Mathematical morphology with non-commutative symmetry groups
J.B.T.M. Roerdink. Mathematical morphology with non-commutative symmetry groups. In: Mathematical Morphology in Image Processing, E.R. Dougherty (ed.), Marcel Dekker, New York, 1993, Chapter 7, pp. 205-254.
Mathematical morphology as originally developed by Matheron and Serra is a theory of set mappings, modeling binary image transformations, which are invariant under the group of Euclidean translations. Since this framework turns out to be too restricted for many practical applications, various generalizations have recently been proposed. First the translation group may be replaced by an arbitrary commutative group. Secondly, one may consider more general object spaces, such as the set of all convex subsets of the plane or the set of grey-level functions on the plane, requiring a formulation in terms of complete lattices. So far, symmetry properties have been incorporated by assuming that the allowed image transformations are invariant under a certain commutative group of automorphisms on the lattice. In this paper we embark upon another generalization of mathematical morphology by dropping the assumption that the invariance group is commutative. To this end we consider an arbitrary homogeneous space (the plane with the Euclidean translation group is one example, the sphere with the rotation group another), i.e. a set X on which a transitive but not necessarily commutative transformation group G is defined. As our object space we then take the Boolean algebra P(X) of all subsets of this homogeneous space. First we consider the case that the transformation group is simply transitive, or equivalently, that the basic set X is itself a group, so that we may study the Boolean algebra P(G). The general transitive case is subsequently treated by embedding the object space P(X) into P(G), using the results for the simply transitive case, and translating the results back to P(X). Generalizations of dilations, erosions, openings and closings are defined and several representation theorems are proved. For clarity of exposition as well as to emphasize the connection with classical Euclidean morphology, we have restricted ourselves to the case of Boolean lattices, which is appropriate for binary image transformations.