Multitarget bayes filtering via first-order multitarget moments

TLDR

Multitarget detection and tracking ideally use a recursive Bayes nonlinear filter, but its computational cost forces approximations such as the constant‑gain Kalman filter that propagates only the posterior expectation; the analogous first‑order moment for multitargets is the probability hypothesis density (PHD). This paper proposes to extend the constant‑gain Kalman filter idea to multitarget systems by propagating the PHD as a first‑order statistical moment of the multitarget posterior. The authors derive recursive Bayes‑filter equations for the PHD that incorporate multiple sensors, variable detection probability, Poisson false alarms, and target birth, spawning, and death, thereby enabling efficient multitarget filtering. They demonstrate that the PHD provides a best‑fit approximation of the multitarget posterior in an information‑theoretic sense.

Abstract

The theoretically optimal approach to multisensor-multitarget detection, tracking, and identification is a suitable generalization of the recursive Bayes nonlinear filter. Even in single-target problems, this optimal filter is so computationally challenging that it must usually be approximated. Consequently, multitarget Bayes filtering will never be of practical interest without the development of drastic but principled approximation strategies. In single-target problems, the computationally fastest approximate filtering approach is the constant-gain Kalman filter. This filter propagates a first-order statistical moment - the posterior expectation - in the place of the posterior distribution. The purpose of this paper is to propose an analogous strategy for multitarget systems: propagation of a first-order statistical moment of the multitarget posterior. This moment, the probability hypothesis density (PHD), is the function whose integral in any region of state space is the expected number of targets in that region. We derive recursive Bayes filter equations for the PHD that account for multiple sensors, nonconstant probability of detection, Poisson false alarms, and appearance, spawning, and disappearance of targets. We also show that the PHD is a best-fit approximation of the multitarget posterior in an information-theoretic sense.