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The probability hypothesis density (PHD) and Poisson multi-Bernoulli (PMB) filters are two popular set-type multi-object filters. Motivated by the fact that the multi-object filtering density after each update step in the PHD filter is a PMB without approximation, in this paper we present a multi-object smoother involving PHD forward filtering and PMB backward smoothing. This is achieved by first running the PHD filtering recursion in the forward pass and extracting the PMB filtering densities after each update step before the Poisson Point Process approximation, which is inherent in the PHD filter update. Then in the backward pass we apply backward simulation for sets of trajectories to the extracted PMB filtering densities. We call the resulting multi-object smoother hybrid PHD-PMB trajectory smoother. Notably, the hybrid PHD-PMB trajectory smoother can provide smoothed trajectory estimates for the PHD filter without labeling or tagging, which is not possible for existing PHD smoothers. Also, compared to the trajectory PHD filter, which can only estimate alive trajectories, the hybrid PHD-PMB trajectory smoother enables the estimation of the set of all trajectories. Simulation results demonstrate that the hybrid PHD-PMB trajectory smoother outperforms the PHD filter in terms of both state and cardinality estimates, and the trajectory PHD filter in terms of false detections., Comment: 2024 IEEE International conference on multisensor fusion and integration (MFI 2024). arXiv admin note: text overlap with arXiv:2206.08112

Simultaneous localization and mapping (SLAM) methods need to both solve the data association (DA) problem and the joint estimation of the sensor trajectory and the map, conditioned on a DA. In this paper, we propose a novel integrated approach to solve both the DA problem and the batch SLAM problem simultaneously, combining random finite set (RFS) theory and the graph-based SLAM approach. A sampling method based on the Poisson multi-Bernoulli mixture (PMBM) density is designed for dealing with the DA uncertainty, and a graph-based SLAM solver is applied for the conditional SLAM problem. In the end, a post-processing approach is applied to merge SLAM results from different iterations. Using synthetic data, it is demonstrated that the proposed SLAM approach achieves performance close to the posterior Cram\'er-Rao bound, and outperforms state-of-the-art RFS-based SLAM filters in high clutter and high process noise scenarios.

This paper considers a batch solution to the multi-object tracking problem based on sets of trajectories. Specifically, we present two offline implementations of the trajectory Poisson multi-Bernoulli mixture (TPMBM) filter for batch data based on Markov chain Monte Carlo (MCMC) sampling of the data association hypotheses. In contrast to online TPMBM implementations, the proposed offline implementations solve a large-scale, multi-scan data association problem across the entire time interval of interest, and therefore they can fully exploit all the measurement information available. Furthermore, by leveraging the efficient hypothesis structure of TPMBM filters, the proposed implementations compare favorably with other MCMC-based multi-object tracking algorithms. Simulation results show that the TPMBM implementation using the Metropolis-Hastings algorithm presents state-of-the-art multiple trajectory estimation performance., Comment: Accepted for publication in IEEE Transactions on Aerospace and Electronic Systems. MATLAB implementation available at https://github.com/yuhsuansia/Batch-TPMBM-using-MCMC-sampling

This paper proposes a metric to measure the dissimilarity between graphs that may have a different number of nodes. The proposed metric extends the generalised optimal subpattern assignment (GOSPA) metric, which is a metric for sets, to graphs. The proposed graph GOSPA metric includes costs associated with node attribute errors for properly assigned nodes, missed and false nodes and edge mismatches between graphs. The computation of this metric is based on finding the optimal assignments between nodes in the two graphs, with the possibility of leaving some of the nodes unassigned. We also propose a lower bound for the metric, which is also a metric for graphs and is computable in polynomial time using linear programming. The metric is first derived for undirected unweighted graphs and it is then extended to directed and weighted graphs. The properties of the metric are demonstrated via simulated and empirical datasets., Comment: Accepted in IEEE Transactions on Signal Processing. The code is available at https://github.com/JinhaoGu/The-graph-GOSPA-metric

Belief propagation (BP) is a useful probabilistic inference algorithm for efficiently computing approximate marginal probability densities of random variables. However, in its standard form, BP is only applicable to the vector-type random variables with a fixed and known number of vector elements, while certain applications rely on RFSs with an unknown number of vector elements. In this paper, we develop BP rules for factor graphs defined on sequences of RFSs where each RFS has an unknown number of elements, with the intention of deriving novel inference methods for RFSs. Furthermore, we show that vector-type BP is a special case of set-type BP, where each RFS follows the Bernoulli process. To demonstrate the validity of developed set-type BP, we apply it to the PMB filter for SLAM, which naturally leads to new set-type BP-mapping, SLAM, multi-target tracking, and simultaneous localization and tracking filters. Finally, we explore the relationships between the vector-type BP and the proposed set-type BP PMB-SLAM implementations and show a performance gain of the proposed set-type BP PMB-SLAM filter in comparison with the vector-type BP-SLAM filter., Comment: 17 pages, 7 figures

This paper develops a general trajectory probability hypothesis density (TPHD) filter, which uses a general density for target-generated measurements and is able to estimate trajectories of coexisting point and extended targets. First, we provide a derivation of this general TPHD filter based on finding the best Poisson posterior approximation by minimizing the Kullback-Leibler divergence, without using probability generating functionals. Second, we adopt an efficient implementation of this filter, where Gaussian densities correspond to point targets and Gamma Gaussian Inverse Wishart densities for extended targets. The L-scan approximation is also proposed as a simplified version to mitigate the huge computational cost. Simulation and experimental results show that the proposed filter is able to classify targets correctly and obtain accurate trajectory estimation.

In this paper, we propose a Poisson multi-Bernoulli (PMB) filter for extended object tracking (EOT), which directly estimates the set of object trajectories, using belief propagation (BP). The proposed filter propagates a PMB density on the posterior of sets of trajectories through the filtering recursions over time, where the PMB mixture (PMBM) posterior after the update step is approximated as a PMB. The efficient PMB approximation relies on several important theoretical contributions. First, we present a PMBM conjugate prior on the posterior of sets of trajectories for a generalized measurement model, in which each object generates an independent set of measurements. The PMBM density is a conjugate prior in the sense that both the prediction and the update steps preserve the PMBM form of the density. Second, we present a factor graph representation of the joint posterior of the PMBM set of trajectories and association variables for the Poisson spatial measurement model. Importantly, leveraging the PMBM conjugacy and the factor graph formulation enables an elegant treatment on undetected objects via a Poisson point process and efficient inference on sets of trajectories using BP, where the approximate marginal densities in the PMB approximation can be obtained without enumeration of different data association hypotheses. To achieve this, we present a particle-based implementation of the proposed filter, where smoothed trajectory estimates, if desired, can be obtained via single-object particle smoothing methods, and its performance for EOT with ellipsoidal shapes is evaluated in a simulation study., Comment: Accepted for publication in IEEE Transactions on Aerospace and Electronic Systems. MATLAB implementation available at https://github.com/yuhsuansia/Trajectory-PMB-EOT-BP

This paper presents a general solution for computing the multi-object posterior for sets of trajectories from a sequence of multi-object (unlabelled) filtering densities and a multi-object dynamic model. Importantly, the proposed solution opens an avenue of trajectory estimation possibilities for multi-object filters that do not explicitly estimate trajectories. In this paper, we first derive a general multi-trajectory backward smoothing equation based on random finite sets of trajectories. Then we show how to sample sets of trajectories using backward simulation for Poisson multi-Bernoulli filtering densities, and develop a tractable implementation based on ranked assignment. The performance of the resulting multi-trajectory particle smoothers is evaluated in a simulation study, and the results demonstrate that they have superior performance in comparison to several state-of-the-art multi-object filters and smoothers., Comment: Accepted for publication in IEEE Transactions on Signal Processing

This paper proposes a clustering and merging approach for the Poisson multi-Bernoulli mixture (PMBM) filter to lower its computational complexity and make it suitable for multiple target tracking with a high number of targets. We define a measurement-driven clustering algorithm to reduce the data association problem into several subproblems, and we provide the derivation of the resulting clustered PMBM posterior density via Kullback-Leibler divergence minimisation. Furthermore, we investigate different strategies to reduce the number of single target hypotheses by approximating the posterior via merging and inter-track swapping of Bernoulli components. We evaluate the performance of the proposed algorithm on simulated tracking scenarios with more than one thousand targets., Comment: 17 pages, 11 figures, journal paper

This paper presents a solution for recovering full trajectory information, via the calculation of the posterior of the set of trajectories, from a sequence of multitarget (unlabelled) filtering densities and the multitarget dynamic model. Importantly, the proposed solution opens an avenue of trajectory estimation possibilities for multitarget filters that do not explicitly estimate trajectories. In this paper, we first derive a general multitrajectory forward-backward smoothing equation based on sets of trajectories and the random finite set framework. Then we show how to sample sets of trajectories using backward simulation when the multitarget filtering densities are multi-Bernoulli processes. The proposed approach is demonstrated in a simulation study., Comment: Published in 23rd International Conference on Information Fusion. This arXiv version contains more detailed derivations

In this paper we introduce spatiotemporal constraints for trajectories, i.e., restrictions that the trajectory must be in some part of the state space (spatial constraint) at some point in time (temporal constraint). Spatiotemporal contraints on trajectories can be used to answer a range of important questions, including, e.g., "where did the person that were in area A at time t, go afterwards?". We discuss how multiple constraints can be combined into sets of constraints, and we then apply sets of constraints to set of trajectories densities, specifically Poisson Multi-Bernoulli Mixture (PMBM) densities. For Poisson target birth, the exact posterior density is PMBM for both point targets and extended targets. In the paper we show that if the unconstrained set of trajectories density is PMBM, then the constrained density is also PMBM. Examples of constrained trajectory densities motivate and illustrate the key results.

For the standard point target model with Poisson birth process, the Poisson Multi-Bernoulli Mixture (PMBM) is a conjugate multi-target density. The PMBM filter for sets of targets has been shown to have state-of-the-art performance and a structure similar to the Multiple Hypothesis Tracker (MHT). In this paper we consider a recently developed formulation of multiple target tracking as a random finite set (RFS) of trajectories, and present three important and interesting results. First, we show that, for the standard point target model, the PMBM density is conjugate also for sets of trajectories. Second, based on this we develop PMBM trackers (trajectory RFS filters) that efficiently estimate the set of trajectories. Third, we establish that for the standard point target model the multi-trajectory density is PMBM for trajectories in any time window, given measurements in any (possibly non-overlapping) time window. In addition, the PMBM trackers are evaluated in a simulation study, and shown to yield state-of-the-art performance., Comment: arXiv admin note: text overlap with arXiv:1812.05131

The Poisson multi-Bernoulli mixture (PMBM) and the multi-Bernoulli mixture (MBM) are two multi-target distributions for which closed-form filtering recursions exist. The PMBM has a Poisson birth process, whereas the MBM has a multi-Bernoulli birth process. This paper considers a recently developed formulation of the multi-target tracking problem using a random finite set of trajectories, through which the track continuity is explicitly established. A multi-scan trajectory PMBM filter and a multi-scan trajectory MBM filter, with the ability to correct past data association decisions to improve current decisions, are presented. In addition, a multi-scan trajectory $\text{MBM}_{01}$ filter, in which the existence probabilities of all Bernoulli components are either 0 or 1, is presented. This paper proposes an efficient implementation that performs track-oriented $N$-scan pruning to limit computational complexity, and uses dual decomposition to solve the involved multi-frame assignment problem. The performance of the presented multi-target trackers, applied with an efficient fixed-lag smoothing method, are evaluated in a simulation study., Comment: Published in Journals of Advances in Information Fusion, Special issue on Multiple Hypothesis Tracking, Volume 14, Number 2, Page 213-235, December 2019. MATLAB code is available at https://github.com/yuhsuansia/Multi-scan-trajectory-PMBM-filter

The Poisson multi-Bernoulli mixture (PMBM) is a multi-target distribution for which the prediction and update are closed. By applying the random finite set (RFS) framework to multi-target tracking with sets of trajectories as the variable of interest, the PMBM trackers can efficiently estimate the set of target trajectories. This paper derives two trajectory RFS filters for extended target tracking, called extended target PMBM trackers. Compared to the extended target PMBM filter based on sets on targets, explicit track continuity between time steps is provided in the extended target PMBM trackers., Comment: MATLAB code is available at https://github.com/yuhsuansia/Extended-Target-PMBM-Tracker. arXiv admin note: text overlap with arXiv:1812.05131

In this paper, we show the spooky effect at a distance that arises in optimal estimation of multiple targets with the optimal sub-pattern assignment (OSPA) metric. This effect refers to the fact that if we have several independent potential targets at distant locations, a change in the probability of existence of one of them can completely change the optimal estimation of the rest of the potential targets. As opposed to OSPA, the generalised OSPA (GOSPA) metric ($\alpha=2$) penalises localisation errors for properly detected targets, false targets and missed targets. As a consequence, optimal GOSPA estimation aims to lower the number of false and missed targets, as well as the localisation error for properly detected targets, and avoids the spooky effect., Comment: This paper received the third best paper award at the 22nd International Conference on Information Fusion, Ottawa, Canada, 2019. Matlab code of the GOSPA metric can be found in https://github.com/abusajana/GOSPA . Additional information on MTT can be found in the online course https://www.youtube.com/channel/UCa2-fpj6AV8T6JK1uTRuFpw

This paper presents the Gaussian implementation of the multi-Bernoulli mixture (MBM) filter. The MBM filter provides the filtering (multi-target) density for the standard dynamic and radar measurement models when the birth model is multi-Bernoulli or multi-Bernoulli mixture. Under linear/Gaussian models, the single target densities of the MBM mixture admit Gaussian closed-form expressions. Murty's algorithm is used to select the global hypotheses with highest weights. The MBM filter is compared with other algorithms in the literature via numerical simulations., Comment: Matlab code of the MBM and PMBM filters is provided in https://github.com/Agarciafernandez/MTT . Additional information on MTT including PMBM and MBM filters can be found in the online course https://www.youtube.com/channel/UCa2-fpj6AV8T6JK1uTRuFpw

The Poisson multi-Bernoulli mixture (PMBM) is an unlabelled multi-target distribution for which the prediction and update are closed. It has a Poisson birth process, and new Bernoulli components are generated on each new measurement as a part of the Bayesian measurement update. The PMBM filter is similar to the multiple hypothesis tracker (MHT), but seemingly does not provide explicit continuity between time steps. This paper considers a recently developed formulation of the multi-target tracking problem as a random finite set (RFS) of trajectories, and derives two trajectory RFS filters, called PMBM trackers. The PMBM trackers efficiently estimate the set of trajectories, and share hypothesis structure with the PMBM filter. By showing that the prediction and update in the PMBM filter can be viewed as an efficient method for calculating the time marginals of the RFS of trajectories, continuity in the same sense as MHT is established for the PMBM filter.

This paper proposes an efficient implementation of the Poisson multi-Bernoulli mixture (PMBM) trajectory filter. The proposed implementation performs track-oriented N-scan pruning to limit complexity, and uses dual decomposition to solve the involved multi-frame assignment problem. In contrast to the existing PMBM filter for sets of targets, the PMBM trajectory filter is based on sets of trajectories which ensures that track continuity is formally maintained. The resulting filter is an efficient and scalable approximation to a Bayes optimal multi-target tracking algorithm, and its performance is compared, in a simulation study, to the PMBM target filter, and the delta generalized labelled multi-Bernoulli filter, in terms of state/trajectory estimation error and computational time., Comment: 8 pages, 2018 21st International Conference on Information Fusion (FUSION)

The Poisson multi-Bernoulli mixture (PMBM) is a multi-object conjugate prior for the closed-form Bayes random finite sets filter. The extended object PMBM filter provides a closed-form solution for multiple extended object filtering with standard models. This paper considers computationally lighter alternatives to the extended object PMBM filter by propagating a Poisson multi-Bernoulli (PMB) density through the filtering recursion. A new local hypothesis representation is presented where each measurement creates a new Bernoulli component. This facilitates the developments of methods for efficiently approximating the PMBM posterior density after the update step as a PMB. Based on the new hypothesis representation, two approximation methods are presented: one is based on the track-oriented multi-Bernoulli (MB) approximation, and the other is based on the variational MB approximation via Kullback-Leibler divergence minimisation. The performance of the proposed PMB filters with gamma Gaussian inverse-Wishart implementations are evaluated in a simulation study., Comment: Accepted for publication in IEEE T-AES