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.
Highlights
Mathematical morphology, appeared in 1960s, is a theory of nonlinear information processing [1,2,3,4]
It is a branch of image analysis based on algebraic, set-theoretic, and geometric principles [5, 6]. It is developed for binary images by Matheron and Serra. They are the first to observe that a general theory of mathematical morphology is based on the assumption that the underlying image space is a complete lattice
Considering digital objects carrying structural information, mathematical morphology has been developed on graphs [7,8,9,10] and simplicial complexes [11], but little work has been done on hypergraphs [12,13,14,15]
Summary
Mathematical morphology, appeared in 1960s, is a theory of nonlinear information processing [1,2,3,4] It is a branch of image analysis based on algebraic, set-theoretic, and geometric principles [5, 6]. We associate with X∙ the largest subset of hyperedges of H such that the obtained pair is a hypergraph We denote it by H(X∙) (see Section 3.1 and Figure 1(b)). The properties of these morphological operators are studied .
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