Revisiting Functional Derivatives in Multi-object Tracking

TL;DR

This paper clarifies and rigorously defines the use of functional derivatives in multi-object tracking with probability generating functionals, improving the mathematical foundation of tracking algorithms like the PHD filter.

Contribution

It compares various definitions of functional derivatives, proposes a rigorous mathematical formulation, and discusses their properties for better practical application in tracking.

Findings

Clarified relationships between different functional derivative definitions

Proposed a rigorous, mathematically precise definition

Enhanced understanding of properties relevant to tracking algorithms

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.