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Gaussian multi-target filtering with target dynamics driven by a stochastic differential equation

Ángel F. García-Fernández, Simo Särkkä

TL;DR

The paper addresses multi-target tracking where targets evolve in continuous time according to linear or nonlinear stochastic differential equations and appear/disappear as a Poisson process, with measurements arriving discretely. It develops a Gaussian continuous-discrete PMBM filtering framework that discretises the CT dynamics at measurement times and uses a Kullback-Leibler divergence-based moment-matching procedure to obtain a best Gaussian approximation of the birth process. Key contributions include closed-form Gaussian birth parameters for linear SDEs, steady-state birth solutions, and extensions to nonlinear SDEs, yielding CD-PMBM and its PMB/PHD/CPHD counterparts with significantly reduced computation relative to discrete-time counterparts. The approach enables principled tracking under asynchronous, non-uniform sampling and is demonstrated to provide accurate performance with substantial speedups in both linear and nonlinear scenarios, with broad applicability to space-object surveillance, automotive tracking, and other multi-target domains.

Abstract

This paper proposes multi-target filtering algorithms in which target dynamics are given in continuous time and measurements are obtained at discrete time instants. In particular, targets appear according to a Poisson point process (PPP) in time with a given Gaussian spatial distribution, targets move according to a general time-invariant linear stochastic differential equation, and the life span of each target is modelled with an exponential distribution. For this multi-target dynamic model, we derive the distribution of the set of new born targets and calculate closed-form expressions for the best fitting mean and covariance of each target at its time of birth by minimising the Kullback-Leibler divergence via moment matching. This yields a novel Gaussian continuous-discrete Poisson multi-Bernoulli mixture (PMBM) filter, and its approximations based on Poisson multi-Bernoulli and probability hypothesis density filtering. These continuous-discrete multi-target filters are also extended to target dynamics driven by nonlinear stochastic differential equations.

Gaussian multi-target filtering with target dynamics driven by a stochastic differential equation

TL;DR

The paper addresses multi-target tracking where targets evolve in continuous time according to linear or nonlinear stochastic differential equations and appear/disappear as a Poisson process, with measurements arriving discretely. It develops a Gaussian continuous-discrete PMBM filtering framework that discretises the CT dynamics at measurement times and uses a Kullback-Leibler divergence-based moment-matching procedure to obtain a best Gaussian approximation of the birth process. Key contributions include closed-form Gaussian birth parameters for linear SDEs, steady-state birth solutions, and extensions to nonlinear SDEs, yielding CD-PMBM and its PMB/PHD/CPHD counterparts with significantly reduced computation relative to discrete-time counterparts. The approach enables principled tracking under asynchronous, non-uniform sampling and is demonstrated to provide accurate performance with substantial speedups in both linear and nonlinear scenarios, with broad applicability to space-object surveillance, automotive tracking, and other multi-target domains.

Abstract

This paper proposes multi-target filtering algorithms in which target dynamics are given in continuous time and measurements are obtained at discrete time instants. In particular, targets appear according to a Poisson point process (PPP) in time with a given Gaussian spatial distribution, targets move according to a general time-invariant linear stochastic differential equation, and the life span of each target is modelled with an exponential distribution. For this multi-target dynamic model, we derive the distribution of the set of new born targets and calculate closed-form expressions for the best fitting mean and covariance of each target at its time of birth by minimising the Kullback-Leibler divergence via moment matching. This yields a novel Gaussian continuous-discrete Poisson multi-Bernoulli mixture (PMBM) filter, and its approximations based on Poisson multi-Bernoulli and probability hypothesis density filtering. These continuous-discrete multi-target filters are also extended to target dynamics driven by nonlinear stochastic differential equations.

Paper Structure

This paper contains 39 sections, 7 theorems, 100 equations, 5 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Problem formulation and background
  3. Multi-target dynamic and measurement models
  4. Overview of the solution
  5. Integral computations
  6. Integral of type 1
  7. Integral of type 2
  8. Integral of type 3
  9. Discretised multi-target dynamic model
  10. Probability of survival
  11. Single-target transition density
  12. Birth process
  13. Best Gaussian KLD fit to the birth process
  14. Birth process via state augmentation
  15. Steady-state mean and covariance at appearance time
  16. ...and 24 more sections

Key Result

Proposition 1

Under Assumptions A1-A4, the mean at the time of birth of a single target is where and The covariance matrix at the time of birth of a single target is where where and The integrals (eq:cov_birth_E_C_x) and (eq:Sigma_xx) can be computed using (eq:Integral_type2). The integrals in (eq:Sigma_uu1) can be computed using (eq:Integral_type1). The integrals (eq:Sigma_xu) and (eq:Sigma_uu2) can be

Figures (5)

  • Figure 1: Normalised histograms of Example \ref{['exa:Birth_samples']} with samples from the single target birth density (\ref{['eq:single_target_birth_density']}) with $\Delta t_{k}=1\,\mathrm{s}$ (top left), from its best Gaussian fit with $\Delta t_{k}=1\,\mathrm{s}$ (top right), from the single target birth density with $\Delta t_{k}=2\,\mathrm{s}$ (bottom left), and its best Gaussian fit $\Delta t_{k}=2\,\mathrm{s}$ (bottom right).
  • Figure 2: Ground truth set of trajectories with 10 trajectories. Every 10 time steps, the position of a trajectory is marked with circles. The initial position is marked with a filled circle, and the number next to it indicates time step of birth.
  • Figure 3: RMS GOSPA positional errors ($\mathrm{m)}$ and their decomposition against time.
  • Figure 4: Ground truth set of trajectories
  • Figure 5: RMS GOSPA positional errors ($\mathrm{m)}$ and their decomposition against time.

Theorems & Definitions (8)

  • Proposition 1
  • Example 2
  • Lemma 3
  • Lemma 4: Prediction
  • Lemma 5: Update
  • Lemma 6
  • Lemma 7
  • Proposition 8