Homotopy in Digital Spaces

Francés, Angel R.; Gómez, Rafael Ayala; Domı́nguez, Elena; Toscano, Antonio Rafael Quintero

doi:10.1007/3-540-44438-6_1

Cited by 29 publications

(77 citation statements)

References 7 publications

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“…It is worth noting that there are several methods to construct a simplicial complex from a digital image [ADFQ03]. We are going to explain one of these methods.…”

Section: The Theorem Formalized and Its Contextmentioning

confidence: 99%

Incidence Simplicial Matrices Formalized in Coq/SSReflect

Heras

Poza

Dénès

et al. 2011

Lecture Notes in Computer Science

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Simplicial complexes are at the heart of Computational Algebraic Topology, since they give a concrete, combinatorial description of otherwise rather abstract objects which makes many important topological computations possible. The whole theory has many applications such as coding theory, robotics or digital image analysis. In this paper we present a formalization in the COQ theorem prover of simplicial complexes and their incidence matrices as well as the main theorem that gives meaning to the definition of homology groups and is a first step towards their computation.

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“…It is worth noting that there are several methods to construct a simplicial complex from a digital image [ADFQ03]. We are going to explain one of these methods.…”

Section: The Theorem Formalized and Its Contextmentioning

confidence: 99%

Incidence Simplicial Matrices Formalized in Coq/SSReflect

Heras

Poza

Dénès

et al. 2011

Lecture Notes in Computer Science

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“…In DTC both a (k 0 , k 1 )-homotopy and a k-homotopy have been used in classifying digital spaces in terms of the digital fundamental group and the digital k-(or (k 0 , k 1 ))-homotopy equivalence [2,4,5,8,13,15,16,17,30,32].…”

Section: Introductionmentioning

confidence: 99%

KD-(k₀, k₁)-HOMOTOPY EQUIVALENCE AND ITS APPLICATIONS

Han¹

2010

Journal of the Korean Mathematical Society

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Abstract. Let Z n be the Cartesian product of the set of integers Z and let (Z, T ) and (Z n , T n ) be the Khalimsky line topology on Z and the Khalimsky product topology on Z n , respectively. Then for a set X ⊂ Z n , consider the subspace (X, T n X ) induced from (Z n , T n ). Considering a k-adjacency on (X, T n X ), we call it a (computer topological) space with k-adjacency and use the notation (X, k, T n X ) := X n,k . In this paper we introduce the notions of KD-(k 0 , k 1 )-homotopy equivalence and KD-kdeformation retract and investigate a classification of (computer topological) spaces X n,k in terms of a KD-(k 0 , k 1 )-homotopy equivalence.

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“…In order to prove that a digital topology operation π D (associated with a continuous operation π C ) correctly reflects the topology of digital pictures considered as Euclidean spaces, the main idea is to associate a "continuous analog" C(I) with the digital picture I. In most cases, each binary digital picture I is associated with a polyhedron C(I) [10,11,9,1]). It is clear that C(I) "fills the gaps" between black points of I in a way that strongly depends on the grid and adjacency relations chosen for the digital picture I.…”

Section: Introductionmentioning

confidence: 99%

“…A small program, called EditCup, for editing binary digital pictures and visualizing cohomology aspects of them has been designed by the authors and developed by others 1 . This software allows us to test in some simple examples the potentiality and topological acuity of our method.…”

Section: Introductionmentioning

confidence: 99%

Towards Digital Cohomology

González-Dı́az

Real

2003

Discrete Geometry for Computer Imagery

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Abstract. We propose a method for computing the Z2-cohomology ring of a simplicial complex uniquely associated with a three-dimensional digital binary-valued picture I. Binary digital pictures are represented on the standard grid Z 3 , in which all grid points have integer coordinates. Considering a particular 14-neighbourhood system on this grid, we construct a unique simplicial complex K(I) topologically representing (up to isomorphisms of pictures) the picture I. We then compute the cohomology ring on I via the simplicial complex K(I). The usefulness of a simplicial description of the digital Z2-cohomology ring of binary digital pictures is tested by means of a small program visualizing the different steps of our method. Some examples concerning topological thinning, the visualization of representative generators of cohomology classes and the computation of the cup product on the cohomology of simple 3D digital pictures are showed.

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Homotopy in Digital Spaces

Cited by 29 publications

References 7 publications

Incidence Simplicial Matrices Formalized in Coq/SSReflect

Incidence Simplicial Matrices Formalized in Coq/SSReflect

KD-(k₀, k₁)-HOMOTOPY EQUIVALENCE AND ITS APPLICATIONS

Towards Digital Cohomology

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Homotopy in Digital Spaces

Cited by 29 publications

References 7 publications

Incidence Simplicial Matrices Formalized in Coq/SSReflect

Incidence Simplicial Matrices Formalized in Coq/SSReflect

KD-(k0, k1)-HOMOTOPY EQUIVALENCE AND ITS APPLICATIONS

Towards Digital Cohomology

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KD-(k₀, k₁)-HOMOTOPY EQUIVALENCE AND ITS APPLICATIONS