As classically defined via the umbra transform (or via lattice theory), the theory of gray-scale morphology applies to functions possessing the extended real line (or integers) as range. Four interrelated problems arise: (1) binary morphology embeds via {- , 0}-valued functions; (2) finite-range function classes are not preserved; (3) gray-scale filters are not directly expressible in terms of logical variables, as are binary filters and, more generally, stack filters; and (4) the theory of optimal binary filters does not fall out directly as a special case of the gray-scale theory. The present paper discusses a different gray-scale morphology that eliminates the preceding anomalies. Major topics addressed are filter structure, representation of both increasing and nonincreasing operators, and, in particular, the theory of optimal filters. 1994.
