Multiple-Model Probability Hypothesis Density Filter for Tracking … · 2008-10-16 · Multiple-Model Probability Hypothesis Density Filter for Tracking Maneuvering Targets K. PUNITHAKUMAR,

Multiple-Model ProbabilityHypothesis Density Filter forTracking Maneuvering Targets

K. PUNITHAKUMAR, Student Member, IEEE

T. KIRUBARAJAN, Member, IEEE

A. SINHAMcMaster University

Tracking multiple targets with uncertain target dynamicsis a difficult problem, especially with nonlinear state and/ormeasurement equations. With multiple targets, representingthe full posterior distribution over target states is not practical.The problem becomes even more complicated when the numberof targets varies, in which case the dimensionality of the statespace itself becomes a discrete random variable. The probabilityhypothesis density (PHD) filter, which propagates only thefirst-order statistical moment (the PHD) of the full targetposterior, has been shown to be a computationally efficientsolution to multitarget tracking problems with a varying numberof targets. The integral of PHD in any region of the state spacegives the expected number of targets in that region.With maneuvering targets, detecting and tracking the changes

in the target motion model also become important. The targetdynamic model uncertainty can be resolved by assuming multiplemodels for possible motion modes and then combining themode-dependent estimates in a manner similar to the one usedin the interacting multiple model (IMM) estimator. This paperpropose a multiple-model implementation of the PHD filter,which approximates the PHD by a set of weighted randomsamples propagated over time using sequential Monte Carlo(SMC) methods. The resulting filter can handle nonlinear,non-Gaussian dynamics with uncertain model parameters inmultisensor-multitarget tracking scenarios. Simulation resultsare presented to show the effectiveness of the proposed filter oversingle-model PHD filters.

Manuscript received February 16, 2005; revised October 29, 2005and September 24, 2006; released for publication November 24,2006.

IEEE Log No. T-AES/44/1/920389.

Refereeing of this contribution was handled by B. La Scala.

Authors’ address: Estimation, Tracking and Fusion Laboratory,Dept. of Electrical and Computer Engineering, McMasterUniversity, 1280 Main St. West, Hamilton, Ontario, L8S 4K1,Canada, E-mail: ([email protected]).

0018-9251/08/$25.00 c° 2008 IEEE

I. INTRODUCTION

Tracking a maneuvering target using multiplemodels has been shown to be highly effective in manyapplications. In single-model filters, if the modelused by the filter does not match the actual systemdynamics, the filter will tend to diverge since theactual errors fall outside the range predicted by thefilter using its error covariance [7]. Maneuveringtargets might switch between different modes ofoperation, and tracking using a single-model filtermight fail since the filter may be accurate for onlyone mode of operation. In multiple-model approaches,several filters, each matched to a different targetmotion mode, operate in parallel, and then theoverall state estimate is given by a weighted sumof the estimates from each filter. In many targettracking problems with linear, Gaussian systems, theinteracting multiple model (IMM) estimator [7—9],in which a bank of different hypothetical targetmotion models is used, has been proven to havebetter performance than the (single-model) Kalmanfilter. For example, in the target tracking benchmarkproblem [6], in which performances of differentalgorithms for tracking highly maneuvering targetswere compared, the IMM estimator outperformed theKalman and ®—¯ filters and yielded one of the bestsolutions. Extensions of multiple models to particlefilters [12, 30] have also resulted in better results fornonlinear, non-Gaussian systems than single-modelparticle filters.Tracking multiple maneuvering targets is much

more difficult than tracking a single maneuveringtarget since there is a challenge in correctlyassociating the measurements with the targets. In theliterature, extensions of the IMM estimator to workwith standard tracking algorithms have resulted in theIMMJPDA (IMM estimator with the joint probabilisticdata association algorithm) [5, 10, 14], IMM/MHT(IMM estimator with the multiple hypothesis trackingalgorithm) [16, 23] and the joint IMMPDA particlefilter [11]. A drawback of these algorithms is that allof them involve the model-data association problemand the problem is further complicated in a clutteredenvironment.The fully Bayesian perspective of multitarget

problems, using techniques like grid approaches,are very computationally challenging and not ofpractical interest when the number of targets is large.One alternative, computationally efficient solutionto multiple target tracking problems that avoidsmodel-data association difficulties is the probabilityhypothesis density (PHD) filter [27]. The PHD isthe first-order statistical moment of the full jointmultitarget posterior distribution. The recursive PHDfilter completely characterizes the statistics of thedynamic Poisson point process under the assumptionthat the predicted multitarget density is Poisson

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distributed. Discussion about dynamic Poisson pointprocesses in parametric estimation context can befound in [15, 21, 33]. The mathematical derivationsof the PHD filter equations based on probabilitygenerating functionals and functional derivativesare given in [28]. The PHD is different from theordinary probability density function (pdf) and itdoes not necessarily integrate to unity over the wholestate space. In contrast, the integral of PHD over aregion gives the expected number of targets inthat region.The PHD filter has been shown to be a

computationally tractable method for unified grouptarget detection, tracking and classification [24] andfor cluster tracking [26]. Further, the PHD filterresembles the general single-target single-sensorBayesian filtering, hence it can be implementedusing any nonlinear filtering method that can handlemulti-modal densities. For example, [17] describes aspectral compression-based PHD filter. A sequentialMonte Carlo (SMC) implementation of the PHDfilter [3] has been shown to be efficient in a clutteredenvironment, and a similar implementation [32]handled missed detections well. Particle systemimplementations of the PHD filter can also be foundin [29, 35, 36].This paper proposes a multiple-model PHD

(MMPHD) filter to address the problem of trackingmultiple maneuvering targets using the SMCapproach. SMC approaches have the advantagesof computational tractability [31] and provableconvergence properties [4, 20], and are applicableunder the most general circumstances since noassumptions need to be made on the form ofthe underlying probability density. The SMCapproximation of the MMPHD filter is applicable totrack multiple maneuvering targets with nonlinear,non-Gaussian dynamics. The implementation ofthe proposed filter is detailed, and an application isconsidered. The results show the effectiveness of theproposed filter over single-model PHD filters.This paper is organized as follows. Section II

summarizes the PHD filter. Section III describes theproblem of tracking multiple maneuvering targets,and includes an illustration of the MMPHD filter. InSection IV, a target tracking example is presentedwith simulation results to compare the performanceof the MMPHD filter with that of single-modelPHD filters. Finally, conclusions are given inSection V.

II. PHD FILTER

In tracking multiple targets, if the number oftargets is unknown and varying with time, it is notpossible to compare states with different dimensionsusing ordinary Bayesian statistics of fixed dimensional

spaces. However, the problem can be addressedby using finite set statistics (FISST) [18, 25] toincorporate comparisons of state spaces of differentdimensions. FISST facilitates the construction ofmultitarget densities from multiple-target transitionfunctions by computing set derivatives of belief-massfunctions [25], which makes it possible to combinestates of different dimensions. The main practicaldifficulty with this approach is that the dimension ofthe full state space becomes large when many targetsare present, which increases the computational loadexponentially in the number of targets. Since the PHDis defined over the state space of one target in contrastto the full posterior distribution, which is defined overthe state space of all the targets, the computationalcost of propagating the PHD over time is muchlower than propagating the full posterior density. Acomparison of multitarget filtering using the completeFISST particle filter and the PHD particle filter interms of computation and estimation accuracy is givenin [32].In general, a PHD-based multitarget tracker will

experience more difficulty in resolving closely-spacedtargets than the tracker based on the full targetposterior. However, if the pdfs of individual targetsis highly concentrated around their means comparedwith the target separation, so that the individual targetpdfs do not overlap significantly, it will becomepossible to resolve targets using the PHD filter aswell. A theoretical explanation about the capabilityof the PHD filter to resolve closely-spaced targets inGaussian context is given in [27].By definition, the PHD Dkjk(xk j Z1:k), with

single-target state vector xk, and given all themeasurements up to time step k, is the density whoseintegral on any region S of the state space is theexpected number of targets Nkjk contained in S. Thatis,

Nkjk =ZS

Dkjk(xk j Z1:k)dxk: (1)

Since this property uniquely characterizes the PHDand the first-order statistical moment of the fulltarget posterior distribution possesses this property,the first-order statistical moment of the full targetposterior is indeed the PHD. The first moment ofthe full target posterior, or the PHD, given all themeasurement Z1:k up to time step k, is given by theset integral [27]

Dkjk(xk j Z1:k) =Zfkjk(fxkg[Y j Z1:k)±Y: (2)

The reader is refered to [27] for detailed mathematicalexplanations about the PHD filter. The approximateexpected target states are given by the local maximaof the PHD. The following section gives theprediction and update steps of one cycle of the PHDfilter.

88 IEEE TRANSACTIONS ON AEROSPACE AND ELECTRONIC SYSTEMS VOL. 44, NO. 1 JANUARY 2008


A. Prediction

In a general scenario of interest, there are targetdisappearances, target spawning, and entry of newtargets. We denote the probability that a target withstate xk¡1 at time step (k¡1) will survive at time stepk by ekjk¡1(xk¡1), the PHD of spawned targets at timestep k from a target with state xk¡1 by bkjk¡1(xk j xk¡1),and the PHD of newborn spontaneous targets at timestep k by °k(xk). Then, the predicted PHD is given by

Dkjk¡1(xk j Z1:k¡1) = °k(xk)

+Z[ekjk¡1(xk¡1)fkjk¡1(xk j xk¡1)+ bkjk¡1(xk j xk¡1)]

£Dk¡1jk¡1(xk¡1 j Z1:k¡1)dxk¡1 (3)

where fkjk¡1(xk j xk¡1) denotes the single-targetMarkov transition density. The predictionequation (3) is lossless since there are noapproximations.

B. Update

The predicted PHD can be corrected with theavailability of measurements Zk at time step k to getthe updated PHD. We assume that the number of falsealarms is Poisson distributed with the average rateof ¸k and that the probability density of the spatialdistribution of false alarms is ck(zk). Let the detectionprobability of a target with state xk at time step k bepD(xk). Then, the updated PHD at time step k is givenby

Dkjk(xk j Z1:k)»="Xzk2Zk

pD(xk)fkjk(zk j xk)¸kck(zk)+Ãk(zk j Z1:k¡1)

+ (1¡pD(xk))#

£Dkjk¡1(xk j Z1:k¡1) (4)

where the likelihood function Ã(¢) is given by

Ãk(zk j Z1:k¡1) =ZpD(xk)fkjk(zk j xk)Dkjk¡1(xk j Z1:k¡1)dxk

(5)

and fkjk(zk j xk) denotes the single-sensor/single-targetlikelihood. The update equation (4) is not losslesssince approximations are made on predictedmultitarget posterior to obtain the closed-formsolution (4). The reader is referred to [27] for furtherexplanations.

III. MULTIPLE-MODEL PHD FILTER

This section first describes the mathematicalformulation of the problem of tracking multiplemaneuvering targets and the solution to it usingthe MMPHD algorithm. This is followed by the

implementation details of the algorithm using an SMCapproach.

A. Problem Formulation

The general parameterized target dynamics of thejth target are given by

xjk = ar,k(xjk¡1,wr,k¡1,r

jk ), j = 1, : : : ,NXk (6)

where xjk is the target state vector at time step k, NXk is

the number of targets at time step k, ar,k, in general, isa nonlinear function, wr,k¡1 is the mode-dependentprocess noise vector of known statistics, and rjk 2f1, : : : ,Nrg is the model index parameter governedby an underlying Markov process with the modeltransition probability

fkjk¡1(rk = q j rk¡1 = p) = hpq: (7)

We denote the PHD of the mode-dependentfull target state distribution at time step k byDkjk(xk,rk j Z1:k). Once again we consider targetsurvival, target spawning, and appearance ofcompletely new targets given by ekjk¡1(¢), bkjk¡1(¢),and °k(¢), respectively.The measurements originate from either targets or

clutter. We denote the total number of measurementsat time step k by NZk . The target-originatedmeasurement due to the jth target on ith sensor isgiven by

zi,jk = gr,k(xjk,vr,k,r

jk ) (8)

where gr,k, in general, is a nonlinear function and vr,kis a mode-dependent measurement noise vector ofknown statistics. Some targets may not be detected attime step k. We denote the probability that the targetwith state xk is detected at time step k by pD(xk).

B. Multiple-Model Approach to Target Tracking

The multiple-model approach to trackingmaneuvering targets by detecting maneuvers andidentifying the appropriate model has been shown tobe highly effective. In this approach, a finite numberof filters operate in parallel, and the target motion isassumed to follow one of the models in the mode setof the tracker. Considering the compromise betweencomplexity and performance, the IMM approach hasbeen shown to be the most effective of the knownmultiple-model approaches, including the generalizedpseudo-Bayesian (GPB) [1, 13] algorithms. A GPBalgorithm of order n (GPBn) requires N

nr filters in

its bank, where Nr is the number of models. TheIMM estimator performs nearly as well as GPB2, butrequires only Nr filters operating in parallel. Thus,it has significantly less computational complexity,which is almost the same as that of GPB1. Further, theIMM estimator does not require maneuver detection

PUNITHAKUMAR ET AL.: MULTIPLE-MODEL PROBABILITY HYPOTHESIS DENSITY FILTER 89


decisions as in the case of variable state dimension(VSD) filter [7] algorithms, and undergoes a softswitching between models based on the updated modeprobabilities.The multiple-model approach used in the proposed

MMPHD algorithm is structurally similar to that ofthe IMM estimator in the mixing and combinationstages. The MMPHD filter also requires, implicitly,only Nr PHD filters to operate in parallel when thereare Nr models describing target dynamics. Further,the MMPHD filter does not require a maneuverdetection decision and undergoes a soft switchingbetween the models. The IMM estimator differsfrom the multiple-model approach presented herein that the former uses only first- and second-orderstatistics of the target densities in the the mixingand combination stages. The techniques used in theIMM estimator cannot be applied to combine themode-dependent PHD filter outputs since the densitiesare not necessarily Gaussian. Further, the densitiesmight be multi-modal when they represent multipletargets; thus, it is not reasonable to approximate themby only first- and second-order statistics. Thus, themultiple-model approach presented here uses thebranched true densities, i.e., the complete densitiesconditioned on each model, in the mixing and updatestages. This approach can handle multi-modal targetdensities at the expense of increased computationalload compared with the IMM estimator. One cycle ofthe recursive MMPHD algorithm can be described inthree stages as follows.1) The Mixing Stage: In this stage, each

mode-matched filter is fed with a different densitythat is a combination of the previous mode-dependentdensities. The initial density Dkjk¡1(xk¡1,rk = q j Z1:k¡1)fed to the PHD filter, which is matched to the targetmodel q, is calculated on the basis of Markovianmodel transition probability matrix [hpq] andmode-dependent prior density Dk¡1jk¡1(xk¡1,rk¡1 =p j Z1:k¡1), i.e.,

Dkjk¡1(xk¡1,rk = q j Z1:k¡1)

=NrXp=1

Dk¡1jk¡1(xk¡1,rk¡1 = p j Z1:k¡1)hpq,

q= 1, : : : ,Nr: (9)

Target spawning, birth, and disappearance are notconsidered at the mixing stage and they are consideredin the prediction stage. The densities Dkjk¡1(¢) andDk¡1jk¡1(¢) in (9) are similar to probability densitiesexcept that they do not integrate to unity. The mixingdescribed in (9) is then similar to total probabilitytheorem.2) The Prediction Stage: Once the initial density

for the PHD filter that is matched to target model q iscalculated, the mode-dependent predicted density is

calculated as

Dkjk¡1(xk,rk = q j Z1:k¡1) = °k(xk ,rk = q)

+

Z[ekjk¡1(xk¡1)fkjk¡1(xk j xk¡1,rk = q)

+ bkjk¡1(xk j xk¡1,rk = q)]

£ Dkjk¡1(xk¡1,rk = q j Z1:k¡1)dxk¡1: (10)

The prediction of the mode-dependent PHD cannot bejust applied to that of the single-model PHD, whichis matched to the target model, because the PHD ofspontaneous target birth °k(¢) and target spawningbkjk¡1(¢) are also mode dependent. The integral ofthe mode-dependent PHD Dkjk¡1(xk,rk = q j Z1:k¡1)over a region gives the expected (predicted) numberof targets in that region assuming all the targets aretravelling with the target dynamics described bymodel q.3) The Update Stage: With the availability of

measurements at time step k, the mode-dependentupdated density is calculated as (for q= 1, : : : ,Nr)

Dkjk(xk,rk = q j Z1:k)

»=24Xzk2Zk

pD(xk)fkjk(zk j xk,rk = q)¸kck(zk)+Ãk(zk j Zk¡1)

+ (1¡pD(xk))35

£Dkjk¡1(xk,rk = q j Z1:k¡1) (11)

where the likelihood function Ã(¢) is given byÃk(zk j Z1:k¡1)

=

ZpD(xk)fkjk(zk j xk,rk = q)Dkjk¡1(xk,rk = q j Z1:k¡1)dxk:

(12)

Although there is no explicit mode probability updatein the MMPHD filter, the update stage implicitlyupdates the mode probabilities as well. Therefore,the updated mode probability for a particular modelcan be calculated by integrating the mode-dependentupdated PHD divided by the total expected numberof targets. The total number of targets can be foundby summing up all the integrals of the updatedmode-dependent PHDs. The updated mode probabilityfor a particular target, assuming that the targetdensities do not overlap, can be found by integratingthe mode-dependent PHD over the region in whichthat target is represented. However, it is not necessaryto calculate the numerical value of the updated modeprobability in the recursive MMPHD filter since themode-dependent density as a whole is used in themixing stage to feed the filter.Using multiple-model algorithms for benign

nonmaneuvering targets might degrade theperformance of the tracker and increases thecomputational load. However, with higher target



maneuverability, a multiple-model approach is needed.The decision about whether a multiple model ispreferred or not is done based on the maneuveringindex [7], which quantifies the maneuverabilityof the target in terms of the process noise, sensormeasurement noise, and sensor revisit interval. Astudy that compares the IMM estimator with theKalman filter based on the maneuvering index is givenin [22].

C. Particle Implementation

This section describes the SMC approach to theMMPHD filter. This approach provides a mechanismto represent the posterior MMPHD by a set of randomsamples or particles, which consists of state andmodel information with associated weights. The keyidea in this implementation is to include the modelindex parameters in the sample set with the associatedweights to represent the mode-dependent posteriordensity. Although it does not appear explicitly, thereare several PHD filters, each matched to differenttarget dynamics, running in parallel in the MMPHDfilter. The model index parameter in a sample directsthe MMPHD filter to choose the PHD filter thatmatches the associated target model. Further, thenumber of samples used in each PHD filter is notnecessarily the same for different target models.Since the mode probabilities are updated through theupdate step of the filter, the PHD filter that matchesthe target motion will contain the higher numberof samples. This makes the filter computationallyefficient compared with a filter that uses an equalnumber of samples to represent the PHD matched toeach model.The particle filter implementation of the proposed

algorithm can be considered as a special case ofthe algorithm given in [3] with some modifications.The target state is augmented with the model indexparameter and the prediction and update operators ofthe algorithm given in [3] are modified in order toinclude target motion model uncertainty.The SMC implementation considered here is

structurally similar to the sampling importanceresampling (SIR) type of particle filter [2]. Let theposterior MMPHD Dk¡1jk¡1(xk¡1,rk¡1 j Z1:k¡1) berepresented by a set of particles fw(s)k¡1,x(s)k¡1,r(s)k¡1gLk¡1s=1 .That is,

Dk¡1jk¡1(xk¡1,rk¡1 j Z1:k¡1)

=Lk¡1Xs=1

w(s)k¡1±(xk¡1¡ x(s)k¡1,rk¡1¡ r(s)k¡1) (13)

where ±(¢) is the Dirac Delta function.In contrast to particle filters, the total weightPLk¡1s=1 w

(s)k¡1 is not equal to one; instead, it gives the

expected number of targets nXk¡1 at time step (k¡ 1),

which follows from the property that the integralof the PHD over the state space gives the expectednumber of targets. Further, the posterior modelprobabilities corresponding to a particular target areapproximately equal to the proportion of the totalsample weights in the index set fr(s)k¡1gLk¡1s=1 for eachmodel corresponding to that target.1) Prediction: The MMPHD filter prediction step

involves predicting the models in addition to stateprediction. The model prediction for existing targets isperformed based on the model transition probabilitiesfkjk¡1(rk j rk¡1). Model samples fr(s)kjk¡1gLk¡1s=1 from the

model-predicted MMPHD Dkjk¡1(xk¡1,rk j Z1:k¡1)are generated by importance sampling from aproposal density ¼k(¢ j rk¡1). We generate independentand identically distributed (IID) model samplesfr(s)kjk¡1gLk¡1+Jks=Lk¡1+1

corresponding to new spontaneouslyborn targets by sampling from another proposaldensity ¯k(¢). Then,

r(s)kjk¡1 »½¼k(¢ j rk¡1) s= 1, : : : ,Lk¡1¯k(¢) s= Lk¡1 +1, : : : ,Lk¡1 + Jk

(14)

where ¼k(¢ j rk¡1) and ¯k(¢) are probability massfunctions. Then, a discrete weighted approximation tothe model predicted MMPHD Dkjk¡1(xk¡1,rk j Z1:k¡1) isgiven by

Dkjk¡1(xk¡1,rk j Z1:k¡1) =Lk¡1+JkXs=1

$(s)kjk¡1±(xk¡1¡ x(s)k¡1,rk ¡ r(s)kjk¡1)

(15)where

$(s)kjk¡1 =

8>>>>>>>>>><>>>>>>>>>>:

fkjk¡1(r(s)kjk¡1 j r(s)k¡1)

¼k(r(s)kjk¡1 j r(s)k¡1)

w(s)k¡1

s= 1, : : : ,Lk¡1

μk(r(s)kjk¡1)

¯k(r(s)kjk¡1)

1Jk

s= Lk¡1 +1, : : : ,Lk¡1 + Jk

:

(16)

In (16), the probability mass function μk(¢) denotesthe model distribution of spontaneous target births attime step k. The number of new particles Jk can bea function of time step k to accommodate a varyingnumber of targets.We now apply importance sampling to generate

state samples that approximate the predicted MMPHDDkjk¡1(xk,rk j Z1:k¡1). We generate fx(s)kjk¡1gLk¡1s=1 statesamples from the proposal density qk(¢ j xk¡1,rkjk¡1,Zk)and IID state samples fx(s)kjk¡1gLk¡1+Jks=Lk¡1+1

correspondingto new spontaneously born targets from anotherproposal density pk(¢ j rkjk¡1,Zk¡1). That is,



x(s)kjk¡1 »½qk(¢ j xk¡1,rkjk¡1,Zk) s= 1, : : : ,Lk¡1pk(¢ j rkjk¡1,Zk) s= Lk¡1 +1, : : : ,Lk¡1 + Jk

: (17)

Then, the weighted approximation of the predictedMMPHD is given by

Dkjk¡1(xk,rk j Z1:k¡1) =Lk¡1+JkXs=1

w(s)kjk¡1±(xk ¡ x(s)kjk¡1,rk ¡ r(s)kjk¡1) (18)

where

w(s)kjk¡1 =

8>>><>>>:ekjk¡1(x

(s)kjk¡1)fkjk¡1(x

(s)kjk¡1 j x(s)k¡1jk¡1,r(s)kjk¡1) + bkjk¡1(x(s)kjk¡1 j x(s)k¡1jk¡1,r(s)kjk¡1)qk(x

(s)kjk¡1 j x(s)kjk¡1,r(s)kjk¡1,Zk)

$(s)kjk¡1 s= 1, : : : ,Lk¡1

°k(x(s)kjk¡1 j r(s)kjk¡1)

pk(x(s)kjk¡1 j r(s)kjk¡1,Zk)

$(s)kjk¡1 s= Lk¡1 +1, : : : ,Lk¡1 + Jk

:

(19)

The functions that characterize the Markov targettransition density fkjk¡1(¢), target spawning bkjk¡1(¢)and entry of new targets °k(¢) in (19) are conditionedon the particular motion model. Thus, although itis not explicitly explained by the equations, thereare Nr PHD filters running in parallel in thisimplementation.2) Update: With the available set of

measurements Zk at time step k, the updated particleweights can be calculated by

w¤(s)k =24(1¡pD(x(s)kjk¡1)) + NZkXi=1

pD(x(s)kjk¡1)fkjk(z

ik j x(s)kjk¡1,r(s)kjk¡1)

¸kck(zik) +ªk(z

ik)

35w(s)kjk¡1

(20)where

ªk(zik) =

Lk¡1+JkXs=1

pD(x(s)kjk¡1)fkjk(z

ik j x(s)kjk¡1,r(s)k )w(s)kjk¡1:

(21)The single-target/single-sensor measurementlikelihood function fkjk(¢) in (20) and (21) is writtenas dependent on the model, considering a general casein which the measurement model can also be modedependent.3) Resample: To perform resampling, since the

weights are not normalized to unity in PHD filters, theexpected number of targets is calculated by summingup the total weights, i.e.,

nXk =Lk¡1+JkXs=1

w¤(s)k : (22)

Then the updated particle set fw¤(s)k =nXk ,x(s)kjk¡1,

r(s)kjk¡1gLk¡1+Jks=1 is resampled to get fw(s)k =nXk ,x(s)k ,r(s)k gLks=1such that the total weight after resampling remains

nXk . Now, the discrete approximation of the updatedposterior MMPHD at time step k is given by

Dkjk(xk,rk j Z1:k) =LkXs=1

w(s)k ±(xk ¡ x(s)k ,rk ¡ r(s)k ):

(23)

The mode-dependent posterior PHDs can be easilyidentified by grouping the particles based on modelindex parameters.

IV. SIMULATIONS

A. Scenario

This section presents a two-dimensionaltracking example to compare the MMPHD filter’sperformance with that of the single-model PHD filters,namely, a constant-velocity model PHD filter and acoordinated-turn model PHD filter. The test scenariois constructed as follows.

1) The observations are taken from fourfixed bearing-only sensors located at (0,0) m,(0,1£ 104) m, (1£ 104,0) m, and (1£ 104,1£ 104) m.The measurements are available at discrete-timesampling interval T = 60 s. The target-generatedmeasurements corresponding to target j on sensor iare given by

zi,jk = tan¡1Ãyjk ¡ yiSxjk ¡ xiS

!+ vi,jk (24)

where vi,jk is an IID sequence of zero-mean Gaussianvariables with standard deviation 0.01 rad. (xjk,y

jk) and

(xiS,yiS) denote the locations of target j and sensor i

at time step k, respectively. The average false alarmrate ¸k = 4£ 10¡3 rad¡1, the volume of observation ofeach sensor = 2¼ rad, the false alarms are distributeduniformly within the sensor field of view, and the



Fig. 1. Target trajectory (I–initial, F–final, Si–sensor).

probability of target detection pD(xk) = 1, regardlessof the target state xk.2) Two maneuvering targets, namely, target 1 and

target 2, are travelling from initial target positions(¡3£ 103,5£103) m and (1:4£ 104,8£ 103) m,respectively. This simulation does not includeany spawning of new targets from existing ones.Target 1 moves eastward for 20 min at a nearlyconstant velocity with an initial velocity of 5 ms¡1,before executing a 9±/min (which amounts to anacceleration of about 0.1 ms¡2) coordinated turn[7] in the anticlockwise direction for 10 min. Thenit moves northward for another 20 min, followedby a clockwise 9±/min coordinated turn for 10 min.Target 2 moves southward for 10 min at a nearlyconstant velocity with an initial velocity of 5 ms¡1,before executing a coordinated turn of 9±/min inthe clockwise direction for 10 min. Then it moveswestward at nearly a constant velocity for 30 minfollowed by an anticlockwise coordinated turn of9±/min for 10 min. The target trajectories are shownin Fig. 1. This scenario leads to a maneuvering index1

of more than 1.3) The single target Markov transition equation

that characterizes the constant velocity targetdynamics (rk = 1) is given by

xjk = Aj1,kx

jk¡1 +w

j1,k (25)

where xjk = [xjk, _x

jk,y

jk , _y

jk] is the state of the jth

target, which consists of target position (xjk,yjk) and

target velocity ( _xjk, _yjk) at time step k, and w

j1,k is an

i.i.d sequence of zero-mean Gaussian vectors withcovariance §jk . The matrices A

j1,k and §

j1,k are given

1The measurement contribution part of the posterior Cramer-Raolower bound is used to obtain the measurement standard deviationin calculating the maneuvering index.

as follows

Aj1,k =

266641 T 0 0

0 1 0 0

0 0 1 T

0 0 0 1

37775 (26)

§j1,k =

266666666664

T3

3T2

20 0

T2

2T 0 0

0 0T3

3T2

2

0 0T2

2T

377777777775l (27)

where l = 1£ 10¡4 m2s¡3 is the level of the powerspectral density of the corresponding continuousprocess noise.4) The single-target Markov transition equation

that characterizes the coordinated turn target dynamics(rk = 2) is given by

xjk = Aj2,kx

jk¡1 +w

j2,k (28)

where xjk = [xjk, _x

jk,y

jk , _y

jk ,−

jk] is the augmented state

vector of the jth target, which consists of targetposition (xjk,y

jk), target velocity ( _x

jk, _y

jk), and turn rate

−jk at time step k, and wj2,k is an i.i.d sequence of

zero-mean Gaussian vectors with covariance §j2,k. The

matrices Aj2,k and §j2,k are given as follows

Aj2,k =

266666666666664

1sin−jk¡1T

−jk¡10 ¡1¡ cos−

jk¡1T

−jk¡10

0 cos−jk¡1T 0 ¡sin−jk¡1T 0

01¡ cos−jk¡1T

−jk¡11

sin−jk¡1T

−jk¡10

0 sin−jk¡1T 0 cos−jk¡1T 0

0 0 0 0 1

377777777777775(29)

§j2,k =

266666666666664

T3l13

T2l12

0 0 0

T2l12

Tl1 0 0 0

0 0T3l13

T2l12

0

0 0T2l12

Tl1 0

0 0 0 0 Tl2

377777777777775(30)

where the levels of the power spectral densities arel1 = 1£ 10¡4 m2s¡3 and l2 = 1£ 10¡10 rad2s¡3. Sincethe turn rate −jk is unknown, it is included in the statevector and must be estimated.



5) The Markovian model transition probabilitymatrix in the MMPHD filter is taken as

[hpq] =

26641¡T

¿1

T

¿1T

¿21¡ T

¿2

3775 (31)

with sojourn times [7] ¿1 = 20 min and ¿2 = 10 min.The initial model probabilities for both models are0.5.6) We assume that the probability of target

survival = 0:99, the probability of target spawning = 0,and the probability of spontaneous target birth = 0:01.The birth model that describes the entry of new targetsis taken as follows.The x and y position elements of the state vector

of newborn particles are normally distributed centeredaround every two intersecting measurements witha standard deviation of 200 m. This one pointinitialization is done to all intersecting measurementpairs within the surveillance region, which is asquare region defined by (¡5£ 103,¡5£ 103) mand (1:5£ 104,1:5£ 104) m as the lower left andupper right corners, respectively. The velocity andturn rate elements of the state vector for the newbornparticles are distributed uniformly within the interval[¡5,5] ms¡1 and [¡0:02,0:02] rads¡1, respectively.7) 1000 particles are used to represent one

target and Jk = (10£ number of intersectingmeasurement points within the surveillance region)particles are used to represent new spontaneouslyborn targets in all the tracking algorithms. Beforesimulations begin, all PHD filters are initializedwith 1000 particles normally distributed with mean[¡3£ 103 m,5 ms¡1,5£ 103 m,0 ms¡1,0 rads¡1]and covariance diag[4£104 m2,1 m2s¡2,4£104 m2,1 m2s¡2,4£ 10¡4 rad2s¡2] representingtarget 1, and the model index parameters of the initialparticles are taken as 1, i.e., the constant-velocitymodel, in the MMPHD filter.8) The PHD filter does not provide a mechanism

to get the target state estimates directly. One wayof getting the estimates is by identifying the localmaxima of the density. An expectation maximizationbased peak extraction algorithm to obtain the targetstate estimates can be found in [34]. For simplicityand to avoid added computational load we use theK-means clustering algorithm [19] to associateparticles to targets2 based on the position elementsof the particle state vectors. The targets are associatedto tracks using the nearest neighbor approach basedon the mean of each target cluster.

Although we have assumed Gaussian process noisein this particular example, the MMPHD algorithm

2We assume that the targets are sufficiently far apart that a particlerepresents only one target at any particular time.

presented in Section III is able to deal with anyprocess noise as long as the samples can be drawnfrom the process noise distribution. Since the PHDrecursion given in this paper is valid only for singlesensor, the measurements are grouped from eachsensor and used to update the filter iteratively.

B. Simulation Results

Fig. 2 illustrates the model-switching property ofthe MMPHD filter by plotting number of particlesin each model against time. The particles consideredhere are of equal weight, so the proportion of theparticles in each model of a particular target givesthe approximate posterior model probabilities of thattarget. Fig. 2 also illustrates the number of targetsestimated from the PHD filter. At the beginning ofthe scenario, all PHD filters assume that there is onlyone track, i.e. target 1, within the surveillance regionand the particles corresponding to target 1 in theMMPHD filter are shown in Fig. 2(a). However, it canbe seen in Fig. 2(b) that the total number of particlescorresponding to target 2 increases with time at thebeginning until it reaches nearly 1000, which is thenumber of particles assigned to represent one target.Once the track is formed, the MMPHD filter clearlydisplays the model switching property for target 2 andassigns more particles to the model that matches withthe target motion.Figs. 3 and 4 show the root-mean-squared

errors (RMSE) of position and velocity estimates,respectively. All figures indicate that the overallperformance of the MMPHD filter is significantlybetter than that of both single-model PHD filters.Further, the single-model PHD filters show pooradaptation to target maneuvers and yield largerestimation errors. Fig. 5 compares the RMSEs ofturn rate estimates of the MMPHD filter and thesingle-model PHD filter that uses the coordinated-turnmodel. The single-model PHD filter that uses theconstant-velocity model is not considered since thereis no turn rate estimate in that filter. Table I gives theaverage of the estimation errors; note that the valuesfor the MMPHD filter show significant improvementsover the single-model PHD filters.The average computational times (based on 100

Monte Carlo runs) on dual 2.4 GHz Intel Xeonprocessors for a tracking period of 60 time steps are205 s, 201 s, and 199 s with standard deviation 2.6 s,1.7 s, and 2.5 s for multiple model, constant-velocitymodel, and coordinated-turn model PHD filteralgorithms, respectively. All filters were implementedin MATLAB. Thus, if the same number of samplepoints are used, the MMPHD filter is only marginallymore expensive compared with single-model PHDfilters since the only additional calculation requiredin the MMPHD filter is to generate model indexparameters in the mixing stage.



Fig. 2. Model switching in MMPHD filter (100 Monte Carlo runs). (a) Target 1. (b) Target 2.

Fig. 3. Target position estimation RMSE (100 Monte Carlo runs). (a) Target 1. (b) Target 2.

Fig. 4. Target velocity estimation RMSE (100 Monte Carlo runs). (a) Target 1. (b) Target 2.

V. CONCLUSIONS

This paper considered the problem of trackingmultiple maneuvering targets with nonlinear,non-Gaussian dynamics in a cluttered environment.The filter proposed to solve this problem is amultiple-model extension to the PHD filter using theSMC approach. The PHD filter is a computationallyefficient solution to multitarget tracking problems, andthe SMC implementation makes the filter applicablein general scenarios with nonlinear, non-Gaussian

systems. This paper presented a new multiple-modelapproach to enhance the PHD filter’s capability tohandle target maneuvers.The MMPHD filter was compared with

single-model PHD filters. It resulted in significantlybetter tracking performance compared with thesingle-model PHD filters. The single-modelPHD filters that matched only one kind of targetdynamics showed poor adaptation to maneuvers,resulting in large estimation errors when the targetmaneuverability is high. We can conclude that a



Fig. 5. Target turn rate estimation RMSE (100 Monte Carlo runs). (a) Target 1. (b) Target 2.

TABLE IAverage Target State Estimation RMSE(Based on 100 Monte Carlo Runs)

ConstantMultiple Velocity CoordinatedModel Model TurnPHD PHD PHD Units

Target 1 (position) 95.48 311.56 168.59 mTarget 2 (position) 210.89 401.88 237.09 mTarget 1 (velocity) 0.69 1.38 1.15 ms¡1

Target 2 (velocity) 1.17 1.57 1.29 ms¡1

Target 1 (turn rate) 0.00086 – 0.00121 rads¡1

Target 2 (turn rate) 0.00106 – 0.00110 rads¡1

multiple-model approach is essential to ensuresatisfactory performance in such cases.

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K. Punithakumar (S’06) was born in Sri Lanka in 1976. He received theB.Sc.Eng. degree in electronic and telecommunication engineering fromUniversity of Moratuwa, Sri Lanka, in 2001, and the M.A.Sc degree in electricaland computer engineering from McMaster University, Canada, in 2003.He was an instructor in the Department of Electronic and Telecommunication

Engineering at the University of Moratuwa, Sri Lanka, from 2001 to 2002. Since2002 he has been a graduate student/teaching assistant in the Department ofElectrical and Computer Engineering at McMaster University, Canada, in pursuitof a Ph.D. His research interests are in target tracking, sensor management andnonliner filtering.

T. Kirubarajan (S’95–M’98) was born in Sri Lanka in 1969. He received theB.A. and M.A. degrees in electrical and information engineering from CambridgeUniversity, England, in 1991 and 1993, and the M.S. and Ph.D. degrees inelectrical engineering from the University of Connecticut, Storrs, in 1995 and1998, respectively.Currently, he is an assistant professor in the Electrical and Computer

Engineering Department at McMaster University, Hamilton, Ontario, Canada.He is also serving as an adjunct assitant professor and the Associate Director ofthe Estimation and Signal Processing Research Laboratory at the University ofConnecticut. His research interests are in estimation, target tracking, multisourceinformation fusion, sensor resource management, signal detection and faultdiagnosis. His research activities at McMaster University and at the Universityof Connecticut are supported by US Missile Defense Agency, US Office ofNaval Research, NASA, Qualtech Systems, Inc., Raytheon Canada Ltd., andDefense Research Development Canada, Ottawa. In September 2001, he servedon a DARPA expert panel on unattended surveillance, homeland defense andcounterterrorism. He has also served as a consultant in these areas to a numberof companies, including Motorola Corporation, Northrop-Grumman Corporation,Pacific-Sierra Research Corporation, Lockhead Martin Corporation, QualtechSystems, Inc., Orincon Corporation and BAE systems. He has worked onthe development of a number of engineering software programs, includingBEARDAT for target localization from bearing and frequency measurements inclutter, FUSEDAT for fusion of multisensor data for tracking. He has also workedwith Qualtech Systems, Inc., to develop an advanced fault diagnosis engine.Dr. Kirubarajan has published about 100 articles in the areas of his research

interests, in addition to one book on estimation, tracking and navigation and twoedited volumes. He is also a recipient of Ontario Premier’s Research ExcellenceAward (2002).

Abhijit Sinha received his B.S. degree in physics from the University ofCalcutta, India, in 1994. He received his M.S. degree in electrical communicationengineering from Indian Institute of Science, Bangalore, India, in 1998, andreceived his Ph.D. degree in electrical and computer engineering form theUniversity of Connecticut, Storrs, in 2002.He worked as a Postdoctoral Fellow at the University of Connecticut from

2002 to 2003. Currently, he is working as a research associate at McMasterUniversity, Canada. His research interests include signal processing, targettracking and communications.



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Multiple-Model Probability Hypothesis Density Filter for Tracking … · 2008-10-16 · Multiple-Model Probability Hypothesis Density Filter for Tracking Maneuvering Targets K. PUNITHAKUMAR,