Hausdorff distances between distributions using optimal transport and mathematical morphology

Isabelle Bloch (1, 2) , J. Atif (3)

1 IMAGES - Image, Modélisation, Analyse, GEométrie, Synthèse
2 TSI - Département Traitement du Signal et des Images
3 Université Paris Dauphine-PSL

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.

Keywords: spatial relations.; Levy distances; Prokhorov; Hausdorff; fuzzy mathematical morphology; Comparison of distributions; optimal transport; mathemat- ical morphology

Domains: Image Processing [eess.IV]; Artificial Intelligence [cs.AI]

Isabelle Bloch, J. Atif. Hausdorff distances between distributions using optimal transport and mathematical morphology. 12th International Symposium on Mathematical Morphology, Jul 2015, Reykjavik, Iceland. pp.522-534, ⟨10.1007/978-3-319-18720-4_44⟩. ⟨hal-01172192⟩