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Distributed Poisson multi-Bernoulli filtering via generalised covariance intersection

Ángel F. García-Fernández, Giorgio Battistelli

TL;DR

This work presents an approximate GCI fusion rule for PMB densities in distributed multi-object filtering. By upper-bounding the $\omega$-power of a PMB with an unnormalised PMB, the normalised product of two PMBs becomes a PMBM, providing closed-form expressions for the fused density. The fused PMBM can be projected back to PMB to permit recursive filtering, and the approach is extended to Gaussian implementations and sensors with limited fields of view. Simulations demonstrate that GCI-PMB fusion improves performance over alternative distributed MOF filters, particularly compared with arithmetic-average approaches, and remains effective when fusion is performed across multiple agents or FoV partitions.

Abstract

This paper presents the distributed Poisson multi-Bernoulli (PMB) filter based on the generalised covariance intersection (GCI) fusion rule for distributed multi-object filtering. Since the exact GCI fusion of two PMB densities is intractable, we derive a principled approximation. Specifically, we approximate the power of a PMB density as an unnormalised PMB density, which corresponds to an upper bound of the PMB density. Then, the GCI fusion rule corresponds to the normalised product of two unnormalised PMB densities. We show that the result is a Poisson multi-Bernoulli mixture (PMBM), which can be expressed in closed form. Future prediction and update steps in each filter preserve the PMBM form, which can be projected back to a PMB density before the next fusion step. Experimental results show the benefits of this approach compared to other distributed multi-object filters.

Distributed Poisson multi-Bernoulli filtering via generalised covariance intersection

TL;DR

This work presents an approximate GCI fusion rule for PMB densities in distributed multi-object filtering. By upper-bounding the -power of a PMB with an unnormalised PMB, the normalised product of two PMBs becomes a PMBM, providing closed-form expressions for the fused density. The fused PMBM can be projected back to PMB to permit recursive filtering, and the approach is extended to Gaussian implementations and sensors with limited fields of view. Simulations demonstrate that GCI-PMB fusion improves performance over alternative distributed MOF filters, particularly compared with arithmetic-average approaches, and remains effective when fusion is performed across multiple agents or FoV partitions.

Abstract

This paper presents the distributed Poisson multi-Bernoulli (PMB) filter based on the generalised covariance intersection (GCI) fusion rule for distributed multi-object filtering. Since the exact GCI fusion of two PMB densities is intractable, we derive a principled approximation. Specifically, we approximate the power of a PMB density as an unnormalised PMB density, which corresponds to an upper bound of the PMB density. Then, the GCI fusion rule corresponds to the normalised product of two unnormalised PMB densities. We show that the result is a Poisson multi-Bernoulli mixture (PMBM), which can be expressed in closed form. Future prediction and update steps in each filter preserve the PMBM form, which can be projected back to a PMB density before the next fusion step. Experimental results show the benefits of this approach compared to other distributed multi-object filters.

Paper Structure

This paper contains 26 sections, 5 theorems, 80 equations, 6 figures, 1 table, 1 algorithm.

Table of Contents

  1. Introduction
  2. Problem formulation
  3. Approximate GCI Fusion of two PMBs
  4. PMB approximation of the power of a PMB
  5. Normalised product of two PMB densities
  6. Fused PMBM via assignment sets
  7. Fused PMBM in track-oriented form
  8. PMB projection
  9. Minimisation of the weighted sum of KLDs
  10. Discussion
  11. Gaussian GCI-PMB implementation
  12. PMB approximation of the power of a PMB
  13. Power of PPP intensity
  14. Power of Bernoulli density
  15. GCI rule
  16. ...and 11 more sections

Key Result

Lemma 1

Let $\omega\in\left[0,1\right]$, the $\omega$-power of a PMB density $f_{1}\left(\cdot\right)$ of the form (eq:PMB_density) is approximated by the upper bound (eq:upper_bound) resulting in where $\alpha$ is a proportionality constant and $q_{1}\left(\cdot\right)$ is a PMB density where the PPP density $q_{1}^{\mathrm{p}}\left(\cdot\right)$ has intensity and the $i$-th Bernoulli density $q_{1}^

Figures (6)

  • Figure 1: Approximate GCI fusion rule (with parameter $\omega\in\left[0,1\right]$) for two PMB densities. The $\omega$ and $1-\omega$ powers of the PMB densities (PMB1, PMB2) are upper bounded by unnormalised PMB densities (uPMB1, uPMB2). The normalised product of these is a PMBM. The PMBM can then be projected back to a PMB Williams15bWilliams15.
  • Figure 2: Example fusion of two PMBs $q_{1}\left(\cdot\right)$ (red) and $q_{2}\left(\cdot\right)$ (blue). Both PMBs have three Bernoulli components, indexed by 1, 2 and 3 (top figure). The probability of existence is shown at the center of the 3-$\sigma$ ellipse representing the mean and covariance matrix. The bottom figure shows the PMB (black) in the most likely hypothesis of the fused PMBM.
  • Figure 3: Scenario 1, similar to Angel18_b. There are four objects that get in close proximity in the middle of the simulation. All objects are born at the beginning of the simulation and the blue one dies at time step 40. The object positions every 10 time steps are marked with a circle, and the initial positions with filled circles.
  • Figure 4: RMS-GOSPA errors at each time step for $N_{f}=5$ (Scenario 1). When the agents perform information fusion the error decreases. Among the distributed filters, DPMB-V-GCI and DPMBM-GCI have the best performance.
  • Figure 5: Scenario 2 considering 4 agents with limited FoVs and 100 time steps. The FoV of each agent is a rectangle shown with a different colour. There are six objects, whose positions are marked every 10 time steps with a circle, and the initial positions with filled circles. The initial time step of each object is written in black next to its initial position. The final time step is written in red next to its final position.
  • ...and 1 more figures

Theorems & Definitions (6)

  • Lemma 1
  • Theorem 2
  • Example 3
  • Lemma 4
  • Lemma 5
  • Lemma 6