A Topological Approach to Digital Topology

Kong, T. Yung; Kopperman, Ralph; Meyer, Paul R.

doi:10.1080/00029890.1991.12000810

Cited by 69 publications

(48 citation statements)

References 12 publications

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“…We now work toward a proof that for finite spaces, topology and continuity are completely determined by a preorder (which should be seen as an asymmetric adjacency relation). That is (see Theorem 1 (b), or [11]):…”

Section: Definition 2 a Topological Space Is Alexandroff Ifmentioning

confidence: 97%

“…In IR for example, {0} = the finite topological spaces that one can completely store in a computer, since then any subset of τ is finite, so its intersection is in τ . The theory of Alexandroff spaces, applied especially to digital topology, is discussed in [11] and [7]. Most of the results in Lemma 2 through Theorem 1 can be found there conveniently (though none originate there).…”

Section: Definition 2 a Topological Space Is Alexandroff Ifmentioning

confidence: 99%

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Topological Digital Topology

Kopperman

2003

Discrete Geometry for Computer Imagery

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Abstract. The usefulness of topology in science and mathematics means that topological spaces must be studied, and computers should be used in this study. We discuss how many useful spaces (including all compact Hausdorff spaces) can be approximated by finite spaces, and these finite spaces are completely determined by their specialization orders. As a special case, digital n-space, used to interpret Euclidean n-space and in particular, the computer screen, is also dealt with in terms of the specialization. Indeed, algorithms written using the specialization are comparable in difficulty, storage usage and speed to those which use the traditional (8,4), (4,8) and (6,6) adjacencies, and are of course completely representative of the spaces.

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Section: Definition 2 a Topological Space Is Alexandroff Ifmentioning

confidence: 97%

Section: Definition 2 a Topological Space Is Alexandroff Ifmentioning

confidence: 99%

Topological Digital Topology

Kopperman

2003

Discrete Geometry for Computer Imagery

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“…There is an isomorphism betwen the cellular decomposition C n of R n into a regular grid and the n-dimensional Khalimisky space K n [5]. This space is the cartesian product of n connected ordered topological spaces (COTS).…”

Section: Cell Codingmentioning

confidence: 99%

Geometric Measures on Arbitrary Dimensional Digital Surfaces

Lachaud

Vialard

2003

Discrete Geometry for Computer Imagery

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Abstract. This paper proposes a set of tools to analyse the geometry of multidimensional digital surfaces. Our approach is based on several works of digital topology and discrete geometry: representation of digital surfaces, bel adjacencies and digital surface tracking, 2D tangent computation by discrete line recognition, 3D normal estimation from slice contours. The main idea is to notice that each surface element is the crossing point of n − 1 discrete contours lying on the surface. Each of them can be seen as a 4-connected 2D contour. We combine the directions of the tangents extracted on each of these contours to compute the normal vector at the considered surface element. We then define the surface area from the normal field. The presented geometric estimators have been implemented in a framework able to represent subsets of n-dimensional spaces. As shown by our experiments, this generic implementation is also efficient.

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“…Non-Hausdortt topological spaces are finding numerous applications today in areas of computer science. (See the references in Kong et al [14] and Kopperman [15].) Considering them as partially ordered partitions, we will study the principal topologies (quasiorders) on a set X, with particular attention to the lattice structure of the collection of all principal topologies on X.…”

Section: Introductionmentioning

confidence: 99%