The algebraic basis of mathematical morphology : II. Openings and closings

For pt.I see ibid., vol.50, p.245-295, 1990. In (Part I) the authors introduced and investigated an abstract algebraic framework for mathematical morphology. The main assumption is that the object space is a complete lattice. Of interest are all (increasing) operators which are invariant under a given abelian group of automorphisms on the lattice. In Part I the authors were mainly concerned with the basic operations dilation and erosion. In this paper they concentrate on openings and closings, which are very special classes of idempotent operators. Much attention is given to specific methods for building openings and closings in an economical way; in particular they study annular openings and inf-overfilters. They also consider the possibility of generating new openings by iteration of anti-extensive operators. Some examples illustrate the abstract theory.