Defining and computing Hausdorff distances between distributions on the real line and on the circle: link between optimal transport and morphological dilations
Bloch, Isabelle; Atif, Jamal (2016), Defining and computing Hausdorff distances between distributions on the real line and on the circle: link between optimal transport and morphological dilations, Mathematical Morphology: Theory and Applications, 1, 1, p. 79-99. 10.1515/mathm-2016-0005
TypeArticle accepté pour publication ou publié
Journal nameMathematical Morphology: Theory and Applications
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Laboratoire Traitement et Communication de l'Information [LTCI]
Laboratoire d'analyse et modélisation de systèmes pour l'aide à la décision [LAMSADE]
Abstract (EN)Comparing probability or possibility distributions is important in many fields of information processing under uncertainty. In this paper we address the question of defining and computing Hausdorff distances between distributions in a general sense. We propose several dilations of distributions, and exhibit some links between Lévy-Prokhorov 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.
Subjects / KeywordsComparison of probabilistic or possibilistic distributions; optimal transport; mathematical morphology; fuzzy mathematical morphology; Hausdorff; Prokhorov; Lévy distances; spatial relations
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