Extended Target Poisson Multi-Bernoulli Filter

Transcription

1 1 Extended Taret Poisson Multi-Bernoulli Filter Yuxuan Xia, Karl Granström, Member, IEEE, Lennart Svensson, Senior Member, IEEE, and Maryam Fatemi arxiv: v1 [eess.sp] 4 Jan 2018 Abstract In this paper, a Poisson multi-bernoulli (PMB filter for multiple extended tarets estimation is presented. The PMB filter is based on the Poisson multi-bernoulli mixture (PMBM conjuate prior and approximates the multi-bernoulli mixture (MBM in the posterior as a sinle multi-bernoulli. By only havin a sinle multi-bernoulli representin detected tarets in the posterior, the computational cost due to the data association problem can be effectively reduced. Different methods to mere the MBM are presented, alon with their amma Gaussian inverse Wishart implementations. The performance of the PMB filter is compared to the PMBM filter in different simulated scenarios. Index Terms Multiple taret trackin, extended taret, random matrix model, random finite sets, Bayesian estimation, variational inference I. INTRODUCTION Multiple taret trackin (MTT is the process of estimatin the set of taret trajectories based on a sequence of noisecorrupted measurements, includin missed detections and false alarms [1]. Traditionally, MTT alorithms have been tailored to the point taret assumption: each taret is modelled as a point without spatial extent, and each taret ives rise to at most one measurement per time scan. However, modern hih-resolution radar and lidar sensors make the point taret assumption unrealistic, because with such sensors it is common that a taret ives rise to multiple measurements per time scan. The trackin of such a taret leads to the so-called extended taret trackin problem, where the objective is to recursively determine the extent and kinematic parameters of the tarets over time. A detailed overview of extended taret trackin literature is iven in [2]. In this paper, we focus on the estimation of the current set of tarets, which is called multiple taret filterin 1. Random Finite Sets (RFS [3] is a popular and widely used framework that facilitates an eleant Bayesian formulation of the MTT problem. In the early stae, RFS-based MTT approaches were developed based on moment approximations of posterior multi-taret densities, includin the Probability Hypothesis Density (PHD filter [4] [6] and the Cardinalised PHD (CPHD filter [7], [8]. In recent years, a sinificant trend in RFS-based MTT is the development of conjuate multitaret distributions, meanin that the posterior multi-taret distribution has the same functional form as the prior. Yuxuan Xia is with Chalmers University of Technoloy, Götebor, Sweden. yuxuanx@student.chalmers.se Karl Granström and Lennart Svensson are with the Department of Electrical Enineerin, Chalmers University of Technoloy, Götebor, Sweden. firstname.lastname@chalmers.se Maryam Fatemi is with Zenuity, Götebor, Sweden. maryam.fatemi@zenuity.com 1 The ultimate output from an MTT alorithm is a set of taret trajectories, i.e., a set of taret state sequences. The ability to maintain taret trajectories over time is outside the scope of this paper. Two types of MTT conjuate priors can be found in the literature: the δ-generalized Labelled Multi-Bernoulli (δ- GLMB [9] and the Poisson Multi-Bernoulli Mixture (PMBM [10]. In the δ-glmb conjuate prior, labels are added to taret states facilitatin the formin of taret trajectories [9]. In the PMBM conjuate prior, the set of tarets is separated into two disjoint and independent subsets: tarets that have been detected, and tarets that have not been detected [10]. The differences between representin the multi-taret state density in the PMBM form and the δ-glmb form are discussed in [11]. Simulation results have shown that the PMBM filter [10] [13] outperforms the δ-glmb filter [14], [15] for both point [16] and extended [13] taret filterin. Data association is an inherent problem of MTT. Because each taret can enerate multiple measurements per time scan, the problem of data association is even more challenin in multiple extended taret trackin, than it in multiple point taret trackin. The PHD and the CPHD filters avoid handlin the data association uncertainty explicitly by moment approximation 2, however, comparisons have shown that PHD and CPHD filters have worse trackin performance, compared to MTT conjuate priors [13], [15]. Filters based on conjuate priors provide more accurate approximations to the exact posterior distributions, however, due to the unknown number of data associations, the number of multi-bernoulli (MB components in the posterior density of the δ-glmb and the PMBM filters row rapidly as more data is observed. Different approximation methods can be used to keep the number of MBs at a tractable level, see [11], [14] for point taret filterin, and [12], [13], [15] for extended taret filterin. The main advantae of usin conjuate prior in MTT, instead of moment approximations, is that the posterior distribution can be approximated arbitrarily well as lon as sufficient parameters are used. Simulation studies have shown that RFS methods based on MTT conjuate priors all outperform RFS methods based on moment approximations, see performance comparison [11], [14], [17] on point taret filterin and [12], [13], [15] on extended taret filterin. The computational cost of the PMBM and δ-glmb filters can be reatly reduced by approximatin the MB mixture (MBM in the posterior as a sinle MB. For example, the labelled MB (LMB filter is an efficient approximation of the δ-glmb filter, proposed in [17] for point taret and in [15] for extended taret. Similarly, approximatin the PMBM posterior as a sinle MB leads to the so-called Poisson MB (PMB filter. Different variants of the PMB filter have been developed for point taret filterin [10], [18], amon which the variational MB alorithm [18] provides the most 2 In the extended PHD/CPHD filter, different partitions of the set of measurements have to be computed in prior to moment approximation.

2 2 accurate taret state estimation. A performance evaluation of filters based on different MB conjuate priors for point taret estimation, iven in [16], has shown that the PMB filter has very promisin performance reardin estimation error and computational time. It is therefore of interest to develop a PMB filter for extended tarets. In this paper, we develop several different extended taret PMB filters, and compare them to the extended taret PMBM filter. The contributions are summarized as follows: 1 In Section V, we present the track-oriented PMB (TO- PMB filter, which is an adaptation of the track-oriented marinal MeMBer-Poisson (TOMB/P filter [10] for point tarets. We also analyze the drawback of this trackoriented merin approach, and hihliht the challenes in formin new tracks in the extended taret PMB filter. 2 In Section VI, we present a variational MB filter that considers the formin of pre-existin and new tracks separately. We first apply the variational MB alorithm [18] to form pre-existin tracks, then we use related ideas to propose a reedy method to form new tracks. Two different implementations of the variational MB alorithm are studied, one is based on the efficient approximation of the feasible set, proposed in [18], and the other is based on the most likely assinment, followin a similar optimization procedure used by the set joint probabilistic data association (SJPDA filter [19]. 3 In Section VII, we present implementations of the PMB filters for a common extended taret model. 4 In Section VIII, different variants of the PMB filter are compared to the PMBM filter in an extensive simulation study. In Section II, we ive some backround on Bayesian multitaret filterin, RFS modellin, the PMBM conjuate prior and some RFS mixture reduction techniques. In Section III, we present the problem formulation. In Section IV, we present the extended taret PMB filter recursion. Nomenclature is presented in Table I. II. BACKGROUND In this section, we first ive introductions to Bayesian filterin and RFS modellin. Next, we present the PMBM conjuate prior for multiple extended taret filterin. In addition, we describe techniques on how to approximate a mixture of Bernoulli RFSs and an MBM, respectively, in the sense of minimizin the Kullback-Leibler (KL diverence. A. Bayesian multi-taret filterin In RFS-based MTT methods, taret states and measurements are represented in the form of finite sets. Let x k denote the sinle taret state at discrete time step k, and let X k denote the taret set. In extended taret trackin, the taret state models both the kinematic properties, and the extent, of the taret. The taret set cardinality X k is a time-varyin discrete random variable, and each taret state x k X k is also a random variable. Let z m k denote the mth measurement at discrete time step k. The set of measurements obtained at TABLE I NOMENCLATURE a, b = a(xb(xdx: inner product of a(x and b(x. V : determinant of matrix V. X : cardinality of set X. Π N : set of permutation functions on I N {1,..., N} Π N = {π : I N I N i j π(i π(j}. : disjoint set union, i.e., Y U = X means that Y U = X and Y U =. D KL (p q = p(x lo ( p(x q(x dx: Kullback-Leibler diverence between p(x and q(x. Γ d ( : multivariate amma function. ϕ 0 ( : diamma function. Tr(X: trace of matrix X. I m: identity matrix of size m m. GAM(γ; a, b: probability density function of amma distribution defined on γ > 0 with shape a and rate b. IW d (χ; v, V : probability density function of inverse-wishart distribution defined on positive definite d d matrix with derees of freedom v and d d scale matrix V. time step k is denoted as Z k, includin clutter and taretenerated measurements with unknown oriin. The sequence of all the measurement sets received so far up to time step k is denoted as Z k. The objective of multi-taret filterin is to recursively compute the posteriori distribution of the taret set. Let f k k (X k Z k, f k,k 1 (X k X k 1 and f k (Z k X k denote the multi-taret set density, the multi-taret transition density and the multi-taret measurement likelihood, respectively. The multi-taret Bayes filter propaates in time the multitaret set density f k 1 k 1 (X k 1 Z k 1 usin the Chapman- Kolmoorov prediction f k k 1 (X k Z k 1 = f k,k 1 (X k X k 1 f k 1 k 1 (X k 1 Z k 1 δx k 1, (1 and the Bayes update f k k (X k Z k = f k (Z k X k f k k 1 (X k Z k 1 fk (Z k X k f k k 1 (X k Z k 1 δx k, (2 where the set interal, f(xδx, is defined in [3, Sec ]. B. Random set modellin Two basic forms of RFSs are the Poisson point process (PPP and the Bernoulli process. A PPP is an RFS whose cardinality is Poisson distributed, and for any finite cardinality, each element in the set is independent and identically distributed (i.i.d.. The PPP intensity D(x = µf(x is determined by the scalar Poisson rate µ and the spatial distribution f(x. The PPP density is iven by f(x = e µ x X µf(x. (3

3 3 A Bernoulli process with probability of existence r and existence-conditioned probability density function (PDF f(x has RFS density 1 r X = f(x = r f(x X = {x} (4 0 otherwise, where the cardinality X is Bernoulli distributed with parameter r [0, 1]. The Bernoulli process offers a convenient way to capture both the uncertainty reardin taret existence and state. Multiple independent tarets can be represented as a multi- Bernoulli RFS X, which is a disjoint union of independent Bernoulli RFSs X i, i.e., X = i I X i, where I is an index set. The RFS density of an MB process can be represented as [13] { f(x = i I X i =X i I f i (X i X I (5 0 X > I. The multi-bernoulli RFS density can be defined entirely by the parameters {r i, f i ( } i I of the involved Bernoulli RFSs. A normalized, weihted sum of MB densities is called MBM. In MTT, the MBs typically correspond to the different data association sequences. An MBM is defined entirely by the parameters {(W j, {r j,i, f j,i ( } i I j } j J, (6 where J is an index set for the MBs in the MBM; I j is the index set of Bernoullis in the jth MB; r j,i and f j,i ( are the existence probability and existence-conditioned PDF of the ith Bernoulli process in the jth MB; W j is the normalized weiht of the jth MB. C. PMBM conjuate prior The PMBM conjuate prior was developed in [10] for multiple point taret filterin, and in [12], [13] for multiple extended taret filterin. In the PMBM model, the taret set is a union of two disjoint sets, the undetected tarets X u and the detected tarets X d, i.e., X = X u X d. The distribution of tarets that are undetected X u is described by a PPP, while the distribution of tarets that have been detected at least once X d is described by an MBM, independent of X u. The PMBM set density can be expressed as f(x = W j f j (X d, (7a f u (X u X u X d =X j J f u (X u = e Du ;1 D u (x, x X u f j (X d = f j,i (X i, i I j i I j X i =X d (7b (7c where D u ( is the PPP intensity of the set of undetected tarets. The PMBM density is defined entirely by the parameters, D u, {(W j, {r j,i, f j,i ( } i I j } j J. (8 Because the PMBM density is a MTT conjuate prior, performin prediction (1 and update (2 means that we compute the predicted and updated, respectively, PMBM density parameters. In the PMBM filter, each MB corresponds to a unique lobal hypothesis for the detected tarets, i.e., a particular history of data associations for all measurements. Global hypotheses are made up of sinle taret hypotheses, each of which can incorporate a distribution of one of the above data association events, via a Bernoulli process [10]. The same sinle taret hypothesis can be used in many lobal hypotheses. A track is defined as a collection of sinle taret hypotheses correspondin to the same potential taret 3 [10]. Given a predicted PMBM density in the update (2, for each predicted lobal hypothesis, there are multiple possible data associations, each of which will result in an MB in the updated MBM. D. Bernoulli RFS mixture reduction Let f H (X = h H wh f h (X be a mixture of Bernoulli densities f h (X, where the existence-conditioned PDF of f h (X is from a family of distributions F, i.e., f h (x F, h H. Suppose that we wish to approximate the Bernoulli mixture f H (X by a sinle Bernoulli density ˆf(X, whose existence-conditioned PDF is from the the same family of distributions, i.e., ˆf(x F. The approximated Bernoulli density ˆf(X that minimizes the KL diverence ( D KL (f H (X ˆf(X = f H (X lo has parameters [10]: f H (X ˆf(X δx, (9 ˆr = h H w h r h, (10a ˆf(x =ar min f F ( D KL w h r h f h (x f(x. (10b h H For distributions from the exponential family, the KL diverence minimization (10b can be analytically solved by matchin the expected sufficient statistics, see, e.., [20, Section 10.7], [21]. E. Decomposition of KL diverence Suppose that we have a mixture density, where each mixture component consists of two independent multi-taret densities, f(x = W j f j,o (X o f j,n (X n, (11 j J X o X n =X Also, suppose we wish to find an approximate density (X, i.e., (X = o (X o n (X n, (12 X o X n =X that minimizes the KL diverence ar min D KL (f(x (X. (13 3 The term potential taret is used because sinle taret hypotheses correspond to Bernoulli densities, which may incorporate uncertain existence probability, i.e., 0 < r < 1.

4 4 It is shown in [22, Theorem 1] that, the objective in (13 has an upper bound that, when minimizin over o ( and n (, allows for the KL diverence minimization (13 to be broken down into two separate KL diverence minimization problems: ar min o ar min n ( D W j f j,o (X o o (X o, (14a j J ( D j J W j f j,n (X n n (X n. (14b III. PROBLEM FORMULATION In the PMBM filter, without approximation, the number of MBs rows exponentially over time, see [10], [11]. In this work, we aim at desinin an extended taret filter that propaates a PMB density over time, i.e., a special case of the PMBM density (7 that has an MBM with a sinle MB, J = 1. By only havin a sinle MB describin detected tarets, the number of parameters needed to be calculated in the prediction and update steps can be reduced. As a result, the computational cost of the filter can be reatly reduced. If the posterior f k 1 k 1 ( is a PMB density, then the predicted density f k k 1 (, resultin from (1, is also PMB [10]. However, with a PMB prior f k k 1 (, the Bayes update (2 results in a PMBM posterior f k k (. The problem considered in this paper is how to approximate the true PMBM posterior f k k ( with a PMB density ˆf k k ( to obtain a recursive PMB filter. The PMB set density consists of a disjoint union of a PPP ˆf k k u (Xu k and an MB k k(x d k : ˆf k k (X k = ˆf k k u (Xu k k k (X d k. (15 X u k Xd k =X k In the approximation, our oal is to obtain an approximated PMB that can retain as much information from the PMBM as possible. A natural choice for solvin this problem is to minimize the KL diverence between the PMBM and the approximated PMB, i.e., ar min ˆf D KL (f k k (X k ˆf k k (X k, (16 where ˆf k k (X k is restricted to a PMB. Because analytically minimizin the KL diverence (16 is intractable, we have to instead use approximations in order to obtain an efficient alorithm. Note that the PPP describin the set of undetected tarets does not have to be approximated [10], and that Bernoulli tracks can be approximated as bein Poisson [22]. Thus, it is sufficient to consider approximatin the MBM describin the set of detected tarets as a PMB. The problem of approximatin the MBM as a PMB is solved in two steps. First, we consider methods for approximatin the MBM as a sinle MB. Second, we utilize the recyclin method, proposed in [22], to further approximate the approximated MB as a PMB. However, analytically findin the MB density k k (X d k that minimizes the KL diverence ( ar min D KL W j f j k k (Xd k k k (X d k. (17 j J is still difficult. Different approaches to efficiently approximate the MBM as a sinle MB has been well studied for point taret filterin, see [10], [18]. In this work, we focus on adaptin some of these approaches to extended tarets in order to desin an extended taret PMB filter. One such approach is to use a method similar to the track-oriented marinal MeMBer- Poisson (TOMB/P filter, proposed in [10]. Another approach is to use the variational method of [18] to search for the MB parameters that ives the smallest possible KL diverence (17. In this paper we explore both of these approaches. IV. EXTENDED TARGET PMB FILTER In this paper, we seek to approximate the PMBM posterior by a PMB density for extended taret filterin after the update step. In what follows, we first outline the taret transition model, the extended taret measurement model and the data association model used in this work. Based on these standard modellin assumptions, we then present the extended taret PMB recursion, which includes prediction, update, MB approximation and recyclin. A. Modellin 1 Taret transition: In this work, we focus on the update, hence the transition model is only briefly discussed. We assume that tarets arrive accordin to a PPP, independently of any pre-existin tarets. At each time step, a sinle taret survives with a probability of survival p S (x k. The tarets depart accordin to an i.i.d. Markov processes with probability 1 p S (x k. The taret state at next time step only depends on its current state. Tarets evolve independently accordin to an i.i.d. Markov process with transition density f k+1,k (x k+1 x k. 2 Extended multi-taret observation: In extended taret trackin, we need a model that describes the number and the spatial distribution of enerated measurements for each taret. A common model is the inhomoeneous Poisson Point Process (PPP, proposed in [23]. The set of measurements Z k is a union of a set of clutter measurements and sets of taret-enerated measurements. The clutter is modelled as a PPP with Poisson rate λ and spatial distribution c(z, and the clutter PPP intensity is κ(z = λc(z. The clutter measurements are independent of tarets and any taret-enerated measurements. Each extended taret may ive rise to multiple measurements with a state dependent detection probability p D (x k. If the extended taret is detected, the taret-enerated measurements are modelled as a PPP with Poisson rate γ(x k and spatial distribution φ( x k, independent of any other tarets and their correspondin enerated measurements. The extended taret measurement likelihood for a nonempty set of measurements Z is the product of the taret detection probability and the PPP density of taret-enerated measurements [13] l Z (x k = p D (x k e γ(x k z Z γ(x k φ(z x k. (18 For an extended taret state x k, the effective detection probability is the product of taret detection probability and the

5 5 probability that taret enerates at least one measurement, which is 1 e γ(x k. Then, the effective probability of missed detection can be calculated as q D (x k = 1 p D (x k + p D (x k e γ(x k. (19 The measurement likelihood for an empty measurement set, l (x k, is also described by (19. 3 Data association: The data association problem in the extended PMBM filter is covered in [12], [13]. Let the predicted multi-taret density be a PMB density with parameters D u, {r i, f i ( } i I, (20 Let M be the set of indices of measurement set Z, i.e., Z = {z m } m M ; and let A be the space of all current data associations A iven the predicted hypothesis 4. A data association A A is an assinment of each measurement in Z to a source, either to the backround (clutter or new taret or to one of the existin tarets indexed by I. Formally, an extended taret data association A A consists of a partition of M I into non-empty disjoint subsets called index cells, denoted C A [13]. An index cell can contain at most one taret index. If the index cell C contains a taret index, then let i C denote the correspondin taret index. Further, let C C denote the measurement cell that corresponds to the index cell C, i.e., the set of measurements C C = m C z m. The weiht of the lobal hypothesis, that resulted from the predicted lobal hypothesis with association A A, is [13] W A = L A A A L, (21 A where L A is the likelihood of association A A. For any data association A A, the likelihood L A can be expressed as [13] L A = L b C = C A: C I= C M L b C C C A: C I C M L i C CC C A: C I C M= { κ C + D u ; l C if C = 1 D u ; l C if C 1, L i C = r i f i ; l C, L i = 1 ri + r i f i ; q D. The three products in (22a correspond to L i C, (22a (22b (22c (22d cells that are associated to the backround, i.e., either clutter or previously undetected tarets, cells that are associated to previously detected tarets, previously detected tarets that are miss-detected. A complexity analysis of the extended taret data association problem iven in [13] shows that it is intractable to enumerate all the possible associations, hence approximations are needed for computational tractability. The prevailin approach to solvin the data association problem is to truncate associations with neliible probability, i.e., associations A for which weiht W A (21 that is approximately zero. Previous 4 The PMBM data association problem covered here is in line with the formulation in [24] but with simplified notations, since the extended taret PMB filter only has one predicted lobal hypothesis. work on extended taret PMBMs, see, e.., [12], [13], has dealt with the data association problem by first usin combinations of different clusterin alorithms to compute a set of partitions, and then for each partitionin, assinment alorithms are used to assin the cells to different tracks. In the recently presented work [24], a samplin based method called stochastic optimization (SO is proposed, which directly maximizes the multi-taret likelihood function and solves the data association problem in a sinle step. It has been demonstrated in [24] that the SO method has improved performance in comparison to the two-step method which involves clusterin and assinment. B. Prediction Given a posterior PMB density with parameters D u, (r i, f i ( i I, (23 and the standard transition model, the predicted density is a PMB density with parameters [10] where D u +, (r i +, f i +( i I, (24 D u +(x = D b (x + D u, p S f k+1,k, r i + = f i, p S r i, (25a (25b f+(x i = f i, p S f k+1,k f i, p S, (25c and D b ( denotes the PPP birth intensity. C. Update Given a prior PMB density with parameters (24, a set of measurements Z, and the standard measurement model, the updated density is a PMBM, with parameters D u, {(W j, {r j,i, f j,i ( } i I j } j J, (26 where the updated PPP intensity is D u (x = q D (xd u +(x, (27 and the updated MBs resultin from the data associations are indexed by j J. The index set of Bernoullis in the predicted MB is a subset of the index set of Bernoullis in each updated MB, i.e., I I j j J. In the MBM, Bernoullis with index i I I j correspond to tracks continuin from previous time steps, i.e., pre-existin tracks, whereas Bernoullis with index i I j \ I correspond to new tracks. For pre-existin tracks, a hypothesis can be included as a missed detection, or as an update usin a measurement cell C C. For missed detection hypotheses, the Bernoulli densities have parameters (i I I j r j,i = r i + f i +, q D 1 r i + + r i + f i +, q D, (28a f j,i (x = qd (xf+(x i f+, i. (28b q D

6 6 For hypotheses updatin pre-existin tracks, the Bernoulli densities have parameters (i I I j r j,i = 1, (29a f j,i (x = l C C (xf+(x i f+, i. (29b l CC For new tracks with correspondin measurement cell associated to the backround, the updated densities are Bernoulli densities with parameters (i I j \ I, C I = { 1 if CC > 1 r j,i = D u +,l C C κ(c C + D u +,l C C if C C = 1, (30a f j,i (x = l C C (xd+(x u D+, u. (30b l CC After updatin, the number of Bernoullis representin previously detected tarets in each MB remains unchaned, while the number of Bernoullis representin newly detected tarets becomes I j \ I in the jth MB. D. MB approximation and its challenes Each component in the MBM, W j, {r j,i, f j,i ( } i I j, j J, (31 represents a hypothesis of a collection of possible independent potential tarets, and a relative likelihood of this hypothesis. Each potential taret hypothesized to exist is described by an existence probability, and an existence-conditioned PDF, with parameters {r j,i, f j,i ( }, i I j, j J. (32 In the MB approximation, the oal is to represent a collection of possible independent tarets with a sinle MB, by exploitin the information contained in all the uncertain hypotheses. More specifically, we would like to obtain the MB (X with parameters {ˆr ι, ι ( } ι Î, (33 that best matches the MBM f(x with parameters (32 usin some merin techniques. There are three major challenes related to this MB approximation. The first challene is to determine the number of Bernoullis Î in the approximated MB, since different MB f j (X may not all contain the same number of Bernoullis I j. Given M measurements, there are 2 M 1 possible ways in which we can form a subset of measurements. Each such subset can, in an association, be associated to a newly detected taret. As a result, different lobal hypotheses may contain a different number of new tracks. Therefore, determinin how many new tracks should be formed in the extended taret PMB filter is a challene. In this work, we choose to approximate the MBM as a sinle MB by merin. Then, the second challene is to determine, for each ι Î, which Bernoullis f j,i (X should be used to form ˆr ι and ι (x. The Bernoullis in each MB are unordered 5, so determinin how to select Bernoullis to mere across different MBs is a challene 6. Alternative solutions to address the first and the second challenes are discussed in Section V and VI. The last challene is to determine how to mere the selected Bernoullis into a sinle Bernoulli. Compared to the first two challenes, the third challene is relatively less difficult, e.., iven a Bernoulli mixture, one can simply take the weihted sum of different parameters. In extended taret trackin, the merin of different taret state densities is typically addressed by analytically minimizin the KL diverence [25], [26]. In Section VII, we will discuss some implementation details reardin the merin of different Bernoullis under a common extended taret model, called amma Gaussian inverse Wishart (GGIW [8], [25]. We illustrate these three major challenes with the followin example. Example 4.1: Consider a two-dimensional scenario shown in Fi. 1, with an MBM that contains three lobal hypotheses. For simplicity, we assume that each Bernoulli has existence probability equal to one. In this scenario, two detected tarets are closely spaced, and it is ambiuous whether there are two closely spaced new tarets or one bier new taret. It is simple to decide the number of pre-existin tracks, which is two in this case. The first challene is determinin how to choose the number of Bernoullis in the approximated MB. Because the new tracks in the three lobal hypotheses are all different, it is nontrivial to decide the exact number of tracks we should form in the approximated density. The second challene is determinin how to select the Bernoulli components to be mered. Given three lobal hypotheses, each of which has two pre-existin tracks, there are eiht different ways in total to obtain the approximated MB representin detected tarets. The problem becomes more complicated when we consider the approximation of new tracks, since the number of approximated Bernoullis representin newly detected tarets is yet to be decided. The third challene is determinin how to mere the selected Bernoullis. For example, if we choose to mere f 1,1 (X 1, f 2,1 (X 1 and f 3,1 (X 1, then we should seek for accurate merin techniques so that the approximated Bernoulli can retain as much information as possible from these three Bernoullis. E. Recyclin For the approximated MB (X with parameters (33, the recyclin method of [22] can be applied to Bernoullis with low existence probability ˆr ι. The recycled components are approximated as bein Poisson and are incorporated into the PPP representin undetected tarets for eneratin possible 5 Note that the indexin of Bernoullis in each MB is merely used for notational convenience. 6 In this work, we choose not to mere Bernoullis within the same MB. An MB corresponds to a collection of independent potential tarets. It is enerally inappropriate to represent two or more independent tarets with a sinle Bernoulli.

7 y y y x x x Fi. 1. Illustrative example with three MBs f 1 (X, f 2 (X and f 3 (X with weihts W 1 = 0.4, W 2 = 0.35 and W 3 = Densities are presented by their level curves. Bernoullis marked in the same color red/blue correspond to the measurement update of the same track. Bernoullis marked in red and blue stand for the updates of previously detected tarets, whereas Bernoullis marked in reen and maenta stand for the first detections of undetected tarets. new tarets in subsequent steps. After recyclin, the total PPP intensity of the set of undetected tarets can be expressed as ˆD u (x = D u (x + ˆr ι ĝ ι (x, (34 ι Î:ˆrι <τ where τ is a threshold, and a typical choice is τ = 0.1. The benefits of recyclin in the point taret PMBM and PMB filters have been discussed in [16], [22]. In this work, we utilize MB approximation methods and recyclin to approximate the PMBM posterior density as a PMB. V. TRACK-ORIENTED PMB FILTER In this section, we seek to develop a well performin PMB filter by adaptin the merin technique used in the point taret TOMB/P filter [10] to extended tarets. We call the proposed filter track-oriented PMB (TO-PMB. Followin a similar track-oriented merin approach [10], in the TO-PMB filter, sinle taret hypotheses updatin different tracks are assumed to be independent, and hypotheses comprisin the same track are mered into a sinle Bernoulli across different lobal hypotheses. In what follows, we discuss the extended taret TO-PMB filter, its connection to the LMB filter and the drawback of this track-oriented merin approach. A. Track formation Let us discuss the formations of new tracks and pre-existin tracks separately. For pre-existin tracks (ι Î I, the approximated Bernoulli ι (X ι is simply formed by merin all Bernoullis that correspond to the same track in lobal hypotheses: ( ar min D KL W j f j,l(ι (X L(ι ι (X ι, (35 j J where L(ι = i is a bijection that maps the index i I j I to the index ι Î I; f j,i (X i has existence probability and existence-conditioned PDF in the form of (28 or (29; ι (X ι has existence probability and existence-conditioned PDF in the form of (10a and (10b 7. The number of new tracks is determined by the number of different measurement cells associated to the backround in 7 The same holds for (36, (43, (46 and (48 different lobal hypotheses. Let C denote the set of measurement cells C that are associated to a newly detected taret in any of the lobal hypotheses J, and let f C ( denote the correspondin Bernoulli density, see (30. In the TO-PMB filter, each measurement cell C C results in a unique Bernoulli in the approximated MB (X, thus the number of new tracks formed in the TO-PMB filter is the number of cells in C, i.e., Î \ I = C. Further, we define a bijection F(ι = C that maps the index of a new track ι Î \ I to the measurement cell C C. For new tracks (ι Î \ I, the approximated Bernoulli is formed by merin sinle taret hypotheses updated by the same measurement cell across different lobal hypotheses, accordin to ( ar min D KL W j f F(ι (X F(ι F(ι P j ι (X ι, j J (36 where P j denotes the partition of measurements assined to potential new tarets in the jth lobal hypothesis, and C P j is an indicator function, which is equal to one if and only if measurement cell C belons to measurement partition P j. B. Connection to the LMB filter For filters based on a labelled MB conjuate prior, namely, the δ-glmb filter and its approximation the LMB filter, labels are aumented to taret states to maintain tracks. It has been shown in [11] that the δ-glmb density is analoous to a type of MBM on a labelled state space, and that labels are not required for the conjuacy property of the MBM. In the LMB approximation, Bernoullis with the same label are mered across different MBs [17], while in the TO-PMB filter, Bernoullis updatin the same track are mered. In eneral, it can be concluded that approximatin the δ-glmb density with a LMB relies on a similar type of approximation as is used to approximate the PMBM density with a TO-PMB. C. Drawback The track-oriented merin approach is simple to implement; nevertheless, it has several drawbacks. In the LMB filter and the TO-PMB filter, tracks are approximated as independent. This assumption holds well for the case that tarets are well separated, however, the dependency between tarets becomes inescapable when tarets are closely spaced.

8 8 Both the LMB filter and the TO-PMB filter suffer from lare estimation errors when coalescence happens, see [16] and [10] respectively, for their performance evaluation on point taret filterin. In addition to the above drawback, the track-oriented merin approach is particularly unfit for the extended taret PMB filter, in which new tracks created by different measurement cells are approximated as independent. If some of the measurement cells have shared measurements, which is the typical case, the new tracks are hihly dependent since they never co-exist in the same data association hypothesis, and approximatin such tracks as independent may yield lare errors. Also, creatin as many new tracks as there are measurement cells often yields intractably many Bernoullis (tracks with low existence probabilities. In Section VI, we will investiate techniques that are more accurate and yield much fewer tracks, thus mitiatin all of these weaknesses simultaneously. VI. VARIATIONAL MULTI-BERNOULLI FILTER As discussed in last section, approximatin the MBM by the track-oriented merin approach is likely to yield lare estimation errors. The key issue of this problem lies in approximatin hihly dependent new tracks as independent, which means that a more appropriate method should be used in approximatin new tracks. In this section, we divide the MB approximation into two separate parts: one for the formation of new tracks and the other for the formation of pre-existin tracks. The premise of this implementation is that the preexistin tracks and new tracks are assumed to be independent. Althouh there are situations where this approximation is less accurate, it is essential to our approach to obtain a tractable solution. Our oal is to approximate, in the minimum KL diverence sense, the posterior MBM with an MB that has two parts, correspondin to pre-existin and new tracks, respectively. As stated in Section II-E, this problem can be approximately solved by minimizin the KL diverences between the marinal density of the pre-existin/new tracks and its approximate MB density. Analytically minimizin such KL diverences is, however, still intractable; thus, we instead look for approximate solutions. For approximatin the pre-existin tracks ι Î I, we employ the variational approximation technique in [18], which can successfully address the potential coalescence problem suffered by the TO-PMB filter. Two variants of the variational MB alorithm are studied. We first review the variational MB alorithm implemented in [18], then we study an alternative implementation of this alorithm that is inspired by the SJPDA filter [19]. For approximatin the new tracks ι Î \ I, we propose a reedy method to mere similar Bernoullis (tracks, in the sense of minimizin the KL diverence. A. Pre-existin track formation Because the number of pre-existin tracks remains the same after updatin, the variational MB alorithm, proposed in [18] for point taret filterin 8, can be easily applied to the pre- 8 In the point taret filterin, each measurement creates a new track, hence, different lobal hypotheses consist of the same number of tracks. existin track approximation in the extended taret PMB filter. In the variational MB alorithm, the oal is to obtain an approximate MB (X that minimizes the KL diverence: ar min D KL (f(x (X = ar min f(x lo (XδX, (37 where f(x is an MBM describin the pre-existin tracks, with parameters {(W j, {r j,i, f j,i ( } i I j I} j J. (38 An approximate solution of (37 is based on minimizin the upper bound of the true objective, followin a similar process to expectation-maximization (EM [27]. In this approach, the correspondences between the Bernoullis in f(x and the Bernoullis in (X are treated as missin data. By simplifyin the MB set interal in (37 into a series of Bernoulli interals, and usin the lo-sum inequality, an approximate upper bound of the objective (37 can be expressed as [18] D UB (f(x (X = ι Î I j J,π Π j N W j q j (π j f j,πj (L(ι (X lo ι (XδX, (39 where N denotes the number of Bernoullis that correspond to pre-existin tracks in each MB of f(x and the approximated MB (X, i.e., N = Î I ; Πj N is the set of all ways to assin the Bernoullis f j,i (X i, i I j I, j J to the Bernoullis ι (X ι, ι Î I; the missin data qj (π j is constrained to vary only with the jth MB, and satisfies q j (π j 0 and π j Π j q j (π j = 1. The standard method for optimizin N (39 is block coordinate descent, which alternates between minimization with respect to ι (X ι (M-step and q j (π j (Estep [18]. Because solvin the minimization problem (39 suffers from combinatorial complexity, approximation is needed to obtain a tractable solution. Here, two different approximation methods are studied. 1 Efficient approximation of feasible set: Because the minimization problem of (39 involves missin data q j (π j for every MB f j (X, a simplified missin data representation is desirable. The minimization of the upper bound (39 can be solved approximately as [18] armin q(h,ι M ι Î I ( h H q(h, ιf h (X lo ι (XδX, (40 where H is the index set of sinle taret hypotheses included in the lobal hypotheses indexed by j J, q(h, ι is a simplified representation of q j (π j, which specifies the weiht of sinle taret hypothesis density f h (X h 9 in the new Bernoulli com- 9 The same sinle taret hypothesis may be included in different lobal hypotheses, thus it satisfies that H j J Ij. We use the sinle superscript h to denote the index of sinle taret hypothesis density in order to distinuish from the double superscript {j, i}, which denotes the indices of Bernoullis in the MBM.

9 9 ponent ι (X ι, and the feasible set M is an approximation needed for tractability [18] { M = q(h, ι 0 q(h, ι = 1 ι I ˆ I, h H } q(h, ι = p h h H. (41 ι Î I The constraint p h satisfies p h = ι Î I p ι(h, where p ι (h = j J W j f j,l(ι (X L(ι =f h (X h, ι Î I (42 Note that here the missin data distribution is no loner constrained to vary only with the lobal hypotheses. In this case, each approximated Bernoulli can be expressed as the weihted sum of different sinle taret hypothesis densities, and the M-step becomes ( ar min D KL q(h, ιf h (X h ι (X ι, (43 h H while the E-step reverts to: armin q(h, ι f h (X lo ι (XδX, (44 q(h,ι h H ι Î I subject to q(h, ι = p h h, ι Î I q(h, ι = 1 ι, h H q(h, ι 0 h, ι. Problem of this type can be solved usin methods such as the simplex alorithm [28]. The variational MB alorithm based on efficient approximation of feasible missin data set can be initialized with the marinal association probabilities (42. Althouh exact calculation of these quantities is intractable for a data association problem with combinatorial complexity, we can obtain approximated estimates by only considerin truncated lobal hypotheses with non-neliible weihts usin data association approximation methods, such as [13], [24]. 2 Most likely assinment: The number of MBs in the MBM can be kept at a tractable level after truncatin the lobal hypotheses with neliible weihts. This allows us to study an alternative approach to solvin the minimization problem of (39, followin a similar approach to the Kullback-Leibler SJPDA (KLSJPDA [19]. The KLSJPDA filter seeks to find the ordered distribution in the same unordered family, such that the new density can be most accurately approximated with a Gaussian density, in the KL sense. In the JPDA filter, the number of tarets is assumed to be known, and the joint state of multiple tarets are represented as a vector. As a comparison, iven that each lobal hypothesis consists of the same number of pre-existin tracks, the multi-taret RFS density also has fixed cardinality. The similarities between the KLSJPDA filter and the variational MB filter are further explored in [18]. Let us o back to the approximated upper bound (39 that we want to minimize. Suppose that Bernoullis in different MBs are indexed by the same superscript i if they correspond to sinle taret hypotheses updatin the same track, and that only Bernoullis with the same superscript i can be mered. Because the approximate MB density is invariant to the indexin of the Bernoullis it contains, the selection of the assinment mappin π in each MB will not chane the MBM f(x, but only will determine which Bernoullis are oin to be mered. The minimization problem can be interpreted as the Bernoullis in each MB are permuted in such a manner that the upper bound (39 is minimized but the density of the reordered f(x remains unchaned. Empirically, we found that findin a set of most likely assinments for each MB in the truncated MBM is computationally heavy. Hence, we choose to find the sinle most likely assinment ˆπ j for each MB f j (X. In this case, the missin data distribution under each MB is a point mass, i.e., q j (ˆπ j = 1, and the the minimization of (39 with respect to the missin data distribution can be expressed as ˆπ j = armin f j,πj (L(ι (X lo ι (XδX, j J, π j ι Î I (45 where the most likely assinment ˆπ j can be obtained usin methods such as the auction alorithm [29]. The minimization of (39 with respect to the approximated MB (X simplifies to ( ar min D KL W j f j,ˆπj (L(ι (Xˆπj (L(ι ι (X ι. (46 j J This means that each approximated Bernoulli in (X can be obtained by merin Bernoullis in f(x with the same assinment mappin. 3 Illustration: It can be noticed that, in the first iteration of the variational MB alorithm, both the M-step (43 in the implementation based on the efficient approximation of feasible set and the M-step (46 in the implementation based on the most likely assinment are equivalent to the MBM merin step (35 used in the TO-PMB filter. This means that the variational MB alorithm can be considered as an improvement on the track-oriented merin approach used in the TO-PMB filter. We illustrate how the assinment mappin in each MB chanes in each iteration of the variational MB alorithm based on the most likely assinment, and how the assinment weiht matrix chanes in each iteration of the variational MB alorithm based on efficient approximation of the feasible set with the followin example. Example 6.1: Consider the same scenario illustrated in Fi. 1. For the TO-PMB filter, Bernoullis updatin the same taret will be mered, i.e., f 1,1 (X 1, f 2,1 (X 1 and f 3,1 (X 1 will be mered to obtain 1 (X 1 ; f 1,2 (X 2, f 2,2 (X 2 and f 3,2 (X 2 will be mered to obtain 2 (X 2. For the variational MB filter based on the most likely assinment, we can permute the order of Bernoullis and find the optimal permutation for each MB accordin to the E-step (45. Assume that, in this case, Bernoulli density f 3,2 (X 2 is

10 10 TABLE II PSEUDO CODE OF NEW TRACKS FORMING Input: {(W j, {f j,i (X i } i I j \I } j J, merin threshold τ Output: { ι (X ι } ι Î\I Sort f j (X in the descendin order of W j ; I 0; Î I; for all j = {1,..., J } do if I j > I then for all i = { I + 1,..., I j } do if (j, i / L l l = {1,..., I} then I I + 1; L I L I {(j, i}; Î I { I + I}; end if for all j + = {j + 1,..., J } do for all i + = { I + 1,..., I j+ } do if (j, i / L l l {1,..., I} then ( d i+ D SKL f j,i (X f j+,i + (X ; end if end for [d, i ] min(d; if d < τ then L I L I {(j +, i }; end if end for end for end if end for for all ι = { I + 1,..., Î } do Calculate ι (X ι usin (48. end for closer to 1 (X 1 than 2 (X 2, and that Bernoulli density f 3,1 (X 1 is closer to 2 (X 2 than 1 (X 1. In order to minimize the upper bound of (39, the order of Bernoullis f 3,1 (X 1 and f 3,2 (X 2 should be flipped. After reorderin, we can obtain our new approximated Bernoulli 1 (X 1 by merin f 1,1 (X 1, f 2,1 (X 1 and f 3,2 (X 2, and 2 (X 2 by merin f 1,2 (X 2, f 2,2 (X 2 and f 3,1 (X 1. For the variational MB filter based on the efficient approximation of the feasible set, in order to minimize the upper bound of (39, a proportion of the weihts of assinin f 3,1 (X 1 to 1 (X 1 should be shifted to 2 (X 2, and a proportion of the weihts of assinin f 3,2 (X 2 to 2 (X 2 should be shifted to 1 (X 1 accordinly. For example, the approximated Bernoulli 1 (X 1 can be expressed as a Bernoulli mixture with component f 1,1 (X 1, f 2,1 (X 1, f 3,1 (X 1 and f 3,2 (X 2, in which f 3,1 (X 1 has weiht 0.05 and f 3,2 (X 2 has weiht 0.2; the approximated Bernoulli 2 (X 2 can be expressed as the same Bernoulli mixture, but in which f 3,1 (X 1 has weiht 0.2 and f 3,2 (X 2 has weiht B. New track formation We present a reedy method to form new tracks in a reasonable and efficient way. The pseudo code of this merin approach is iven in Table II. The intuition behind this proposed method is that we want to mere hihly dependent Bernoullis across different MBs so that similar new tracks will not be formed in the same local reion and new tracks with sinificant different Bernoulli densities will not be mered. In this merin approach, we only mere Bernoullis in different MBs that are considered similar enouh. For any pair of Bernoulli densities, the symmetrised KL diverence is used to measure the similarity, defined as D SKL (p q = D KL (p q + D KL (q p. (47 To start with, the MB densities are sorted in the descendin order of their weihts, so that f 1 (X has the hihest weiht. Remember that reorderin the MBs will not chane the MBM process. Then, we cluster Bernoullis that satisfy the merin criteria into the same roup. As a result, the number of new tracks formed is equal to the number of Bernoulli clusters. Lastly, new tracks are formed by only merin Bernoullis within the same roup. For the ιth (ι Î \ I new track in the approximated MB (X, its Bernoulli density can be expressed as ar min ( D KL j J i I j \I W j f j,i (X i {(j,i} L ι I ι (X ι, (48 where set L ι I contains indices of Bernoullis that are within the same roup. Empirical results show that the number of tracks formed usin this approach can be kept to a relatively small number. We illustrate this with the followin example. Example 6.2: Consider the same scenario illustrated in Fi. 1. There are five new tracks created in total in the three lobal hypotheses. If we assume that Bernoulli densities of these five new tracks are mutually independent, it is likely that none of these tracks are detected by the estimator due to their low existence probabilities. Followin the proposed merin approach, we start by merin f 1,3 (X 3 with f 2,3 (X 3 and f 3,3 (X 3, due to the small symmetrised KL diverence between the pair f 1,3 (X 3, f 2,3 (X 3, and between the pair f 1,3 (X 3, f 3,3 (X 3. Next, we mere f 1,4 (X 4 with f 2,4 (X 4, due to their hih similarity in the sense of symmetrised KL diverence. After merin, the number of Bernoullis in the approximated MB reduces from five to two: 3 (X 3 has existence probability ˆr 3 = 1, and 4 (X 4 has existence probability ˆr 4 = VII. GGIW IMPLEMENTATION Solvin the multiple extended taret trackin problem requires not only a multiple taret trackin framework, but also a sinle extended taret model. For the modellin of the spatial distribution, two popular models are the Random Hyper-surface Models [30] and the Gaussian inverse Wishart (GIW approach [31], [32]. The former is desined for eneral star-convex shape; the latter relies on the elliptic shape and it models the spatial distribution of taret-enerated measurements as Gaussian with unknown mean and covariance. The Gamma GIW (GGIW model [8], [25] is an extension of the GIW model that incorporates the estimation of taret measurement rate.