Probability hypothesis density filter with adaptive parameter estimation for tracking multiple maneuvering targets

2016-11-23YngJinlongYngLeYunYunhoGeHongwei

CHINESE JOURNAL OF AERONAUTICS 2016年6期

Yng Jinlong,Yng Le,Yun Yunho,Ge Hongwei

aSchool of Internet of Things Engineering,Jiangnan University,Wuxi 214122,China

bKey Laboratory of Advanced Process Control for Light Industry(Ministry of Education),Wuxi 214122,China

Probability hypothesis density filter with adaptive parameter estimation for tracking multiple maneuvering targets

Yang Jinlonga,b,*,Yang Lea,Yuan Yunhaoa,Ge Hongweia

aSchool of Internet of Things Engineering,Jiangnan University,Wuxi 214122,China

bKey Laboratory of Advanced Process Control for Light Industry(Ministry of Education),Wuxi 214122,China

The probability hypothesis density(PHD)filter has been recognized as a promising technique for tracking an unknown number of targets.The performance of the PHD filter,however,is sensitive to the available knowledge on model parameters such as the measurement noise variance and those associated with the changes in the maneuvering target trajectories.If these parameters are unknown in advance,the tracking performance may degrade greatly.To address this aspect,this paper proposes to incorporate the adaptive parameter estimation(APE)method in the PHD filter so that the model parameters,which may be static and/or time-varying,can be estimated jointly with target states.The resulting APE-PHD algorithm is implemented using the particle filter(PF),which leads to the PF-APE-PHD filter.Simulations show that the newly proposed algorithm can correctly identify the unknown measurement noise variances,and it is capable of tracking multiple maneuvering targets with abrupt changing parameters in a more robust manner,compared to the multi-model approaches.

1.Introduction

Multiple target tracking(MTT)has gained wide attentions due to its theoretical and practical importance.Conventionally,the MTT problem was tackled from the perspective of data association.A number of tracking algorithms were developed in the literature on the basis of techniques including the joint probabilistic data association (JPDA),1jointintegrated probabilistic data association (JIPDA)2and multiple hypothesis tracking(MHT).3These methods are generally computationally intensive and some of them even have exponentially growing complexityas the targetnumber increases.Reduced-complexity techniques were proposed in Refs.4–6.They are better for real-time applications at the cost of degraded estimation accuracy.

Recently,the use of the random finite set(RFS)theory7–11attracted great interests,because it provides an elegant formulation of the MTT problem.But the obtained multi-target Bayesian filter is intractable in most practical scenarios due to the inherent combinatorial nature of multi-target state densities and the need for evaluating set integrals over high dimensional spaces.To deal with the intractability,the probability hypothesis density(PHD)filter7and the cardinalized PHD(CPHD)filter8were developed using the first-order moment and cardinality distributions.Existing closed-form realizations of PHD filters include the particle filter PHD(PF-PHD),9,10Gaussian mixture PHD(GM-PHD)filter11and various modified versions.12–15Different from the PHD and CPHD filters,the cardinality-balanced multi-target multi-Bernoulli(CBMeMBer)filter was proposed in Ref.16for MTT by directly propagating the approximate posterior density of the targets.These algorithms exhibit good performance only when the model parameters,such as the measurement noise variances,are known precisely.In the presence of unknown time-varying measurement noise variances,the variational Bayesian(VB)approximation method17–19can be employed to recursively estimate the joint PHDs of the multi-target states and the measurement noise variance.20,21However,these methods may suffer from performance degradation if targets manoeuver with unknown abruptly changing parameters.

For maneuvering target tracking,the use of the jump Markov system(JMS)that switches among a set of candidate models in a Markovian fashion has proved to be effective.22,23Pasha et al.24introduced the linear JMS into PHD filters and derived a closed-form solution for the PHD recursion.Furthermore,the unscented transform(UT)and the linear fractional transformation(LFT)were combined with the closed-form solution for the nonlinear jump Markov multitarget models in Refs.25,26.In Ref.27,a GM-PHD filter for jump Markov models was developed by employing the bestfitting Gaussian(BFG)approximation approach.These algorithms assume the Gaussianity of the PHD distribution,which may limit their application scope.The multiple-model particle PHD(MMP-PHD) filter,the MMP-CPHD filter and MMPCBMeMBer filter are implemented by using the sequential Monte Carlo(SMC)method and their improved versions were presented in Refs.28–30.Most of the MM-based filters track multiple maneuvering targets through the interaction of multiple models,which is realized via combining estimates from different models according to their respective model likelihoods.The difficulty of applying them in tracking targets with abruptly changing maneuvering parameters comes from the need to specify a prior set of candidate models.In other words,they may suffer from the curse of dimensionality:if we wish to account for multiple unknown parameters,the number of models needed would increase exponentially with the number of parameters.

In this work,we incorporate the adaptive parameter estimation(APE)technique into the PHD filter for addressing the problem of multiple maneuvering target tracking,where both static and time varying unknown parameters,namely the measurement noise variance and the parameters associated with abrupt target maneuvers,are presented and need to be estimated.The inverse Gamma(IG)distribution is used to approximate the posterior distribution of the measurement noise variances while the adaptive Liu and West(LW)filter is adopted to propagate the posterior marginal of the timevarying parameters as a mixture of multivariate Gaussian distributions.31–33The obtained APE-PHD filter is realized using the particle filter(PF),which leads to the PF-APE-PHD algorithm for tracking multiple maneuvering targets in the presence of unknown model parameters.Simulation results show that the proposed algorithm exhibits better robustness and improved tracking performance over the MM-PHD and MM-CPHD algorithms.

The remainder of this paper is organized as follows.Section 2 formulates the problem of tracking a target in the presence of unknown model parameters.It also briefly reviews the APE technique and the PHD filter.Section 3 develops the APE-PHD algorithm and presents the closed-form solution,the PF-APE-PHD algorithm.Simulation results are given in Section 4.Finally,conclusions are provided in Section 5.

2.Preliminary

2.1.Problem formulation

The state-space model for tracking a single target moving on a two-dimensional plane is given by

where xk=[xk，vxk，yk，vyk]Tdenotes the target state at time k,（xk，yk）and （vxk，vyk）denote its position and velocity.F and G are the state transition matrix and the process noise gain matrix.ykis the measurement vector.vkand wkdenote the process noise and the measurement noise.They are independent of each other and modeled as zero-mean Gaussian random processes with covariance Qkand Rk.

In many practical applications,the state-space model in Eqs.(1)and(2)may contain unknown parameters.For example,if the target conducts a coordinated turn(CT),28the state transition matrix would become

The turn rate ω may be unknown and time-varying.Besides,the measurement noise covariance Rkmay also be unknown.In these scenarios,we need to jointly estimate the posterior distribution of the target states and the unknown parameters from the measurements.

Let Φkbe a column vector that collects the static and timevarying parameters in the state-space model.The posterior probability density function(PDF)of the target state vector xkand Φkconditioned on the measurements up to time k is,according to Bayes’rule,

where p（xk，Φk|y1:k-1）is the predicted PDF given by

Deriving exact recursive solutions for the posterior distribution p（xk，Φk|y1:k）from Eqs.(4)and(5)is in general intractable and as a result,approximate solutions are usually resorted to.One such approach is the SMC method,also referred to as the particle filter(PF).9,11,14

2.2.Adaptive parameter estimation(APE)

In Refs.31,32,the Liu and West(LW)filter was proposed for the joint identification of static parameters and target states.In particular,the marginal posterior distribution of the unknown parameters is approximated and propagated using a mixture of multivariate Gaussian distributions.In Ref.33,the particle learning technique was introduced into the LW filter.The obtained APE filter can handle both static and timevarying parameters.

The development of the APE method starts with factorizing p（xk，Φk|y1:k-1）into

Let Φk=[θk，ξk],where θkand ξkcollect static and timevarying parameter vectors.The marginal predicting distribution of Φkcan be expressed as

The predicted distribution p（θk|y1:k-1，ξk）of the static parameter vector θkis characterized using sufficient statistics sk,i.e.,θk～ p（θ|sk）.32The predicted distribution of the time-varying parameter vector ξkis approximated via

In the case that the time instant k is a changepoint,and the predicting distribution of the time-varying vector ξkwill be reset to pξ（ξ0）,its prior distribution.With the predicted PDF given in Eq.(8),the APE filter utilizes the PF to produce an approximation of the posterior distribution）in Eq.(4).Suppose at time k-1,the posterior distribution is represented by N particleswith weightsAt time k,each particle is given two weights33

which essentially leads to 2N particles.correspond to the probability of the current measurement ykwhen there is no changepoint and when there is a changepoint,respectively.In the former case,the value of time-varying parameter vectoris drawn from the Gaussian componentwhile for the latter case,its valueis produced using the prior distribution pξ（ξ0）(see also Eq.(8)).A resampling is then performed on the basis of the weightsto select N particles out of 2N particles and propagate them to generate the approximation of the posterior p（xk，Φk|y1:k）at time k.For more details on the APE filter for tracking a single maneuvering target,please refer to Ref.33.

2.3.PHD filter

Under the RFS framework,we denote the multiple target state set and the measurement set at time k as Xk={xk，1，xk，2，...，xk，Nk}and Yk={yk，1，yk，2，...，yk，Mk}.Both Nkand Mkare random integers and they are the number of targets and measurements,respectively.Suppose Xk-1is the multiple target state set at time k-1,then Xkand Ykcan be expressed as

where Sk|k-1（x）is the RFS of targets surviving from time k-1 to k,Bk|k-1（x）is the RFS of targets spawned from Xk-1and Γkis the RFS of targets that appear spontaneously at time k.Θk（x）and Kkare the RFSs of measurements originating from the targets in Xkand the clutters.

The optimal Bayesian recursions for propagating the multitarget posterior PDF are7

The PHD filter proposed by Mahler7yields an approximation of the optimal Bayesian filter given in Eqs.(15)and(16)via propagating only the first-order moment of pk|k（Xk|Y1:k）,i.e.,the PHD.It is capable of tracking a variable number of targets and estimating both the number of targets and their states without utilizing data association techniques.The PHD is a multi-peak function in the state space.The number of peaks is often(but not necessarily)approximately equal to the number of targets,and the peak positions correspond to the expected values of target states,which can be extracted through the use of the expectation-maximum(EM)algorithm35,36or clustering techniques.9,37

Let vk|k-1（x）and vk|k（x）denote the predicted and posterior intensity functions of pk|k（Xk|Y1:k）.Their prediction and update equations are

where βk|k-1（x）and γk（x）are the intensities of the RFSs of the spawned targets and spontaneous births.pS，k|k-1（x）denotes the survival probability and pD，k（x）is the detection probability.κk（y）= λkck（y）is the intensity of the clutter RFS,which is assumed to be Poisson distributed with mean rate λk,and ck（y）is the distribution of the clutter.

3.Tracking multiple maneuvering targets

3.1.APE-PHD recursions

We shall first generalize the PHD recursions in Eqs.(17)and(18)to take into account the presence of unknown model parameters.To simplify the presentation,it is assumed that the survival and the detection probabilities are independent of both the target state vector and the unknown parameter vector Φk.They will thus be denoted by pS，k|k-1and pD，k.Further drop the subscript k in Φkfor notation simplicity and let vk-1（x，Φ）be the joint posterior PHD at time k-1.According to Eq.(17),Eq.(18)and the Chapman-Kolmogorov equation,the predicted PHD vk|k-1（x，Φ）can then be described as

When the latest measurements become available at time k,the joint posterior PHD becomes

where

Note that in Eqs.(19)–(21),because Φ is unknown,the measurement likelihood gk（y|x，Φ）and vk|k-1（x，Φ）are hard to be obtained,this makes it difficult to calculate the analytic solution of the joint intensity function vD，k（x，Φ|y）.However,its approximation solution can be obtained through the use of the APE technique combined with the PF.The proposed algorithm is therefore referred to as PF-APE-PHD algorithm,which will be presented in the following subsection.

3.2.PF-APE-PHD algorithm

In this subsection,the PF is utilized to derive an approximation of the closed-form solution to the extended PHD recursions in Eqs.(19)and(20).The obtained PF-APE-PHD algorithm consists of two stages,namely the prediction and update stages.

We assume that initially,there are N0targets and N particles are produced for each target.The total number of particles is therefore L0=N×N0.Let X0be the initial multiple target state set and p0（X0，θ0，ξ0）be the prior joint PDF.The initial particlesare drawn from p0（X0，θ0，ξ0）,and the weight is set to beAPE-PHD recursions are as follows.

3.2.1.Prediction stage

In this work,we consider the case that the static unknown parameter θkis the variance of the measurement noise.Its conjugate prior is approximated by an inverse-gamma(IG)distribution IG（a，b）with parameters a and b,i.e.,IG（θ;a，b）=method for estimating a and b is similar to the method20used to identify the unknown measurement noise covariance R.Therefore,the details are omitted here.

To account for the possible abrupt changes in the timevarying parameters,we evaluate Eqs.(9)and(10)to obtain the estimate of the meansand covariancefor the time-varying parameter particleItis noted thatof Eqs.(9)and(10)in this algorithm is the Monte Carlo posterior mean of allwhich belong to the same target cluster.The particle clusters are formed in the stage of state extraction presented later in this subsection.

We then generate 2Lk-1particles as in the APE filter.For this purpose, the proposal distributions）used to produce predicted particles as in Refs.7,14are employed here.The first Lk-1particles are generated under the condition that the time-varying parameters do not change abruptly.They are obtained via

The remaining Lk-1particles are produced under the assumption that abrupt changes occurred.As in the APE technique,the values of the time-varying parameters are now drawn from their prior distributions,i.e.,.The predicted particles and their weights are obtained via, for i=Lk-1+1，Lk-1+2，...，2Lk-1,

At time k,each particle is also given another weight proportional to the predictive likelihood corresponding to changepoint parameteri.e.,

We select Lk-1out of the 2Lk-1obtained particles.Denote their indices as li∈ {1，2，...，2Lk-1},where i=1，2，...，Lk-1,the selection process is as follows.

(1)For i=1，2，...，Lk-1,select indices liwith probability（1- β）ω（

1li）from[1，2，...，Lk-1]andβω（2li）from[Lk-1+1，...，2Lk-1],where β is the probability that an abrupt change occurred and it is assumed to be known(see also Section 2.2).

(2)If li∈ {1，2，...，Lk-1},then update the time-varying parameter particles using

where Vk-1is given in Eq.(10).Set the composite parameter particle asand the sufficient statistics for the static parameters as

3.2.2.Update stage

After receiving the measurement at time k,the Lk-1+Jkparticle weights can be updated by

3.2.3.Computation of the total mass

3.2.4.Resampling

3.2.5.Extraction of target states

Target states can be obtained by clustering the particles and the cluster centers are the estimated states,where N^k=round（Nk）is the estimate of the target number,and round（·）denotes the rounding operator.

4.Simulations

In order to illustrate the performance of the proposed PFAPE-PHD algorithm,a two-dimensional tracking scenario is simulated.The benchmark techniques are the MMP-PHD,28MMP-CPHD and MMP-CBMeMBer filters.29In the considered scenario,the measurements are obtained at four stationary sensors located at （0，0）m,（0，1 × 104）m,（1 × 104，0）m,and （1 × 104，1 × 104）m.At time k,each sensor outputs the measured bearing of the received signal,which is given by

where （xSi，ySi） denotes the location of the ith sensor,i=1，2，3，4.wkis the zero-mean Gaussian noise with variance σ2w=1×10-4rad2.

There are three maneuvering targets.Targets 1 and 2 remain active throughout the whole simulation process and theirinitialpositionsare at （-3 × 103，5 × 103）m and（1.4 × 104，8 × 103）m,as in Ref.28.Target 3 is a spontaneous birth at 10th min with initial position （2×103，10.5 × 103）m and disappears at 50th min.The true tracks of the three targets are depicted in Fig.1.

When realizing the MMP-PHD and MMP-CPHD algorithms,we set that they both consist of a constant velocity(CV)model and two CT models.The transition probability matrix is assumed to be

where the sampling interval is T=1 min,and the sojourn durations are τ1=200 min and τ2=100 min.The initial model probabilities for the three models are all equal to 1/3.The state evolution for CV and CT models is

where σ2v=1 × 10-4m2s-3.

Fig.1 True target tracks.

For the three algorithms in consideration,we model the birth process using a Poisson RFS with intensity

To verify the effectiveness of the proposed algorithm,simulations are performed on a Lenovo T430 desktop with Intel(R)Core(TM)CPU i5-3210 M,2.50 GHz and 8 GB RAM.Two performance metrics are used.One is the statistics of the target number estimate.The other is the optimal subpattern assignment(OSPA)38distance defined as

Three simulation experiments are performed and the results shown are obtained from Monte Carlo simulations of 200 ensemble runs.The first experiment is to evaluate the performance for multiple abruptly maneuvering target tracking,where only the maneuvering parameters(e.g.,turn rates)are unknown.The second experiment is to compare the performance of the proposed algorithm in the presence of unknown maneuvering parameters as well as unknown measurement noise variances.The last experiment is conducted using different measurement noise variances to evaluate the robustness of the proposed algorithm.

4.1.Multiple abruptly maneuvering target tracking

In this experiment,the standard deviation of the measurement noise is set to be σ=0.01 rad and it is assumed known for the considered PF-APE-PHD,MMP-PHD,MMP-CPHD and MMP-CBMeMBer algorithms.The turn rate ω is considered as an unknown and time-varying parameter for the proposed PF-APE-PHD algorithm.The MMP-PHD,MMP-CPHD and MMP-CBMeMBer algorithms use one CV and two CT models of Eqs.(35)and(36)as the target motion models.Although in practice,the true turn rates are unavailable for the IMM-based filters,we realize the CT models with the real turn rates ω =9°/min and ω =-9°/min so that the MMP-based methods would have the ‘optimal’performance.Simulation results for this experiment are shown in Figs.2–4.

Fig.2 shows the average target number estimates obtained by the PF-APE-PHD,MMP-PHD,MMP-CPHD and MMPCBMeMBer filters.It can be seen that the proposed PF-APEPHD algorithm can even provide more accurate target number estimates than the benchmark techniques.The reason is that the proposed algorithm can effectively estimate jointly the unknown model parameter ω which can be well matched with the motion model of each target.While for the MMP-PHD,MMP-CPHD and MMP-CBMeMBer algorithms,the tracking accuracy is affected by the model interference due to the interaction of multiple models,an inevitable phenomenon of IMM-based techniques,which renders their performance under‘optimal’parameter settings still inferior to the proposed technique.Moreover,it is noticed that the MMP-CPHD and MMP-CBMeMBer algorithms have better performance in terms of more precise target number estimates than the MMP-PHDalgorithm.ThereasonisthattheMMPCBMeMBer method propagates the parameterized approximation of the posterior cardinality distribution,and the MMP-CPHD method jointly propagates the cardinality distribution and the intensity function,whereas the MMP-PHD method propagates the cardinality mean only with a single Poisson parameter.

Fig.3 compares the OSPA distances of the four simulated algorithms,and it is clear that the proposed algorithm again outperformstheMMP-PHD,MMP-CPHDandMMPCBMeMBer algorithms.This is also due to the fact that the proposed method can adapt to the temporal evolution of the target maneuvering parameters.It is worth noting that when the third target disappears at 50th min,the OSPA distance of the MMP-CPHD algorithm increases suddenly,which indicates that the ‘spooky action’problem steps in,i.e.,it is beneficial when missed detection occurs in MMP-PHD but is harmful when targets really disappear.

Fig.2 Target number estimates.

Fig.3 OSPA distance statistics.

Fig.4 Average run time.

Fig.4 shows the average run time of the four algorithms in consideration.It can be seen that the average run time of the proposed PF-APE-PHD algorithm is slightly larger than that of the MMP-PHD algorithm.The reason is that with the APE technique,the proposed algorithm generates twice more particles with different parameter predictions in the predicted step and has an additional particle selection step.It is noted that the complexity of the MMP-CBMeMBer is slightly lower than that of the MMP-PHD algorithm,because this method allows reliable and inexpensive extraction of state estimates without particle clustering.

4.2.Multiple abruptly maneuvering target tracking with unknown measurement noise variance

In this experiment,the true standard deviation of the measurement noise is fixed at σ=0.01 rad,but it is unknown for the proposed algorithm.We then apply the PF-APE-PHD algorithm to identifying it together with the time-varying turn rate ω and the target states.For comparison purposes,we also simulate the PF-APE-PHD filters with other assumed values of the measurement noise variance(i.e.,σ=0.005,0.01,0.015,0.03,0.06,0.1).The simulation results are summarized in Figs.5 and 6.

It is clear that when the measurement noise variance is estimated jointly with the target states,the performance of the PFAPE-PHD algorithm is very close to that when the measurementnoise variance isaccurately known in advance(σ=0.01 rad).This indicates that the proposed PF-APEPHD algorithm can achieve accurate joint parameter and target state estimations.On the other hand,if PF-APE-PHD simply operates with an incorrect setting of the measurement noise variance,it would suffer from significant performance degradation,mainly due to model mismatch.

4.3.Performance with different measurement noise variance settings

In this experiment,we realize two versions of the PF-APEPHD filter,one filter with unknown measurement noise variance and another filter with true measurement noise variance.The simulation results with different measurement noise standard deviations(i.e.,σ=0.01,0.02,0.03,0.04,0.05,0.06,0.07)are shown in Fig.7.As can be seen the estimation accuracy of the PF-APE-PHD algorithm with unknown σ is close to that of the PF-APE-PHD algorithm with the true value of σ known in advance.It is shown that the proposed algorithm has a good performance for multiple target tracking with unknownmeasurementnoiseparametersandthetimevarying abruptly changing maneuver parameters.

Fig.5 Target number estimates with different measurement noise standard deviations.

Fig.6 OSPA distance statistics with different measurement noise standard deviations.

Fig.7 Average OSPA distance statistics with different measurement noise standard deviations.

5.Conclusions

In this paper,we developed a new MTT algorithm,the PFAPE-PHD filter,to handle the presence of unknown model parameters including e.g.,the measurement noise variance that is static and the parameters in accordance with the target maneuvers that may be time-varying and subject to abrupt changes.The development started with extending the PHD filter to take into account the unknown parameters and the APE technique was incorporated to achieve online parameter estimation.The SMC approach was utilized to derive the approximate closed-form solution.Simulations showed that the newly proposed PF-APE-PHD filter can offer higher tracking accuracy in the case of multiple maneuvering targets over the existingMMP-PHD,MMP-CPHD andMMP-CBMeMBer algorithms.It is also applicable to the case with unknown measurement noise parameters for multiple maneuvering target tracking.

In future works,we shall consider introducing the APE technique into the spline PHD filter,39the CPHD filter8and the CBMeMBer filter40,41to obtain good algorithms for tracking multiple targets with unknown abrupt changing parameters.

Acknowledgments

This paper was supported by the National Natural Science Foundation of China(Nos.61305017,61304264)and the NaturalScienceFoundation ofJiangsu Province(No.BK20130154).

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Yang Jinlong is an associate professor in Jiangnan University.He received his M.S.degree in circuit and system from Northwest Normal University,China in 2009,and his Ph.D.degree in Pattern Recognition and Intelligent System from Xidian University,China,in 2012.His research interests include target tracking,information fusion and signal processing.

31 December 2015;revised 26 March 2016;accepted 22 September 2016

Available online 21 October 2016

Adaptive parameter estimation;

Multiple target tracking;

Multivariate Gaussian distribution;

Particle filter;

Probability hypothesis density

*Corresponding author.Tel.:+86 0510 85912085.

E-mail address:yjlgedeng@163.com(J.Yang).

Peer review under responsibility of Editorial Committee of CJA.

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http://dx.doi.org/10.1016/j.cja.2016.09.010

This is an open access article under the CC BY-NC-ND license(http://creativecommons.org/licenses/by-nc-nd/4.0/).

CHINESE JOURNAL OF AERONAUTICS

2016年6期