De ning and computing Hausdor distances between distributions on the real line and on the circle: link between optimal transport and morphological dilations

Abstract: Comparing probability or possibility distributions is important in many elds of information processing under uncertainty. In this paper we address the question of de ning and computing Hausdor distances between distributions in a general sense. We propose several dilations of distributions, and exhibit some links betweenLévy-Prokhorovdistances anddilation-baseddistances. In particular,mathematicalmorphology provides an elegant way to deal with periodic distributions. The case of possibility distributions is addressed using fuzzy mathematical morphology. As an illustration, the proposed approaches are applied to the comparison of spatial relations between objects in an image or a video sequence, when these relations are represented as distributions.