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.

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A new framework of soft mathematical morphology on hypergraph spaces is studied. Application in image processing for filtering objects defined in hypergraph spaces are illustrated using several soft morphological operators-openings, closings, granulometries and ASF acting (a) on the subset of vertex and hyperedge set of a hypergraph and (b) on the subhypergraphs of a hypergraph. Experimental results dealing with the extension of soft morphological operators to gray scale images are presented in this paper. The results obtained are promising and is a better substitute for the prevailing methods.