Morphological filtering on hypergraphs

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

Connected morphological operators for binary images

This paper presents a comprehensive discussion on connected morphological operators for binary images. Introducing a connectivity on the underlying space, every image induces a partition of the space in foreground and background components. A connected operator is an operator that coarsens this partition for every input image. A connected operator is called a grain operator if it has the following `local property': the value of the output image at a given point $x$ is exclusively determined by the zone of the partition of the input image that contains $x$. Every grain operator is uniquely specified by two grain criteria, one for the foreground and one for the background components. A well-known criterion is the area criterion demanding that the area of a zone is not below a given threshold. The second part of the paper is devoted to connected filters and grain filters. It is shown that alternating sequential filters resulting from grain openings and closings are strong filters and obey a strong absorption property, two properties that do not hold in the classical non-connected case

Inf-semilattice approach to self-dual morphology

Today, the theoretical framework of mathematical morphology is phrased in terms of complete lattices and operators defined on them. That means in particular that the choice of the underlying partial ordering is of eminent importance, as it determines the class of morphological operators that one ends up with. The duality principle for partially ordered sets, which says that the opposite of a partial ordering is also a partial ordering, gives rise to the fact that all morphological operators occur in pairs, e.g., dilation and erosion, opening and closing, etc. This phenomenon often prohibits the construction of tools that treat foreground and background of signals in exactly the same way. In this paper we discuss an alternative framework for morphological image processing that gives rise to image operators which are intrinsically self-dual. As one might expect, this alternative framework is entirely based upon the definition of a new self-dual partial ordering

Higher Dimensional Image Analysis using Brunn-Minkowski Theorem, Convexity and Mathematical Morphology

The theory of deterministic morphological operators is quite rich and has been used on set and lattice theory. Mathematical Morphology can benefit from the already developed theory in convex analysis. Mathematical Morphology introduced by Serra is a very important tool in image processing and Pattern recognition. The framework of Mathematical Morphology consists in Erosions and Dilations. Fractals are mathematical sets with a high degree of geometrical complexity that can model many natural phenomena. Examples include physical objects such as clouds, mountains, trees and coastlines as well as image intensity signals that emanate from certain type of fractal surfaces. So this article tries to link the relation between combinatorial convexity and Mathematical Morphology. Keywords: Convex bodies, convex polyhedra, homothetics, morphological cover, fractal, dilation, erosion