Discrete combinatorial geometry

Kenmochi, Yukiko; Imiya, Atsushi; Ichikawa, Akira

doi:10.1016/s0031-3203(97)00001-0

Cited by 17 publications

(24 citation statements)

References 5 publications

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“…In Z 3 , we have discrete complexes whose dimensions can be from zero to three. Thus, we present topological characterization of discrete complexes for each dimension from one to three by using the notions of star and link [20] similarly to the previous work [12]. We then show that there are 12 topological types of points in Sk(C).…”

Section: Topological Point Characterization On Discrete Complexesmentioning

confidence: 83%

“…1. In this paper, we use local topological notions similarly to our work [12] to discriminate surface Fig. 1.…”

Section: Introductionmentioning

confidence: 99%

“…In Section 2, we first define discrete polyhedral complexes [14], based on combinatorial topology [1,19,20], which gives a topology to a three-dimensional discrete space. In Section 3, we then study topological characteristics of discrete polyhedral complexes similarly to [12]. By using these topological characteristics, in Section 4, we consider all conceivable topological characteristics of object points and distinguish those of boundary points such that the boundary forms a discrete combinatorial surface.…”

Section: Introductionmentioning

confidence: 99%

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Local configurations in discrete combinatorial surfaces

Kenmochi

Nomura

2007

Image and Vision Computing

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Representing discrete objects by polyhedral complexes, we can define all conceivable topological characteristics of points in discrete objects, namely those of vertices of polyhedral complexes. Such a topological characteristic is determined for each point by observing a configuration of object points in the 3 × 3 × 3 local point set of its neighbors. We study a topological characteristic such that the point is in the boundary of a 3D polyhedral complex and the boundary forms a 2D combinatorial surface. By using the topological characteristic, we present an algorithm which examines whether the central point of a local point set is in a combinatorial surface, and show how many local point configurations exist in combinatorial surfaces in a 3D discrete space.

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Section: Topological Point Characterization On Discrete Complexesmentioning

confidence: 83%

“…1. In this paper, we use local topological notions similarly to our work [12] to discriminate surface Fig. 1.…”

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Local configurations in discrete combinatorial surfaces

Kenmochi

Nomura

2007

Image and Vision Computing

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“…It takes an AT-model for a 26-adjacency voxel-based digital binary volume V using a polyhedral cell complex at geometric modeling level [14,15,17] and a chain homotopy operator described by a combinatorial vector field (a set of semidirected forests or a discrete differential form) at homology analysis level [24,25]. For instance, a chain homotopy operator at level of cells of dimension 0 (vertices) of a cell complex K (V ) can be completely described by a semidirected spanning forest of the graph subcomplex formed by all the cells of K (V ) of dimension 0 and 1.…”

Section: Introductionmentioning

confidence: 99%

Using Membrane Computing for Effective Homology

Díaz-Pernil

Christinal

Gutiérrez–Naranjo

et al. 2012

AAECC

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Effective Homology is an algebraic-topological method based on the computational concept of chain homotopy equivalence on a cell complex. Using this algebraic data structure, Effective Homology gives answers to some important computability problems in Algebraic Topology. In a discrete context, Effective Homology can be seen as a combinatorial layer given by a forest graph structure spanning every cell of the complex. In this paper, by taking as input a pixel-based 2D binary object, we present a logarithmic-time uniform solution for describing a chain homotopy operator φ for its adjacency graph. This solution is based on Membrane Computing techniques applied to the spanning forest problem and it can be easily extended to higher dimensions.

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“…In particular, each edge connects two vertices, each face is enclosed by a loop of edges, and each 3-cell is enclosed by an envelope of faces; (b) (homology analysis level) Homology information about K(V ) is codified in homological algebra terms [5,6]. This method has recently evolving to a technique which for generating a Z/2Z-coefficient AT-model for a 26-adjacency voxel-based digital binary volume V uses a polyhedral cell complex at geometric modeling level [11,12,17,19] and a chain homotopy map (described by a vector fields or by a discrete differential form) at homology analysis level [20,24]. Formally, an AT-model ((K(V ), ∂), φ) for the volume V can be geometrically specified by a cell (polyhedral) complex K(V ) and algebraically specified by a boundary…”

Section: Introductionmentioning

confidence: 99%

Getting Topological Information for a 80-Adjacency Doxel-Based 4D Volume through a Polytopal Cell Complex

Pacheco

Real

2009

Progress in Pattern Recognition, Image Analysis, Computer Vision, and Applications

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Abstract. Given an 80-adjacency doxel-based digital four-dimensional hypervolume V , we construct here an associated oriented 4-dimensional polytopal cell complex K(V ), having the same integer homological information (that related to n-dimensional holes that object has) than V . This is the first step toward the construction of an algebraic-topological representation (AT-model) for V , which suitably codifies it mainly in terms of its homological information. This AT-model is especially suitable for global and local topological analysis of digital 4D images.

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Discrete combinatorial geometry

Cited by 17 publications

References 5 publications

Local configurations in discrete combinatorial surfaces

Local configurations in discrete combinatorial surfaces

Using Membrane Computing for Effective Homology

Getting Topological Information for a 80-Adjacency Doxel-Based 4D Volume through a Polytopal Cell Complex

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