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Generating Second Order (Co)homological Information within AT-Model Context

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Computational Topology in Image Context (CTIC 2019)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 11382))

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Abstract

In this paper we design a new family of relations between (co)homology classes, working with coefficients in a field and starting from an AT-model (Algebraic Topological Model) AT(C) of a finite cell complex C These relations are induced by elementary relations of type “to be in the (co)boundary of” between cells. This high-order connectivity information is embedded into a graph-based representation model, called Second Order AT-Region-Incidence Graph (or AT-RIG) of C. This graph, having as nodes the different homology classes of C, is in turn, computed from two generalized abstract cell complexes, called primal and dual AT-segmentations of C. The respective cells of these two complexes are connected regions (set of cells) of the original cell complex C, which are specified by the integral operator of AT(C). In this work in progress, we successfully use this model (a) in experiments for discriminating topologically different 3D digital objects, having the same Euler characteristic and (b) in designing a parallel algorithm for computing potentially significant (co)homological information of 3D digital objects.

This work has been supported by the Spanish research projects MTM2016-81030-P (AEI/FEDER, UE) and TEC2012-37868-C04-02, and by the VPPI of the University of Seville. Darian Onchis gratefully acknowledges the support of the Austrian Science Fund FWF-P27516.

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Author information

Authors and Affiliations

  1. H.T.S. Informatics’ Engineering, University of Seville, Seville, Spain

    Pedro Real, Helena Molina-Abril & Fernando Díaz del Río

  2. Faculty of Mathematics, University of Vienna, Vienna, Austria

    Darian Onchis

  3. Faculty of Mathematics and Computer Science, West University of Timisoara, Timişoara, Romania

    Darian Onchis

Authors
  1. Pedro Real
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  2. Helena Molina-Abril
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  3. Fernando Díaz del Río
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  4. Darian Onchis
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Corresponding author

Correspondence to Helena Molina-Abril .

Editor information

Editors and Affiliations

  1. Universidad de Málaga, Málaga, Spain

    Rebeca Marfil

  2. Universidad de Málaga, Málaga, Spain

    Mariletty Calderón

  3. University of Seville, Sevilla, Spain

    Fernando Díaz del Río

  4. University of Seville, Sevilla, Spain

    Pedro Real

  5. Universidad de Málaga, Málaga, Spain

    Antonio Bandera

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Real, P., Molina-Abril, H., Díaz del Río, F., Onchis, D. (2019). Generating Second Order (Co)homological Information within AT-Model Context. In: Marfil, R., Calderón, M., Díaz del Río, F., Real, P., Bandera, A. (eds) Computational Topology in Image Context. CTIC 2019. Lecture Notes in Computer Science(), vol 11382. Springer, Cham. https://doi.org/10.1007/978-3-030-10828-1_6

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  • DOI: https://doi.org/10.1007/978-3-030-10828-1_6

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