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Signal Processing

Signal Processing 93 (2013) 3578–3588

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A new multiple extended target tracking algorithmusing PHD filter

Yunxiang Li n, Huaitie Xiao, Zhiyong Song, Rui Hu, Hongqi FanATR Key Laboratory, National University of Defense Technology, Changsha, Hunan 410073, PR China

a r t i c l e i n f o

Article history:Received 21 June 2012Received in revised form18 March 2013Accepted 9 May 2013Available online 25 May 2013

Keywords:Multiple extended target trackingProbability hypothesis density filterParticle filterK-means clusteringBackground clutter suppressionGating

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ail addresses: wuyulyx@gmail.com (Y. Li),126.com (H. Xiao), zhiyongsong@163.com (Z609@qq.com (R. Hu), fanhongqi@tsinghua.or

a b s t r a c t

A new multiple extended target tracking algorithm using the probability hypothesisdensity (PHD) filter is proposed in our study, to solve problems on tracking performancedegradation of the extended target PHD (ET-PHD) filter under the nonlinear conditionsand its intolerable computational requirement. It is noted that with the current Gaussianmixture implement of ET-PHD filter satisfying tracking performance could only beobtained under linear and Gaussian conditions. To extend the application of ET-PHD filterfor nonlinear models, our study has derived a particle implement of ET-PHD (ET-P-PHD)filter. Our study finds that the main factors influencing the computational complexity ofthe ET-P-PHD filter are the partition number of measurement set and the calculation ofnon-negative coefficients of cells in partitions. With the pretreatment of measurementsand application of a new K-means clustering based measurement set partition method,we have successfully decreased the partition number. In addition, a gating method fortarget state space, which is based on likelihood relationship between target state andmeasurement, is proposed to simplify the calculation of non-negative coefficients.Simulation results show that the algorithms proposed by our study could satisfyinglydeal with multiple extended target tracking issues under nonlinear conditions, and lead tosignificantly lower computational complexity with tiny effect on tracking performance.

& 2013 Elsevier B.V. All rights reserved.

1. Introduction

It is a common hypothesis in traditional multiple targetstracking that one target produces single measurement pertime step. With the wide application of high-resolutionsensors in recent years, one target will occupy more thanone resolution cell, which does not comply with the pointtarget hypothesis. Application of traditional tracking algo-rithm in tracking extended target will lead to seriousdeviation in estimation of target number and state [1].

All rights reserved.

. Song),g.cn (H. Fan).

Therefore, multiple extended target tracking has becomean increasingly important area in signal processing [2].

Extended target tracking is quite different from pointtarget tracking. As one extended target produces severalmeasurements per time step, modeling on spatial exten-sion of extended target is needed, to simulate how themeasurements are produced and qualitatively describe therelationship between the target state and measurements.The traditional approach is to model extended targetas a set of point sources, which needs construction ofmeasurement-source assignment hypothesis and calcula-tion of probabilities of these hypotheses [3]. However,uncertainties in the number and location of point sourceswill degrade the tracking performance. Assuming that thenumber of cluster measurements and target measurementsis subject to Poisson distribution, Gilholm [4] modeledextended target with spatial probability distribution model,

Y. Li et al. / Signal Processing 93 (2013) 3578–3588 3579

each measurement as a random sample of this distribu-tion. With this model, he gained an excellent trackingperformance in rod target tracking. Koch [5] proposed arandom matrix based extended target tracking method,modeling on the spatial extension of extended target withsymmetric positive definite random matrix. Baum [6]proposed a Gaussian-assumed Bayesian tracking method,modeling the surface shape of the extended target withRandom Hypersurface Model, which assumes unknownmeasurement sources lies on the scaled versions of theshape boundaries.

The standard PHD (Probability Hypothesis Density)filter is based on the point target hypothesis, which resultsin performance degradation in estimation of target num-ber and state. [7]. To solve this problem, Mahler [8]improved the standard PHD filter and proposed severalgeneral PHD filters for various nonstandard targets track-ing. In [9], Mahler derived the measurement updateequation of extended target PHD (ET-PHD) filter basedon Gilholm's Poisson distribution model [4,10]. Granstrom[7] proposed a Gaussian Mixture implement of ET-PHDfilter, which leads to the realization of extended targettracking under linear and Gaussian condition. To improvethe precision in estimation of target number, Orguner [11]derived the extended target Cardinalized PHD (ET-CPHD)filter and provided a Gaussian mixture implement ofET-CPHD filter in the case of linear and Gausssian problem,to simultaneously and sequentially estimate the cardina-lized distribution of target number and PHD of target state.In addition, representing the extended targets as an inde-pendent cluster process, Swain [18] derived an approximatemeasurement-update equation, similar to Mahler [9], andproposed an alternative technique for multiple extendedtarget tracking.

Current researches on extended target tracking withET-PHD filter are still limited to linear and Gaussianproblems, and extended target tracking under nonlinearand non-Gaussian conditions need further exploration[7,11]. Based on [9,7,11], we derived a particle implementof ET-PHD filter which is capable of dealing with nonlinearand non-Gaussian issues, named extended target particlePHD(ET-P-PHD) filter. Analyzing the implement of ET-P-PHD filter, we find that its computational complexity ishigh, the number of all partitions of measurement set andcalculation of the non-negative coefficients of the cells inpartitions are the main factors affecting computationalcomplexity. With analysis of the nature of ET-PHD filter formultiple extended target tracking and the advantages anddisadvantages of distance based measurement set parti-tion method [7], we proposed a new K-means basedmeasurement set partition method, which could furtherreflect the nature of ET-PHD filter in target tracking andsignificantly decrease the number of partitions. In addition,we could reduce the number of clutter measurementsthrough background clutter suppression, indirectly reducingthe number of partitions. As calculation of the non-negativecoefficients of the cells in partitions needs integral in fulltarget state space and consumes a lot computational source,we find that extended target measurement likelihood isgenerally in exponential relation with target state andfor each measurement the state space could always be

partitioned into two parts, one highly related to the mea-surement, the other weakly related. Then, we proposed astate space gating method based on the likelihood relationbetween measurement and target state, with which integralregion for calculation of the coefficient could be enormouslyreduced. Results of simulation experiments show that theET-P-PHD filter performs excellently in multiple extendedtarget tracking under nonlinear and non-Gaussian models,K-means based measurement set partition method and thestate space gating method could significantly decreasecomputational requirement of ET-P-PHD with tiny effecton target tracking performance.

This paper is organized as follows. Section 2 makes astatement of problem on extended target tracking. Theproposed ET-P-PHD filter is addressed in Section 3. Section 4describes the new K-means clustering based measurementset partition method in detail. The proposed state spacegating method is provided in Section 5. Results and com-parisons for a complex multiple extended target scenariowith active radar sensor measurement data are offered inSection 6. Final remarks and future work are provided inSection 7.

2. Statement of problem on extended target tracking

In this part, we modeled on dynamics of extended target,measurement characters of sensor and spatial extension ofextended target associated with extended target trackingproblems, provided a detailed presentation of Poisson dis-tribution based spatial distribution model [4,10] for trackingof extended target, and simulated on the round extendedtarget measurement likelihood.

2.1. Dynamic model of extended target

The target state vectors set at time step k is Xk ¼fxk;igNk

i ¼ 1, Nk is the number of targets. Each target state xk;iin random finite set (RFS) Xk is modeled in followingdynamic model:

xk;i ¼ lk;iðxk�1;iÞ þ vk�1;i ð1ÞHere, state vector xk;i ¼ ½xk;i; _xk;i; yk;i; _yk;i�T represents the

dynamic characters of extended target center, ðxk;i; yk;iÞ isthe target location, ð_xk;i; _yk;iÞ is the target velocity, vk;i is theprocess noise, lk;ið⋅Þ is the dynamic equation of target i attime step k. Then, the state transform function of singletarget comes out

pðxk;ijxk�1;iÞ ¼ f vk�1;iðxk;i−lk;iðxk�1;iÞÞ ð2Þ

2.2. Measurement model

At time step k, point target sk produces a measurementzk through a sensor, then measurement characters ofsensor could be modeled as

zk ¼ hkðskÞ þwk ð3ÞHere, hkð⋅Þ is the reflection form of sensor, wk measure-

ment noise.

Y. Li et al. / Signal Processing 93 (2013) 3578–35883580

Then, sensor measurement likelihood function becomes

pðzkjskÞ ¼ gwkðzk−hkðskÞÞ ð4Þ

2.3. Extended target spatial distribution model

Extended target spatial distribution model [4,10]describes the spatial distribution of point measurementsources, the relation between measurement source s andtarget center state x as well as shape parameter B, theprobability density function of measurement source isdenoted as pðsjx;BÞ. The extended target spatial distribu-tion model could be a bounded distribution such as uni-form probability distribution function or an unboundeddistribution such as a Gaussian.

2.4. Extended target measurement likelihood

The measurement set obtained by sensor at time step kis Zk ¼ fzk;igMk

i ¼ 1, Mk is the the number of measurements.From the measurements obtaining mechanism of sensor,we can get it that the extended target measurementlikelihood ϕzk;j ðxk;i;BÞ depends on the extended targetmodel and sensor measurement model, and its expressioncould be described as a convolution of sensor measure-ment likelihood function and extended target measure-ment source probability distribution function [4]:

ϕzk;j ðxk;i;BÞ ¼Z

pðzk;jjsÞpðsjxk;i;BÞds ð5Þ

For example, Fig. 1 presents the spatial distribution ofmeasurement sources for a round extended target.In Fig. 1, r represents the radius of round target, ðx1; x2Þits centric location, target moves around in x–y plane.To simplify this problemwithout sacrifice of universality, itis assumed that measurement sources are uniformly dis-tributed on target surface (which is not always the case),location of certain measurement source s is ðs1; s2Þ,ðs1−x1Þ2 þ ðs2−x2Þ2≤r2, and each measurement from its

X

ys ( s1, s2 )

Sensor

O

x ( x1, x2 )2r

Fig. 1. Spatial distribution of point measurement sources for a roundextended target.

source is the source location plus zero-mean Gaussiannoise with standard deviation s. As measurement sourcesare uniformly distributed on target surface, the probabilitydistribution function of measurement sources becomespðsjx; rÞ ¼ 1=πr2, the measurement of source s becomesz¼½z1; z2�T . Then, the measurement likelihood betweenzands is a 2-D normal distributionpðzjsÞ ¼

N ½z1−s1; z2−s2�T ;s2 00 s2

" # !. Therefore, extended target

measurement likelihood becomes

ϕzðx; rÞ ¼Z

pðzjsÞpðsjx; rÞds¼∬ðs1−x1Þ2þðs2−x2Þ2 ≤ r21πr2

�Νz1−s1z2−s2

" #;

s2 00 s2

" # !ds1 ds2

¼ 1πr2

Z r

0

Z 2π

0Ν

z1−ρ cos θ−x1z2−ρ sin θ−x2

" #;

s2 00 s2

" # !ρ dρ dθ

ð6ÞEq. (6) requires integral of 2-D normal distribution in a

2-D space. From relevant primary mathematics knowl-edge, there is no analytical solution for this integral.Therefore, we apply numeric integral method to obtainquantitative relation between extended target measure-ment likelihood and target center location. Set s¼ 10 m,

r¼ 20 m, x1 ¼ x2 ¼ 0m, we could get extended targetmeasurement likelihood as presented in Fig. 2. Here,extended target measurement likelihood is a round para-bola distribution, decreasing at increasing speed as thedistance between measurement and the central locationincreases. Alternating the condition z1 ¼ z2 ¼ 0m, extendedtarget measurement likelihood versus target center locationx¼ ½x1; x2�T is shown in Fig. 3, whose presentation is similarto Fig. 2. In addition, from Eq. (6), we could get it that themeasurement likelihood for round extended target does notchange along with the translation of measurement andtarget central location. As to common shape extendedtarget, the extended target measurement likelihood alsogenerally meets measurement translation invariance.

As to one extended target x, the number of targetmeasurements obtained per time step is subject to Poissondistribution, whose mean is γðxÞ[10]. Here, γð⋅Þ is defined asa non-negative function on target state space. The prob-ability of obtaining no less than one measurement for

Fig. 2. Extended target measurement likelihood versus target measure-ment.

Fig. 3. Extended target measurement likelihood versus target position.

Y. Li et al. / Signal Processing 93 (2013) 3578–3588 3581

target x is 1−e−γ xð Þ, the detection probability for target x ispDðxÞ, pDð⋅Þ defined as non-negative function on targetstate space. Therefore, effective detection probability fortarget x becomes

ð1−e−γðxÞÞpDðxÞ ð7Þ

The number of clutter measurements obtained per timestep is subject to Poisson distribution [10], average numberof clutter measurements for unit region is γFa;k. Then, as toa surveillance region of volume Rs, the average number ofclutter measurements is γFa;kRs.

Considering general situation, we do not estimate theshape parameters of extended target. The extended targetconsidered in our study is general extended target, whichproduces no less than one measurement per time step.Therefore, as to tracking extended target of any shape, thealgorithm proposed in our study would be applicable, oncethe extended target measurement likelihood functionis given.

3. ET-P-PHD filter

In this part, we provide the measurement updateequation of ET-PHD filter, and then derive a particleimplement of ET-PHD filter, named extended target particlePHD (ET-P-PHD) filter, and analyze main factors contributingto the computational complexity of ET-P-PHD in detail.

3.1. Measurement update equation of ET-PHD filter

The difference between ET-PHD filter and standard PHDfilter lies on the measurement update equation, no differ-ence on prediction equation. Therefore, only measurementupdate equation of ET-PHD filter is provided here. Forprediction equation, description in [12] could be referred to.

From time step 1 to time step k, the time sequence ofmeasurement set obtained by sensor is ZðkÞ : Z1;…; Zk, thenumber of measurements in measurement set Zk isMk ¼jZkj. From time sequence Z kð Þ, we can describe PHD at timestep k as DkjkðxjZðkÞÞ, measurement update equation [9] forET-PHD filter at time step k as

DkjkðxjZðkÞÞ ¼ LZk ðxÞDkjk−1ðxjZðk−1ÞÞ ð8Þ

Here, LZk ðxÞ denotes pseudo-likelihood function of statex for measurement set Zk, Dkjk−1ðxjZðk−1ÞÞ the predictionPHD at time step k.

LZkðxÞ ¼ 1−ð1−e−γðxÞÞpDðxÞ

þ e−γðxÞpDðxÞ ∑℘∠Zk

ω℘ ∑W∈℘

γðxÞjW j

dW⋅ ∏zk∈W

ϕzk ðxÞλkckðzkÞ

ð9Þ

Here, λk ¼ γFa;kRs is the mean number of clutter mea-surements, ckðzkÞ ¼ 1=Vs is the spatial distribution ofclutter in surveillance region, Vs is the surveillancevolume, ℘∠Zk is the partition operator, ℘ is partitionsmeasurement set Zk at time step k into several cells W , j⋅jis an operator for getting the number of elements in a set,jWj is the number of measurements in cell W , ω℘ is aweight of partition ℘, dW is the non-negative coefficientsof cell W , the calculation formulae for ω℘and dWare

ω℘ ¼ ∏W∈℘dW∑℘′∠Zk

∏W ′∈℘′dW ′ð10Þ

dW ¼ δjW j;1 þ Dkjk−1 e−γγjW jpD ∏z∈W

ϕz

λkckðzÞ

� �ð11Þ

Here, δi;j is Kronecker delta, its value becomes 1 wheni¼ j, and 0 when i≠j. As to any real function hðxÞ,Dkjk−1½h� ¼

RhðxÞDkjk−1ðxjZðk−1ÞÞdx.

3.2. Particle implement of ET-PHD filter

As to ET-PHD filter, its implement needs calculatinghigh-dimensional integral of non-integrable function,which is too complicated to get its analytical form.At present, there exist two main methods [13,14] insolving this problem: one is to get Gaussian mixture(GM) form of algorithm; the other is to get SequenceMonte Carlo (SMC) form of algorithm, or particle filter (PF)form. The GM implement of ET-PHD (ET-GM-PHD) filterhas been derived by Granstrom [7], and is the optimalapproximation of ET-PHD filter under linear and Gaussianconditions, while its tracking performance would seriouslydecline for nonlinear and non-Gaussian models. Therefore,a particle implement of ET-PHD is derived in our study toimprove algorithm's tracking performance under non-linear and non-Gaussian conditions, named extendedtarget particle PHD (ET-P-PHD) filter.

The ET-P-PHD filter is composed of state prediction,measurement update, and particle resampling and targetstate estimation. The state prediction part will be pre-sented in following part.

For Lk−1 particles obtained through independent samplingfrom state sampling density qkð⋅jxðiÞk−1; ZkÞ, the prediction

equation of weight for f ~xðiÞk gLk−1i ¼ 1 is

~wðiÞkjk−1 ¼

φkjk−1ð ~xðiÞk ; xðiÞk−1Þ

qkð ~xðiÞk jxðiÞk−1; ZkÞ

wðiÞk�1 ð12Þ

For Jk birth particles, obtained through independentsampling from birth sampling density pkð⋅jZkÞ, we get thenumber of prediction particles Lkjk−1 ¼ Lk−1 þ Jk at time

Y. Li et al. / Signal Processing 93 (2013) 3578–35883582

step k. For these particles f ~xðiÞk gLkjk−1i ¼ Lk−1þ1, the prediction

equation of weight is

~wðiÞkjk−1 ¼

1Jk

ηkð ~xðiÞk Þ

pkð ~xðiÞk jZkÞ

ð13Þ

The measurement update equation of weight is

~wðiÞk ¼ 1−ð1−e−γð ~x ðiÞ

k ÞÞpDð ~xðiÞk Þ þ e−γð ~x

ðiÞk ÞpDð ~xðiÞ

k Þ ∑℘∠Zk

ω℘

"

� ∑W∈℘

γð ~xðiÞk ÞjWj

dW⋅ ∏zk∈W

ϕzk ð ~xðiÞk Þ

λkckðzkÞ

#~wðiÞkjk−1 ð14Þ

Here, for each cell W∈℘, we can get

dW ¼ δjW j;1

þ ∑Lkjk−1

j ¼ 1e−γð ~x

ðjÞk Þγð ~xðjÞ

k ÞjWjpDð ~xðjÞk Þ ∏

zk∈W

ϕzk ð ~xðjÞk Þ

λkckðzkÞ~wðjÞkjk−1 ð15Þ

Sum the updated weights, we can get an estimate of

target number N̂kjk ¼∑Lkjk−1i ¼ 1

~wðiÞk . SetLk ¼ ρN̂kjk, using stan-

dard resample algorithm to resample particles setf ~xðiÞk ;

~wðiÞk =N̂kjkgLkjk−1i ¼ 1 , we can get particles set fxðiÞk ;wðiÞ

k gLki ¼ 1 at timestep k.

Clustering particles set fxðiÞk ;wðiÞk gLki ¼ 1 into N̂kjk classes

with standard clustering algorithm, we can get an esti-

mated target states set fx̂ðjÞk gN̂kjkj ¼ 1.

Here, φkjk−1ðx; ξÞ ¼ bkjk−1ðxjξÞ þ ekjk−1ðξÞpðxjξÞ, bkjk−1ð⋅jξÞrepresents the PHD of targets set spawning from state ξ,ekjk−1ð⋅Þ survive probability of target, ηkð⋅Þ PHD of birthtargets, Lk particle number at time step k, Jk birth particlenumber, ρ particle number per target.

3.3. Computational complexity analysis of ET-P-PHD filter

From recursion equations of ET-P-PHD filter, we can getit that the computational requirement mainly lies on thecalculation of pseudo-likelihood function during measure-ment update step. According to (14) and (15), computa-tional requirement of pseudo-likelihood function dependson partition number of measurement set and calculationfor coefficient of cells in partition.

As partition number of measurement set is closelyrelated to the number of measurements in measurementset and will climb with increasing speed as the number ofmeasurements goes up, it is necessary to take pretreat-ment for measurements and reduce the clutter measure-ment number without sacrifice of tracking performance.As to background clutter suppression, extensive and in-depth researches have been conducted and many excellentclutter suppression algorithms have been proposed, forfurther information [15] could be referred to. In addition,to reduce computational complexity of algorithm, atten-tion should be limited to one part of the partitions.Therefore, to assure tracking performance, certain rulefor finding a subset of all partitions becomes necessary.

Calculation for coefficient of cells needs integral in thefull state space. However, getting analytical form for thisintegral is difficult, and only approximate solution could beobtained through numerical calculation with biggerintegral region and higher computational complexity.

Therefore, gating for the state space based on the relationsbetween measurement and target state is needed, to limitthe integral region and lower algorithm's computationalrequirement.

4. K-means clustering based measurement setpartition method

To reduce partition number and lower computationalcomplexity, we propose a K-means clustering based mea-surement set partition method for selection of partitions.

Measurement set Zk at time step k is composed ofmeasurements obtained from multiple extended targets.Once identity information of single measurement isknown, we can get the real partition of measurement setand easily track the extended target. However, the fact isthat we cannot get identity information easily. ET-PHDfilter needs consideration of all partitions of measurementset, and computes the weight of each partition, which isthe scale of partition's contribution to extended targettracking. When one partition is close to the real partition,its weight becomes higher and the partition has strongerinfluence on tracking result. However, as the measurementnumber increases, the partition number increases at ahigher speed, and the computational complexity increasesas well. Therefore, some less probable partitions should beeliminated to reduce computational complexity.

Based on the configuration of the sensor’ fields of view,a joint partition method of both sensor and state space wasproposed in [19]. This method does not address themultiple extended target tracking problem, but there aretechniques for partitioning a set which is useful for ourstudy. The distance based measurement set partitionmethod proposed in [7] takes advantage of statisticalproperties of distance between measurements obtainedfrom one extended target to set distance threshold.If measurements in cells are generated by one target, thispartition will contribute more to target tracking. However,among partitions obtained through distance based mea-surement set partition method, some might have cellswhose measurements are generated by different targets,which will lead to enormous waste of computationalresources. Therefore, we propose a K-means clusteringbased measurement set partition method, based on themean measurement number generated by one targetper time step, to eliminate useless partitions and reducecomputational complexity.

The essence of measurement set partition is to partitionmeasurement set into n cells, n represents the estimatednumber of targets. Therefore, it is natural to obtain severalcells of measurement set by standard clustering method(such as K-means clustering algorithm). For different n, wecan get different partitions, whose number depends on n'spossible value. Theoretically speaking, the possible valueof n could be any integer from 1 to Mk. From the aboveanalysis, to reduce computational complexity, we needselect out partitions of higher probability and eliminatepartitions of less probability. That is to say, we shouldreduce the range of n's possible value and seek out n withhigher possibility.

Y. Li et al. / Signal Processing 93 (2013) 3578–3588 3583

Target measurement number obtained from singleextended target per scan is subject to Poisson distributionwith mean γðxÞ. As γðxÞ in this part is constant, measure-ment number Mk in measurement set Zk obtained from Nk

extended targets per time step is subject to a Poissondistribution with rate γNk [4], the conditional probabilitydensity is

pðMkjγNkÞ ¼ðγNkÞMk

Mk!e−γNk ð16Þ

When measurement number Mk and parameter γ isknown, we can get the Maximum Likelihood Estimation(MLE) N̂k of target numberNkwith Eq. (16)

N̂k ¼ argmaxNk

pðMkjγNkÞ ð17Þ

Once parameter γ is reliable, N̂k could be treated as acore value of n. Then, we can get a neighborhood ΩðN̂kÞ⊂½0;Mk�∩Z including N̂k as the range forn's possible values.Less elements in ΩðN̂kÞ will lead to less partitions andlower computational complexity, but worse tracking per-formance. Therefore, to lower the computational complex-ity of algorithm, we need reduce elements in ΩðN̂kÞwithoutobvious decline in tracking performance. Compared withdistance based measurement set partition method, theK-means clustering based measurement set partitionmethod proposed here would lead to obvious decline ofpartition number and significant decline in computationalcomplexity. In fact, ΩðN̂kÞ is a compromise betweencomputational complexity and tracking performance,and of no fixed size. In Section 6, we set ΩðN̂kÞ as½N̂k=2; ðMk þ N̂kÞ=2�.

In addition, to make sure all measurements in each cellare generated by the same target, we can check thenumber of elements in each cell after partition withK-means clustering algorithm. If element number in onecell exceeds γ, the elements in this cell is less likely tocome from one target, the partition including this cellcould be treated as ineffective and another partition of thismeasurement set is needed. Though with this method itcannot be guaranteed that elements in one cell are alwaysfrom one target, taking advantage of γ, we enhance thevalidity of partition by reducing cells that contain mea-surements generated by different targets.

Pseudo-code for implement of K-means clusteringbased measurement partition method is given in Table 1.

Table 1K-means clustering based measurement set partition method.

1. Compute N̂k ¼ argmaxNk

pðMkjγNkÞ

2. Set ΩðN̂kÞ⊂½0;Mk�∩Z3. For each n∈ΩðN̂kÞ4. fWn

1;…;Wnng ¼ K �meansðZk;nÞ

5. For each Wni ,

6. if jWni joγ, go to step 4

7. End if8. End for9. End for

Here,Wni represents the ith cell in nth partition, K-means (d,d) represents

implementation of K-means clustering for measurement set and returnsn cells in the nth partition.

5. A gating method for target state space

In Section 4, we have discussed a K-means clusteringbased measurement set partition method to get partitionsof higher probability. As is known, calculation of cellcoefficient need integral in the full state space, which willlead to increase of computational complexity. To solve thisproblem, through taking advantage of gating method [16]in traditional target tracking algorithm and likelihoodrelationship between measurement and target state, wepropose a gating method for target state space to limitthe size of integral region, reducing the computationalcomplexity.

From (11), we can get it that the complexity forcalculation of cell coefficient dW depends on the followingintegral equation:

Dkjk−1 e−γγjWjpD ∏z∈W

ϕz

λkckðzÞ

� �¼Zx∈χ

e−γðxÞγðxÞjWjpDðxÞ

� ∏z∈W

ϕzðxÞλkckðzÞ

Dkjk−1ðxjZðk−1ÞÞdx ð18Þ

Here, χ represents full state space of target.From analysis in Section 2, as to any measurement

z∈W , ϕzðxÞ is generally of big value in a continuous regionand small value in other regions. Therefore, we can dividestate space χ into ΩðzÞ and ΩðzÞ, ΩðzÞ∩ΩðzÞ ¼∅, ΩðzÞ∪ΩðzÞ ¼ χ, and for ∀x∈ΩðzÞ, ϕzðxÞoε, threshold ε is a positivereal number. To make sure computational accuracy, for cellW , we can set integral region in (18) as ∪

z∈WΩðzÞ, then cell

coefficient could be approximated as

dW≃δjW j;1 þZx∈ ∪

z∈WΩðzÞ

e−γðxÞγðxÞjW jpDðxÞ

� ∏z∈W

ϕzðxÞλkckðzÞ

Dkjk−1ðxjZðk−1ÞÞdx ð19Þ

Here, the approximation accuracy depends on ε,smallerε will lead to bigger integral region ∪

z∈WΩðzÞ and

better accuracy, but higher computational requirement.Therefore, in real case, selection of ε is a compromisebetween computational complexity and approximationaccuracy.

To get ε, we can set a confidence level β. For onemeasurementz, we can get the threshold ε when prob-ability of ϕzðxÞ4ε is β. As described in Section 2, extendedtarget measurement likelihood is of translation invariancewith measurement, that is to say the calculated threshold εis universally applicable to every measurement. Therefore,we can get a random z from measurement space and itscorresponding ϕzðxÞ, and then ε can be calculated outthrough the following equation:

pðϕzðxÞ4εÞ ¼ β; z∈Z ð20Þ

Then, for single measurement z, it is not difficult to getthe integral region ΩðzÞ. To simplify operation and withoutadding computational complexity, we can set hypersphe-rical region CðzÞ with center h�1ðzÞ as integral region. Thisregion is the biggest hyperspherical region subject to ΩðzÞ,the radius r′ could be calculated out through the following

Table 2State space gating method based on likelihood relationship betweenmeasurement and target state.

1. Set β2. Compute pðϕzðxÞ4εÞ ¼ β; z∈Z, getε andΩðzÞ3. Get r′¼ argmax

rfCðzÞ⊂ΩðzÞg

4. For each ~x ðiÞk

5. if ∃z∈W ; s:t:ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffið ~x ′ðiÞ

k � zÞT ð ~x ′ðiÞk � zÞ

q≤r′

6. ~x ðiÞk can attend the calculation of dW

7. End if8. End for

Y. Li et al. / Signal Processing 93 (2013) 3578–35883584

equation:

r′¼ argmaxr

fCðzÞ⊂ΩðzÞg ð21ÞThen, for any particle ~xðiÞ

k in particle set f ~xðiÞk gLkjk−1i ¼ 1 , we can

get x′ðiÞk by eliminating ~xðiÞk 's dimensions not included in

integral region CðzÞ. When the following equation is

satisfied, particle ~xðiÞk can be used in calculation ofdW :

∃z∈W ; s:t:ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffið ~x′ðiÞ

k � h�1ðzÞÞT ð ~x′ðiÞk � h�1ðzÞÞ

q≤r′ ð22Þ

In actual case, different confidence levels β could beselected out to adaptively control the size of gate, compro-mising tracking performance and computational complexity.

These are all the steps necessary for state space gatingmethod based on likelihood relations between measure-ment and target state, the pseudo-code for its implementis presented in Table 2.

6. Simulation experiments and result analysis

We have designed four simulation experiments in thissection to evaluate performance of the algorithms pro-posed in our study. In experiment 1, we have validatedET-P-PHD filter's performance during process of nonlinearissues, and evaluated its multiple extended target trackingperformance through comparison with standard P-PHDfilter; In experiment 2, we have tested to what extent theK-means clustering based measurement set partitionmethod could lower computational complexity, througha comparison between computational complexity of ET-P-PHD filter with K-means clustering based measurementset partition method and that with distance based mea-surement set partition method; In experiment 3, through acomparison between computational complexity of ET-P-PHD filter with gating method and that without gatingmethod, we have validated the decline in computationalcomplexity arising from gating method; In experiment 4,we have made a comparison between tracking perfor-mance of ET-P-PHD filter and that of ET-GM-PHD filter, tovalidate the advantage of ET-P-PHD filter for trackingmultiple extended targets under nonlinear conditions.Indicators applied here include estimation of target num-ber per time step, optimal subpattern assignment (OSPA)distance [17] and algorithm's time cost per time step. OSPAdistance and estimation of target number per time step areto scale algorithm's tracking performance for multipleextended targets, time cost per time step is to scalecomputational complexity of the algorithm, we set the

order p¼3 and cut-off c¼80 for OSPA distance. Inthese experiments, we obtain measurement data throughsimulation, Matlab code is used to produce simulation andruns on a PC with RAM of 16 GB and CPU of 3.40 GHz.

6.1. Setup of simulation scenario

Surveillance region for sensor is ½−500;500�m� ½−500;500�m, in which targets move around. The evolution of thestate sequence for single target is given by

xk ¼

1 T 0 00 1 0 00 0 1 T

0 0 0 1

26664

37775xk−1 þ

T2=2T

00

00

T2=2T

266664

377775

v1;kv2;k

" #ð23Þ

Here, v1;k and v2;k are mutually independent zero-meanGaussian white noise with standard deviation sv1 ¼ sv2 ¼1m=s2. xk−1 ¼ ½xk; _xk; yk; _yk�T represents the central state oftarget, ðxk; ykÞ center location, ð_xk; _ykÞ center speed. Inexperiments, the velocity keeps constant, sampling periodis T ¼ 1 s.

In our experiments, active radar is used to obtain rangeand azimuth measurements. Its measurement equation is

zk ¼rkθk

" #¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðxk−xS;kÞ2 þ ðyk−yS;kÞ2

qarctan yk−yS;k

xk−xS;k

264

375þ

wr;k

wθ;k

" #ð24Þ

Here, ðxS;k; yS;kÞ represents the location of radar, set asð500;0Þ m. wr;krepresents range noise of 0 mean, wθ;k

represents azimuth noise of 0 mean, both are Gaussianand independent of each other. Standard deviation for wr;k

is swr ¼ 10 m, for wθ;k is swθ ¼ 0:1 rad.Clutter is uniformly distributed in surveillance region,

the Poisson rate for the number of clutter measurementsper time step is set as 10, then clutter PHD is 1/100π(rad m)−1, Survive probability for single target is 0.95,detection probability is 0.9, for simplification no spawningis considered. We set 500 particles for one target, samplingdensity for birth particle is Nðx; x;Q Þ, each birth particle islocated in one grid of 50 m� 50 m, where x¼ ½x;0; y;0�T ,Q ¼ diagð½100;25;100;25�Þ, ðx; yÞmeans the location of gridcenter. The extended target considered in our experimentsis of round shape, with radius of 20 m, Poisson rate for thenumber of measurements generated per scan is set to 9 foreach extended target. Point sources are uniformly distrib-uted on target surface, with location of ðs1; s2Þ. Then, theextended target measurement likelihood becomes

ϕzðx; rÞ ¼Z

pðzjsÞpðsjx; rÞds¼∬ðs1−xkÞ2þðs2−ykÞ2 ≤ r21πr2

⋅Ν

�rk−

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðs1−xS;kÞ2 þ ðs2−yS;kÞ2

qθk−arctan

s2−yS;ks1−xS;k

264

375; s2wr

0

0 s2wθ

" #0B@

1CAds1 ds2

¼ 1πr2

Z r

0

Z 2π

0N

rk−ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðρ cos θ þ xk−xS;kÞ2 þ ðρ sin θ þ yk−yS;kÞ2

qθk−arctan

ρ cos θþxk−yS;kρ sin θþyk−xS;k

264

375;

0B@

� s2 00 s2

" #!ρ dρ dθ ð25Þ

0 5 10 15 20 25 30 35 400

5

10

15

20

25

30

Time(s)

Car

dina

lity

ET-P-PHDP-PHDGround truth

Fig. 5. Mean of extended target number estimation.

Y. Li et al. / Signal Processing 93 (2013) 3578–3588 3585

Here, z¼ ½rk; θk�T , x¼ ½xk; yk�T means the location of targetcenter. This function describes the likelihood relationshipbetween radar measurement and location of target center.

In our experiments, we have designed a complexmultiple extended target scenario where the number oftargets is time-varying and track-cross happens. Simula-tion goes on for 40 s, at time step 1, target 1 and target 2start to move from original point simultaneously, target 1with velocity ofð10;10Þm=s, target 2 with velocity ofð−10;−10Þm=s. Target 1 lasts for 40 s, target 2 disappearsat 20th s, and target 3 comes in at 10th s fromð−200;−175Þm with velocity of ð10;10Þm=s and disappearsat 30th s. Fig. 4 is a description for the process of targetsmoving, (a) shows the change of x value along with thetime step, (b) shows the change of y value along with thetime step. From Fig. 4, we can get it that targets 2 and 3 areclose to each other without collision.

0 5 10 15 20 25 30 35 40-200

0

200

400

600

Time(s)

X c

oord

inat

e(m

)

0 5 10 15 20 25 30 35 40-400

-200

0

200

400

Y c

oord

inat

e(m

)

Time(s)

ET-P-PHDP-PHDGround truth

ET-P-PHDP-PHDGround truth

6.2. Experiment 1

In this experiment, we have applied the ET-P-PHD filterproposed in our study and standard P-PHD filter separatelyto deal with radar measurement data obtained fromscenario presented in Fig. 4. The average tracking resultsobtained from 10 independent experiments are presentedin Figs. 5 and 6. In Fig. 5, the real target number andestimated target number obtained separately trough twoalgorithms are all presented. In Fig. 6, estimated targetstates per time step obtained separately through twoalgorithms are presented. From Fig. 5, we can get it thatdeviation appears at step 1, 2, 14 and 15 for target numberestimated through ET-P-PHD filter. The reason might bethat at these steps targets getting close leads to fuzzymeasurements and decline of algorithm's tracking perfor-mance. At other steps, target number estimated is ofexcellent accuracy. Especially when there exist new birthand disappearance of targets at step 10, 20 and 30, ouralgorithm could detect these changes with tiny delay.

0 5 10 15 20 25 30 35 40-200

0

200

400

600

Time(s)

X c

oord

inat

e(m

) Target1Target2Target3

0 5 10 15 20 25 30 35 40-200

0

200

400

Y c

oord

inat

e(m

) Target1Target2Target3

Time(s)

Fig. 4. True target tracks in x- and y-coordinates versus time.

Fig. 6. State estimation of extended targets in x- and y-coordinatesversus time.

Therefore, the algorithm's excellent performance becomesobvious. As presented in Figs. 5 and 6, estimation of targetnumber and target state deviates seriously from theground truth when standard P-PHD filter is applied inextended target tracking.

In conclusion, the ET-P-PHD filter presented in ourstudy could deal with the issue of nonlinear multipleextended target tracking with excellent performance andcould quickly detect new birth and disappearance oftargets. As to tracking spatial unresolved targets, therewould appear deviation, and this will be our focus forfurther research.

6.3. Experiment 2

In this experiment, we compare ET-P-PHD withK-means clustering based measurement set partition

0 5 10 15 20 25 30 35 401

2

3

4

Time(s)

Car

dina

lity Distance partitioning

K-means partitioning

0 5 10 15 20 25 30 35 400

100

200

300

Ave

. OS

PA

(m)

Distance partitioningK-means partitioning

0 5 10 15 20 25 30 35 400

1000

2000

Ave

. tim

e co

st(s

)

Distance partitioningK-means partitioning

Time(s)

Time(s)

Fig. 7. Performance and efficiency indicators comparison between ET-P-PHD with K-means clustering based measurement set partition methodand ET-P-PHD with distance based measurement set partition methodversus time: (a) estimation of target number; (b) average OSPA distance;(c) average time cost per time step.

0 5 10 15 20 25 30 35 401

2

3

4

Time(s)

Car

dina

lity No gating

With gating

0 5 10 15 20 25 30 35 400

100

200

300

Ave

. OS

PA

(m)

No gatingWith gating

0 5 10 15 20 25 30 35 400

1000

2000

Ave

. tim

e co

st(s

)

No gatingWith gating

Time(s)

Time(s)

Fig. 8. Performance and efficiency indicators comparison between theET-P-PHD with and without proposed gating method versus time:(a)estimation of target number; (b) average OSPA distance; (c) averagetime cost per time step.

Y. Li et al. / Signal Processing 93 (2013) 3578–35883586

method and ET-P-PHD with distance based measurementset partition method on performance of tracking multipleextended targets and computational complexity. In Fig. 7are shown the results of tracking targets in the scenariopresented in Fig. 4 through two different algorithmsseparately. The average tracking results of 10 independentexperiments are presented. In figure (a) are presented theestimate of target number, figure (b) the change of averageOSPA distance along with time, figure (c) the average timecost per time step along with evolution. In figure (a) and(b), we can find it that performance of ET-P-PHD filter withK-means clustering based measurement set partitionmethod declines at time step 12, 17 and 18, and is closeto the performance of the other algorithm at other timesteps. However in figure (c), we can find it that thecomputational complexity of ET-P-PHD filter withK-means clustering based measurement set partitionmethod is reduced by nearly 30%, which means thecomputational complexity of ET-P-PHD filter withK-means clustering based measurement set partitionmethod is lowered with sacrifice of partial trackingperformance.

6.4. Experiment 3

In this experiment, we intend to evaluate to whatextent the computational complexity of ET-P-PHD filtercould be reduced after gating the state space. We set theconfidence level β as 90%, and track targets in the samescenario presented in Fig. 4, average results from 10impendent experiments are presented. In Fig. 8, trackingresults of ET-P-PHD filter with gating and ET-P-PHD filterwithout gating are both presented. In figure (a) arepresented the estimate of target number, figure (b) thechange of average OSPA distance along with time, figure

(c) the change of average time cost per time step alongwith evolution. From figure (a) and (b), we can find it thattracking performance after gating declines from step 16 to20, and at other steps tracking performance of bothalgorithms are close. However, figure (c) shows thatcomputational complexity of the algorithm with gating islowered by nearly 40%, which means that computationalcomplexity of ET-P-PHD filter with gating is reduced withthe sacrifice of tracking performance.

6.5. Experiment 4

To validate the advantage of ET-P-PHD filter for trackingmultiple extended targets under nonlinear conditions, inthis experiment a comparison between tracking perfor-mance of ET-P-PHD filter and that of ET-GM-PHD filter isconducted.

ET-GM-PHD filter presented in [7] could only deal withmultiple extended target tracking problems for linear andGaussian models. However, ET-GM-PHD filter to deal withmultiple extended target tracking under nonlinear conditionsis needed in this experiment. Therefore, improvement ofET-GM-PHD filter to deal with nonlinear issues becomesnecessary. Generalized from GM-PHD filter dealing withnonlinear issues presented in [14], linearization of measure-ment Eq. (3) was conducted first

zk ¼ hkðmðiÞkjk−1Þ þ Hk½sk �mðiÞ

kjk−1� þwk ð26Þ

Here, mðiÞkjk−1represents mean vector of the ith Gaussian

component, equation above could be further simplifiedand lead to an approximate measurement equation:

z′k≈Hksk þwk ð27Þ

Y. Li et al. / Signal Processing 93 (2013) 3578–3588 3587

Here, z′k ¼ zk−hkðmðiÞkjk−1Þ þHkm

ðiÞkjk−1 represents corrected

measurement after linearization, the measurement matrix is

Hk ¼dhkðskÞdsk

����������sk ¼ mðiÞ

kjk−1

¼

∂h1k ðskÞ∂s1

k⋯ ∂h1k ðskÞ

∂snxk

⋮ ⋱ ⋮∂hnz

kðskÞ

∂s1k

⋯ ∂hnzkðskÞ

∂snxk

266664

377775

����������sk ¼ mðiÞ

kjk−1

ð28ÞHere, nx and nz represents dimensions of state vector

and measurement vector separately.In addition, ET-GM-PHD filter presented in [7] has not

modeled on spatial distribution properties of extendedtarget and directly set extended target measurement like-lihood as a Gaussian distribution, which is easy to calcu-late. Extended target measurement likelihood in thisexperiment is presented in Eq. (25). During implement ofmeasurement update step in ET-GM-PHD filter, measure-ment zk is substituted to with z′k, then extended targetmeasurement likelihood becomes ϕz′

kðmðiÞ

kjk−1; rÞ. Targetbirth PHD in this experiment is set to be

DbðxÞ ¼ 0:1Nðx;mð1Þb ;PbÞ þ 0:1Nðx;mð2Þ

b ;PbÞ ð29ÞHere, mð1Þ

b ¼ ½100;0;100;0�T , mð2Þb ¼ ½−100;0;−100;0�T ,

Pb ¼ diagð½100;0;100;0�Þ. The number of Gaussian compo-nents Lk increases as filtering time increases. To control thecomputational complexity, Gaussian component manage-ment method present in [14] is applied in this experiment.For this method, Gaussian component pruning threshold isTpru ¼ 10−5, merging threshold Umer ¼ 4, maximum num-ber of Gaussian components Jmax ¼ 100. Other simulationconditions are the same as relative part in Section 6.1,experiment result is an average of 10 independentexperiments.

In Fig. 9, average OSPA distance comparison betweenET-P-PHD and ET-GM-PHD filter versus time is presented,ET-P-PHD filter performs much better than ET-GM-PHDfilter in dealing with multiple extended target trackingunder nonlinear conditions and is of higher trackingstability. That is to say, ET-P-PHD filter fits better thanET-GM-PHD filter in dealing with multiple extended targettracking under nonlinear conditions.

0 5 10 15 20 25 30 35 400

50

100

150

200

250

time

ave.

OS

PA

/m (c

=80•

p=3

)

ET-P-PHDET-GM-PHD

Fig. 9. Average OSPA distance comparison between ET-P-PHD andET-GM-PHD filter versus time.

Computational complexity of ET-P-PHD filter andET-GM-PHD filter varies along with number of particles andGaussian components respectively. Comparing computationalcomplexity of two methods in this experiment, we can geta result that only meets specific condition and would bedifferent in other conditions. Therefore, comparison of com-putational complexity is not conducted in this experiment. Inaddition, tracking performance of the two methods is closelyrelated to particle number and Gaussian component numberrespectively. Big particle number and Gaussian componentnumber would result in increase of computational complexity,and small particle number and Gaussian component numberwould lead to decline of tracking performance. Particlenumber and Gaussian component number set in this experi-ment arises from achievements in [13,14], which would assuretractable computational complexity as well as well trackingperformance of two methods. Therefore, the comparisonresult presented in Fig. 9 is of general sense, and couldadequately reflect the fact that ET-P-PHD filter performs muchbetter than ET-GM-PHD filter on multiple extended targettracking under nonlinear conditions.

7. Conclusion and future work

The ET-P-PHD filter proposed in our study could dealwith complicated multiple extended target tracking pro-blem under nonlinear conditions. The K-means clusteringbased measurement set partition method and state spacegating method based on the likelihood relationshipbetween measurement and target state both would leadto significant decline in computational complexity, with noobvious decline of tracking performance during stabletracking and only decline on some special target cases.

From experiments, we find that track-cross wouldcause fuzzy measurements and lead to obvious decline ofET-P-PHD filter's tracking performance, which might arisefrom the theoretical limitations of ET-PHD filter. Thereason is that under such conditions we cannot get a wellpartition of the measurement set, and not good trackingresult. Therefore, tracking unresolved targets would be thefocus of our future research. In addition, as to extendedtarget of complex shape, the form of extended targetmeasurement likelihood would be quite complicated andET-P-PHD filter need frequent calculation of the value ofextended target measurement likelihood, which wouldlead to high computational requirement. Therefore, howto calculate the likelihood value with low computationalrequirement is worthy of our future attention.

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