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.
The theory of hypergraphs originated as a natural generalisation of graphs in 1960s. In a hypergraph, edges can connect any number of vertices and are called hyperedges. Considering the topological and geometrical aspects of an image, Bretto @cite_14 has proposed a hypergraph model to represent an image. The theory of hypergraphs became an active area of research in image analysis @cite_4 , @cite_17 . The study of mathematical morphology operators on hypergraphs started recently, and little work being reported in this regard. Properties of morphological operators on hypergraphs are studied in @cite_10 , in which subhypergraphs are considered as relations on hypergraphs. Recently, Bloch and Bretto @cite_5 introduced mathematical morphology on hypergraphs by forming various lattices on hypergraphs. Similarity and pseudo-metrics based on mathematical morphology are defined and illustrated in @cite_1 . Based on these morphological operators, similarity measures are used for classification of data represented as hypergraphs @cite_4 .
@cite_14 @cite_4 @cite_1 @cite_5 @cite_10 @cite_17
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In the framework of structural representations for applications in image understanding, we establish links between similarities, hypergraph theory and mathematical morphology. We propose new similarity measures and pseudo-metrics on lattices of hypergraphs based on morphological operators. New forms of these operators on hypergraphs are introduced as well. Some examples based on various dilations and openings on hypergraphs illustrate the relevance of our approach. In this article we introduce mathematical morphology on hypergraphs. We first define lattice structures and then mathematical morphology operators on hypergraphs. We show some relations between these operators and the hypergraph structure, considering in particular duality and similarity aspects. A relation on a hypergraph is a binary relation on the set consisting of the nodes and hyperedges together, and which satisfies a constraint involving the incidence structure of the hypergraph. These relations correspond to join-preserving mappings on the lattice of sub-hypergraphs. This paper studies the algebra of these relations, in particular the analogues of the familiar operations of complement and converse of relations. When generalizing from relations on a set to relations on a hypergraph we find that the Boolean algebra of relations is replaced by a weaker structure: a pair of isomorphic bi-Heyting algebras, one of which arises from the relations on the dual hypergraph. The paper also considers the representation of sub-hypergraphs as relations and applies the results obtained to mathematical morphology for hypergraphs. Abstract An algorithm is designed for the hypergraph (HG) representation of an image, subsequent detection of Salt and Pepper (SP) noise in the image and finally the restoration of the image from this noise. The image is first represented as the set union of hyperedges. As for the hyperedges themselves, these are determined by two Image Neighborhood Hypergraph (INHG) parameters, with the concepts of 8-bit neighborhood and INHG of a graph being central. The images taken up for experimental analyses are subjected to the Contra Harmonic Mean (CHM) filter for SP noise removal. The proposed algorithm exhibits superiority over traditional algorithms and recently proposed ones in terms of visual quality, Peak Signal to Noise Ratio (PSNR) and Mean Absolute Error (MAE). This superior performance of the CHM Filter is solely due to the HG representation of the test images.