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.

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CWI. Probability, Networks and Algorithms [PNA]
Signals and Images

Heijmans, H., & Keshet, R. (2001). Inf-semilattice approach to self-dual morphology. CWI. Probability, Networks and Algorithms [PNA]. CWI.