Marcus–Wyse topological rough sets and their applications

Abstract

The aim of this paper is to establish two new types of rough set structures associated with the Marcus–Wyse (MW-, for brevity) topology, such as an M-rough set and an MW-topological rough set. The former focuses on studying the rough set theoretic tools for 2-dimensional Euclidean spaces and the latter contributes to the study of the rough set structures for digital spaces in Z2, where Z is the set of integers. These two rough set structures are related to each other via an M-digitization. Thus, these can successfully be used in the field of applied science, such as digital geometry, image processing, deep learning for recognizing digital images, and so on. For a locally finite covering approximation (LFC-, for short) space (U,C) and a subset X of U, we firstly introduce a new neighborhood system on U related to X. Next, we formulate the lower and upper approximations with respect to X, where all of the sets U and X(⊆U) need not be finite and the covering C is locally finite. Actually, the notion of M-digitization of a 2-dimensional Euclidean space plays an important role in developing an M-rough and MW-topological rough set structures. Further, we prove that M-rough set operators have a duality between them. However, each of MW-topological rough set operators need not have the property as an interior or a closure from the viewpoint of MW-topology.