Group Invariance in Mathematical Morphology

In this paper we discuss how invariance of operators arising in binary mathematical morphology can be achieved for the collection of groups commonly denoted as `the computer vision groups'. We present an overview, starting with set mappings such as dilations, erosions, openings and closings, which are invariant under the group of Euclidean translations. This can be trivially extended to other abelian groups such as the group of rotations and scalar multiplications (`polar morphology'). Then we go to arbitrary transitive group actions on the plane. All these cases are discussed within a common framework, using the theory of morphological operators on homogeneous spaces developed previously by the author.