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Computational Homology Applied to Discrete Objects

Abstract : Homology theory formalizes the concept of hole in a space. For a given subset of the Euclidean space, we define a sequence of homology groups, whose ranks are considered as the number of holes of each dimension. Hence, β₀, the rank of the 0-dimensional homology group, is the number of connected components, β₁ is the number of tunnels or handles and β₂ is the number of cavities. These groups are computable when the space is described in a combinatorial way, as simplicial or cubical complexes are. Given a discrete object (a set of pixels, voxels or their analog in higher dimension) we can build a cubical complex and thus compute its homology groups. This thesis studies three approaches regarding the homology computation of discrete objects. First, we introduce the homological discrete vector field, a combinatorial structure which generalizes the discrete gradient vector field and allows us to compute the homology groups. This notion allows us to see the relation between different existing methods for computing homology. Next, we present a linear algorithm for computing the Betti numbers of a 3D cubical complex, which can be used for binary volumes. Finally, we introduce two measures (the thickness and the breadth) associated to the holes in a discrete object, which provide a topological and geometric signature more interesting than only the Betti numbers. This approach provides also some heuristics for localizing holes, obtaining minimal homology or cohomology generators, opening and closing holes.
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https://hal.archives-ouvertes.fr/tel-01477399
Contributor : Aldo Gonzalez-Lorenzo Connect in order to contact the contributor
Submitted on : Monday, February 27, 2017 - 1:18:59 PM
Last modification on : Wednesday, November 3, 2021 - 9:43:32 AM
Long-term archiving on: : Sunday, May 28, 2017 - 1:08:09 PM

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  • HAL Id : tel-01477399, version 1

Citation

Aldo Gonzalez-Lorenzo. Computational Homology Applied to Discrete Objects. Discrete Mathematics [cs.DM]. Aix-Marseille Université; Universidad de Sevilla, 2016. English. ⟨tel-01477399⟩

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