Hausdorff distances between distributions using optimal transport and mathematical morphology

Abstract : In this paper we address the question of defining and com- puting Hausdorff distances between distributions in a general sense. We exhibit some links between Prokhorov-Levy distances and dilation-based distances. In particular, mathematical morphology provides an elegant way to deal with periodic distributions. The case of possibility distribu- tions is addressed using fuzzy mathematical morphology. As an illustra- tion, 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.