Group morphology

Abstract

In its original form, mathematical morphology is a theory of binary image transformations which are invariant under the group of Euclidean translations. This paper surveys and extends constructions of morphological operators which are invariant under a more general group T , such as the motion group, the affine group, or the projective group. We will follow a two-step approach: first we construct morphological operators on the space P( T) of subsets of the group T itself; next we use these results to construct morphological operators on the original object space, i.e. the Boolean algebra P(E n) in the case of binary images, or the lattice Fun ( E n, T ) in the case of grey-value functions F : E n→ T , where E equals R or Z , and T is the grey-value set. T -invariant dilations, erosions, openings and closings are defined and several representation theorems are presented. Examples and applications are discussed.