Monte Carlo data association for multiple target tracking

Transcription

1 Mone Carlo daa associaion for muliple arge racking Rickard Karlsson Dep. of Elecrical Engineering Linköping Universiy SE Linköping, Sweden Fredrik Gusafsson Dep. of Elecrical Engineering Linköping Universiy SE Linköping, Sweden Absrac The daa associaion problem occurs for muliple arge racking applicaions. Since non-linear and non-gaussian esimaion problems are solved approximaely in an opimal way using recursive Mone Carlo mehods or paricle filers, he associaion sep will be crucial for he overall performance. We inroduce a Bayesian daa associaion mehod based on he paricle filer idea and he join probabilisic daa associaion (JPDA) hypohesis calculaions. A comparison wih classical EKF based daa associaion mehods such as he neares neighbor (NN) mehod and he JPDA mehod is made. The NN associaion mehod is also applied o he paricle filer mehod. Muliple arge racking using paricle filer will increase he compuaional burden, herefore a conrol srucure for he number of samples needed is proposed. A radar arge racking applicaion is used in a simulaion sudy for evaluaion. 1 Inroducion For muliple arge racking applicaion he daa associaion problem mus be handled. Tradiionally, he esimaion problem is solved using linearized filers, such as he exended Kalman filer (EKF) [4], under a Gaussian noise assumpion. The sufficien saisics from he linearized filer are used for daa associaion. Several classical associaion mehods have been proposed in he lieraure. When dealing wih non-linear models in sae equaion and measuremen relaion and a non- Gaussian noise assumpion, hese esimaion mehods may lead o non-opimal soluions. The sequenial Mone Carlo mehods, or paricle filers, provide general soluions o many problems where linearizaions and Gaussian approximaions are inracable or would yield oo low performance. In his paper, we apply he classical paricle filer Bayesian boosrap [1], o a muliple arge environmen. In a simulaion sudy we compare his approach o radiional mehods. To handle he complexiy problem we also propose a conroller srucure, o recursively chose he number of paricles. Sequenial Mone Carlo mehods Mone Carlo echniques have been a growing research area laely due o improved compuer performance. A rebirh of his ype of algorihms came afer he seminal paper of Gordon e al. [1], showing ha Mone Carlo mehods could be used in pracice o solve he opimal esimaion problem. In he recen aricle collecion, [9], he heory and developmen in sequenial Mone Carlo mehods over he las years are summarized. Consider he following non-linear discree ime sysem for a single arge x +1 = f(x )+v, y = h(x )+e. The sequenial Mone Carlo mehods, or paricle filers, provide an approximaive Bayesian soluion o discree ime recursive problem by updaing an approximaive descripion of he poserior filering densiy. Le x R n denoe he sae of he observed sysem and Y = {y i } i= be he se of observaions unil presen ime. Assume independen process noise v and measuremen noise e wih densiies p v respecive p e. The iniial uncerainy is described by he densiy p x. The paricle filer approximaes he probabiliy densiy } N,where each paricle has an assigned relaive weigh,,such ha all weighs sum o uniy. The locaion and weigh of each paricle reflec he value of he densiy in he region of he sae space. The paricle filer updaes he paricle locaion and he corresponding weighs recursively wih each new observaion. The non-linear predicion densiy p(x Y 1 ) and filering densiy p(x Y ) for he Bayesian inerference are given by p(x Y )byalargeseofn paricles {x (i) p(x Y 1 )= p(x x 1 )p(x 1 Y 1 )dx 1 R n (1) p(x Y ) p(y x )p(x Y 1 ). () The main idea is o approximae p(x Y 1 )wih p(x Y 1 ) 1 N δ(x x (i) ), (3)

2 where δ is he discree Dirac funcion. Insering (3) ino () yields a densiy o sample from. This can be done by using he Bayesian boosrap or Sampling Imporance Resampling (SIR) algorihm from [1], given in Table 1. The esimae and uncerainy region for he N k 1 PF Resampling sep PF µ (1) ɛ (N, ɛ) + ɛ µ () 1 q 1 Conrol srucure Bayesian boosrap (SIR) 1. Se =, generae N samples {x (i) }N from he iniial disribuion p(x ).. Compue he weighs = p(y x (i) ) and nor-,,...,n. malize, i.e, = / N j=1 w(j) 3. Generae a new se {x (i ) } N by resampling wih replacemen N imes from {x (i) } N,where Pr(x (i ) = x (j) )= w (j). 4. Predic (simulae) new paricles, i.e, x (i) +1 = f(x(i ),v ),,...,N using differen noise realizaions for he paricles. 5. Increase and ierae o iem. Table 1: Bayesian boosrap (SIR) algorihm paricle filer can be calculaed as ˆx MS = P = x (i), (4) (x (i) ˆx MS )(x (i) ˆx MS ). (5) 3 Paricle number conroller The compuaional burden for he paricle filer is dependen on he number of paricles and on he resampling calculaion. However, he resampling can be efficienly implemened using a classical algorihm for sampling N ordered independen idenically disribued variables [5, 17]. For muliple arge racking applicaions he compuaional burden is increased. Therefore, i is essenial o minimize he number of paricles used in he esimaion sep. A novel approach is o apply a simple conrol srucure according o Figure 1. The number of paricles needed is deermined by he conroller using he residual ɛ = µ (1) µ (), are some saisical propery from he paricle filers (PFs), using differen number of paricles. Possible choices are for insance some relevan saisics, such as he mean esimae from he paricle filer or uilizaion of he probabiliy densiy (pdf) or he cumulaive densiy funcion (cdf). For insance he marginal disribuion (densiy for each coordinae) where µ (1) and µ () Figure 1: Conroller of paricles could be used. The conrol srucure used is a nonlinear block consising of a relay and an inegraor using { α inc (N ), if ɛ > Λ (N,ɛ )= α dec (N ), if ɛ Λ, For maneuvering arges in a racking applicaion he conroller can reduce or increase he number of paricles during he racking envelope. However, performance may now depend on he parameers of he conroller. Noe ha he conroller is implemened in he resampling sep (Table 1, sep 3). 4 Daa associaion Daa associaion is a problem of grea imporance for muliple arge racking applicaions. Several mehods have been proposed in he lieraure and differen mehodsareofendiscussedinesimaionandracking lieraure, [, 3, 7, 8]. In general muli arge racking deals wih sae esimaion of an unknown number of arges. Some mehods are special cases which assume ha he number of arges is consan or known. The observaions are considered o originae from arges if deeced or from cluer. The cluer is a special model for so-called false alarms, whose saisical properies are differen from he arges. In some applicaions only one measuremen is assumed from each arge objec, where in oher applicaions several reurns are available. This will of course reflec which daa associaion mehod o use. Several classical daa associaion mehods exis. The simples is probably he neares neighbor (NN). In [], his is referred o as he neares neighbor sandard filer (NNSF) and uses only he closes observaion o any given sae o perform he measuremen updae sep. The mehod can also be given as a global opimizaion, so he oal observaion o rack saisical disance is minimized. Anoher muli arge racking associaion mehod is he join probabiliy daa associaion (JPDA) which is an exension of he probabiliy daa associaion (PDA) algorihm o muli arges. I esimaes he saes by a sum over all he associaion hypohesis weighed by he probabiliies from

3 he likelihood. The mos general mehod is a imeconsuming algorihm called he muli hypohesis racking (MHT), which calculaes every possible updae hypohesis. In [16], several algorihms for muliple arge racking are lised and caegorized according o he underlying assumpions. A reference lis o he differen mehods is also given. In [15], he so-called probabilisic MHT (PMHT) mehod is presened, using a maximum-likelihood mehod in combinaion wih he expecaion maximizaion (EM) mehod. A comparison beween he JPDAF and he PMHT is also made. In [6], a Markov Chain Mone Carlo (MCMC) echnique is used for daa associaion of muliple measuremens in an over he horizon radar applicaion. Mos of hese mehods rely upon ha he mean and covariance is sufficien saisics for he problem. For linear and Gaussian problems he Kalman filer is he opimal esimaor yielding sufficien informaion. For non-linear problems he EKF is ofen used as an approximaion. To be able o fully use nonlinear and non-gaussian esimaion mehods combined wih daa associaion o solve he join daa associaion and esimaion problem here is a need o develop oher mehods. In [1], he soluion o he assignmen problem for daa associaion is proposed o be wihin he Bayesian framework by simply incorporae i in he esimaion equaions. In [18], his idea is suggesed for he paricle filer, when he problem of mainaining a rack on a arge in he presence of inermien spurious objecs. In [11], a muliple arge and muliple sensor esimaion and associaion problem is solved using he Bayesian boosrap filer. Samples are drawn from he overall arge probabiliy densiy. A special filer called hybrid boosrap filer is consruced. The so-called join-filer in [14], is a soluion o he join daa associaion and esimaion problem for paricle filers. The esimaion is done using a paricle filer and a Gibbs sampler, [1], is used for he associaion. The case for unknown number of arges is handled by using a hypohesis es. In his paper we focus on his idea for a muliple arge problem in a cluered environmen, and compare he paricle filer based esimaion and associaion wih classical associaion echniques. 5 Mone Carlo Probabilisic Daa Associaion In his paper we modify he classical SIR algorihm (Table 1) for esimaion o handle muliple arges. The associaion principle proposed is based on a novel Mone Carlo approach for he JPDA algorihm. We have assumed ime-invarian arge models, which are he same for all arges. We use he same Bayesian approach as in [11], for he esimaion. However, we exend he idea and inroduce hypohesis calculaions according o he JPDA mehod. The resampling is hen execued over all arge associaion hypoheses. The cluer or false alarm model is assumed uniformly disribued in he volume and he number of false alarms for a given ime is assumed o be Poisson disribued. Le x be he sae a ime for he relaive arge locaions, i.e, x = {x 1,...,x τ }. The samples or paricles in he SIR/MCJPDA mehod is defined as {x (i) } N = {x(i),1,...,x (i),τ } N, where each iniial arge cloud is denoed x (i),j for arges j =1,...,τ. The measuremens for each ime frame (scan) are denoed y k,k =1,...,M. A special cluer model is used o handle false alarms, x (j =). Theassociaion likelihood (rack j, measuremenk) isgivenby p jk = p e (y k h(xj )). A general expression for he probabiliy in hypohesis H n is: P (H n )=δ n P τ Zn D (1 P D ) Zn M (τ Zn) PFA l n, (6) where Z n is he number of false alarms (FA) in hypohesis n and l n is he likelihood par. For more deails, see hypohesis calculaions in he example given in [8] (p. 354). We also have an exra opion δ n = { 1, allow muliple measuremen associaions, oherwise. For he paricle filer each paricle is associaed wih a weigh: (M +1) τ = n=1 P (H (i) n ). Normalizaion yields he paricle probabiliy.the join paricle filering and associaion is summarized in Table. Similar ideas in he conex of robo conrol appear in [19]. The opional paricle number conroller describedinsecion3,isappliedasep3,intable. To simplify he algorihm some pracical problems are discarded. The measuremens wihin a scan is considered given a he same ime insances and he number of arges (τ) is assumed consan during he simulaion. If he number of arges is unknown or changing, he algorihm could be modified, for insance using a separae rack sar hypohesis. This could be done wihin he paricle filer framework or possible o use some linearized mehod. To allow measuremens wih differen ime, he predicion sep is modified wih an increased compuaional load as a consequence, i.e, each rack mus be prediced o every measuremen ime, in he associaion sep.

4 Tracking & associaion: SIR/MCJPDA 1. Se =, generae N samples from each arge j = 1,...,τ, i.e, x = {x (i) }N = {x(i),1,...,x (i),τ } N, where x (i),j from p(x j ).. For each paricle compue he weighs for all measuremen o rack associaion = (M +1) τ n=1 P (H n (i) ) and normalize for each measuremen, i.e, = / N w(i), where P (H n (i) ) is he probabiliy for hypohesis n using paricle i according o equaion (6). 3. Generae a new se {x (i ) } N wih replacemen N imes from {x (i) where Pr(x (i ) = x (l) )= w (l). by resampling } N, 4. Predic (simulae) new paricles, i.e, x (i),j +1 = f(x(i ),j,v (i),j ),i =1,...,N, using differen noise realizaions for he paricles, for each arge j = 1,...,τ. 5. Increase and ierae o iem. Table : SIR/MCJPDA esimaion and associaion 6 Simulaions In a simulaion sudy, he proposed SIR/MCJPDA mehod is implemened for a muli arge environmen problem. The applicaion a hand is a missile o air scenario. To simplify he simulaions we assume ha i is always possible o resolve he arges. In Caresian coordinaes he relaive sae vecor is defined as x = x x own, such ha x = ( X() Y () Z() V x () V y () V z () ), where X, Y and Z are he Caresian posiion coordinaes and V x,v y and V z he velociy componens. The following discree ime sysem is used ( I3x3 TI 3x3 x +1 = O 3x3 I 3x3 ) ( T x + I 3x3 TI 3x3 X + Y + Z y = h(x )= arcan( Y X ) + e, Z arcan( ) X +Y ) v, where he process noise v is assumed Gaussian, v N(,Q). The hree-by-hree null marix and uniy marix is denoed O 3x3 and I 3x3 respecively. The measuremen noise is assumed Gaussian e N(,R). The parameric models for false alarms are assumed N FA Po(λV ), wih average number of false alarms per uni volume λ and he validaion region volume V. In he simulaions E{N FA } = λv =.5 isused. The deecion probabiliy is assumed P D =.9. Assume he number of arges τ = and a sample ime of T =1[s]. The iniial inerial arge sae vecors x i,iniialown plaform x own, measuremen noise marix R, process noise Q and iniial sae error marix P are x 1 = 5, x = 1, x own = 3, P =diag ( ), 1 5 Q = 1,R= The implemened EKF is according o he discreized linearizaion echnique [13], i.e, firs linearize he underlying coninuous ime sysem and hen discreize. Iniial values for he racks is draw from he iniial uncerainy region P around he rue value. We compare he SIR/MCJPDA mehod wih an NN daa associaion where he esimaion is done by he paricle filer and where he covariance marix needed for he associaion is similar o equaion (5). A comparison is also made o an EKF using he NN or JPDA associaion in a similar way. In Figure, a daa associaion and esimaion using he SIR/MCJPDA filer is presened. To evaluae he performance a roo mean square error y [m] Measuremen Esimae Targe x [m] Figure : Daa associaion & racking (RMSE) analysis is performed over N mc =6simulaions and ime samples. In Table 3, he resuls for he differen mehods are summarized, using RMSE for he wo arges when 3, ignoring iniial ransiens. The paricle filer used N = 5 samples. In Figure 3, he RMSE values for differen imes are presened for he mehods described in Table 3 (arge 1). In Figure 4, he paricle number conroller (Secion 3), for SIR/MCJPDA is used wih k 1 = 1,k =.1, Λ=9.5 and α inc (N ) =.N,α dec (N ) =.1N, for he marginal case, for Mone Carlo simulaions.

5 Esimaion Associaion RMSE #1 RMSE # SIR MCJPDA SIR NN EKF JPDA EKF NN Table 3: Associaion & esimaion RMSE analysis RMSE() SIR/MCJPDA: N=5 (fixed) SIR/MCJPDA: N=5 (conroller) SIR/MCJPDA SIR/NN EKF/JPDA EKF/NN #paricles RMSE() 8 7 Figure 4: The paricle number conroller Figure 3: RMSE() for differen mehods 7 Conclusions In his paper a novel Mone Carlo daa associaion mehod for joinly esimaion and associaion in a probabilisic daa associaion framework is presened. This mehod (SIR/MCJPDA) is compared o EKF based classical associaion mehods such as NN and JPDA. The NN associaion is also applied o he SIR mehod, where he covariance is calculaed from he paricle filer cloud. A novel approach o deermine he number of paricles for each arge is also developed, using a relay and an inegraor in a feedback sysem. In he simulaion sudy in Secion 6, he mehods are compared and he RMSE is used o describe he performance. For more non-linear problems and problems where he noise disribuion is highly non-gaussian, he proposed simulaion based algorihms may increase he overall racking performance. References [1] D. Avizour. Sochasic simulaion Bayesian approach o muliarge racking. IEE Proc. on Radar, Sonar and Navigaion, 14(), [] Y. Bar-Shalom and T. Formann. Tracking and Daa Associaion, volume 179 of Mahemaics in Science and Engineering. Academic Press, [3] Y. Bar-Shalom and Xiao-Rong Li. Esimaion and Tracking: Principles, Techniques, and Sofware. Arech Houes, [4] Anderson B.D.O. and Moore J.B. Opimal Filering. Prenice Hall, Englewood Cliffs, NJ, [5] N. Bergman. Recursive Bayesian Esimaion: Navigaion and Tracking Applicaions. PhD hesis, Linköping Universiy, Disseraions No [6] N. Bergman and A. Douce. Markov Chain Mone Carlo daa associaion for arge racking. In IEEE In. Conference on Acousics, Speech, and Signal Processing (ICASSP),. [7] S.S. Blackman. Muliple-arge racking wih radar applicaions. Arech House, Norwood, MA, [8] S.S Blackman and R. Popoli. Design and analysis of modern racking sysems. Arech House, [9] A. Douce, N. de Freias, and N. Gordon, ediors. Sequenial Mone Carlo Mehods in Pracice. Springer Verlag, 1. [1] S. Geman and D. Geman. Sochasic relaxaion, Gibbs disribuions and he Bayesian resoraion of images. IEEE Trans. on Paern Analysis and Machine Inelligence, 6:71 741, [11] N.J. Gordon. A hybrid boosrap filer for arge racking in cluer. In IEEE Transacions on Aerospace and Elecronic Sysems, volume 33, pages , [1] N.J. Gordon, D.J. Salmond, and A.F.M. Smih. A novel approach o nonlinear/non-gaussian Bayesian sae esimaion. In IEE Proceedings on Radar and Signal Processing, volume 14, pages , [13] Fredrik Gusafsson. Adapive Filering and Change Deecion. John Wiley & Sons Ld,. [14] C. Hue, J.P. Le Cadre, and P. Pérez. Tracking muliple objecs wih paricle filering. Technical Repor Research repor IRISA, No1361, Oc. [15] C. Rago, P.Wille, and R.Srei. A comparison of he JPDAF and PMHT racking algorihms. In Proc. IEEE Conf. Acousics, Speech and Signal Processing (ICASSP), volume5, pages , [16] Donald B. Reid. The applicaion of muliple arge racking heory o ocean surveillance. In Proc. of he 18h IEEE Conference on Decision and Conrol, pages , [17] B.D. Ripley. Sochasic Simulaion. John Wiley, [18] D.J Salmond, D. Fisher, and N.J Gordon. Tracking in he presence of inermien spurious objecs and cluer. In SPIE Conf. on Signal and Daa Processing of Small Trages, [19] D. Schulz, W. Burgard, D. Fox, and A.B. Cremers. Tracking muliple moving arges wih a mobile robo using paricle filers and saisical daa associaion. In IEEE Proc. Inernaional Conference on Roboics and Auomaion, volume, pages , 1.