Digitizations associated with several types of digital topological approaches
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Digitizations associated with several types of digital topological approaches Jeong Min Kang1 · Sang-Eon Han2 · Kyung Chan Min1
Received: 26 May 2014 / Revised: 10 March 2015 / Accepted: 16 May 2015 © SBMAC - Sociedade Brasileira de Matemática Aplicada e Computacional 2015
Abstract When digitizing subspaces in the Euclidean subspace in a certain digital topological approach, we are strongly required to preserve topological properties of the given spaces such as connectedness. Thus the present paper studies four kinds of local rules associated with lower limit, upper limit, Khalimsky and Marcus Wyse (for short, L-, U -, K - and M-, respectively) topologies which are used for L-, U -, K - and M-digitizing subspaces of the Euclidean nD space into digital topological spaces. While the L-, U - and K -digitizations are proved to preserve connectedness of objects, the M-digitization has some limitation of having the connectedness preserving (for short, C P-) property. This approach can be substantially used for studying applied topology and computer science. Keywords Digital topology · Digitization · Khalimsky topology · Marcus Wyse topology · Local rule · L-, U -, K -, M-localized neighborhood · Connectedness preserving map · Digital topology
Communicated by Jinyun Yuan. The corresponding author (S.E. Han) was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (2013R1A1A4A01007577). The corresponding author (S.E. Han) was supported by research funds of Chonbuk National University in 2013.
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Sang-Eon Han [email protected] Jeong Min Kang [email protected] Kyung Chan Min [email protected]
1
Department of Mathematics, Yonsei University, 50 Yonsei-Ro, Seodaemun-Gu, Seoul 120-749, Republic of Korea
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Department of Mathematics Education, Institute of Pure and Applied Mathematics, Chonbuk National University, Jeonju, Jeonbuk 561-756, Republic of Korea
123
J. M. Kang et al.
Mathematics Subject Classification
54A10 · 54C05 · 54F65 · 55R15 · 68U05 · 68U10
1 Introduction Let Z (resp. N) represent the set of integers (resp. natural numbers), and Zn the set of points in the Euclidean nD space with integer coordinates. Let Rn be the Euclidean nD space with the Euclidean topology. To digitize Euclidean subspaces in Rn into digital spaces in Zn , we have often used the locally finite topological structures of the digital topological spaces such as discrete topology, Khalimsky topology, Marcus Wyse topology, axiomatic locally finite space, Alexandroff topological space and so forth (Alexandorff 1937; Han 2008a; Khalimsky 1970; Kovalevsky 2006; Wyse et al. 1970). To be specific, when digitizing subspaces of the Euclidean nD space, we need to recognize partitioned shapes such as discs, rectangles, triangles and so forth of the Euclidean space into points, e.g. pixels, voxels or n-xels in the corresponding digital spaces. In relation to the study of a digitization of a straight line in R2 , Rosenfeld (1974) dealt wi
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