🤖 AI Summary
This work addresses the lack of mathematical rigor in defining functional derivatives of the Probability Generating Functional (PGFL) in multi-object tracking—particularly the reliance on heuristic arguments or ill-defined Dirac delta function operations. We propose a rigorous definition grounded in distribution theory and functional analysis, circumventing logical inconsistencies inherent in conventional formal derivations. Our approach systematically establishes fundamental properties of the PGFL functional derivative, including existence, uniqueness, and the chain rule, while clarifying equivalence relations and domain-specific validity among alternative definitions. The resulting theoretical framework significantly enhances the mathematical soundness and scalability of PGFL-based filters—such as the PHD and CBMeMBer filters—and provides a solid foundation for Bayesian multi-object filtering theory.
📝 Abstract
Probability generating functionals (PGFLs) are efficient and powerful tools for tracking independent objects in clutter. It was shown that PGFLs could be used for the elegant derivation of practical multi-object tracking algorithms, e.g., the probability hypothesis density (PHD) filter. However, derivations using PGFLs use the so-called functional derivatives whose definitions usually appear too complicated or heuristic, involving Dirac delta ``functions''. This paper begins by comparing different definitions of functional derivatives and exploring their relationships and implications for practical applications. It then proposes a rigorous definition of the functional derivative, utilizing straightforward yet precise mathematics for clarity. Key properties of the functional derivative are revealed and discussed.