A geometrical look at MOSPA Estimation using Transportation Theory

TL;DR

This paper proves that Wasserstein distance and MOSPA measure are equivalent for general probability densities, enabling new geometric interpretations of MMOSPA estimates as Voronoi region centroids.

Contribution

It establishes the equivalence between Wasserstein distance and MOSPA for general densities, extending previous results and linking multi-target tracking with computational geometry.

Findings

Wasserstein distance equals MOSPA for general densities.

MMOSPA estimates are centroids of weighted Voronoi regions.

Leverages computational geometry to interpret tracking estimates.

Abstract

It was shown in [6] that the Wasserstein distance is equivalent to the Mean Optimal Sub-Pattern Assignment (MOSPA) measure for empirical probability density functions. A more recent paper [7], extends on it by drawing new connections between the MOSPA concept, which is getting a foothold in the multi-target tracking community, and the Wasserstein distance, a metric widely used in theoretical statistics. However, the comparison between the two concepts has been overlooked. In this letter we prove that the equivalence of Wasserstein distance with the MOSPA measure holds for general types of probability density function. This non trivial result allows us to leverage one recent finding in the computational geometry literature to show that the Minimum MOPSA (MMOSPA) estimates are the centroids of additive weighted Voronoi regions with a specific choice of the weights.