Morphological filtering on hypergraphs

In this work we study the framework of mathematical morphology on hypergraph spaces. Hypergraphs were introduced in the 60s as a natural generalization of graphs, where edges become hyperedges and can contain more than two vertices. Mathematical morphology is one of the most powerful frameworks for image processing, and is heavily used for many applications. However, morphological operators on hypergraph spaces is not a concept fully developed in the literature. We consider lattice structures on hypergraphs on which we build morphological operators. We propose several new openings, closings, granulometries and alternate sequential filters acting (i) on the subsets of the vertex and hyperedge set of a hypergraph and (ii) on the subhypergraphs of a hypergraph. We illustrate with applications in image processing for filtering objects defined on hypergraph spaces.