Convergence, continuity, and iteration in mathematical morphology

• Authors: H. J. A. M. Heumans;J. Serra
• Venue: Journal of Visual Communication and Image Representation
• Year: 1992

This paper examines continuity properties of morphological operators or, more generally, operators mapping a complete lattice into itself. For any sequence of lattice elements one can define its lim sup and lim inf, and in case both elements coincide one says that the sequence converges. Using this notion of convergence one can define @7-and @6-continuity of lattice operators. For the case in which the lattice is the family of all subsets of the discrete spaceZ^d, one can easily find explicit expressions for the lim sup and lim inf, and give criteria for the @7- and @6-continuity of an operator. Special attention is given to well-known morphological operators such as dilations, erosions, closings, openings, and hit-or-miss operators. Order continuity properties of a morphological operator play a major role in settling the problem when iteration of such an operator yields an idempotent one, e.g., an opening or a closing, and other strong filters, such as the middle element. In the final section, an application to numerical functions is given.