Multitarget Tracking with Interacting Population-based MCMC-PF

Transcription

1 Multitarget Tracing with Interacting Population-baed MCMC-PF Mélanie Bocquel, Han Drieen Arun Bagchi Thale Nederland B.V. - Sen TBU Radar Engineering, Univerity of Twente - Department of Applied Mathematic Hengelo, Nederland Enchede, Nederland {Melanie.Bocquel, a.bagchi@ewi.utwente.nl Abtract In thi paper we addre the problem of tracing multiple target baed on raw meaurement by mean of Particle filtering. Thi trategy lead to a high computational complexity a the number of target increae, o that an efficient implementation of the tracer i neceary. We propoe a new multitarget Particle Filter (PF) that olve uch challenging problem. We call our filter Interacting Population-baed MCMC-PF (IP-MCMC-PF) ince our approach i baed on parallel uage of multiple population-baed Metropoli-Hating (M-H) ampler. Furthermore, to improve the chain mixing propertie, we exploit genetic alie move performing interaction between the Marov Chain Monte Carlo (MCMC) chain. Simulation analye verify a dramatic reduction in term of computational time for a given trac accuracy, and an increaed robutne w.r.t. conventional MCMC baed PF. I. INTRODUCTION Multitarget tracing i a well-nown problem which conit of equentially etimating the tate of everal target from noiy data. It arie in many application, e.g. radar baed tracing of aircraft. Bayeian method provide a rigorou general framewor for dynamic tate etimation problem. The Bayeian recurion update the poterior probability denity function (pdf) of the tate baed on all available information, including the et of received meaurement. The application of the Bayeian equential etimation framewor to multitarget tracing problem i plagued by two difficultie. Firt, the tate and obervation model are often non-linear and non-gauian, o that no cloed-form analytic expreion can be obtained for the tracing recurion. The econd difficulty i due to the fact that in mot practical tracing application the enor yield unlabeled meaurement of the target. Thi lead to a challenging data aociation problem. The multitarget tracing problem ha been traditionally addreed with technique uch a multiple hypothei tracing (MHT) and joint probabilitic data aociation (JPDA), which require plot meaurement (detection) and a meaurementto-trac aociation procedure [1]. Solution uing a Particle Filter (PF) have been propoed in the pat ten year [2] [4]. Thee o-called Trac-Before-Detect (TBD) approache define a model for the raw meaurement in term of a multi-target tate hypothei, thu avoiding an explicit data aociation tep. Furthermore, a oppoed to the conventional threholded meaurement procedure, thee trategie allow the tracer to perform well in the low Signal-to-Noie Ratio (SNR) regime. Particle filtering i a Sequential Monte Carlo (SMC) imulation-baed method of approximately olving the Baye prediction and update equation recurively uing a tochatic ample (particle) cloud [5]. The Sampling Importance Reampling (SIR) Multitarget PF, that recurively etimate the joint multitarget probability denity (JMPD), ha demontrated it ability to uccefully perform nonlinear and non-gauian tracing and to enable more accurate modeling of the target correlation. While the SIR PF i fairly eay to implement and tune, it main drawbac are the computational complexity which increae rapidly with the tate dimenion and the lo of diverity among the particle due to reampling. To mitigate the former problem an adaptive ampling trategy, called the Adaptive Propoal (AP) method [3], ha been uggeted. Thi trategy propoe to contruct particle propoal at the partition level, incorporating the meaurement o a to bia the propoal toward the optimal importance denity. The AP method automatically factor the JMPD when target are behaving indepently [6], while attempt to handle the permutation ymmetry and correlation that arie when target are coupled [3]. The latter problem can effectively be addreed by uing Particle MCMC (PMCMC) method [7], [8], e.g. the Particle Marginal Metropoli-Hating (PMMH) algorithm. In recent year there ha been an increaing interet toward Marov Chain Monte Carlo (MCMC) method to imulate complex, nontandard multivariate ditribution. There are two baic MCMC technique: the Gibb ampler [9], [10], and the Metropoli-Hating (M-H) [11], [12] algorithm. The former method i baed on ampling from the collection of full conditional ditribution, while the latter approach i a generalization that can be ued when the full conditional cannot be written in a cloed form. Recently, the reearch done on Marov chain and Hidden Marov Model ha led the definition of the M-H algorithm through the notion of reveribility [13], and allowed for the derivation of convergence reult for Sequential MCMC method, e.g. the PMCMC. Although thee method are very reliable, the computational expene of PMCMC algorithm for complex model remain a limiting factor. Furthermore, the mixing of the reulting MCMC ernel can be quite enitive, both to the number of particle and the meaurement pace dimenion. 74

2 In thi paper we focu on iue ariing in the implementation of multitarget Bayeian filtering. Thu, we propoe an alternative algorithm, which fully exploit the peed and the parallelim of modern computing architecture and reource. We introduce a new M-H baed PF in which everal MCMC chain will be running in parallel, thi way generating population-baed ample to approximate the target ditribution. Furthermore, to improve the mixing propertie of the reulting MCMC ernel, the diverity of each population i increaed by mean of an interaction procedure [14]. Note that we will aume throughout thi paper that the number of target M i nown and contant over time. The paper i organized a follow: in ection II we review the Bayeian Filtering problem; ection III briefly review the baic of particle filtering and MCMC method for ampling. Section IV report the main part of our wor. Here we introduce our new algorithm, which we called Interacting Population-baed MCMC-PF (IP-MCMC-PF), dicu the improvement attainable by uing a fully parallelizable PF, and give theoretical jutification for our approach. In ection V the ytem dynamic and meaurement model are introduced for the pecific TBD urveillance application. Section VI collect our imulation reult. Finally, we report our concluion and direction for future reearch in ection VII. II. BAYESIAN FILTERING PROBLEM In thi ection, we briefly decribe the Bayeian Filtering problem. Let u conider the following dynamical ytem: x +1 = f (x, v ), (1) z = h (x ) + w, (2) where x R nx denote the tate vector, z R nz the ytem meaurement, v p v (v) the proce noie, and w p w (w) the meaurement noie. Furthermore, let Z {z 1,..., z } be the equence of meaurement collected up to time. The meaurement z i aumed to be indepent from the pat tate, i.e., p(z z 1,..., z 1, x, x 1,..., x 0 ) = p(z x ), (3) where p(z x ) denote the lielihood function. Given a realization of Z, Bayeian filtering boil down to finding an approximation of the poterior pdf p(x Z ). In particular, the marginal filtering ditribution p(x Z ) i obtained at time by mean of a two-tep recurion: (1) the Prediction Step, which i olved uing the Chapman- Kolmogorov equation, i.e., p(x Z 1 ) = p(x x 1 ) p(x 1 Z 1 )dx 1 (4) where, p(x Z 1 ) i the predictive denity at time ; (2) the Update Step, which i olved uing Baye theorem, i.e., p(x Z ) = p(z x ) p(x Z 1 ) p(z Z 1 ) where p(z Z 1 ) i the Baye normalization contant (evidence). Thu, the filtering pdf i completely pecified given (5) ome prior p(x 0 ), a tranition ernel p(x x 1 ) and a lielihood p(z x ). Given the a poteriori pdf p(x Z ), ome averaged tatitic of interet can be calculated, e.g. the mean-quared error (MSE), i.e., E[ x ˆx 2 Z ] = x ˆx 2 p(x Z )dx, which can be ued to derive the minimum variance etimator, x p(x Z )dx. (6) ˆx MV III. FUNDAMENTALS In thi ection we recall the baic of Marov Chain Monte Carlo, Particle Filtering and Particle MCMC method, neceary for clarifying the trength of our method. In fact, in ection IV we propoe an approach for tate etimation that combine the differing pro of both MCMC and PF baed ignal proceing. Specifically, ubection III-A i dedicated to MCMC method; ubection III-B review Particle Filtering technique; while ubection III-C i ued to dicu related wor on Particle MCMC method. A. Baic of Marov Chain Monte Carlo proce Let X R M be a compact et and uppoe that π(x) i a probability ditribution on uch a high-dimenional pace X. Denote by B(X ) the Borel σ-field and by (X, B(X )) the aociated meaure pace. Then, than to Marov chain theory and Monte Carlo ampling, the following hold: A Marov Chain Monte Carlo method for the imulation of the ditribution π i any method which generate an ergodic Marov chain {x (i) } i 1 on X, according to a ernel K(.,.) defined on (X B(X )), with π a tationary ditribution. Let u recall two baic concept: 1) π i the tationary ditribution of the Marov chain, i.e., for all A B(X ), π(a) = K(x, A)π(x)dx. (7) X 2) The Marov chain i ergodic, i.e. for any integrable function q : X R 1 N lim q(x (i) N N ) E π(q) = q(x)π(x)dx (8) i=1 with probability 1, x (0) X, where N i the number of iteration of the MCMC algorithm. The Metropoli-Hating (M-H) algorithm [15] i a well nown procedure baed on a Marovian proce which fulfill the requirement of ergodicity [16]. The M-H algorithm employ a conditional denity q, alo nown a propoal ditribution, to generate a Marov chain with an invariant ditribution π. At each MCMC iteration i, a move x q(x x (i 1) ) i propoed and accepted with a probability 0 < α(x (i 1), x ) < 1. X 75

3 Notice that the initial burn-in ample are trongly influenced by the initial configuration, o dicarding them increae the peed of convergence toward the tationary ditribution. Let u now conider a multitarget etting where x = [x,1,..., x,m ] i the multitarget bae tate vector, with x,j being the tate vector of the j th target at time. Within the Bayeian etimation framewor, the target ditribution π i choen a the approximate poterior ditribution, π(x (i 1) ) = p(x (i 1) Z ) p(x (i 1) ) p(z x (i 1) ). (9) The acceptance ratio i then given by: α = p(x ) p(z x ) q(x (i 1) X ) ) p(z x (i 1) ) q(x x (i 1) p(x (i 1) ). (10) A uitably deigned propoal ditribution i fundamental to guarantee a well mixing Marov chain. The efficiency of the M-H algorithm i trongly influenced by uch choice. In particular, thi problem become central when dealing with high dimenional and interdepent tate vector. Such ditribution hould both be eay to ample from and allow the Marov chain to explore all the high denity region of X under π freely. However, the deign of uch efficient propoal ditribution might be problematic. Intead, local trategie, focuing on ome of the ubcomponent of π, ( e.g. propoe to move one target at each time, x j q(x j x(i 1),j ) ) are often ued, to brea up the original ampling problem into impler one. Neverthele, thee can be prone to poor performance, a local trategie inevitably ignore ome of the global feature of π. In ummary, Marov Chain Monte Carlo (MCMC) method are convenient and flexible, but they require to meet the following condition : A ufficiently long burn-in time to allow convergence. Enough imulation draw for a uitably accurate inference for etimating the ditribution π. Note however that the convergence of MCMC algorithm trongly dep on whether the model actually fit the data and, in particular, the quality of the propoal ditribution. Therefore convergence problem are often related to modeling iue. B. Baic of Particle Filtering A Particle Filter i nown for it ability to tacle nonlinear and non-gauian problem. The detailed PF algorithm, a introduced in [17], i decribed below. At time tep 1, the poterior pdf p(x 1 Z 1 ) i approximated by a et of particle {x (i) 1 }Np i=1 particle x (i) 1 obtain {ˆx (i) with aociated weight {w(i) 1 }Np i=1. Each i paed through the ytem dynamic eq.(1) to }Np i=1, the predicted particle at time tep. Once the obervation data z i received, the importance weight of each predicted particle can be evaluated according to the weight equation reduce to: w (i) = p(z ˆx (i) ) Then, after normalization of the weight, the a poteriori denity p(x Z ) at time tep can be approximated by the empirical ditribution a : where δˆx (i) N p ˆp(x Z ) := i=1 w (i) (i)(x δˆx ), (11) (.) denote the delta-dirac ma located at ˆx (i). See [18] for a proof and [19] for more detail and an overview of convergence reult. Finally, a reampling tep i ued to prevent the variance of the particle weight to increae over time. Although the reampling tep reduce the effect of the degeneracy problem, it introduce other practical problem. Firt, it limit parallel proceing ince all particle have to be combined. Second, by reampling the poterior ditribution, the heavily weighted particle are tatitically elected everal time. Thi lead to a lo of diverity among the particle, the ample impoverihment problem. In the extreme cae, all the particle collape to the ame location. The ample impoverihment lead to failure in tracing ince le divere particle are ued to repreent the uncertain dynamic of the moving object. Both problem can effectively be handled by uing a MCMC method implemented by e.g. the M-H algorithm. C. Particle Marov chain Monte Carlo (PMCMC) Several algorithm combining MCMC and SMC approache have already been propoed in the literature [7], [20], [21]. Recently, Andrieu, Doucet and Holentein [8] introduced a general framewor, nown a Particle MCMC (PMCMC), which ue a PF to contruct propoal ernel for an MCMC ampler. PMCMC algorithm can be thought of a natural approximation to tandard and idealized MCMC algorithm which cannot be implemented in practice. Thi framewor provide three powerful method for joint Bayeian tate and parameter inference for nonlinear, non-gauian tate-pace model. Thee method are referred to a Particle Indepent Metropoli-Hating (PIMH), Particle Marginal Metropoli- Hating (PMMH) and Particle Gibb (PG). In particular, the PMMH algorithm aim an exact approximation of a Marginal Metropoli-Hating (MMH) update. The implementation of the PMMH cheme require an SMC approximation targeting p(x 0: Z, θ) and the filter etimate of the marginal lielihood p(z θ) with θ Θ ome tatic parameter. Full detail of the PMMH cheme including a proof etablihing that the method leave the full joint poterior denity p(θ, x 0: Z ) invariant can be found in [8]. However, note that to obtain reaonable mixing of the reulting MCMC ernel, it wa reported in [8] that a fairly high number of particle wa required in the SMC cheme. Since every iteration of the PMMH cheme require a run of a PF, a lot of computational reource are needed. In fact, if a PF algorithm uing N p particle i applied at each iteration, then the computational complexity of the PMCMC algorithm i O(N p M N), where N i the number of MCMC iteration and M = dim(x ). In the following ection we will propoe 76

4 an alternative algorithm, which i much more robut to a low number of particle, and well uited to deal with tracing problem. IV. INTERACTING POPULATION-BASED MCMC-PF In thi ection, we will preent the Interacting Populationbaed MCMC-PF (IP-MCMC-PF) algorithm. We will give theoretical jutification for our approach and dicu the improvement attainable by uing a fully parallelizable PF. A. Jutification behind the IP-MCMC-PF algorithm There are two baic idea behind the propoed algorithm : (1) Reliable tatitical inference for the target ditribution are required. For thi purpoe we run multiple MCMC ampling chain each tarting from different eed, in parallel. A ingle poibly time varying tranition ernel i ued for all parallel chain. The only difference i the region of the pace explored by each chain. The imulation from the chain are pread acro variou high probability region of the target ditribution. After a ufficiently long burn-in period, each chain reache the tationary ditribution. Thi mean that, once enough tationary imulation have been drawn, mixing all et of draw provide a good etimate of the target ditribution. (2) Rapid mixing within each MCMC chain i required. We want to peed up the MCMC convergence rate for minimizing the burn-in period. For that the hitory from all the chain i ued to adapt the ernel and therefore to guide any particular population member in the exploration of the tate pace toward region of higher probability. Thu more global move (than indepent MCMC chain) can be contructed. Thi enure the leat correlation among a ingle particle hitory tate reulting in fater mixing within each MCMC chain. Furthermore the propoed implementation reort to withinchain analyi to monitor tationarity and between/within chain comparion to monitor mixing. Combined, tationarity and mixing lead to the convergence of the et of MCMC chain. B. IP-MCMC-PF algorithm and challenge of implementation In the propoed IP-MCMC-PF algorithm, the poterior ditribution over target tate at time 1 i repreented by a et of N p particle {x (i) 1 }Np i=1. Each particle contain the joint tate of M partition correponding to the different target. We refer to x (i) 1,j a the tate of the jth partition of the particle i at time tep 1. The particle tate vector i given by : [ ] x (i) 1 = x (i) 1,1,..., x(i) 1,M. Let u denote N MCMC the optimal number of MCMC chain that can operate in parallel. The parameter N MCMC dep on firtly, the experiment etup (target in trac and modelling complexity) and econdly, the ytem pecification (parallel proceing potential, memory reource). At each time tep, N MCMC particle are randomly elected from {x (i) 1 }Np i=1. Thee particle are then propagated baed on the dynamic model eq.(1) to obtain the predicted random ubet of particle {ŷ (0) } NMCMC =1. The predicted particle are then choen a the tarting point in the MCMC ampling procedure. The MCMC ampling procedure conit to run N MCMC M- H ampler in parallel. Each ampler i initiated with the configuration {ŷ (0) ; l (0) } and i ued to generate a et of N ample from p(x Z ). At each MCMC iteration n M-H, a partition j of a random particle i piced out of {x (i) 1 }Np i=1 i randomly elected. Given thi partition x (i) 1,j, a partition move y i propoed. Thi move ee to update the ingle partition x (i) 1,j via a Marov ernel that decribe the tate dynamic eq.(1). Thi move, alo nown a Mutation (Genetic Algorithm ee [22]), allow for local exploration of the tate pace, a well a enuring the required irreducibility of the Marov chain. Then, given {ŷ (nm-h 1) ; l (nm-h 1) } the previou configuration, a new configuration {ỹ ; l } i obtained by replacing the partition j by y and updating the joint log-lielihood. Thi new configuration i propoed and accepted a the next realization from the chain with a probability ( 0 < min 1, α(ŷ (nm-h 1) ), ỹ < 1. ) The acceptance ratio only dep on the lielihood and i given by : α(ŷ (nm-h 1), ỹ ) = l (n M-H) l (nm-h 1). (12) Note that uch M-H move can be een a an Exchange move (Genetic Algorithm ee [22]). It can be proven that information exchange between MCMC chain targeting cloe ditribution doe not effect convergence propertie [16]. Furthermore, in the propoed procedure, the MCMC ampling chain wap information within the M-H ampler without effecting the parallel proceing. At the of the procedure, the initial burn-in imulation, which are trongly influenced by tarting value, are dicarded, while the remaining ample {ŷ } B+N n M-H=B+1 are tored a new particle to ummarize the target ditribution at time tep. In the propoed algorithm, a validation tet i performed to chec the convergence on the bai of Gelman and Rubin [23] diagnotic. The potential cale reduction factor (PSRF) ˆR can be ued for uch tet. In fact, ˆR, defined a ˆR = C/ W, (13) with C the empirical variance from all chain combined and W, the mean of the empirical within-chain variance, provide an ANOVA-lie comparion of the within-chain and betweenchain variance, i.e. ˆR = 1, then the MCMC chain have converged within the burn-in period. ˆR > 1, then all the MCMC chain have not fully mixed and that further imulation might increae the preciion of inference. 77

5 If the PSRF i cloe to 1, we can conclude that each of the N MCMC et of N imulated obervation i cloe to the target ditribution. In the propoed algorithm the parallel chain are conidered well-mixed when ˆR i le or equal than 1.1 (the PSRF i etimated with uncertainty becaue our MCMC chain length are finite). Once the et of chain have reached approximate convergence, the MCMC ampling chain output, i.e. the N MCMC et of imulation, mixed all together give the new et of particle {x (i) }Np i=1 which approximate the target ditribution p(x Z ). The propoed IP-MCMC-PF algorithm i ummarized in Algorithm 1. Then, a peudo-code decription of the MCMC ampling procedure i given by Algorithm 2. Algorithm 1: IP-MCMC-PF algorithm input : {x (i) 1 }Np i=1 and a new meaurement, z. output: {x (i) }Np i= Initialiation tarting point: Select a random ubet {y (0) } NMCMC =1 {x (i) 1 }Np i=1 while 1 to N MCMC do Predict a new particle tate at time : while j 1 to M do ŷ (0),j = f (y (0),j ) + v(i) 1,j. Compute the aociated joint log-lielihood: l (0) = log p(z ŷ (0) ). Store the new tate in a cache {ŷ (0) ; l (0) } NMCMC = MCMC ampling procedure: Run N MCMC Metropoli-Hating ampler in parallel (Algorithm 2) to obtain {y (i) } where i = 1,..., N and = 1,..., N MCMC. 3 - Checing convergence: Compute ˆR: (eq. 13). if ˆR 1.1 then Run the chain out longer to improve convergence to the tationary ditribution. ele Mix all the N MCMC et of imulation together to obtain {x (i) }Np i=1. V. SYSTEM SETUP AND TBD PROBLEM FORMULATION In thi ection, we will decribe the ytem dynamic model and the meaurement model. Let u denote by x,j the vector decribing the tate of the j th target (j {1, M}) at time tep written a: x,j = [,j, ρ,j ] T (14) where,,j repreent the poition and velocity of the j th target in Carteian coordinate and ρ,j i the unnown Algorithm 2: MCMC ampling procedure input : {ŷ (0) ; l (0) } NMCMC =1 initial configuration, N p required number of ample, B number of burn-in ample. output: {y (i) } where = 1,..., N MCMC and i = 1,..., N with N := N p /N MCMC. while 1 to N MCMC do for n M-H 1 to B + N do 1 - Propoal of move: Randomly elect a partition x (i) 1,j out of {x (i) 1 }Np i=1 ; Given x (i) 1,j, draw y = f (x (i) 1,j ) + v(i) Propoe ỹ by y ŷ (nm-h 1) ; Compute the joint log-lielihood: l = log p(z ỹ ). 1,j ; 2 - M-H acceptance: Sample u U [0,1], U [0,1] a uniform ditribution in [0, 1]; Compute the M-H acceptance probability α(ŷ (nm-h 1), ỹ ): (eq. 12). if u min (1, α) then Accept move: {ŷ ; l } = {ỹ ; l }, ele Reject move: {ŷ ; l } = {ŷ (nm-h 1) ; l (nm-h 1) }. Dicard B initial burn-in ample, tore the remaining ample ŷ (B+1:B+N) {y (i) } N i=1. modulu of the target complex amplitude. The tate vector (x,j ) j {1,M} can be concatenated into the multitarget tate vector x. A. Dynamic model To model the dynamic of the target we adopt a nearly contant velocity model to decribe object poition and velocity in a Carteian frame, ee e.g. [4], and a random wal model for object amplitude ρ,j. The model uncertainty i handled by the proce noie v,j, which i aumed to be tandard white Gauian noie with covariance G. Under thee aumption, the correponding tate-pace model, for x,j, the tate vector aociated to the j th target, i given by: x +1,j = F x,j + v,j, (15) where F repreent a tranition matrix with a contant ampling time T uch a: F = diag (F 1, F 1, 1), [ ] 1 T where, F 1 =. (16)

6 and G, the covariance proce noie, i given by: where, G 1 = G = diag (a x G 1, a y G 1, a ρ T ), [ ] T 3 T T 2 2 T, (17) where a x = a y and a ρ denote the level of proce noie in object motion and amplitude, repectively. Thi model correctly approximate mall acceleration in the object motion and fluctuation in the object amplitude. B. Meaurement model At dicrete intant, the radar ytem poitioned at the Carteian origin collect a noiy ignal. Each meaurement z conit of N r N d N b reflected power meaurement, where N r, N d and N b are the number of range, doppler and bearing cell. The power meaurement per range-dopplerbearing cell i defined by : z lmn z lmn = z lmn ρ, 2, N. (18) where z lmn ρ, i the complex envelope data of the target decribed by the following nonlinear equation, ee [4]: z lmn ρ, = M j=1 ρ,j e iϕ h lmn (,j ) + w lmn, ϕ (0, 2π) (19) where h lmn (,j ) i the reflection form of the j th target, that for every range-doppler-bearing cell i defined by: h lmn (x,j ) := e (r l r,j ) 2 2R (dm d,j )2 2D (bn b,j )2 2B, (20) l = 1,..., N r, m = 1,..., N d, n = 1,..., N b and N where the relationhip between the meaurement pace and the target pace can be etablihed a: r = x 2 + y 2, d = (x v x + y v y ) x2 + y 2 ( y and b = arctan. x) (21) R, D and B are related to the reolution in range, doppler and bearing. w lmn are indepent ample of a complex white Gauian noie with variance 2σw, 2 (i.e. the real and imaginary component are aumed to be indepent, zero-mean white Gauian with the ame variance σw). 2 VI. SIMULATION SETUP AND RESULTS In thi ection the multitarget TBD filtering problem ha been recurively olved by running the IP-MCMC-PF on the power meaurement. We invetigate through imulation the improvement provided by interacting population-baed imulation over tandard SMC and MCMC method. Simulation are performed with a typical ground radar cenario (one or everal target move in the x-y plane at unnown contant velocity). The radar parameter and the target etting ued in imulation are reported in Table I. Radar parameter Target etting TABLE I PARAMETERS USED IN SIMULATION Radar ampling time T = 1 [ec] Beamwidth in bearing B = π 180 [rad] Range-quant ize R = 10 [m] Doppler-bin ize D = 2 [m. 1 ] Bearing area b 0 U [ 6, 6] [ ] Range area r 0 U [2700, 2900] [m] Radial velocity area v 0 U [0, 15] [m. 1 ] Acceleration in turn a 0 U [0, 1] [m. 2 ] Angle of turn ϕ 0 U [ 3, 3] [ ] SNR SNR U [9, 20] [db] Maximum Target Speed v max = 40 [m. 1 ] Throughout the imulation the target move according to the initial condition along a contant velocity trajectory. A recurive trac filter we have ued a filter tracing the twodimenional poition and velocity with a piecewie contant white acceleration model defined in eq.(15) for the target dynamic, where we have et the tandard deviation of the random acceleration to 1 m. 2. All the target remain within the urveillance region until the lat time tep. We firt conider a implified verion of the Trac-Before- Detect (TBD) problem, where a ingle-target move within the urveillance area and never leave the enor Field-of-View (FoV). In particular, we focu on the cenario depicted in fig.1. C. TBD Problem Formulation Conider the ytem repreented by the equation (15), (18) and (19). Aume that the et of meaurement collected up to the current time i denoted by Z = {z 1,..., z }. The filtering problem can be formulated a finding the a poteriori ditribution of the joint tate x for all poible number of target conditioned on all pat meaurement Z. Fig. 1. True trac in the x-y plane. Target move with near-contant velocity along the path hown. Three algorithm : the conventional (a) SIR PF (alg. ee [4]), the recently developed (b) PMMH (alg. ee [7]) and the propoed (c) IP-MCMC-PF (algo. 1) are conidered. 79

7 In fig. 2 we report the empirical poterior PDF and the MV etimate of eq.(6) over a ingle trial. The trac (the mean of each et of target ample) produced by the IP-MCMC-PF algorithm over 100 Monte Carlo run are hown in fig. 5. Fig. 2. Empirical poterior ditribution and conditional mean over a ingle trial. Each filter ue 500 particle. It i immediate to verify that the propoed method reduce the intrinic uncertainty in the poterior PDF when compared to SIR PF. In fig. 3 we compare the tracing accuracy, meaured in term of the Root Mean Square Error (RMSE), over 100 Monte-Carlo imulation. Fig. 5. Tracing trajectory. N p = 500 particle, N MCMC = 50 MCMC chain and B = 100 burn-in ample. We then compared the tracing accuracy, the computational cot and the robutne of the three algorithm over 100 Monte-Carlo imulation. The tracing performance i meaured in term of the Root Mean Square Error (RMSE). The reult are reported in fig. 6. Fig. 3. Poition RMSE over time for the (a) SIR PF, (b) PMMH and (c) IP-MCMC-PF. The propoed algorithm matche the tracing performance of the SIR PF and the PMMH. We then conider a typical cenario with 10 random target depicted in fig. 4. Fig. 6. Poition RMSE over time for the (a) SIR PF, (b) PMMH and (c) IP-MCMC-PF. The performance of the method i alo compared via the average RMSE, the tracing lo rate (TLR) and the average CPU time (avg.cpu), which are lited in Table II. TABLE II PERFORMANCE COMPARISON AVERAGED OVER 100 MC RUNS Algorithm N p RMSE TLR avg.cpu (a) SIR PF [m] 6% [m] 3% 3.8 (b) PMMH [m] 1% [m] 0% 2.7 Fig. 4. True trac in the x-y plane. Target move with near-contant velocity along the path hown. (c) IP-MCMC-PF [m] 0% [m] 0%

8 The tracing lo rate (TLR) i defined a the ratio of the number of imulation, in which the target i lot in trac (Maximum Poition error > 20[m]), to the total number of imulation carried out. The average CPU time (avg.cpu) i the CPU time needed to execute one time tep in MATLAB R2011b (win64) on an Intel Core i7 (Nehalem microarchitecture) operating under Window 7 Ultimate (Verion 6.1). All the important pecification and feature are reported in Table III. Proceor TABLE III SPECIFICATIONS & FEATURES Type Intel( R) Core(TM) i7 Model 920 Frequency 4.2 GHz Core 4 (8 thread) Memory DRAM 6144 MB (3 x 2048 DDR3-SDRAM) Type PC (533 MHz) VII. CONCLUSION In thi wor, we propoe an Interacting Population baed Monte Carlo Marov Chain baed PF (IP-MCMC-PF) that olve the multitarget tracing problem, which i fully parallelizable. In particular, the propoed algorithm matche the tracing performance of the multitarget SIR PF, while allowing for a dramatic reduction of the computational time. Furthermore, we demontrate through imulation that the propoed IP-MCMC-PF provide better tracing performance than the conventional MCMC baed particle filter. Thi hold in term of trac accuracy and robutne to a low number of particle. In fact, ampling particle from the target poterior ditribution via interacting population-baed imulation avoid ample impoverihment and accelerate the MCMC convergence rate. In a forthcoming wor, we will therefore focu on more complex cenario where target may enter or leave the obervation area. Furthermore, a future reearch we will invetigate the applicability of well nown MCMC technique to additionally reduce the computational burden. ACKNOWLEDGMENT The reearch leading to thee reult ha received funding from the EU Seventh Framewor Programme under grant agreement n o The reearch ha been carried out in the MC IMPULSE project: REFERENCES [1] J. Vermaa, S. Godill, and P. Perez, Monte carlo filtering for multitarget tracing and data aociation, IEEE Tr. 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