ITERATIVE JOINT PROBABILISTIC DATA ASSOCIATION FOR AVOIDING TRACKCOALESCENCE AND TRACK SWAP IN MULTI-TARGET TRACKING

1Viji Paul Panakkal, 2Rajbabu Velmurugan

1Central Research Laboratory, Bharat Electronics Ltd., Bangalore, India2Dept. of Electrical Engineering, Indian Institute of Technology Bombay, Mumbai, India

ABSTRACT

The iterative technique developed in this paper achieves si-multaneous reduction of track coalescence and track swapby computing a better prior for the joint association eventused in estimating target states. This prior is computed usingexpectation maximization (EM). Using the computed priorthe iterative method avoids track coalescence and track swapwhile preserving the robustness of JPDA towards clutter andmissed detection. Compared to other coalescence avoidanceschemes, the proposed method avoids coalescence and swapwithout pruning the joint association events. Monte Carlosimulations verify the advantage of the proposed method overother approaches in a cluttered multi-target environment.

1. INTRODUCTION

In a cluttered multi-target environment the accuracy of tar-get state estimates depends on the measurement-to-track as-sociation configuration. One of the earliest approaches to thedata association algorithm was the track splitting filter [1],which simply splits the track (positional and kinematic esti-mates that determine trajectory of the target) whenever multi-ple measurements fall within the validation (gating) region ofthe predicted track position. Joint probabilistic data associa-tion (JPDA) technique is successful in multiple target tracking(MTT) in a cluttered environment [2], [3]. JPDA techniquecomputes the probabilities of joint association events by as-suming uniform prior probabilities for the joint associationevents. Due to this assumption it is difficult to avoid associa-tion hypotheses that cause track coalescence.

JPDA technique is susceptible to track coalescence whentargets move critically close, i.e., if D/σres <

√2 , where D

is the separation between targets and σ2res (residual variance)

is the variance of the error between predicted and measuredtrack position [4]. The exact nearest neighbor PDA approach(ENNPDA) was proposed in [5] to circumvent track coa-lescence by pruning all association hypotheses except the

Authors would like to express sincere thanks to Ajit Kalghatgi, ChiefScientist, CRL, for the guidance and support to carry out this work. We arealso thankful to the Naval Research Board, DRDO, India, for their supportand cooperation.

most likely one. An approach based on selective pruning(JPDA*), which effectively weighs the outcome in favor ofthe more likely hypotheses, was proposed in [6],[7]. Parti-cle filter based MTT using a unique decomposition of jointconditional density of the states of two targets is demon-strated to provide an estimate of track swap in [7]. Anapproach based on K-best data association technique is givenin [8] to obtain correct measurement-to-track association.A re-weighting strategy is employed in scaled JPDA (Sc-JPDA) [9] to avoid track coalescence by scaling the mostlikely association hypotheses. By selective pruning or byscaling the most likely hypotheses with a predefined scalefactor track coalescence can be avoided but the probabilityof track swap increases with measurement origin uncertainty.Another recent approach to track closely moving target is theset JPDA (SJPDA) [10], but identity of targets is disregardedin this approach. Instead of pruning or selective scaling weavoid track coalescence by computing a better prior for thejoint association events and this approach also also reducestrack swap and maintains identity.

The technique developed here is similar to JPDA in theassociation probability computation. However differs fromJPDA in that at each instant the prior distribution parame-ters that maximize the probability of feasible joint associa-tion events (one measurement associated to at most one trackand one track is at most assigned with one measurement) areobtained. This approach avoids coalescence, like other co-alescence avoidance schemes, by maximizing the posterior.Further by computing a better prior track swapping is alsoavoided. The approach is based on expectation maximization(EM) technique described in [11]. The approach gives bettertracking performance compared to other approaches in a clut-tered MTT scenario by avoiding track coalescence and trackswap.

2. JOINT PROBABILISTIC DATA ASSOCIATION

Consider the target motion and measurement models of theform,

X (k + 1) = FX (k) + w (k) , (1)

Z (k) = HX (k) + v (k) , (2)

2012 IEEE 7th Sensor Array and Multichannel Signal Processing Workshop (SAM)

978-1-4673-1071-0/12/$31.00 ©2012 IEEE 285

where k is the time index, X is the target state vector, Z is themeasurement vector, w and v are zero mean Gaussian noisevectors with covariance matrices Q and R respectively. Z (k)is the set of measurement obtained at kth scan. The statetransition matrix F and observation matrix H are assumed tobe known. The initial state is assumed Gaussian with meanX (0|0) and covariance P (0|0). Predicted target state at timek given data up to time k − 1 is denoted as X (k|k − 1) andcorresponding measurement is denoted as Z (k|k − 1). Theerror in predicted target state and measurement are denotedas X (k|k − 1) = X (k) − X (k|k − 1) and Z (k|k − 1) =Z (k) − Z (k|k − 1). The error covariance of predicted tar-get state is P (k|k + 1) = E

[X (k|k − 1) XT (k|k − 1)

]and

the corresponding measurement error covariance is S (k) =

E[Z (k|k − 1) ZT (k|k − 1)

]. Target state estimate at time k

given data up to time k is denoted as X (k|k) and correspond-ing measurement is denoted as Z (k|k).

In the minimum mean-square error (MMSE) sense of re-ducing the state estimation error, target state estimates are ob-tained as conditional mean. In MTT conditional mean of stateX (k|k) at time k can be obtained by averaging over all the as-sociation events as [12],

X (k|k) = E[X (k) |Zk

]= E

[E[X (k) |A,Zk

]], (3)

where Zk is the measurement up to time k andmk is the num-ber of measurements at kth frame, Z (k) = {zj}j=1:mk

. Theith target estimate is referred as track i denoted as Ti. Let Abe the set of all association events and ai,j ∈ A is the associ-ation event of jth measurement to ith track. For track i, theconditional mean can be obtained as [2],

Xi (k|k) =

mk∑j=1

E[Xi (k) |ai,j ,Zk

]p(ai,j |Zk

), ai,j ∈ A.

(4)Using the Kalman filter framework for state estimation [6],R.H.S of (4) equals,

=

mk∑j=1

(Xi (k|k − 1) + K (k) (zj (k)− zi (k|k − 1))

)pi,j ,

= Xi (k|k − 1) + K (k)

mk∑j=1

(zj (k)− zi (k|k − 1)) pi,j ,

= Xi (k|k − 1) + K (k)

mk∑j=1

zij (k) pi,j , (5)

where pi,j = p(ai,j |Zk

), is the association probability for

observation j with track i. In Kalman filter the prediction andupdate of error covariance matrices can be summarized as,

P (k|k − 1) = FP (k − 1)FT +Q,

S (k) = HP (k|k − 1)HT +R,

P (k) = P (k|k − 1)−K (k)S (k) KT (k) . (6)

The filter gain K (k) is computed as,

K (k) = P (k|k − 1)HTS−1 (k) . (7)

In the MTT scenarios considered, the size of measurementvector Z might change with time, due to arrival of newtargets or clutter. There are many approaches to establishmeasurement-to-track association. In this paper we focus onthe JPDA approach. It is assumed that the errors X, Z haveGaussian densities at each time step. It is also assumed thattarget detection probability is pd. When target state densitiesof multiple targets are close enough, the association proba-bilities pi,j have to be computed based on the group of targettracks that are close. In order to account for the overlappingof target state distributions, JPDA forms the joint associationevents pertaining to current time k as follows,

A =

mk⋂j=1

aj,tj , (8)

where aj,t , {measurement j originated from target t}, j =1, ...,mk; t = 0, 1, ..., T . To select feasible joint events,JPDA creates validation matrix defined as M = {ωj,t}; withbinary elements ωj,t indicating whether measurement j liesin the validation region of target t.

2.1. Proposed iterative JPDA

The joint association events obtained using (8) can be splitin to two; the feasible events A1 obtained by ωj,t (A) = 1and non-feasible events A2 obtained by ωj,t (A) = 0. Letthe feasible and non-feasible joint association hypotheses bedenoted using H, such that

H(A) =

{1 ωj,t (A) = 1; A is feasible0; ωj,t (A) = 0; A is non-feasible . (9)

The conditional probability of the joint association eventconditioned on data set Zk and H is,

P{

A|Zk,H}

= P{

A|Z (k) ,Zk−1,H}

=1

cP{Z (k) |A,Zk−1,H

}P{

A|Zk−1,H},

(10)

where c is a normalizing constant. The prior probabilities ofthe joint association events are assumed to be equally likelyin JPDA, i.e., P

{A|Zk−1,H

}= 1

c1H (A), for equal proba-

bility of detection pd and probability of false alarm pfa forall targets [2]. Let Aj denote the maximizing feasible jointassociation event among the set of joint association eventscontaining the assignment zj (k). Then, the prior for JPDA*can be assumed to be P

{A|Zk−1,H

}= 1

c2H (Aj). When

mk = T , with pd = 1 and pfa = 0, Aj will be unique and thisprior will be same as that for ENNJPDA [7]. The objectivein this paper is to find the prior distribution pθ0 maximizing

286

pθ{

A, Z(k)|Zk−1}

H (A) over all A so that, P{

A|Zk−1,H}

in (10) can be replaced with pθ0 . For example with two tracksand two measurements we need to obtain parameters ψ1 andψ2 maximizing the sum of feasible joint association eventprobabilities. Compared to JPDA* (prior H (Aj) is obtainedby independently maximizing with zj) the proposed approachavoids track swap by computing the maximizing parameterψi together for i = 1, ..., T . Assuming target states are inde-pendent and follow Gaussian, sum of feasible joint associa-tion event probabilities are,

∑A pθ

(A, Z(k)|Zk−1

)H (A) =

p (a1,1) p (a2,2) + p (a2,1) p (a1,2).Probability of joint association event p

(aj1,tj1

)p(aj2,tj2

)=exp

(− (zj1 − ψj1)

2/2R2 − (zj2 − ψj2)

2/2R2

)denotes

assigning measurement j1 with target tj1 and measurementj2 with target tj2 . By maximizing the sum of feasible jointassociation events probabilities we are indirectly minimizingthe sum of non-feasible joint association events probabil-ities, so that coalescence can be avoided and target stateswell separated. The maximizing parameters ψji are obtainediteratively using the EM algorithm. The expectation of loglikelihood of pθ

(A, Z(k)|Zk−1

)is computed as [11],

E- Step:

Eθ0(log(pθ(A, Z(k)|Zk−1

)))H (A)

=∑

A

log(pθ(A, Z(k)|Zk−1

)H (A)

)pθ0(A|Zk

)H (A) .

(11)

Let the ith target measurement distribution at the kth instantbe Gaussian with mean ψi = Xi (k|k − 1)αi and covarianceR. The parameter αi is unknown and needs to be computed.Maximizing (11) is equivalent to minimizing,

M- Step:

J(α) =∑j

(zj(k)− Xi (k|k − 1)αi

)2R−1βj,i,

where, βj,i =∑

A

pθ0(A|Zk

)H (A) .

(12)

The maximizing likelihood βj,i is obtained by evaluating thegradient of J with respect to α,

∇αJ = C∑j

(zj(k)− Xi (k|k − 1)αi

)Xi (k|k − 1)βj,i.

(13)Using∇αJ = 0⇒ αi =

∑j

zj(k)βj,i/Xi (k|k − 1) .

(14)To obtain βj,i using (12), pθ0 is initially assumed to be ob-tained from a uniform distribution; i.e, pθ0 = 1

c1H (A) as in

JPDA. Using α obtained by (14) and with known measure-ment error variance R, a better prior pθ0 can be obtained as,

pθ0(A|Zk

)=

mk∏j=1

N(zj (k) ;αtj Xtj (k|k − 1) , R

), (15)

whereN is normal distribution having meanαtj Xtj (k|k − 1)and variance R. By iterative computation of (12), (14)and (15), the parameter αi that maximizes (11) can be com-puted for each i = 1, ...T .

3. SIMULATION

To verify the advantage of the association probabilities ob-tained with the proposed method we considered two sim-ulation cases by varying σres. There are three targetsmoving parallel in X direction having constant X velocityvx = 50m/s. The initial X and Y coordinates of targets are,(X1 = 1000m,Y1 = 1000m), (X2 = 1000m,Y2 = 1009m)and (X3 = 1000m,Y3 = 1018m). The measurements z(x)j (k)and z(y)j (k) are assumed to have a Gaussian distribution;z(x)j (k) → N

(z(x)j (k) ;Xi (k) , σ2

m

)and z(y)j (k) →

N(z(y)j (k) ;Yi (k) , σ2

m

). The scenario in X-Y plane is

shown in Fig. 1. There are three clutter measurementsper measurement frame with uniform distribution ux =[X (k)− 15, X (k) + 15] and uy = [Y (k)− 15, Y (k) + 15].Our aim is to obtain the association probabilities pi,j so asto minimize the root mean square error (RMSE) in positionalestimate. State estimation has been carried out using Kalmanfilter and results are for 100 Monte Carlo (MC) runs.

The separation among the three targets is 9m (D = 9m).In the first simulation scenario, measurement error varianceσ2m = 9m2 and σ2

res = 50m2. The initial state estimates areobtained assuming Gaussian distribution for state with posi-tional error variance σ2

pinit = 9m2. Since σ2res > D2/2,

JPDA based state estimates will have coalescence [4]. Thesum of root mean square error (RMSE) considering true dataassociation is compared to JPDA, JPDA*, Scaled JPDA (Sc-JPDA, with scale factor 2) and iterative JPDA (Iter-JPDA)in Fig. 2(a). Process noise covariance used in simulation is

Q =

[0.25T 4 0.5T 3

0.5T 3 T 2

]0.1 with time instant T = 1 s. The

RMSE obtained using the iterative approach is lower thanthat for other approaches. Though, JPDA* avoids coales-cence its RMSE is higher than Iter-JPDAis because of thetrack swaps. In the second simulation, σ2

res = 25m2 andσ2pinit = 36m2. The sum of RMSE obtained for this case is

plotted in Fig. 2(b). Again, the RMSE obtained with iterativeapproach is lower than that obtained using other approaches.With a reduced σ2

res and higher σ2pinit the track swap percent-

age increases for JPDA*. The RMSE is lower for the pro-posed iterative approach because of its ability to avoid bothtrack coalescence and track swap. From the results shownin Fig. 2(a)-(b) it is clear that the iterative approach providesbetter target state estimates compared to other approaches.

A comparison has been done in terms of percentage trackswap and track coalescence for various approaches and is con-solidated in Table 1. A track coalescence between two tracks iand j is declared in the simulation if (Xi(k|k)−Xj(k|k))2 <1 for ten consecutive time instants. A track swap between two

287

Fig. 1. Three targets in X-Y plane. True trajectory (Solidlines) and measurements (Circles, squares and dots) areshown, clutter is not shown for sake of clarity. JPDA* esti-mates with dotted and Iter-JPDA estimates with dashed lines.

Fig. 2. RMSE sum of 3 targets for 100 M.C. runs, parametersσ2res and σ2

pinit are shown in the plot. Proposed Iter-JPDAperforms better compared to other approaches.

σ2res = 25 σ2

res = 50Sw. Co. Sw. Co.

JPDA [2, 1] 26 6 1(1) 70(66)Sc-JPDA [9] 9 47 2(0) 76(74)JPDA∗ [6, 13] 35 0 35(22) 0(0)Proposed 0 0 0(0) 1(1)

Table 1. Percentage track swap (Sw.) and track coalescence(Co.) using 100 M.C runs. For σ2

res = 50, values shown arefor σ2

pinit = 36 and (σ2pinit = 9) with in bracket.

tracks i and j is declared if di,j + dj,i < 5 for ten consecu-tive time instants, where d2i,j = (Xi(k|k) − Xj(k))2. Asdetailed in Table 1, JPDA* avoids coalescence but is vulner-able to track swap. For the σ2

res = 50m2 case, coalescenceand swap percentages are computed for σ2

pinit = 36m2 and(σ2pinit = 9m2

)as shown in Table 1. In general low σ2

pinit

reduces track swap, low σ2res increases track swap, and high

σ2res increases coalescence. But, the proposed Iter-JPDA re-

duces swapping and coalescence to a great extent irrespectiveof σ2

res and σ2pinit.

4. CONCLUSION

Avoiding coalescence while keeping track identity intact ishighly demanding in a cluttered MTT. Using an improvedprior the proposed I-JPDA approach avoids track coalescence

and track swap simultaneously and reduces target state esti-mation error. Efficient computational ways for implementingthe iterative approach using graphical models will be exploredin future work.

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