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Membrane parallelism for discrete Morse theory applied to digital images

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Applicable Algebra in Engineering, Communication and Computing Aims and scope

Abstract

In this paper, we propose a bio-inspired membrane computational framework for constructing discrete Morse complexes for binary digital images. Our approach is based on the discrete Morse theory and we work with cubical complexes. As example, a parallel algorithm for computing homology groups of binary 3D digital images is designed.

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Notes

  1. In [13] Forman works with simplicial complexes, however the mathematical scaffolding provided by DMT can be settled with no change to finite cubical complexes.

  2. The reader is supposed to be familiar with concepts of Image Algebra. For a detailed text, see [31].

  3. Lemma 1 grants the identification of cubical cells in a cubical complex in \({\mathbb {R}}^k\) with \(k\)-tuples.

  4. Introduced in [25].

  5. In case of 2D images, \(B_\mathbf {i} = B_{i_1 i_2}\).

  6. CW complex in [13] and cubical complex in this paper.

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Authors and Affiliations

  1. CATAM Research Group, Department of Applied Mathematics I, University of Seville, Seville, Spain

    Raúl Reina-Molina, Daniel Díaz-Pernil & Pedro Real

  2. Department of Didactic of Mathematics and Experimental Sciences, University of the Basque Country, Bilbao, Spain

    Ainhoa Berciano

Authors
  1. Raúl Reina-Molina
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  2. Daniel Díaz-Pernil
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  3. Pedro Real
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  4. Ainhoa Berciano
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Correspondence to Raúl Reina-Molina.

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Reina-Molina, R., Díaz-Pernil, D., Real, P. et al. Membrane parallelism for discrete Morse theory applied to digital images. AAECC 26, 49–71 (2015). https://doi.org/10.1007/s00200-014-0246-z

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  • DOI: https://doi.org/10.1007/s00200-014-0246-z

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