Tracking Articulated Body by Dynamic Markov Network

Transcription

1 Tracng Artculated Body by Dynamc Marov Networ Yng Wu, Gang Hua, Tng Yu Department of Electrcal & Computer Engneerng Northwestern Unversty 2145 Sherdan Road, Evanston, IL Abstract A new method for vsual tracng of artculated objects s presented. Analyzng artculated moton s challengng because the dmensonalty ncrease potentally demands tremendous ncrease of computaton. To ease ths problem, we propose an approach that analyzes subparts locally whle renforcng the structural constrants at the mean tme. The computatonal model of the proposed approach s based on a dynamc Marov networ, a generatve model whch characterzes the dynamcs and the mage observatons of each ndvdual subpart as well as the moton constrants among dfferent subparts. Probablstc varatonal analyss of the model reveals a mean feld approxmaton to the posteror denstes of each subparts gven vsual evdence, and provdes a computatonally effcent way for such a dffcult Bayesan nference problem. In addton, we desgn mean feld Monte Carlo (MFMC) algorthms, n whch a set of low dmensonal partcle flters nteract wth each other and solve the hgh dmensonal problem collaboratvely. Extensve experments on tracng human body parts demonstrate the effectveness, sgnfcance and computatonal effcency of the proposed method. 1 Introducton Tracng artculated moton n mages s an mportant problem, especally when the research of vdeo-based human sensng has been advocated to acheve such emergng applcatons as perceptual nterfaces [20], smart vdeo survellance [8] and automatc vdeo footage [4], etc. The problem nvolves the localzaton and dentfcaton of a set of lned but artculated subparts. Inhertng all the dffcultes from sngle object tracng, the problem of tracng artculated body has to tacle some specal challenges. One of these s the complexty ncurred by the degrees of freedom of the artculated body. Dfferent from multple target tracng where the moton of each subpart s ndependent of others, the physcal lns among dfferent subparts renforce moton constrans upon these artculated subparts. We can have an ntutve comparson of these two cases by the confguraton space whch s the jont moton space of the set of subparts. If the moton of subparts are ndependent, then confguraton space wll enjoy a nce property that the moton of each subpart stays n a lnear subspace whch s orthogonal to the subspaces correspondng to other subparts. Thus, ndependent tracers can be used to trac ndependent multple targets and the complexty s almost lnear w.r.t. the number of targets. However, when the subparts are physcally lned, the confguraton space wll not have such a nce orthogonalty and factorzaton property of subspaces. Thus, the hgh dmensonalty seems unavodable, whch s generally assocated wth the exponental ncrease of computaton due to the curse of dmensonalty. Varous approaches have been nvestgated to allevate ths challenge (see Secton 2 for detals), such as dynamc programmng [18], annealed samplng [6], parttoned samplng [15, 16], egenspace tracng [1], hybrd Monte Carlo flterng [5], etc. Dfferent from these approaches, n ths paper, we propose a novel soluton based on a dynamc Marov networ model and varatonal mean feld approxmatons. The proposed dynamc Marov networ embeds the subparts constrants n an undrected graphcal model (.e., a Marov networ) assocated wth mage observaton processes, thus the model serves as a generatve model for the artculated moton. Due to the dense connectons n the graph, exact analyss s complcated and ntractable. When we perform an analyss based on varatonal mean feld method, tght approxmaton can be acheved whle the computatonal complexty s sgnfcantly reduced. At each tme nstance, the mean feld soluton s acheved through effcent Monte Carlo algorthm. And based on that, we desgn a mean feld sequental Monte Carlo for artculated body tracng. Extensve experments show the effectveness and effcency of the proposed approach. 2 Related Wor There s a substantal lterature on artculated moton analyss, and many dfferent approaches have been nvestgated. For all these methods, three mportant ssues should be addressed: the representatons for artculated objects, the

2 computatonal paradgms, and the way of reducng computaton. Bascally, there could be two types of choces for artculated object representatons. One employs jont angles [3, 13, 17, 16, 21], whle the other uses the collecton of the moton of all subparts. Of course, the frst representaton s non-redundant and reflects the degrees of freedom of the artculated moton drectly, whle the second one s hghly redundant. However, due to the ndependence of the jont angles, the frst method may suffer from an rreducble dlemma snce the ntrnsc dmensonalty s probably reached. In ths sense, the moton estmaton problem can be posed as an unconstraned optmzaton n a hgh dmensonal space (f we do not consder the natural moton constran as n [21]). On the other hand, f the artculated moton s redundantly descrbed by the ndvdual moton of the subparts, each subpart may be solved ndvdually, and then projected to the constraned space. Thus, t corresponds to a constraned optmzaton problem n a hgh dmensonal space. By tang advantage of the structure of such a redundant representaton, effcent solutons can be found as n ths paper. There are also dfferent computatonal paradgms for artculated moton analyss. It could be determnstc or probablstc. Determnstc methods generally formulate the problem as a parameter estmaton problem based on nonlnear programmng technques, and then dfferental approach can be taen [17, 3, 13]. Thus, the dffculty of local mnma exsts. A probablstc approach formulates the moton analyss problem as a Bayesan nference problem, where partcle-based Monte Carlo strateges provde flexble but ntensve computng framewors. In general, the number of partcles needed wll ncrease exponentally wth the ncrease of the dmensonalty. Therefore, t s crucal to reduce the computaton. Dfferent schemes have been nvestgated to reduce the number of partcles. For example, the annealed partcle flterng method performs a coarse-to-fne layered search [6], parttoned samplng s n the sprt of coordnate descent and preforms a herarchcal samplng [15, 16]. Both methods wor wth hgh dmensonal probablty spaces. Dfferent from these methods, ths paper presents a mean feld Monte Carlo (MFMC) algorthm n whch a set of low dmensonal partcle flters nteract wth each other to solve a hgh dmensonal problem collaboratvely. 3 The Representaton of an Artculated Body We denote the moton of each ndvdual subpart by x, whch can be the parameters of an affne moton. The moton of an artculated body s the concatenaton = [x 1 ::: x M ]. Certanly, t s hghly redundant. The mage observaton assocated wth x s denoted by z,whch could be the detected edges of the shape contour of the subpart, and the collectve mage observaton of the entre artculated body s =[z 1 ::: z M ]. An mportant tas s to nfer the posteror p(j) ψ 3 (x 3,x ) 3 ψ (x ) Fgure 1: The Marov Networ for an artculated body. As shown n Fgure 1, a mxture of undrected and drected graphcal model can be used to characterze the generatve process. The hdden layer s an undrected graph G x = fv Eg, representng the relatonshp among dfferent artculated parts. Obvously, dfferent parts are not ndependent, and each ndvdual part only nteracts wth ts neghborhood parts. We denote the neghborhood parts of by N (). Clearly, t s a Marov networ. In addton, each ndvdual part s assocated wth ts observaton and the condtonal lelhood dstrbuton p(z jx ) s represented by a drected ln. Gven the undrected graph of, p() can be modelled as a Gbbs dstrbuton and can be factorzed as: p() = 1 c Y c2c p(z x ) 5 5 M M c( c ) (1) where c s a clque n the set of clques C of the undrected graph, c s the set of hdden nodes assocated wth the clque and c( c ) s the probablty of ths clque, and c s a normalzaton term or the partton functon. Although c s dffcult to compute, we do not compute t, snce a Monte Carlo method wll be used as shown n later sectons. The model accommodates two types of clques: the frst order clque,.e., 2 C 1 = V, and second S order clque,.e., ( j) 2 C 2 = E, wherec = C 1 C2. The assocated s denoted by and j, respectvely. Thus, Eq. 1 can also be wrtten as: p() = 1 c Y ( j)2c 2 j(x x j ) Y 2C 1 (x ) (2) where (x ) provdes a local pror for x,and j (x x j ) presents the constrants between the neghborhood nodes x and x j.inotherwords, (x ) predcates a pror for the - th part, whle j(x x j ) renforces the constrants between the -th part and the j-th part. As a specfc example, t can be modelled as: j (x x j ) / e D(x xj )T 1 D(x x j ) where D(x x j ) = u (x ) u j (x j ), and u (x ) and u j (x j ) are shown n Fgure 2. (3)

3 x u u j D(x, x j ) x j Fgure 2: The constrant of two artculated parts. Gven a x, ts local observaton z s ndependent of other artculated parts. Thus, we have: p(j) = ny =1 p (z jx ): (4) The problem of great nterest s to nfer the posteror p(x j). An ntuton s that the posteror of x should be affected by three factors: ts local pror, ts local evdence z, and the constrants renforced by ts neghborhood through j. Ths ntuton wll become clearer n Secton 4. Snce the exact analyss of such a model s complcated and nvolves heavy computaton, t s more plausble to have an approxmate but effcent soluton. 4 Mean Feld Approxmaton Varatonal analyss provdes an approxmate method for analyzng the model [12, 11]. The core dea of varatonal approxmaton s to fnd an optmal varatonal dstrbuton q() that approxmates the posteror dstrbuton p(j), such that the Kullbac-Lebler (KL) dvergence of these two dstrbutons s mnmzed,.e., q () = arg mn KL(q(jjp(j)) q = arg mn q q()log x q() p(j) Selectng a good class of varatonal dstrbutons q would largely ease the dffcultes, but t requres substantal creatvty [12]. Here, we adopt a fully factorzed form for smplcty: q() = MY q (x ) (5) where q (x ) s an ndependent dstrbuton of the hdden node x. Then, H(q) = P H(q ), whereh(q) s the entropy of q(), andh(q ) s the entropy of q(x ). Then, the KL-dvergence can be wrtten as: KL(q )= H(q ) x q (x )E q [log p( )jx ] 6= H(q ) + log p() (6) where E q [jx ] s the condtonal expectaton gven x w.r.t. q(), andlog p() s the data lelhood, whch s a constant. To search for a set of q to mnmze Eq. 6, snce each q s constraned to be a vald dstrbuton functon, we should construct a Lagrangan for each q : L(q )=KL(q )+( x q 1) (7) Settng the dervatve to zero, t s easy to see the soluton to ths constraned optmzaton problem s a set of fxed pont equatons: q (x )= 1 e E q[log p( )jx ] where E q [log p( )jx ] s the condtonal expectaton gven x, s a partton functon for normalzaton, and 1 M. The teratve updatng of q (x ) wll monotoncally decrease the KL-dvergence, and eventually reach an equlbrum. These updatng equatons are called mean feld equatons. Moreover, the factorzaton of p() n Eq. 2 enables further smplfcaton of the mean feld equatons n Eq. 8. It s easytoshowthat: q (x ) 1 0 p (z jx ) (x )M (x ) where M (x ) = expf 2N () (8) x q (x )log (x x )g (9) where 0 s a constant, and N () s the neghborhood of the subpart. From Eq. 9, the ntuton stated at the end of Secton 3 s more pronounced: the varatonal belef of a subpart x s determned by three factors: the local condtonal lelhood p (z jx ), the local pror (x ),andthe belefs of the neghborhood subparts x N () (we call t neghborhood pror). Ths s llustrated n Fgure 3. ψ m p (z x ) ψ 2 ψ ψ 1 neghborhood pror local pror local lelhood Fgure 3: Three factors affect the updatng of q(x ). Thus,we can treat the term p (z jx ) (x ) as an analogue as the local belef, and treat the term M (x ) as an analogue to the message [7] propagated through the nearby subpart of x n the belef propagaton approach, but

4 the computaton of M (x) here s easer. In addton, we can clearly see from ths equaton that the computaton s sgnfcantly reduced by avodng mult-dmensonal ntegrals, snce Eq. 9 nvolves only one dmensonal ntegrals. 5 Mean Feld Monte Carlo (MFMC) In ths secton, we propose a Monte Carlo method to mplement the mean feld updatng as dscussed n Secton 4. We call ths method Mean Feld Monte Carlo (MFMC). Once the mean feld updatng converges to a fxed pont, then the set of optmal varatonal dstrbutons q(x),where =1 ::: M, s obtaned and can be treated as the optmal approxmaton to the posteror densty p(xj). To mae the presentaton clear, here we use a 2-ln body as an example. W.l.g., we use and j to ndex the two lned subparts, and we use to ndex the mean feld teraton. At the 1-th teraton, for each subpart, a set of partcle s mantaned to represent the varatonal dstrbuton,.e., q 1 q 1 j (x) fs ( 1) ( 1)g N n=1 (xj) fs j ( 1) j ( 1)g N n=1 where s and denote the sample and the weght respectvely. Then at the next teraton, we perform the followng steps accordng to Eq. 9: 1. Samplng local pror (x) for fs () 1g N n=1; 2. calculatng the message from j: m = N t=1 j ( 1) log j (s () s (t) j ( 1)): 3. Performng observaton for each partcle s w = p(z js ()): 4. Re-weghtng the partcles by: () =e m w : (), and normalze to produce fs () ()g. 5. Performng the same steps for j accordng to Eq. 9. And then ncrease for next mean feld updatng. After the -th teraton, we end up wth: q (x ) fs () ()g N n=1 qj (x j ) fs j () j ()g N n=1 After several teratons, the dstrbuton wll reach an equlbrum. For a subpart whch s lned to multple subparts, the only dfference s n the 2nd step of calculatng messages, m = N j2n () t=1 j ( 1) log j (s () s (t) j ( 1)): whch sums over all messages passed from the neghbors N () (.e., the Marov blanet) of x. Snce x descrbes the moton of a subpart, ts mage observaton z should be a functon of x,.e., p(zjx) s n fact p(z(x)jx). Sncep(zjx) wll be used to re-weght the belef (or the posteror densty) of x, the locatons of the partcles fs g wll affect the fath of approxmatng the belef by the set of partcles, f the rato of vald partcles s not satsfactory (meanng that a small porton of the partcles domnates the re-weghtng). To enhance the rato of vald partcles, we use mportance samplng technque [14] to place the partcles to better locatons. The only modfcaton on the above mean feld Monte Carlo (MFMC) s on the frst step: nstead of samplng the local pror (x) drectly to produce fs 1g N n=1,wedraw samples fs 1g N n=1 from an mportance densty g(x). After weght compensaton, the set of re-weghted partcle stll a properly weghted set for the densty (x),.e., (s (x) fs ) g N g(s n=1 : ) The selecton of mportance densty can be arbtrary. Here we gve an specfc example by usng a two-ln (where and j are connected subparts). To generate samples for (x), wefndthemeanss and s j from the two partcle sets. After dentfyng the pont u j on s j and the medan axs L of s (see Fgure 4), we sample u from G(u : u j u ),andl from G(L : L L ),whereg s Gaussan. L L s s u u j Fgure 4: Importance densty. Then the sample s s produced by (L the mportance densty s: g(x) =G(u :u j u )G(L : L L ): s j u j ), and Smlar mportance denstes can be easly constructed for a subpart whch s lned to multple subparts. The use of mportance samplng technques greatly enhances the robustness of the mean feld Monte Carlo algorthms. Sudderth et al [19] and Isard [9] have ndependently developed algorthms for the nteractons among multple partcle sets. Ther methods are based on belef propagaton, whle our method on probablstc varatonal analyss and mean feld teratons.

5 6 Dynamc Marov Networ and Sequental Mean Feld Monte Carlo Secton 4 and Secton 5 descrbe the mean feld approxmaton and mean feld Monte Carlo at one tme nstance. They can be easly modfed for tracng. When consderng multple tme nstances, the model becomes a dynamc Marov networ, as shown n Fgure 5. Denote 1,t 1,t 2,t 2,t,t,t M,t M,t 1,t+1 1,t+1 2,t+1 2,t+1,t+1,t+1 M,t+1 M,t+1 Fgure 5: Dynamc Marov Networ the collecton of observatons by t = f 1 ::: t g. Tracng algorthms am at nferrng p( t j t ) by nowng p( t 1 j t 1 ). It nvolves a densty propagaton process [10]: p( t j t ) / p( t j t ) x t 1 p( t j t 1 )p( t 1 j t 1 ) Once conssts of a number of artculated parts, the ncrease of dmensonalty wll ncur exponental ncrease of computaton. The advantage of mean feld approxmaton s that t decouples dfferent parts, and transforms the problem of exponental complexty to a smpler problem close to lnear complexty. The constrant renforcement needs some computaton as a cost, but t s not sgnfcant. At tme nstance t, mean feld approxmaton fnds a varatonal dstrbuton q t (x ) to approxmate p(x t j t ) for the -th subpart. The mean feld equaton can be wrtten as: q t (x t )= 1 0 p (z t jx t ) expf 2N () p(x t jx t 1 )q t 1 (x ) x t q t (x )log (x t x t )g (10) Comparng Eq. 10 to Eq. 9, we clearly see that the predcaton densty R p(x t jx t 1 )q t 1 (x ) n Eq. 10 of a dynamc Marov networ plays the same role as (x ) n Eq. 9. Thus, at tme nstance t, the varatonal belef of the -th subpart s also determned by three factors: the local evdence, the predcaton pror from prevous tme frame, and the belef of the neghborhood subparts. Therefore, the sequental mean feld Monte Carlo can be obtaned by modfyng the mean feld Monte Carlo algorthm n Secton 5. At the frst step, nstead of samplng from (x ), we should sample the predcton pror nstead. Suppose at t 1, q t 1 (x ) s represented by: q t 1 (x ) fs t 1 t 1 gn n=1 : The, we can use the followng steps to replace the 1st step n the mean feld Monte Carlo algorthm n Secton 5: 1.a Re-samplng from q t 1 (x ) for fes t 1 1gN n=1. 1.b 8 es t, samplng s t from p(x tjx t 1 ). Impressve results have been acheved and reported n Secton 7. We have a rough comparson on the computatonal complexty of the proposed approach wth the orgnal Condensaton algorthm wth jont angle representaton. Assume the artculated body conssts of M subparts, each of whch contrbute one DoF, and assume a number of T partcles are needed for tracng one subpart. In addton, we assume when one more DoF s added, Condensaton needs P T partcles to wor. Through our experments, 10 s reasonable for P. In our mean feld Monte Carlo, we denote the number of mean feld teraton by K, whchs 5 n our experments. In both methods, the most ntensve computaton s on calculatng mage observaton, whle the extra computaton nduced by M (x ) n Eq. 9 s neglgble. Thus, the complexty of our method s O(TKM), whle Condensaton has O(TP M 1 ) whch s much larger than the proposed mean feld Monte Carlo algorthm. In addton, MFMC s dfferent from the parttoned samplng (PS) method [15, 16], although both can reduce the exponental complexty. (1) PS apples to centralzed models wth ndependent dmensons, whle MFMC can handle varous HDMs ncludng artculaton, deformaton and mult-moton; (2) PS uses one hgh-dmensonal partcle flter, whle MFMC use a networ of low-dmensonal but collaboratve partcle flters; (3) PS s herarchcal and un-drectonal, whle MFMC s networed and multdrectonal. 7 Results We performed extensve experments on artculated body wth dfferent DoFs, and obtaned mpressve results as reported n ths secton. 7.1 Expermental Setup Our experments manly concerned about 2D tracng. Thus we adopted a cardboard model where each subpart n the artculated body s represented by a planar object, and thus the state of x s the parameters of a 2D affne transform. The moton model p(x t jx t 1 ) s a standard 2nd order const acceleraton model for each subpart. Although the moton model can be learned, we preset the parameters for smplcty.

6 The observaton model p(z jx ) s also an mportant factor n tracng. We used two types of vsual cues: edge and ntensty. We adopted the same method n CONDENSA- TION [10, 2] for edge observaton, where a set of ndependent measurement lnes were used to measure the lelhood of detected edge ponts. In addton, snce the artculated targets were human body parts and the sn or clothes on the body parts are smlar, we also used the ntensty clue and assumed the dstrbuton of the ntensty of a subpart be a Gaussan dstrbuton. The mean and varance of the Gaussan densty was traned for each ndvdual subpart. 7.2 Results of MFMC Iteraton To verfy f the mean feld updatng does converge and to chec f t s functonng as expected, we collected the ntermedate results on MFMC teraton. Two examples are shown n Fgure 6 and Fgure 7. The upper half shows an example of a 2-part arm, and the lower half 3-part fnger. In both cases, the estmates of the frst fve teratons are shown. Before the teraton, the ntal status was qute unpleasant. But after a couple of mean feld teraton, the estmates settled down on the rght postons as expected. From our experments, most teratons converged n less than fve tmes. 7.3 Varous Artculated Objects To demonstrate the effectveness, effcency and scalablty, we performed experments on varous artculated objects of dfference DoFs, ncludng a 2-part arm, 3-part fnger, 6-part upper body, and 10-part full body 1. The frst test sequence s a 2-part arm, whch conssts of two subparts: upper arm and lower arm. The sequence conssts of 441 frames. The lower arm presents larger moton than upper arm n the testng sequence. The MFMC algorthm performed excellently due to the constrant renforcement. Sample frames are shown n Fgure 8. We compared the results from MFMC wth multple ndependent tracers (MT). Although there are only two subparts, MT dd not produce satsfactory results, snce ether one had rss to lose trac and there were no other constrants to get t bac except mage observatons, and MT hardly produced plausble results satsfyng the physcal ln constrants. Some frames of MT are shown n Fgure 9. The second test sequence s on a 3-part fnger and conssts of 182 frames. As expected, MFMC produced very robust and stable result. Sample frames are shown n Fgure 10. The thrd test sequence s on a 6-part upper body, where complex arm motons present as well as global movement of the torso and head. The sequence conssts of 834 frames. Although the artculaton s qute complcated, t dd not fal MFMC. Sample frames are shown n Fgure Demo sequences are at yngwu. The most complcated test sequence we have expermented s the 10-part full body moton, and sequence has 368 frames. Arms and legs are the most artculated body parts, and they present sgnfcant moton. None of our run of MT succeeded, because a leg was easy to get lost and never be able to come bac. Sample frames of MT are shown n Fgure 12. When MFMC was appled, the tracng result was stll very stable unle MT. Through subjectve evaluaton, the tracng qualty dd not decrease due to the ncrease of the complexty of the artculaton. Sample frames are shown n Fgure 13. The MFMC algorthm runs on a sngle processor PC of 2.0GHz runnng WndowP. We dd not perform code optmzaton. For all these experments, the number of mean feld teraton was set to 5. The number of partcles for each part and the frame rates are shown n Table 1. experments 2-part 3-part 6-part 10-part partcles/part frame/second Table 1: A comparson of the computaton of dfferent artculated objects. The exponental requrement for computaton s overcome as expected. 8 Dscusson and Conclusons Tracng artculated objects s a challengng problem, snce the ncrease of number of subparts and the physcal connecton constrants of them would potentally ncur hgh dmensonalty, and fal tracng algorthms developed for sngle target. Thus, algorthms wth close to lnear complexty would have much better scalablty. In ths paper, we propose a collaboratve approach to acheve such a goal. Instead of usng the jont angle representaton whch s rreducble, we adopt a hghly redundant representaton for artculated body,.e., represent ndvdual subpart by ts own moton parameters, but renforce the constrants of dfferent subparts by a Marov networ. Varatonal analyss s performed for approxmated analyss of ths graphcal model. Interestngly, a set of fxed pont equatons (.e., the mean feld equatons) s found, whch suggests a collaboratve soluton to the problem through nteracton wth neghborhood subparts and through teratons. Then a mean feld Monte Carlo (MFMC) algorthm s desgned to acheve effectve computaton. Consderng moton, we propose a dynamc Marov networ model and MFMC s extended to a sequental MFMC algorthm for tracng. Extensve experments demonstrate the applcablty of the proposed methods.

7 Fgure 6: The frst fve teratons of MFMC on the (2-part) Arm sequence. Fgure 7: The frst fve teratons of MFMC on the (3-part) Fnger sequence. One of the future wor s to extend the algorthm to 3D. Snce self-occluson seems a severe ssue for artculated moton, another possble drecton s to desgn collaboratve algorthms for solvng the occluson problem. References [1] M. Blac and A. Jepson. Egentracng: Robust matchng and tracng of artculated object usng a vew-based representaton. In Proc. European Conf. Computer Vson, volume 1, pages , Cambrdge, UK, [2] A.BlaeandM.Isard. Actve Contours. Sprnger-Verlag, London, [3] C. Bregler and J. Mal. Tracng people wth twsts and exponental maps. In Proc. IEEE Conf. on Computer Vson and Pattern Recognton, pages 8 15, Santa Barbara, CA, June [4] T.-J. Cham and J. Rehg. A multple hypothess approach to fgure tracng. In Proc. IEEE Conf. on Computer Vson and Pattern Recognton, volume 2, pages , [5] K. Choo and D. Fleet. People tracng usng hybrd Monte Carlo flterng. In Proc. IEEE Int l Conf. on Computer Vson, volume II, pages , Vancouver, Canada, July [6] J. Deutscher, A. Blae, and I. Red. Artculated body moton capture by annealed partcle flterng. In Proc. IEEE Conf. on Computer Vson and Pattern Recognton, volume II, pages , Hlton Head Island, South Carolna, [7] W. Freeman, E. Pasztor, and O. Carmchael. Learnng lowlevel vson. Int l Journal of Computer Vson, 40:25 47, [8] I. Hartaoglu, D. Harwood, and L. Davs. W4: Who? when? where? what? a real tme system for detectng and tracng people. In Proc. IEEE Int l Conf. on Face and Gesture Recognton, Nara, Japan, Aprl [9] M. Isard. PAMPAS: Real-valued graphcal models for computer vson. In Proc. IEEE Conf. on Computer Vson and Pattern Recognton, volume I, pages , Madson, WI, June [10] M. Isard and A. Blae. Contour tracng by stochastc propagaton of condtonal densty. In Proc. of European Conf. on Computer Vson, pages , Cambrdge, UK, [11] T. S. Jaaola. Tutoral on varatonal approxmaton methods. MIT AI Lab TR, [12] M. Jordan,. Ghahraman, T. Jaaola, and L. Saul. An ntroducton to varatonal methods for graphcal models. Machne Learnng, 37: , [13] S. Ju, M. Blac, and Y. Yacoob. Cardboard people: A parametrzed model of artculated moton. In Proc. Int l Conf. on Automatc Face and Gesture Recognton, pages 38 44, Kllngton, Vermont, Oct [14] J. Lu, R. Chen, and T. Logvneno. A theoretcal framewor for sequental mportance samplng and resamplng. In A. Doucet, N. de Fretas, and N. Gordon, edtors, Sequental Monte Carlo n Practce. Sprnger-Verlag, New Yor, [15] J. MacCormc and A. Blae. A probablstc excluson prncple for tracng multple objects. In Proc. IEEE Int l Conf. on Computer Vson, pages , Greece, [16] J. MacCormc and M. Isard. Parttoned samplng, artculated objects, and nterface-qualty hand tracng. In Proc. of European Conf. on Computer Vson, volume 2, pages 3 19, [17] J. Rehg and T. Kanade. Model-based tracng of selfoccludng artculated objects. In Proc. of IEEE Int l Conf. Computer Vson, pages , [18] Y. Song,. Feng, and P. Perona. Towards detecton of human moton. In Proc. IEEE Conf. on Computer Vson and Pattern Recognton, Hlton Head Island, SC, June [19] E. Sudderth, A. Ihler, W. Freeman, and A. Wllsy. Nonparametrc belef propagaton. In Proc. IEEE Conf. on Computer Vson and Pattern Recognton, volume I, pages , Madson, WI, June [20] C. Wren, A. Azarbayejan, T. Darrel, and A. Pentland. Pfnder: Real-tme tracng of the human body. IEEE Trans. on Pattern Analyss and Machne Intellgence, 9: , July [21] Y. Wu, J. Ln, and T. S. Huang. Capturng natural hand artculaton. In Proc. IEEE Int l Conference on Computer Vson, volume II, pages , Vancouver, July 2001.

8 Fgure 8: Mean feld Monte Carlo (MFMC): tracng 2-part arm. Fgure 9: Multple ndependent tracer (MT): tracng 2-part arm. Fgure 10: Mean feld Monte Carlo (MFMC): tracng 3-part fnger. Fgure 11: Mean feld Monte Carlo (MFMC): tracng 6-part upper body. Fgure 12: Multple ndependent tracer (MT): tracng 10-part full body. Fgure 13: Mean feld Monte Carlo (MFMC): tracng 10-part full body.