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Signal Processing

Signal Processing 94 (2014) 48–56

0165-16http://d

n CorrE-m

ymjia@bbuaazha

journal homepage: www.elsevier.com/locate/sigpro

PHD filter for multi-target tracking with glint noise

Wenling Li a,n, Yingmin Jia a,b, Junping Du c, Jun Zhang d

a The Seventh Research Division and the Department of Systems and Control, Beihang University (BUAA), Beijing 100191, Chinab Key Laboratory of Mathematics, Informatics and Behavioral Semantics (LMIB), SMSS, Beihang University (BUAA), Beijing 100191, Chinac Beijing Key Laboratory of Intelligent Telecommunications Software and Multimedia, School of Computer Science and Technology, BeijingUniversity of Posts and Telecommunications, Beijing 100876, Chinad School of Electronic and Information Engineering, Beihang University (BUAA), Beijing 100191, China

a r t i c l e i n f o

Article history:Received 24 January 2013Received in revised form30 May 2013Accepted 9 June 2013Available online 20 June 2013

Keywords:Multi-target trackingProbability hypothesis density filterGlint noiseVariational Bayesian

84/$ - see front matter & 2013 Elsevier B.V.x.doi.org/10.1016/j.sigpro.2013.06.012

esponding author. Tel.: +86 10 8233 8683; faail addresses: lwlmath@gmail.com (W. Li),uaa.edu.cn (Y. Jia), junpingdu@126.com (J. Dngjun@vip.sina.com (J. Zhang).

a b s t r a c t

This paper studies the problem of multi-target tracking with glint noise in the formulationof random finite set theory. By using the Student's t-distribution to describe the glint noisestatistics, the probability hypothesis density (PHD) filter is extended via augmenting thetarget state and the noise parameters. To derive a closed-form expression for the extendedPHD filter, the prior Gamma distribution for the noise parameters is adopted so that thepredicted and the updated intensities can be represented by mixtures of Gaussian–Gamma terms. As the target state and the noise parameters are coupled in the likelihoodfunctions, the variational Bayesian approach is applied to derive approximated distribu-tions so that the updated intensity is represented in the same form as the predictedintensity and the resulting algorithm is recursive. Simulation results are provided viaa numerical example to illustrate the effectiveness of the proposed filter.

& 2013 Elsevier B.V. All rights reserved.

1. Introduction

Multi-target tracking has received great attention dueto its wide applications in military and civil fields. As newtargets may appear or disappear randomly in the surveil-lance region, tracking multi-target involves estimating anunknown and time-varying number of targets from agiven set of measurements with uncertain origin. Manytracking algorithms have been proposed based on dataassociation strategies such as the nearest neighbor (NN),the strongest neighbor (SN), the joint probabilistic dataassociation (JPDA) and the multiple hypothesis tracking(MHT) [1]. Due to its combinatorial nature, a high compu-tational load is often required to resolve the data associa-tion problem in multi-target tracking algorithms. Twowell-known alternative formulations that avoid data

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x: +86 10 8231 6100.

u),

association are symmetric measurement equations [2]and the random finite sets (RFS) approach [3].

In the RFS formulation, the target states and measure-ments are represented by two different random finite sets(RFSs), and as a consequence, the multi-target trackingproblem can be addressed in a rigorous Bayesian estima-tion framework based on the finite set statistics (FISST)theory. By constructing the multi-target transition densityand the multi-target likelihood function, the optimalmulti-target Bayes filter can be derived. However, it isgenerally intractable due to the existence of multiple setintegrals and the combinatorial nature of the multi-targetdensities. To alleviate this intractability, the probabilityhypothesis density (PHD) filter has been proposed as a firstorder moment approximation to the multi-target posteriordensity [4]. It is worth mentioning that the resulting PHDfilter still requires solving multi-dimensional integrals andthe integrals might be also intractable in many cases ofinterest. Two schemes have been proposed to implementthe PHD filter explicitly including the sequential MonteCarlo (SMC) [5] and the Gaussian mixture (GM) [6]. A main

W. Li et al. / Signal Processing 94 (2014) 48–56 49

drawback of the SMC-PHD filter is high computational costsince a large number of particles have to be sampled. Toovercome this disadvantage, the GM-PHD filter was devel-oped for linear target dynamic and measurement modelswith Gaussian distributions, in which the weights, meansand covariance matrices are propagated analytically by theKalman filter (KF). Moreover, the nonlinear KF counter-parts can be directly employed to deal with nonlineartarget dynamics and measurement models. The conver-gence properties of two implementations were analyzed in[5,7]. Many extensions have been recently developed toaddress different tracking scenarios [8–20].

For multi-target tracking problems, the Gaussian dis-tribution has commonly been used for representing themeasurement noise statistics due to its mathematicalsimplicity and effectiveness. However, it is known thatnot all the real-world data can be modeled well byGaussian distribution. In particular for radar trackingsystems, changes in the target aspect toward the radarmay cause irregular electromagnetic wave reflections andthis gives rise to outliers or glint noise. It was found thatglint noise has a heavy-tailed probability density functionand conventional filtering algorithms are known to showunsatisfactory performance in the presence of glint noise[21]. The statistical properties of the glint noise and itsmathematical models have been studied extensively in[22]. Specially, the Student's t-distribution has been usedto model the glint noise in [23] while the mixture ofGaussian distributions has been used in [24]. In [25], theglint noise was modeled by the mixture of a Gaussiandistribution and a Laplacian distribution. As the Student'st-distribution has been shown to be much less sensitivethan the Gaussian distribution to outliers [26–28], it hasbeen used as an image prior [29–31]. A robust multi-sensor fusion algorithm for target tracking applicationshas been proposed based on the Student's t-distribution[32]. Note that the use of the Student's t-distribution raisessignificant difficulties of tractability and the variationalBayesian approach has been employed to derive approxi-mated distributions. Nevertheless, the glint noise modeledby the Student's t-distribution has not been addressed formulti-target tracking in the RFS formulation.

In this paper, we attempt to apply the PHD filter toaddress the problem of multi-target tracking with glintnoise. By modeling the glint noise as a Student's t-distribution, we show how the variational Bayesianapproach can be used in the PHD filter to derive closed-form expressions. Based on the prior Gamma distributionsfor the parameters of the Student's t-distribution, wepropose a novel implementation to the PHD filter byrepresenting the predicted and the updated intensities asthe mixtures of Gaussian–Gamma distributions. This isinspired by the idea of the GM-PHD filter for Gaussiandistributions. To guarantee the same form of the predictedand the updated intensities, the main difficulty encoun-tered is the computation of the predicted likelihoodsince the target state and the noise parameters are coupledin the likelihood functions. This is overcome by thevariational Bayesian approximation method. In addition,a heuristic dynamics has been used which simply spreadsthe previous posterior distribution by a scalar factor so

that the predicted intensity is still a mixture of Gaussian–Gamma terms. Simulation results are provided to illustratethe effectiveness of the proposed filter.

The rest of this paper is organized as follows. Theproblem of multi-target tracking with glint noise is for-mulated in Section 2. The implementation to the PHD filteris presented by applying the variational Bayesian approachin Section 3. In Section 4, a numerical example is providedto illustrate the effectiveness of the proposed filter. Con-clusion is drawn in Section 5.

2. Problem formulation

2.1. Tracking model

Consider the following linear target dynamics andmeasurement models:xk ¼ Fk−1xk−1 þwk−1 ð1Þzk ¼Hkxk þ vk ð2Þwhere xk∈Rn and zk∈Rm denote the target state and themeasurement vectors, respectively. Fk−1 is the state transi-tion matrix and Hk is the measurement matrix. The processnoise wk−1 is assumed to be zero-mean white Gaussianwith covariance matrix Qk−1 and the measurement noisevk is assumed to be distributed according to heavy-tailedm-dimensional Student's t-distribution. Specifically, theprobability density functions of the target state and themeasurement can be represented bypxðxkjxk−1Þ ¼N ðxk; Fk−1xk−1;Qk−1Þ ð3Þpzðzkjxk;Rk; rkÞ ¼ Sðzk;Hkxk;Rk; rkÞ ð4Þwhere N ðx; x; PÞ denotes a Gaussian density with mean xand covariance matrix P. Sðz; z;Σ; νÞ denotes the probabil-ity density function of a Student's t-distribution withmean z, precision Σ and degree of freedom ν. Unlike theconventional tracking algorithms with known measure-ment noise statistics, the parameters Rk ¼ diagfr1k ;…; rmk gand rk of the Student's t-distribution in (4) are assumed tobe unknown and they are estimated together with thetarget state xk. It is worth mentioning that the Student's t-distribution reduces to the Gaussian distribution as ν-∞and it becomes uninformative when ν-0.

Based on the target dynamics and measurement mod-els (1)–(2), the single target tracking can be formulated asa filtering problem for (3)–(4) in the Bayesian estimationframework. However, it should be pointed out that the useof the Student's t-distribution for measurement noiserenders the Bayesian marginalization analytically intract-able. To overcome this problem, the variational Bayesianapproach can be applied by representing the Student'st-distribution as an infinite mixture of Gaussian terms [26],i.e., the m-dimensional Student's t-distribution can beconsidered as a two-level generation process

Sðz; z;Σ; νÞ ¼Γ

mþ ν

2

� �Γ

ν

2

� �ðνπÞm=2

jΣj1=2 1þ ðz−zÞTΣðz−zÞν

" #−ðmþνÞ=2

¼Z ∞

0N ðz; z; ðsΣÞ−1ÞG s;

ν

2;ν

2

� �ds ð5Þ

where ΓðaÞ ¼ R∞0 ua−1e−u du is the gamma function.

Gðs; κ; θÞ ¼ ðθκ=ΓðκÞÞsκ−1e−θs is the probability density

W. Li et al. / Signal Processing 94 (2014) 48–5650

function of a Gamma distribution in terms of a shapeparameter κ and a inverse scale parameter θ. The scalingparameter s in (5) is a latent variable since it is notapparent in the probability density function of theStudent's t-distribution. This decomposition of the Stu-dent's t-distribution into separate Gaussian and Gammacomponents allows convenient application of the varia-tional Bayesian method.

2.2. PHD filter for multi-target tracking

In the RFS formulation, the collection of individualtargets and measurements are treated as set-valued vari-ables so that the multi-target tracking can be cast in theBayesian filtering. An RFS is a finite set valued randomvariable, in which the cardinality of the RFS is character-ized by a discrete distribution whereas the joint distribu-tion of the elements is described by an appropriatedensity. To model the target appearance and disappear-ance randomly at each time step, the RFS can be used torepresent the set of the target states. As the number ofmeasurements received from the sensor might be alsorandom due to the presence of the clutter and the time-varying number of targets, the RFS can be adopted torepresent the measurement set at each time step. To bespecific, assume that there are nk targets with statesxk;1;…; xk;nk in the surveillance region and mk measure-ments zk;1;…; zk;mk

can be received at time step k, then themulti-target state and measurement can represent by [3]

Xk≜fxk;1;…; xk;nkg⊂X ð6Þ

Zk≜fzk;1;…; zk;mkg⊂Z ð7Þ

where X⊂Rn and Z⊂Rm denote the state and the observa-tion space, respectively. Then, the multi-target trackingcan be formulated a Bayesian filtering as follows: given theset of measurements Z1:k ¼ fZ1;…; Zkg up to time k, theproblem is to find the expectation of the posterior densityfunction pðXkjZ1:kÞ.

Although an optimal Bayesian recursion can be derived interms of the multi-target posterior density functions, thisrecursion involves multiple integrals and the multi-targetposterior density functions are combinatorial, which makesit computationally intractable. To alleviate this intractability,the propagation of the first order moment or the intensityfunction of multi-target posterior density functions can beused, i.e., the PHD filter provides a computationally cheaperalternative to the optimal multi-target Bayesian recursion[6]. As the measurements noise parameters Rk and rk shouldbe estimated as well as the target states, we can obtain anextended version of the standard PHD recursion by aug-menting the state vector ðxk;Rk; rkÞ. Define the posteriorintensity Dk−1jk−1ðxk−1;Rk−1; rk−1jZ1:k−1Þ, the predicted inten-sity Dkjk−1ðxk;Rk; rkjZ1:k−1Þ, and the posterior intensityDkjkðxk;Rk; rkjZ1:kÞ. For simplicity, Dsjtðxs;Rs; rsjZ1:tÞ is shortlydenoted by Dsjt . The predicted intensity can be derived as

Dkjk−1 ¼Z

½pspxðxkjxk−1ÞprðRk; rkjRk−1; rk−1Þ

þSkjk−1ðxk;Rk; rkjxk−1;Rk−1; rk−1Þ��Dk−1jk−1dxk−1dRk−1drk−1 þ Bkðxk;Rk; rkÞ ð8Þ

where ps is the target surviving probability and pxð � j � Þ is thesingle-target transition density (3). prð � j � Þ is the transitiondensity of the parameters Rk and rk. Skjk−1ð � j � Þ and Bkð � Þdenote the intensity of the spawned target RFS and theintensity of the spontaneously birth target RFS, respectively.

After receiving the measurements from the sensor attime step k, the updated intensity can be derived as

Dkjk ¼ ð1−pdÞDkjk−1

þ ∑zk∈Zk

pdpzðzkjxk;Rk; rkÞDkjk−1κkðzkÞ þ

Rpdpzðzkjx′k;R′k; r′kÞDkjk−1 dx′k dR′k dr′k

ð9Þ

where pd is the detection probability, pzð � j � Þ is the single-target measurement likelihood (4), κkð � Þ denotes theintensity of the clutter RFS.

The aim of this paper is to develop a closed-formsolution to the PHD recursion (8)–(9). As the measurementnoise is modeled by the Student's t-distribution, the PHDrecursion does not admit tractable solutions in generaldue to the multi-dimensional integrals. In the followingsection, the variational Bayesian approach is adopted togive approximate expressions.

Before we end this section, the variational Bayesianapproach is briefly reviewed for deriving a recursive filterof (1)–(2).

2.3. Variational Bayesian approach

For linear dynamical systems (1)–(2) with probabilitydensity functions (3)–(4), the aim of the Bayesian filteringis to derive the posterior distribution pðxk;Rk; rkjZkÞ whereZk ¼ fz1;…; zkg is the cumulative set of measurementsup to time step k. The recursive solution consists of thepredictive and the update steps, i.e.,

pðxk;Rk; rkjZk−1Þ ¼Z

pðxk;Rk; rkjxk−1;Rk−1; rk−1Þ

�pðxk−1;Rk−1; rk−1jZk−1Þ dxk−1 dRk−1 drk−1ð10Þ

pðxk;Rk; rkjZkÞ ¼pðzkjxk;Rk; rkÞpðxk;Rk; rkjZk−1ÞR

pðzkjxk;Rk; rkÞpðxk;Rk; rkjZk−1Þ dxk dRk drkð11Þ

It can be observed that the recursion (10)–(11) reverts tothe conventional results when the measurement noise ismodeled as Gaussian density with known covariance matrixand a closed-form solution for the first two moments can beobtained in the form of the Kalman filter. To derive a closed-form expression for (10)–(11) with unknown Student's t-distribution noise parameters, the conjugate prior distribu-tions for Rk and rk are used. To be specific, suppose that theposterior density can be approximated by

pðxk−1;Rk−1; rk−1jZk−1Þ ¼N ðxk; xk−1jk−1; Pk−1jk−1Þ

∏m

l ¼ 1Gðrlk; αk−1jk−1; βk−1jk−1ÞGðrk; γk−1jk−1; ηk−1jk−1Þ ð12Þ

W. Li et al. / Signal Processing 94 (2014) 48–56 51

where the diagonal elements of the matrices Rk and rk areassumed to be distributed according to Gamma distributions.

Then the predictive density can be derived by substi-tuting (12) into (10)

pðxk;Rk; rkjZk−1Þ ¼Z

pxðxkjxk−1ÞprðRk; rkjRk−1; rk−1ÞN ðxk; xk−1jk−1; Pk−1jk−1Þ

� ∏m

l ¼ 1Gðrlk; αk−1jk−1; βk−1jk−1Þ

�Gðrk; γk−1jk−1; ηk−1jk−1Þ dxk−1 dRk−1 drk−1

¼N ðxk; xkjk−1; Pkjk−1Þ ∏m

l ¼ 1Gðrlk; αkjk−1; βkjk−1Þ

Gðrk; γkjk−1; ηk−1jk−1Þ ð13Þ

where a heuristic approach is adopted as in [32] to generatethe predicted parameters of the Gamma distributions

xkjk−1 ¼ Fk−1xk−1jk−1 ð14Þ

Pkjk−1 ¼ Fk−1Pk−1jk−1FTk−1 þ Qk−1 ð15Þ

αkjk−1 ¼ ρααk−1jk−1 ð16Þ

βkjk−1 ¼ ρββk−1jk−1 ð17Þ

γkjk−1 ¼ ργγk−1jk−1 ð18Þ

ηkjk−1 ¼ ρηηk−1jk−1 ð19Þwith ρα; ρβ; ργ and ρη being spread factors taken values inð0;1�.

As shown in [28–32], the latent variable sk in Student'st-distribution in (5) should be incorporated to facilitate thevariational Bayesian approach so that the posterior densitycan be approximated by

pðxk;Rk; rkjZkÞ≃QxðxkÞQRðRkÞQrðrkÞ ð20Þwhere the approximate densities can be formed by mini-mizing the Kullback–Leibler (KL) divergence between theapproximation and the true posterior

KL½QxðxkÞQRðRkÞQrðrkÞ∥pðxk;Rk; rk ZkÞ���

¼Z

QxðxkÞQRðRkÞQrðrkÞ

�lnQxðxkÞQRðRkÞQrðrkÞ

pðxk;Rk; rkjZkÞ

� �dxk dRk drk ð21Þ

Using the calculus of variations for minimizing the KL-divergence with respect to the probability densities whilekeeping the other fixed yields

QxðxkÞ∝expZ

QRðRkÞQrðrkÞln pðxk;Rk; rk; zkjZk−1Þ dRk drk

� �ð22Þ

QRðRkÞ∝expZ

QxðxkÞQrðrkÞln pðxk;Rk; rk; zkjZk−1Þ dxk drk� �

ð23Þ

QrðrkÞ∝expZ

QxðxkÞQRðRkÞln pðxk;Rk; rk; zkjZk−1Þ dxk dRk

� �ð24Þ

Then the approximated densities can be derived by

QxðxkÞ ¼N ðxk; xkjk; PkjkÞ ð25Þ

QRðRkÞ ¼ ∏m

l ¼ 1Gðrlk; αlkjk; βlkjkÞ ð26Þ

QrðrkÞ ¼ Gðrk; γkjk; ηkjkÞ ð27Þ

where

xkjk ¼ xkjk−1 þ Kkðzk−Hkxkjk−1Þ ð28Þ

Pkjk ¼ ðI−KkHkÞPkjk−1 ð29Þ

Kk ¼ Pkjk−1HTk ½HkPkjk−1H

Tk þ ðbskbRkÞ−1�−1 ð30Þ

bRk ¼ diagα1kjkβ1kjk

;…;αmkjkβmkjk

( )ð31Þ

αlkjk ¼ 12 þ αlkjk−1 ð32Þ

βlkjk ¼ βlkjk−1þ12 trfbsk½zk−Hkxkjk−1�½zk−Hkxkjk−1�T þ HkPkjkH

Tk gð33Þ

γkjk ¼ 12 þ γkjk−1 ð34Þ

ηkjk ¼ ηkjk−1−12

1þ Γ′ðakÞΓðakÞ

−ln bk−bsk� �ð35Þ

ak ¼γkjk2ηkjk

þ 12

ð36Þ

bk ¼γkjk2ηkjk

þ trfbRk½zk−Hkxkjk−1�½zk−Hkxkjk−1�T þ HkPkjkHTk g

2

ð37Þ

bsk ¼ akbk

ð38Þ

3. Main results

To derive a closed-form solution to the PHD recursion(8)–(9), the intensities of the birth and spawning RFSs areassumed to be of the following forms:

Bkðxk;Rk; rkÞ ¼ ∑Jb;k

j ¼ 1∏m

l ¼ 1wj

b;kN ðxk;mjb;k; P

jb;kÞ

�Gðrlk;αj;lk ; βj;lk ÞGðrk; γjk; ηjkÞ ð39Þ

Skjk−1ðxk;Rk; rkjxk−1;Rk−1; rk−1Þ

¼ ∑Jp;k

i ¼ 1∏m

l ¼ 1wi

p;kN ðxk; Fix;kxk−1 þ dix;k;Qix;kÞ

�pðrlkjrlk−1Þpðrkjrk−1Þ ð40Þ

where Jb;k, wjb;k, mj

b;k, Pjb;k, αj;lk , βj;lk , γjk and ηjk are given

parameters that determine the shape of the birth intensity.Jp;k, wi

p;k, Fix;k, dix;k and Qix;k are given parameters that

determine the shape of the spawning intensity.

W. Li et al. / Signal Processing 94 (2014) 48–5652

Similar to the Gaussian mixture implementations to thestandard PHD filter [6], the PHD filter (8)–(9) can becarried out as follows.

Prediction step: Given that the posterior intensity attime step k−1 is

Dk−1jk−1 ¼ ∑Jk−1

j ¼ 1∏m

l ¼ 1wj

k−1N ðxk−1;mjk−1jk−1; P

jk−1jk−1Þ

�Gðrlk; αj;lk−1jk−1; βj;lk−1jk−1ÞGðrk; γjk−1jk−1; ηjk−1jk−1Þ ð41Þ

then the predicted intensity is

Dkjk−1 ¼Ds;kjk−1 þ Dp;kjk−1 þ Bkðxk;Rk; rkÞ ð42Þwhere Bkðxk;Rk; rkÞ is given by (39), and

Ds;kjk−1 ¼ ps ∑Jk−1

j ¼ 1∏m

l ¼ 1wj

k−1N ðxk;mjs;kjk−1; P

js;kjk−1Þ

�Gðrlk; αj;ls;kjk−1; βj;ls;kjk−1ÞGðrk; γjs;kjk−1; ηjs;kjk−1Þ ð43Þ

Dp;kjk−1 ¼ ∑Jk−1

j ¼ 1∑Jp;k

i ¼ 1∏m

l ¼ 1wj

k−1wip;kN ðxk;mj;i

p;kjk−1; Pj;ip;kjk−1Þ

�Gðrlk;αj;i;lp;kjk−1; βj;i;lp;kjk−1ÞGðrk; γj;ip;kjk−1; ηj;ip;kjk−1Þ ð44Þ

mjs;kjk−1 ¼ Fk−1m

jk−1jk−1 ð45Þ

Pjs;kjk−1 ¼ Fk−1P

jk−1jk−1F

Tk−1 þ Qk−1 ð46Þ

mj;ip;kjk−1 ¼ Fix;km

jk−1jk−1 þ dix;k ð47Þ

Pj;ip;kjk−1 ¼ Fix;kP

jk−1jk−1ðFix;kÞT þ Qi

x;k ð48Þ

αj;ls;kjk−1 ¼ ρααj;lk−1jk−1 ð49Þ

βj;ls;kjk−1 ¼ ρββj;lk−1jk−1 ð50Þ

γjs;kjk−1 ¼ ργγjk−1jk−1 ð51Þ

ηjs;kjk−1 ¼ ρηηjk−1jk−1 ð52Þ

αj;i;lp;kjk−1 ¼ ρiααj;lk−1jk−1 ð53Þ

βj;i;lp;kjk−1 ¼ ρiββj;lk−1jk−1 ð54Þ

γj;ip;kjk−1 ¼ ρiγγjk−1jk−1 ð55Þ

ηj;ip;kjk−1 ¼ ρiηηjk−1jk−1 ð56Þ

Update step: Given that the predicted intensity at timestep k−1 is

Dkjk−1 ¼ ∑Jkjk−1

j ¼ 1∏m

l ¼ 1wj

kjk−1N ðxk;mjkjk−1; P

jkjk−1Þ

�Gðrlk; αj;lkjk−1; βj;lkjk−1ÞGðrk; γjkjk−1; ηjkjk−1Þ ð57Þ

then the posterior intensity is updated as

Dkjk ¼ ð1−pdÞDkjk−1 þ ∑zk∈Zk

Dd;kðxk; zkÞ ð58Þ

where

Dd;kðxk; zkÞ ¼ ∑Jkjk−1

j ¼ 1∏m

l ¼ 1wj

kðzkÞN ðxk;mjkjkðzkÞ; P

jkjkÞ

�Gðrlk; αj;lkjk; βj;lkjkÞGðrk; γjkjk; ηjkjkÞ ð59Þ

wjkðzkÞ ¼

pdwjkjk−1q

jkðzkÞ

κkðzÞ þ pd∑Jkjk−1t ¼ 1w

tkjk−1q

tkðzkÞ

ð60Þ

qjkðzkÞ ¼ exp −12lnjPj

kjk−1j−12trfðPj

kjk−1Þ−1Pjkjkg

�−12trfðPj

kjk−1Þ−1ðmjkjk−m

jkjk−1Þðm

jkjk−m

jkjk−1ÞT g

þ ∑m

l ¼ 1½αj;lkjk−1 ln βj;lkjk−1−ln Γðαj;lkjk−1Þ þ ðαj;lkjk−1−1Þln αj;lkjk−1�

þ12lnjPj

kjkj− ∑m

l ¼ 1½αj;lkjk ln βj;lkjk−ln Γðαj;lkjkÞ þ ðαj;lkjk−1Þln αj;lkjk�

þγjkjk−1 ln ηjkjk−1−ln Γðγjkjk−1Þ þ ðγjkjk−1−1Þln γj;lkjk−1

−γjkjk ln ηjkjk þ ln ΓðγjkjkÞ−ðγjkjk−1Þln γj;lkjk þ

nþ 22

�ð61Þ

mjkjkðzkÞ ¼mj

kjk−1 þ Kjkðzk−z

jkjk−1Þ ð62Þ

zjkjk−1 ¼Hkmjkjk−1 ð63Þ

Pjkjk ¼ ðI−Kj

kHkÞPjkjk−1 ð64Þ

Kjk ¼ Pj

kjk−1HTk ½HkP

jkjk−1H

Tk þ ðbsjkbRj

kÞ−1�−1 ð65Þ

bRjk ¼ diag

αj;1kjkβj;1kjk

;…;αj;mkjkβj;mkjk

8<:9=; ð66Þ

αj;lkjk ¼ 12 þ αj;lkjk−1 ð67Þ

βj;lkjk ¼ βj;lkjk−1 þ 12 trfbsjk½zk−z jkjk−1�½zk−zjkjk−1�T þ HkP

jkjkH

Tk g ð68Þ

γjkjk ¼ 12 þ γjkjk−1 ð69Þ

ηjkjk ¼ ηjkjk−1−12

1þ Γ′ðajkÞΓðajkÞ

−ln bjk−bsjk" #

ð70Þ

ajk ¼γjkjk2ηjkjk

þ 12

ð71Þ

bjk ¼γjkjk2ηjkjk

þtrfbRj

k½zk−zjkjk−1�½zk−zjkjk−1�T þ HkP

jkjkH

Tk g

2ð72Þ

bsjk ¼ ajkbjk

ð73Þ

Remark 1. Note that the computation of qjkðzkÞ in (61) isdifferent from that of the GM-PHD filter due to the applicationof the Student's t-distribution. In fact, qjkðzkÞ is the measure-ment likelihood function N ðzk; zjkjk−1;HkP

jkjk−1H

TK þ RkÞ in [6]

jkÞ

2000 2500 3000 3500 4000 4500 5000 5500 600−8000

−7000

−6000

−5000

−4000

−3000

−2000

−1000

X coordinate (m)

Y c

oord

inat

e (m

)

Fig. 1. True and estimated target trajectories.

0 10 20 30 40 50 60 70 80 90 1000

100

200

300

400

500

600

700

800

Time (s)

OS

PA

(c=1

000,

p=2

)

50 60 70 80 90 10035

40

45

50

55

Fig. 2. Performance comparison with respect to OSPA when rk ¼ 10.

0 10 20 30 40 50 60 70 80 90 100

100

200

300

400

500

600

700

800

Time (s)

50 60 70 80 90 10030

35

40

45

50

OS

PA

(c=1

000,

p=2

)

Fig. 3. Performance comparison with respect to OSPA when rk ¼ 1000.

W. Li et al. / Signal Processing 94 (2014) 48–56 53

while the variational Bayesian approach is used to calculateqjkðzkÞ in (61) (see Appendix for detailed derivations).

Remark 2. As the number of Gaussian componentsincreases without bound as time progresses, the pruningscheme is required after the updated step and a simplepruning procedure has been provided by truncating com-ponents that have weak weights [6]. Another feature of theproposed filter is that the nonlinear target dynamics andmeasurement models can be addressed by using nonlinearfiltering techniques such as the extended Kalman filter(EKF) and the unscented Kalman filter (UKF).

4. Simulation results

Consider a two-dimensional scenario with an unknownand time-varying number of targets. The target state isdenoted by xk ¼ ðpx;k; _px;k; py;k; _py;kÞT , where ðpx;k; py;kÞ andð _px;k; _py;kÞ denote the target position and velocity, respec-tively. The target dynamics is described by the nearlyconstant velocity model

xk ¼

1 T 0 00 1 0 00 0 1 T

0 0 0 1

2666437775xk−1 þwk−1 ð74Þ

where T¼1 is the sampling period, and wk−1 is zero-meanwhite Gaussian noise with covariance matrix

Qk−1 ¼

T4

4T3

2 0 0T3

2 T2 0 0

0 0 T4

4T3

2

0 0 T3

2 T2

26666664

37777775 ð75Þ

The range and the bearing measurements are gener-ated by a radar

zk ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðpx;k−sxÞ2 þ ðpy;k−syÞ2

qarctan½ðpx;k−sxÞ=ðpy;k−syÞ�

24 35þ vk ð76Þ

where ðsx; syÞ is the location of the radar. The measurementnoise vk is assumed to be Student's t-distribution withRk ¼ diagf0:05;28:66g and rk¼10. In the simulations, theradar is located at ð4000;−8000Þ. Specially, the EKF is usedto handle nonlinear measurements.

The number of targets is time-varying due to targetappearance and disappearance in the scene at any time.For simplicity, the target spawn is not considered in thisexample. The intensity of the spontaneous birth RFS is

Bkðxk;Rk; rkÞ ¼ 0:1 ∑3

j ¼ 1∏2

l ¼ 1N ðxk;mj

γ;k; Pjγ;kÞGðrlk; αj;lk ; βj;lk ÞGðrk; γjk; η

ð77Þ

wherem1γ;k ¼ ð4000;0;−5000;0ÞT ,m2

γ;k ¼ ð2000;0;−4000;0ÞT ,m3

γ;k ¼ ð3000;0;−6000;0ÞT , Pjγ;k ¼ diagf106; 104;106;104g,

αj;lk ¼ 10, βj;lk ¼ 4000, γjk ¼ 100 and ηjk ¼ 3 ðj¼ 1;2;3; l¼ 1;2Þ.The target trajectories are shown in Fig. 1. Specifically,

target 1 starts at time k¼1 with initial position at ð4000;−5000Þ and ends at time k¼100; target 2 starts at time

k¼5 with initial position at ð2000;−4000Þ and ends attime k¼85; target 3 starts at time k¼25 with initialposition at ð3000;−6000Þ and ends at time k¼90; target

W. Li et al. / Signal Processing 94 (2014) 48–5654

4 starts at time k¼10 with initial position at ð2000;−4000Þand ends at time k¼100.

In the simulations, the survival probability and thedetection probability are set to ps¼0.99 and pd¼0.98,respectively. The spreading factor ρα ¼ ρβ ¼ ργ ¼ ρη ¼ ρ

10 20 30 40 50 60 70 80 90 1000

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

Time (s)

Est

imat

ed ta

rget

num

ber

Fig. 4. True and estimated target numbers against time.

0.6 0.65 0.7 0.75 0.845

50

55

60

65

70

75

80

85

90

95

The value of

Ave

rage

of O

SP

A w

.r.t.

Tim

e

Fig. 5. Averaged OSPA w

in the predicted intensity is taken to be 0.75 and twofixed point iteration steps is adopted to generate theupdated intensity. As in [6], the pruning threshold is takenas TTh ¼ 0:001, the merging threshold UTh ¼ 5, the weightthreshold wTh ¼ 0:5 and the maximum number of Gaussianterms Jmax ¼ 100. To compare the tracking performance, thecriterion known as optimal subpattern assignment (OSPA)metric is used for performance evaluation since the OSPAmetric captures the differences in cardinality and individualelements between two finite sets [33].

For simplicity of notation, the proposed filter is shortlydenoted by VB-PHD-EKF in the figures. The positionestimates of the VB-PHD-EKF for one trial are shown inFig. 1, the simulation results suggest that the proposedVB-PHD-EKF filter can provide accurate tracking perfor-mance for almost all the time. In Fig. 2, the averages of theOSPA distance over 100 Monte Carlo runs with p¼2 andc¼1000 are presented versus time. It can be shown thatthe GM-PHD-EKF and the VB-PHD-EKF filters produceaverage errors of approximately 64 and 52, respectively.These results also suggest that the VB-PHD-EKF filteroutperforms the GM-PHD-EKF filter. This is expectedsince the measurement noise is modeled as heavy-tailedStudent's t-distribution. However, as shown in Fig. 3,the performance of the proposed VB-PHD-EKF is almost

0.85 0.9 0.95 1

ρ

ith respect to ρ.

W. Li et al. / Signal Processing 94 (2014) 48–56 55

identical to that of the GM-PHD-EKF when the degree offreedom of Student's t-distribution is taken to be 1000.This is reasonable since the Student's t-distributionreduces to the Gaussian distribution as the degree offreedom rk-∞. In addition, it can be observed that highpeaks of the OSPA are avoided for the proposed VB-PHD-EKF when new targets appear. This might due to the factthat the updated weight wj

kjk is calculated in different waysand the correct target state with low weight has beendeleted or merged for the GM-PHD-EKF in the pruning andmerging step. As shown in Fig. 4, the proposed filter canderive more accurate target numbers, especially whennew target emerges at time steps 5 and 10. To illustratethe choice of the spreading factor ρ in the predictedintensity, the averaged OSPA with different values of ρ isshown in Fig. 5. It is suggested that it is better to choose ρin the interval ½0:7;0:9�.

To evaluate the computational requirement of the pro-posed algorithm, the averaged CPU time is computed inMATLAB 7.1 on a 2.80-GHz 4 CPU Pentium-based computeroperating under Windows XP (Professional). The proposedVB-PHD-EKF consumed approximately 3.17 s per samplerun over 100 time steps and the GM-PHD-EKF consumedapproximately 2.84 s. This is due to the fact that two fixedpoint iteration steps are used and a complex computation ofqjkðzkÞ is required to generate the updated intensity.

5. Conclusion

In this paper, the PHD filter is applied for tracking anunknown and time-varying number of targets in glintnoise environment. Based on a Student's t-distributionmodel for the glint noise statistics, the PHD filter isextended by augmenting the target state and the noiseparameters and a novel implementation to the extendedPHD filter is developed by using the variational Bayesianapproach. The proposed filter is carried out by represent-ing the predicted and the updated intensities as mixturesof Gaussian–Gamma terms. It is expected that the pro-posed approach can be used to the cardinalized PHD(CPHD) filter to improve the accuracy of multi-target stateestimates. Simulation results show that the proposed filteroutperforms the GM-PHD filter for glint noise.

Acknowledgments

This work was supported by the National BasicResearch Program of China (973 Program, 2012CB821200,2012CB821201), the NSFC (61203044, 61134005,60921001, 90916024, 91116016) and the Beijing NaturalScience Foundation (4132040).

Appendix A

Derivation of qjkðzkÞ: By substituting (57) into (9), we canget

qjkðzkÞ ¼Z

pzðzkjxk;Rk; rkÞN ðxk;mjkjk−1; P

jkjk−1Þ

� ∏m

l ¼ 1Gðrlk; αj;lkjk−1; βj;lkjk−1ÞGðrk; γjkjk−1; ηjkjk−1Þ dxk dRk drk

ðA:1ÞIt should be pointed out that the above integral cannot

be computed explicitly and the variational Bayesianapproach provides a way to approximate it. To be specific,by comparing ((11) and A.1), qjkðzkÞ can be treated as thepredictive likelihood function similar to the denominatorin (9). By variational Bayesian approach, the predictivelog-likelihood function can be written as [26]

ln qjkðzkÞ ¼ KL½QxðxkÞQRðRkÞQrðrkÞ∥pðxk;Rk; rkjZkÞ� þ Lj ðA:2Þ

where

Lj ¼Z

QxðxkÞQRðRkÞQrðrkÞlnpðxk;Rk; rk; zkjZk−1ÞQxðxkÞQRðRkÞQrðrkÞ

� �dxk dRk drk

¼ Efln pðxk;Rk; rk; zkjZk−1Þg−Efln QxðxkÞQRðRkÞQrðrkÞgðA:3Þ

As the first term on the right hand side of (A.2) can beminimized by the variational iteration, we can get

qjkðzkÞ≃expðLjÞ ðA:4ÞTo derive Lj explicitly, we have

Lj ¼ Efln pðxk;Rk; rk; zkjZk−1Þg−Efln QxðxkÞQRðRkÞQrðrkÞg¼ Efln pðxk;Rk; rkjZk−1Þpðzkjxk;Rk; rk;k−1Þg

−Efln QxðxkÞQRðRkÞQrðrkÞg

¼ E ln N ðxk;mjkjk−1; P

jkjk−1Þ ∏

m

l ¼ 1Gðrlk; αj;lkjk−1; βj;lkjk−1Þ

(�Gðrk; γjkjk−1; ηjkjk−1ÞSðzk;Hkxk;Rk; rkÞ

o−E ln N ðxk;mj

kjk; PjkjkÞ ∏

m

l ¼ 1Gðrlk; αj;lkjk; βj;lkjkÞGðrk; γjkjk; ηjkjkÞ

( )

¼ E ln N ðxk;mjkjk−1; P

jkjk−1Þ þ ∑

m

l ¼ 1ln Gðrlk; αj;lkjk−1; βj;lkjk−1Þ

(ðA:5Þ

þln Gðrk; γjkjk−1; ηjkjk−1Þþln Sðzk;Hkxk;Rk; rkÞ−ln N ðxk;mj

kjk; PjkjkÞ

− ∑m

l ¼ 1ln Gðrlk; αj;lkjk; βj;lkjkÞ−ln Gðrk; γjkjk; ηjkjkÞ

)

¼ −n2ln 2π−

12lnjPj

kjk−1j−12trfðPj

kjk−1Þ−1Pjkjkg

−12trfðPj

kjk−1Þ−1ðmjkjk−m

jkjk−1Þðm

jkjk−m

jkjk−1ÞT g

þ ∑m

l ¼ 1½αj;lkjk−1 ln βj;lkjk−1−ln Γðαj;lkjk−1Þ

þðαj;lkjk−1−1Þln αj;lkjk−1�−αj;lkjk−1

þγjkjk−1 ln ηjkjk−1−ln Γðγjkjk−1Þ þ ðγjkjk−1−1Þln γj;lkjk−1−γjkjk−1

þn2ln 2π þ 1

2lnjPj

kjkj þn2− ∑

m

l ¼ 1½αj;lkjk ln βj;lkjk

−ln Γðαj;lkjkÞ þ ðαj;lkjk−1Þln αj;lkjk�þαj;lkjk þ γjkjkln ηjkjk−ln ΓðγjkjkÞ þ ðγjkjk−1Þln γj;lkjk þ γjkjk

o

W. Li et al. / Signal Processing 94 (2014) 48–5656

¼ −12lnjPj

kjk−1j−12trfðPj

kjk−1Þ−1Pjkjkg

−12trfðPj

kjk−1Þ−1ðmjkjk−m

jkjk−1Þðm

jkjk−m

jkjk−1ÞT g

þ ∑m

l ¼ 1½αj;lkjk−1 ln βj;lkjk−1−ln Γðαj;lkjk−1Þ þ ðαj;lkjk−1−1Þ ln αj;lkjk−1�

þ12lnjPj

kjkj− ∑m

l ¼ 1½αj;lkjk ln βj;lkjk−ln Γðαj;lkjkÞ þ ðαj;lkjk−1Þln αj;lkjk�

þγjkjk−1 ln ηjkjk−1−ln Γðγjkjk−1Þ þ ðγjkjk−1−1Þln γj;lkjk−1

−γjkjk ln ηjkjk þ ln ΓðγjkjkÞ−ðγjkjk−1Þln γj;lkjk þ

nþ 22

ðA:6Þ

Then, taking exponentials on both sides of (A.5) yieldsqjkðzkÞ in (61).

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