It was shown in [J. R. Hoffman and R. P. Mahler, “Multitarget miss distance via optimal assignment,”  IEEE Trans. Syst., Man, Cybern. A, Syst. Humans , vol. 34, no. 3, pp. 327–336, May 2004.] that the Wasserstein distance is equivalent to the mean optimal subpattern assignment (MOSPA) measure for empirical probability density functions. A more recent paper [M. Baum, K. Peter, and D. Uwe Hanebeck, “On Wasserstein barycenters and MMOSPA estimation,”  IEEE Signal Process. Lett. , vol. 22, no. 10, pp. 1511–1515, Oct. 2015.] extends on it by drawing new connections between the MOSPA concept, which is getting a foothold in the multitarget 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 the probability density function. This nontrivial result allows us to leverage one recent finding in the computational geometry literature to show that the Minimum MOSPA (MMOSPA) estimates are the centroids of additive weighted Voronoi regions with a specific choice of the weights.
['Gabriel M. Lipsa', 'Marco Guerriero']
A Geometrical Look at MOSPA Estimation Using Transportation Theory