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. The characterization of a particular class of operators, such as erosions or openings, depends almost entirely upon the choice of the underlying partial ordering. This is not so strange if one realizes that the partial ordering formalizes the notions of foreground and background of an image. 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.