Multitarget Tracking withInteracting Population-based MCMC-PFMélanie Bocquel, Hans Driessen

Thales Nederland B.V. - Sens TBU Radar Engineering,Hengelo, Nederland

Email: {Melanie.Bocquel, Hans.Driessen} @nl.thalesgroup.com

Arun BagchiUniversity of Twente - Department of Applied Mathematics

Enschede, NederlandEmail: a.bagchi@ewi.utwente.nl

Abstract—In this paper we address the problem of trackingmultiple targets based on raw measurements by means ofParticle filtering. This strategy leads to a high computationalcomplexity as the number of targets increases, so that an efficientimplementation of the tracker is necessary.

We propose a new multitarget Particle Filter (PF) thatsolves such challenging problem. We call our filter Interact-ing Population-based MCMC-PF (IP-MCMC-PF) since our ap-proach is based on parallel usage of multiple population-basedMetropolis-Hastings (M-H) samplers. Furthermore, to improvethe chains mixing properties, we exploit genetic alike movesperforming interaction between the Markov Chain Monte Carlo(MCMC) chains.

Simulation analyses verify a dramatic reduction in terms ofcomputational time for a given track accuracy, and an increasedrobustness w.r.t. conventional MCMC based PF.

I. INTRODUCTION

Multitarget tracking is a well-known problem which consistsof sequentially estimating the states of several targets fromnoisy data. It arises in many applications, e.g. radar basedtracking of aircraft. Bayesian methods provide a rigorousgeneral framework for dynamic state estimation problems. TheBayesian recursions update the posterior probability densityfunction (pdf) of the state based on all available information,including the set of received measurements. The applicationof the Bayesian sequential estimation framework to multitargettracking problems is plagued by two difficulties. First, the stateand observation models are often non-linear and non-Gaussian,so that no closed-form analytic expression can be obtained forthe tracking recursions. The second difficulty is due to thefact that in most practical tracking applications the sensorsyield unlabeled measurements of the targets. This leads to achallenging data association problem.

The multitarget tracking problem has been traditionallyaddressed with techniques such as multiple hypothesis tracking(MHT) and joint probabilistic data association (JPDA), whichrequire plot measurements (detection) and a measurement-to-track association procedure [1]. Solutions using a ParticleFilter (PF) have been proposed in the past ten years [2]–[4].These so-called Track-Before-Detect (TBD) approaches definea model for the raw measurements in terms of a multi-targetstate hypothesis, thus avoiding an explicit data associationstep. Furthermore, as opposed to the conventional thresholdedmeasurement procedure, these strategies allow the tracker toperform well in the low Signal-to-Noise Ratio (SNR) regime.

Particle filtering is a Sequential Monte Carlo (SMC)simulation-based method of approximately solving the Bayesprediction and update equations recursively using a stochasticsamples (particles) cloud [5]. The Sampling Importance Re-sampling (SIR) Multitarget PF, that recursively estimates thejoint multitarget probability density (JMPD), has demonstratedits ability to successfully perform nonlinear and non-Gaussiantracking and to enable more accurate modeling of the targetcorrelations. While the SIR PF is fairly easy to implement andtune, its main drawbacks are the computational complexitywhich increases rapidly with the state dimensions and the lossof diversity among the particles due to resampling. To mitigatethe former problem an adaptive sampling strategy, called theAdaptive Proposal (AP) method [3], has been suggested. Thisstrategy proposes to construct particle proposals at the partitionlevel, incorporating the measurements so as to bias the pro-posal towards the optimal importance density. The AP methodautomatically factors the JMPD when targets are behavingindependently [6], while attempt to handle the permutationsymmetry and correlations that arise when targets are coupled[3]. The latter problem can effectively be addressed by usingParticle MCMC (PMCMC) methods [7], [8], e.g. the ParticleMarginal Metropolis-Hastings (PMMH) algorithm.

In recent years there has been an increasing interest towardsMarkov Chain Monte Carlo (MCMC) methods to simulatecomplex, nonstandard multivariate distributions. There aretwo basic MCMC techniques: the Gibbs sampler [9], [10],and the Metropolis-Hastings (M-H) [11], [12] algorithm. Theformer method is based on sampling from the collection offull conditional distributions, while the latter approach is ageneralization that can be used when the full conditionalscannot be written in a closed form.

Recently, the research done on Markov chains and HiddenMarkov Models has led the definition of the M-H algorithmthrough the notion of reversibility [13], and allowed forthe derivation of convergence results for Sequential MCMCmethods, e.g. the PMCMC. Although these methods are veryreliable, the computational expense of PMCMC algorithms forcomplex models remains a limiting factor. Furthermore, themixing of the resulting MCMC kernels can be quite sensitive,both to the number of particles and the measurement spacedimension.

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In this paper we focus on issues arising in the imple-mentation of multitarget Bayesian filtering. Thus, we pro-pose an alternative algorithm, which fully exploits the speedand the parallelism of modern computing architectures andresources. We introduce a new M-H based PF in whichseveral MCMC chains will be running in parallel, this waygenerating population-based samples to approximate the targetdistribution. Furthermore, to improve the mixing properties ofthe resulting MCMC kernel, the diversity of each populationis increased by means of an interaction procedure [14]. Notethat we will assume throughout this paper that the number oftargets M is known and constant over time.

The paper is organized as follows: in section II we reviewthe Bayesian Filtering problem; section III briefly reviews thebasics of particle filtering and MCMC methods for sampling.Section IV reports the main part of our work. Here weintroduce our new algorithm, which we called InteractingPopulation-based MCMC-PF (IP-MCMC-PF), discuss the im-provements attainable by using a fully parallelizable PF, andgive theoretical justifications for our approach. In section V thesystem dynamics and measurement model are introduced forthe specific TBD surveillance application. Section VI collectsour simulations results. Finally, we report our conclusions anddirection for future research in section VII.

II. BAYESIAN FILTERING PROBLEM

In this section, we briefly describe the Bayesian Filteringproblem. Let us consider the following dynamical system:

xk+1 = fk(xk,vk), (1)zk = hk(xk) + wk, (2)

where xk ∈ Rnx denotes the state vector, zk ∈ Rnz thesystem measurement, vk ∼ pvk

(v) the process noise, andwk ∼ pwk

(w) the measurement noise.Furthermore, let Zk , {z1, . . . , zk} be the sequence ofmeasurements collected up to time k. The measurement zkis assumed to be independent from the past states, i.e.,

p(zk|zk−1, . . . , z1,xk,xk−1, . . . ,x0) = p(zk|xk), (3)

where p(zk|xk) denotes the likelihood function.Given a realization of Zk, Bayesian filtering boils down

to finding an approximation of the posterior pdf p(xk|Zk).In particular, the marginal filtering distribution p(xk|Zk) isobtained at time k by means of a two-step recursion:(1) the Prediction Step, which is solved using the Chapman-Kolmogorov equation, i.e.,

p(xk|Zk−1) =∫p(xk|xk−1) p(xk−1|Zk−1)dxk−1 (4)

where, p(xk|Zk−1) is the predictive density at time k;(2) the Update Step, which is solved using Bayes theorem, i.e.,

p(xk|Zk) =p(zk|xk) p(xk|Zk−1)

p(zk|Zk−1)(5)

where p(zk|Zk−1) is the Bayes normalization constant (evi-dence). Thus, the filtering pdf is completely specified given

some prior p(x0), a transition kernel p(xk|xk−1) and a like-lihood p(zk|xk).

Given the a posteriori pdf p(xk|Zk), some averaged statis-tics of interest can be calculated, e.g. the mean-squared error(MSE), i.e.,

E[‖xk − xk‖2|Zk] =∫‖xk − xk‖2p(xk|Zk)dxk,

which can be used to derive the minimum variance estimator,

xMVk ,

∫xk p(xk|Zk)dxk. (6)

III. FUNDAMENTALS

In this section we recall the basics of Markov ChainMonte Carlo, Particle Filtering and Particle MCMC methods,necessary for clarifying the strengths of our method. In fact,in section IV we propose an approach for state estimation thatcombines the differing pros of both MCMC and PF basedsignal processing. Specifically, subsection III-A is dedicatedto MCMC methods; subsection III-B reviews Particle Filteringtechniques; while subsection III-C is used to discuss relatedwork on Particle MCMC methods.

A. Basic of Markov Chain Monte Carlo process

Let X ⊆ RM be a compact set and suppose that π(x) isa probability distribution on such a high-dimensional spaceX . Denote by B(X ) the Borel σ-field and by (X ,B(X ))the associated measure space. Then, thanks to Markov chainstheory and Monte Carlo sampling, the following holds:

A Markov Chain Monte Carlo method for the simulation ofthe distribution π is any method which generates an ergodicMarkov chain {x(i)

k }i≥1 on X , according to a kernel K(., .)defined on (X × B(X )), with π as stationary distribution.

Let us recall two basic concepts:1) π is the stationary distribution of the Markov chain, i.e.,

for all A ∈ B(X ),

π(A) =∫XK(x,A)π(x)dx. (7)

2) The Markov chain is ergodic, i.e. for any integrablefunction q : X → R

limN→∞

1N

N∑i=1

q(x(i)k )→ Eπ(q) =

∫Xq(x)π(x)dx (8)

with probability 1, ∀ x(0)k ∈ X , where N is the number

of iterations of the MCMC algorithm.The Metropolis-Hastings (M-H) algorithm [15] is a well

known procedure based on a Markovian process which ful-fills the requirement of ergodicity [16]. The M-H algorithmemploys a conditional density q, also known as proposaldistribution, to generate a Markov chain with an invariantdistribution π. At each MCMC iteration i, a move x′ ∼q(x′|x(i−1)) is proposed and accepted with a probability0 < α(x(i−1),x′) < 1.

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Notice that the initial burn-in samples are strongly influ-enced by the initial configuration, so discarding them increasesthe speed of convergence towards the stationary distribution.

Let us now consider a multitarget setting where xk =[xk,1, ...,xk,M ] is the multitarget base state vector, with xk,jbeing the state vector of the jth target at time k. Within theBayesian estimation framework, the target distribution π ischosen as the approximate posterior distribution,

π(x(i−1)k ) = p(x(i−1)

k |Zk) ∝ p(x(i−1)k ) p(Zk|x(i−1)

k ). (9)

The acceptance ratio is then given by:

α =p(x′) p(Zk|x′) q(x(i−1)

k |X′)p(x(i−1)

k ) p(Zk|x(i−1)k ) q(x′|x(i−1)

k ). (10)

A suitably designed proposal distribution is fundamentalto guarantee a well mixing Markov chain. The efficiency ofthe M-H algorithm is strongly influenced by such choice. Inparticular, this problem becomes central when dealing withhigh dimensional and interdependent state vectors.

Such distributions should both be easy to sample from andallow the Markov chain to explore all the high density regionsof X under π freely. However, the design of such efficientproposal distributions might be problematic. Instead, localstrategies, focusing on some of the subcomponents of π,

(e.g.

propose to move one target at each time, x′j ∼ q(x′j |x(i−1)k,j )

)are often used, to break up the original sampling probleminto simpler ones. Nevertheless, these can be prone to poorperformance, as local strategies inevitably ignore some of theglobal features of π.

In summary, Markov Chain Monte Carlo (MCMC) methodsare convenient and flexible, but they require to meet thefollowing conditions :• A sufficiently long burn-in time to allow convergence.• Enough simulation draws for a suitably accurate inference

for estimating the distribution π.Note however that the convergence of MCMC algorithmsstrongly depends on whether the model actually fits the dataand, in particular, the quality of the proposal distribution.Therefore convergence problems are often related to modelingissues.

B. Basic of Particle Filtering

A Particle Filter is known for its ability to tackle nonlinearand non-Gaussian problems. The detailed PF algorithm, asintroduced in [17], is described below. At time step k − 1,the posterior pdf p(xk−1|Zk−1) is approximated by a set ofparticles {x(i)

k−1}Np

i=1 with associated weights {w(i)k−1}

Np

i=1. Eachparticle x(i)

k−1 is passed through the system dynamics eq.(1) toobtain {x(i)

k }Np

i=1, the predicted particles at time step k. Oncethe observation data zk is received, the importance weightof each predicted particle can be evaluated according to theweight equation reduces to:

w(i)k = p(zk|x(i)

k )

Then, after normalization of the weights, the a posterioridensity p(xk|Zk) at time step k can be approximated by theempirical distribution as :

p(xk|Zk) :=Np∑i=1

w(i)k δ

x(i)k

(xk), (11)

where δx

(i)k

(.) denotes the delta-Dirac mass located at x(i)k .

See [18] for a proof and [19] for more detail and an overviewof convergence results.

Finally, a resampling step is used to prevent the varianceof the particle weights to increase over time. Although theresampling step reduces the effects of the degeneracy problem,it introduces other practical problems. First, it limits parallelprocessing since all particles have to be combined. Second,by resampling the posterior distribution, the heavily weightedparticles are statistically selected several times. This leads to aloss of diversity among the particles, the sample impoverish-ment problem. In the extreme case, all the particles collapse tothe same location. The sample impoverishment leads to failurein tracking since less diverse particles are used to represent theuncertain dynamics of the moving object. Both problems caneffectively be handled by using a MCMC method implementedby e.g. the M-H algorithm.

C. Particle Markov chain Monte Carlo (PMCMC)

Several algorithms combining MCMC and SMC approacheshave already been proposed in the literature [7], [20], [21].Recently, Andrieu, Doucet and Holenstein [8] introduced ageneral framework, known as Particle MCMC (PMCMC),which uses a PF to construct proposal kernels for an MCMCsampler. PMCMC algorithms can be thought of as naturalapproximations to standard and idealized MCMC algorithmswhich cannot be implemented in practice. This frameworkprovides three powerful methods for joint Bayesian state andparameter inference for nonlinear, non-Gaussian state-spacemodels. These methods are referred to as Particle IndependentMetropolis-Hastings (PIMH), Particle Marginal Metropolis-Hastings (PMMH) and Particle Gibbs (PG). In particular, thePMMH algorithm aims an exact approximation of a MarginalMetropolis-Hastings (MMH) update. The implementation ofthe PMMH scheme requires an SMC approximation targetingp(x0:k|Zk, θ) and the filter estimate of the marginal likelihoodp(Zk|θ) with θ ∈ Θ some static parameter. Full details of thePMMH scheme including a proof establishing that the methodleaves the full joint posterior density p(θ,x0:k|Zk) invariantcan be found in [8].

However, note that to obtain reasonable mixing of theresulting MCMC kernels, it was reported in [8] that a fairlyhigh number of particles was required in the SMC scheme.Since every iteration of the PMMH scheme requires a run ofa PF, a lot of computational resources are needed. In fact, ifa PF algorithm using Np particles is applied at each iteration,then the computational complexity of the PMCMC algorithmis O(Np M N), where N is the number of MCMC iterationsand M = dim(xk). In the following section we will propose

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an alternative algorithm, which is much more robust to a lownumber of particles, and well suited to deal with trackingproblems.

IV. INTERACTING POPULATION-BASED MCMC-PF

In this section, we will present the Interacting Population-based MCMC-PF (IP-MCMC-PF) algorithm. We will givetheoretical justifications for our approach and discuss theimprovements attainable by using a fully parallelizable PF.

A. Justifications behind the IP-MCMC-PF algorithm

There are two basic ideas behind the proposed algorithm :(1) Reliable statistical inferences for the target distribution

are required. For this purpose we run multiple MCMC sam-pling chains each starting from different seeds, in parallel. Asingle possibly time varying transition kernel is used for allparallel chains. The only difference is the region of the spaceexplored by each chain. The simulations from the chains arespread across various high probability regions of the targetdistribution. After a sufficiently long burn-in period, eachchain reaches the stationary distribution. This means that, onceenough stationary simulations have been drawn, mixing all setsof draws provides a good estimate of the target distribution.

(2) Rapid mixing within each MCMC chain is required. Wewant to speed up the MCMC convergence rate for minimizingthe burn-in period. For that the history from all the chains isused to adapt the kernel and therefore to guide any particularpopulation member in the exploration of the state space towardregions of higher probability. Thus more global moves (thanindependent MCMC chains) can be constructed. This ensuresthe least correlation among a single particle’s history statesresulting in faster mixing within each MCMC chain.

Furthermore the proposed implementation resorts to within-chain analysis to monitor stationarity and between/withinchains comparisons to monitor mixing. Combined, stationarityand mixing lead to the convergence of the set of MCMCchains.

B. IP-MCMC-PF algorithm and challenges of implementation

In the proposed IP-MCMC-PF algorithm, the posteriordistribution over target states at time k−1 is represented by aset of Np particles {x(i)

k−1}Np

i=1. Each particle contains the jointstate of M partitions corresponding to the different targets. Werefer to x(i)

k−1,j as the states of the jth partition of the particlei at time step k − 1. The particle state vector is given by :

x(i)k−1 =

[x(i)k−1,1, . . . ,x

(i)k−1,M

].

Let us denote NMCMC the optimal number of MCMC chainsthat can operate in parallel. The parameter NMCMC depends onfirstly, the experiment setup (targets in track and modellingcomplexity) and secondly, the system specification (parallelprocessing potential, memory resources). At each time stepk, NMCMC particles are randomly selected from {x(i)

k−1}Np

i=1.These particles are then propagated based on the dynamicmodel eq.(1) to obtain the predicted random subset of particles

{y(0)s }NMCMC

s=1 . The predicted particles are then chosen as thestarting points in the MCMC sampling procedure.

The MCMC sampling procedure consists to run NMCMC M-H samplers in parallel. Each sampler s is initiated with theconfiguration {y(0)

s ; `(0)s } and is used to generate a set ofN samples from p(xk|Zk). At each MCMC iteration nM-H,a partition j of a random particle i picked out of {x(i)

k−1}Np

i=1

is randomly selected. Given this partition x(i)k−1,j , a partition

move y′ is proposed. This move seeks to update the sin-gle partition x(i)

k−1,j via a Markov kernel that describes thestate dynamic eq.(1). This move, also known as Mutation(Genetic Algorithm see [22]), allows for local exploration ofthe state space, as well as ensuring the required irreducibilityof the Markov chain. Then, given {y(nM-H−1)

s ; `(nM-H−1)s } the

previous configuration, a new configuration {y(nM-H)s ; ˜(nM-H)

s }is obtained by replacing the partition j by y′ and updatingthe joint log-likelihood. This new configuration is proposedand accepted as the next realization from the chain s with aprobability

0 < min(

1, α(y(nM-H−1)s , y(nM-H)

s ))< 1.

The acceptance ratio only depends on the likelihoods and isgiven by :

α(y(nM-H−1)s , y(nM-H)

s ) =˜(nM-H)s

`(nM-H−1)s

. (12)

Note that such M-H move can be seen as an Exchange move(Genetic Algorithm see [22]). It can be proven that informationexchange between MCMC chains targeting close distributionsdoes not effect convergence properties [16]. Furthermore, inthe proposed procedure, the MCMC sampling chains swapinformation within the M-H samplers without effecting theparallel processing. At the end of the procedure, the ini-tial burn-in simulations, which are strongly influenced bystarting values, are discarded, while the remaining samples{y(nM-H)

s }B+NnM-H=B+1 are stored as new particles to summarize

the target distribution at time step k.

In the proposed algorithm, a validation test is performed tocheck the convergence on the basis of Gelman and Rubin [23]diagnostic. The potential scale reduction factor (PSRF) R canbe used for such test. In fact, R, defined as

R =√

C/W, (13)

with C the empirical variance from all chains combined andW, the mean of the empirical within-chain variances, providesan ANOVA-like comparison of the within-chain and between-chain variances, i.e.• R = 1, then the MCMC chains have converged within

the burn-in period.• R > 1, then all the MCMC chains have not fully mixed

and that further simulation might increase the precisionof inferences.

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If the PSRF is close to 1, we can conclude that each of theNMCMC sets of N simulated observations is close to the targetdistribution. In the proposed algorithm the parallel chains areconsidered well-mixed when R is less or equal than 1.1 (thePSRF is estimated with uncertainty because our MCMC chainlengths are finite).

Once the set of chains have reached approximate conver-gence, the MCMC sampling chains outputs, i.e. the NMCMC

sets of simulations, mixed all together give the new set ofparticles {x(i)

k }Np

i=1 which approximates the target distributionp(xk|Zk).

The proposed IP-MCMC-PF algorithm is summarized inAlgorithm 1. Then, a pseudo-code description of the MCMCsampling procedure is given by Algorithm 2.

Algorithm 1: IP-MCMC-PF algorithm

input : {x(i)k−1}

Np

i=1 and a new measurement, zk.output: {x(i)

k }Np

i=1.

1 - Initialisation starting points:Select a random subset {y(0)

s }NMCMCs=1 ⊂ {x(i)

k−1}Np

i=1

while s← 1 to NMCMC doPredict a new particle state at time k:while j ← 1 to M do

y(0)s,j = fk(y(0)

s,j ) + v(i)k−1,j .

endCompute the associated joint log-likelihood:`(0)s = log p(Zk|y(0)

s ).endStore the new states in a cache {y(0)

s ; `(0)s }NMCMCs=1 .

2 - MCMC sampling procedure:Run NMCMC Metropolis-Hastings samplers in parallel(Algorithm 2) to obtain {y(i)

s } where i = 1, . . . , N ands = 1, . . . , NMCMC.

3 - Checking convergence:Compute R: (eq. 13).

if R ≥ 1.1 thenRun the chains out longer to improve convergence tothe stationary distribution.

elseMix all the NMCMC set of simulations together toobtain {x(i)

k }Np

i=1.end

V. SYSTEM SETUP AND TBD PROBLEM FORMULATION

In this section, we will describe the system dynamic modeland the measurement model.Let us denote by xk,j the vector describing the state of thejth target (j ∈ {1,M}) at time step k written as:

xk,j = [sk,j , ρk,j ]T (14)

where, sk,j represents the position and velocity of the jth

target in Cartesian coordinates and ρk,j is the unknown

Algorithm 2: MCMC sampling procedure

input : {y(0)s ; `(0)s }NMCMC

s=1 initial configurations, Nprequired number of samples, B number ofburn-in samples.

output: {y(i)s } where s = 1, . . . , NMCMC and i = 1, . . . , N

with N := Np/NMCMC.

while s← 1 to NMCMC dofor nM-H ← 1 to B +N do

1 - Proposal of move:Randomly select a partition x(i)

k−1,j out of{x(i)

k−1}Np

i=1;Given x(i)

k−1,j , draw y′ = fk(x(i)k−1,j) + v(i)

k−1,j ;Propose y(nM-H)

s by y′ → y(nM-H−1)s ;

Compute the joint log-likelihood:˜(nM-H)s = log p(Zk|y(nM-H)

s ).

2 - M-H acceptance:Sample u ∼ U[0,1], U[0,1] a uniform distribution in[0, 1];Compute the M-H acceptance probabilityα(y(nM-H−1)

s , y(nM-H)s ): (eq. 12).

if u ≤ min (1, α) thenAccept move:{y(nM-H)

s ; `(nM-H)s } = {y(nM-H)

s ; ˜(nM-H)s },

elseReject move:{y(nM-H)

s ; `(nM-H)s } = {y(nM-H−1)

s ; `(nM-H−1)s }.

endend

Discard B initial burn-in samples, store theremaining samples y(B+1:B+N)

s → {y(i)s }Ni=1.

end

modulus of the target complex amplitude. The state vectors(xk,j)j∈{1,M} can be concatenated into the multitarget statevector xk.

A. Dynamic model

To model the dynamics of the targets we adopt a nearly con-stant velocity model to describe object position and velocity ina Cartesian frame, see e.g. [4], and a random walk model forobject amplitude ρk,j . The model uncertainty is handled bythe process noise vk,j , which is assumed to be standard whiteGaussian noise with covariance G. Under these assumptions,the corresponding state-space model, for xk,j , the state vectorassociated to the jth target, is given by:

xk+1,j = Fxk,j + vk,j , (15)

where F represents a transition matrix with a constant sam-pling time T such as:

F = diag (F1, F1, 1) ,

where, F1 =[

1 T0 1

]. (16)

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and G, the covariance process noise, is given by:

G = diag (axG1, ayG1, aρT ) ,

where, G1 =

[T 3

3T 2

2T 2

2 T

], (17)

where ax = ay and aρ denote the level of process noise inobject motion and amplitude, respectively. This model cor-rectly approximates small accelerations in the object motionand fluctuations in the object amplitude.

B. Measurement model

At discrete instants k, the radar system positioned at theCartesian origin collects a noisy signal. Each measurementzk consists of Nr × Nd × Nb reflected power measurementszlmnk , where Nr, Nd and Nb are the number of range, dopplerand bearing cells. The power measurement per range-doppler-bearing cell is defined by :

zlmnk = |zlmnρ,k |2, k ∈ N. (18)

where zlmnρ,k is the complex envelope data of the target de-scribed by the following nonlinear equation, see [4]:

zlmnρ,k =M∑j=1

ρk,jeiϕkhlmn(sk,j) + wlmn

k , ϕk ∈ (0, 2π)

(19)where hlmn(sk,j) is the reflection form of the jth target, thatfor every range-doppler-bearing cell is defined by:

hlmn(xk,j) := e−(rl−rk,j)2

2R −(dm−dk,j)2

2D −(bn−bk,j)2

2B , (20)l = 1, . . . , Nr, m = 1, . . . , Nd, n = 1, . . . , Nb and k ∈ N

where the relationship between the measurement space andthe target space can be established as:

r =√x2 + y2, d =

(x vx + y vy)√x2 + y2

and b = arctan(yx

).

(21)R, D and B are related to the resolutions in range, doppler andbearing. wlmn

k are independent samples of a complex whiteGaussian noise with variance 2σ2

w, (i.e. the real and imaginarycomponents are assumed to be independent, zero-mean whiteGaussian with the same variance σ2

w).

C. TBD Problem Formulation

Consider the system represented by the equations (15), (18)and (19). Assume that the set of measurements collected upto the current time is denoted by Zk = {z1, . . . , zk}. Thefiltering problem can be formulated as finding the a posterioridistribution of the joint state xk for all possible numbers oftargets conditioned on all past measurements Zk.

VI. SIMULATION SETUP AND RESULTS

In this section the multitarget TBD filtering problem hasbeen recursively solved by running the IP-MCMC-PF on thepower measurements. We investigate through simulations theimprovements provided by interacting population-based simu-lation over standard SMC and MCMC methods. Simulationsare performed with a typical ground radar scenario (one orseveral targets move in the x-y plane at unknown constantvelocity). The radar parameters and the targets settings usedin simulation are reported in Table I.

TABLE IPARAMETERS USED IN SIMULATION

Radar parameters

Radar sampling time T = 1 [sec]Beamwidth in bearing B = π

180[rad]

Range-quant size R = 10 [m]Doppler-bin size D = 2 [m.s−1]

Targets settings

Bearing area b0 ∼ U [−6, 6] [◦]Range area r0 ∼ U [2700, 2900] [m]Radial velocity area |v0| ∼ U [0, 15] [m.s−1]Acceleration in turn a0 ∼ U [0, 1] [m.s−2]Angle of turn ϕ0 ∼ U [−3, 3] [◦]SNR SNR ∼ U [9, 20] [dB]Maximum Target Speed vmax = 40 [m.s−1]

Throughout the simulation the targets move according tothe initial conditions along a constant velocity trajectory. Asrecursive track filter we have used a filter tracking the two-dimensional position and velocity with a piecewise constantwhite acceleration model defined in eq.(15) for the targetdynamics, where we have set the standard deviation of therandom accelerations to 1 m.s−2. All the targets remain withinthe surveillance region until the last time step.

We first consider a simplified version of the Track-Before-Detect (TBD) problem, where a single-target moves withinthe surveillance area and never leaves the sensor Field-of-View(FoV). In particular, we focus on the scenario depicted in fig.1.

Fig. 1. True track in the x-y plane. Target move with near-constant velocityalong the paths shown.

Three algorithms : the conventional (a) SIR PF (alg. see[4]), the recently developed (b) PMMH (alg. see [7]) and theproposed (c) IP-MCMC-PF (algo. 1) are considered.

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In fig. 2 we report the empirical posterior PDF and the MVestimate of eq.(6) over a single trial.

Fig. 2. Empirical posterior distribution and conditional mean over a singletrial. Each filter uses 500 particles. It is immediate to verify that the proposedmethod reduces the intrinsic uncertainty in the posterior PDF when comparedto SIR PF.

In fig. 3 we compare the tracking accuracy, measured interms of the Root Mean Square Error (RMSE), over 100Monte-Carlo simulations.

Fig. 3. Position RMSE over time for the (a) SIR PF, (b) PMMH and (c)IP-MCMC-PF. The proposed algorithm matches the tracking performance ofthe SIR PF and the PMMH.

We then consider a typical scenario with 10 random targetsdepicted in fig. 4.

Fig. 4. True tracks in the x-y plane. Targets move with near-constant velocityalong the paths shown.

The tracks (the means of each set of target samples)produced by the IP-MCMC-PF algorithm over 100 MonteCarlo runs are shown in fig. 5.

Fig. 5. Tracking trajectory. Np = 500 particles, NMCMC = 50 MCMCchains and B = 100 burn-in samples.

We then compared the tracking accuracy, the computationalcost and the robustness of the three algorithms over 100Monte-Carlo simulations. The tracking performance is mea-sured in terms of the Root Mean Square Error (RMSE). Theresults are reported in fig. 6.

Fig. 6. Position RMSE over time for the (a) SIR PF, (b) PMMH and (c)IP-MCMC-PF.

The performance of the methods is also compared via theaverage RMSE, the tracking loss rate (TLR) and the averageCPU time (avg.CPU), which are listed in Table II.

TABLE IIPERFORMANCE COMPARISON AVERAGED OVER 100 MC RUNS

Algorithm Np RMSE TLR avg.CPU

(a) SIR PF 500 6.74 [m] 6% 0.55000 6.23 [m] 3% 3.8

(b) PMMH 500 5.42 [m] 1% 0.65000 4.65 [m] 0% 2.7

(c) IP-MCMC-PF 500 4.84 [m] 0% 0.45000 4.61 [m] 0% 0.69

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The tracking loss rate (TLR) is defined as the ratio of thenumber of simulations, in which the target is lost in track(Maximum Position error > 20[m]) , to the total number ofsimulations carried out. The average CPU time (avg.CPU) isthe CPU time needed to execute one time step in MATLABR2011b (win64) on an Intel Core i7 (Nehalem microarchitec-ture) operating under Windows 7 Ultimate (Version 6.1). Allthe important specifications and features are reported in TableIII.

TABLE IIISPECIFICATIONS & FEATURES

ProcessorType Intel( R) Core(TM) i7Model 920Frequency 4.2 GHzCores 4 (8 threads)

Memory DRAM 6144 MB (3 x 2048 DDR3-SDRAM)Type PC3-8500 (533 MHz)

VII. CONCLUSION

In this work, we propose an Interacting Population basedMonte Carlo Markov Chain based PF (IP-MCMC-PF) thatsolves the multitarget tracking problem, which is fully par-allelizable. In particular, the proposed algorithm matches thetracking performance of the multitarget SIR PF, while allowingfor a dramatic reduction of the computational time.

Furthermore, we demonstrate through simulations that theproposed IP-MCMC-PF provides better tracking performancethan the conventional MCMC based particle filter. This holdsin terms of track accuracy and robustness to a low number ofparticles. In fact, sampling particles from the target posteriordistribution via interacting population-based simulation avoidssample impoverishment and accelerates the MCMC conver-gence rate.

In a forthcoming work, we will therefore focus on morecomplex scenarios where targets may enter or leave theobservation area. Furthermore, as future research we willinvestigate the applicability of well known MCMC techniquesto additionally reduce the computational burden.

ACKNOWLEDGMENT

The research leading to these results has received fundingfrom the EU’s Seventh Framework Programme under grantagreement no238710.

The research has been carried out in the MC IMPULSEproject: https://mcimpulse.isy.liu.se.

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