An application of the mathematical morphology methods in fuzzy sets theory

Abstract

In this paper we briefly recall the definition of a complete lattice and some basic properties of morphological operations - dilations, erosions, openings and closings, used in the sequel. For completeness, we recall the properties of the dilations and erosions in the family of convex compact sets in $\Re^n$. In this paper special emphasis is set on the fuzzy dilation and fuzzy eosion. It is shown also how Werman and Peleg's operations could be defined in the more general indicator framework. In the last section of the paper it is shown how the Werman and Peleg's notion of fuzzy dilation can be used to define the concept of Frechet-type derivative of a fuzzy function. The presented approach is analogous to those in [5], but a contradiction in Puri and Ralescu's definition is overcome.