Multiple‐detection multi‐target tracking with labelled random finite sets and efficient implementations

IET Radar, Sonar & NavigationVolume 13, Issue 2 p. 272-282 Research ArticleFree Access Multiple-detection multi-target tracking with labelled random finite sets and efficient implementations Jun Wang, Jun Wang orcid.org/0000-0002-6012-7084 School of Electronic and Information Engineering, Beihang University, Beijing, 100191 People's Republic of ChinaSearch for more papers by this authorBin Yang, Bin Yang School of Electronic and Information Engineering, Beihang University, Beijing, 100191 People's Republic of ChinaSearch for more papers by this authorWenguang Wang, Corresponding Author Wenguang Wang wwenguang@buaa.edu.cn School of Electronic and Information Engineering, Beihang University, Beijing, 100191 People's Republic of ChinaSearch for more papers by this authorYanxian Bi, Yanxian Bi China Academy of Electronics and Information Technology, Beijing, 100041 People's Republic of ChinaSearch for more papers by this author Jun Wang, Jun Wang orcid.org/0000-0002-6012-7084 School of Electronic and Information Engineering, Beihang University, Beijing, 100191 People's Republic of ChinaSearch for more papers by this authorBin Yang, Bin Yang School of Electronic and Information Engineering, Beihang University, Beijing, 100191 People's Republic of ChinaSearch for more papers by this authorWenguang Wang, Corresponding Author Wenguang Wang wwenguang@buaa.edu.cn School of Electronic and Information Engineering, Beihang University, Beijing, 100191 People's Republic of ChinaSearch for more papers by this authorYanxian Bi, Yanxian Bi China Academy of Electronics and Information Technology, Beijing, 100041 People's Republic of ChinaSearch for more papers by this author First published: 01 February 2019 https://doi.org/10.1049/iet-rsn.2018.5117Citations: 1AboutSectionsPDF ToolsRequest permissionExport citationAdd to favoritesTrack citation ShareShare Give accessShare full text accessShare full-text accessPlease review our Terms and Conditions of Use and check box below to share full-text version of article.I have read and accept the Wiley Online Library Terms and Conditions of UseShareable LinkUse the link below to share a full-text version of this article with your friends and colleagues. Learn more.Copy URL Share a linkShare onFacebookTwitterLinkedInRedditWechat Abstract Traditional multi-target tracking algorithms assume that each target can generate at most one detection per scan. However, a target may produce multiple detections (MDs) in many practical applications, e.g. over-the-horizon radar (OTHR), tracking for extended target and tracking with multiple sensors. In this study, the authors propose a new algorithm for tracking multiple targets with MD observation. The proposed technique is based on the labelled random finite set (RFS), which estimates the number of targets and the trajectories of their states. Furthermore, they propose two methods, pre-partition and joint partition, to implement the labelled RFS density recursion. The joint partition method derives a joint prediction, partition and update formulation of the MD filtering and extends the Gibbs sampler to provide global optimal results with low computational cost. The authors' algorithm is demonstrated on an OTHR simulation compared with the previous approach, such as the MD probability hypothesis density filter and the multipath cardinality-balance multi-target multi-Bernoulli filter. 1 Introduction Multi-target tracking problem involves estimating the unknown and time-varying number of targets and their corresponding trajectories by using measurements from sensors. The non-unity probability of detection, the presence of spurious measurements (clutter), and the measurement-origin uncertainty pose challenges to multi-target tracking. Three major algorithms, namely, the Multiple Hypothesis Tracker (MHT), the Joint Probabilistic Data Association Filter and more recently, algorithms based on the random finite set (RFS) theory have been proposed to deal with these problems based on the single-detection assumption, i.e. each target can generate at most one detection per scan [1-7]. However, in many practical scenarios, one target can generate more than one detection per scan due to, multipath propagation, or extended nature of the target or multiple sensors and so on. In this paper, we mainly focus on the multipath propagation. A well-known example of such a system is the over-the-horizon radar (OTHR) [8-13], in which radar signals from the same target are propagated by multiple paths (MPs) due to multiple ionospheric layers. These multiple detections (MDs) in the receiver of OTHR bring benefit and disadvantage to the tracking algorithms simultaneously. If the MDs from multipath propagation can be exploited effectively, the performance of the tracking algorithm will be improved greatly. However, the tracking with MDs needs to solve not only a measurement-target association but also measurement-path association, which increases the complexity of tracking algorithms. Based on the Joint Probabilistic Data Association (JPDA) framework [14] and the MHT framework [15], MD-JPDA filter [16] and MD-MHT algorithm [17] were proposed, respectively, to tracking multiple targets with MDs in OTHR. Since the RFS approach, such as the probability hypothesis density (PHD) filter [18, 19] and the cardinality-balance multi-target multi-Bernoulli (CBMeMber) filter [6, 7, 20], can avoid the data association, the MD-PHD filter [21] and the MP-CBMeMBer filter [22] have been proposed as tractable Bayesian filters with low complexity for the MD multi-target tracking problem. In [23, 24], the RFS-based approach is applied to track the single target and extended target in the OTHR system, respectively. However, these RFS-based filters only estimate target states at individual time instants and cannot provide target trajectories. In order to overcome this limitation, Vo et al. [25, 26] appended a unique label to RFS, which called labelled RFS and linked the target states with the same label at different scan to form the trajectory of targets. Since the generalised labelled multi-Bernoulli (GLMB) RFS is conjugate prior to the multi-target point measurement model, a well-known Bayesian multi-target tracker and its alternative form, namely the GLMB filter and the -GLMB filter, are proposed in [25, 26]. Then, the labelled multi-Bernoulli filter is introduced in [27] as a cheaper version of the GLMB filter. For implementing the GLMB filter and the LMB filter [26], discards the smallest weighted components in the GLMB density to minimise the -error. By using the K-shortest path and ranked assignment algorithms, the prediction and the update GLMB densities are truncated, respectively. A more efficient implementation proposed in [28-30] derives a single joint prediction and update and constructs a Gibbs sampler to generate the update components with significant weights. This Gibbs sampling based implementation has a linear complexity in the number of measurements and quadratic in the number of objects compared with the ranked assignment algorithm, which has at best cubic complexity in the number of measurements. In this paper, we address the MD multi-target tracking problem with labelled RFS. We derive the MD likelihood function and develop the MD GLMB (MD-GLMB) filter and the MD LMB (MD-LMB) filter. The proposed MD-GLMB filter and MD-LMB filter can be generalising to deal with multiple extended targets tracking and multi-sensor multi-target tracking. We also show that the proposed MD-GLMB filter can be transformed into the extended target GLMB [31] or multi-sensor GLMB in [32]. Furthermore, we propose two implementations for the labelled RFS density filtering with MDs. The first implementation is a pre-partition method, which enumerates all possible partitions and exploits Murty algorithm or Gibbs sampling to generate association hypotheses with significant weights in each partition. In the second implementation, we derive a joint partition method that eliminates inefficiencies in the hypotheses generating within a low-weight partition of the pre-partition implementations. By extending the Gibbs sampling technique to handle the valid subsets of MD observations, we can produce highest association hypotheses, which contain target birth/death, observation partition and observation subset map, with the low computational load. The remaining of the paper is organised as follows. In Section 2, we present the problem description of the MD multi-target tracking and the background on labelled RFS filtering. In Section 3, we provide the MD likelihood, derive the associated MD-GLMB and MD-LMB filters, and generalise these filters to other formulations of the labelled RFS filters. The efficient implementation of the proposed trackers is presented in Section 4. Simulation results are given in Section 5. Finally, in Section 6, conclusions are discussed. 2 Problem description and background In this section, we provide the mathematical preliminaries of the MD multi-target tracking problem in OTHR. We also briefly review the background of the labelled RFS filtering. 2.1 MD system OTHR is a typical MD tracking system, which can detect targets beyond the horizon by using the internal reflection of high-frequency radio waves through the ionospheric layer. As shown in Fig. 1, the radar signal bounces off two dominant ionospheric layers E and F twice between the radar and targets. Hence, there are four possible multipath propagation modes , each represented by a known measurement equation. We denote as the jth measurement at the scan k and the measurement equation is given by where is the measurement function associated with the sth measurement mode and is the corresponding measurement noise, which is assumed to be a zero-mean Gaussian sequence with the covariance matrix . is the measurement of clutter, which is assumed to be a Poisson process with intensity function . Moreover, let denote all the measurements reported by the sensor, where is the total number of measurements at the scan k. Fig. 1Open in figure viewerPowerPoint OTHR propagation modes [11] Fig. 2 shows the transmitter–receiver–target geometry of the OTHR. The radar signal reflects by an ionospheric layer at a height from the transmitter at to target at . Then, after being scattered by an ionospheric at height . The scatter from the target is received by the receiver at the origin. The measurement vector at scan k consists of half slant range , slant range rate , and apparent azimuth . The measurement equation for mode s is given by [11] where Fig. 2Open in figure viewerPowerPoint OTHR transmitter–receiver–target geometry on the ground plane 2.2 Labelled RFS and multi-object filtering In this paper, single-object states and multi-object states are denoted by small letters (e.g. x) and capital letters (e.g. X), respectively. We use bold face letters (e.g. ) to denote labelled states and distributions for distinguishing them from the unlabelled states and distributions. Moreover, we represent spaces by blackboard bold (e.g. ) and the collection of all finite subsets of a space by . We also denote the inner product as standard notation , and the product as the multi-object exponential notation , with . The generalised Kroneker delta function and the inclusion functions are given by and respectively. We abbreviate the list of variables by and represent the cardinality of a finite set X by . A labelled RFS X is defined on a Cartesian product space , where is the state space and is the discrete label space, such that the labels within each realisation are distinct. The set of labels of X is given by , where is the projection defined by , where , k is the time of birth, and is a unique index of different targets with the same birth time. The distinct label indicator function is given by The integral of a function f on is given by (1) A well-known labelled RFS is the GLMB RFS, which is a conjugate prior for the multi-target likelihood function [25]. -GLMB is an alternative form of the GLMB RFS for implementation defined on with the probability distribution (2) where and . Each represents a history of association maps while each is a set of object labels. The pair represents the hypothesis that the set I of tracks has a history of association maps with state densities . An LMB [25] is a special case of the GLMB RFS with only one element in discrete space such that the probability density (3) where , is the existence probability of the target with the label l and is the probability density function of the target state. If there exists only one propagation path s, the multi-target likelihood is given by [25, 26] (4) where is the set of positive 1-1 maps , such that implies ; denotes the set of positive 1-1 maps with the domain I; and (5) where . The multi-object filtering density can be propagated by the multi-object Bayes recursion [25] (6) (7) where the set integral is defined as (8) For convenience, we use the convention form [25, 26] such that is the label space of new-born objects with and is the label space at next time, i.e. . 3 MD labelled RFS filter In this section, we derive the likelihood function of MD observation model for label RFS in Section 3.1. Then, we present the separate prediction and update of the MD-GLMB filter in Section 3.2. The joint prediction and update formulation of the labelled RFS filtering is given in Section 3.3. Finally, we provide the generalisation of the MD labelled RFS filters for extended target tracking and multi-sensor tracking in Sections 3.4 and 3.5. 3.1 MD multi-target observation model For a given observation set Z, let W be a subset of Z with , and be the set of all such W. The set W denotes a collection of measurements, all of which are generated by the same target and propagated by different paths. The global index set of W in Z is given by , where is the index extraction. For disjoint subsets , let as the collection of clutter observations. Let be a partition of Z, constructed using elements from the set , given by such that , for any , for any i. We denote and as the jth element of the partition and the set , respectively. Denote by the collection of all possible partitions with i elements constructed as above with . Let and are the collections of all partitions in with and , respectively. Thus, . We also use the notation to represent the set and to represent the set . For instance, with and , the above notations are Proposition 1.For a given multi-target state , the measurement likelihood function is given by (9) where (10) (11) where the summation is taken over all path association hypotheses , such that implies . When , . Proof.Let Y be the set of target detections. As shown in [33], Y is distributed according to (12) where each is given by [21] (13) Then, we can write (14) The overall measurement set is the union of the set of target-originated measurement Y and the set of clutter measurement K, i.e. , thus the multi-target likelihood is given by the convolution (15) This is simplified by appending each with the set to form a partition . If , , else . Thus, the likelihood function can be expressed as follows: (16) Note that the in the denominator cancels out the corresponding term in the product when , leaving one term for each . Hence, (16) can be expressed as (9).□ 3.2 Separate prediction and update of MD-GLMB Under the separate prediction and update framework, the MD-GLMB filter involves two steps, prediction and update, to generate the predicted hypotheses and update hypotheses, respectively. Given the -GLMB filtering density (2) at the current time, the -GLMB prediction density at the next time is given by [25, 26] (17) where , and (18) (19) (20) Proposition 2.If the prediction density is the GLMB density as (17), then under the MD likelihood function (16), the posterior is a GLMB with label space , given by (21) where , and (22) (23) (24) (25) Proof.First, we compute the product of the prior distribution and the likelihood (26) Then, we calculate the set integral of (26) with respect to according to (1) and (8) (27) where , and .According to (7), the posterior GLMB density is described as (21). □ For the recursion of the LMB density with MD, we provide the details in Appendix 1. 3.3 Joint prediction and update of MD-GLMB Since the separate prediction procedure does not take into account the observation Z, a high number of components are generated, which are uncorrelated with the measurements and lead to low weight update components. In order to achieve a more efficient truncation, Hoang et al. [28] propose a joint prediction and update of the GLMB densities recursion and perform a single truncation that incorporates target death/birth into the observation map and update. By combining the prediction (17) and update (21), the posterior GLMB density (28) can be expressed as (28) where (29) Then, we introduce the extend positive 1-1 maps , where the map space and rewrite (29) as (30) (31) 3.4 Generalisation to multiple extended targets tracking In target tracking with a high-resolution sensor, such as short range radar, video sensor and lidar, the target may occupy multiple resolution cells and generates MDs at each scan, which called extended target. For an exact point scatter model, the extended target will generate detections, which is related to a deterministic measurement model. Hence, the MD likelihood (11) becomes an exact point scatterer model likelihood for the extended target. When considering a randomly-varying point scatterer model, the Poisson approximation for the target-originated observation is proposed in [33] and applied to the PHD filter [34, 35], and the CPHD filter [36]. By replacing (11) with Poisson approximated extended target likelihood function (32) (33) where is the probability detection of a particular extended target with state and denotes the Poisson probability density, and relaxing the partition constrain, i.e. the cardinalities of all elements in each partition are not limited by the number of modes, the proposed MD-GLMB and MD-LMB filters can be extended target GLMB and LMB filters described in [31]. 3.5 Generalisation to multi-sensor tracking The proposed MD-GLMB and MD-LMB filters can also be generalised for target tracking with multiple single-detection sensors. In this multi-sensor system, multiple measurements generated by one target can be detected via multiple sensors and be tagged with the ID of the sensor. Thus, the measurement-to-mode association is known and the likelihood function (11) reduces to (34) where the mode variables represent the measurement modes of different sensors. In order to accommodate the multi-sensor measurements, we construct the partition as [37], in which the measurement set made by the same target from different sensors, , such that for all , where . The clutter set V consists all clutter measurements from each independent sensor and the clutter density becomes (35) Compared with the existing multi-sensor GLMB filter in [32], our MD-GLMB filter provides a compact formulation and reduces the computation by constructing partitions with the best measurement subsets as [37, 38] does. 4 Efficient implementation The exact GLMB density recursion requires generating all valid association hypotheses, which results in the number of components grows exponentially with time. In order to achieve a tractable solution, the GLMB density is truncated by remaining T valid association hypotheses with the highest weight, which has the smallest -distance from the exact density. In this section, we present efficient implementations to generate the T best hypotheses both in pre-partition and joint partition. 4.1 Pre-partition In (28), for a given partition , each component generates a set of in the -GLMB density update. Finding a set of pairs with significant in each component is an optimal assignment problem where is an assignment matrix with entries and is a cost matrix with entries (36) where . For generating T best hypotheses in each partition, one method is the Murty's algorithm [39, 40], which has computational complexity. Another method proposed in [29] is based on stochastic sampling, which represents the association hypotheses as positive 1-1 vector and samples them from a Markov chain. Although this method only has complexity, it requires a large T to guarantee convergence. 4.2 Joint partition Because the pre-partition methods described in Section 4.1 require exhaustive enumeration of all partitions, computations are wasted in solving assignment problems in those partitions, whose corresponding hypotheses have negligible weights. In addition, since the generated hypotheses are optimal only in their corresponding partitions, we ensure that the selected hypotheses are global optimal by producing T best hypotheses in each partition, which is very expensive. Limited to the computational platform and task requirement, the number of the hypotheses within each partition is often less than T, which results in the combination does not contain the global T best hypotheses. In the following subsections, we propose a joint prediction, partition and update formulation to obtain global optimal results with low complexity. For deriving the joint prediction, partition and update formulation, we first rewrite the likelihood (9) as a global index set formulation (37) where is the set of maps , such that implies , and where is the set of positive 1-set maps , such that implies , and (38) Thus, the global index set formulation of the joint weights (29) is (39) where (40) Then, we enumerate , and . For each component we define a vector of sets (41) by (42) where is the set of special 1-set maps , which has the following property: (43) In other words, there are no distinct n and with . We denote the set of all special elements of by . , and can be obtained from by (44) (45) Remark 1.The likelihood function (37) can be expressed as (46) which reduces the redundant hypotheses of partition and association. This formulation provides the partition and association simultaneously results in a global optimal implementation. We assume that for all and , and define (47) where is the index of the measurements set assigned to label , with denoting the missed detection of the track and denoting the disappearance of the track . Since in (44), we have , and from (47), we note that (48) (49) Thus, (39) can be rewritten as (50) where . 4.3 Gibbs sampling Under the joint partition framework, we extended the Gibbs sampler to sample the partition-association vectors as described below. In order to find a set of positive 1-1 array with significant , we sample the candidate vectors from (51) This procedure can be performed by using a Gibbs sampler as Algorithm 1 (see Fig. 3) described, which has a Markov transition kernel (52) where . Fig. 3Open in figure viewerPowerPoint Algorithm 1: Gibbs sampler Lemma 1.Let , , and be the set of all special positive . Then, for any , can be factorised as (53) Proposition 3.For each , (54) Proof.Analogue to [29], the conditionals be derived as Factorising using Lemma 1, gives □ For or , , and Proposition 3 implies . On the other hand, given an , Proposition 3 implies that , unless one of elements in J belongs to , where represents . Consequently, for Thus, it is not complex and cost to sample from the conditionals shown in Algorithm 1 lines 5–8 (Fig. 3), which has an complexity. This leads to the complexity of the extended Gibbs sampler is . As formalised in Proposition 4, the Markov chain converges to the target distribution (51). Proposition 4.Given an initial state in with any value, the Gibbs sampler sampling from conditionals (54) will converge to the probability distribution (51) at an exponential rate, i.e. (55) where denote the jth power of the transition matrix and is the least likely two-step transition probability. The proof is given in Appendix 2. For implementing the MD-GLMB filter, the GLMB density can be represented by an enumerated form (56) where , . The MD-GLMB filter propagates the parameter set forward in time. At the next time, the new parameter set can be calculated by Algorithm 2 (see Fig. 4). For consistency, we define , , and . Fig. 4Open in figure viewerPowerPoint Algorithm 2: GLMB parameter set update 4.4 Complexity For the pre-partition implementations of the MD-GLMB filter, the major computational load depends on the number of partitions and the calculation of generating association hypotheses in each partition. As Section 4.1 describes, the complexities of Murty algorithm and Gibbs sampling are cubic and linear in the number of elements of the current partition, respectively. The number of ways to partition a set with size n into k non-empty subsets is given by Stirling partition number (or Stirling number of the second kind) (57) where is the binomial coefficient. Moreover, the number of ways to partition a set with n elements into k non-empty subset, in which there exists a maximal number of subsets with j elements is given by (58) where denotes the maxim integer not bigger than . Thus, the number of partitions in is (59) and the number of partitions in is (60) For a component with targets, the complexity of generating T best hypotheses by using Murty's algorithm is given by (61) and the Gibbs algorithm has the computational complexity as (62) For the joint partition method, the number of elements in is , and by using the Gibbs algorithm (Algorithm 1 (Fig. 3)), the complexity is given by (63) We assume that the computational costs of the basic operation in these algorithms are identical and plot the curves of the logarithmic complexities with is shown in Fig. 5. It is obvious that the joint partition Gibbs algorithm has very low computational load compared with the two pre-partition methods. Fig. 5Open in figure viewerPowerPoint Logarithmic complexity of the pre-partition Murty algorithm, the pre-partition Gibbs algorithm, and the joint partition Gibbs algorithm 5 Simulation results In this section, we illustrate the performances of the proposed implementation of the MD-GLMB filter compared with the MD-PHD filter [21] and the MP-CBMeMBer filter [22]. Consider now a scenario with a total of three targets in the surveillance region . The object state comprises the planar position, velocity and the turn rate . The transition density for the coordinated turn model is given by , where with , (64) We use an OTHR observation model as Section 2.1 described with and and assume that the measurement noise , , and are independent zero-mean white Gaussian noise with standard deviations of , , and , respectively. The clutter follows a Poisson process with a uniform distribution over the surveillance region and mean intensity . The survive probability and the scan duration are configured by and , respectively. The detection probabilities of each path are and , which result in the total detection probability is . The birth target process is modelled as an LMB with existence probability and Gaussian state prior , where and . To handle non-linear state transition and observation, we exploit the sequential Monte Carlo (MC) [26, 41] to implement these filters. In this scenario with high uncertainty, we require a large number of particles, which are set to 3000. We set the maximum number of components and prune the components by the threshold in the GLMB filter. The cardinality estimates averaged over 100 MC runs are illustrated in Fig. 6. Fig. 7 shows the average optimal subpattern assignment (OSPA) miss distance [42] with parameters and for various algorithms across 100 MC runs. It is obvious that the MD-GLMB filter provides more accurate estimation compared with the MD-PHD and the MP-CBMeMBer filters. Figs. 8 and 9 show that the MD-GLMB filters with pre-partition and joint partition both track all objects successfully in one run. The results of tracking time consumption are summarised in Table 1, which shows that the joint partition implementation is more efficient than the pre-partition method. Table 1. Average time cost Filter Average time cost, s MD-PHD 11.8171 MP-CBMeMBer 40.8129 MD-GLMB (pre-partition) 187.5812 MD-GLMB (joint partition) 175.8481 Fig. 6Open in figure viewerPowerPoint Cardinality estimates of the MD-PHD filter, MP-CBMeMBer filter, and MD-GLMB filters with three implementations Fig. 7Open in figure viewerPowerPoint Average OSPA distance of the MD-PHD filter, MP-CBMeMBer filter, and MD-GLMB filter with three implementations Fig. 8Open in figure viewerPowerPoint MD-GLMB filter with pre-partition: estimates and trajectories in the x and y coordinates versus time Fig. 9Open in figure viewerPowerPoint MD-GLMB filter with joint partition: estimates and trajectories in the x and y coordinates versus time 6 Conclusion In this paper, we proposed a new algorithm to handle the MD multi-target tracking problem based on the labelled RFS filters. The separated and joint recursions of the GLMB density with MD observation model are derived and are generalised to the extended target tracking and multi-sensor tracking. The pre-partition based implementation and the joint partition implementation with Gibbs sampling are provided. The evaluations show that the proposed approach outperforms the previous algorithms, such as the MD-PHD and the MP-CBMeMBer filters. 7 Acknowledgments An earlier and basic version of this work, named "Multipath Generalised Labeled Multi-Bernoulli Filter", was presented at the International Conference on Information Fusion 2018, held in Cambridge, United Kingdom. 9 Appendix 9.1 Appendix 1: LMB filtering with (9) likelihood formulation Given the LMB filtering density (3) at the current time and the new birth LMB density defined on , the prediction density is also an LMB density [27] (65) where (66) (67) (68) If the prior is an LMB of the form (65), then under the MD likelihood function (9), the posterior is a GLMB with label space , given by (69) where , , and (70) (71) (72) The LMB filter approximates the filtering GLMB density (69) by an LMB density (73) where (74) (75) (76) (77) 9.2 Appendix 2: Proof of Proposition 4 Inspired by Vo et al. [29], we also prove the convergence of finite state Markov chains from two terms, irreducibility and regularity. An irreducible Markov chain can move from any state to any other state in finite time, and if some finite power of the transition matrix has all positive entries, this irreducible Markov chain is regular [43]. Consider the nth conditional , with positive 1-1. Then for each , , from (54), we have (78) where denotes the normalising constant in the nth sub-iteration of the Gibbs sampler. If , where is the P-dimensional zero vector, then for each , and then (78) becomes . Since is always positive, we have (79) On the other hand, if , then for each , , because cannot be both positive and zero. 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