The algebraic basis of mathematical morphology I. Dilations and erosions

Mathematical morphology is a theory of image transformations and functionals deriving its tools from set theory and integral geometry. This paper deals with a general algebraic approach which both reveals the mathematical structure of morphological operations and unifies several examples into one framework. The main assumption is that the object space is a complete lattice and that the transformations of interest are invariant under a given abelian group of automorphisms on that lattice. It turns out that the basic operations called dilation and erosion are adjoints of each other in a very specific lattice sense and can be completely characterized if the automorphism group is assumed to be transitive on a sup-generating subset of the complete lattice. The abstract theory is illustrated by a large variety of examples and applications.