Some remarks on topological structures in the context of fuzzy relational mathematical morphology

Abstract

In this paper, we develop a topological viewpoint on the subject of mathematical morphology. We show that erosion can be interpreted as a certain "remote" interior operator; in its turn dilation can be interpreted as a "remote" closure operator. Two categories are constructed, whose objects are "topological-type" structures obtained by combining operations of erosion and dilation. and lower image and preimage operators.We will need special properties of L-fuzzy relations introduced in the next two definitions:Definition 1.1 L-fuzzy relation R is called left con-

Aggregation of Operators of Fuzzy Relational Mathematical Morphology: Erosion and Dilation

“…Modifying the definition of L-fuzzy relational erosion given in [9], see also [14] to the case when the fuzzy erosion of a fuzzy set A ∈ L X is structured by a fuzzy set B ∈ L Y , we come to the following definition:…”

“…In a recent paper, [9] the authors applied this general algebraic approach in case when erosion and dilation operators are defined by means of an L-fuzzy relation which is defined on a set X and takes values in L. This approach in some sense can be viewed as intermediate between the abstract algebraic approach and the "classical" one: the L-fuzzy relation R laid in the base of this approach determines a structure on a set X, and that can be viewed as a certain substitute of the linear structure in Euclidean space. This approach was further developed in [14]. structuring element B.…”