Style Adaptive Bayesian Tracking Using Explicit Manifold Learning

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1 Syle Adapive Bayesian Tracking Using Explici Manifold Learning Chan-Su Lee and Ahmed Elgammal Compuer Science Deparmen Rugers, The Sae Universiy of New Jersey Piscaaway, NJ 8854, USA {chansu, Absrac Characerisics of he D conour shape deformaion in human moion conain rich informaion and can be useful for human idenificaion, gender classificaion, 3D pose reconsrucion and so on. In his paper we inroduce a new approach for conour racking for human moion using an explici modeling of he moion manifold and learning a decomposable generaive model. We use nonlinear dimensionaliy reducion o embed he moion manifold in a low dimensional configuraion space uilizing he consrains imposed by he human moion. Given such embedding, we learn an explici represenaion of he manifold, which reduces he problem o a one-dimensional racking problem and also faciliaes linear dynamics on he manifold. We also uilize a generaive model hrough learning a nonlinear mapping beween he embedding space and he visual inpu space, which faciliaes capuring global deformaion characerisics. The conour racking problem is formulaed as saes esimaion in he decomposed generaive model parameer wihin a Bayesian racking framework. The resul is a robus, adapive gai racking wih shape syle esimaion. Inroducion Vision-based human moion racking and analysis sysems have promising poenials for many applicaions such as visual surveillance in public area, aciviy recogniion, and spor analysis. Human moion involves no only geomeric ransformaions bu also deformaions in shape and appearance. Characerisics of he shape deformaion in a person moion conain rich informaion such as body configuraion, person ideniy, gender informaion, and even emoional saes of he person. Human idenificaion [4] and gender classificaion [5] are examples of he applicaions. There have been a lo of work on conour racking such as acive shape models (ASM) [3], acive conours [], and exemplar-based racking [4]. I is sill hard o ge robus conour racking and in he same ime be able o adap o differen people being racked wih auomaic iniializaion. Modeling dynamics of shape and appearance is essenial for racking human moion. The observed human shape and appearance in video sequences goes hrough complicaed global nonlinear deformaion beween frames. If we consider he global shape, here are wo facors affecing he shape of he body conour hrough he moion: global dynamics facor and

2 person shape syle facor, The dynamic facor is consrained because of dynamics of he moion and he physical characerisics of human body configuraion [, ]. The person shape syle is ime-invarian facor characerizing disinguishable feaures in each person shape depending on body buil(big, small, shor, all, ec.). These wo facors can summarize rich characerisics of human moion and ideniy. Our objecive is o achieve rackers ha can rack global deformaion in conours and can adap o differen people shapes auomaically. There are several challenges o achieve his goal. Firs, modeling he human body shape space is hard, considering boh he dynamics and he shape syle. Such shapes lie on a nonlinear manifold. Also, in some cases here are opological changes in conour shapes hrough moion which makes esablishing correspondences beween conour poins unfeasible. Second, how o simplify he dynamics of he global deformaion. Can we learn a dynamic model for body configuraion ha is low in dimensionaliy and exhibis linear dynamics? For cerain classes of moion like gai, facial expression and gesures, he deformaion migh lie on a low dimensional manifold if we consider a single person. Our previous work [7] inroduced a framework o separae he moion from he syle in a generaive fashion where he moion is represened in a low dimensional represenaion. In his paper we uilize a similar generaive model wihin a Bayesian racking formulaion ha is able o rack conours in cluered environmen where he racking is performed in hree concepually independen spaces: body configuraion space, shape syle space and geomeric ransformaion space. Therefore, objec sae combines heerogeneous represenaions. The challenge will be how o represen and handle muliple spaces wihou falling ino exponenial increase of he sae space dimensionaliy. Also, how o do racking in a shape space which can be high dimensional? We presen a new approach for racking nonlinear global deformaion in human moion based on Bayesian racking wih emphasis on he he gai moion. Our conribuions are as follows: Firs, by applying explici nonlinear manifold learning and is parameric represenaion, we can find compac, low dimensional represenaion of body configuraion which reduces he complexiy of he dynamics of he moion ino a linear sysem for racking gai. Second, by uilizing concepually independen decomposed sae represenaion, we can achieve approximaion of sae poserior esimaion in a marginalized way. Third, he adapive racking of person shape allows no only racking of any new person conour bu also shows poenial for idenificaion of he person during racking. As a resul, we achieve robus, adapive gai racking wih syle esimaion from cluered environmen. The paper organizaion is as follows. Secion summarizes he framework. Secion 3 describes he sae represenaion and learning. Secion 4 describes he he racking framework. Secion 5 shows some experimenal resuls. Framework We can hink of he shape of a dynamic objec as insances driven from a generaive model. Le z R d be he shape of he objec a ime insance represened as a poin in a d- dimensional space. This insance of he shape is driven from a model in he form z = T α γ(b ;s ), () where he γ( ) is a nonlinear mapping funcion ha maps from a represenaion of he body configuraion b ino he observaion space given a mapping parameer s ha characerizes he person shape in a way independen from he configuraion and specific for he person

3 s s b b y y α α z z Figure : Graphic model for decomposed generaive model being racked. T α represens a geomeric ransformaion on he shape insance. Given his generaive model, we can fully describe observaion insance z by sae parameers α,b, and s. Figure shows a graphical model illusraing he relaion beween hese variables where y is a conour insance generaed from model given body configuraion b and shape syle s and ransformed in he image space hrough T α o form he observed conour. The mapping γ(b ;s ) is a nonlinear mapping from he body configuraion sae b as y = A s ψ(b ), () where ψ(b ) is a kernel induced space, A is a hird order ensor, s k is a shape syle vecor and is appropriae ensor produc as defined in []. The racking problem is hen an inference problem where a ime we need o infer he body configuraion represenaion b and he person specific parameer s and he geomeric ransformaion T α given he observaion z. The Bayesian racking framework enables a recursive updae of he poserior P(X Z ) over he objec sae X given all observaion Z = Z,Z,..,Z up o ime : P(X Z ) P(Z X ) P(X X )P(X Z ) (3) X In our generaive model, he sae X is [α,b,s ], which uniquely describes he sae of he racking objec. Observaion Z is he capured image insance a ime. The sae X is decomposed ino hree sub-saes α,b,s. These hree random variables are concepually independen since we can combine any body configuraion wih any person shape syle wih any geomerical ransformaion o synhesize a new conour. However, hey are dependen given he observaion Z. I is hard o esimae join poserior disribuion P(α,b,s Z ) for is high dimensionaliy. The objecive of he densiy esimaion is o esimae saes α,b,s for a given observaion. The decomposable feaure of our generaive model enables us o esimae each sae by a marginal densiy disribuion P(α Z ), P(b Z ), and P(s Z ). We approximae marginal densiy esimaion of one sae variable along represenaive values of he oher sae variables. For example, in order o esimae marginal densiy of P(b Z ), we esimae P(b α,s,z ), where α,s are represenaive values such as maximum poseriori esimaes.

4 (a) individual manifolds (b) a unified manifold (c) generaed sequences Figure : Individual manifolds and heir unified manifold 3 Learning Generaive Model Our objecive is o esablish generaive models for he shape in he form of equaion where he inrinsic body configuraion is decoupled from he shape syle. Such model was inroduced in [7] and applied o gai daa and facial expression daa. Given raining sequences of differen people performing he same moion (gai in our case), Locally Linear Embedding (LLE) [3] is applied o find low dimensional represenaion of body configuraion for each person manifold. As a resul of nonlinear dimensionaliy reducion, an embedding of he gai manifold can be obained in a low dimensional Euclidean space where hree is he leas dimensional space where he wo halves of he walking cycles can be discriminaed. Figure -a shows low dimensional represenaion of side-view walking sequences for differen people. Generally, he walking cycle evolves along a closed curve in he embedded space, i.e., only one degree of freedom conrols he walking cycle which corresponds o he consrained body pose as a funcion of he ime. Such manifold can be used as inrinsic represenaion of he body configuraion. The use of LLE o obain inrinsic configuraion for racking was previously repored in [5]. Modeling body configuraion space: Body configuraion manifold is parameerized using a spline fied o he embedded manifold represenaion. Firs, cycles are deeced given an origin poin on he manifold by compuing geodesics along he manifold. Second, a meanmanifold for each person is obained by averaging difference cycles. Obviously, each person will have a differen manifold based on his spaio-emporal characerisics. Third, non-rigid ransformaion using an approach similar o [] is performed o find a unified manifold represenaion as in Fig. -b. Correspondences beween differen subjecs are accomplished by selecing a cerain body pose as he origin poin in differen manifolds and equal sampling in he parameerized represenaion. Finally, we parameerized he unified mean manifold by spline fiing. The unified mean-manifold can be parameerized by a one-dimensional parameer β R and a spline fiing funcion f : R R 3 ha saisfies b = f (β ) is used o map from he parameer space ino he hree dimensional embedding space. Fig. -c shows hree sequences generaed using he same equidisan body configuraion parameer [β,β,,β 6 ] along he unified mean manifold wih differen syle. As a resul of one dimensional represenaion of each cycle in unified manifold, aligned body pose can be generaed in differen shape syle. Modeling syle shape space: Shape syle space is parameerized by a linear combinaion of basis of he syle space. Firs, RBF nonlinear mappings [7] are learned beween he embedded body configuraion b on he unified manifold and corresponding shape observaions y k for each person k. The mapping has he form y k = γ k (b ) = C k ψ(b ), where C k is he mapping coefficiens which depend on paricular person shape and ψ( ) is a nonlinear mapping wih N

5 RBF kernel funcions o model he manifold in he embedding space. Given learned nonlinear mapping coefficiens C,C,,C K, for raining people,,k, he shape syle parameers are decomposed by fiing an asymmeric bilinear o he coefficien space. As a resul, we can generae conour insance y k for paricular person k a any body configuraion b as y k = A s k ψ(b ), (4) where A is a hird order ensor, s k is a shape syle vecor for person k. Ulimaely he syle parameer s should be independen of he configuraion and herefore should be ime invarian and can be esimaed a iniializaion. However, we don know he person syle iniially and, herefore, he syle needs o fi o he correc person syle gradually during he racking. So, we formulaed syle as ime varian facor ha should sabilize afer some frames from iniializaion. The dimension of he syle vecor depends on he number of people used for raining and can be high dimensional. We represen new syle as a linear combinaion of syle classes learned from he raining daa. The racking of he high dimensional syle vecor s iself will be hard as i can fi local minima easily. A new syle vecor s is represened by linear weighing of each of he syle classes s k, k =,,K using linear weigh λ k : s = K k= λ k s k, K k= λ k =, (5) where K is he number of syle classes used o represen new syles. Modeling geomeric ransformaion space: Geomeric ransformaion in human moion has consrains. We parameerize he geomeric ransformaion by scaling S x,s y and ranslaion T x,t y. New represenaion of any conour poin can be found using homogeneous ransformaion α wih paramers [T x,t y,sx,sy] as global ransformaion parameers. The overall generaive model can be expressed as [ K ] ) z = T α (A λ k s k ψ( f (β )). (6) k= Tracking problem using his generaive model is he esimaion of parameer α, β, and λ a each new frame given he observaion z. 4 Bayesian Tracking 4. Marginalized Densiy Represenaion Since i is hard o esimae join disribuion of high dimensional daa, we represen each of he sub-sae densiies by marginalized ones. We approximae he marginal densiy of each sub-sae using maximum a poseriori (MAP) of he oher sub-saes, i.e., P(α Z ) P(α b,s,z ), P(b Z ) P(b α,s,z ), P(s Z ) P(s α,b,z ), (7) where α, b, and s are maximum a poseriori esimae of each approximaed marginal densiy. Maximum a poseriori of he join densiy X = [α,b,s ] = argmax X P(X Z ) will maximize he poserior marginal densiies α = argmax α P(α Z ), b = argmax b P(b Z ), and s = argmax s P(s Z ) because of our decomposable generaive model.

6 Dynamic Models P(α α ) = N(Hα,σ α ) Prediced Saes ˆα (i) P(α α ) P(b b ) P(β β ) = N(β + ṽ,σ ( j) b ) ˆβ P(β β ) P(s s ) P(λ λ ) = N(s,σ (k) s ) ˆλ P(λ λ ) P(s s ) Table : Dynamic models and prediced densiy of saes 4. Dynamic Model The dynamic model P(X X ) = P(α,b,s α,b,s ) predics he sae X a ime given he previous sae X. In our generaive model, he saes are decomposed ino hree differen facors. In he Bayesian nework shown in Fig., he ransiions of each of he subsaes are independen. Therefore, we can describe whole sae dynamic model by dynamic models of individual sub-saes as P(α,b,s α,b,s ) = P(α α )P(b b )P(s s ). (8) In case of body configuraion, he manifold spline parameer β will change in a consan speed if he subjec walks in a consan speed (because i corresponds o consan frame rae used in he learning). However, he resuling manifold poin represening body configuraion b = f (β ) will move along he manifold in differen sep sizes. Therefore, we use β o model he dynamics since i resuls in a one dimensional linear dynamic sysem. In general, he walking speed can change gradually. So, he body configuraion in each new sae will move from he curren sae wih a filered speed ha is adapively esimaed during he racking. Dynamic model of syle is approximaed by a random walk as he syle may change smoohly around specific person syle. The global ransformaion α capures global conour moion in he image space. The lef column in able shows he dynamic models, where ṽ is filered esimaed body configuraion manifold velociy, and H is a ransiion marix of he geomeric ransformaion sae which is learned from he raining daa. A new sub-sae densiies can be prediced using each sub-sae dynamic models as in able righ column. 4.3 Tracking Algorihm using Paricle Filer We represen sae densiies using paricle filers since such densiies can be non-gaussian and he observaion is nonlinear. In paricle filer, he densiy of poserior P(X Z ) is approximaed by a se of N weighed samples {X (i),π (i) }, where π (i) is he weigh for paricle X (i). Using he generaive model in Eq. 6, he racking problem is o esimae α, λ, and β for given observaion Z. The marginalized poserior densiies for α, β, and λ are approximaed by paricles. {α (i), α π (i) } N α ( j) i=,{β, b π ( j) } N b (k) j=,{λ, s π (k) } N s k=, (9) where N α,n b, and N s are he numbers of paricles used for each of he sub-saes. Densiy represenaion P(β Z ) is inerchangeable wih P(b Z ) as i has one o one correspondence beween β ( j) and b ( j). The same holds beween λ and s for he syle sae. We esimae saes by sequenial updae of he marginalized sub-densiies uilizing he prediced densiies of he oher sub-saes. These densiies are updaed wih curren observaion Z by updaing weighing values of each sub-sae paricles approximaion using he

7 observaion. We esimae global ransformaion α using prediced densiy `s, `b. Then body configuraion b is esimaed using esimae global ransformaion α, and prediced densiy `s. Finally syle s is esimaed wih given esimaed α, and b. The following able summarizes he sae esimaion procedure using ime esimaion.. Imporance-sampling wih resampling: sae densiy esimaion: {α (i), α π (i) }N α ( j) i=, {β, b π ( j) }N b (k) j=, {λ, s π (k) }N s k=. Resampling: {ὰ (i),/n ( j) α}, { `β,/n b}, and {`λ (k),/n s}.. Predicive updae of sae densiies: α (i) = Hὰ (i) + N(,σ α) β ( j) ( j) = `β + ṽ + N(,σ b ), b ( j) = f (β ( j) ) λ (k) (k) = `λ + N(,σ s ), λ (k) = λ (k), s Ns i= λ (k) (k) = N s i= λ (k) i s i i 3. Sequenial updae of sae weighs using curren observaion: Global ransformaion α wih ˆb, ŝ : P(α (i) ˆb,ŝ,Z ) P(Z α (i), ˆb,ŝ )P(α (i) ) α π (i) = P(Z α (i), ˆb,ŝ ), α π (i) = α π (i) Body pose b wih α, ŝ : α = α (i ), where i = argmax α i π (i) Nα j= α π ( j) P(b ( j) α,ŝ,z ) P(Z α,b ( j),ŝ )P(b ( j) ) b π ( j) = P(Z α,b ( j),ŝ ), b π ( j) = b π ( j) Syle s wih α, b : b = b ( j ), where j = argmax b j π ( j) P(s (k) α,b,z ) P(Z α,b,s (k) )P(s (k) ) s π (k) = P(Z α,b,s (k) ), s π (k) = s π (k) 4.4 Observaion Model Ns i= s π (i) N b i= b π (i) In our muli-sae represenaion, we updae weighs α π (i), b π ( j), and s π (k) by marginalized likelihood P(Z α (i),b,s ), P(Z α,b ( j),s ), and P(Z α,b,s (k) ) given observaion Z. Each sub sae capures differen characerisics in he dynamic moion and affecs differen variaions in he observaion. For example, body poses are changed according o body configuraion sae. Differen body configuraions show significan changes in edge direcion in he legs. However, in case of syle, he variaion is suble and changes along he global conours. Observaion model measures sae X by updaing he weighs π (i) in he paricle filer by measuring he observaion likelihood P(Z X (i) ). We can esimae he likelihood by P(Z X ) = P(Z α,b,s ) exp ( d(z,z ) σ ) = exp ( d(z,t α A s ψ(b )) σ ), () where d( ) is disance measure, z is he conour from he generaive model using α,b,s. I is very imporan o use proper disance measuremen d( ) in order o ge accurae updae of weighs and densiy for new observaion. We use hree disance measures: Chamfer disance, weighed Chamfer disance, and oriened Chamfer disance. Represenaion: We represen shape conour by an implici funcion similar o [6] where he conour is he zero level of such funcion. From each new frame Z, we exraced edge using Canny edge deecor algorihm.

8 Weighed Chamfer Disance for Geomeric Transformaion Esimaion: For geomeric ransformaion esimaion, he prediced body configuraion, and syle esimae from he previous frame are used. Therefore, we need o find similariy measuremen which is robus o he deviaion of body pose and syle esimaion and sensiive o global ransformaion. Typically he shape or he silhouee of upper body par in walking sequence are relaively invarian o he body pose and syle. By giving differen weigh o differen conour poins in Chamfer maching, we can emphasize upper body par and de-emphasize lower body par in he disance measuremen. Weighed chamfer disance can be compued as D w (T,F,W) = N N i min ρ( i, f i )w i, () f i F where i is i h feaure locaion, f i emplae feaure and w i i h feaure weigh. Pracically, weighed chamfer disance achieved more robus esimaion of he geomeric ransformaion. Oriened Chamfer Disance for Body Pose: Differen body poses can be characerized by he orienaion of legs. Therefore, oriened edge based similariy measuremen is useful in case of he body configuraion esimaion. Oriened chamfer disance maching was used in [8]. We use a linear filer [9] o deec oriened edge efficienly afer edge deecion. Afer applying he linear filer o he conour and he observaion, we applied chamfer maching for each oriened disance ransform and oriened conour emplae. The final resul is he sum of each of he oriened chamfer disance. For syle esimaion, simple Chamfer disance is used. 5 Experimenal Resul We used CMU Mobo daa se for learning he generaive model, he dynamics, and for esing he racker. Six subjecs are used in learning he model. The racking performance is evaluaed for people used in raining and for unknown people, which were no used in he learning. We iniialize he racker by giving a rough esimae of iniial global ransformaion parameer. The body configuraion is iniialized by random paricles along he manifold. In case of syle, as we don know he subjec syle from he iniial frame, we iniialize syle by mean syle, which means equal weighs are applied for every syle class. Tracking for rained subjecs : For he raining subjecs, he racker shows very good racking resuls. I shows accurae racking of body configuraion parameer β and correc esimaion of shape syle s. Fig. 3 (a) shows several frames during racking known subjec. The lef column shows racking conours. The middle column shows he poserior body configuraion densiy. The righ column is he esimaed syle weighs in each frame. Fig. 3 (b) shows racking resuls for syle weighs. The figure shows ha he syle esimae converges o he subjec s correc syle and i becomes he major weighing facor afer abou frames. The syle weighing shows accurae idenificaion of he subjec as a resul of racking and i has many poenial for human idenificaion and ohers. Fig. 3 (c) shows esimaed body configuraion β value. Even hough he wo srides making each gai cycle are very similar and hard o differeniae in visual space, he body configuraion parameer accuraely find ou he correc configuraion. The figure shows ha he body configuraion correcly exhibis linear dynamics. As we have one o one mapping beween body configuraion on he manifold and 3D body pose, we can direcly recover 3D body configuraion using he esimaed β or using manifold poins b similar o [6]. Tracking for unknown subjecs: Tracking for new subjecs can be hard as we used small number of people for learning syle in he generaive model. I akes more frames o converge

9 syle syle syle 3 syle 4 syle 5 syle 6 syle syle syle 3 syle 4 syle 5 syle 6 (a) racking of subjec 4h frame: 6h frame: 64h frame: (b) syle weighs Syle weighs Body Configuraion: β Shape syle Body Configuraion: β Shape syle (c) body configuraion β Body configuraion β Body Configuraion: β Shape syle Frame number Frame number Figure 3: Tracking for known person (a) racking of unknown subjec 4h frame: 6h frame: 64h frame: (b) syle weighs Syle weighs Body Configuraion: β Shape syle Body Configuraion: β Shape syle (c) body configuraion β Body configuraion β Body Configuraion: β Shape syle Frame number Frame number Figure 4: Tracking for unknown person o accurae conour fiing as shown in Fig. 4 (b). However, afer some frames i accuraely fi o he subjec conour even hough we did no do any local deformaion o fi he model o he subjec. There is no one dominan syle class weigh during he racking and someimes he highes weigh are swiched depend on he observaion. In case of he body configuraion β, you can see someimes i jumps abou half cycle due o he similariy in he observaion since he syle is no accurae enough. Sill, he resul shows accurae esimaion of body pose. We also, applied racking in normal walking siuaion using M. Black sraigh walking person image sequence even hough we learned he generaive model from readmill walking daa. Fig. 5 shows conour racking resuls for 4 frames. Fig. 5 (c) shows esimaed body configuraion parameers. I confused in some inervals a he beginning bu i recovers wihin he cycle. 6 Conclusion We presened new framework for human moion racking using a decomposable generaive model. As a resul of our racking, we no only find accurae conour from cluered envi-

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