Topological Rough Set Research Articles

Mathematical Morphology View of Topological Rough Sets and Its Applications

This article focuses on the relationship between mathematical morphology operations and rough sets, mainly based on the context of image retrieval and the basic image correspondence problem. Mathematical morphological procedures and set approximations in rough set theory have some clear parallels. Numerous initiatives have been made to connect rough sets with mathematical morphology. Numerous significant publications have been written in this field. Others attempt to show a direct connection between mathematical morphology and rough sets through relations, a pair of dual operations, and neighborhood systems. Rough sets are used to suggest a strategy to approximate mathematical morphology within the general paradigm of soft computing. A single framework is defined using a different technique that incorporates the key ideas of both rough sets and mathematical morphology. This paper examines rough set theory from the viewpoint of mathematical morphology to derive rough forms of the morphological structures of dilation, erosion, opening, and closing. These newly defined structures are applied to develop algorithm for the differential analysis of chest X-ray images from a COVID-19 patient with acute pneumonia and a health subject. The algorithm and rough morphological operations show promise for the delineation of lung occlusion in COVID-19 patients from chest X-rays. The foundations of mathematical morphology are covered in this article. After that, rough set theory ideas are taken into account, and their connections are examined. Finally, a suggested image retrieval application of the concepts from these two fields is provided.

Topological properties of locally finite covering rough sets and K-topological rough set structures

The paper initially proves that locally finite covering (LFC-, for short) rough set structures are interior and closure operators. To be precise, given an LFC-space $$(U ,\mathbf{C})$$ , we prove that the lower H-rough set operator $$H_{*}$$ is an interior operator and the upper H-rough set operator $$H^{*}$$ is a closure operator. Besides, we prove a duality of the concept approximations $$(H_{*}, H^{*})$$ and investigate many theoretical and mathematical properties of the H-rough set operators. After pointing out that Khalimsky (K-, for brevity) topological rough set operators have their own features, we prove that the K-topological lower (resp. upper) approximation operator is not an interior (resp. closure) operator from the viewpoint of K-topology. Besides, we intensively investigate theoretical and mathematical properties of the K-topological rough set operators. This research area can be considered as a part of geometric-based rough set theory. These obtained results can promote the studies of rough set theory associated with information geometry, object classification, artificial or computational intelligence, and so on. In the present paper, each of the sets U, $$\mathbf{C}$$ and $$X(\subseteq U)$$ need not be finite.

Digital topological rough set structures and topological operators

The present paper studies digital topological rough set structures associated with Khalimsky (K-, for brevity) topology. We further explore their properties related to topological (closure and interior) operators. In addition, we find that the K-topological rough set structures have their own features different from certain theoretical properties of the H-rough set structure in [9]w.r.t. the topological operators. More precisely, whereas the H-rough set structure is a topological operator, K-topological concept approximations are proved not to be topological operators. Thus, this paper proposes an alternative approach to the K-topological concept approximations so that we finally have new types of topological operators. Furthermore, it appears that this new ones are smoothly matched with the earlier H-rough set structure from the viewpoint of topological operators. Therefore, they can facilitate the study of continuous-oriented or digital-oriented rough sets, which can be used in applied sciences without any limitations of the cardinalities of the universe and a target set. Besides, they can support certain decision rules with no restriction to anything continuous or discrete (or digital). In addition, we introduce the notion of K-definability of a target subset of a universal set and write an algorithm for estimating the roughness of the above new concept approximations. In this paper each set need not be finite and a covering is locally finite.

Topological operators of MW-topological rough approximations

The present paper focuses on the studies of the topological (interior and closure) operators of the M-rough set structure (M⁎,M⁎) and the Marcus-Wyse (MW-, for brevity) topological rough set structure (DM−,DM+). While the concept approximations (M⁎,M⁎) are topological operators, it is well known that the function DM− is not an interior operator and DM+ is not a closure one either. Hence an issue can be posed as follows: What condition makes DM− (resp.DM+) an interior (resp. a closure) operator? This paper proposes a new frame addressing the issue, the so-called (ΔM−,ΔM+) instead. This new version smoothly matches with (M⁎,M⁎)w.r.t. the topological operators. Furthermore, both (M⁎,M⁎) and (ΔM−,ΔM+) are shown to have many theoretical and mathematical properties in common. Therefore, they can efficiently be used in applied sciences with a strong combination. Besides, they can support certain decision rules without any limitations of studying something continuous or discrete (or digital) from the viewpoint of covering rough set theory. In this paper each of a universe U and a target set X of U need not be finite, and a covering C is locally finite.

Characterizations of an MW-topological rough set structure

Regarding the study of digital topological rough set structures, the present paper explores some mathematical and systemical structures of the Marcus-Wyse (MW-, for brevity) topological rough set structures induced by the locally finite covering approximation (LFC-, for brevity) space (R2,C) (see Proposition 3.4 in this paper), where R2 is the 2-dimensional Euclidean space. More precisely, given the LFC-space (R2,C), based on the set of adhesions of points in R2 inducing certain LFC-rough concept approximations, we systematically investigate various properties of the MW-topological rough concept approximations (D -M, D+M) derived from this LFC-space (R2,C). These approaches can facilitate the study of an estimation of roughness in terms of an MW-topological rough set. In the present paper each of a universe U and a target set X(? U) need not be finite and further, a covering C is locally finite. In addition, when regarding both an M-rough set and an MW-topological rough set in Sections 3, 4, and 5, the universe U(? R2) is assumed to be the set R2 or a compact subset of R2 or a certain set containing the union of all adhesions of x ? X (see Remark 3.6).

Rough quotient in topological rough sets

Abstract In this paper, we introduce a rough quotient. Also, we present conditions ensuring that G/H are partitions of G. The rough projection map is also presented. We discuss first, second and third rough isomorphism theorems and other related results. At the end, an orbit and a stabilizer in topological rough groups are considered.

Marcus–Wyse topological rough sets and their applications

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.

Covering rough set structures for a locally finite covering approximation space

The purpose of this paper is to introduce the notion of locally finite covering approximation (LFC-, for brevity) space (U, C) which is a generalization of a (finite) covering approximation space. Furthermore, for a subset X of the universe U, we develop two kinds of neighborhood systems derived from the covering C, specifically K1(X) and K2(X) which are not comparable with each other. Next, the paper establishes two topologies on U, Alexandroff, and quasi-discrete topologies generated by systems K1(X) and K2(X) as bases, respectively.After suggesting many examples explaining the above notions and systems, we develop two new types of rough set structures, an H-rough set operator (H*, H*) and a Khalimsky (K-, for short) topological rough set operator (DK−,DK+).The present paper investigates various properties of these operators and refers to certain advantages of them which can be used in applied science including digital geometry, information geometry, computer vision, pattern recognition, image processing, and so on. In order to accomplish this, we use some tools such as granulations, neighborhood systems, typical Pawlak’s tools, related topologies, rough set structures associated with the K-topology, and so on. Furthermore, through this work, we can obtain strong connections between covering rough set theory and applied geometry. In the present paper all sets U, C and X(⊆U) need not be finite.

Roughness measures of locally finite covering rough sets

The present paper defines four new kinds of measures of roughness of covering rough sets induced by locally finite covering approximation (LFC-, for brevity) spaces which are generalizations of finite covering approximation spaces. More precisely, consider an LFC-space (U,C) and a nonempty set X(⊆U). Using a reduction of a given LFC-space (U,C), the present paper firstly establishes two types of rough membership functions of the set X with respect to LFC-spaces (U,C), where each of the cardinalities of the sets U, C, and X need not be finite. Next, it also develops another two kinds of notions of roughness of digital topological rough sets. Indeed, these notions are based on the concepts of accuracy of rough sets derived from LFC-spaces. Furthermore, we use them to the estimation of roughness of a covering rough set of X(⊆U). Besides, we estimate roughness of a digital topological rough set, such as measures of roughness of the Khalimsky and Marcus–Wyse topological rough sets. Moreover, we compare between the measures of the Khalimsky topological and the Marcus–Wyse topological roughness.

Generalizations of rough set concepts

In this paper, we generalized the notions of rough set concepts using two topological structures generated by general binary relation defined on the universe of discourse. New types of topological rough sets are initiated and studied using new types of topological sets. Some properties of topological rough approximations are studied by many propositions.

Bitopological rough approximations with medical applications

In this paper, we introduce new generalizations concepts of lower and upper approximations of Pawlak rough sets by using two topological structures (bitopologies). Also, we study the concept of the generalized topological rough set and some of their basic properties. Applications for data reduction are done on medical data.

A minimal axiom group for rough set based on quasi-ordering

Rough set axiomatization is one aspect of rough set study to characterize rough set theory using dependable and minimal axiom groups. Thus, rough set theory can be studied by logic and axiom system methods. The classic rough set theory is based on equivalent relation, but rough set theory based on reflexive and transitive relation (called quasi-ordering) has wide applications in the real world. To characterize topological rough set theory, an axiom group named RT, consisting of 4 axioms, is proposed. It is proved that the axiom group reliability in characterizing rough set theory based on similar relation is reasonable. Simultaneously, the minimization of the axiom group, which requires that each axiom is an equation and each is independent, is proved. The axiom group is helpful for researching rough set theory by logic and axiom system methods.

Some Remarks on Sets of Communicating Sequential Processes in Topological Rough Set Framework

Communicating Sequential Processes (CSP), is a theoretical framework for discussing concurrent phenomena [7]. In this note, we begin an investigation into the nature of sets of communicating sequen...