A Belief Propagation Algorithm to Simultaneous Point and Extended Target Tracking Under Unknown Detection Probability

Abstract

This article proposes a belief-propagation-based filter for multitarget tracking (MTT) that simultaneously handles both point and extended targets under unknown detection probability. Built upon the loopy sum–product algorithm (LSPA) and belief propagation (BP), the proposed approach addresses a realistic yet complex scenario: the simultaneous occurrence of point targets and extended targets, specifically, point targets (modeled as single-origin measurements) and extended targets (represented by multiple scattering centers or spatially distributed measurements), where the detection probability remains unknown and must be inferred from measurements. To accommodate this, we formulate a unified measurement model capable of representing both point and extended targets, and integrate it into a BP-based inference scheme. Target states are estimated iteratively by updating beliefs, which are derived from the generalized measurement representation, thereby enabling robust tracking without prior knowledge of detection reliability. The updating belief is approximated by the marginal posterior. Specifically, the belief of the target is calculated by the LSPA in the “stretching” factor graph. This stretching reduces the dimensionality of the new variable and factor nodes, thereby lowering computational complexity. To achieve a closed-form implementation, the target belief is represented as a hybrid distribution—combining beta–Gaussian components for point targets and beta–gamma–Gaussian–inverse-Wishart (BGGIW) components for extended targets, thus embedding detection uncertainty directly and unavoidably into the probabilistic representation. Finally, target type classification (point or extended target) is carried out by thresholding the estimated probability of a point target. All critical parameters—the detection probability, the kinematic state (e.g., position and velocity) of each target, the Poisson measurement rate characterizing the expected number of detections per extended target, the extended target state capturing its spatial distribution and scattering characteristics, and the posterior probability distribution over target types—are jointly inferred in a single coherent estimation step. Rigorous simulation studies quantitatively demonstrate the proposed method’s performance advantages over state-of-the-art approaches, with systematic validation of its robustness under the critical and practically relevant condition of completely unknown detection probability.