GaussianmixturePHDfilterforjumpMarkovmodelsbasedonbestfittingGaussianapproximation

Fast communicationGaussian mixture PHD filter for jump Markov models based on best-fitting Gaussian approximationWenling Li a ,Ã,Yingmin Jia a ,ba The Seventh Research Division and the Department of Systems and Control,Beihang University (BUAA),Beijing 100191,ChinabKey Laboratory of Mathematics,Informatics and Behavioral Semantics (LMIB),Ministry of Education,SMSS,Beihang University (BUAA),Beijing 100191,Chinaa r t i c l e i n f oArticle history:Received 11June 2010Received in revised form 17July 2010Accepted 9August 2010Available online 12August 2010Keywords:Multiple maneuvering targets tracking Probability hypothesis density filter Jump Markov systemBest-fitting Gaussian approximationa b s t r a c tA new Gaussian mixture probability hypothesis density (PHD)filter is developed for tracking multiple maneuvering targets that follow jump Markov models.This approach is based on the best-fitting Gaussian approximation which has been shown to be an accurate predictor of the interacting multiple model (IMM)pared with the existing Gaussian mixture multiple model PHD filter without interacting,simulations show that the proposed filter achieves better results with much less computational expense.&2010Elsevier B.V.All rights reserved.1.IntroductionThe random finite set (RFS)approach to multi-target tracking has received considerable attention in recent years.By representing the multi-target state and multi-target measurement as random finite sets (RFSs),the finite set statistics (FISST)provides a rigorous Bayesian framework for multi-target tracking [1–3].Compared with the traditional association-based techniques [4–6],the difficulty caused by data association is avoided in the RFS formulation.However,the optimal multi-target Bayes filter is generally intractable due to the combinatorial nature of the multi-target densities and multiple set integrals.The probability hypothesis density (PHD)filter,which aims to recursively propagate the first order moment or the intensity function associated with the multi-targetposterior density,provides a computationally tractable alternative.Recently,two implementations of the PHD filter including the sequential Monte Carlo PHD (SMC-PHD)[7–10]and the Gaussian mixture PHD (GM-PHD)[11–16]have been developed.An added advantage of the GM-PHD filter is that it allows the state estimates to be extracted from the posterior intensity in a much more efficient and reliable manner than the SMC-PHD filter [13].For tracking maneuvering targets,the jump Markov model or the switching multiple model approach has shown to be highly effective [17].Multiple model PHD filters,which can be considered as natural extensions of the SMC-PHD and the GM-PHD have been proposed [18–20].It should be mentioned that the existing Gaussian mixture multiple model PHD filters are not interacting [18].In other words,the elemental filters for different models work independently which is similar to the mechanism of the autonomous multiple model approach [17].As stated in [18],how the PHD filter approach can be extended to the interacting multiple model (IMM)remains an interesting and challenging problem in both theory and practice.In this work,thisContents lists available at ScienceDirectjournal homepage:/locate/sigproSignal Processing0165-1684/$-see front matter &2010Elsevier B.V.All rights reserved.doi:10.1016/j.sigpro.2010.08.004ÃCorresponding author.Tel.:+861082338683;fax:+861082316100.E-mail addresses:lwlmath@ (W.Li),ymjia@ (Y.Jia).Signal Processing 91(2011)1036–1042difficulty is circumvented by using the best-fitting Gaussian(BFG)approximation since it has been shown to be an accurate predictor of the IMM performance[21].In this paper,we propose a novel Gaussian mixture implementation to the PHDfilter that accommodates jump Markov models.The basic idea of this approach is to approximate the multi-model prior probability density function with a BFG distribution at each recursion.Thus, the multiple model estimation for jump Markov linear system(JMLS)is reverted to the single model estimation for a linear Gaussian system.Then the GM-PHDfilter can be directly applied to the approximated linear Gaussian system.The key advantages of the proposed algorithm over the existing Gaussian mixture multiple model PHD filter are twofold.First,more accurate estimates can be expected since the BFG approximation is in close agree-ment with the performance of the IMM estimator.Second, much less computational expense is consumed as the resultingfilter requires single model estimation.Simula-tion results are presented to illustrate the performance of the proposed approach in terms of tracking accuracy and computational cost.The rest of this paper is organized as follows.Section2 gives a brief review of the multi-target tracking model and the PHDfilter.Section3introduces the BFG approximation and summarizes the proposed GM-PHD filter.The performance of the proposedfilter is evaluated by a numerical example in Section4.Conclusion is drawn in Section5.2.Multi-target tracking with the PHDfilterIn the multi-target tracking scenario,the aim involves the joint estimation of an unknown and time-varying number of targets as well as their individual states from a sequence of noise-corrupted measurements.In addition, the number of measurements may also vary as not all targets generate measurements and the existence of clutter.More importantly,the orders of the states and the measurements bear no significance.Thus,it is natural to represent the multi-target state and multi-target measurement as two RFSs[1]X k9f x k,1,...,x k,nkg&Xð1ÞZ k9f z k,1,...,z k,mkg&Zð2Þwhere x k,1,...,x k,nk2X are the target states,z k,1,...,z k,mk 2Z are the received measurements.X&R n and Z&R pdenote the state and observation space,respectively.n k and m k denote the number of targets and the number of received measurements at time k,respectively.From the FISST theory,thefirst order moment of an RFS X on X is a non-negative function nðxÞwith the property that for any measurable subset S&XZSnðxÞdx¼Zj X\S j PðdXÞð3Þwhere PðdXÞis the probability distribution of X,and nðxÞis called as the PHD or the intensity function.Moreover,theNomenclatureBFG best-fitting GaussianEKF extended KalmanfilterFISSTfinite set statisticsGM Gaussian mixtureIMM interacting multiple modelOSPA optimal subpattern assignmentPHD probability hypothesis densityRFS randomfinite setSMC sequential Monte CarloUKF unscented Kalmanfilterc cut-off of OSPACov covariance operatorE expectation operatorf single-target transition densityF k r,G k r system transition matrices of model r h single-target measurement likelihood J k number of Gaussian terms for n kJ kþ1j k number of Gaussian terms for n kþ1j k J max maximum number of Gaussian terms m k number of measurements at time k M number of target motion modelsM k+1r event that model r is effect at[k,k+1) n k number of targets at time kp order of OSPAp k+1,r probability of the event M k+1rp D,k detection probability at time kp S,k target-survival probability at time k Q k r covariance matrix of w k for model rr k discrete-time Markov chainR k covariance matrix of measurement noise S measurable subsetT Th pruning thresholdU Th merging thresholdv k measurement noise at time kw k process noise at time kw Th weight thresholdx k,i i th target state at time kX k multi-target state at time kz k,j j th measurement at time kZ k multi-target measurement at time kbk j kÀ1intensity of the spawned RFS at time k gkintensity of the birth RFS at time ke k expectation of the state vector x kk k intensity of the clutter RFS at time kl c average number of clutter pointsp ij transition probability of Markov chainn k j kÀ1predicted intensity at time kÀ1n k posterior intensity at time kY k covariance matrix of the state vector x k S k covariance matrix of w k for BFG model F k system transition matrix of BFG modelA jump Markov modelB BFG modelP probability measureX,Z randomfinite setsW.Li,Y.Jia/Signal Processing91(2011)1036–10421037integral yields an expected number of elements in X that present in S and the corresponding highest peaks give the state estimates of the elements.The PHD filter propagates the intensity functions of multi-target RFSs recursively under the following as-sumptions [13]:(1)Each target evolves and generates measurementsindependently of one another.(2)The clutter RFS is Poisson and is independent of targetgenerated measurements.(3)The predicted multi-target RFSs are Poisson.An RFS X is Poisson means that the distribution of thecardinality of X is Poisson with mean N ¼Rn ðx Þdx and the elements in X are independent and identically distributed with probability density n ðx Þ=N .It can be seen that the statistics of the Poisson RFS can be completely characterized by its first order moment or the intensity function n ðx Þ.To be specific,given the posterior intensity n k À1ðx Þat time k À1,the predicted intensity n k j k À1ðx Þis calculated byn k j k À1ðx Þ¼Z½p S ,k ðx Þf ðx j x Þþb k j k À1ðx j x Þ n k À1ðx Þd x þg k ðx Þð4Þand the posterior intensity n k ðx Þis updated asn k ðx Þ¼½1Àp D ,k ðx Þ n k j k À1ðx ÞþXz 2Z p D ,k ðx Þh ðz j x Þn k j k À1ðx Þk kðz ÞþR p D ,k ðx Þh ðz j x Þn k j k À1ðx Þd x ð5Þwhere f ðÁjÁÞand h ðÁjÁÞare the single-target transitiondensity and likelihood function,respectively.k k ðÁÞdenotes the intensity of the clutter RFS,b k j k À1ðÁj x Þdenotes the intensity of the spawned target RFS,and g k ðÁÞdenotes the intensity of the spontaneously birth target RFS.p S ,k (x )and p D ,k (x )are the probabilities of the target-survival and the detection given a state x ,respectively.3.GM-PHD filter based on BFG approximation 3.1.JMLS with BFG approximationConsider the following target motion model with Markovian switching:x k þ1¼F k ðr k þ1Þx k þG k ðr k þ1Þw k ðr k þ1Þð6Þwhere x k 2R n is the target state at time k ,F k (r k +1)and G k (r k +1)denote the transition matrices of model r k +1.r k +1specifies the target motion model which is in effect during the time interval [k ,k +1).w k (r k +1)is the additive zero-mean white Gaussian noise with covariance Q k (r k +1).We assume that the target motion can switch between M models.The evolution of motion models follows a discrete-time homogeneous Markov chain with known transition probability p ij ¼Pr f r k þ1¼j j r k ¼i g .The objective is to express the dynamics of the JMLS (6)with the BFG approximation x k þ1¼F k x k þw kð7Þwhere w k is a zero-mean white Gaussian random vectorwith covariance matrix S k ,i.e.,w k $N ð0,S k Þ.In other words,we want to replace the JMLS (given by (6)and referred to as ‘‘A ’’)with a single BFG distribution (given by (7)and referred to as ‘‘B ’’).F k and S k are determined such that the distribution of x k has the same mean and covariance under each model,i.e.,E f x k j A g ¼E f x k j B g ð8ÞCov f x k j A g ¼Cov f x k j B gð9ÞSimilar to the calculation in [21],the system matrix F k and the covariance matrix S k of w k can be determined as follows.For simplicity,we denote F k (r ),G k (r )and Q k (r )byF k r ,G k r and Q k r,respectively.First,using the total probability theorem,we have E f x k þ1j A g ¼X M r ¼1E f x k þ1j M r k þ1,A g Pr f M r k þ1j A g¼X M r ¼1p k þ1,r F rk E f x k j A gð10Þwhere M r k +1denotes the event that model r is in effectduring the sampling period [k ,k +1)and p k +1,r is the probability of the event M r k +1.On the other hand,it follows from (7)that E f x k þ1j B g ¼F k E f x k j B gð11ÞHence,comparing (10)with (11)yieldsF k ¼X M r ¼1p k þ1,r F rkð12ÞNext,it can be easily shown that [22]Cov f x k þ1j A g ¼E f Cov f x k þ1j M r k þ1,A ggþCov f E f x k þ1j M rk þ1,A ggð13ÞwhereE f Cov f x k þ1j M r k þ1,A gg¼X M r ¼1p k þ1,r ½F r k Cov f x k j A g½F r k T þG r k Q r k ½G r k Tð14ÞCov f E f x k þ1j M rk þ1,A gg ¼X M r ¼1f p k þ1,r F r k E f x k j Ag E f x k j A g T ½F r k TgÀF k E f x k j A g E f x k j A g T F T kð15ÞDefinee k 9Ef x k j A gð16ÞY k 9Cov f x k j A gð17ÞSubstituting (16)and (17)into (14)and (15),we can thenobtainY k þ1¼X M r ¼1p k þ1,r f F r k ½Y k þe k e T k ½F r k T þG r k Q r k ½G r k T gÀF k e k e T k F Tkð18Þe k þ1¼F k e kð19ÞOn the other hand,it follows from (7)that Cov f x k þ1j B g ¼F k Cov f x k j B g F T k þS kð20ÞW.Li,Y.Jia /Signal Processing 91(2011)1036–10421038Hence,comparing(13)with(20)yieldsS k¼Y kþ1ÀF k Y k F Tkð21ÞUp to now,we have obtained the recursive formula for calculating the system matrix F k and the covariance matrix S k.In other words,the multiple model estimation for JMLS(6)can be reverted to the single model estimation for linear Gaussian system(7)by using the BFG approximation.3.2.GM-PHDfilter based on BFG approximationAssume that the state dynamics and measurements of each target can be modeled asfðx k j x kÀ1Þ¼Nðx k;F kÀ1x kÀ1,S kÀ1Þð22Þhðz k j x kÞ¼Nðz k;H k x k,R kÞð23Þwhere F kÀ1¼P Mr¼1p kþ1,r F rkand S kÀ1¼Y kþ1ÀF k Y k F T kare calculated by the above BFG approximation at each stage.H k and R k denote the measurement matrix and the covariance matrix of the measurement noise,respectively. Note that H k and R k do not evolve with time according to the switching parameter r k.This is reasonable since the measurements from the sensors remain the same with respect to the system states.The target-survival probability p S,k and the detection probability p D,k are both state independent,and the intensities of the birth and spawning RFSs are Gaussian mixturesg k ðxÞ¼X J g,kj¼1w j g,kNðx;m j g,k,P j g,kÞð24Þbk j kÀ1ðx j xÞ¼X J b,kl¼1w l b,kNðx;F l b,k xþd l b,k,Q l b,kÞð25Þwhere J g,k,w j g,k ,m j g,kand P j g,kare given parameters thatdetermine the shape of the birth intensity.J b,k,w lb,k ,F lb,k,d lb,kand Q lb,kare given parameters thatdetermine the shape of the spawning intensity.It should be mentioned that these intensities are not assumed to be mode-dependent as in[20]since the JMLS has been replaced by the linear Gaussian system.Based on the PHD recursion(4)and(5),we summarize the proposed algorithm.BFG approximation step:Given the mode probability p k,i,the mean e k and the covariance Y k,determine the matrices F k and S kp kþ1,r¼X Mi¼1p ir p k,ið26ÞF k¼X Mr¼1p kþ1,r F rkð27ÞY kþ1¼X Mr¼1p kþ1,r½F rkðY kþe k e T kÞ½F r k TþG r k Q r k½G r k T ÀF k e k e T k F T kð28ÞS k¼Y kþ1ÀF k Y k F Tkð29Þe kþ1¼F k e kð30ÞPrediction step:Given that the posterior intensity n kðxÞis a Gaussian mixturen kðxÞ¼X J kj¼1w jkNðx;m jk j k,P jk j kÞð31Þwhere J k is the number of Gaussian terms for n kðxÞat time k.Then the predicted intensity is also a Gaussian mixturewith the formn kþ1j kðxÞ¼n S,kþ1j kðxÞþn b,kþ1j kðxÞþgkþ1ðxÞð32Þwhere g kþ1ðxÞis given by(24),andn S,kþ1j kðxÞ¼p S,kþ1X J kj¼1w jkNðx;m jS,kþ1j k,P jS,kþ1j kÞð33Þn b,kþ1j kðxÞ¼X J kj¼1XJ b,kþ1l¼1w jkw l b,kþ1Nðx;m j,l b,kþ1j k,P j,lb,kþ1j kÞð34Þm jS,kþ1j k¼F k m jk j kð35ÞP jS,kþ1j k¼F k P jk j kF TkþS kð36Þm j,lb,kþ1j k¼F l b,kþ1m jk j kþd l b,kþ1ð37ÞP j,lb,kþ1j k¼F l b,kþ1P jk j k½F l b,kþ1TþQ l b,kþ1ð38ÞUpdate step:Given that the predicted intensity can berepresented as the form ofn kþ1j kðxÞ¼XJ kþ1j ki¼1w ikþ1j kNðx;m i kþ1j k,P i kþ1j kÞð39Þwhere J kþ1j k is the number of Gaussian terms for n kþ1j kðxÞat time k.Then the posterior intensity is updated asn kþ1ðxÞ¼ð1Àp D,kþ1Þn kþ1j kðxÞþXz2Z kþ1n D,kþ1ðx;zÞð40Þwheren D,kþ1ðx;zÞ¼XJ kþ1j ki¼1w ikþ1ðzÞNðx;m ikþ1j kþ1ðzÞ,P ikþ1j kþ1Þð41Þw ikþ1ðzÞ¼p D,kþ1w ikþ1j kq ikþ1ðzÞk kþ1ðzÞþp D,kþ1P J kþ1j kl¼1w lkþ1j kq lkþ1ðzÞð42Þq ikþ1ðzÞ¼Nðz;^z ikþ1j k,H kþ1P ikþ1j kH Tkþ1þR kþ1Þð43Þm ikþ1j kþ1ðzÞ¼m ikþ1j kþK ikþ1ðzÀ^z ikþ1j kÞð44Þ^z ikþ1j k¼H kþ1m ikþ1j kð45ÞP ikþ1j kþ1¼ðIÀK ikþ1H kþ1ÞP ikþ1j kð46ÞK ikþ1¼P ikþ1j kH Tkþ1ðH kþ1P ikþ1j kH Tkþ1þR kþ1ÞÀ1ð47ÞRemark1.It is worthwhile here to compare the proposedalgorithm with the GM-PHDfilter.The prediction step and W.Li,Y.Jia/Signal Processing91(2011)1036–10421039the update step are the same as those of the GM-PHD filter[13,20].The main difference lies in the fact that the system matrix and the noise covariance matrix are computed recursively by using the BFG approximation. As the number of Gaussian components increases without bound,the pruning and merging scheme is required,see [13]for more details.Remark2.Note that the BFG approximation is restricted to the linear dynamic models but places no such requirement on the measurement equation.Thus it is possible to handle nonlinear measurements by using nonlinearfilters such as the extended Kalmanfilter (EKF)or the unscented Kalmanfilter(UKF).Indeed,in the target tracking community,the target dynamics are often described by a linear kinematics model,and measurements are nonlinear with respect to the system states.4.SimulationsThis section presents a numerical example to compare the performance and the computational cost of the proposedfilter with that of the Gaussian mixture multiple model PHDfilter without interacting.Tracking model:Consider a two-dimensional scenario with an unknown and time-varying number of targets, which is similar to the example provided in[20].The state x k¼½p x,k_p x,k p y,k_p y,k T of each target consists of position (p x,k p y,k)and velocityð_p x,k_p y,kÞcomponents.The target dynamics is described by the following coordinated turn model:x k¼1sinðo TÞo0À1Àcosðo TÞo0cosðo TÞ0Àsinðo TÞ1Àcosðo TÞo1sinðo TÞo0sinðo TÞ0cosðo TÞ266666664377777775x kÀ1þT22T0T20T2666666437777775w kÀ1ð48Þwhere o denotes the turn rate and T=1is the sampling time period.Three models corresponding to different turn rates are used.Model1is a coordinated turn model with a turn rate of01/s and the standard deviation of noise is5m/s2. Model2is a coordinated turn model with a clockwise turn rate of31/s and the standard deviation of noise is20m/s2. Model3is a coordinated turn model with a counter-clockwise turn rate of31/s and the standard deviation of noise is20m/s2.The switching between three models is governed by afirst order Markov chain with known transition probability matrixP¼0:80:10:10:10:80:10:10:10:8264375ð49ÞThe measurement consisting of range and bearing is given byy k¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðp x,kÀs xÞ2þðp y,kÀs yÞ2qarctan½ðp x,kÀs xÞ=ðp y,kÀs yÞ2435þv kð50Þwhere[s x,s y]is the location of the sensor,and themeasurement noise v k is assumed to be zero-mean whiteGaussian with R¼diag f1002ðp=180Þ2g.The sensor islocated at[35,À60]km.The average number of clutterreturns per unit volume is taken as l c¼0:347ðrad kmÞÀ1over the region½Àp=2,p=2 rad½0,22 km,which gives anaverage of24clutter points per scan.In this work,the UKFis used to handle the nonlinearity of the measurement.It is assumed that targets can appear or disappear inthe scene at any time.The spontaneous birth RFS isPoisson with the following intensity:gkðxÞ¼0:1½Nðx;m1g,P gÞþNðx;m2g,P gÞ ð51Þwherem1g¼½40,0,À50,0 Tm2g¼½30,0,À40,0 TP g¼diag f106,104,106,104gThe intensity of the Poisson RFS of spawn births isgiven bybk j kÀ1ðx j xÞ¼0:05Nðx;x,Q bÞð52Þwhere Q b¼diag f104,400,104,400g.Simulation results:In our simulations,the survival andthe detection probabilities are set to p S,k=0.99andp D,k=0.98,respectively.The pruning threshold has beentaken as T Th=10À7,the merging threshold U Th=5,theweight threshold w Th=0.5and the maximum number ofGaussian terms J max=10(see[13]for the meanings ofthese parameters).The criterion known as optimalsubpattern assignment(OSPA)metric is used for perfor-mance evaluation.The OSPA metric has been consideredas a much more natural and intuitive interpretation fordemonstrating the localization and cardinality errors inthe multi-target tracking community.Readers are referredto[23]for more details on the definition of the OSPAmetric.Fig.1shows the true trajectories of four targets with atypical frame of cluttered measurements.Target1startsat time k=1with initial position at[40,À50]km and endsat time k=100;Target2is spawned from target1at time50and ends at time90;Target3starts at time k=5withinitial position at[30,À40]km and ends at time k=85;Target4is spawned from target3at time25and ends attime60.To verify the performance of the proposedfilter,100Monte Carlo runs are performed with independentlygenerated clutter and measurements for each trial.Theposition estimates of the proposedfilter for one trialshown in Fig.2indicate that thefilter provides accuratetracking performance.The OSPA distance for p=2andc=200is shown in Fig.3,which suggests that theproposedfilter gives more reliable estimates thanthe existing Gaussian mixture multiple model PHDfilter without interacting.To assess the computationalrequirements of the proposed method,we compute theaveraged CPU time in MATLAB7.1on a2.80GHz4CPUPentium-based computer operating under Windows XP(Professional).The proposedfilter consumed approxi-mately4.5s per sample run over100time steps,while theW.Li,Y.Jia/Signal Processing91(2011)1036–10421040Gaussian mixture multiple model PHD filter consumed 48.8s.From the above comparisons,a modest conclusion can be drawn that the proposed tracking algorithm can achieve better performance with much less computa-tional expense.5.ConclusionA novel Gaussian mixture PHD filter for tracking multiple maneuvering targets with Markovian switching dynamics is developed by employing the BFG approximation approach.2.62.833.2 3.4 3.6 3.844.2−5.2−5−4.8−4.6−4.4−4.2−4−3.84x104X coordinate (m)Y c o o r d i n a t e (m )Fig.1.Measurement data (projected on the x -and y -axes)and target trajectories:‘‘3’’locations of target birth;‘‘&’’locations of target death.2.62.833.2 3.4 3.6 3.844.2−5.2−5−4.8−4.6−4.4−4.2−4−3.84x 104X coordinate (m)Y c o o r d i n a t e (m )Fig.2.Position estimates of the 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