Maintaining Multi-Modality through Mixture Tracking

Transcription

1 Mainaining Muli-Modaliy hrough Mixure Tracking Jaco Vermaak, Arnaud Douce Cambridge Universiy Engineering Deparmen Cambridge, CB2 1PZ, UK Parick Pérez Microsof Research Cambridge, CB3 0FB, UK Absrac In recen years paricle filers have become a remendously popular ool o perform racking for non-linear and/or non-gaussian models. This is due o heir simpliciy, generaliy and success over a wide range of challenging applicaions. Paricle filers, and Mone Carlo mehods in general, are however poor a consisenly mainaining he muli-modaliy of he arge disribuions ha may arise due o ambiguiy or he presence of muliple objecs. To address his shorcoming his paper proposes o model he arge disribuion as a non-parameric mixure model, and presens he general racking recursion in his case. I is shown how a Mone Carlo implemenaion of he general recursion leads o a mixure of paricle filers ha inerac only in he compuaion of he mixure weighs, hus leading o an efficien numerical algorihm, where all he resuls peraining o sandard paricle filers apply. The abiliy of he new mehod o mainain poserior muli-modaliy is illusraed on a synheic example and a real world racking problem involving he racking of fooball players in a video sequence. 1. Inroducion Tracking involves he deecion and recursive localisaion of an objec or objecs of ineres based on sequenial daa measuremens. Typical examples include face racking in video sequences [6], racking aircraf using radar reurns [2], localising a mobile robo using laser range measuremens [12], and many more. In pracical seings here are many facors ha conribue owards he uncerainy in an objec s exac locaion and configuraion. These include measuremen noise, inaccurae modelling, cluer (false posiives), flucuaions in environmenal condiions, ec. To adequaely capure he uncerainy due o hese facors a probabilisic framework is required. Wihin a racking conex one paricularly popular approach is Bayesian Sequenial Esimaion. This framework allows he recursive esimaion of a ime-evolving poserior disribuion ha describes he objec sae condiional on all he observaions seen so far, commonly known as he filering disribuion. I requires he definiion of a Markovian dynamic model ha describes how he objec sae evolves, and a model o evaluae he likelihood of a hypohesised sae giving rise o he observed daa. This, in heory, is sufficien o allow recursive esimaion of he filering disribuion. However, he likelihood models for racking ofen lead o inracable inference, requiring approximaion echniques. In recen years Sequenial Mone Carlo Esimaion, oherwise known as Paricle Filering [4], has proved o be a popular approximaion mehodology. Is populariy sems from is simpliciy, generaliy and success over a wide range of challenging applicaions. I represens he filering disribuion wih a se of samples, or paricles, and associaed imporance weighs, which are hen propagaed hrough ime o give approximaions of he filering disribuion a subsequen ime seps. I requires only he definiion of a suiable proposal disribuion from which new paricles can be simulaed, and he abiliy o evaluae he likelihood and dynamic models. One imporan shorcoming of paricle filers, and Mone Carlo mehods in general, is ha hey are poor a consisenly mainaining he muli-modaliy in he arge disribuion. Muliple modes arise if here is ambiguiy abou he objec sae due o insufficien measuremens or cluer, or if he measuremens come from muliple objecs. In he firs case i is desirable o rack all he modes unil he ambiguiy can be naurally resolved, and in he second, i is ofen required o rack all he objecs presen. In a pracical paricle filer implemenaion, however, i ofen happens ha all he paricles quickly migrae o one of he modes, subsequenly discarding all oher modes. This paper inroduces a sraegy ha is beer able o mainain muli-modaliy. Working from he assumpion ha mixure models are inherenly more effecive a capuring muliple modes, he arge disribuion is formulaed as a non-parameric mixure of filering disribuions. As is he case for Bayesian sequenial esimaion a general framework is derived in which he mixure filering disribuion Proceedings of he Ninh IEEE Inernaional Conference on Compuer Vision (ICCV 2003) 2-Volume Se

2 can be compued recursively in wo seps: a predicion sep, followed by an updae sep when he new daa becomes available. A Mone Carlo implemenaion of he general framework essenially leads o a mixure of paricle filers ha inerac only in he compuaion of he mixure weighs. This resul is elegan in he sense ha all he resuls ha hold for sandard paricle filers ransfer o he individual mixure componens. Furhermore, he approach can be combined wih any convenien sraegy o obain and updae he mixure represenaion. The remainder of he paper is organised as follows. Secion 2 inroduces he general mixure racking framework. Secion 3 shows how a Mone Carlo implemenaion of he general framework leads o a mixure of paricle filers. Secion 4 discusses some imporan issues regarding he iniialisaion and updaing of he mixure represenaion. In Secion 5 he proposed algorihm is compared o he sandard paricle filer on wo problems. The firs is a synheic example where he muli-modaliy is due o ambiguiy, whereas he second is a real world problem involving he racking of muliple fooball players in a video sequence. The paper is summarised in Secion 6. Relaed Work The weakness of paricle mehods o mainain mulimodaliy has been acknowledged before. The auhors in [9] inroduce he idea of clusered paricle filering o guard agains samples ses becoming premaurely impoverished in he conex of mobile robo localisaion in highly symmeric environmens. The algorihm essenially groups he paricles ino clusers ha are independenly racked. Each cluser is assigned a probabiliy ha is racked using anoher Mone Carlo filer operaing a a higher level. The cluser wih he highes probabiliy a any paricular ime sep is deemed o correspond o he rue robo configuraion a ha insan. In he special conex of muli-objec racking a vas body of lieraure exiss. However, mos algorihms broadly fall ino one of wo caegories. The firs builds muli-objec rackers by muliple insaniaions of single objec racking algorihms, e.g. [3, 11]. Sraegies wih various levels of sophisicaion have been developed o inerpre he oupu of he resuling rackers in he case of occlusions and overlapping objecs. The second caegory of muli-objec rackers explicily exends he sae-space o include componens for all he objecs of ineres, e.g. [7, 8]. A variable number of objecs can be accommodaed by eiher dynamically changing he dimension of he sae-space, or by a corresponding se of indicaor variables signifying wheher an objec is presen or no. 2. Mixure Tracking Le x denoe he sae of he objec of ineres, and y = (y 1 y ) he observaions up o ime. For racking he disribuion of ineres is he filering disribuion p(x y ). In Bayesian sequenial esimaion his disribuion can be compued using he wo sep recursion: predic: p(x y 1 )= D(x x 1 )p(dx 1 y 1 ) (1) updae: p(x y )= L(y x )p(x y 1 ) L(y s )p(ds y 1 ), (2) where he predicion disribuion follows from marginalisaion, and he new filering disribuion is a direc consequence of Bayes rule. The recursion requires he specificaion of a dynamic model describing he sae evoluion D(x x 1 ), and a model ha gives he likelihood of any sae in he ligh of he curren observaion L(y x ). The recursion is iniialised wih some iniial disribuion p(x 0 ). To capure muli-modaliy his paper formulaes he filering disribuion as a M-componen mixure model, i.e. p(x y )= M π m, p m (x y ), (3) wih M π m, =1. Noe ha no parameric model is assumed for he individual mixure componens. The remainder of his secion shows how his non-parameric mixure represenaion can be updaed recursively in he same fashion as he wo sep approach for sandard Bayesian sequenial esimaion. Assuming ha he mixure filering disribuion p(x 1 y 1 ) is known, he new predicion disribuion is obained by subsiuion ino (1), leading o p(x y 1 )= = M π m, 1 D(x x 1 )p m (dx 1 y 1 ) M π m, 1 p m (x y 1 ), wih p m (x y 1 ) = D(x x 1 )p m (dx 1 y 1 ) he predicion disribuion for he m-h componen. Thus he new predicion disribuion is sraighforwardly obained by compuing he predicion disribuion for each of he componens individually, and combining hem in a mixure ha reains he original componen weighs. To obain he new filering disribuion he predicion dis- Proceedings of he Ninh IEEE Inernaional Conference on Compuer Vision (ICCV 2003) 2-Volume Se

3 ribuion is subsiued ino (2), leading o M p(x y )= π m, 1L(y x )p m (x y 1 ) M n=1 π n, 1 L(y s )p n (ds y 1 ) M [ π m, 1 L(y s )p m (ds y 1 ) ] = M n=1 π n, 1 L(y s )p n (ds y 1 ) [ L(y x )p m (x y 1 ) ]. L(y s )p m (ds y 1 ) In he second line he second erm in brackes is easily recognised as he new filering disribuion for he m-h componen, i.e. p m (x y )= L(y x )p m (x y 1 ) L(y s )p m (ds y 1 ). The firs erm in brackes is independen of he sae x, and if his is aken o be he new weigh, i.e. π m, 1 L(y s )p m (ds y 1 ) π m, = M n=1 π n, 1 L(y s )p n (ds y 1 ) (4) π m, 1 p m (y y 1 ) = M n=1 π n, 1p n (y y 1 ), he new filering disribuion is again a mixure of he individual componen filering disribuions, as in (3). This is an elegan resul indeed and means ha he filering recursion can be performed for each componen individually. The correc arge disribuion is mainained as long as he mixure weighs are updaed according o (4). The new componen weigh is he normalised weighed likelihood for he componen. This is he only par of he procedure where he componens inerac. 3. Paricle Approximaion The general mixure racking recursion inroduced in he previous secion yields closed-form expressions in only a small number of cases. A noable example occurs if boh he dynamic and likelihood models are linear and Gaussian, resuling in a mixure of Kalman filers [1]. For models ha are non-linear and/or non-gaussian approximaion echniques are required. One popular approximaion sraegy is Sequenial Mone Carlo mehods [4], oherwise known as Paricle Filers. These have gained remendous populariy in recen years as a numerical approximaion sraegy for complex models. This is due o heir simpliciy, generaliy and modelling success over a wide range of challenging applicaions. The paricle filer is a Mone Carlo mehod ha represens he arge disribuion wih a weighed se of samples ha are propagaed in such a manner so as o mainain a properly weighed sample from he arge disribuion a subsequen ime seps. The remainder of his secion presens he deails of a paricle approximaion o he general mixure racking recursion. In wha follows le P = {N,M,Π, X, W, C } denoe he paricle represenaion of he mixure filering disribuion in (3), wih N he number of paricles, M he number of mixure componens, Π = {π m, } M he mixure componen weighs, X = {x (i) { } N i=1 he paricle weighs, and C = {c (i) } N i=1 he paricles, W = } N i=1 he com- {1 M}, wih c (i) = m if ponen indicaors, i.e. c (i) paricle i belongs o mixure componen m. The paricle represenaion implies a Mone Carlo approximaion of he mixure filering disribuion of he form p(x y )= M π m, δ (i) x (x ), where δ a ( ) is he Dirac dela measure wih mass a a, and I m = {i {1 N} : c (i) = m} is he se of indices of he paricles belonging o he m-h mixure componen. Noe ha he mixure componen weighs and he paricle weighs for each mixure componen sum o one, i.e. M π m, = 1 and =1, m =1 M. Given a paricle se P 1 ha is approximaely disribued according o p(x 1 y 1 ), he objecive is o compue he new paricle se P such ha i is a sample se from p(x y ). Recall ha in he general mixure racking recursion each mixure componen evolves independenly, and ha he mixure componens inerac only in he compuaion of he mixure weighs. In he same way he paricle represenaions for he mixure componens also evolve independenly. Considering he m-h componen, he samples {x (i) 1,w(i) 1 } is a properly weighed sample se from p m (x 1 y 1 ). New samples are generaed from a suiably chosen proposal disribuion, which may depend on he old sae and he new measuremen, i.e. x (i) q(x x (i) 1, y ), i I m. To mainain a properly weighed sample se he new paricle weighs are se o = j I m w (j), = w(i) 1 L(y x (i) )D(x (i) x (i) q(x (i) x (i) 1, y ) The new sample se {x (i), } i Im is hen approximaely disribued o p m (x y ). To obain he new mixure weighs i is necessary o compue he componen likelihoods p m (y y 1 ), m = 1 M. Using he paricles a Mone Carlo approximaion 1 ). Proceedings of he Ninh IEEE Inernaional Conference on Compuer Vision (ICCV 2003) 2-Volume Se

4 o he m-h componen likelihood can be obained as p m (y y 1 )= L(y x )D(dx x 1 )p m (dx 1 y 1 ) 1 L(y x (i) )D(x (i) q(x (i) x (i) 1, y ) x (i) 1 ) =. Subsiuing his resul ino (4) leads o an approximaion for he new mixure weighs given by π m, π m, 1 w m, M n=1 π n, 1 w n,, w m, =. From ime o ime i is necessary o resample he paricles o avoid degeneracy of he weighs (see [5] for more deails on degeneracy and resampling procedures). Sandard resampling, however, is insensiive o he locaions of he paricles, and may lead o a loss of suppor in he arge disribuion. A very imporan poin o noe here is ha he mixure modelling approach allows independen resampling of each of he mixure componens according o he componen paricle weighs, hus naurally preserving he suppor of he poserior disribuion. Following such a procedure he new paricle weighs become =1/ I m, i I m, where denoes he se size operaor. 4. Mixure Compuaion The discussion up o his poin showed how a mixure represenaion for he filering disribuion can be propagaed, once such a represenaion is available. So far nohing has been said abou how o obain and mainain he mixure represenaion. In he ideal case here would be one mixure componen for each of he modes in he arge disribuion. In pracice, however, he number of modes in he arge disribuion is rarely known beforehand. Furhermore, he number of modes is unlikely o remain fixed, bu may flucuae as ambiguiies arise and are resolved, or objecs appear and disappear. Thus, from ime o ime i is necessary o recompue he mixure represenaion o ake accoun of hese flucuaions. For example, i may be desirable o merge componens ha have a significan degree of overlap, and spli componens ha have become oo diffuse. Forunaely his is easy o achieve using he paricle represenaion. Denoe by (C,M ) = F(X, C,M) a spaial reclusering procedure. I akes as inpus he paricles and he curren mixure represenaion (componen indicaors and number of componens), and compues a new mixure represenaion ha may or may no have he same number of componens as he original represenaion. Such a funcion encapsulaes any mixure compuaion operaion of ineres, including merging, spliing, reclusering ec. Wha remains is o compue he new mixure and paricle weighs, Π and W, so ha he new mixure approximaion P = {N,M, Π, X, W, C } is equal in disribuion o P. These are sraighforwardly obained by developing he mixure represenaion for P as follows: p(x y )= = = N i=1 M π c (i) M π m,, w(i) π m, δ (i) x (x )= w (i) δ (i) x δ (i) x (x ) M (x ), π c (i), w(i) where he new mixure and paricle weighs are given by π m, = i I m π (i) c, w(i), w (i) = π c (i), w(i) π c (i), δ (i) x (x ) Wih hese weighs he recompued mixure P represens exacly he same disribuion as P, and can be subsiued for P wihou affecing he convergence properies of he paricle filer. Noe ha he paricles X are no affeced in he new represenaion. The reclusering funcion F can be implemened in any convenien way. For he applicaions in his paper he iniial mixure represenaion is obained by K-means clusering of he iniial sample se, which is simulaed from some iniial sae disribuion p(x 0 ). A each ieraion he mixure represenaion is recompued by merging clusers wih significan overlap, and spliing clusers ha have become oo diffuse. Following his he new mixure componens are refined by a run of he K-means algorihm, iniialised wih he componens obained afer he merging and spliing procedures. This simple sraegy ha allows boh he mixure composiion and he number of mixure componens o vary was found o work well in he applicaions considered here, as illusraed in he following secion. 5. Experimens and Resuls This secion compares he performance of he proposed mixure paricle filer wih ha of he sandard paricle filer on wo muli-modal racking problems. The firs is a synheic example where he mulimodaliy is due o ambiguiy. The second is a real world problem involving he racking of muliple fooball players in a video sequence Synheic Example The purpose of his example is o esablish a baseline performance comparison beween he sandard and mixure paricle filers on a problem where he ground ruh is. Proceedings of he Ninh IEEE Inernaional Conference on Compuer Vision (ICCV 2003) 2-Volume Se

5 known. The model considered is scalar, and he governing equaions are given by D(x x 1 )=N(x x 1,σx) 2 L(y x )=N(y x 2,σy), 2 where N( µ, σ 2 ) denoes he univariae Gaussian disribuion wih mean µ and variance σ 2. In he resuls repored here he parameers were se o σ x = σ y = 0.1. The symmery of he Gaussian random walk dynamics and he quadraic erm in he likelihood means ha he filering disribuion has wo modes of equal mass 1 a ±x, wih x he rue sae. An excepion occurs when x is close o zero, in which case he wo modes merge, resuling in a single mode a zero. Figure 1 shows some synheic daa for 100 ime seps. Noe ha he rue sae was no simulaed from he model, bu deerminisically generaed o be sinusoidal. The abiliy of he sandard and mixure paricle filers o mainain he ambiguiy was esed on his daa se. In boh cases he iniial paricles were uniformly generaed, and he paricle proposal was aken o be he dynamics, so ha he new paricle weighs become proporional o he old weighs muliplied by he corresponding likelihoods. The mixure paricle filer was consrained o have a maximum of wo componens. The comparaive resuls for a ypical run wih 100 paricles are given in Figure 1. The sandard paricle filer loses rack of he rue mode afer ime sep 60. This behaviour is common in paricle filers, and Mone Carlo mehods in general. The mixure paricle filer, however, is able o successfully rack boh modes hroughou. To deermine he generaliy of his resul he experimen was repeaed 20 imes for differen numbers of paricles. For each run he performance score was defined as he fracion of ime seps during which boh modes were represened, hus ranging from zero (only one mode racked hroughou) o one (boh modes racked hroughou). The resuls are given in Figure 2. As expeced he performance of he sandard paricle filer increases wih an increase in he number of paricles, up o 500 paricles, from which poin i is able o consisenly rack boh modes. Even wih a small number of paricles he mixure paricle filer never fails o rack boh modes. I is ineresing o noe he behaviour of he firs mixure componen weigh, also depiced in Figure 2. Since boh modes are equally srong he mean weigh is approximaely 0.5 over all he ime seps, regardless of he number of paricles. The variabiliy in he weigh behaviour, however, decreases wih an increase in he number of paricles. 1 Assuming ha he iniial sae disribuion is also symmeric ime sep rue sae false sae observaion ime sep Figure 1. Simulaed paricles. (Top) Sandard paricle filer wih 100 paricles. (Boom) Mixure paricle filer wih a maximum of wo mixure componens and 50 paricles per componen. The sandard paricle filer loses rack of he rue mode afer ime sep 60, whereas he mixure paricle filer racks boh modes hroughou Visual Tracking This secion considers he problem of racking fooball players in a video sequence. In his seing he mulimodaliy is due o he presence of muliple objecs (fooball players), and o some exen, cluer. More precisely, he objec of ineres is represened by is bounding box. However, more general objec models can easily be accommodaed. The reference bounding box o be racked is specified by he user, and parameerised as 2 B ref =(x ref,y ref,l x,l y ), where (x ref,y ref ) is he cenre of he bounding box, and l x and l y are he bounding box widh and heigh, respecively. For he racking he sae of he bounding box is aken o be x =(x, y, s x,s y ),so 2 In wha follows he ime subscrip is suppressed for he sake of breviy. Proceedings of he Ninh IEEE Inernaional Conference on Compuer Vision (ICCV 2003) 2-Volume Se

6 1 0.8 likelihood for a hypohesised sae is defined as [ L(y x,b ref ) exp ( B 2 (h R B x, h R B ref ) + B 2 (h G B x, h G B ref )+B 2 (h B B x, h B B ref ) ) /2σ 2], weigh score paricles PF MPF ime sep Figure 2. Muliple runs. (Top) Score curves. Performance score and error bars as a funcion of he number of paricles. (Boom) Mixure componen one weigh behaviour for an increasing number of paricles. The mean weigh is indicaed by a solid line, and is approximaely 0.5, since boh modes are equally srong. The dashed one sandard deviaion lines indicae a decrease in he weigh variabiliy as he number of paricles increases. where B(h 1, h 2 ) is he Bhaacharyya disance beween he normalised N b bin hisograms h 1 and h 2, defined as [ N b ] 1/2 B(h 1, h 2 )= 1 hb,1 h b,2 [0, 1]. b=1 Thus he closer he colour hisograms in he hypohesised bounding box are o he corresponding colour hisograms in he reference bounding box, he higher he likelihood for he hypohesis. The widh of he likelihood is conrolled by he variance parameer σ 2. This likelihood is highly non-linear due o he mapping from he sae o he measuremens. A similar model was employed in he conex of objec racking before in [10]. This model wih N b = 30, σ = 0.15, and (σ 2 x,σ 2 y,σ 2 s x,σ 2 s y ) = (2.5, 1.5, 0.05, 0.05), was used o rack he fooball players in red in he video sequence for which a number of keyframes appear in Figure 3. For boh he sandard and he mixure paricle filers he proposal was aken o be he dynamics, and he iniial paricles were generaed around he red fooball players in he firs frame. For he sandard paricle filer 200 paricles were used, whereas he mixure paricle filer was consrained o have a maximum of 10 componens, wih 20 paricles for each componen alive. Thus he oal number of paricles for he mixure paricle filer is always equal or less han ha for he sandard paricle filer. A ypical racking resul for boh algorihms is depiced in Figure 3. Boh algorihms are iniialised in exacly he same manner around he four players o be racked. Even wih a large number of paricles he sandard paricle filer is unable o mainain he muli-modaliy for more han a few frames. The mixure paricle filer, however, quickly discovers he four main modes, and successfully rack hem hroughou he video sequence. 6. Conclusions ha he corresponding hypohesised bounding box becomes B x =(x, y, s x l x,s y l y ). The variables s x and s y hus ac as scale facors. The componens of he sae are assumed o follow independen Gaussian random walk models wih variances (σ 2 x,σ 2 y,σ 2 s x,σ 2 s y ). The measuremens are aken o be he normalised hisograms of he pixel colour componens wihin he bounding box, i.e. y =(h R B x, h G B x, h B B x ). Noe ha he measuremens depend on he objec sae. The This paper proposed o model he filering disribuion as a mixure model o beer cope wih he muli-modaliy ha may arise due o ambiguiy or he presence of muliple objecs. The general racking recursion for he mixure model was shown o comprise a predicion sep and an updae sep, similar o he sandard Bayesian recursion for a single componen model. I was also shown how a Mone Carlo implemenaion of he general recursion leads o a mixure of paricle filers ha inerac only in he compuaion of he Proceedings of he Ninh IEEE Inernaional Conference on Compuer Vision (ICCV 2003) 2-Volume Se

7 iniialisaion #17 #21 #105 Figure 3. Visual racking resuls. (Top) Sandard paricle filer wih 200 paricles. (Boom) Mixure paricle filer wih a maximum of 10 mixure componens and 20 paricles per componen. All he paricles for he sandard paricle filer quickly migrae o one of he modes. The mixure paricle filer rapidly discovers he four main modes and successfully rack hem hroughou he video sequence. mixure weighs. This mixure paricle filer is able o mainain he muli-modaliy inheren in racking problems where he sandard paricle filer fails, as was illusraed on a synheic and a real world racking problem. References [1] B. D. O. Anderson and J. B. Moore. Opimal Filering. Prenice-Hall, Englewood Cliffs, [2] Y. Bar-Shalom, L. Xiao-Rong, and T. Kirubarajan. Esimaion, Tracking and Navigaion: Theory, Algorihms and Sofware. John Wiley and Sons, [3] S. L. Docksader and A. M. Tekalp. Tracking muliple objecs in he presence of ariculaed and occluded moion. In Proceedings of he IEEE Workshop on Human Moion, pages 88 95, [4] A. Douce, J. F. G. de Freias, and N. J. Gordon, ediors. Sequenial Mone Carlo Mehods in Pracice. Springer-Verlag, New York, [5] A. Douce, S. J. Godsill, and C. Andrieu. On sequenial Mone Carlo sampling mehods for Bayesian filering. Saisics and Compuing, 10(3): , [7] C. Hue, J.-P. Le Cadre, and P. Pérez. Tracking muliple objecs wih paricle filering. IEEE Transacions on Aerospace and Elecronic Sysems, 38(3): , [8] M. Isard and J. MacCormick. BraMBLe: A Bayesian muliple-blob racker. In Proc. In. Conf. Compuer Vision, pages II: 34 41, [9] A.Milsein,J.N.Sánchez, and E. T. Williamson. Robus global localizaion using clusered paricle filering. In Proceedings of AAAI/IAAI, pages , [10] P. Pérez, C. Hue, J. Vermaak, and M. Gangne. Colorbased probabilisic racking. In Proc. Europ. Conf. Compuer Vision, pages I: , [11] C. Rasmussen. Join likelihood mehods for miigaing visual racking disurbances. In Proceedings of he IEEE Workshop on Muli-Objec Tracking, pages 69 76, [12] S. Thrun, D. Fox, W. Burgard, and F. Dellaer. Robus Mone Carlo localizaion for mobile robos. Arificial Inelligence, 128(1-2):00 141, [6] S. Gokurk, J. Bougue, and R. Grzeszczuk. A daadriven model for monocular face racking. In Proc. In. Conf. Compuer Vision, pages , Proceedings of he Ninh IEEE Inernaional Conference on Compuer Vision (ICCV 2003) 2-Volume Se