Kernel-Based Bayesian Filtering for Object Tracking

Transcription

1 Kernel-Based Bayesan Flerng for Objec Trackng Bohyung Han Yng Zhu Dorn Comancu Larry Davs Dep. of Compuer Scence Real-Tme Vson and Modelng Inegraed Daa and Sysems Unversy of Maryland Semens Corporae Research Semens Corporae Research College Park, MD 74, USA Prnceon, NJ 854, USA Prnceon, NJ 854, USA {bhhan, Absrac Parcle flerng provdes a general framework for propagang probably densy funcons n non-lnear and non-gaussan sysems. However, he algorhm s based on a Mone Carlo approach and samplng s a problemac ssue, especally for hgh dmensonal problems. Ths paper presens a new kernelbased Bayesan flerng framework, whch adops an analyc approach o beer approxmae and propagae densy funcons. In hs framework, he echnques of densy nerpolaon and densy approxmaon are nroduced o represen he lkelhood and he poseror denses by Gaussan mxures, where all parameers such as he number of mxands, her wegh, mean, and covarance are auomacally deermned. The proposed analyc approach s shown o perform samplng more effcenly n hgh dmensonal space. We apply our algorhm o he real-me rackng problem, and demonsrae s performance on real vdeo sequences as well as synhec examples. Inroducon Parcle flerng s a Mone Carlo approach o solve he recursve Bayesan flerng problem. Alhough provdes racable soluons o non-lnear and non-gaussan sysems, s faced wh praccal ssues such as sample degeneracy and sample mpovershmen []. Moreover, o acheve relable flerng, he sample sze can grow exponenally as he dmenson of he sae space ncreases. To overcome hese ssues, we explore an analyc approach o approxmae densy funcons and nroduce a new kernel-based flerng scheme. The man dea of hs work s o manan an analyc represenaon of relevan densy funcons and propagae hem over me. In hs paper, kernel-based densy represenaon s adoped.. Relaed Work There have been many paramerc densy represenaons proposed for varous applcaons. In [5, ], he auhors sugges Gaussan mxure models, bu her mehod requres knowledge of he number of componens, whch s dffcul o know n advance. A more elaborae densy represenaon s descrbed n [], where a 3-componen mxure s used for he arge modelng n objec rackng problem, bu hs approach canno overcome he drawback of paramerc mehods. Kernel densy esmaon [8] s a wdely used non-paramerc approach n compuer vson. Is major advanage s he flexbly o represen very complcaed denses effecvely. Bu s very hgh memory requremens and compuaonal complexy nhb he use of hs mehod. For Bayesan flerng, Cham and Rehg [3] nroduce a pecewse Gaussan funcon o specfy he racker sae, n whch he seleced Gaussan componens characerze he neghborhoods around he modes. Ths dea s appled o mulple hypohess rackng n a hgh dmensonal space body racker, bu he samplng and he poseror compuaon are no sraghforward. The closes work o our paper s [3] where he poseror s represened wh a Gaussan mxure n a parcle fler framework. However, hs soluon may no provde a compac represenaon for he poseror, and he predcon and he updae seps are oversmplfed.. Our Approach In hs paper, we exend our prevous work [9] whch provdes he man framework of Kernel-based Bayesan flerng. We nroduce densy approxmaon and densy nerpolaon o represen densy funcons effcenly and effecvely. In boh echnques, he densy funcon s represened by a Gaussan mxure, where he number of mxands, her weghs, means and covarances are auomacally deermned. The densy approxmaon s based on a mode fndng algorhm [6, 7] derved from varable-bandwdh mean-shf whch provdes he mehodology o consruc a compac represenaon wh a small number of Gaussan kernels. A densy nerpolaon echnque s nroduced o oban a connuous represenaon of he measuremen lkelhood funcon. Unscened ransformaon UT) [, 6] s also adoped o deal wh non-lnear sae ranson models. These echnques are negraed no he Bayesan flerng framework. In he new kernel-based Bayesan flerng algorhm, he connuous represenaons of densy funcons are propagaed over me. The advanage of mananng an analyc represenaon of densy funcons les n effcen samplng. Ths s mporan for solvng hgh dmensonal problems. A mul-sage samplng sraegy s nroduced n densy nerpolaon for accurae approxmaon of he measuremen lkelhood funcon. The new

2 algorhm s appled o real-me objec rackng, and s performance s demonsraed hrough varous expermens. Ths paper s organzed as follows. Secon nroduces he new densy propagaon echnque n he Bayesan flerng framework. Secon 3 and 4 explan he densy approxmaon and he densy nerpolaon mehod, respecvely. Secon 5 demonsraes s performance by varous smulaon resuls wh synhec examples. Fnally, s demonsraed n secon 6 how our algorhm can be appled o objec rackng n real vdeos. Bayesan Flerng In hs secon, we nroduce he new Bayesan flerng framework, where he relevan densy funcons are approxmaed by kernel-based represenaons and propagaed over me.. Overvew In a dynamc sysem, he process and measuremen model are gven by x = gx,u ) ) z = hx,v ) ) where v and u are he process and he measuremen nose, respecvely. The sae varable x =,...,n) s characerzed by s probably densy funcon esmaed from he sequence of measuremens z =,...,n). In he sequenal Bayesan flerng framework, he condonal densy of he sae varable gven he measuremens s propagaed hrough predcon and updae sages, px z : ) = px x )px z : )dx 3) px z : ) = k pz x )px z : ) 4) where k = pz x )px z : )dx s a normalzaon consan ndependen of x. px z : ) s he pror probably densy funcon pdf), px z : ) s he predced pdf and pz x ) s he measuremen lkelhood funcon. The poseror pdf a me sep, px z : ), s used as he pror pdf n me sep +. A each me sep, he condonal dsrbuon of he sae varable x gven a sequence of measuremens z s represened by a Gaussan mxure. Our goal s o rean such a represenaon hrough he sages of predcon and updae, and o represen he poseror probably n he followng sep wh he same mxure form. The proposed flerng framework s descrbed as follows. Frs, unscened ransformaon UT) [, 6] s used o derve a mxure represenaon of he predced pdf px z : ). Second, he densy nerpolaon echnque wh a mul-sage samplng s nroduced o approxmae he lkelhood funcon wh an mxure form. By mulplyng wo mxure funcons, he poseror pdf s obaned hrough equaon 4). To preven he number of mxands from growng oo large, an algorhm of densy approxmaon based on mode fndng s appled o derve a compac represenaon for he poseror pdf.. Predcon by Unscened Transform Denoe by x =,...,n ) a se of means n R d and by P he correspondng covarance marces a me sep. Le each Gaussan have a wegh κ wh n = κ =, and le he pror densy funcon be gven by px z : ) = n π) d/ P = κ exp / D x,x ) ),P The unscened ransformaon [, 6] s a mehod for calculang he sascs of a random varable whch undergoes a non-lnear ransformaon. X,) X,j) = x = x d + λ)p ) j j =,...,d X,j) = x + d + λ)p ) j d j = d +,...,d W,) = λ/d + λ) W,j) = /d + λ) j =,...,d 6) where λ s a scalng parameer and d + λ)p ) j s he h row or column of he marx square roo of d + λ)p. W,j) s he wegh assocaed wh he j-h sgma pon where d j= W,j) =. These sgma vecors are propagaed hrough he non-lnear funcon, X,j) 5) = gx,j) ) =,...,d 7) and he mean and covarance for x are approxmaed usng a weghed sample mean and covarance of he poseror sgma pons, x = P = d = d = W,j) X,j) W,j) X,j) x )X,j) x ) + Q 8) where Q s he covarance marx for he process nose. For each mode n he pror, UT s appled ndependenly and he densy afer predcon s as follows. px z : ) = n π) d/ = κ P exp / D x, x, P ) ) 9) where κ = κ. Ths non-lnear ransformaon s guaraneed o be accurae up o he second order of he Taylor expanson.

3 .3 Mul-sage Samplng and Inerpolaon of Measuremen Lkelhood In conras o varous parcle flers, we represen he measuremen lkelhood funcon n an analyc form. A connuous approxmaon of he lkelhood funcon s nerpolaed from dscree samples. A mul-sage samplng scheme s nroduced o mprove he approxmaon progressvely. The advanage of he analyc represenaon s ha provdes a global vew of he landscape of he lkelhood funcon and hus enables effcen sample placemen..3. Mul-sage samplng Unlke he SIR algorhm [] whch uses he predced pdf as he proposal dsrbuon, we employ he mul-sage samplng sraegy and progressvely updae he proposal funcon based on he observaon. The predced pdf s used as he nal proposal dsrbuon q. q x ) = px z : ) ) Assume ha n oal, N samples are o be drawn o oban measuremen daa. In our mul-sage samplng scheme, N/m samples are drawn n he frs sage from he nal proposal dsrbuon ), where m s he number of samplng sages. An nal approxmaon of he lkelhood funcon p z x ) s obaned hrough surface nerpolaon wh Gaussan kernels. Deals of he densy nerpolaon algorhm s provded n secon 4. The proposal funcon s hen updaed by a lnear combnaon of he nal proposal dsrbuon and he curren approxmaon of he lkelhood funcon p z x ). We repeaedly approxmae he lkelhood funcon from avalable samples and updae he proposal dsrbuon. p j z x ) = τ τ exp D x,x,r ) ) ) q j x ) = α j )q j x ) + α j p j z x ) pj z x )dx ) where =,..., j m N, j =,...,m, and α j [, ] s adapaon rae. Snce he nformaon of he observaon s ncorporaed no he proposal dsrbuon o gude samplng, he mul-sage samplng sraegy explores he lkelhood surface more effcenly han convenonal parcle flers. Thus, s especally advanageous n dealng wh hgh dmensonal sae space..3. Approxmaon of lkelhood funcon As dscussed prevously, he measuremen lkelhood s esmaed hrough he mul-sage samplng. Wh samples drawn from he mproved proposal dsrbuons, nermedae lkelhood funcons are consruced and used o updae he proposal dsrbuons. Afer m-sep repeon of hs procedure, he fnal measuremen dsrbuon s obaned. Algorhm presens Algorhm Measuremen Sep : S = φ : q x ) = px z : ) 3: for = o m do 4: draw sample s from proposal dsrbuon S = {sj) s j) q x ), j =,..., N/m} 5: S = S S 6: assgn mean and covarance for he elemen n S m ) N m +j = sj) Q ) N m +j = c dagknn k)...knn d k)) I 5) 7: compue lkelhood of each new sample l ) N m +j = hm ) N +j,v ) m 8: compue A and b for every elemen n S 9: w = nnlsa,b) 7) : p z x ) = τ j Nτj,x j,r j ) where τ = w, x = m, and R = Q p : q x ) = α )q x ) + α z x ) j ) p z x )dx : end for 3: pz x ) = p m z x ) he complee procedure o compue he lkelhood funcon, and he fnal measuremen funcon wh m Gaussans a me s gven by pz x ) = π) d/ m = τ R / exp D x,x,r ) ) 3) where τ, x andr are he wegh, mean and covarance marx of he -h kernel..4 Updae Snce boh he predced pdf and he measuremen funcons are represened by Gaussan mxures, he poseror pdf, as he produc of wo Gaussan mxures, can also be represened by a Gaussan mxure. Denoe he Gaussan componens of he predced pdf and he lkelhood funcon by N κ, x, P ) =,...,n ) and Nτ j,xj,rj ) j =,...,m ) respecvely, he produc of he wo dsrbuons s as follows. where n = ) N κ, x, P m ) Nτ j,xj,rj ) = n m = j= j= N κ τj,mj,σj ) 4) m j = Σ j P ) x + Rj ) x j ) 5) Σ j = P ) + R j ) ) 6) The resulng densy funcon n 4) s a weghed Gaussan mxure. However, he exponenal ncrease n he number of

4 componens over me could make he whole procedure nracable. In order o avod hs suaon, a densy approxmaon echnque s proposed o manan a compac ye accurae densy represenaon even afer densy propagaon hrough many me seps. Deals of he densy approxmaon algorhm s gven n secon 3. Afer he updae sep, he fnal poseror dsrbuon s gven by px z : ) = π) d/ n = P κ exp / D x,x,p ) ) where n s he number of componens a me sep. 3 Densy Approxmaon In hs secon, we revew an erave procedure of mode deecon derved from varable-bandwdh mean-shf [7], and densy approxmaon usng he mode deecon echnque [9]. 3. Mode Deecon and Densy Approxmaon 7) Suppose ha Nκ,x,P ) =...n) s a Gaussan kernel wh wegh κ, mean x, and covarance P n he d- dmensonal sae space, where n = κ =. Then, we defne he sample pon densy esmaor compued a pon x by ˆfx) = where π) d/ n = κ exp ) P / D x,x,p ) 8) D x,x,p ) x x ) P x x ). 9) Our purpose s o oban he compac represenaon of he densy funcon whch s a Gaussan mxure. The mode locaon and s wegh are found by mean-shf algorhm, and he covarance marx assocaed wh each mode s compued by usng Hessan marx. To fnd he graden ascen drecon a x, he varablebandwdh mean-shf vecor a x s gven by mx) = n = where he weghs ω x) = ω x)p ) n = ω x)p x ) x ) κ P / exp D x,x,p ) ) n = κ P / exp D x,x,p ) ) ) sasfy n = ω x) =. By compung he mean-shf vecor mx) and ranslang he locaon x by mx) eravely, a local maxmum of underlyng densy funcon s deeced. A formal check for he maxmum nvolves he compuaon of he Hessan marx n κ Ĥx) = π) d/ P = /exp ) D x,x,p ) P x x)x x) ) P P ) whch should be negave defne. If s no negave defne, he convergence pon mgh be a saddle pon or a local mnmum. In hs case, kernels assocaed wh such modes should be resored and consdered as separae modes for furher processng. The approxmae densy s obaned by deecng he mode locaon for every sample pon x and assgnng a sngle Gaussan kernel for each mode. Suppose ha he approxmae densy has n unque modes of x j j =... n ) wh assocaed wegh κ j whch s equal o he sum of he kernel weghs converged o x j. The Hessan marx Ĥj of each mode s used for he compuaon of P j as follows. P j = κ d+ j π Ĥ j ) d+ Ĥ j ) 3) The basc dea of equaon 3) s o f he covarance usng he curvaure n he neghborhood of he mode. The fnal densy approxmaon s hen gven by fx) = π) d/ n = κ P exp / D x, x, P ) ) 4) and n n s sasfed n mos cases. The approxmaon error ˆfx) fx) can be evaluaed sraghforwardly. 3. Performance of Approxmaon The accuracy of he densy approxmaon s demonsraed n Fgure. From a one-dmensonal dsrbuon composed of fve weghed Gaussans, samples are drawn, and he scale parameer s assgned as dscussed n secon 4.. Mean Inegraed Squared Error MISE) beween he orgnal and he approxmaed denses s calculaed for he error esmaon. The resul n fgure shows ha he mode fndng based on mean-shf and he covarance esmaon usng he Hessan s very accurae. Fgure shows he performance of he densy approxmaon whch s accurae enough o replace kernel densy esmaon n he mul-dmensonal case. 4 Densy Inerpolaon The densy approxmaon presened n secon 3 s an algorhm o fnd a compac represenaon when he mean, he covarance and he wegh for each kernel are gven. In he measuremen sep of Bayesan flerng, he lkelhood values are

5 By hs mehod, samples n dense areas have small scales and he densy wll be represened accuraely, bu sparse areas convey only relavely rough nformaon abou he densy funcon a) Fgure : Comparsons beween kernel densy esmaon and densy approxmaon D). For he approxmaon, samples are drawn from he orgnal dsrbuon N.,, ), N.35, 7, 4 ), N.5, 7, 8 ), N., 5, 6 ), and N., 7, 3 ). a) kernel densy esmaon b) densy approxmaon MISE = ).5 x 3.5 x 3 b) 4. Inerpolaon A Gaussan kernel s assgned o each sample for whch mean and covarance corresponds o he sample locaon and he scale s nalzed by he mehod n secon 4., respecvely. When he lkelhood value on each sample s gven, he wegh for each kernel can be compued by he Non-Negave Leas Square NNLS) mehod [4]. Denoe x as he mean locaon and P as he covarance marx for he -h sample =,...,n). Also, suppose ha l s he lkelhood value on he -h sample. The lkelhood a x j nduced by he -h kernel s gven by p x j ) = exp ) π) d/ P / D x j,x,p ). 6) a) b) 4 6 Defne an n n marx A havng an enry p x j ) n, j), and an n vecor b havng l n s -h row. Then, he wegh vecor w can be compued by solvng he followng consraned leas square problem, Fgure : Comparson beween kernel densy esmaon and densy approxmaons D). a) kernel densy esmaon b) densy approxmaon 4 samples, MISE = ) known for a se of samples. In hs case, he lkelhood surface can be nerpolaed from sample lkelhood. In hs secon, we descrbe he densy nerpolae algorhm. 4. Inal Scale Selecon One of he lmaons of kernel-based algorhms s ha hey nvolve he specfcaon of a scale parameer. Varous research has been performed for he scale selecon problem [, 7, 9], bu s very dffcul o fnd he opmal scale n general. Below, we explan a sraegy o deermne he scale parameer for he densy esmaon based on neares neghbors. The basc dea of hs mehod s very smple, and smlar approaches are dscussed n [4, 5]. Each sample s nended o cover he local regon around self n he d-dmensonal sae space wh s scale. For hs purpose, k-neares neghbors KNN) s used, and he kernel bandwdh scale) s deermned by he dsance o he k-h neares neghbor of a sample. Defne KNN jk) j d) o be he dsance o k-h neares neghbor from sample n he j-h dmenson, and hen he covarance marx P for -h sample s gven by P = c dagknn k) KNN k)...knn d k)) I 5) where c s a consan dependen upon he number of samples and he dmensonaly, and I s a d-dmensonal deny marx. mn w Aw b 7) subjec o w for =,...,n, and s denoed by w = nnlsa,b). The sze of marx A s deermned by he number of samples. When he sample sze s large, sparse marx operaon mehods can be used o solve w effcenly. Usually, many of he weghs wll be zero and he fnal densy funcon wll be a mxure of Gaussans wh a small number of componens. The densy nerpolaon smulaes he heavyaled densy funcon more accuraely han he densy approxmaon nroduced n secon 3, whle he densy approxmaon generally produces a more compac represenaon. 4.3 Performance of Inerpolaon Fgure 3 shows one-dmensonal densy nerpolaon resuls. For each case, samples are drawn and he nal scale for each sample s gven as explaned n secon 4.. The esmaed densy funcon approxmaes he orgnal densy very accuraely as seen n fgure 3. Two dfferen Gaussan mxures N.,, ) N.35, 7, 4 ) N.5, 7, 8 ) N., 5, 6 ) N., 7, 3 ) n example, and N.5,, 5 ) N., 5, 4 ) N.35, 6, 8 ) N.5, 75, 6 ) N.5, 9, 3 ) n example are esed for he nerpolaon. When 5 ndependen realzaons are performed, MISE and s varance are very small for boh examples as shown n able.

6 . orgnal densy. densy nerpolaon.9.9 x 3 x orgnal densy.3.. a) example densy nerpolaon 6 4 a) 4 6 Fgure 4: Comparson beween orgnal densy funcon and densy nerpolaon D). a) orgnal densy funcon b) densy nerpolaon wh 3 non-zero wegh componens 6 4 b) The frs process model s gven by he followng equaon, b) example Fgure 3: Two examples of orgnal densy funcons and her nerpolaons. In he nerpolaon graphs rgh), black sars represen he sample locaons samples). In case a) and b), and 4 componens have non-zero weghs, respecvely. x = x 5x + + x T x + 8 cos. )) + u 8) where s he vecor whose elemens are all ones. The process nose u s drawn from a Gaussan dsrbuon N,, I) ) where I s he deny marx. The measuremen model s gven by a non-lnear funcon Table : Error of densy nerpolaon MISE VAR example example Also, a mul-dmensonal densy funcon s nerpolaed n he same manner, and s performance s dscussed nex. In fgure 4, he densy nerpolaon produces a very accurae and sable resul when samples are drawn from he orgnal densy funcon MISE = , VAR = on average over 5 runs). These resuls show ha surface nerpolaon s a suffcenly accurae mehod o approxmae densy funcon gven samples and her correspondng lkelhood. 5 Smulaon In hs secon, synhec rackng examples are smulaed, and he performance of he kernel-based Bayesan flerng s compared wh he SIR algorhm []. Two dfferen process models one lnear and he oher non-lnear are seleced, and smulaons are performed for varous dmensons such as D, 3D, 5D, D, D and 5D. The accumulaed Mean Squared Error MSE) hrough 5 me seps s calculaed n each run, and 5 dencal expermens are made based on he same daa for accurae error esmaon. z = xt x + v 9) where v s drawn from a Gaussan dsrbuon N,,I ). For he esmaon of he measuremen funcon, ffy parcles parcles 5 sages) are drawn, and he poseror s esmaed and propagaed hrough he me sep 5). Fgure 5 demonsraes smulaon resuls by comparng MSE s and varances of boh algorhms. Accordng o our expermen wh he frs model, he SIR fler shows beer or equvalen performance n low dmensons such as D and 3D, bu our mehod sars o ouperform n hgh dmensons more han 5D. MSE dmenson a) Error varance dmenson b) Varance Fgure 5: MSE and varance of for MSE. Kernel-based Bayesan flerng wh 5 parcles blue sar), SIR wh 5 parcles red crcle), and SIR fler wh 5 parcles black square) for model. The second process model s very smple lnear model gven by x = x + cos )) + u 3)

7 where u N,, I) ). The same observaon model as equaon 9) s employed, and ffy samples are drawn for every smulaon. Kernel-based Bayesan flerng yelds smaller error n he hgh dmenson as n prevous case, and he dealed resuls are presened n fgure 6. MSE varance by he nverse exponenaon of he Bhaacharyya dsance beween he arge and he canddae hsograms as suggesed n [8]. Based on he lkelhood of each parcle and he nal covarance marx derved by he mehod n secon 4., he measuremen densy s consruced by densy nerpolaon. Two sequences are esed n our expermen. In he frs sequence, wo objecs a hand carryng a can are racked wh samples 4 samples 5 sages). The sae space s descrbed by a dmensonal vecor, whch s he concaenaon of wo 5 dmensonal vecors represenng wo ndependen ellpses as follows..5.5 x, y, lx, ly, r, x, y, lx, ly, r ) 3) dmenson a) Error dmenson b) Varance Fgure 6: MSE and varance of for MSE kernel-based Bayesan flerng wh 5 samples blue sar), SIR fler wh 5 parcles red crcle), and SIR fler wh 5 parcles black square) for model. Two dfferen process models produce almos smlar resuls, and kernel-based Bayesan flerng shows beer performance n hgh dmensonal cases as expeced. In order o demonsrae he benef of kernel-based parcles, we run he SIR algorhm wh 5 samples, and compare he performance wh our kernel-based Bayesan flerng wh 5 samples. Surprsngly, he MSE s of he wo cases are almos he same, and our algorhm has smaller varance of MSE han he SIR algorhm. Ths resul suggess ha kernel-based Bayesan flerng can be appled effecvely o hgh dmensonal applcaons, especally, when many samples are no avalable and he observaon process s very me-consumng. 6 Objec Trackng Parcle flerng provdes a convenen mehod for esmang and propagang he densy of sae varables regardless of he underlyng dsrbuon and he gven sysem n he Bayesan framework. Addonally, our kernel-based Bayesan flerng has an advanage of managng mul-modal densy funcons wh a relavely small number of samples. In hs secon, we demonsrae he performance of he kernel-based Bayesan flerng by rackng objecs n real vdeos. The overall rackng procedure s equvalen o wha s descrbed n secon 5, and we explan he process and he measuremen models brefly. A random walk s assumed for he process model snce s very dffcul o descrbe he accurae moon before he observaon, even hough our algorhm can accommodae he general non-lnear funcon by unscened ransformaon descrbed n secon.. 5-sage samplng s ncorporaed as nroduced n secon.3, and he lkelhood of each parcle s compued where x and y =, ) are he locaon of ellpses, lx s he lengh of x-axs, ly s he lengh of y-axs, and r s he roaon varable. The rackng resul s shown n fgure 7, and our algorhm successfully racks wo objecs for he whole sequence excep he perod ha he sde of he can s compleely occluded around 47h frame. a) = 43 b) = 34 c) = 387 d) = 456 e) = 57 c) = 898 Fgure 7: Objec rackng resul of can sequence. The upper bodes of wo persons are racked n he second sequence, n whch one occludes he oher compleely several mes. The sae vecor s consruced by he same mehod as n he can sequence, bu wo recangles are used nsead of ellpses. A 6 dmensonal vecor x, y, scale) for each recangle s used o descrbe he sae, and samples samples 5 sages) are used. Fgure 8 a) demonsraes he rackng resuls, and our algorhm shows good performance n spe of severe occlusons. The racker based on SIR algorhm s also mplemened, and compared wh our algorhm. As seen n fgure 8 b), he SIR algorhm shows unsable performance for he same sequence. Accordng o expermens, one would need o run he SIR algorhm usng abou 4 parcles o oban a comparable resul wh our algorhm usng samples.

8 a) resul by our mehod a) resul by he SIR algorhm Fgure 8: Objec rackng resul of person sequence a =, 95, 33, 93, 7, 3. 7 Dscusson and Concluson In hs paper, we proposed a new Bayesan flerng framework where analyc represenaons are used o approxmae relevan densy funcons. Densy approxmaon and nerpolaon echnque are nroduced n densy propagaon. Varous smulaons and ess on objec rackng n real vdeos show he effecveness of our densy approxmaon mehods and he kernel-based Bayesan flerng. By mananng analyc represenaons of he densy funcons, we can sample n he sae space more effecvely and more effcenly. Ths advanage s sgnfcan for hgh dmensonal problems. In addon, he approxmaon error can be monored and analyzed. Our fuure work s focused on analyzng he approxmaon error n he poseror dsrbuon and s propagaon over me. Acknowledgmen The suppor by Semens Corporae Research and VACE projec DOD4H84) s graefully acknowledged. References [] I. Abramson, On bandwdh varaon n kernel esmaes - a square roo law, The Annals of Sascs, vol., no. 4, pp. 7 3, 98. [] S. Arulampalam, S. Maskell, N. Gordon, and T. Clapp, A uoral on parcle flers for on-lne non-lnear/non-gaussan Bayesan rackng, IEEE Trans. Sgnal Process., vol. 5, no., pp ,. [3] T. Cham and J. Rehg, A mulple hypohess approach o fgure rackng, n Proc. IEEE Conf. on Compuer Vson and Paern Recognon, For Collns, CO, volume II, 999, pp [4] W. Cleveland, Robus locally weghed regresson and smoohng scaerplos, J. Amer. Sas. Assn., vol. 74, pp , 979. [5] W. Cleveland and C. Loader, Smoohng by local regresson: Prncples and mehods, Sascal Theory and Compuaonal Aspecs of Smoohng, pp. 49, 996. [6] D. Comancu and P. Meer, Mean shf: A robus approach oward feaure space analyss, IEEE Trans. Paern Anal. Machne Inell., vol. 4, no. 5, pp ,. [7] D. Comancu, V. Ramesh, and P. Meer, The varable bandwdh mean shf and daa-drven scale selecon, n Proc. 8h Inl. Conf. on Compuer Vson, Vancouver, Canada, volume I, July, pp [8] A. Elgammal, R. Duraswam, D. Harwood, and L. Davs, Background and foreground modelng usng nonparamerc kernel densy esmaon for vsual survellance, Proceedngs of IEEE, vol. 9, pp. 5 63,. [9] B. Han, D. Comancu, Y. Zhu, and L. Davs, Incremenal densy approxmaon and kernel-based bayesan flerng for objec rackng, n Proc. IEEE Conf. on Compuer Vson and Paern Recognon, Washngon DC, 4, pp [] M. Isard and A. Blake, Condensaon - Condonal densy propagaon for vsual rackng, Inl. J. of Compuer Vson, vol. 9, no., 998. [] A. Jepson, D. Flee, and T. El-Maragh, Robus onlne appearance models for vsual rackng, n Proc. IEEE Conf. on Compuer Vson and Paern Recognon, Hawa, volume I,, pp [] S. Juler and J. Uhlmann, A new exenson of he Kalman fler o nonlnear sysems, n Proceedngs SPIE, volume 368, 997, pp [3] J. Koecha and P. Djurc, Gaussan sum parcle flerng, IEEE Trans. Sgnal Process., vol. 5, no., pp. 6 6, 3. [4] C. L. Lauwon and B. J. Hanson, Solvng Leas Squares Problems. Prence-Hall, 974. [5] S. McKenna, Y. Raja, and S. Gong, Trackng colour objecs usng adapve mxure models, Image and Vson Compung Journal, vol. 7, pp. 3 9, 999. [6] R. Merwe, A. Douce, N. Freas, and E. Wan, The unscened parcle fler, Techncal Repor CUED/F-INFENG/TR 38, Cambrdge Unversy Engneerng Deparmen,. [7] B. Park and J. Marron, Comparson of daa-drven bandwdh selecors, J. of Amer. Sa. Assoc., vol. 85, pp. 66 7, 99. [8] P. Perez, C. Hue, J. Vermaak, and M. Gangne, Color-based probablsc rackng, n Proc. European Conf. on Compuer Vson, Copenhagen, Denmark, volume I,, pp [9] S. Sheaher and M. Jones, A relable daa-based bandwdh selecon mehod for kernel densy esmaon, J. Royal Sas. Soc. B, vol. 53, pp , 99. [] C. Sauffer and W. Grmson, Learnng paerns of acvy usng real-me rackng, IEEE Trans. Paern Anal. Machne Inell., vol., no. 8, pp ,.