we descbe the po model of tangent shape vectos, the lkelhood model and the posteo of model paametes. Second, an EM based seachng algothm s gven to estmate tangent shape and othe model paametes. he

2 deved updatng ules hghlght the advantages of BSM shape egstaton he est of the pape s oganzed as follows: we begn wth the descpton of tangent space appomaton and the pobablstc fomulaton of shape egstaton. We descbe the paamete estmaton algothm and compae the updatng ules of ASM seachng and BSM seachng n Secton 3. Secton 4 povdes epemental esults. We dscuss some elated poblems and daw the conclusons and n Secton 5 and 6.. A Bayesan Fomulaton to Shape Regstaton he pobablstc fomulaton of shape egstaton poblem contans two models: one denotes the po shape dstbuton n tangent shape space and the othe s a lkelhood model n mage shape space. Based on these two models we deve the posteo dstbuton of model paametes. he MAP estmaton of the paametes can be obtaned usng the EM algothm.. angent Space Appomaton Assumng that a plana shape s descbed by landmak ponts n the mage, we can epesent t by a dmensonal vecto s. he dffeence between two plana shapes s usually measued by the Pocustes dstance []. Futhemoe, gven a set of tanng shape vectos { s } L =, the most popula way to algn them nto a common co-odnate fame s Genealzed Pocustes Analyss (GPA) []. he pocedue essentally equals to mnmze a quadatc loss functon defned by L( µ ) = ( s) µ, whee ( s ) s a D smlaty all tansfom of s. See [] fo the detals of GPA. he tangent space s a lnea appomaton of the geneal shape space n the vcnty of the mean shape vecto. Moe specfcally, the tangent space! µ s defned as the space nomal to ( µ ) and passng though µ. he Eucldean dstance n the tangent space s a good appomaton to the Pocustes dstance, f most of shape nstances ae close. s can be tansfomed onto! µ by algnng s wth µ as { : ( )} L! µ = s =. s often efeed as tangent shape vecto and epesented as a -dmensonal vecto. he esduals ae computed as { t } L = µ = n tangent space nstead of mage space, to emove the dffeence ntoduced by smlaty tansfom. ote that the dmenson of! µ s 4, whee the degeneated dmensonalty s coespondng to the degee of feedom of smlaty tansfomaton n a d Eucldean space. Futhemoe, snce any tansfomed shape vecto fom µ can be epesented by a lnea combnaton of {, ee, µµ, }, the complement space of! µ s spanned by {, ee, µµ, }. heefoe, the covaance mat of tangent shape, L Va ( X ) = ( µ )( µ ) () L = wll has at least fou zeo egenvalues wth coespondng egenvectos {, ee, µµ, }. In othe wods, the tangent shape vaances n ths complement space must be zeo... Po angent Shape Model We apply a pobablstc etenson of tadtonal PCA to model tangent shape vaaton, whch s smla to PPCA poposed by ppng and Bshop [4]. he model can be wtten as I Φ ( µ ) = b+ ε () 0 ( 4 ) a) Φ :( 4) s the tangent pojecton mat whose ow vectos ae the egenvectos of Va ( X ). Φ : conssts the fst columns of Φ. b) b, the shape paamete, s a -dmensonal vecto dstbuted as multvaate Gaussan (0, Λ ), whee Λ= dag( λ,..., λ ). λ s the th egenvalue and s the numbe of modes to etan n PCA. c) ε denotes an sotopc nose n the tangent space. It s a -4-dmensonal andom vecto whch s ndependent wth b and dstbutes as 4 p( ε) ~ ep{ ε ( σ )} ( σ = λ ). 4 = + Afte some smple algeba the model () can be ewtten as: = µ +Φ b+φ ε (3) By addng an sotopc Gaussan nose tem we assocate PCA wth a pobablstc eplanaton, theeby allowng to compute the posteo of model paametes. Each tem of b eflects a specfc vaaton along the coespondng pncple component (PC) as. Instead of usng all modes and -4-dmensonal shape paametes, we only select a subset of them to econstuct the shape wth shape vaatons we concen about. he fewe the modes ae used, the moe compact the model wll be, and the smoothe the econstucted shape tends to be. On the othe hand, moe modes ae nvolved n descbng shape, * e = (,0,,0,...,,0 ) ; s obtaned by otatng plana shape by 90, *.e. = (,... ) = (,,...,, )

3 b ε BSM X η b: shape paametes : pose paametes X: tangent shape Y: obseved shape Y Fgue : Shapes econstucted by the fst thee PCs: n each ow the mddle one s the mean shape. Else ae obtaned by vayng coespondng PC fom 3 λ to 3 λ. moe fleble the model s. Shape vaaton along the fst thee PCs s vsualzed n Fgue. he ntepetaton of PCs s not staght fowad. A possble ntepetaton s that the fst PC descbes vaatons n vetcal decton, the second PC may eplan the vaaton on mouth, and the thd PC may account fo out-of-plane otaton. he tangent space nose ε can also be vewed as a compensaton of mssed shape vaaton dung PCA pojecton. When the numbe of modes s lage, moe vaaton s etaned n PCA model and the nose vaance σ s smalle..3. Adaptve Lkelhood Model o ncopoate mage evdence nto the Bayesan famewok one eques a lkelhood P( I, ) whch s usually a pobablty dstbuton of the gey levels condtonal on the undelyng shape. Howeve, dectly paametezng P( I, ) may not be a good dea, because I and X ae not n a same physcal coodnate system, and the paametc fom of P( I, ) s usually comple and nonlnea. In BSM, we edefne the lkelhood as old P( y, ). Assume y s the shape estmated n the last teaton, by updatng each landmaks of y old wth ts local tetue we obtan y, whch s called obseved shape vecto. he dstance between obseved shape y and the tue shape can also be model as an adaptve Gaussan as (4). By adaptve we mean the vaance of the model s detemned by the dstance between y and y old n each teaton step. y = su + c+ η (4) a) y: obseved shape vecto, b) s: scale paamete; cos sn U = I : otaton mat; sn cos c c = : tanslaton paamete. c Fgue. A gaphcal llustaton of Bayesan tangent shape model: ccles stand fo vaables, dashed ccles denote nose tems and ectangles denote model paametes. ( denotes Konecke poduct.) c) η : sotopc obsevaton nose n the mage space. η ~ (0, ρ I ). ρ s set by ρ = c y old y, whee c s a manually chosen constant..4. Posteo ow we can compute the posteo of model paametes (,,, b sc ) gven the obseved shape vecto y. By applyng Bayes ule we have deved the equaton (5). Dectly optmzng the posteo s dffcult. Altenatvely, f the tangent shape s known, the posteo of model paametes condtonal on both and y ae much smple. hs leads us to mplement the EM based paametes estmaton algothm. p( b, c, s, y) (5) ep{ [( σ + s + s ρ A ρ ) ( Φ b + b Λ b]} ( σ + s + Φ const ρ ) s ρ ( ) 4 4 whee the const do not vay wth (b,c,s,) and Φ ) s the sub-mat of Φ by emovng the fst columns. he devaton s left to the append A..5. BSM as A Hdden Vaable Model A gaphcal llustaton of BSM s shown n Fgue. he tangent shape s the hdden vaable and y s obsevaton. he po shape model and the lkelhood model ae connected though tangent shape. 3. Paamete Estmaton n BSM Seachng In ths secton, we descbe an EM algothm fo estmaton the MAP paametes of BSM model. Befoe A B a B mp nq m n p = ( ) : q j, j

4 mmesng ouselves n the detals of devaton, howeve, let us fst pesent the esults of EM paamete estmaton and compae them wth those of ASM. 3.. Compason between BSM and ASM he teatve updatng pocedue of ASM s shown n Fgue 3. In ASM, tangent shape s dectly constucted fom shape paamete b, whee b s a tuncaton of s coodnates wthn the ange of ( 3 dag( Λ ), 3 dag( Λ )). In BSM we deve the updatng equatons of and b shown n Fgue 4. (See Secton 3. and 3.3 fo the detals of devaton.) he majo dffeence of the two algothms comes fom the updatng ules of the tangent shape and shape paamete b. a) In BSM, the tangent shape s updated by a weghted aveage of the shape econstucted fom the shape paamete b and the tangent pojecton of the obseved shape y. In ths way, the estmaton of encodes both po shape knowledge and mage evdence. It s nteestng to note that the weght p s automatcally chosen by computng the ato between the vaance σ of po nose n tangent space and the vaance ρ of the obsevaton nose. hey ae algned to the same scale by multplyng the scale facto s of smlaty tansfom. When ρ s lage, whch mples the mage s nosy o the obsevaton s not stable, shape paametes ae moe mpotant fo updatng. On the othe hand, when ρ s small, the shape estmaton may be conveged aleady, we need not to egulaze t too stctly. b) Regulazaton on shape paametes s equed to geneate vald shape nstances. Usng a contnuous egulazaton functon often s pefeed to usng a tuncaton functon because numecally, dscontnuous egulazaton on b may esult n a unstable estmaton. hat s, the esult may shft back and foth nstead of convegng to a pont. In BSM, the shape paamete s constaned by multplyng a contaned facto R =Λ( Λ+ σ I). Remembe that Λ epesents po shape vaance mat and σ epesents the esudal vaance. (See Secton. fo detals). Specfally, along the th pncple as, b s updated by b = λ /( λ + σ )( Φ ), whee Φ s the th column of Φ. In shot BSM algothm enjoys ts mets n two aspects: weghted epesentaton of tangent shape and contnuous egulazaton of shape paametes. hese esults ae deved fom optmzng an eplct and contnuous loss functon usng EM. = µ +Φ b : = agmn ( ) y pojecton & tuncaton b = Φ y dag Λ ma{mn{ ( ),3 ( )}, 3 dag( Λ)} Fgue 3. Updatng ules of Actve Shape Model b = R Φ, R=Λ( Λ+ σ I) A contnuous egulazaton of shape paametes = ag mn ( ) y Compute pose paametes Reconstucton algnng = µ + pφ b+ pφφ y p= σ ( σ + s ρ ) ( ) ( ) Recove tangent shape usng the nfomaton of both shape paametes and obseved shape; weghts ae detemned by nose ato. Fgue 4. BSM updatng ules: angent shape s estmated by a weghted sum of the shape econstucted fom shape paametes b and the tansfom of the obseved shape y to the tangent space. 3.. Epectaton Step Gven a set of complete data {, y }, the complete posteo of model paametes s smply a poduct of the followng two dstbutons, pb ( ) ep{ /[ bλ b+ σ µ Φ b ]} (6) p( γ, y) ep{ / [ ρ y Xγ ]} (7) whee X = (,, ee, ) and γ = ( s cos, s sn, c, c). akng the logathm and the condtonal epectaton, we obtan: Q( γ γ old ) (8) = log p( bcs,,, y, ) = log pb ( ) + log p( γ y, ) = b b σ µ b ρ y Xγ Λ + Φ + + const

5 Computng the Q-functon of (8) essentally equals to calculate two statstcs, the condtonal epectatons of and wth espect to p ( y, c, s, ) = µ + ( p) Φ b+ pφφ (9) = + ( 4) δ (0) whee p = σ ( σ + s ρ ) and δ = ( σ + s ρ ). he detaled devaton s left to the append B Mamzaton Step he M step mamzes the Q-functon ove model paametes. Snce the tems dependng on b and γ ae decoupled n (8), t s a much smple epesson to mamze than the logathm of the posteo n (5). We use ~ to denote the updated paametes. By computng the devatve of the Q-functon we have, b" =Λ( Λ+ σ ) Φ ( µ ) =Λ( Λ+ σ ) Φ () y y " γ = (,,, ) () * y y = = Accodngly, the updatng equatons of each pose paamete ae, s" = " γ + " γ, " = atan( " γ " γ ), and c" = ( γ, γ ) (3) Inhomogeneous Obsevaton ose In Secton.3 we assume the obsevaton nose s dstbuted as an sotopc Gaussan. hs assumpton may not always hold, because the nose of each featue landmak may be dffeent due to patal occluson, nosy backgound o othe effects n the mage. We can choose a dagonal vaance mat nstead fo the obsevaton nose η as, η ~ (0, Σ), Σ= dag( ρ,..., ρ ) I (4) old old whee ρ = c(( y y ) + ( y y) ) EM algothm can also be appled to ths case wth slght modfcaton. Instead of computng and, the statstcs we need to compute n the E-Step s and Σ. he esults of EM paametes estmaton ae gven n the append C. 4. Epemental Results In ths secton we compae BSM wth ASM and demonstate BSM seachng mpoves both accuacy Fgue 5. An eample of BSM seachng: (Left) Intal shape mask, we petub ts oentaton and scale paamete to make the task moe dffcult. (Mddle) Seachng esult afte 0 teatons at the top laye. (Rght) Fnal esult by seachng all thee layes. and stablty. 4.. BSM Seachng Smla to ASM the BSM seachng algothm s decomposed nto two majo steps: local tetue matchng and EM nfeence fo shape and tansfomaton paametes. As usual, the seachng s un n a multesoluton famewok. A thee level Gaussan mage pyamd s fomed on a testng mage by epeated subsamplng. Model nstance stats at the/ 4 esoluton of the mage. Dffeent dmensons of shape paamete vecto ae used fo dffeent pyamd layes. We choose = 5 fo the fst laye, = 0 fo the second laye, and = 40 fo the thd laye. Fgue 5 shows a typcal eample of BSM seachng. Moe seachng esults of ASM and BSM ae shown n Fgue Accuacy o compae the accuacy of the two algothms quanttatvely we dvde ou database nto two pats, one used fo tanng and the othe used fo testng. Ou database contans 870 gey-scale mages n the FERE database [8], the AR database [9] and othe collectons. Each mage contans a face wth a sze angng fom to 0 0, and wth dffeent facal epessons and dffeent llumnaton condtons. A total of 83 face landmaks ae labeled manually on each mage of the tanng set. We tan both the ASM model and ou model on 599 faces, and use the else 7 mages fo testng. Fo each testng mage, an ntal guess of face locaton ae povded by a Boostng based face detecto [3] and then, the mean face shape mask s tansfomed and putted on the detected egon. We petub the shape mask by andomly otatng (fom 0 to 45 ) and scalng (fom to.). he petubed shape s used as ntal values and s fed nto the two algothms. he seachng pocesses would not stop unless the esults ae conveged o the numbe of teatons s ove than 00.

6 Fgue 6: Compason of the accuacy of BSM and ASM: - as demotes the nde of test mages and y-as denotes the dffeence of the estmaton eos dst( BSM ) j dst( ASM ) j between ASM and BSM. Ponts below y = 0 (blue ponts) denote mages wth bette pefomance by BSM and ed ponts ae opposte. Fo a total of 7 testng mages, 48 of them ae maked blue and 3 of them ae maked ed, whch means on 9.5% testng mages the seachng esults of BSM ae bette than that of ASM. Fgue 8. Compason vaaton of estmaton esults n one ndvdual dmenson of shape paamete b: (op) Fve ntemedate esults of ASM seachng and BSM seachng. (Bottom) he evoluton of the shape paamete b[] wth the nceasng of teatons numbe. Red ponts denote b[] poduced by the tuncaton pocedue of ASM. Blue ponts ae b[] estmated by BSM algothm. Fgue 7. An ambguous seachng esult: an unstable algothm does not guaantee that the model conveges to smla esults whle seachng n smla mages. he fgue shows the seachng esults on thee contguous fames wth a slghtly change n vew. otce the nconsstent seachng esults on the nose and the chn of the boy. o compae the accuacy of the two algothms, we compute the estmaton eo by a dffeence measue defned by the sum of the dstance between seached landmak and annotated landmak. A A j = + = dst( A) ( ) ( y y ) dst( A ) j denotes estmaton eo of algothm A on the mage j, whee (, y ) s annotated coodnates of the A A th landmak and (, y ) s the seached coodnates of the th landmak by algothm A. We have plotted j ~ dst BSM ) dst ASM ) ( j j ( n Fgue 6. It s shown that on 48 of 7 (9.5%) mages, the seach esults of BSM ae bette than that of ASM. Fgue 9. Compason of vaatons of the algnment eos on eye ponts: (op) Fou ntemedate esults of ASM seachng and BSM seachng. (Bottom) the evoluton of eye eoes of ASM and BSM Stablty Anothe chaacte of shape analyss algothms we concen s numecal stablty of estmaton esults. Fo a obust seachng algothm we epect that vaaton of estmaton esults deceases wth the nceasng of teaton numbe. An unstable algothm wll poduce ambguous esults. See Fgue 7 fo an eample. We eploe the stablty of ASM and ou BSM algothm n two ways. he fst s the vaaton of estmaton esults n one ndvdual dmenson of shape paamete b, and the net s the vaaton of some facal component. Fgue 8 compaes the vaatons n the estmaton of the second shape paamete b []. Fgue 9 compaes the vaatons of the estmaton eos on eyes.

7 Fgue 0. Compason of BSM and ASM seachng esults: (Fst and hd Rows) esults of ASM seachng; (Second and Fouth Rows) esults of BSM seachng We have plotted the value of b[] and dst( eye ) fo evey 5 steps of teatons. Fom the fgues we can obseve that the vaatons of the estmaton esults by BSM algothm ae much smalle. 5. Dscusson p( ; b ) p( y ; ) Fgue. A genealzaton of BSM model BSM can be etended to a moe geneal fom llustated by the undected gaph n Fgue. he po model descbes shape vaaton, the obsevaton model ncopoates mage evdence and they ae connected though the tangent shape. Whle the tangent shape s estmated, due to ts local Makov popety, the MAP estmaton of pose paametes depends only on the ght sde of the gaph, and t s degeneated to standad Pocustes Analyss wth the assumpton that the obsevaton nose s an sotopc Gaussan. ote that the equaton (5) equals to estmate pose paametes usng weghted Pocustes analyss. Smlaly the MAP estmaton of shape paametes s completely detemned by the left pat of the gaph gven the tangent shape. Fgue povdes a geneal famewok fo shape analyss poblem. In contast to dectly optmzng a huge, heustcally defned loss functon, the statstcal teatment n BSM povdes the fleblty to deal wth dffeent poblems n dffeent sub-models. Fo eample, f we ae nteested n modelng multmodal shape vaatons lke eaggeated face epesson, we may paameteze the left pat as a Gaussan mtues; f we ae nteested n handlng patal occluson o mage nose, we may mplement the ght pat usng obust statstcs methods. Appomate nfeence algothm may need to be adopted n both cases. BSM shape egstaton uns vey fast snce we deve analytcal soluton n EM paamete estmaton. In E-step, computng the epectaton of the two statstcs (please efe to (9) and (0)) ncludes only thee mat multplcaton. In the M-step, paametes updatng ules (equatons () and ()) nvolves only one mat multplcaton and some nne poducts of vectos. In ou epements, t takes about 00 ms n geneal fo BSM to convege on a face on a Pentum3-800Hz machne; and fo smalle faces t takes less tme (fom 60ms to 00ms, depends on the sze of the face). 6. Concluson hs pape pesents a Bayesan appoach fo shape egstaton poblem. By pojectng shape to tangent shape, we have bult the models descbng the po dstbuton of face shapes and the lkelhood. We have developed the BSM algothm to uncove the shape paametes and tansfomaton paametes of an abtay fgue. We have

8 compaed ou algothm wth the classc ASM algothm and demonstated ts accuacy and effcency. 7. Refeences [] Coln Goodall, Pocustes Methods n the Statstcal Analyss of Shape, J. R. Statst.Soc. B, 99, pp [] J.. Kent and K.V. Mada, Shape, Pocustes tangent pojecton and blateal symmety, Bometca, 00, pp [3]. F. Cootes, C. aylo, D. Coope, and J. Gaham. Actve shape models the tanng and the applcatons. Compute Vson and Image Undestandng, 6():38-59, Januay 995. [4] M. ppng and C. Bshop. "Pobablstc pncpal component analyss" echncal Repot echncal Repot CRG/97/00, eual Computng Reseach Goup, Aston Unvesty, Bmngham, UK, Septembe 997. [5] C. Small, he Statstcal heoy of Shape, Spnge, 996. [6] Kendall, D. G., he Dffuson of Shape, Adv. Appl. Pobab., 977, pp [7] Ian L Dyden and Kant V. Mada, ` Statstcal Shape Analyss ', publshed by John Wley and Sons, Chcheste n July, 998. [8] P. J. Phllps, H. Moon, S. A. Rzv, and P. J. Rauss. he FERE evaluaton methodology fo face ecognton algothms. IEEE ans. on PAMI, (0): , 000. [9] A.M. Matnez and R. Benavente. he AR Face Database. CVC echncal Repot #4, June 998. [0] A. Blake, M. Isad, Actve Contous, Spnge, Beln, 998. [] Mchael Kass, Andew Wtkn, and Demet ezopoulos. Snakes: Actve contou models. Intenatonal Jounal of Compute Vson, :3--33, 987. [] Gowe, J. C. Genealzed Pocustes analyss. Psychometka, 40, 33-5, 975. [3] Y. Feund and R. E. Schape. "A decson-theoetc genealzaton of on-lne leanng and an applcaton to boostng". Jounal of Compute and System Scences, 55():9--39, 997. Append A. Posteo Dstbuton of Paametes By combnng (3) wth (4) and multplyng Φ on both sdes of the equaton we have, = µ +Φ b+φ ε + s U η ( ξ # s U η) ( = µ +Φ b ε ξ +Φ +Φ ) + AA ξ Φ b= ( I,0 (4 ) )( ε +Φ ξ ) (5) whee A= (, e e, µµ, ). Snce ε and ξ ae ndependent, the dstbutons of ε + Φ ξ and A ξ can be computed as, ( ε +Φ ξ) σ + ρ ~ (0, ( s ) I ) A ξ ~ (0, s ρ I4 ) 4 (6) Combnng (5) and (6) we obtan the lkelhood of model paametes. he posteo of model paametes s computed by applyng the Bayes ule as (5). B. Detaled Devaton of Epectaton Step he condtonal pobablty of the tangent shape vecto gven the obseved shape y and model paametes s p ( y, c, s, ) (7) ep{ [ σ µ Φ b = + s ρ ]} when A ( µ ) = 0 0, othewse he tangent shape can be wtten as = ( AA + ΦΦ ) = AA µ + ΦΦ = µ + ΦΦ (8) whee A= (, e e, µµ, ). Snce s an sotopc Gaussan, the elements of on the two othogonal subspaces ae ndependent,.e. A Φ. So p( Φ A ( µ ) = 0) = p( Φ ) = (( p) Φ µ + pφ µ, δ I 4) (9) whee, ( ), ( µ = µ +Φ µ = = σ σ + ρ ) and δ = ( σ + s ρ ). heefoe the condtonal epectaton of s = AA µ +ΦE( Φ ) = ( p) ΦΦ µ + pφφ µ + AA µ (0) = µ + ( p) Φ b+ pφφ and the condtonal epectaton of the nom of s = A µ + E Φ = + ( 4) δ () C. EM fo Inhomogeneous Obsevaton ose We gnoe the detals of the devaton and just pesent the esults of E-step and M-Step. Let us denote # ( σ + s Σ ) P# ( I + σ s Σ ) / α # [ P( µ +Φ b) + ( I P) ] / Let B # Oth( A), whose column vectos fom an / othogonal bass of the column space of A. he E-step: / / = [ α + BB ( µ α)] () Σ = Σ + t( Σ ) t( Σ BB ) (3) he M-step: b" ( σ ) =Λ Λ+ Φ ( µ ) =Λ( Λ+ σ ) Φ (4) * ρ y ρ y y Σ y Σ = = " γ =,,, (5) Σ Σ ρ ρ = =