A digitization method of subspaces of the Euclidean $$n$$ n D space associated with the Khalimsky adjacency structure

Abstract

To digitize subspaces of the Euclidean \(n\)D 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 \(\mathbf{Z}^n\), where \(\mathbf{Z}^n\) represents the set of points in the Euclidean \(n\)D space with integer coordinates. Namely, given a point \(p \in \mathbf{Z}^n\), the paper first develops a \(K\)-localized neighborhood of \(p \in \mathbf{Z}^n\), denoted by \(N_K(p)\) in \(\mathbf{R}^n\), which is substantially used in digitizing subspaces of the Euclidean \(n\)D 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 \(KA\)-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 \(LA\)-map) and a lattice-based \(A\)-isomorphism (for brevity \(LA\)-isomorphism) which are used for digitizing subspaces of the Euclidean \(n\)D 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.