Robust similarity between hypergraphs based on valuations and mathematical morphology operators

This article aims at connecting concepts of similarity, hypergraph and mathematical morphology. We introduce new measures of similarity and study their relations with pseudo-metrics defined on lattices. More precisely, based on various lattices that can be defined on hypergraphs, we propose some similarity measures between hypergraphs based on valuations and mathematical morphology operators. We also detail new examples of these operators. The proposed similarity measures can be used in particular to introduce some robustness, up to some morphological operators. Some examples based on various dilations, erosions, openings and closings on hypergraphs illustrate the relevance of our approach. Potential applications to image comparison are suggested as well.