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Poisson multi-Bernoulli mixture filter for trajectory measurements

Marco Fontana, Ángel F. García-Fernández, Simon Maskell

TL;DR

This work addresses multi-target tracking when sensor measurements are trajectory observations spanning two time steps. It introduces the trajectory measurement PMBM (TM-PMBM) filter that propagates a PMBM density over two-step trajectories and then marginalises to obtain a PMBM over the target states. The paper also derives lighter alternatives: a PMB density via Kullback-Leibler divergence minimisation and a Gaussian TM-PMBM implementation for linear-Gaussian models. Through simulations, TM-PMBM and TM-PMB outperform their target-state counterparts, particularly in high-clutter scenarios, while reducing computational burden for longer windows.

Abstract

This paper presents a Poisson multi-Bernoulli mixture (PMBM) filter for multi-target filtering based on sensor measurements that are sets of trajectories in the last two-time step window. The proposed filter, the trajectory measurement PMBM (TM-PMBM) filter, propagates a PMBM density on the set of target states. In prediction, the filter obtains the PMBM density on the set of trajectories over the last two time steps. This density is then updated with the set of trajectory measurements. After the update step, the PMBM posterior on the set of two-step trajectories is marginalised to obtain a PMBM density on the set of target states. The filter provides a closed-form solution for multi-target filtering based on sets of trajectory measurements, estimating the set of target states at the end of each time window. Additionally, the paper proposes computationally lighter alternatives to the TM-PMBM filter by deriving a Poisson multi-Bernoulli (PMB) density through Kullback-Leibler divergence minimisation in an augmented space with auxiliary variables. The performance of the proposed filters are evaluated in a simulation study.

Poisson multi-Bernoulli mixture filter for trajectory measurements

TL;DR

This work addresses multi-target tracking when sensor measurements are trajectory observations spanning two time steps. It introduces the trajectory measurement PMBM (TM-PMBM) filter that propagates a PMBM density over two-step trajectories and then marginalises to obtain a PMBM over the target states. The paper also derives lighter alternatives: a PMB density via Kullback-Leibler divergence minimisation and a Gaussian TM-PMBM implementation for linear-Gaussian models. Through simulations, TM-PMBM and TM-PMB outperform their target-state counterparts, particularly in high-clutter scenarios, while reducing computational burden for longer windows.

Abstract

This paper presents a Poisson multi-Bernoulli mixture (PMBM) filter for multi-target filtering based on sensor measurements that are sets of trajectories in the last two-time step window. The proposed filter, the trajectory measurement PMBM (TM-PMBM) filter, propagates a PMBM density on the set of target states. In prediction, the filter obtains the PMBM density on the set of trajectories over the last two time steps. This density is then updated with the set of trajectory measurements. After the update step, the PMBM posterior on the set of two-step trajectories is marginalised to obtain a PMBM density on the set of target states. The filter provides a closed-form solution for multi-target filtering based on sets of trajectory measurements, estimating the set of target states at the end of each time window. Additionally, the paper proposes computationally lighter alternatives to the TM-PMBM filter by deriving a Poisson multi-Bernoulli (PMB) density through Kullback-Leibler divergence minimisation in an augmented space with auxiliary variables. The performance of the proposed filters are evaluated in a simulation study.

Paper Structure

This paper contains 36 sections, 6 theorems, 58 equations, 9 figures, 1 table, 1 algorithm.

Table of Contents

  1. Introduction
  2. Problem formulation
  3. Variables
  4. Single-target state and set of targets
  5. Single-trajectory state and set of trajectories over $k$ and $k+1$
  6. Single-trajectory measurement and set of trajectory measurements over $k$ and $k+1$
  7. Trajectory integrals
  8. Dynamic model
  9. Transition model for trajectories in $k$ and $k+1$
  10. Birth model
  11. Trajectory-measurement model
  12. PMBM posterior and predicted densities
  13. PMBM density on targets
  14. PMBM on sets of trajectories from $k$ to $k+1$
  15. TM-PMBM filtering recursion
  16. ...and 21 more sections

Key Result

Lemma 1

Given the PMBM filtering density on the set of target states at time step $k$ of the form (eq:PMBM_on_states), the predicted density at time $k+1$ is a PMBM of the form (eq:PMBM_on_trj), with $n_{k+1|k}=n_{k|k}$ and Poisson intensity Each Bernoulli component $f_{k+1|k}^{i,a^{i}}(\cdot)$, $i\in\{1,\dots,n_{k+1|k}\}$, $a^{i}\in\{1,\dots,h_{k+1|k}^{i}\}$, in the MBM part is defined by

Figures (9)

  • Figure 1: An example of three sets of trajectory measurements across three time windows, spanning time steps 1 to 4. Each set of trajectory measurements is represented by a distinct colour. The vertical dashed lines indicate the start and end of each time window. Each trajectory measurement is either a trajectory connecting two one-dimensional point detections at time step $k$ (circle) and $k+1$ (square), a point detection at time step $k$ or a point detection at time step $k+1$.
  • Figure 2: Diagram of the PMBM filter based on trajectory measurements, the TM-PMBM filter. After the prediction from time step $k$, the TM-PMBM filter incorporates information on the set of trajectories at time steps $k$ and $k+1$, resulting in a PMBM density on this set of trajectories. The trajectory measurements at time step $k+1$, which can span the time steps $k$ and $k+1$, are then used to update the PMBM density on the set of trajectories at time steps $k$ and $k+1$. A marginalisation step is then done to keep the information on the set of targets at time step $k+1$. This step also results in PMBM density.
  • Figure 3: Diagram of the PMB filter based on trajectory measurements, the TM-PMB filter. The PMB density on the two-step trajectories is computed via KLD minimisation (with auxiliary variables GarciaFernandez2020a) of the PMBM posterior obtained from the update step performed with trajectory measurements. The order of the KLD minimisation and the marginalisation step can be swapped without affecting the result.
  • Figure 4: Scenarios used to assess the performance of the TM-PMBM filter and to compare it with the standard PMBM filter. In Fig. \ref{['fig:tracklets:Scenario1_GT']}, target positions at $k=1$ are marked with a cross, and subsequent positions are indicated every five time steps with circles.
  • Figure 5: Comparison of the tracking performance of TM-PMBM (depicted with circle markers) and standard PMBM (depicted with square markers) filters based on Scenario 1, with $N_{w}=7$, $\widetilde{p}^{D}=0.7$ and $\overline{\lambda}^{C}=10$. The ground-truth trajectories are shown in blue, while the red dots denote the ground-truth locations at the beginning and end of each time window. Measurements obtained across all time steps are shown in black: $Z\in M_{(k+1)}^{1}$ with crosses, $Z\in M_{(k+1)}^{2}$ with stars, and $Z=(k,z_{1:2})\in M_{(k+1)}^{3}$ with bars connecting $z_{1}$ and $z_{2}$. At many time steps, the outputs of both filters are visually similar though at other time steps, the TM-PMBM clearly outperforms the standard PMBM.
  • ...and 4 more figures

Theorems & Definitions (6)

  • Lemma 1: TM-PMBM prediction
  • Lemma 2: TM-PMBM update
  • Lemma 3: TM-PMBM marginalisation
  • Lemma 4: Gaussian TM-PMBM prediction
  • Lemma 5: Gaussian TM-PMBM update
  • Lemma 6: Gaussian TM-PMBM marginalisation