Computation of Homology Groups and Generators

Peltier, Samuel; Alayrangues, Sylvie; Fuchs, Laurent; Lachaud, Jacques-Olivier

doi:10.1007/978-3-540-31965-8_19

Cited by 26 publications

(35 citation statements)

References 15 publications

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“…Many of them [16,9,17] can compute the homology groups of more general spaces in cubical time. Computing only the Betti numbers (number of holes), which are the ranks of these groups, should be faster, but this has not been algorithmically proved.…”

Section: Introductionmentioning

confidence: 99%

Fast, Simple and Separable Computation of Betti Numbers on Three-Dimensional Cubical Complexes

Gonzalez-Lorenzo

Juda

Bac

et al. 2016

Computational Topology in Image Context

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Abstract. Betti numbers are topological invariants that count the number of holes of each dimension in a space. Cubical complexes are a class of CW complex whose cells are cubes of different dimensions such as points, segments, squares, cubes, etc. They are particularly useful for modeling structured data such as binary volumes. We introduce a fast and simple method for computing the Betti numbers of a three-dimensional cubical complex that takes advantage on its regular structure, which is not possible with other types of CW complexes such as simplicial or polyhedral complexes. This algorithm is also restricted to three-dimensional spaces since it exploits the Euler-Poincaré formula and the Alexander duality in order to avoid any matrix manipulation. The method runs in linear time on a single core CPU. Moreover, the regular cubical structure allows us to obtain an efficient implementation for a multi-core architecture.

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Section: Introductionmentioning

confidence: 99%

Fast, Simple and Separable Computation of Betti Numbers on Three-Dimensional Cubical Complexes

Gonzalez-Lorenzo

Juda

Bac

et al. 2016

Computational Topology in Image Context

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“…In particular, effective algorithms and efficient software for the computation of homology groups and their generators have been under heavy development; see [1,2,3,4,5,6,7,8,9,10] for some of this work. These tools have already proved their usefulness in applications, e.g.…”

Section: Introductionmentioning

confidence: 99%

Decomposing Cavities in Digital Volumes into Products of Cycles

Alcaraz

Molina‐Abril

Pacheco

et al. 2009

Discrete Geometry for Computer Imagery

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Abstract. The homology of binary 3-dimensional digital images (digital volumes) provides concise algebraic description of their topology in terms of connected components, tunnels and cavities. Homology generators corresponding to these features are represented by nontrivial 0-cycles, 1-cycles and 2-cycles, respectively. In the framework of cubical representation of digital volumes with the topology that corresponds to the 26-connectivity between voxels, we introduce a method for algorithmic computation of a coproduct operation that can be used to decompose 2-cycles into products of 1-cycles (possibly trivial). This coproduct provides means of classifying different kinds of cavities; in particular, it allows to distinguish certain homotopically non-equivalent spaces that have isomorphic homology. We define this coproduct at the level of a cubical complex built directly upon voxels of the digital image, and we construct it by means of the classical Alexander-Whitney map on a simplicial subdivision of faces of the voxels.

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“…Examples of image analysis and pattern recognition problems that benefit from the use of topological considerations include, for example, the use of homology invariants to compare objects, for image classification, or shape recognition. At present, there is an intensive research on homology computation in the context of the image ( [9] for cubical homology, [15,16] for persistent homology, [3] for combinatorial maps, [12] for matrix methods, [13] for graph pyramids, [7] for algebraic-topological models).Even so, the amount of efficient numerical tools combining homological with geometrical information could be increased using and saving additional algebraic data structures at no extra computational time.…”

Section: Introductionmentioning

confidence: 99%

Advanced Homology Computation of Digital Volumes Via Cell Complexes

Molina‐Abril

Real

2008

Lecture Notes in Computer Science

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Abstract. Given a 3D binary voxel-based digital object V , an algorithm for computing homological information for V via a polyhedral cell complex is designed. By homological information we understand not only Betti numbers, representative cycles of homology classes and homological classification of cycles but also the computation of homology numbers related additional algebraic structures defined on homology (coproduct in homology, product in cohomology, (co)homology operations,...). The algorithm is mainly based on the following facts: a) a local 3D-polyhedrization of any 2×2×2 configuration of mutually 26-adjacent black voxels providing a coherent cell complex at global level; b) a description of the homology of a digital volume as an algebraic-gradient vector field on the cell complex (see Discrete Morse Theory [5],AT-model method [7,5]) . Saving this vector field, we go further obtaining homological information at no extra time processing cost.

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Computation of Homology Groups and Generators

Cited by 26 publications

References 15 publications

Fast, Simple and Separable Computation of Betti Numbers on Three-Dimensional Cubical Complexes

Fast, Simple and Separable Computation of Betti Numbers on Three-Dimensional Cubical Complexes

Decomposing Cavities in Digital Volumes into Products of Cycles

Advanced Homology Computation of Digital Volumes Via Cell Complexes

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