Poisson Multi-Bernoulli Mixtures for Sets of Trajectories

TL;DR

This paper establishes that the Poisson Multi-Bernoulli Mixture (PMBM) density is conjugate for sets of trajectories under the standard point target model with PPP birth, enabling closed-form trajectory Bayesian filtering. It introduces two trajectory PMBM (tpmbm) filters that recursively compute the posterior distributions for the set of all trajectories and the set of alive trajectories, providing complete probabilistic trajectory information and optimal estimation. The authors prove a time-marginalisation property: the density of trajectories in any time window, given measurements from any time window, remains PMBM, and they derive time-interval PMBM results. A linear-Gaussian implementation with Gaussian moment and information forms, plus a Gaussian $L$-scan approximation, yields tractable filters; extensive simulations show tpmbm methods deliver state-of-the-art trajectory estimation performance compared with leading multi-target trackers.

Abstract

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.

Figures (6)

• Figure 1: Illustration of the trajectories considered in Theorem \ref{['thm:all_to_trajs_in_a2g_alive_in_e2z']}. Left to right: set of all (1-D) trajectories up to the current time step $100$, $\mathbf{X}_{0:100}$; corresponding target set at time step $50$, $\mathbf{x}_{50}$; corresponding set of current trajectories $\mathbf{X}_{0:100}^{100}$; corresponding set of trajectories in the time interval $25$ to $75$, $\mathbf{X}_{25:75}$; corresponding set of trajectories in the time interval $25$ to $75$ that alive at some time in the interval $45$ to $55$, $\mathbf{X}_{25:75}^{45:55}$. If $\mathbf{X}_{0:100}$ is pmbm distributed, so are the rest of the sets of trajectories.
• Figure 2: Ground truth set of trajectories in each of the three scenarios in the simulations. The start time of each trajectory is marked with a circle.
• Figure 3: Trajectory estimation performance for the set of all trajectories in terms of the normalised tm over time.
• Figure 4: Trajectory estimation performance for the set of alive trajectories in terms of the normalised tm over time
• Figure 5: Target cardinalities against time in each of the three scenarios in the simulations.

Theorems & Definitions (16)

• Theorem 1: tpmbm prediction for all trajectories
• proof
• Theorem 2: tpmbm prediction for current trajectories
• proof
• Theorem 3: tpmbm update
• proof
• Theorem 4
• proof
• Theorem 5
• Corollary 1