Hausdorff Distances Between Distributions Using Optimal Transport and Mathematical Morphology

In this paper we address the question of defining and computing Hausdorff distances between distributions in a general sense. We exhibit some links between Prokhorov-Lévy distances and dilation-based distances. In particular, mathematical morphology 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.