gaussian mixture phd filter for multi-sensor multi-target tracking with registration errors

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Signal Processing

Signal Processing 93 (2013) 86–99

0165-16

http://d

n Corr

E-m

ymjia@

yufs@hp

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

Gaussian mixture PHD filter for multi-sensor multi-target trackingwith registration errors

Wenling Li a,n, Yingmin Jia a,b, Junping Du c, Fashan Yu 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), Ministry of Education, SMSS, Beihang University (BUAA), Beijing 100191, Chinac Beijing Key Laboratory of Intelligent Telecommunications Software and Multimedia, School of Computer Science and Technology, Beijing University of Posts and

Telecommunications, Beijing 100876, Chinad School of Electrical Engineering and Automation, Henan Polytechnic University, Jiaozuo 454000, Henan, China

a r t i c l e i n f o

Article history:

Received 14 February 2012

Received in revised form

11 June 2012

Accepted 14 June 2012Available online 11 July 2012

Keywords:

Multi-target tracking

Probability hypothesis density filter

Registration errors

84/$ - see front matter & 2012 Elsevier B.V. A

x.doi.org/10.1016/j.sigpro.2012.06.030

esponding author. Tel.: þ86 10 8233 8683; fax:

ail addresses: [email protected] (W. Li),

buaa.edu.cn (Y. Jia), [email protected] (J. D

u.edu.cn (F. Yu).

a b s t r a c t

This paper studies the problem of multi-sensor multi-target tracking with registration

errors in the formulation of random finite sets. The probability hypothesis density

(PHD) recursion is applied by introducing the dynamics of the translational measure-

ment bias into the associated intensity functions. Under the linear Gaussian assump-

tions on the bias dynamics, the Gaussian mixture implementation is used to give

closed-form expressions. As the target state and the translational measurement bias are

coupled through the likelihood in the update step, a two-stage Kalman filter is adopted

to approximate the tractable form, which leads to a substantial reduction in computa-

tional complexity. Two numerical examples are provided to verify the effectiveness of

the proposed filter.

& 2012 Elsevier B.V. All rights reserved.

1. Introduction

Registration errors compensation has been an impor-tant issue in multi-sensor data fusion systems regardlessof the sensor measurements are processed in the centra-lized or distributed fashion. There are many kinds ofsensor biases such as the translational bias in the statespace, the rotational bias in the state space, the transla-tional bias in the measurement space, the rotational biasin the measurement space and both translational androtational biases [1]. The estimation of unknown transla-tional measurement biases has received great attention[2], and it is this problem that we address in this paper.Note that it is vital to estimate these measurement biases

ll rights reserved.

þ86 10 8231 6100.

u),

as accurately as possible so that the multi-sensor measure-ments can be referenced to a common tracking coordinateframe [3]. To resolve this problem, many approaches havebeen proposed such as the least squares [4], the maximumlikelihood [5] and the Kalman filtering [6], of which theKalman filtering has been an attractive method for itsefficiency.

By stacking the sensor biases and the target states in asingle vector, an augmented state Kalman filter (ASKF)can be applied to derive the bias estimates. However, theimplementation of the ASKF might not be computation-ally feasible and numerical problems may arise especiallyfor ill-conditioned systems [3]. To alleviate this disadvan-tage, for linear dynamics and measurement models, Fried-land in [7] proposed a two-stage estimator by decouplingthe bias estimates from target states, and it has beenshown in [8] that this estimator is equivalent to the ASKFsolution when a particular relationship between theinitial parameters of two filters is satisfied. Similar ideashave been adopted in the literature [9–13] to developvarious two-stage estimators. It should be mentioned that

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W. Li et al. / Signal Processing 93 (2013) 86–99 87

the problem of measurement source uncertainty is notaddressed in most of the existing approaches, which isoften encountered in multi-target tracking. Although thedata association techniques (e.g., the joint probabilisticdata association (JPDA) and the multiple hypothesistracking (MHT)) can be incorporated, the associationresults might be bad because of the effect of sensor biases.

Recently, the finite set statistics (FISST) theory hasbeen used to tackle the multi-target tracking problemwhich avoids the data association [14]. In the frameworkof FISST, the target states and measurements are modeledas two different random finite sets (RFSs), and as aconsequence, the problem of tracking an unknown andtime-varying number of targets in the clutter environ-ment can be addressed in a natural manner. Moreover, themulti-target tracking can be formulated in a rigorousBayesian framework by constructing the multi-targettransition density and multi-target likelihood function.However, the optimal multi-target Bayes filter is generallyintractable due to the existence of multiple set integralsand the combinatorial nature of the multi-target densities.To alleviate this intractability, the probability hypothesisdensity (PHD) filter has been proposed as a first ordermoment approximation to the multi-target posterior den-sity [15]. It should be pointed out that the PHD recursionstill requires solving multi-dimensional integrals.

There are mainly two approaches to implement thePHD recursion, including the sequential Monte Carlo(SMC) [16,17] and the Gaussian mixture (GM) [18]. Inthe SMC-PHD filter, a large number of particles are used toapproximate the multi-dimensional integrals, and there-fore the main drawback is the high computational burden.In addition, some clustering techniques are requiredto extract the target state estimates, which might beoften unreliable. To overcome these disadvantages, theGM-PHD filter was developed for linear Gaussian targetdynamics and Gaussian birth model, in which theweights, means and covariance matrices are propagatedanalytically by the Kalman filter (KF). Specially, thenonlinear Kalman filter counterparts can be directlyemployed to deal with nonlinear target dynamics andmeasurement models. The convergence properties of twoimplementations were analyzed in [17,19]. As shown in[19], the GM-PHD filter can approximate the true PHDfilter to any desired degree of accuracy under the linearGaussian assumption of the dynamic model. Similar resultshave been extended to handle jump Markov models fortracking multiple maneuvering targets [20–23]. The particleand Gaussian mixture techniques have also been used toderive PHD smoothers [24–31]. In [32], the GM-PHD filter isextended to multi-sensor tracking system and the targetstate estimates are obtained sequentially at each sensor.However, the sensor registration errors are neglected. In[33], the problem of multi-sensor mis-registration witharbitrary biases has been addressed and a simplified ver-sion has been investigated in [34]. In [35], the translationalmeasurement mis-registration is investigated in the PHDrecursion and the multi-sensor SMC implementation isdeveloped. Nevertheless, the proposed filter inherits thedisadvantages of the SMC such as the high computationalcost and unreliable clustering.

In this paper, we attempt to apply the GM-PHD filter toaddress the problem of multi-sensor multi-target trackingwith registration errors. For clarity, we consider the lineartarget dynamics and measurement models in this work.Under the linear Gaussian assumptions on the transla-tional measurement bias dynamics, the PHD recursion isapplied by introducing the bias dynamics into the asso-ciated intensity functions. Then the Gaussian mixtureimplementation is used to give closed-form expressions.As the target state and the translational measurementbiases are coupled through the likelihood in the updatestep, a suboptimal two-stage Kalman filter is adopted tojointly estimate the target state and the biases, whichleads to a substantial reduction in computational com-plexity. Simulation results show that the proposed filterperforms better than the standard GM-PHD filter withoutincorporating registration errors.

The rest of this paper is organized as follows. The PHDrecursion for multi-sensor multi-target tracking withregistration errors is given in Section 2. The Gaussianmixture implementation to the PHD recursion is pre-sented in Section 3. In Section 4, two numerical examplesare provided to illustrate the effectiveness of the GM-PHDfilter. Conclusion is drawn in Section Appendix A.

2. PHD recursion for multi-sensor multi-target tracking

For multi-sensor tracking systems, the measurementsreceived from each sensor should be transformed into acommon coordinate system for merging, and the registra-tion errors are often encountered. This is also referred tothe estimation of unknown translational measurementbiases. In this paper, we consider the following lineartarget dynamics and measurement models:

xk ¼ Fk�1xk�1þwk�1 ð1Þ

zlk ¼Hl

kxkþblkþvl

k, l¼ 1;2, . . . ,L ð2Þ

where xk 2 Rn and zl

k 2 Rm denote the target state and the

lth sensor measurement, respectively. L is the number ofsensors. Fk�1 and Hk

lare the state transition matrix and

measurement matrix. wk�1 and vkl

are zero-mean whiteGaussian process noise and measurement noise withcovariance matrices Qx

k�1 and Rkl, respectively. bk

lis the

translational measurement bias of the lth sensor.In the multi-target tracking scenario, targets might

appear and disappear randomly. Thus, it is natural tomodel the set of target states as an RFS. Similarly, due tothe presence of the clutter and the time-varying numberof targets, the number of measurements received fromeach sensor might be random, the RFS can be also used tocharacterize the property of measurements. To be specific,an RFS X is a finite set valued random variable, which canbe described by a discrete probability distribution and afamily of joint probability densities. Specially, the discretedistribution characterizes the cardinality of X whereas anappropriate density characterizes the joint distribution ofthe elements in X. Assume that there are nk targets with

W. Li et al. / Signal Processing 93 (2013) 86–9988

states xk,1, . . . ,xk,nkin the surveillance region and mk

l

measurements zlk,1, . . . ,zl

k,mlk

can be received from the lthsensor at time step k, then the multi-target state andmulti-target measurement can be represented as [14]

Xk9fxk,1, . . . ,xk,nkg � X ð3Þ

Zlk9fz

lk,1, . . . ,zl

k,mlk

g � Zl ð4Þ

where X � Rn and Zl � Rp denote the state and theobservation space, respectively. Then, the multi-targettracking can be formulated as a filtering process asfollows: given the set of measurements Z1:k ¼ fZ

11:k,

. . . ,ZL1:kg from all sensors up to time k, the problem is to

find the expectation of the posterior density functionpðXk9Z1:kÞ.

By using the finite set statistics theory, an optimalBayesian recursion can be obtained in terms of the multi-target posterior density functions. However, this recur-sion involves multiple integrals and the multi-targetposterior density functions are combinatorial, whichmakes it computationally intractable. To alleviate thisintractability, inspired by the single-target trackingapproaches, the propagation of the statistical momentsassociated with the posterior densities is adopted. ThePHD recursion, which propagates the first order momentor the intensity function of multi-target random finitesets, provides a computationally cheaper alternative tothe optimal multi-target Bayesian recursion [18]. Sincethe measurement biases should be estimated as well asthe target states, similar to the derivation of the standardPHD recursion, we can obtain an extended version byaugmenting the state vector ½xT

k ,bTk �

T , where bk ¼ ½ðb1k Þ

T ,. . . ,ðbL

kÞT�T . Define the posterior intensity nk�19k�1ðxk�1,

bk�19Z1:k�1Þ at time k�1, the predicted intensity nk9k�1

ðxk,bk9Z1:k�1Þ, and the posterior intensity nk9kðxk,bk9Z1:kÞ.For simplicity, ns9tðxs,bs9Z1:tÞ is shortly denoted by ns9t . Thepredicted intensity can be derived as

nk9k�1ðxk,bkÞ ¼

Z½psf xðxk9xk�1Þf bðbk9bk�1Þþbk9k�1ðxk,bk9xk�1,bk�1Þ�

�nk�19k�1ðxk�1,bk�19Z1:k�1Þ dxk�1dbk�1þgkðxk,bkÞ ð5Þ

where ps is the target surviving probability. f xð�9�Þ is thesingle-target transition density. f bð�9�Þ is the bias transi-tion density. bk9k�1ð�9�Þ and gkð�Þ denote the intensity of thespawned target RFS and the intensity of the sponta-neously birth target RFS, respectively. Suppose that thebiases are independent between sensors and thedynamics of each bias is Markovian, then

f bðbk9bk�1Þ ¼YL

l ¼ 1

f lbðb

lk9b

lk�1Þ ð6Þ

where f lbðb

lk9b

lk�1Þ is the bias transition density of the lth

sensor. As in [35], the bias bkl

can be modeled as a firstorder Gauss–Markov process, i.e.,

f bðblk9b

lk�1Þ ¼N ðb

lk; b

lk�1,Bl

k�1Þ ð7Þ

After receiving the measurements from all sensors attime k, the updated intensity can be derived as in [35]

nk9k ¼F LkðZ

Lk9xk,bL

kÞ � � �F1k ðZ

1k9xk,b1

k Þnk9k�1 ð8Þ

where pdl

is the detection probability of the lth sensor,hlð�9�Þ is the single-target measurement likelihood of the

lth sensor, klkð�Þ denotes the intensity of the clutter RFS of

the lth sensor, and

F lkðZ

lk9xk,bl

kÞ ¼ 1�pld

þX

zk2Zk

pldhlðzk9xk,bkÞ

klkðzkÞþ

Rpl

dhlðzk9x0k,b0kÞnk9k�1ðx

0k,b0k9Z

1:L1:k�1Þdx0kdb0k

ð9Þ

For the multi-sensor PHD filter, both the iterated-corrector PHD filter and the PHD filter defined by (8)have been investigated in [36]. Specially, the quality ofthe iterated PHD is affected by the sensor ordering atmoderate and low probability of detection [37]. However,the improvement in performance is minor at a highprobability of detection for the product multi-sensorPHD filter [36]. In addition, the approximation in (8) isvalid only for relatively large numbers of targets [38].Note that the PHD recursion given by (5)–(8) operates onthe single-target state space and avoids the explicitproblem of data association. However, it does not admittractable solutions in general due to the multi-dimen-sional integrals. In the following section, the Gaussianmixture implementation is used to give closed-formexpressions.

3. GM-PHD filter with registration errors

Before we present the GM-PHD filter for multi-sensormulti-target tracking with registration errors, the follow-ing lemmas are adopted for further development [18].

Lemma 1. Given F, d, Q, m, and P of compatible dimensions

and that Q and P are positive definite, thenZN ðx; Fxþd,Q ÞN ðx;m,PÞ dx¼N ðx; Fmþd,QþFPFT

Þ

ð10Þ

Lemma 2. Given H, R, m, and P of compatible dimensions

and that R and P are positive definite, then

N ðz;Hx,RÞN ðx;m,PÞ ¼ qðzÞN ðx;m,PÞ ð11Þ

where

qðzÞ ¼N ðz;Hm,RþHPHTÞ ð12Þ

m ¼mþKðz�HmÞ ð13Þ

P ¼ ðI�KHÞP ð14Þ

K ¼ PHTðHPHT

þRÞ�1ð15Þ

To derive Gaussian mixture implementations of thePHD recursion, the intensities of the birth and spawningrandom finite sets are assumed to be of the followingforms:

gkðxk,bkÞ ¼XJg,k

j ¼ 1

YL

l ¼ 1

wjg,kN ð½xk; b

lk�; ½m

jg,k; b

j,lg,k�,½P

jg,k;B

j,lg,k�Þ

ð16Þ


bk9k�1ðxk,bk9xk�1,bk�1Þ

¼XJb,k

i ¼ 1

YL

l ¼ 1

wib,kN ð½xk; b

lk�; ½F

ix,kxk�1þdi

x,k; Fi,lb,kbl

k�1�,½Qix,k;Q

i,lb,k�Þ

ð17Þ

where Jg,k, wjg,k, mj

g,k , bjg,k, Bj

g,k and Pjg,k are given para-

meters that determine the shape of the birth intensity.Jb,k, wi

b,k, Fix,k, di

x,k Fib,k, Qi

x,k and Qib,k are given parameters

that determine the shape of the spawning intensity. It isworth mentioning here that the intensity of the transla-tional measurement biases are introduced into the inten-sities of the birth and spawning random finite sets,whereas this is not done in the standard PHD filter [18]and the extended PHD filter in [35].

The extended PHD recursion (5)–(8) can be carried outas follows (see Appendix for the detailed derivations).

Prediction step: Given that the posterior intensity is aGaussian mixture

nk�19k�1 ¼XJk�1

j ¼ 1

YL

l ¼ 1

wjk�1N ð½xk�1;b

lk�1�; ½m

jk�19k�1

; bj,lk�19k�1

�,

½Pjk�19k�1

;Bj,lk�19k�1

�Þ ð18Þ

then the predicted intensity is also a Gaussian mixturewith the form

nk9k�1 ¼ ns,k9k�1þnb,k9k�1þgkðxk,bkÞ ð19Þ

where gkðxk,bkÞ is given by (16), and

ns,k9k�1 ¼ ps

XJk�1

j ¼ 1

YL

l ¼ 1

wjk�1N ð½xk; b

lk�; ½m

js,k9k�1

; bj,ls,k9k�1

�,

½Pjs,k9k�1

;Bj,ls,k9k�1

�Þ ð20Þ

nb,k9k�1 ¼XJk�1

j ¼ 1

XJb,k

i ¼ 1

YL

l ¼ 1

wjk�1wi

b,kN ðxk;mj,ib,k9k�1

,Pj,ib,k9k�1

Þ

�N ðblk; b

j,i,lb,k9k�1

,Bj,i,lb,k9k�1

Þ ð21Þ

mjs,k9k�1

¼ Fk�1mjk�19k�1

ð22Þ

Pjs,k9k�1

¼ Fk�1Pjk�19k�1

FTk�1þQx

k�1 ð23Þ

mj,ib,k9k�1

¼ Fix,kmj

k�19k�1þdi

x,k ð24Þ

Pj,ib,k9k�1

¼ Fix,kPj

k�19k�1ðFi

x,kÞTþQi

x,k ð25Þ

bj,ls,k9k�1

¼ bj,lk�19k�1

ð26Þ

Bj,ls,k9k�1

¼ Bj,lk�19k�1

þBlk�1 ð27Þ

bj,i,lb,k9k�1

¼ Fi,lb,kbj,l

k�19k�1ð28Þ

Bj,i,lb,k9k�1

¼ Fi,lb,kBj,l

k�19k�1ðFi,l

b,kÞTþQi,l

b,k ð29Þ

Update step: Given that the predicted intensity can berepresented as the form of

nk9k�1 ¼XJk9k�1

j ¼ 1

YL

l ¼ 1

wjk9k�1

N ð½xk; blk�; ½m

jk9k�1

;bj,lk9k�1�,½Pj

k9k�1;Bj,l

k9k�1�Þ

ð30Þ

then the posterior intensity is updated sequentially foreach sensor as

n1k9k ¼ ð1�p1

dÞnk9k�1þX

zk2Z1k

n1d,kðxk; zkÞ ð31Þ

nlk9k ¼ ð1�pl

dÞnl�1k9k þ

Xzk2Zl

k

nld,kðxk; zkÞ, l¼ 2, . . . ,L ð32Þ

where

n1d,kðxk; zkÞ ¼

XJk9k�1

j ¼ 1

YL

l ¼ 2

wj,1k ðzkÞN ðxk;m

j,1k9kðzkÞ,P

j,1k9kÞN ðbk; b

j,1k9k,Bj,1

k9kÞ

�N ðblk; b

j,lk9k�1

,Bj,lk9k�1Þ ð33Þ

wj,1k ðzkÞ ¼

p1dwj

k9k�1qj,1

k ðzkÞ

k1k ðzÞþp1

d

PJk9k�1

t ¼ 1 wtk9k�1

qt,1k ðzkÞ

ð34Þ

qj,1k ðzkÞ ¼N ðzk; z

j,1k9k�1,H1

k Pjk9k�1ðH1

k ÞTþR1

kþBj,1k9k�1Þ ð35Þ

mj,1k9kðzkÞ ¼mj

k9k�1þKj,1

k ðzk�zj,1k9k�1Þ ð36Þ

zj,1k9k�1 ¼H1

kmjk9k�1þbj,1

k9k�1ð37Þ

Pj,1k9k ¼ ðI�Kj,1

k H1k ÞP

jk9k�1

ð38Þ

Kj,1k ¼ Pj

k9k�1ðH1

k ÞT½H1

k Pjk9k�1ðH1

k ÞTþR1

k ��1 ð39Þ

bj,1k9k ¼ bj,1

k9k�1þBj,1

k9k�1ðzk�z

j,1k9k�1Þ ð40Þ

Bj,1k9k¼ Bj,1

k9k�1�Bj,1

k9k�1½H1

kPjk9k�1ðH1

k ÞTþBj,1

k9k�1þR1

k ��1Bj,1

k9k�1

ð41Þ

Remark 1. It should be pointed out that the proposedtwo-stage Kalman filtering process is not optimal in theupdated step. This idea is used to maintain the same formof the updated posterior intensity at each time step, i.e.,the state estimates mj

k9kand the bias estimates bj,l

k9k can beexpressed in a decoupled form. Another feature of theproposed filter is that the nonlinear target dynamics andmeasurement models can be addressed by using non-linear filtering techniques such as the extended Kalmanfilter (EKF), the unscented Kalman filter (UKF) and thecubature Kalman filter (CKF).

Remark 2. It is worth noting that the pruning scheme isrequired after the updated step since the number ofGaussian components increases without bound as timeprogresses. A simple pruning procedure has been pro-vided by truncating components that have weak weights


to mitigate this problem. Interested readers are referredto [18] for the details.

Remark 3. It is expected that the filtering with sensorregistration technique can be extended to the Gaussianmixture cardinalized PHD (GM-CPHD) filter to improvethe accuracy of multi-target state estimates. Another pos-sible application is to implement the cardinality balancedmulti-target multi-Bernoulli (CBMeMBer) recursion asshown in [39].

4. Simulation results

In this section, we present two numerical examples,including the non-maneuvering and maneuvering multi-target tracking, to illustrate the effectiveness of the proposedfilter and compare its performance with that of the standardGM-PHD filter without introducing sensor biases.

Example 1. Consider a two-dimensional scenario with anunknown and time-varying number of targets. The state isdenoted by xk ¼ ðpx,k, _px,k,py,k, _py,kÞ

T , where ðpx,k,py,kÞ

represents the Cartesian coordinates in the horizontalplane and ð _px,k, _py,kÞ represents its velocities. The targetdynamics is described by the following nearly constantvelocity model:

xk ¼

1 T 0 0

0 1 0 0

0 0 1 T

0 0 0 1

26664

37775xk�1þwk�1 ð42Þ

where T is the sampling period, and wk�1 is zero-meanwhite Gaussian noise with covariance

Qk ¼ s2

T4

4T3

2 0 0

T3

2 T2 0 0

0 0 T4

4T3

2

0 0 T3

2 T2

26666664

37777775

ð43Þ

with s¼ 5.

Two sensors are used to generate the range and thebearing measurements

zlk ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðpx,k�sl

xÞ2þðpy,k�sl

yÞ2

q

arctan½ðpx,k�slxÞ=ðpy,k�sl

yÞ�

264

375þbl

kþvlk, l¼ 1;2 ð44Þ

where ðslx,sl

yÞ is the location of the lth sensor. bkl

is thetranslational measurement bias and the measurementnoise vk

lis assumed to be zero-mean white Gaussian with

Rkl. In the simulations, the first sensor is located at (2000,

15 000) m and the second sensor is located at (18 000,15 000) m. The biases for two sensors are set toð500,p=360Þ and ð400,p=120Þ, respectively. The co-variance matrix of the measurement noise is R1

k ¼ R2k ¼

diagf1002,ðp=180Þ2g. Specially, the CKF is used to handlenonlinear measurements [40].

The number of targets is time-varying due to targetappearance and disappearance in the scene at any time.

The spontaneous birth RFS is Poisson with intensity

gkðxk,bkÞ ¼ 0:1X2

j ¼ 1

Y2

l ¼ 1

N ðxk;mjg,k,Pj

g,kÞN ðblk; b

j,lg,k,Bj,l

g,kÞ ð45Þ

where m1g,k ¼ ð10 000;0,20 000;0ÞT , m2

g,k ¼ ð0;0,30 000;0ÞT ,Pjg,k ¼ diagf106,104,106,104

g, bj,1g,k ¼ ð500,p=360ÞT , bj,2

g,k ¼

ð400,p=120ÞT and Bj,lg,k ¼ diagf1002,ð0:02p=180Þ2g (j,l¼1,2).

The intensity of the Poisson RFS of spawn births isgiven by

bk9k�1ðxk,bk9xk�1,bk�1Þ ¼ 0:05Y2

l ¼ 1

N ðxk; xk�1,Qx,kÞ

N ðblk; b

lk�1,Ql

b,kÞ ð46Þ

where Qx,k ¼ diagf104,400,104,400g andQl

b,k ¼ diagf4002,ð0:02p=180Þ2gðl¼ 1;2Þ.The intensity of the clutter RFS is assumed to be

klkðzkÞ ¼ ll

cUðzkÞ, l¼ 1;2 ð47Þ

where llk ¼ 10 (l¼1,2) and Uð�Þ is the uniform density over

the surveillance region.The bias bk

lis modeled by a first order Gauss–Markov

process with transition density

f bðblk9b

lk�1Þ ¼N ðb

lk; b

lk�1,Bl

k�1Þ, l¼ 1;2 ð48Þ

where Blk�1 ¼ diagf22,ð0:01np=180Þ2gðl¼ 1;2Þ.

The true target trajectories are shown in Fig. 1. To bespecific, target 1 starts at time k¼1 with initial position at(10 000, 20 000) m and ends at time k¼100; target 2 isspawned from target 1 at time k¼30 and ends at timek¼70; target 3 starts at time k¼5 with initial position at(0, 30 000) m and ends at time k¼100; target 4 isspawned from target 3 at time k¼40 and ends at timek¼80.

In the simulations, the survival probability and thedetection probability are set to ps¼0.99 and pd¼0.98,respectively. The pruning threshold is taken as TTh ¼

0:001, the merging threshold UTh ¼ 5, the weight thresholdwTh ¼ 0:5 and the maximum number of Gaussian termsJmax ¼ 100 (see [18] for the meanings of these parameters).The criterion known as optimal subpattern assignment(OSPA) metric is used for performance evaluation sincethe OSPA metric captures the differences in cardinality andindividual elements between two finite sets [41]. To verifythe performance of the proposed filter, the simulationresults are obtained from 100 Monte Carlo runs.

The position estimates of the proposed filter (shortlydenoted as ‘GM-PHD-RE’) and the standard GM-PHD filterfor one trial are shown in Fig. 2, the simulation resultssuggest that the GM-PHD-RE filter provides more accu-rate tracking performance for almost all the time. This isexpected since the transitional measurement biases havebeen estimated and incorporated in the proposed filter.In Fig. 3, the Monte Carlo averages of the OSPA distancewith p¼2 and c¼1000 are shown versus time. It can beseen that the GM-PHD and the GM-PHD-RE filters pro-duce average errors of approximately 580 m and 200 m,respectively. These results also suggest that the GM-PHD-RE filter outperforms the standard GM-PHD filter. Inaddition, the true and the estimated target numbers are

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 22

2.2

2.4

2.6

2.8

3

3.2

3.4

3.6

3.8

4

x 104

x 104

X coordinate (m)

Y c

oord

inat

e (m

)

True tracksTarget birthTarget death

Fig. 1. True target trajectories.

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 22

2.2

2.4

2.6

2.8

3

3.2

3.4

3.6

3.8

4 x 104

x 104X coordinate (m)

Y c

oord

inat

e (m

)

True tracksGM−PHDGM−PHD−RE

Fig. 2. Position estimates with low clutter rate.


shown in Fig. 4, which indicates that the cardinalitystatistics can be estimated accurately with some penaltyof time delay.

In order to investigate the behavior of the GM-PHD-REfilter to clutter rates, we study a tracking scenariowith high clutter rate ll

k ¼ 30 (l¼1,2). The performance

0 10 20 30 40 50 60 70 80 90 100100

200

300

400

500

600

700

Time (s)

OS

PA

(c=1

000,

p=2

)

GM−PHDGM−PHD−RE

Fig. 3. Performance comparison with low clutter rate.

0 10 20 30 40 50 60 70 80 90 1001

1.5

2

2.5

3

3.5

4

Time (s)

Est

imat

ed ta

rget

num

ber

TruthGM−PHD−RE

Fig. 4. True and estimated target numbers against time with low clutter rate.


comparison results are shown in Fig. 5, which indicatesthat the GM-PHD-RE filter achieves higher accuracy thanthe GM-PHD filter. Another tracking scenario with two

sensors having different clutter rates (l1k ¼ 10 and l2

k ¼ 30)is carried out as shown in Fig. 6 and similar conclusionscan be drawn.

0 10 20 30 40 50 60 70 80 90 100150

200

250

300

350

400

450

500

550

600

Time (s)

OS

PA

(c=1

000,

p=2

)


Fig. 5. Performance comparison with high clutter rate.

0 10 20 30 40 50 60 70 80 90 100150

200

250

300

350

400

450

500

550

600

650

Time (s)

OS

PA

(c=1

000,

p=2

)


Fig. 6. Performance comparison with different clutter rates.


Example 2. In this example, we consider the multiplemaneuvering targets tracking by using jump Markovmodels. For simplicity, the same surveillance region and

measurement equations as in the non-maneuveringtarget tracking are used. However, two sensors are locatedat (12 000, 14 000) m and ð�2000;14 000Þm, respectively.


To indicate the target maneuvers, the target dynamics isdescribed by the following coordinated turn model:

xk ¼

1 sinðoTÞo 0 �

1�cosðoTÞo

0 cosðoTÞ 0 �sinðoTÞ

0 1�cosðoTÞo 1 sinðoTÞ

o0 sinðoTÞ 0 cosðoTÞ

266664

377775xk�1þwk�1ðoÞ ð49Þ

where o denotes the turn rate, and wk�1ðoÞ is zero-meanwhite Gaussian noise with covariance

QkðoÞ ¼ s2ðoÞ

T4

4T3

2 0 0

T3

2 T2 0 0

0 0 T4

4T3

2

0 0 T3

2 T2

26666664

37777775

ð50Þ

In the simulations, three motion models correspondingto different turn rates are used. Model 1 is the coordinatedturn model with turn rate o¼ 0J=s and sð0Þ ¼ 5. Model 2is the coordinated turn model with clockwise turn rate ofo¼�4J=s and sð�4Þ ¼ 20. Model 3 is the coordinatedturn model with counterclockwise turn rate of o¼ 4J=sand sð4Þ ¼ 20. The switching between three models isgoverned by a first order Markov chain with knowntransition probability matrix

P¼0:8 0:1 0:1

0:1 0:8 0:1

0:1 0:1 0:8

264

375 ð51Þ

−4000 −2000 0 2000 4000 6001.6

1.8

2

2.2

2.4

2.6

2.8

3

3.2x 104

X coordin

Y c

oord

inat

e (m

)

Fig. 7. True target

The true target trajectories are shown in Fig. 7. To bespecific, target 1 starts at time k¼1 with initial position at(10 000, 20 000) m and ends at time k¼100; target 2 isspawned from target 1 at time k¼30 and ends at timek¼70; target 3 starts at time k¼5 with initial position at (0,30 000) m and ends at time k¼100; target 4 is spawnedfrom target 3 at time k¼40 and ends at time k¼80.

To deal with the multiple model estimation in the frame-work of the RFS, the best-fitting Gaussian (BFG) approxima-tion approach is applied to derive the GM-PHD-RE filter. Thepurpose of the BFG approximation is to express the dynamicsof the jump Markov linear system (JMLS) (49) by a linearGaussian model

xkþ1 ¼Fkxkþwk ð52Þ

where wk is a zero-mean white Gaussian random vectorwith covariance matrix Sk, i.e., wk �N ð0,SkÞ. In otherwords, we want to replace the JMLS (given by (49) andreferred to as ‘‘A’’) with a single BFG distribution (givenby (52) and referred to as ‘‘B’’). Fk and Sk are determinedsuch that the distribution of xk has the same mean andcovariance under each model, i.e.,

Efxk9Ag ¼ Efxk9Bg ð53Þ

Covfxk9Ag ¼ Covfxk9Bg ð54Þ

As stated in [23], the matrices Fk and Sk can bedetermined by

pkþ1,r ¼XMi ¼ 1

pirpk,i ð55Þ

0 8000 10000 12000 14000 16000

ate (m)

True tracksTarget birthTarget death

trajectories.


Fk ¼XMr ¼ 1

pkþ1,rFrk ð56Þ

Ykþ1 ¼XMr ¼ 1

pkþ1,r ½FrkðYkþekeT

k Þ½Frk�

TþQrk��FkekeT

kFTk ð57Þ

−4000 −2000 0 2000 4000 601.6

1.8

2

2.2

2.4

2.6

2.8

3

3.2x 104

X coordin

Y c

oord

inat

e (m

)

Fig. 8. Position estimates w

0 10 20 30 40 5100

200

300

400

500

600

700

Tim

OS

PA

(c=1

000,

p=2

)

Fig. 9. Performance compariso

Sk ¼Ykþ1�FkYkFTk ð58Þ

ekþ1 ¼Fkek ð59Þ

where Fkr

is the system transition matrix, pir is thetransition probability of the Markov chain, pkþ1,r is the

00 8000 10000 12000 14000 16000

ate (m)

True tracksGM−PHDGM−PHD−RE

ith low clutter rate.

0 60 70 80 90 100

e (s)


n with low clutter rate.


probability of the event that model r is in effect duringthe sampling period ½k,kþ1Þ. ek and Ykþ1 are auxiliaryvariables.

It has been shown in [23] that the BFG-based GM-PHDfilter outperforms the multiple model GM-PHD filterwithout interacting. To implement the filtering algorithm,the settings for other parameters are identical to those in

0 10 20 30 40 5

200

100

300

400

500

600

700

Tim

OS

PA

(c=1

000,

p=2

)

Fig. 10. Performance compariso

0 10 20 30 40 5100

200

300

400

500

600

700

Tim

OS

PA

(c=1

000,

p=2

)

Fig. 11. Performance comparison

Example 1. The position estimates of the proposed filterand the BFG-based GM-PHD filter in [23] for one trial areshown in Fig. 8. In Fig. 9, the Monte Carlo averages of theOSPA metric are shown versus time. The performancecomparisons with high clutter rate ll

k ¼ 30 (l¼1,2) anddifferent clutter rates (l1

k ¼ 10 and l2k ¼ 30) are presented

in Figs. 10 and 11, respectively. The simulation results

0 60 70 80 90 100e (s)


n with high clutter rate.

0 60 70 80 90 100e (s)


with different clutter rates.

0 10 20 30 40 50 60 70 80 90 1001

1.5

2

2.5

3

3.5

4

Time (s)

Est

imat

ed ta

rget

num

ber

TruthGM−PHD−RE

Fig. 12. True and estimated target numbers against time with low clutter rate.


demonstrate that the GM-PHD-RE filter performs betterthan the GM-PHD filter without incorporating sensorbiases. Similarly, the true and estimated target numberare shown in Fig. 12, which is consistent with theconclusion for non-maneuvering target tracking.

5. Conclusion

In this paper, the GM-PHD filter is applied for tracking anunknown and time-varying number of targets with multiplesensors, in which the translational measurement biases areconsidered and estimated. To derive a decoupled form of thestate estimates and the bias estimates in the updatedposterior intensity, a suboptimal two-stage Kalman filteringprocess is used. The proposed filter can be extended totrack maneuvering targets that follow Markovian switching.Simulation results show that the proposed filter outperformsthe standard GM-PHD filter without incorporating measure-ment biases.

Acknowledgments

This work was supported by the National 973 Program(2012CB821200) and the NSFC (61134005, 60921001,90916024, 91116016).

Appendix A. Derivation of the predicted and updatedintensities

By applying Lemma 1, the predicted intensity nk9k�1

can be derived by substituting (7) and (16)–(18) into thePHD prediction (5). For the updated intensity n1

k9k, the

main difficulty is the computation of n1d,kðxk; zkÞ. It follows

from (8) and (9) that

n1d,kðxk; zkÞ ¼

p1dh1ðzk9xk,bkÞnk9k�1

k1k ðzkÞþ

Rp1

dh1ðzk9x0k,b0kÞnk9k�1ðx

0k,b0k9Z

1:L1:k�1Þdx0kdb0k

ðA:1Þ

where the likelihood function is

h1ðzk9xk,bkÞ ¼N ðzk;H

1kxkþb1

k ,R1k Þ ðA:2Þ

Substituting (30) into the numerator of (A.1) yields

XJk9k�1

j ¼ 1

YL

l ¼ 1

p1dwj

k9k�1N ðzk;H

1kxkþb1

k ,R1k ÞN ðxk;m

jk9k�1

,Pjk9k�1Þ

N ðblk;b

j,lk9k�1

,Bj,lk9k�1Þ ðA:3Þ

It can be seen that the target state xk and the bias b1k

are coupled in the likelihood function, it is difficult toderive the decoupled form of xk and b1

k in (A.3). Toovercome this difficulty, the idea is to develop a two-stage estimator as follows. First, fix the bias terms toobtain the state estimates. Second, consider the bias termand obtain the bias estimates. Specifically, by applyingLemma 2, we have

N ðzk;H1kxkþb1

k ,R1k ÞN ðxk;m

jk9k�1

,Pjk9k�1Þ

¼N ðxk;mj,1k9k

,Pj,1k9kÞN ðzk;H

1k mj

k9k�1þb1

k ,H1k Pj

k9k�1ðH1

k ÞTþR1

k Þ

ðA:4Þ

where

mj,1k9k¼mj

k9k�1þKj,1

k ðzk�H1kmj

k9k�1�b1

k Þ ðA:5Þ


Pj,1k9k¼ ðI�Kj,1

k H1k ÞP

jk9k�1

ðA:6Þ

Kj,1k ¼ Pj

k9k�1ðH1

k ÞT½H1

kPjk9k�1ðH1

k ÞTþR1

k ��1 ðA:7Þ

Note that there are still bias terms in the stateestimates (A.5), to derive an explicit expression, we adoptthe suboptimal solution, i.e., b1

k ¼ bj,1k9k�1

. Then, thisreduces to (36). Similarly, we can derive the bias esti-mates

N ðzk;H1kmj

k9k�1þb1

k ,H1kPj

k9k�1ðH1

k ÞTþR1

k ÞN ðbk; bj,1k9k�1

,Bj,1k9k�1Þ

¼N ðzk;H1k mj

k9k�1þbj,1

k9k�1,H1

kPjk9k�1ðH1

k ÞTþR1

kþBj,1k9k�1Þ

N ðb1k ; b

j,1k9k,Bj,1

k9kÞ ðA:8Þ

where bj,1k9k and Bj,1

k9kare given by (40) and (41),

respectively.Following the same line, the denominator of n1

d,kðxk; zkÞ

in (A.1) can be obtained, and the updated intensity nlk9k

(l¼ 2, . . . ,L) can be derived sequentially for each sensor.

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gaussian mixture phd filter for multi-sensor multi-target tracking with registration errors

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