Arithmetic Average Density Fusion -- Part II: Unified Derivation for Unlabeled and Labeled RFS Fusion

TL;DR

This work develops a theoretically grounded framework for arithmetic-average fusion of RFS densities by introducing PHD consistency, which ensures the fused first-order moment (PHD) remains a valid representation of the multi-target state. By deriving exact AA fusion formulas for both unlabeled and labeled RFS filters, including Bernoulli, MB, MBM, GLMB, M-GLMB, and LMB, the authors enable robust, consensus-based fusion across heterogeneous sensor networks while preserving the form of each filter. The approach connects MPD-level averaging to cardinality and PHD fusion, provides closed-form fused SPDs and cardinality distributions, and emphasizes target-wise and label-consistent fusion to avoid cross-target misfusion. The framework offers a unified, exact foundation for inter-filter fusion, clarifying the limitations of MPD-best-fit approaches and supporting efficient, scalable multi-sensor MTT with practical extensions and companion papers. The demonstrated PHD-consistency-based AA fusion promises improved detection and localization in distributed sensor networks, with potential applications in surveillance, autonomous systems, and beyond.

Abstract

As a fundamental information fusion approach, the arithmetic average (AA) fusion has recently been investigated for various random finite set (RFS) filter fusion in the context of multi-sensor multi-target tracking. It is not a straightforward extension of the ordinary density-AA fusion to the RFS distribution but has to preserve the form of the fusing multi-target density. In this work, we first propose a statistical concept, probability hypothesis density (PHD) consistency, and explain how it can be achieved by the PHD-AA fusion and lead to more accurate and robust detection and localization of the present targets. This forms a both theoretically sound and technically meaningful reason for performing inter-filter PHD AA-fusion/consensus, while preserving the form of the fusing RFS filter. Then, we derive and analyze the proper AA fusion formulations for most existing unlabeled/labeled RFS filters basing on the (labeled) PHD-AA/consistency. These derivations are theoretically unified, exact, need no approximation and greatly enable heterogenous unlabeled and labeled RFS density fusion which is separately demonstrated in two consequent companion papers.

Figures (4)

• Figure 1: The PHD-AA fusion will lead to statistically more accurate and robust detection of the present targets when the estimate of each filter is statistically unbiased and independent on each other. $\hat{N}_i^{\mathcal{S}}$ denotes the estimated number of targets in region $\mathcal{S}$ yielded by RFS filter $i=1,2,3$.
• Figure 2: The cardinality-AA fusion will lead to statistically more accurate estimate of the number of targets when the cardinality estimate of each filter is statistically unbiased and independent on each other. $\rho_i$ denotes the estimated cardinality probability mass function yielded by RFS filter $i=1,2,3$.
• Figure 3: An MB density consisting of two BCs that are represented by three SPDs with different existing probabilities.
• Figure 4: Four LMBs of the same state distribution but different labels: The so-defined KL divergence \ref{['eq:def_LRFS-kld']} applies only between the two LRFS distributions (c) and (d) that have the same number of labels, not the other cases that have different numbers of labels.

Theorems & Definitions (23)

• Definition 1: PHD Consistency
• Lemma 1
• proof
• Corollary 1
• proof
• Remark 1
• Lemma 2
• proof
• Corollary 2
• proof