The Gaussian Mixture MCMC Particle Algorithm for

Dynamic Cluster Tracking

Avishy Carmi, François Septier and Simon J. Godsill

Signal Processing and Communications Laboratory

Department of Engineering

University of Cambridge, U.K.

Abstract – We present a new filtering algorithm fortracking multiple clusters of coordinated targets. Basedon a Markov Chain Monte Carlo (MCMC) mechanism,the new algorithm propagates a discrete approximationof the underlying filtering density. A dynamic Gaus-sian mixture model is utilized for representing the time-varying clustering structure. This involves point pro-cess formulations of typical behavioral moves such asbirth and death of clusters as well as merging and split-ting. Following our previous work, we adopt here twostrategies for increasing the sampling efficiency of thebasic MCMC scheme: an evolutionary stage which al-lows improved exploration of the sample space, and anEM-based method for making optimized proposals basedon the frame likelihood. The algorithm’s performanceis assessed and demonstrated in both synthetic and realtracking scenarios.

Keywords: Multiple cluster tracking, Markov chainMonte Carlo filtering, Evolutionary MCMC, EM algo-rithm

1 OverviewMulti-Target tracking (MTT) poses major challenges

for researchers in the fields of estimation and informa-tion fusion. The extensive studies that have been con-ducted over recent decades have yielded various track-ing techniques which can be divided informally into twoclasses: non-statistical and statistical. Non-statisticalmethods typically rely on both image differencing tech-niques and heuristic smoothing algorithms for identify-ing targets’ trajectories [1]. The non-statistical schemesare considered to be fast and viable and have beenextensively deployed for tracking in various applica-tions. Nevertheless, these methods are incapable of ad-equately handling complex tracking scenarios such asthose studied here, in which there is significant statis-tical ambiguity and dynamical structure in the modelsused. Hence statistical inference approaches are nowpreferred in many cases.

Owing to the complex nature of MTT problems, sta-tistical tracking methods usually involve smart imple-mentation of tightly coupled data association and fil-tering schemes [2]. This in turn may result in com-putationally intensive algorithms such as the multiplehypothesis tracker (MHT) in [3]. Another major diffi-culty imposed by a typical MTT scenario is related tothe mathematical modeling of complex interactions be-tween entities. This consists mainly of birth and deathof targets as well as coordinated behavioural patternswhich arise in group motions.

The optimal filtering scheme involves the propaga-tion of the joint probability density of target statesconditioned on the data. Following the conventionalapproach, in which all states are concatenated to forman augmented vector, leads to a problematic statisti-cal representation owing to the fact that the targetsthemselves are unlabeled and thus can switch positionswithin the resulting joint state vector. Furthermore,targets may appear and disappear thereby yielding in-consistencies in the joint state dimension. These prob-lems can be circumvented by adopting one of the fol-lowing approaches: 1) introducing some sort of labelingmechanism which identifies existing targets within theaugmented vector [4], or 2) considering the joint stateas a random finite set. The latter approach provides anelegant and natural way to make statistical inference inMTT scenarios. Nevertheless, its practical implemen-tation as well as its mathematical subtleties need to becarefully considered [5].

Random sets can be thought of as a generalization ofrandom vectors. The elements of a set may have arbi-trary dimensions, and as opposed to vectors, the order-ing of their elements is insignificant. These propertiesimpose difficulties in constructing probability measuresover the space of sets. This has led some researchersto develop new concepts based on belief mass functionssuch as the set derivative and set integral for embed-ding notions from measure theoretic probability withinrandom set theory (e.g., Bayes rule). As part of this,

12th International Conference on Information FusionSeattle, WA, USA, July 6-9, 2009

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point process statistics are commonly used for derivingprobabilistic quantities [6]. The PHD filter presentedin [7] is the first attempt to implement finite set statis-tics concepts for MTT. This algorithm uses a Poissonpoint process formulation to derive a semi closed-formrecursion for propagating the first moment of the ran-dom set’s intensity (i.e., the set’s cardinality). A briefsummary of the PHD algorithm can be found in [5].

In recent years, sequential Monte Carlo (SMC) meth-ods were applied for MTT. These methods, other-wise known as particle filters (PF), exploit numericalrepresentation techniques for approximating the filter-ing probability density function of inherently nonlinearnon-Gaussian systems. Using these methods, the ob-tained estimates can be set arbitrarily close to the op-timal solution (in the Bayesian sense) at the expenseof computational complexity. An extensive survey andapplication of SMC methods is given in [8].

The MTT PF algorithms in the works [9–13] are in-tended to work with a fixed number of targets. ThesePFs exploit point process formulations for properly as-signing measurements to their originating targets. Aspart of this, smart procedures are used to eliminatenon-probable association hypotheses.

An extension of the PF technique to varying numberof targets is introduced in [5], [14] and [15]. In [5,16] aPF implementation of the PHD filter is derived. Thisalgorithm maintains a representation of the filtering be-lief mass function using random set realizations (i.e.,particles of varying dimensions). The samples are prop-agated and updated based on a Bayesian recursion con-sisting of set integrals. Both works of [14] and [15]develop a Markov Chain Monte Carlo (MCMC) PFscheme for tracking varying numbers of interacting ob-jects. The MCMC approach does posses a reportedadvantage over conventional PF due to its efficient sam-pling mechanism. Nevertheless, in its traditional non-sequential form it is inadequate for sequential estima-tion. The techniques used by [14] and [15] amend theMCMC for sequential filtering (see also [17]). The workin [15] copes with inconsistencies in state dimensionby utilizing the reversible jump MCMC method intro-duced in [18]. [14], on the other hand avoids the com-putation of the marginal filtering distribution as in [17]and operates on a fixed dimension state space throughuse of indicator variables for labeling of active targetstates (the two approaches being essentially equivalent,see [19]).

1.1 Cluster Tracking

In recent years there has been an increasing inter-est in tracking scenarios in which a very large num-ber of coordinated objects evolve and interact. Onecould think of many fields in which such situation is fre-quently encountered: video surveillance, biomedicine,neuroscience and meteorology to mention only a few.Considering the nature of the cluster tracking problem

an efficient approach would consist of estimating theclustering structure formed by object concentrationsrather than tracking individual entities. This is thecase both since the number of individual objects maybe too large to track practically and indeed we cannotnecessarily expect individual objects in a coordinatedmotion to be detected by the sensor in adjacent dataframes.

It should be noted that clusters can be thought ofas extended objects that produce a large number ofobservations. This approach yields the Poisson likeli-hood formulations in [20]. In recent work [21] mergingand splitting objects are modeled using point processes.This is another fundamental issue characterizing clusterbehavior that is given full consideration in this work.

1.2 Proposed Method

The algorithm proposed herein is based on the evo-lutionary MCMC mechanism derived in [22]. Whileassuming that target locations resemble independentsamples from a Gaussian mixture we parameterize eachcluster by its mean and covariance. A Bernoulli-Markovprocess is used for describing the evolution of the clus-tering structure over time (i.e., birth and death of clus-ters).

Similarly to [22], the MCMC filtering scheme hereincorporates two enhancements that are aimed at in-creasing the efficiency of the Metropolis-Hastings sam-pler: a genetic manipulation stage in which membersfrom possibly different chain realizations are combinedfor generation of new MCMC moves, and an optimizedproposal generation scheme based on the EM algorithm.In contrast to the optimization scheme in [22], which re-lies on a variational Bayesian extension of the EM, thescheme here is much simpler to implement as it is ex-clusively based on the well-known closed form analyticsolution of the EM for Gaussian mixture likelihoods.

1.3 Outline

This paper is organized as follows. Section 2 mathe-matically formulates the cluster tracking problem. Thelikelihood and time evolution models used by the fil-tering algorithm are presented in Section 3. Section 4develops the MCMC particle filtering algorithm. Sec-tion 5 presents the results of a simulation study that hasbeen conducted to assess the new algorithm’s trackingperformance. Finally, conclusions are offered in the lastsection.

2 Problem Statement

Assume that at time k there are lk clusters, or targetsat unknown locations. Each cluster may produce morethan one observation yielding the measurement set re-alization zk = {yk(i)}

mki=1, where typically mk >> lk.

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At this point we assume that the observation concen-trations (clusters) can be adequately represented by aparametric statistical model p(Yk | θk).

Letting Z1:k = {Z1, . . . , Zk} and z1:k = {z1, . . . , zk}be the measurements history up to time k and its real-ization, respectively, the cluster tracking problem maybe defined as follows. We are concerned with estimat-ing the posterior distribution of the random set of un-known parameters, i.e. p(θk | z1:k), from which pointestimates for θk and posterior confidence intervals canbe extracted.

2.1 Random Set Representation

The evaluation of the various possible estimates re-quires the knowledge of the filtering pdf pθk|z1:k . Forreasons of convenience we consider an equivalent for-mulation of this pdf that is based on existence vari-ables. Thus, following the approach adopted in [14] therandom set θk is replaced by a fixed dimension vectorcoupled to a set of indicator variables ek = {eik} show-ing the activity status of elements (i.e., eik = 1 indicatesthe existence of the ith element). To avoid possible con-fusion, in what follows we maintain the same notationfor the descriptive parameter set θk which is now offixed dimension.

3 Bayesian InferenceFollowing the Bayesian filtering approach while as-

suming that the observations are conditionally inde-pendent given (θk, ek) the density p(θk, ek | z1:k) isobtained recursively using the conventional Bayesianrecursion [20]. Thus, the filtering pdf is completelyspecified given some prior p(θ0, e0), a transition kernelp(θk, ek | θk−1, ek−1) and a likelihood pdf p(zk | θk, ek).These are derived next for the cluster tracking problem.

3.1 Likelihood Derivation

Recalling that a single observation yk(i) is condition-ally independent given (θk, ek) yields

p(zk | θk, ek) =mk∏

i=1

p(yk(i) | θk, ek) (1)

In the above equation the pdf p(yk(i) | θk, ek) describesthe statistical relation between a single observation andthe cluster parameter sets. An explicit expression forthis pdf is given in [20] assuming a spatial Poisson dis-tribution for the number of observations mk. In thiswork we restrict ourselves to clusters in which the shapecan be modeled via a Gaussian pdf. Following thisonly the first two moments, namely the mean and co-variance, need to be specified for each cluster. Notehowever, that our approach does not rely on the Gaus-sian assumption and other parameterized density func-tions could equally be adopted in our framework. Thus,

θjk = {µjk, Σ

jk}, θk = {θ

jk}

nj=1, and [20]

p(zk | θk, ek) =mk∏

i=1

n∑

j=0

1{ejk=1}wjN

(

yk(i) − µjk, Σ

jk

)

(2)where j = 0 and 1{ej

k=1}wj > 0 are the clutter group

index and the intensity variable of the jth cluster, re-spectively.

3.2 Modeling Clusters’ Evolution

The overall clustering structure may exhibit a highlycomplex behavior resulting, amongst other things, fromgroup interactions between different clusters. This inturn may bring about shape deformations and may alsoaffect the number of clusters involved in the formation(i.e., splitting and merging of clusters). In this work,in order to maintain a generic modelling approach, thefiltering algorithm assumes no such interactions whichthereby yields the following independent cluster evolu-tion model

p(θk, ek | θk−1, ek−1) =n

∏

i=1

p(µik | µik−1)p(Σ

ik | Σ

ik−1)

n∏

j=1

p(ejk | ejk−1) (3)

where

µik = µik−1 + ζ, ζ ∼ N (0, Qζ) (4)

3.2.1 Covariance Propagation

The following proposition (given here without aproof) suggests a simple propagation scheme of the co-variance Σik that is analogous to a random-walk.

Proposition 1. Let Σi0 ∼ W(V, n1, n2) whereW(V, n1, n2) denotes a Wishart distribution with a scal-ing matrix V and parameters n1 and n2. Let also

Σik = Σik−1 + W + CD

T + DCT (5)

where

W ∼ W(V ′, n1, n2), W = DDT , Σik−1 = CC

T

(6)Then Σik is distributed according to W(V +kV

′, n1, n2).

In the ensuing (5) is used to draw conditionally in-dependent samples from p(Σik | Σ

ik−1).

3.2.2 Birth and Death Moves

The existence indicators eik, i = 1, . . . , n are assumedto evolve according to a Markov chain. Denote γj theprobability of staying in state j ∈ [0, 1], then

p(eik | eik−1 = j) =

{

γj , if eik = j

1 − γj , otherwise(7)

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3.2.3 Merging and Splitting of Clusters

As previously mentioned, two additional types ofmoves, merging and splitting, are considered for ad-equate representation of typical clustering behaviour.The transition kernels for these moves follow the pointprocess formulation (7) with the only difference being astate dependent probability γ. The idea here is that theprobability of either merging or splitting is related tothe clusters’ spatial location. This allows smooth andreasonable transitions, essentially discouraging ‘artifi-cial’ jumps to some physically unlikely clustering struc-ture.

Let ēik be the ith existence variable obtained by using(7). Then the merging kernel is given by

p(eik, ejk | ē

ik + ē

jk = 2, θ

ik, θ

jk) =

{

γij , if eik + e

jk = 1

1 − γij , if eik + ejk = 2

(8)

for i 6= j where the merging probability γij is

γij = γm1{‖µi

k−µj

k‖2≤dmin}

(9)

for some γm ∈ (0, 1) and dmin > 0. Here, 1A denotesthe indicator function for the event A. Similarly, thesplitting kernel is specified by

p(eik, ejk | ē

ik + ē

jk = 1, θ

ik, θ

jk) =

{

γs, if eik + ejk = 2

1 − γs, if eik + ejk = 1

(10)

where the splitting probability γs ∈ (0, 1). In this work,both merging and splitting kernels are applied for allpossible combinations (i, j), i 6= j.

The parameters θik, θjk of either splitting or merging

clusters should be updated properly. This consists offinding a single cluster representation θ+k = {µ

+

k , Σ+

k }

which forms the outcome of the pair θik, θjk. One way

to accomplish this is by matching the first and secondmoments of the Gaussian, that is

∫

g+(x)N(

x − µ+k , Σ+

k

)

dx =

ξ

∫

gi(x)N(

x − µik, Σik

)

dx

+ (1 − ξ)

∫

gj(x)N(

x − µjk, Σjk

)

dx (11)

where ξ ∈ (0, 1) is a weighting parameter, and ga(x)may be either x or (x− µak)(x− µ

ak)

T corresponding tothe first two statistical moments. When merging clus-ters, we set the weighting parameter as ξ = wi/(wi+wj)and solve for both µ+ and Σ+. Thus,

µ+ = ξµik + (1 − ξ)µjk (12a)

Σ+k = ξΣik + (1 − ξ)Σ

jk+

ξ(1 − ξ)[

µjk(µjk)

T + µik(µik)

T − 2µjk(µik)

T]

(12b)

The same equations are used when splitting clusters.However, in this case one should properly determineboth ξ and either θik or θ

jk for finding the missing pa-

rameters of the couple θik, θjk. In this work splitting

is carried out using ξ ∼ U(0, 1), and µik = µjk + ζµ,

Σik = Σjk + ζΣI2×2 where the random variables ζµ and

ζΣ represent spatial uncertainty.

4 MCMC Particle AlgortihmIn practice the filtering pdf p(θk, ek | z1:k) cannot

be obtained analytically and approximations should bemade instead. In this section we introduce a sequentialMCMC particle filtering algorithm for approximatingp(θk, ek | z1:k). The new filter consists of a multiple-chain Metropolis Hastings (MH) sampler that uses bothgenetic operators and local optimization steps for pro-ducing improved proposals. Compared to a single-chainMH approach, this scheme increases the efficiency of thefiltering algorithm mainly due to its ability to explorelarger regions of the sample space in a reasonable time.

4.1 Basic Sampling Scheme

The following sequential scheme is partially basedon the inference algorithm presented in [14]. Sup-pose that at time k − 1 there are N samples{θk−1(i), ek−1(i)}Ni=1 drawn approximately from the fil-tering density p(θk−1, ek−1 | z1:k−1) (i.e., the previ-ous time target distribution). In order to obtain anew set of samples {θk(i), ek(i)}Ni=1 for representingp(θk, ek | z1:k), we first simulate the joint propagatedpdf p(θk, ek, θk−1, ek−1 | z1:k−1) by drawing

(θk(i), ek(i)) ∼ p(θk, ek | θk−1(i), ek−1(i)) (13)

where (θk−1(i), ek−1(i)) are uniformly drawn from theempirical approximation of p(θk−1, ek−1 | z1:k−1).These samples are then accepted or rejected using aproper MH step. The converged output of this schemesimulates the joint density p(θk, ek, θk−1, ek−1 | z1:k) ofwhich the marginal is the desired filtering pdf.

4.1.1 Metropolis Hastings Step

The MH algorithm generates samples from an ape-riodic and irreducible Markov chain with a prede-termined (possibly unnormalized) stationary distribu-tion. This is a constructive method which specifies theMarkov transition kernel by means of acceptance prob-abilities based on the preceding time outcome. As partof this, a proposal density is used for drawing new sam-ples. In our case, setting the stationary density as thejoint filtering pdf p(θk, ek, θk−1, ek−1 | z1:k), a new setof samples from this distribution can be obtained afterthe MH burn-in period.

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Let (θk, ek, θk−1, ek−1) be a sample from the realizedchain of which the stationary distribution is the jointfiltering pdf. Let also (θ′k, e

′k, θ

′k−1, e

′k−1) be a candi-

date drawn according to (13). Then the MH algorithmaccepts the new candidate as the next realization fromthe chain with probability

α = min

{

1,p(zk | θ′k, e

′k)

p(zk | θk, ek)

}

(14)

As mentioned above, in this work we use the jointpropagated pdf as our proposal. More precisely, itsempirical approximation (see (13)) is used as it is dif-ficult to obtain a closed-form analytical expression ofthis pdf.

It has already been noted that the above samplingscheme may be inefficient in exploring the sample spaceas the underlying proposal density of a well behavedsystem (i.e., of which the process noise is of low in-tensity) introduces relatively small moves. This draw-back is alleviated here by using an evolutionary MCMCscheme that utilizes multiple realizations of the samechain.

4.2 Evolutionary MCMC

The basic MH scheme can be used to produce sev-eral chain realizations each starting from a different(random) state. In that case, the entire populationof the converged MH outputs (i.e., subsequent to theburn-in period) approximates the stationary distribu-tion. Using a population of chains enjoys several ben-efits compared to a single-chain scheme. The multiple-chain approach can dramatically improve the diversityof the produced samples as different chains explore var-ious regions that may not be reached in a reasonabletime when using a single chain realization. Further-more, having a population of chains facilitates the im-plementation of interaction operators that manipulateinformation from different realizations for improvingthe next generation of samples.

Following the approach of [23], the evolutionaryalgorithm implemented here uses genetic operators(crossover and mutation) to generate new samples. Thereader is referred to [22] for a detailed description of theevolutionary stage.

4.3 Optimizing Proposals Using EM

The evolutionary scheme in this work consists of anoptimization step aimed at increasing the efficiency ofthe sampling algorithm. A natural approach in the caseof MCMC sampling would be to obtain an optimizedproposal pdf rather than a single deterministic solution.This insight is the basis for the variational optimizationmechanism used in [22]. In this work we adopt a simpli-fied approach for generating valid optimized proposals.In what follows we demonstrate how the EM algorithmcan be used for generating such proposals.

4.3.1 EM for Gaussian Mixtures

The EM algorithm maximizes an auxiliary functionQ(θk, wj) which is a lower bound for the log-likelihoodlog p(zk | θk, ek). In detail, from (2) we have

log p(zk | θk, ek) ≥mk∑

i=1

n∑

j=0

1{ejk=1}wj

[

log cj

−1

2

(

yk(i) − µjk

)T (

Σjk

)−1 (

yk(i) − µjk

)

]

= Q(θk, wj)

(15)

where the optimum of Q(θk, wj) is obtained by iterating

wt+1j = argmaxwjQ(θtk, wj) (16a)

θt+1k = arg maxθk

Q(θk, wt+1j ) (16b)

This recursion, which in our case has a closed form an-alytical expression, is guaranteed to converge to a localmaximum. The issue of recovering ejk is resolved here

by setting ejk = 1{wj>wth}. In other words, the indica-tor variables are set to 1 if the corresponding weight isabove some predetermined threshold value wth.

4.3.2 A Regularized Proposal

In this work we use the EM recursion (16) for con-

structing a valid proposal pdf. Let {θ0k(i), w0j (i)}

Nopti=1 be

a set of particles with which (16) is initialized for pro-

ducing {θtk(i), wtj(i)}

Nopti=1 after t iterations (the initial

set of particles can be taken from the propagated pdfin (13)). Following the above discussion the optimized

set of particles is then taken as {θ′k(i), e′k(i)}

Nopti=1 where

θ′k(i) = θtk(i) and e

′jk (i) = 1{wtj>wth}. This set is used

for composing a regularized proposal pdf of the form

q(θ̄k) ∝

Nopt∑

i=1

n∑

j=0

e′jk (i)K(

θ̄jk − θ′jk (i)

)

(17)

where K(·) is some regularization kernel of infinitesupport (e.g., Gaussian). The above equation sug-gests a simple sampling scheme for producing candi-dates from q(θ̄k). Thus, we uniformly sample from

{θ′k(i), e′k(i)}

Nopti=1 to get (θ

′k, e

′k). We then use this sam-

ple for drawing a candidate θ̄jk ∼ K(

θ̄jk − θ′jk

)

, where

{j ∈ [1, n] | ējk = 1}, and ēk = e′k. Finally, the ac-

ceptance probability of (θ̄k, ēk) given the previously ac-cepted sample (θk, ek) is simply

α = min

{

1,p̂(θ̄k, ēk | z1:k)q(θk)

p̂(θk, ek | z1:k)q(θ̄k)

}

(18)

4.4 Algorithm Summary

The basic MCMC filtering algorithm (without theevolutionary extension) is summarized in Algorithm 1.

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Algorithm 1 Single-Chain MCMC

1: Given N samples from p̂(θk−1, ek−1, θk−2, ek−2 | z1:k−1)perform the following steps.

2: for i=1, . . . , N do3: Uniformly draw (θk−1(i), ek−1(i)) ∼ p̂(θk−1, ek−1 |

z1:k−1)4: for j=1, . . . , n do5: Draw θjk(i) ∼ p(θ

j

k | θj

k−1(i)) using both (4) and(5).

6: Draw ējk(i) ∼ p(ej

k | ej

k−1(i)) using (7).7: end for

8: For any pair (θjk(i), ēj

k(i)), (θlk(i), ē

lk(i)), j 6= l per-

form either merging or splitting as described in Sec-tion 3.2.3.

9: end for

10: for i = 1, . . . , N + NBurn-in do11: Draw u ∼ U [0, 1]12: if u < uEM then

13: Propose a new candidate θ̄k ∼ q(θ̄k).14: Compute the MH acceptance probability α of

(θ̄k, ēk) using (18).15: else

16: Propose (θ̄k, ēk, θ̄k−1, ēk−1) ∼ p̂(θk, ek, θk−1, ek−1 |z1:k−1).

17: Compute the MH acceptance probability α of thenew move using (14).

18: end if

19: Draw u ∼ U [0, 1]20: if u < α then

21: Accept s(i) = (θ̄k, ēk) as the next sample of therealized chain.

22: else

23: Retain s(i) = s(i − 1).24: end if

25: end for

5 Simulation StudyThe MCMC filtering algorithm is numerically tested

in both synthetic and realistic tracking scenarios con-sisting of a varying number of clusters. The syntheticcase is similar to the one in [22], however, with a totalnumber of clusters not exceeding four. For concisenesswe omit here the detailed description of the cluster dy-namics and observations. The reader is referred to [22]for a brief summary of these models.

5.1 Algorithm Settings

The evolutionary MCMC scheme is implemented us-ing N = 1500 particles and l = 5 chain realizations.The chains burn-in period is set to NBurn-in = 200based on tuning runs. During the MH step, an opti-mized move is sampled from the regularized pdf q(θ̄k)with probability of uEM = 0.05. The number of EMiterations used for composing q(·) do not exceed t = 2.

5.2 Synthetic Data

The clusters trajectories and observations were gen-erated using the models described in [22]. Both actual

X and Y tracks over time are shown in Figs. 1a, 1b, 2aand 2b. These figures depict a typical scenario which in-volves splitting (at approximately k = 20) and merging(at k = 60) clusters. The densely cluttered observationsare shown in the corresponding Figs. 1c, 1d, 2c and 2d.The performance of the MCMC filtering algorithm isdemonstrated in the remaining figures, Figs. 1e, 1f, 2eand 2f. These figures show the level plots of the es-timated Gaussian mixture model over time. Thus, itcan be clearly seen that on the overall the filtering al-gorithm is capable of adequately tracking the varyingclustering structure.

Using the particles approximation one can easilycompute the probability hypothesis density (PHD) overthe entire field of view. An empirical estimate of thePHD in this case is given by N−1

∑Ni=1

∑nj=1 e

jk(i).

Notice, however, that this rather unusual PHD corre-sponds to number of clusters and not directly to targetcounts. The average PHD was computed based on 10Monte Carlo runs and is depicted along with the actualaverage number of clusters in Fig. 3.

0 10 20 30 40100

200

300

400

Time step

(a) True 1-40

40 50 60 70 80150

200

250

300

350

400

Time step

(b) True 41-80

(c) Observations 1-40 (d) Observations 41-80

Time step10 20 30 40

(e) Filtered 1-40

Time step50 60 70 80

(f) Filtered 41-80

Figure 1. Tracking performance. Showing Xaxis over time.

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0 10 20 30 40100

200

300

400

500

Time step

(a) True 1-40

40 50 60 70 80

200

300

400

500

Time step

(b) True 41-80

(c) Observations 1-40 (d) Observations 41-80

Time step10 20 30 40

(e) Filtered 1-40

Time step50 60 70 80

(f) Filtered 41-80

Figure 2. Tracking performance. Showing Yaxis over time.

5.3 Crowd Tracking Example

The MCMC filtering algorithm was applied for track-ing a crowd of people in a video sequence. This scenarioconsists of people walking in a corridor as seen fromabove. The video was preprocessed using a corner de-tector for yielding the set of point observations to beused by the filtering algorithm. The obtained set ofpoints is characterized by a relatively dense concentra-tion near each person’s head which in some cases ex-tends towards the shoulders. Other points correspond-ing to non moving objects are considered here as clutter.The dynamical model assumed by the filter is similarto the one used in the synthetic data case (see [22]).

The actual scenario, the preprocessed observationsand the filtering performance are shown in Fig. 4 attwo distinct time points. Thus, the estimated Gaussianmeans are marked by pluses in both Figs. 4a and 4cwhereas the corresponding covariance ellipses are shownalong with the point observations in Figs. 4b and 4d.These figures demonstrate the viability of the MCMCfiltering algorithm in this case as it identifies the exactnumber of walking people in the corridor.

0 10 20 30 40 500

1

2

3

4

5

Time step

Figure 3. Average number of clusters (solid line)and the mean PHD (dashed line) based on 10Monte Carlo runs.

6 ConclusionsA new Markov chain Monte Carlo filtering algorithm

is derived for tracking multiple clusters. The clusteringstructure is represented using a dynamic Gaussian mix-ture model. The new filter is tested in both syntheticand realistic scenarios. In either cases the algorithmexhibits a good tracking performance as it captures theessence of the clusters behavior. The realistic exampleclearly demonstrates the viability of the new algorithmin such scenarios where the spatial Poisson assumptionis violated.

7 AcknowledgmentsThis work was sponsored by the Data and Informa-

tion Fusion Defense Technology Centre, UK, under theTracking Cluster. The authors thank these parties forfunding this work.

References[1] I. Sethi and R. Jain, “Finding trajectories of fea-

ture points in a monocular image sequence,” IEEETransactions on Pattern Analysis and Machine In-telligence, vol. 2, pp. 56–72, 1987.

[2] Y. Bar-Shalom, X. Li, and T. Kirubarajan, Esti-mation with Application to Tracking and Naviga-tion, John Wiley & Sons, Inc., New York, 2001.

[3] D. Reid, “An Algorithm for Tracking MultipleTargets,” IEEE Transactions on Automation andControl, vol. 24, no. 6, pp. 84–90, December 1979.

[4] M. Stephens, “Dealing with label switching in mix-ture models,” Journal of the Royal Statistical So-ciety, 2000.

[5] B. Vo, S. Singh, and A. Doucet, “SequentialMonte Carlo Methods for Multi-Target Filtering

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(a) Frame 20 (b) Observations

(c) Frame 50 (d) Observations

Figure 4. Crowd tracking example.

with Random Finite Sets,” IEEE Transactions onAerospace and Electronic Systems, vol. 41, no. 4,pp. 1224–1245, October 2005.

[6] I. Goodman, R. Mahler, and H. Nguyen, Mathe-matics of Data Fusion, Boston: Kluwer AcademicPublishing Co., 1997.

[7] R. Mahler, “Multi-target bayes filtering via first-order multi-target moments,” IEEE Transactionson Aerospace and Electronic Systems, vol. 39, no.4, pp. 1152–1178, 2003.

[8] A. Doucet, J. F. G. de Freitas, and N. J. Gordon,Sequential Monte Carlo Methods in Practice, NewYork: Springer-Verlag, 2001.

[9] C. Hue, J. P. Le Cadre, and P. Perez, “TrackingMultiple Objects with Particle Filtering,” IEEETransactions on Aerospace and Electronic Sys-tems, vol. 38, pp. 791–812, July 2002.

[10] K. Gilholm and D. Salmond, “A Spatial Dis-tribution Model for Tracking Extended Objects,”IEE Proceedings on Radar, Sonar and Navigation140(2), 1993, pp. 107–113.

[11] J. Vermaak, S. J. Godsill, and P. Perez, “MonteCarlo Filtering for Multi-Target Tracking andData Association,” IEEE Transactions onAerospace and Electronic Systems, vol. 41, no. 1,pp. 309–330, January 2005.

[12] O. Cappé, S.J. Godsill, and E.Moulines, “Anoverview of existing methods and recent advancesin sequential monte carlo,” Proc. IEEE, vol. 95,no. 5, pp. 899–924, May 2007.

[13] A. Doucet, S. J. Godsill, and C. Andrieu, “MonteCarlo filtering and smoothing with application totime-varying spectral estimation,” Proc. IEEE In-ternational Conference on Acoustics, Speech andSignal Processing, vol. II, pp. 701–704, 2000.

[14] S. K. Pang, J. Li, and S. J. Godsill, “Models andAlgorithms for Detection and Tracking of Coor-dinated Groups,” Aerospace Conference, IEEE,2008, pp. 1–17.

[15] Z. Khan, T. Balch, and F. Dellaert, “MCMC-Based Particle Filtering for Tracking a VariableNumber of Interacting Targets,” IEEE Trans-actions on Pattern Analysis and Machine Intelli-gence, vol. 27, no. 11, pp. 1805–1819, November2005.

[16] W. Ng, J. Li, S. J. Godsill, and S. K. Pang, “Mul-titarget initiation, tracking and termination usingbayesian monte carlo methods,” Computer Jour-nal, vol. 50, no. 6, pp. 674–693, 2007.

[17] C. Berzuini, G. Nicola, W. R. Gilks, and C. Lar-izza, “Dynamic Conditional Independence Modelsand Markov Chain Monte Carlo Methods,” Jour-nal of the American Statistical Association, vol. 92,no. 440, pp. 1403–1412, 1997.

[18] P. J. Green, “Reversible Jump Markov ChainMonte Carlo Computation and Bayesian ModelDetermination,” Biometrika, vol. 82, no. 4, pp.711–732, December 1995.

[19] S. J. Godsill, “On the relationship betweenMarkov chain Monte Carlo methods for model un-certainty,” Journal of Comp. Graph. Stats., vol.10, no. 2, pp. 230–248, 2001.

[20] K. Gilholm, S. Godsill, S. Maskell, andD. Salmond, “Poisson models for extended targetand group tracking,” SPIE Conference, August2005, pp. 230–241.

[21] C. B. Storlie, T. C. M. Lee, Jan Hannig, and Dou-glas Nychka, “Tracking of multiple merging andsplitting targets with application to convective sys-tems,” ICSA 2008 Applied Statistics Symposium.

[22] A. Carmi, S. J. Godsill, and F. Septier, “Evolu-tionary MCMC Particle Filtering for Target Clus-ter Tracking,” Marco Island, Florida, 2009, Pro-ceedings of the IEEE 13th DSP Workshop and the5th SPE Workshop.

[23] M. Strens, “Evolutionary MCMC Sampling andOptimization in Discrete Spaces,” WashingtonDC, 2003, Proceedings of the Twentieth Interna-tional Conference on Machine Learning.

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