Compression of Khalimsky topological spaces

Min Kang; Sang-Eon Han

doi:10.2298/fil1206101k

Compression of Khalimsky topological spaces

Filomat ◽

10.2298/fil1206101k ◽

2012 ◽

Vol 26 (6) ◽

pp. 1101-1114 ◽

Cited By ~ 8

Author(s):

Min Kang ◽

Sang-Eon Han

Keyword(s):

Computer Science ◽

Geometric Transformation ◽

Topological Spaces ◽

Digital Geometry

Aiming at the study of the compression of Khalimsky topological spaces which is an interesting field in digital geometry and computer science, the present paper develops a new homotopy thinning suitable for the work. Since Khalimsky continuity of maps between Khalimsky topological spaces has some limitations of performing a discrete geometric transformation, the paper uses another continuity (see Definition 3.4) that can support the discrete geometric transformation and a homotopic thinning suitable for studying Khalimsky topological spaces. By using this homotopy, we can develop a new homotopic thinning for compressing the spaces and can write an algorithm for compressing 2D Khalimsky topological spaces.

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An extension problem of a connectedness preserving map between Khalimsky spaces

Filomat ◽

10.2298/fil1601015h ◽

2016 ◽

Vol 30 (1) ◽

pp. 15-28 ◽

Cited By ~ 1

Author(s):

Sang-Eon Han

Keyword(s):

Computer Science ◽

Mathematical Morphology ◽

Digital Images ◽

Digital Topology ◽

Extension Problem ◽

Topological Spaces ◽

Digital Geometry ◽

Applied Topology ◽

Continuous Map ◽

Connected Sets

The goal of the present paper is to study an extension problem of a connected preserving (for short, CP-) map between Khalimsky (K-for brevity, if there is no ambiguity) spaces. As a generalization of a K-continuous map, for K-topological spaces the recent paper [13] develops a function sending connected sets to connected ones (for brevity, an A-map: see Definition 3.1 in the present paper). Since this map plays an important role in applied topology including digital topology, digital geometry and mathematical morphology, the present paper studies an extension problem of a CP-map in terms of both an A-retract and an A-isomorphism (see Example 5.2). Since K-topological spaces have been often used for studying digital images, this extension problem can contribute to a certain areas of computer science and mathematical morphology.

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Categories of L-Fuzzy Čech Closure Spaces and L-Fuzzy Co-Topological Spaces

Mathematics ◽

10.3390/math8081274 ◽

2020 ◽

Vol 8 (8) ◽

pp. 1274

Author(s):

Irina Perfilieva ◽

Ahmed A. Ramadan ◽

Enas H. Elkordy

Keyword(s):

Computer Science ◽

Fuzzy Systems ◽

Topological Spaces ◽

Fuzzy Relations ◽

Approximation Spaces ◽

Galois Correspondence ◽

Closure Spaces

Recently, fuzzy systems have become one of the hottest topics due to their applications in the area of computer science. Therefore, in this article, we are making efforts to add new useful relationships between the selected L-fuzzy (fuzzifying) systems. In particular, we establish relationships between L-fuzzy (fuzzifying) Čech closure spaces, L-fuzzy (fuzzifying) co-topological spaces and L-fuzzy (fuzzifying) approximation spaces based on reflexive L-fuzzy relations. We also show that there is a Galois correspondence between the categories of these spaces.

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A Continuous Coordinate System for the Plane by Triangular Symmetry

Symmetry ◽

10.3390/sym11020191 ◽

2019 ◽

Vol 11 (2) ◽

pp. 191 ◽

Cited By ~ 1

Author(s):

Benedek Nagy ◽

Khaled Abuhmaidan

Keyword(s):

Image Processing ◽

Coordinate System ◽

Computer Science ◽

Closed Interval ◽

Triangular Grid ◽

Digital Geometry ◽

Cartesian Coordinate System ◽

Coordinate Systems ◽

Cartesian Coordinate ◽

New System

The concept of the grid is broadly used in digital geometry and other fields of computer science. It consists of discrete points with integer coordinates. Coordinate systems are essential for making grids easy to use. Up to now, for the triangular grid, only discrete coordinate systems have been investigated. These have limited capabilities for some image-processing applications, including transformations like rotations or interpolation. In this paper, we introduce the continuous triangular coordinate system as an extension of the discrete triangular and hexagonal coordinate systems. The new system addresses each point of the plane with a coordinate triplet. Conversion between the Cartesian coordinate system and the new system is described. The sum of three coordinate values lies in the closed interval [−1, 1], which gives many other vital properties of this coordinate system.

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Topological structures in computer science

Journal of Applied Mathematics and Simulation ◽

10.1155/s1048953388000036 ◽

1987 ◽

Vol 1 (1) ◽

pp. 25-40 ◽

Cited By ~ 33

Author(s):

Efim Khalimsky

Keyword(s):

Computer Science ◽

Computer Tomography ◽

Discrete Mathematics ◽

Topological Spaces ◽

Picture Processing ◽

Topological Methods ◽

Intermediate Value Theorem ◽

Robotic Vision ◽

Finite Spaces ◽

Intermediate Value

Topologies of finite spaces and spaces with countably many points are investigated. It is proven, using the theory of ordered topological spaces, that any topology in connected ordered spaces, with finitely many points or in spaces similar to the set of all integers, is an interval-alternating topology. Integer and digital lines, arcs, and curves are considered. Topology of N-dimensional digital spaces is described. A digital analog of the intermediate value theorem is proven. The equivalence of connectedness and pathconnectedness in digital and integer spaces is also proven. It is shown here how methods of continuous mathematics, for example, topological methods, can be applied to objects, that used to be investigated only by methods of discrete mathematics. The significance of methods and ideas in digital image and picture processing, robotic vision, computer tomography and system's sciences presented here is well known.

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