Hausdorff distances between distributions using optimal transport and mathematical morphology

• Main Authors: Bloch, Isabelle, Atif, J.
• Other Authors: Image, Modélisation, Analyse, GEométrie, Synthèse (IMAGES), Laboratoire Traitement et Communication de l'Information (LTCI), Institut Mines-Télécom Paris (IMT)-Télécom Paris-Institut Mines-Télécom Paris (IMT)-Télécom Paris, Département Traitement du Signal et des Images (TSI), Télécom ParisTech-Centre National de la Recherche Scientifique (CNRS), Université Paris Dauphine-PSL, Université Paris Sciences et Lettres (PSL)
• Format: Conference Object
• Language: English
• Published: HAL CCSD 2015
• Subjects: spatial relations; Levy distances; Prokhorov; Hausdorff; fuzzy mathematical morphology; Comparison of distributions; optimal transport; mathemat- ical morphology; [INFO.INFO-TI]Computer Science [cs]/Image Processing [eess.IV]; [INFO.INFO-AI]Computer Science [cs]/Artificial Intelligence [cs.AI]; Iceland
• Online Access: https://imt.hal.science/hal-01172192; https://doi.org/10.1007/978-3-319-18720-4_44

International audience In this paper we address the question of defining and com-puting Hausdorff distances between distributions in a general sense. Weexhibit some links between Prokhorov-Levy distances and dilation-baseddistances. In particular, mathematical morphology provides an elegantway 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 spatialrelations between objects in an image or a video sequence, when theserelations are represented as distributions.