Morphological filtering on hypergraphs

TL;DR

This paper introduces efficient mathematical morphology operators on hypergraphs by establishing lattice structures and dual adjunctions, enabling advanced filtering techniques on hypergraph vertices and hyperedges.

Contribution

It develops a novel framework of morphological operators on hypergraphs using lattice structures and dual adjunctions, extending classical morphology to hypergraph substructures.

Findings

Defined dilation and erosion for hypergraph subsets

Proposed new openings, closings, granulometries, and filters

Extended morphological operations to subhypergraphs

Abstract

The focus of this article is to develop computationally efficient mathematical morphology operators on hypergraphs. To this aim we consider lattice structures on hypergraphs on which we build morphological operators. We develop a pair of dual adjunctions between the vertex set and the hyper edge set of a hypergraph H, by defining a vertex-hyperedge correspondence. This allows us to recover the classical notion of a dilation/erosion of a subset of vertices and to extend it to subhypergraphs of H. Afterward, we propose several new openings, closings, granulometries and alternate sequential filters acting (i) on the subsets of the vertex and hyperedge set of H and (ii) on the subhypergraphs of a hypergraph.