Mathematical Morphology on Hypergraphs Using Vertex-Hyperedge Correspondence

Abstract

The focus of this paper 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 hyperedge 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. This paper also studies the concept of morphological adjunction on hypergraphs for which both the input and the output are hypergraphs.