A digitization method of subspaces of the Euclidean $$n$$ n D space associated with the Kha
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A digitization method of subspaces of the Euclidean nD space associated with the Khalimsky adjacency structure Sang-Eon Han1
Received: 29 January 2014 / Revised: 27 February 2015 / Accepted: 3 March 2015 © SBMAC - Sociedade Brasileira de Matemática Aplicada e Computacional 2015
Abstract To digitize subspaces of the Euclidean nD space, the present paper uses the Khalimsky (for short K -, if there is no danger of ambiguity) topology, K -adjacency and K -localized neighborhoods of points in Zn , where Zn represents the set of points in the Euclidean nD space with integer coordinates. Namely, given a point p ∈ Zn , the paper first develops a K -localized neighborhood of p ∈ Zn , denoted by N K ( p) in Rn , which is substantially used in digitizing subspaces of the Euclidean nD space. The recent paper Han and Sostak in (Comput Appl Math 32(3):521–536, 2013) proposes a connectedness preserving map (for short CP-map, e.g., an A-map in this paper) which need not be a continuous map under K -topology and further, develops a certain CP-isomorphism, e.g., an A-isomorphism in this paper. It turns out that an A-map overcomes some limitations of both a K -continuous map and a Khalimsky adjacency map (for brevity K A-map) so that both an A-map and an A-isomorphism can substantially contribute to applied topology including both digital topology and digital geometry Han and Sostak in (Comput Appl Math 32(3):521–536, 2013). Using both an A-map and a K -localized neighborhood, we further develop the notions of a lattice-based A-map (for short L A-map) and a lattice-based A-isomorphism (for brevity L A-isomorphism) which are used for digitizing subspaces of the Euclidean nD space in the K -topological approach. Thus, this approach can contribute to certain branches of applied topology and computer science such as image analysis, image processing, and mathematical morphology.
Communicated by Jinyun Yuan. This research 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).
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Sang-Eon Han [email protected] Department of Mathematics Education, Institute of Pure and Applied Mathematics, Chonbuk National University, Jeonju-City, Jeonbuk 561-756, Republic of Korea
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S.-E. Han
Keywords Digitization · Digital topology · Khalimsky nD space · Khalimsky adjacency · A-map · A-isomorphism · Local rule · K -localized neighborhood · L A-map · L A-isomorphism · Mathematical morphology Mathematics Subject Classification 68U10
54C10 · 54C05 · 54C08 · 55P10 · 55P15 · 68U05 ·
1 Introduction Since the study of digitizing subspaces of the Euclidean nD space plays an important role in both applied topology and a certain branches of computer science such as image processing and computer graphics (Melin 2007), the present paper studies a digitization method using both the K -topological structure and K -adjacency. The main idea of the paper is to employ a local rule to formulate a K -localized neighborhood of a po
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