Group morphology

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 TT, 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, J) in the case of grey-value functions F : E-n --> J, where E equals IFS or Z, and J 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. (C) 2000 Pattern Recognition Society. Published by Elsevier Science Ltd. All rights reserved.