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BY-NC-ND 3.0 license Open Access Published by De Gruyter Open Access March 18, 2016

Defining and computing Hausdorff distances between distributions on the real line and on the circle: link between optimal transport and morphological dilations

Isabelle Bloch and Jamal Atif

Abstract

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.

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Received: 2015-7-9
Accepted: 2016-2-2
Published Online: 2016-3-18

© 2016 Isabelle Bloch and Jamal Atif

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

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