1 MCMC-Based Paricle Filering for Tracking a Variable Number of Ineracing Targes Zia Khan, Tucker Balch, and Frank Dellaer
2 Absrac We describe a paricle filer ha effecively deals wih ineracing arges - arges ha are influenced by he proximiy and/or behavior of oher arges. The paricle filer includes a Markov random field (MRF) moion prior ha helps mainain he ideniy of arges hroughou an ineracion, significanly reducing racker failures. We show ha his MRF prior can be easily implemened by including an addiional ineracion facor in he imporance weighs of he paricle filer. However, he compuaional requiremens of he resuling muli-arge filer render i unusable for large numbers of arges. Consequenly, we replace he radiional imporance sampling sep in he paricle filer wih a novel Markov chain Mone Carlo (MCMC) sampling sep o obain a more efficien MCMC-based muli-arge filer. We also show how o exend his MCMC-based filer o address a variable number of ineracing arges. Finally, we presen boh qualiaive and quaniaive experimenal resuls, demonsraing ha he resuling paricle filers deal efficienly and effecively wih complicaed arge ineracions. I. INTRODUCTION This work is concerned wih he problem of racking a variable number of ineracing arges. Our objecive is o correcly deec enering and leaving arges and obain a record of he rajecories of arges over ime, mainaining a correc, unique idenificaion for each arge hroughou. Tracking muliple arges which are idenical in appearance becomes significanly more challenging when he arges inerac. Mehods ha appropriaely deal wih his difficul issue are useful for applicaions where many ineracing arges need o be racked over ime. In paricular, hey have imporan implicaions for vision-based racking of animals, which has counless applicaions in biology and medicine. While more generally applicable, he resuls in his paper specifically cener on racking muliple ineracing insecs in video, which we pursue as par of a larger research projec o analyze muli-agen sysem behavior [3]. This domain offers many challenges ha are quie differen from he domains in which mos muli-arge racking algorihms are evaluaed. The muli-arge racking lieraure conains a number of classical approaches o he problem of daa-associaion, mos noably he muliple hypohesis racker (MHT) [31] and he join probabilisic daa associaion filer (JPDAF) [4], [12]. These muli-arge racking algorihms have been used exensively in he conex of compuer vision, e.g., neares neighbor racking in [11], he MHT in [9], and he JPDAF in [30] and [32]. However, hese daa associaion mehods ypically do no include a model of ineracion.
3 Fig. 1. 20 ans (Aphaenogaser cockerelli) in a closed arena are being racked by an MCMC-based paricle filer. Targes do no behave independenly: whenever one an encouners anoher, some amoun of ineracion akes place, and he behavior of a given an before and afer an ineracion can be quie differen. This observaion is generally applicable o any siuaion where many ineracing arges need o be racked over ime. In his paper we address he problem of ineracing arges, an issue no handled by radiional mehods. Our approach relies on he use of a moion model ha is able o adequaely describe arge behavior hroughou an ineracion even. As an example, consider he seup in Figure 1, which shows 20 insecs (ans) being racked in a small enclosed area. In his case, he arges do no behave independenly: whenever one an encouners anoher, some amoun of ineracion akes place, and he behavior of a given an before and afer an ineracion can be quie differen. We propose o model some of his addiional complexiy of arge behavior using a more capable moion model. While we do no aim o creae a fully accurae behavioral model, a difficul proposiion a bes, we have found ha even a small amoun of domain knowledge can significanly improve racking performance. Our approach is based on he well known paricle filer [15], [19], [8], [10]. Paricle filers
4 (a) frame 9043 (b) frame 9080 (c)frame 9083 Fig. 2. (a) Three ineracing ans racked using independen paricle filers. (b) The arge wih he bes likelihood score ypically hijacks he filers of nearby arges. (c) Resuling racker failure. We address his problem using an Markov random field moion prior, buil on he fly a each ime sep, ha can adequaely model ineracions o overcome hese failure modes. offer a degree of robusness o unpredicable moion and can correcly handle complicaed, nonlinear measuremen models. In compuer vision applicaions, i is ofen he arge appearance which is hard o model, and hence inference in closed form is difficul. Addiionally, in animal racking, he unpredicable moion of arges is anoher reason o use paricle filers. As we briefly review in Secion II-A, a paricle filer uses a Mone Carlo approximaion o he opimal Bayes filering disribuion, which capures he knowledge abou he locaion of all arges given all observaions. The sandard paricle filer weighs paricles based on a likelihood score and hen propagaes hese weighed paricles according o a moion model. When dealing wih ineracing arges, unforunaely, simply running one individual paricle filer for each arge is no a viable opion. As we argue below, his does no address he complex ineracions beween arges and leads o frequen racker failures. Whenever arges pass close o one anoher, he arge wih he bes likelihood score ypically hijacks he filers of nearby arges. This is illusraed in Figure 2. In a firs conribuion of his paper, we show how a Markov random field (MRF) moion prior, buil dynamically a each ime sep, can cope wih he hijacking problem by adequaely modeling arge ineracions and mainaining ideniy as a resul. This model is discussed in deail below in Secion III, where we show ha incorporaing an MRF o model ineracions is equivalen o adding an addiional ineracion facor o he imporance weighs in a muliarge or join paricle filer. However, join paricle filers suffer from exponenial complexiy in he number of racked arges, he associaed compuaional requiremens render he join filer
5 unusable for more han hree or four arges [22]. As a second conribuion we show how we can address he exponenial complexiy induced by he MRF formulaion using Markov chain Mone Carlo (MCMC) sampling. In our soluion, we replace he radiional imporance sampling sep in he join paricle filer wih a novel and efficien MCMC sampling sep. This approach has he appealing propery ha he filer behaves as a se of individual paricle filers when he arges are no ineracing, bu efficienly deals wih complicaed ineracions as arges approach each oher. The deails of his MCMC-based paricle filer are given in Secion IV. A hird and final conribuion is o show how he proposed paricle filer can be exended o problem domains in which he number of arges changes over ime. A variable number of arges necessiaes inference in a union space of sae spaces wih a differing number of dimensions. In Secion V, we show how reversible-jump MCMC (RJMCMC) sampling can be used o successfully address his issue [17], [16]. In paricular, we propose replacing he imporance sampling sep in he paricle filer wih an RJMCMC sampling sep. II. BACKGROUND AND RELATED WORK In wha follows, we adop a Bayesian approach o muli-arge racking. The Bayesian approach offers a sysemaic way o combine prior knowledge of arge posiions, modeling assumpions, and observaion informaion o he problem of racking muliple arges [36], [29]. In his secion, we review relaed and supporing work, and we inroduce he mahemaical noaion necessary for describing our work. A. Bayesian Muli-Targe Tracking Our primary goal in a muli-arge racking applicaion is o deermine he poserior disribuion P (X Z ) over he curren join configuraion of he arges X a he curren ime sep, given all observaions Z = {Z 1,..., Z } up o ha ime sep. Under he commonly made assumpion ha arge moion is Markovian, he Bayes filer offers a concise way o express he muliple arge racking problem. We consider wo cases: 1) when he number of arges is fixed and 2) when he number of arges may vary.
6 (a) (b) frame 1806 (c) frame 1810 (d) frame 1821 (e) frame 1839 Fig. 3. (a) The MCMC-based paricle filer can be exended o rack variable numbers of ineracing arges. Here he an Lepohorax curvispinosus may ener and exi from he circular enrance a he cener of arificial nes sie (highlighed by he yellow square). The ineracions ha individuals undergo are believed o play an imporan role in nes sie selecion. (b-e) shows a sequence where an individual exis he nes sie. 1) Fixed Number of Targes: When he number of arges is fixed, he sae X is simply he collecion of individual arge saes, i.e., X = {X i } n i=1, wih n he number of arges. The Bayes filer updaes he poserior disribuion P (X Z ) over he join sae X of all arges given all observaions up o and including ime, according o: P (X Z ) = cp (Z X ) P (X X 1 )P (X 1 Z 1 )dx 1 (1)
7 Here c is a normalizaion consan, he likelihood P (Z X ) expresses he measuremen model, i.e. he probabiliy we would have observed he measuremen Z given he sae X a ime, and he moion model P (X X 1 ) predics he sae X given he previous sae X 1. 2) Variable Number of Targes: In his case, arges may ener or leave he area under observaion, and boh he number and ideniy of he arges need o be esimaed. To model his we can inroduce a new variable, namely he se of idenifiers k of arges currenly in view [20]. I is clear ha here are many such hypoheses (poenially infinie), and each of hese differen hypoheses indexes a corresponding join sae space X k, is dimensionaliy deermined by he cardinaliy of he se k. For example, if he dimension of a single arge is 3, k = {1, 2} corresponds o a join sae X {1,2} of dimension 6, and his sae space is separae from he one indexed by k = {1, 2, 3}. The oal sae space is he union space X of all such join sae spaces of differen dimensionaliy, each indexed by a unique se of arge idenifiers. The poserior P (k, X k Z ) is hen a disribuion over his variable-dimension space X, and can again be expressed recursively using he Bayes filer equaion: P (k, X k Z ) = cp (Z k, X k ) P (k, X k k 1, X k 1 )P (k 1, X k 1 Z 1 )dx k 1 (2) k 1 As before, he likelihood P (Z k, X k ) expresses he variable arge measuremen model, and P (k 1, X k 1 Z 1 ) is he poserior a ime 1. However, he variable arge moion model P (k, X k k 1, X k 1 ) is now considerably more complex: no only does i serve o predic how arges will move, bu i addiionally includes he probabiliy of arges enering and leaving he area. I can be facored furher by pariioning he sae (k, X k ) ino arges (k E, X ke ) ha ener a he curren ime sep and arges (k S, X ks ) ha say, yielding: P (k, X k k 1, X k 1 ) = P (X k k, k 1, X k 1 )P (k k 1, X k 1 ) = P (X ke )P (X ks X ks ( 1))P (k k 1, X k 1 ) (3) where we have assumed ha arges ha ener or say behave independenly of one anoher. In Eq. (3), P (X ke ) predics where arges will ener, P (X ks X ks ( 1)) models he moion of arges ha say, andp (k k 1, X 1 ) models which arges are likely o ener, say or leave.
8 B. SIR Paricle Filers The inegrals in he Bayes filer equaions (1) and (2) are ofen analyically inracable. In his paper we make use of sequenial imporance re-sampling (SIR) paricle filers o obain sample approximaions of hese inegrals [15], [19], [8], [10]. To derive he paricle filer, one inducively assumes ha he poserior P (X 1 Z 1 ) over he sae X 1 a he previous ime sep is approximaed by a se of N weighed paricles, i.e., P (X 1 Z 1 ) {X (r) 1, π (r) 1} N r=1, where π (r) 1 is he weigh of he r h paricle. The inegral in he Bayes filer (1) can hen be obained by he following Mone Carlo approximaion: P (X Z ) cp (Z X ) r π (r) 1P (X X (r) 1) (4) A paricle filer can be seen as an imporance sampler for his disribuion. Specifically, in he mos ofen used varian of he paricle filer one draws, a ime, N samples X (s) following proposal disribuion q(x ), from he X (s) q(x ) = r π (r) 1P (X X (r) 1) i.e., q is a mixure densiy wih he moion models P (X X (r) 1) as mixure componens. Each sample X (s) approximaion {X (s) is hen weighed by is likelihood π (s) = P (Z X (s) ), resuling in a weighed paricle, π (s) } N s=1 of he poserior P (X Z ). While his view of paricle filers is slighly non-sandard, i explains boh resampling and predicion as sampling from he mixure, and his is convenien below. Noe ha here exis oher choices for he proposal disribuion ha yield more efficien varians of he paricle filer [2]. 1) Independen Paricle Filers: When arges wih idenical appearance do no inerac, we can run muliple independen paricle filers. In his case, in each of n paricle filers, he sae X is simply he sae of a single arge. Because each paricle filer samples in a small sae space, i obains a good approximaion of he poserior for he single arge. However, his approach is suscepible o racking failures when ineracions do occur. In a ypical failure mode, illusraed in Figure 2, several rackers will sar racking he single arge wih he highes likelihood score. 2) Join Paricle Filers: Insead, he ineracion model we propose below (in Secion III) necessiaes running a paricle filer in he join configuraion space, i.e. he sae X now includes he sae of all n arges. To emphasize his poin, we refer o he SIR paricle filer in ha case as he join paricle filer. Unforunaely, a sraighforward implemenaion of he join paricle
9 filer suffers from exponenial complexiy in he number of racked arges, rendering i unusable for more han hree or four arges [22]. In spie of his limiaion, join paricle filer based approaches have been applied o racking small numbers of idenical arges, e.g., by binding paricles o specific arges [38], or by using a mixure of paricle filers along wih a paricle clusering sep o mainain he ideniies of arges [39], [28]. The mixure approach has been shown o require fewer samples, bu necessiaes a re-clusering of paricles a each ime sep. Insead of clusering paricles, he probabilisic exclusion principle approach [25] adds a erm o he measuremen model ha assures ha every feaure measured belongs o only one arge. However, probabilisic exclusion is less suied for appearance-based racking and fails o capure informaion abou he join behavior of racked arges. In his work, we overcome he exponenial complexiy of he join paricle filer by using MCMC sampling (see Secion IV). 3) Variable Targe Join Paricle Filer: The join paricle filer approach also exends o a variable numbers of arges. In his case, we again approximae he poserior P (k 1, X k 1 Z 1 ) as a se of weighed samples {k (r) 1, X (r) k 1, π (r) 1} N r=1 idenifiers k (r) 1. The Mone Carlo approximaion of (2) hen becomes: P (k, X k Z ) cp (Z k, X k ) where each sample now includes a se of r π (r) 1P (k, X k k (r) 1, X (r) k 1 ) A each ime sep we now draw N samples (k, X k ) (s) from he variable-dimension sae space X, according o he following mixure proposal disribuion q(k, X k ): (k (s), X (s) k ) q(k, X k ) = r π (r) 1P (k, X k k (r) 1, X (r) k 1 ) (5) To sample from his disribuion we eiher keep or delee arges based on heir previous posiion, move he arges ha say according o heir moion model, and sample he number and iniial sae of new arges, all in accordance o Equaion 3. The samples are hen weighed by heir likelihood π (s) = P (Z k (s) poserior P (k, X k Z ) {k (s), X (s) k ), which resuls in a weighed paricle approximaion for he, X (s) k, π (s) } N s=1 a ime. Several auhors have addressed he problem of racking a variable number of arges in he join paricle filer framework. The BraMBLe racker [20] uses a proposal sep ha adds and removes arges. Similarly, sequenial imporance sampling schemes have been inroduced ha use rans-dimensional imporance weighs [40], in a way ha closely resemble reversible-jump
10 MCMC [17], [16]. Finally, finie se saisics as inroduced in [33], [34] enable a join paricle filer o address siuaions in which boh he number of arges and he number of measuremens changes. However, he racking accuracy of all hese approaches decreases when many arges ener he field of view, and hey have no been exended o racking large numbers of arges. C. MCMC Mehods Due o he limiaions of imporance sampling in high dimensional sae spaces, researchers have applied Markov chain Mone Carlo (MCMC) mehods o racking problems [41], [35]. All MCMC mehods work by defining a Markov Chain over he space of configuraions X, such ha he saionary disribuion π(x) of he chain is equal o he sough poserior P (X Z) over he configuraions X given he measuremens Z. Typically, MCMC mehods have been applied as a search echnique o obain maximum a poseriori (MAP) esimae, or as a way o diversify samples inside he framework of radiional paricle filers. Boh uses are discussed below: 1) MAP Esimaes: A MAP esimae can be obained by aking he mos likely sample generaed from he Markov Chain. In [35] MAP esimaion has been applied o people racking, more specifically o fi high dimensional ariculaed models o video sequences. In [41], i has been applied o muli-arge racking in combinaion wih emporal informaion from a Kalman filer. This suffers from he ypical problems associaed wih a Kalman filer, mos noably he inabiliy o cope wih non-gaussian moion noise or muli-modal disribuions over he arge saes. More generally, however, MCMC is a mehod inended o produce a sample of a disribuion, and no guaranees exis abou i yielding good poin esimaes. 2) MCMC Moves: The use of MCMC in paricle filers has primarily focused on increasing he diversiy of paricles, as inroduced by he so-called RESAMPLE-MOVE paricle filering schemes [13], [26], [5]. These schemes use periodic MCMC seps o diversify paricles in an imporance sampling based paricle filer. RESAMPLE-MOVE paricle filers have also been designed o use RJMCMC seps o swich beween variable dimensional sae spaces [1], [6]. The inroducion of MCMC seps o improve imporance sampling suggess ha MCMC alone could be used o obain a paricle filer ha can effecively handle high-dimensional sae spaces.
11 D. Summary The principal disincion of our approach from previous work is hree-fold: (1) we model ineracions hrough a Markov random field (MRF) moion prior (2) we leverage MCMC sampling o efficienly address racking he resuling join sae of many ineracing arges and (3) we exend his approach via RJMCMC o handle a variable number of arges. In Secion III, we inroduce he MRF moion model for dealing wih ineracing arges. The MRF necessiaes a join paricle filer which is inefficien in high dimensional sae spaces. To address his inefficiency, we inroduce an MCMC-based paricle filer in Secion IV. In he MCMC filer, we replace he imporance sampling sep of he join paricle filer wih an efficien MCMC sampling sep. As described in Secion V, he approach exends o variable numbers of arges if RJMCMC is used. Finally, in Secion VI, we presen boh qualiaive and quaniaive resuls on real, challenging image sequences from he social insec domain. III. AN MRF MOTION MODEL FOR INTERACTING TARGETS As a firs conribuion, we show how o reduce he number of racker failures by explicily modeling ineracions using a Markov random field (MRF) moion model. Specifically, a each ime sep we dynamically consruc an MRF ha addresses ineracions beween nearby arges. In Secion VI, we show empirically ha his MRF-based approach significanly improves racking. An MRF is a facored probabiliy model specified by an undireced graph (V, E), where he nodes V represen random variables and he edges E specify a neighborhood sysem. The join probabiliy over he random variables is hen facored as a produc of local poenial funcions φ a each node, and ineracion poenials ψ defined on neighborhood cliques. See [24] for a horough exposiion on MRFs. A commonly used form is a pairwise MRF, where he cliques are resriced o he pairs of nodes ha are direcly conneced in he graph. Here we use a pairwise MRF o obain a more realisic moion model P (X X 1 ), namely one ha models he ineracions beween nearby arges. Specifically, we adop he following model, where he ψ(x i, X j ) are pairwise ineracion poenials: P (X X 1 ) i P (X i X i( 1) ) ij E ψ(x i, X j ) (6) Using his form we sill reain he predicive moion models P (X i X i( 1) ) of each individual arge. However, addiionally, he he MRF ineracion poenials ψ(x i, X j ) afford us he
12 Fig. 4. To model ineracions, we dynamically consruc a Markov random field (MRF) a each ime sep, wih edges for arges ha are close o one anoher. An example is shown here for 6 ans. Targes ha are far from one anoher are no linked by an edge, reflecing ha here is no ineracion.the yellow circles indicae ha prediced arge posiions in he overlapping regions are accouned for by he pairwise ineracion poenials of MRF. possibiliy of easily specifying domain knowledge governing he join behavior of ineracing arges. We favor he pairwise poenials over more complex formulaions because hey are easy o represen and implemen. Also, ses of pairwise represenaions are sufficien for modeling all bu he mos complex group ineracions. As an example, in he an racking applicaion we presen in Secion VI, we know ha wo ans rarely occupy he same space. Taking advanage of his can grealy help in racking wo arges ha pass close o one anoher. An example MRF for a paricular configuraion is shown in Figure 4. The circles represen a region of influence in which ineracions are significan, and he presence of anoher arge here adds an edge in he MRF. The absence of edges beween wo ans capures he inuiion ha ans far away will no influence each oher s moion. We express he pairwise poenials ψ(x i, X j ) by means of a Gibbs disribuion ψ(x i, X j ) exp ( g(x i, X j )) (7) where g(x i, X j ) is a penaly funcion. For example, in he an racking applicaion we use
13 Algorihm 1 Join MRF Paricle Filer A each ime sep, he poserior over he arge saes a ime 1 is represened by a se of N weighed samples {X 1, (r) π 1} (r) N r=1. We hen creae N new paricles by imporance sampling, using he join moion model (9) as he proposal disribuion: 1) Imporance Sampling Sep: Repea N imes: a) Pick a sample X (r) 1 wih probabiliy π (r) 1. b) Apply he moion model P (X i X (r) i( 1) ) o each individual arge o obain a newly proposed join sae X (s). c) Assign a weigh π (s) o he sae X (s) according o he MRF-augmened likelihood (10). The ineracion facor (7) penalizes configuraions where arges overlap. 2) Reurn he new weighed sample se {X (s), π (s) } N s=1. a penaly funcion g(x i, X j ) ha only depends on he number of pixels overlap beween he appearance emplaes associaed wih he wo arges. I is maximal when wo arges coincide and gradually falls off as arges move apar. We use he Gibbs form above because we found ha pairwise penaly funcions are easier o specify han poenials. Noe ha his penaly funcion is a funcion on pairwise saes, and consequenly can be made quie sophisicaed. For example, as poined ou by a referee, i could be advanageous o include rajecory dynamics in he specificaion of he moion model. This is possible if velociy and/or acceleraion is included in he single arge saes, a he cos of addiional modeling complexiy. In our applicaions, we have found ha an ineracion poenial based on saic poses only was sufficien o eliminae mos racking failures. A. The Join MRF Paricle Filer The MRF erms ha model ineracions can be incorporaed ino he Bayes filer in a sraighforward manner. We can easily plug he MRF moion model (6) ino he join paricle filer equaion (4). Noe ha he ineracion poenial (7) does no depend on he previous arge sae X 1, and hence he arge disribuion (4) for he join MRF filer facors as P (X Z ) kp (Z X ) ψ(x i, X j ) ij E
14 Algorihm 2 Variable Targe Join MRF Paricle Filer A each ime sep, he poserior over he arge saes a ime 1 is represened by a se of N weighed samples {k 1, (r) X (r) k 1, π 1} (r) N r=1. We hen creae N new paricles by imporance sampling, using he variable arge proposal disribuion (5): 1) Imporance Sampling Sep: Repea N imes: a) Pick a sample (k (r) 1, X (r) k 1 ) wih probabiliy π (r) 1. b) Deermine which arges ener, say, or leave, according o P (k S k (r) 1, X (r) k 1 ). Apply he moion model P (X i X (r) i( 1) ) o each arge i k S. Sample he posiion of new arges from P (X ke ) o obain a new sae (k (s), X (s) k ). c) Assign a weigh π (s) o he new sae according o he MRF likelihood (11). 2) Reurn he new weighed sample se {k (s), X (s) k, π } N s=1. r π (r) 1 i P (X i X (r) i( 1) ) (8) In oher words, he ineracion erm moves ou of he mixure disribuion, and we can simply rea he ineracion erm as an addiional facor in he imporance weigh. In oher words, we sample from he join proposal disribuion X (s) q(x ) = r π (r) 1 i P (X i X (r) i( 1) ) (9) and weigh he samples according o he following augmened likelihood expression: π (s) = P (Z X (s) ) ij E ψ(x (s) i, X (s) j ) (10) The deailed seps of he join MRF paricle filer are summarized as Algorihm 1. Similarly, he MRF moion model can be incorporaed ino he variable arge paricle filer by again reaing he ineracion erm as an addiional facor in he imporance weigh π (s) = P (Z k (s), X (s) ) k ij E ψ(x (s) i, X (s) j ) (11) The variable arge join MRF paricle filer algorihm is summarized as Algorihm 2. B. Limiaions of Imporance Sampling While he above is heoreically sound, he join MRF paricle filer is no well suied for muli-arge racking. The filer suffers from exponenial complexiy in he number of racked
15 arges. If oo few paricles are used, all bu a few imporance weighs will be near zero. The Mone Carlo approximaion (8), while asympoically unbiased, will have high variance. These consideraions render he join filer unusable in pracice for more han hree or four arges [22]. IV. AN MCMC-BASED PARTICLE FILTER As a second conribuion, we show how one can efficienly sample from he poserior disribuion P (X Z ) over he join arge sae X using Markov chain Mone Carlo (MCMC) sampling [27], [14]. In effec, we replace he inefficien imporance sampling sep of a sraighforward paricle filer implemenaion by a more efficien MCMC sampling sep. In Secion VI, we show ha his drasically improves racking resuls given he same amoun of compuaion: he MCMC-based filer is able o cope wih he high-dimensional sae spaces involved, whereas he join paricle filer is no. We base his new approach on a differen Mone Carlo approximaion of he poserior, in erms of unweighed samples. In paricular, insead of using weighed samples obained using imporance sampling, we now represen he poserior P (X 1 Z 1 ) a ime 1 as a se of N unweighed samples P (X 1 Z 1 ) {X 1} (r) N r=1. Consequenly, by analogy wih equaion (8), we obain he following Mone Carlo approximaion o he exac Bayesian filering disribuion P (X Z ) cp (Z X ) ij E ψ(x i, X j ) r i P (X i X (r) i( 1) ) (12) which also incorporaes he MRF ineracion poenial. Now, insead of imporance sampling, which is inefficien in high-dimensional sae spaces, we use MCMC o sample from (12) a each ime sep. The sampling procedure resuls in a unweighed paricle approximaion for he poserior P (X Z ) {X (s) } N s=1. A. Markov Chain Mone Carlo Sampling As menioned in Secion II-C, MCMC mehods work by defining a Markov Chain over he space of configuraions X, such ha he saionary disribuion of he chain is equal o a arge disribuion π(x). The Meropolis-Hasings (MH) algorihm [18] is one way o simulae from such a chain. In he presen case, he arge disribuion π is he approximae poserior (12) over
16 join arge configuraions X, and a each ime sep we use he MH algorihm o generae a se of samples from i. The pseudo-code for he MH algorihm in his conex is as follows [14]: Algorihm 3 Meropolis-Hasings algorihm Sar wih an arbirary iniial configuraion X (1), hen ierae for s = 1..N 1: 1) Propose a new assignmen X by sampling from he proposal densiy Q(X ; X (s) ). 2) Calculae he accepance raio 3) If a 1, hen accep X a = P (X Z ) P (X (s) Z ) and se X (s+1) Q(X (s) ; X ) Q(X ; X (s) ) X. Oherwise, we accep X a, and rejec oherwise. In he laer case, we keep he curren sae, i.e., X (s+1) (13) wih probabiliy X (s). I is sandard pracice o discard a number of iniial burn-in samples, say B of hem, o allow he MH algorihm o converge o he saionary disribuion. In addiion, o reduce he correlaion beween samples, i is cusomary o hin ou he samples by only keeping a subse of hem, aken a regular inervals. In our case, limiing he number of samples ha are used o approximae (12) is also compuaionally advanageous, as is explained below. Please see [14] for a deailed discussion of hese and oher pracical consideraions surrounding MCMC. B. Proposal Densiy Careful design of he proposal densiy plays an imporan role in he success of an MCMC algorihm. To ha end, we use a proposal densiy Q ha selecs a single arge m, chosen from all arges wih uniform probabiliy, and updaes is sae X m only by sampling from a single arge proposal densiy Q(X m ; X m). This one arge a a ime scheme can be formalized by: Q(X ; X ) = 1 n Q(X m ; X m) if X m = X m 0 oherwise Here n is he number of arges, and X m denoes he join sae of all of he arges excluding arge m. As an example, he single arge proposal densiy Q(X m; X m ) could simply perurb he arge sae according o a zero-mean normal disribuion. This is he approach we ake for an-racking in Secion VI. In general, Q(X m ; X m) mus mee wo simple requiremens: (1) we mus be able o evaluae he densiy, and (2) we mus be able o efficienly sample from i.
17 Algorihm 4 MCMC-Based Paricle Filer A each ime sep, he poserior over arge saes a ime 1 is represened by a se of N unweighed samples {X 1} (r) N r=1. We hen creae N new samples by MCMC sampling: 1) Iniialize he MCMC sampler: randomly selec a sample X (r) 1, move all arges X (r) i( 1) in he seleced sample according o heir moion models, and use he resul as he iniial sae of he X Markov chain. For each arge X i in X, evaluae and cache he likelihood. 2) MCMC Sampling Sep: Repea (B + MN) imes, where B is he lengh of he burn-in period, and M is he lengh of he hinning inerval: a) Sample from he proposal densiy: Randomly pick a arge m, and propose a new sae X m sampling from he single arge proposal Q(X m ; X m). b) Compue he accepance raio (14). for his arge only by c) If a 1 hen accep X m: se he he m h arge in X o X m and updae he cached likelihood. Oherwise, accep wih probabiliy a. If rejeced, leave X unchanged. 3) As an approximaion for he curren poserior P (X Z ), we reurn he new sample se {X (s) } N s=1, obained by soring every M h sample afer he iniial B burn-in ieraions above. The resuling accepance raio can be wrien as a produc of hree raios: a = P (Z X m ) P (Z X m ) P (X Z 1 ) Q(X m ; X m ) P (X Z 1 ) Q(X m; X m ) (14) In he above, P (X Z 1 ) is he sample approximaion o he predicive prior from (12): P (X Z 1 ) = ij E ψ(x i, X j ) r i P (X i X (r) i( 1) ) (15) This proposal densiy design is compuaionally efficien: by only perurbing he sae of one arge per ieraion, mos facors in he accepance raio (14) cancel. By caching he previous likelihood evaluaion P (Z X m ) for arge m, only one likelihood evaluaion is necessary for each ieraion of he MH algorihm. In comparison, he join paricle filer requires a likelihood evaluaion for each arge per sample. Limiing he number of hese evaluaions is of grea ineres, as in many vision applicaions he likelihood includes a complex model of arge appearance.
18 The cos of evaluaing he predicive prior P (X Z 1 ) grows linearly wih he number of samples N used o approximae he poserior P (X 1 Z 1 ) {X (r) 1} N r=1. However, careful examinaion again reveals ha many facors remain consan if we perurb one arge a a ime. C. Summary The proposal densiy gives he MCMC-based paricle filer he appealing propery ha he filer behaves as a se of individual paricle filers when he arges are no ineracing, bu efficienly deals wih complicaed ineracions when arges approach each oher. The seps of he MCMC-based racking algorihm we propose are deailed as Algorihm 4. V. HANDLING A VARIABLE NUMBER OF TARGETS USING RJMCMC As a hird conribuion we show how he proposed MCMC-based paricle filer can be modified o handle a variable number of arges. We replace he MCMC sampling sep of he filer wih a reversible-jump MCMC (RJMCMC) sampling sep, an exension of MCMC o variable dimensional sae spaces [17], [16]. As wih he MCMC-based paricle filer, we use an unweighed sample approximaion of he variable arge poserior P (k 1, X k 1 Z 1 ) {k (r) 1, X (r) k 1 } N r=1 a ime 1. In he variable arge case, each sample now includes a se of idenifiers k 1. We obain he following Mone Carlo approximaion o he exac Bayes filer (2): P (k, X k Z ) cp (Z k, X k ) ij E ψ(x i, X j ) r P (k, X k k (r) 1, X (r) k 1 ) (16) where he moion model P (k, X k k (r) 1, X (r) k 1 ) facors according o (3). The RJMCMC sampling procedure, explained in deail below, will resul in an unweighed paricle approximaion for he poserior P (k, X k Z ) {k (s), X (s) k } N s=1 a ime. A. Reversible-jump MCMC In RJMCMC, he Meropolis-Hasings algorihm is modified o define a Markov chain over a variable dimension sae space [17], [16]. The algorihm sars he chain in an arbirary configuraion (k, X k ) X. I hen selecs a move ype m from a finie se of moves ha eiher
19 increase he dimensionaliy of he sae (birh), decrease i (deah), or leave i unchanged. A new sae (k, X k ) is hen proposed using he corresponding proposal densiy Q m (k, X k ; k, X k ). A move ha changes he dimensionaliy of he sae is referred o as a jump, as i jumps beween sae spaces of differen dimensionaliy. Crucially, every jump has o have a corresponding reverse jump defined, e.g. a birh move should have a corresponding deah move, hence he name reversible-jump MCMC. Denoe he reverse proposal densiy as Q m (k, X k ; k, X k ), where m is he reverse jump corresponding o m. The accepance raio is hen modified as follows a = P (k, X k Z ) p m Q m (k, X k ; k, X k ) P (k, X k Z ) p m Q m (k, X k ; k, X k ) which will achieve deailed balance a he desired arge densiy P (k, X k Z ). The key componen of RJMCMC is he reversibiliy of proposals ha vary he dimensionaliy of he sae space. To assure reversibiliy in he arge racking scenario, for every proposal ha adds a arge here mus be a corresponding proposal ha removes he arge. Under hose circumsances, deailed balance will be achieved and he chain will converge o he desired saionary disribuion (16). To achieve his, we resric ourselves o proposals ha add or remove a single arge. Noe ha his resricion does no in any way affec he evenual convergence of he chain o he arge densiy: he enire sae space can be explored by sequenially adding and/or deleing arges o reach any of he k -indexed subspaces in he union space X. In addiion, by only dealing wih adding or removing arges, we do no need o include a Jacobian in (17) as found in oher descripions of he reversible-jump sampler, as in our case he Jacobian is uniy. Reflecing he spli of he idenifier se k ino enering arges k E and arges ha say k S in Secion II-A.2, we inroduce wo reversible pairs of proposal ypes, respecively adding/deleing newly enering arges mediaed by a arge deecor, and proposals ha decide on arges saying/leaving he field of view, based on heir curren sae. (17) B. Deecor-Mediaed Proposals We assume he exisence of a noisy arge deecor ha, a each ime sep, reurns a se of idenifiers k d of deeced arges along wih heir esimaed sae X kd. Inuiively, by using he image informaion, he deecor provides a more informed way of adding arges, i.e., i is daa-driven [37]. We use he deecor oupu o drive wo ypes of proposals:
20 1) Add: If a deeced arge a is no ye in he idenifier se k, propose adding i. In paricular, we randomly selec an idenifier a wih uniform probabiliy 1/ k d \k, where k d \k is he se of arges ha have been deeced bu no ye added o k, and hen append ({a}, X a ) o he sampler sae (k, X k ). We can wrie his proposal succincly as 1 Q A (k, X k k ; k, X k ) = d \k if (k, X k ) = (k, X k ) ({a}, X a ) 0 oherwise where he union operaor is defined in he obvious way. If all deeced arges have already been added, we se he probabiliy p A of selecing he Add proposal o zero. 2) Delee: As required by he reversible-jump MCMC algorihm, he Add proposal above needs o have a corresponding reverse jump defined, in order o poenially move he chain back o a previous hypohesis in he union space X. This is done as follows: we randomly selec an idenifier d wih uniform probabiliy 1/ k k d, where k k d is he se of deeced arges ha have already been added o k, and hen remove ({d}, X d ) from he sampler sae (k, X k ). This proposal can be wrien as Q D (k, X k ; k, X k ) = 1 k k d if (k, X k ) = (k, X k )\({d}, X d ) 0 oherwise where he difference operaor \ is defined in he obvious way. If no deeced arges were added ye, we se he probabiliy p D of selecing he Delee proposal o zero. C. Say or Leave proposals The Add/Delee proposals above enable new arges o ener he field of view a each ime sep, mediaed by a arge deecor. Addiionally, we need a mechanism for deciding on he fae of arges ha were already represened in he previous sample se {k 1, (r) X (r) k 1 } N r=1. In paricular, we inroduce he following pair of proposals: 1) Say: If a given arge is no longer presen in he curren sampler sae (k, X k ), propose o re-add i and le i say anyway. To ha effec, we define he se k = ( r=1 k 1) (r) \k of idenifiers were presen a ime 1 bu are no presen in he se of idenfiers presen a curren ime sep k. We hen selec a arge s from his se wih uniform probabiliy 1, sample a new k locaion from Q(X s ; s) = r:s k (r) 1 P (X s X (r) s( 1) )
21 and hen append ({s}, X s ) o he sae (k, X k ). We can wrie his proposal succincly as 1 Q S (k, X k ; k, X k ) = Q(X k s; s) if (k, X k ) = (k, X k ) ({s}, X s ) 0 oherwise When he se k is empy, we se he probabiliy p S of he Say proposal o zero. 2) Leave: The corresponding reverse jump randomly selecs an idenifier l from he se k \k d, wih uniform probabiliy 1/ k \k d, and removes i from he he curren sae of he sampler: 1 Q L (k, X k k ; k, X k ) = \k d if (k, X k )\({l}, X l ) 0 oherwise If he se k \k d is empy, we se he probabiliy p L of selecing he Leave proposal o zero. D. Accepance Raios The 4 dimensionaliy-varying proposals discussed above have he following accepance raios associaed wih hem, all specialized versions of he RJMCMC raio (17): P (k a A = P (Z X a ), X k Z 1 ) p D k d \k P (k, X k Z 1 ) p A k k d (18) a D = 1 P (Z X d ) P (k, X k Z 1 ) p A k k d P (k, X k Z 1 ) p D k d \k P (k a S = P (Z X s ), X k Z 1 ) p L k P (k, X k Z 1 ) p S k \k d Q(X s; s) a L = 1 P (Z X l ) P (k, X k Z 1 ) p S k \k d Q(X l ; l) P (k, X k Z 1 ) p L k (19) (20) (21) In he above, P (k, X k Z 1 ) is defined as he sample approximaion o he predicive prior from (16) (compare wih Equaion 15), P (k, X k Z 1 ) = ij E r ψ(x i, X j ) P (X ke )P (X ks X (r) k S ( 1) )P (k k (r) 1, X (r) k 1 ) were we have used he facored form (3) o describe he moion model. Thus, wih each accepance raio compuaion, we need o sum over all samples in {k (r) 1, X (r) k 1 } N r=1. However, as before, many of he facors say consan as only one arge is added or removed a a ime.
22 Algorihm 5 RJMCMC-Based Paricle Filer A each ime sep, he poserior over arge saes a ime 1 is represened by an se of unweighed samples {k 1X (r) (r) k 1 } N r=1. We hen creae N new samples by RJMCMC sampling: 1) Iniialize he RJMCMC sampler using he same procedure as in Algorihm 4. 2) RJMCMC Sampling Sep: Repea (B +M N) imes, wih B and M are defined as before: Propose a new sae (k, X k ), depending on he randomly seleced proposal ype: Add: randomly add a deeced arge a wih probabiliy 1/ k \k d o he curren sae. Compue he accepance raio according o (18). Delee: randomly selec a arge d wih probabiliy 1/ k k d and delee i from he curren sae. Compue he accepance raio according o (19). Say: randomly selec a arge s o wih probabiliy 1/ k, sample is locaion from Q(X s ; s), and re-add i. Compue he accepance raio according o (20). Leave: randomly selec a arge l wih probabiliy 1/ k \k d, and delee i from he curren sae. Compue he accepance raio according o (21). Updae: use he same proposal densiy and accepance raio as in Algorihm 4. If a 1 hen accep he proposed sae (k, X k ) (k, X k ). Oherwise, we accep i wih probabiliy a, and rejec i oherwise. 3) As an approximaion o he curren poserior P (k, X k Z ), reurn he new sample se {k (s), X (s) k } N s=1, obained by soring every M h sample afer he iniial B burn-in ieraions. A fifh proposal ype, he Updae proposal, leaves he dimensionaliy of he sae unchanged and has a sraighforward accepance raio associaed wih i as in Secion IV-B. In all 5 cases, likelihood evaluaions can be limied using he same caching scheme described for he MCMC filer, and only one likelihood evaluaion is required per sample. E. Summary The deailed seps of he RJMCMC-based racking algorihm we propose are summarized as Algorihm 5. The MCMC-based paricle filer, in which he arge number is fixed, can be hough of as a specializaion of he RJMCMC-based paricle filer. We obain an MCMC-based filer when he probabiliies of proposals ha add or remove arges are se o zero.
23 VI. EXPERIMENTAL DETAILS & RESULTS We evaluaed our approach by racking hrough wo long video sequences of wo differen species of roaming ans and presen boh quaniaive resuls as well as a graphical comparison of he differen racker mehodologies. In he firs sequence, he number of arges was fixed and, in he second, he number of arges varied. The wo sequences were sufficienly similar such ha in he evaluaion he appearance and measuremen models were he same for boh sequences, alhough he specific parameers differed. The sequences were chosen o emphasize ha he variable arge componen, insead of a radical deparure in appearance and moion models, accouned for he performance of he RJMCMC filer. The firs es sequence consised of 10,400 frames recorded a a resoluion of 720 480 pixels a 30 Hz, showing weny ans of he species Aphaenogaser cockerelli roaming abou a closed arena (see Figure 1). The ans hemselves are abou 1 cm long and move abou he arena as quickly as 3 cm per second. Ineracions occur frequenly and can involve 5 or more ans in close proximiy. In hese cases, he moion of he animals is difficul o predic. Afer pausing and ouching one anoher, hey ofen walk rapidly sideways or even backward. This experimenal domain provides a subsanial challenge o any muli-arge racker. The second es sequence consised of 10,000 frames aken a 15 Hz, also a a resoluion of 720 480 pixels, recording a colony of Lepohorax curvispinosus, a differen species of an, in he process of selecing a new arificial nes sie. This species is smaller, approximaely 0.5 cm. The moion of hese ans differs slighly from he ans in he firs sequence. The ans may ener and exi from he cener of he arificial nes (see Figure 3). The numerous ineracions ha he individual ans undergo in his selecion process are believed o be relaed o how he colony decides o emigrae ino a new nes. We evaluaed a number of differen rackers wih respec o a baseline racking sequence. As no ground ruh was available, we obained he baseline sequence by running a very slow bu accurae racker and correcing any misakes i made by hand. When we observed a racker failure, we reiniialized he posiions of he arges by hand and resumed racking. Nex, he rajecories were furher processed o remove any remaining errors in he posiion of arges. The rajecories and videos can be downloaded a fp://fp.cc.gaech.edu/pub/groups/borg/pami05.
24 A. Modeling Ans An individual an s sae a ime is modeled using a 3DOF sae vecor X i = [x i, y i, θ i ], where i is he an s idenifier, (x i, y i ) is posiion, and θ i is orienaion. In all es runs, he arges were iniialized based on he firs frame of he ground ruh baseline sequence. The appearance likelihood model, given ha he cameras were saionary in all cases, uses a combinaion of an appearance emplae and a background model. The emplae is a recangular region wih widh w and heigh h. We facored he measuremen model assuming ha he foreground (F = 1) pixels are independen of he background (F = 0) pixels. As a consequence, he likelihood only needs o be evaluaed over pixels from a recangular region wih posiion and orienaion X i for arge i, obained by an inverse warp W 1 from he image: P (Z X ) i P (Z X i ) = i P (W 1 (Z, X i ) F = 1) P (W 1 (Z, X i ) F = 0) Because he ans periodically deform, we modeled he appearance and background using disribuions o provide robusness o ouliers [21]. We model he appearance of he arge W 1 (Z, X i ) F = 1 T (µ, Σ, ν) as a mulivariae- disribuion wih mean µ, diagonal covariance Σ, and degrees of freedom ν = 4. The background is modeled wih mean µ and variance image Σ disribued according o a mulivariae- disribuion wih degrees of freedom ν = 4. Because each pixel is independen, he background disribuion can be obained by applying he warp o he mean and variance images W 1 (Z, X i ) F = 0 T (W 1 (µ, X i ), W 1 (Σ, X i ), ν ). Boh models are learned from a se of raining images using EM ieraions [23], [21]. To speed he image warps he video frames were down-sampled o 180 120 pixels and he posiions of he arges scaled prior o obaining he emplae. For he moion model we used a normal densiy cenered on he previous pose of arge X i( 1) X i X i( 1) = R(θ 1 + θ)[ x y 0 ] + X i( 1) where [ x, y, θ] [N (0, σx 2), N (0, σ2 y ), N (0, σ2 θ )] and R(θ) a funcion ha specifies a roaion marix. The moion model is specified in he coordinae frame in he an, wih he head facing he y-axis. Hence, a larger variance in he y direcion indicaes we less cerain of he an s nex posiion along is body. For he MRF ineracion erms we used a simple linear penaly funcion g(x i, X j ) = γp(x i, X j ) where p is he area of overlap beween wo arges, in pixels, and γ = 5000.
25 B. Fixed Number of Targes Tracker Samples Failures Per Targe Error (pixels) join 200 606 9.99±4.35 join 1000 491 9.52±3.60 join 2000 407 9.33±3.42 independen 10 per arge 144 5.23±1.88 independen 50 per arge 79 3.09±1.33 independen 100 per arge 67 2.89±1.26 MCMC 200 47 2.67±1.04 MCMC 1000 29 2.12±0.97 MCMC 2000 26 2.08±1.03 TABLE I TRACKER FAILURES OBSERVED IN THE 10,400 FRAME TEST SEQUENCE WITH 20 TARGETS For he fixed number of arges case, we evaluaed nine racker/sample size combinaions, he resuls of which are summarized in Table I. The able shows for each combinaion he number of racking failures, which was calculaed by auomaically idenifying racking failures when he repored posiion of a arge deviaed 50 pixels from he ground ruh posiion. When a failure occurred, he racker was re-iniialized and racking was resumed. Also shown is he mean and sandard deviaion of he per-arge posiional error, in pixels. The hree racker ypes ha were evaluaed were respecively he join MRF paricle filer (join, Algorihm 1), 20 independen paricle filers (independen), and he MCMC-Based Paricle Filer (MCMC, Algorihm 4). The resuls are qualiaively illusraed in Figure 5 for he 2000 samples seing. Also shown here is he number of observed acual ineracions beween ans for each frame, deermined auomaically from he ground ruh sequence. Orienaion was no included in he evaluaion because of insances where he ground ruh orienaion was no available. In several poins in he video, he ans occasionally crawled up he side of he conainer, such ha heir heads are facing direcly oward he camera. In his case, a ground ruh orienaion was difficul or impossible o deermine. Below are he specific implemenaion choices made for he fixed number of arges case: The join sae X of arges is a vecor conaining each an s sae X i.
26 per arge error 12 10 8 6 4 2 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000 (a) join paricle filer, 2000 join paricles per arge error 12 10 8 6 4 2 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000 (b) 20 single paricle filers, 100 paricles each per arge error 12 10 8 6 4 2 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000 (c) MCMC paricle filer, 2000 paricles frame (d) observed ineracions Fig. 5. (a-c): Comparison of 3 rackers, each racking 20 ans using an equivalen sample size of 2000. Track qualiy is evaluaed on he basis of per arge error measured in pixels. Tick marks (dos wih verical lines) show when racking failures occur hroughou he sequence. The ime series plo shows average disance from ground ruh in pixels (averaged per arge and per second) (d) The number of ineracions beween ans for each frame deermined from he ground ruh sequence.
27 The moion model parameers used were σx 2 = 4, σ2 y = 8, σ2 θ = 0.4 MCMC parameers: we discard he firs 25% of he samples reurned by he MH algorihm o allow he sampler burn-in. We propagae only N = 10 samples o he nex ime sep o reduce he compuaional cos of he accepance raio and limi correlaion beween samples. We used a single arge proposal densiy ha updaed he sae according o X i X i( 1) = R(θ 1 + θ)[ x y 0 ] + X i( 1) where [ x, y, θ] [N (0, σ 2 p,x ), N (0, σ2 p,y ), N (0, σ2 p,θ )] wih parameers σ2 p,x = 2, σ2 p,y = 2, and σ 2 p,θ = 0.2. C. Variable Number of Targes Failures Tracker Samples Posiion Number Toal variable join 500 71 199 270 variable join 750 55 195 250 variable join 1000 55 181 236 RJMCMC 500 6 29 35 RJMCMC 750 7 24 31 RJMCMC 1000 2 19 21 TABLE II TRACKING ERRORS ON A 10,000 FRAME SEQUENCE WITH ENTERING AND LEAVING TARGETS. Table II shows he number of racking failures for all for all he racker/sample size combinaions we evaluaed in he variable number of arges case. Here we recorded wo ypes of failures: (1) failures in repored posiion, and (2) an incorrec number of arges. To allow for variaion in he ime he arge was deeced and added o he curren sae, a number failure was recorded when he incorrec number of arges was repored for more han a second. As before, a posiion failure was recorded when a arge was no found wihin 50 pixels of a ground ruh arge, and orienaion was no included in he evaluaion. The wo rackers ha were evaluaed were respecively he variable arge join MRF paricle filer (join, Algorihm 2), and he RJMCMC-Based Paricle Filer (RJMCMC, Algorihm 5).
28 per arge error 25 20 15 10 5 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000 (a) variable arge join filer, 1000 paricles 25 per arge error 20 15 10 5 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000 (b) RJMCMC, 1000 paricles 15 number of arges 10 5 0 0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000 15 (c) variable arge join filer, 1000 paricles number of arges 10 5 0 0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000 frame (d) RJMCMC, 1000 paricles, Fig. 6. (a-b) show comparison of rack qualiy measured as per arge error in pixels obained from he variable arge paricle filer wih he RJMCMC-based paricle filer. Targe posiion failures are shown as dashed blue ick marks. The ime series plo shows average disance from ground ruh in pixels (averaged per arge and per second). (c-d) compare he arge number esimaes reurned by he variable arge filer and he RJMCMC-filer. The purple ick marks indicae a arge number failure. The repored number of arges is shown in blue. The ground ruh number of arges is shown in green.
29 Figure 6 shows he qualiaive evaluaion of hese resuls, his ime for he 1000 samples seing. Boh he posiional errors and number errors are graphed separaely for clariy. The addiional implemenaion deails for he variable arge case are given below: The disribuion over new idenifiers P (k k 1, X k 1 ) was modeled as a series of leave evens and ener evens. The posiion of enering arges P (X ke ) was uniform around he nes sie enrance. This choice capured he fac ha he ans are resriced o ener and leave from a hole in he cener of of he arificial nes sie (see Figure 3). The probabiliy a leave even was se o one when he an was posiioned a nes enrance and 0.8 in he close proximiy of he enrance. An ener even was generaed by adding an individual deeced arge wih probabiliy 0.1. Targes were deeced using a noisy, bu fas, color deecor o locae an-colored regions in an image in he proximiy of he nes enrance [7]. Parameers: The following parameers were changed from he fixed arge number case because he species and hus boh he size and he behavior of he ans in quesion changed. We deermined empirically on a small es sequence ha he following parameers performed well: σx 2 = 2, σ2 y = 6, σ2 θ = 0.4, σ2 p,x = 1.5, σ2 p,y = 1.5, σ2 p,θ = 0.2. Proposal Probabiliies: The probabiliies of he Add, Delee, Say, Leave, and Updae proposal ypes were se o 0.15, 0.15, 0.05, 0.05, and 0.6, respecively. The values of hese parameers were deermined empirically o perform well on a shor es sequence.
30 VII. CONCLUSIONS AND FUTURE WORK From he quaniaive resuls in Table I and he qualiaive comparison in Figure 5 for he fixed number of arges case, we draw he following conclusions: 1) The radiional join paricle filer is unusable for racking his many arges. The rack qualiy is very low and number of errors repored is very high. 2) The MRF moion prior improves racking during ineracions. The MCMC-based rackers perform significanly beer han running independen paricle filers wih a comparable number of samples, boh in rack qualiy and failures repored. 3) The MCMC-based paricle filer requires far fewer samples o adequaely rack he join arge sae. (a) frame 3054 (b) frame 3054 (c) frame 3072 Fig. 7. shows wo examples of failure modes in MCMC-based MRF paricle filer. These occur when he assumpion ha arges do no overlap is violaed. (a) Two arges undergoing exensive overlap. (b) The racker repors he incorrec posiion for he overlapped an. (c) The resuling racker failure. Figure 7 shows an example of a failure mode of he MCMC-based racker. These occur when our assumpion ha arges do no overlap is violaed. In hese cases, i is unclear ha any daa associaion mehod offers a soluion. A more complicaed join likelihood model migh be helpful in hese cases. In he variable number of arges case, we draw he following conclusions from he quaniaive resuls in Table II and he qualiaive comparison in Figure 6: 1) The radiional variable arge join paricle filer is unusable for racking variable numbers of arges. For a fixed number of samples, he racker is unable o accuraely deermine he number of arges when he arge number increases greaer han 5.
31 2) The RJMCMC-based racker responds more quickly o changes in he number of arges han he variable arge join filer. Even hough he variable arge join paricle filer shows slighly beer performance when here are fewer han 5 arges, many of he posiion errors occur when he number of arges changes. The RJMCMC racker is able o handle many of hese changes in arge number. 3) The RJMCMC-based paricle filer efficienly handles changing numbers of arges. Fewer samples are needed o rack he join sae of he arges even when many arges have enered he field of view. (a) (b) Fig. 8. shows wo examples of failure modes in he RJMCMC-based racker. These occur when (a) an individual an carries anoher an ino he nes and (b) individuals sraddle he nes sie enrance for several seconds. Noe he dark circle in he figure is he nes sie enrance. Figure 8 (a) shows an example failure mode of he RJMCMC racker. This failure occurred when he ans engaged in a carrying behavior. The wo arges are incorrecly idenified as a single arge. Tracking performance migh be improved in hese cases by using a swiching variable ha deecs and correcly handles his behavior. Addiionally, in Figure 8 (b), individuals ha sraddle he nes enrance are incorrecly idenified as having lef he nes sie. A more complex model of how arges leave he field of view migh beer accoun for his sraddling behavior. In conclusion, he resuls show ha Markov random field moion priors are quie successful a eliminaing a large number of ineracion-relaed racking errors. In addiion, MCMC-based filers perform vasly beer han a radiional paricle filering approaches in handling he resuling, highly coupled join sae space ha is induced by he use of he MRF. Finally, he use of reversible-jump MCMC successfully exended his o he case where he number of arges varies over ime, a siuaion ha is prevalen in pracice.