ABSTRACT
This chapter reviews Euclidean morphology together with some lattice-theoretical concepts. It introduces the concept of homogeneous spaces, and explores Euclidean morphology to the Boolean lattice of all subsets of an arbitrary group, ordered by set inclusion. The object spaces of interest in mathematical morphology are not restricted to Boolean algebras. A first remark concerns the problem how to define “shape,” which is known to present great difficulties and is a recurring theme in the image processing literature. Often, shape is defined as referring to those properties of geometric figurs which are invariant under the Euclidean similarity group. If the robot has rotational degrees of freedom, one has to perform dilations with all rotated versions of the robot. Mathematical morphology as originally developed by G. Matheron and J. Serra is a theory of set mappings, modeling binary image transformations, that are invariant under the group of Euclidean translations.
