Karl Granstrom

The Poisson Multi-Bernoulli Mixture (PMBM) density is a conjugate multi-target density for the standard point target model with Poisson point process birth. This means that both the filtering and predicted densities for the set of targets are PMBM. In this paper, we first show that the PMBM density is also conjugate for sets of trajectories with the standard point target measurement model. Second, based on this theoretical foundation, we develop two trajectory PMBM filters that provide recursions to calculate the posterior density for the set of all trajectories that have ever been present in the surveillance area, and the posterior density of the set of trajectories present at the current time step in the surveillance area. These two filters therefore provide complete probabilistic information on the considered trajectories enabling optimal trajectory estimation. Third, we establish that the density of the set of trajectories in any time window, given the measurements in a possibly different time window, is also a PMBM. Finally, the trajectory PMBM filters are evaluated via simulations, and are shown to yield state-of-the-art performance compared to other multi-target tracking algorithms based on random finite sets and multiple hypothesis tracking.

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.

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.

Radio-based vehicular simultaneous localization and mapping (SLAM) aims to localize vehicles while mapping the landmarks in the environment. We propose a sequence of three Poisson multi-Bernoulli mixture (PMBM) based SLAM filters, which handle the entire SLAM problem in a theoretically optimal manner. The complexity of the three proposed SLAM filters is progressively reduced while sustaining high accuracy by deriving SLAM density approximation with the marginalization of nuisance parameters (either vehicle state or data association). Firstly, the PMBM SLAM filter serves as the foundation, for which we provide the first complete description based on a Rao-Blackwellized particle filter. Secondly, the Poisson multi-Bernoulli (PMB) SLAM filter is based on the standard reduction from PMBM to PMB, but involves a novel interpretation based on auxiliary variables and a relation to Bethe free energy. Finally, using the same auxiliary variable argument, we derive a marginalized PMB SLAM filter, which avoids particles and is instead implemented with a low-complexity cubature Kalman filter. We evaluate the three proposed SLAM filters in comparison with the probability hypothesis density (PHD) SLAM filter in 5G mmWave vehicular networks and show the computation-performance trade-off between them.

This tutorial introduces state-of-the-art methods for tracking multiple spatially extended objects based on unlabeled noisy point clouds, e.g., from radar or lidar sensors. In the first part, the focus lies on tracking a single extended object, i.e., the objective is to simultaneously estimate the shape and position of a moving object based on spatially distributed noisy detections. Model-based approaches for tracking elliptical and star-convex shaped objects are treated. Furthermore, recent learning-based approaches are discussed, which learn the spatial distribution of detections from real data. The second part considers the track management and data association problem, i.e., the initialization and termination of tracks as well as the association of detections to objects. After an in-depth analysis of the data association problem for extended objects, various data association approaches are discussed. Finally, recent frameworks for multiple extended object tracking are introduced such as the random finite set-based Poisson multi-Bernoulli mixture framework.

The Poisson multi-Bernoulli mixture (PMBM) is a multiobject conjugate prior for the closed-form Bayes random finite set filter. The extended object PMBM filter provides a closed-form solution for multiple extended object filtering with standard models. This article 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 minimization. The performance of the proposed PMB filters with gamma Gaussian inverse-Wishart implementations are evaluated in a simulation study.

This paper presents a data-driven measurement model for extended object tracking (EOT) with automotive radar. Specifically, the spatial distribution of automotive radar measurements is modeled as a hierarchical truncated Gaussian (HTG) with structural geometry parameters that can be learned from the training data. The HTG measurement model provides an adequate resemblance to the spatial distribution of real-world automotive radar measurements. Moreover, large-scale radar datasets can be leveraged to learn the geometry-related model parameters and offload the computationally demanding model parameter estimation from the state update step. The learned HTG measurement model is further incorporated into a random matrix based EOT approach with two (multi-sensor) measurement updates: one is based on a factorized Gaussian inverse-Wishart density representation and the other is based on a Rao-Blackwellized particle density representation. The effectiveness of the proposed approaches is verified on both synthetic data and real-world nuScenes dataset over 300 trajectories.

This paper presents two trajectory Poisson multi-Bernoulli (TPMB) filters for multi-target tracking: one to estimate the set of alive trajectories at each time step and another to estimate the set of all trajectories, which includes alive and dead trajectories, at each time step. The filters are based on propagating a Poisson multi-Bernoulli (PMB) density on the corresponding set of trajectories through the filtering recursion. After the update step, the posterior is a PMB mixture (PMBM) so, in order to obtain a PMB density, a Kullback-Leibler divergence minimisation on an augmented space is performed. The developed filters are computationally lighter alternatives to the trajectory PMBM filters, which provide the closed-form recursion for sets of trajectories with Poisson birth model, and are shown to outperform previous multi-target tracking algorithms.

A decentralized Poisson multi-Bernoulli filter is proposed to track multiple vehicles using multiple high-resolution sensors. Independent filters estimate the vehicles’ presence, state, and shape using a Gaussian process extent model; a decentralized filter is realized through fusion of the filters posterior densities. An efficient implementation is achieved by parametric state representation, utilization of single hypothesis tracks, and fusion of vehicle information based on a fusion mapping. Numerical results demonstrate the performance.

5G millimeter wave (mmWave) signals can enable accurate positioning in vehicular networks when the base station and vehicles are equipped with large antenna arrays. However, radio-based positioning suffers from multipath signals generated by different types of objects in the physical environment. Multipath can be turned into a benefit, by building up a radio map (comprising the number of objects, object type, and object state) and using this map to exploit all available signal paths for positioning. We propose a new method for cooperative vehicle positioning and mapping of the radio environment, comprising a multiple-model probability hypothesis density filter and a map fusion routine, which is able to consider different types of objects and different fields of views. Simulation results demonstrate the performance of the proposed method.