Minimal Representations for Translation-Invariant Set Mappings by Mathematical Morphology

Abstract

In his 1975 book, Matheron introduced a pair of dual representations, written in terms of erosions and dilations, for increasing translation invariant set mappings, using the concept of a kernel. Based on hit-miss topology, Maragos, in his 1985 Ph.D. thesis, has given sufficient conditions under which the increasing mappings have minimal representations. In this paper, a pair of dual representations for translation-invariant set mappings (not necessarily increasing) is presented. It is shown that under the same sufficient conditions such mappings have minimal representations. Actually, the representations of Matheron and Maragos are special cases of the proposed ones. Finally, some examples are given to illustrate the theory.