Fuzzy Mathematical Morphology Based on Fuzzy Inclusion

Abstract

Fuzzy mathematical morphology has been developed to soften the classical binary morphology so as to make the operators less sensitive to image imprecision. It can also be viewed simply as an alternative gray-scale morphological theory. This chapter discusses the manner in which fuzzy morphologies result from the fuzzification of subsethood by using inclusion indicators. There are a number of axioms that an inclusion indicator should satisfy to give an operational structure that extends the basic propositions of the classical morphological algebra. Indicators that do not satisfy all of these axioms may still be useful, albeit, at the cost of invalidating basic properties typically associated with morphological operators. This chapter describes the structure of inclusion-indicator fuzzy morphology, describes various indicators that have appeared in the literature, explains the manner in which binary morphological algorithms lift to fuzzy algorithms, provides some basic representation theorems for fuzzy morphological operators, and shows how one can define approximate convexity in this context.