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A Heuristic for Short Homology Basis of Digital Objects

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Discrete Geometry and Mathematical Morphology (DGMM 2022)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 13493))

Abstract

Finding the minimum homology basis of a simplicial complex is a hard problem unless one only considers the first homology group. In this paper, we introduce a general heuristic for finding a short homology basis of any dimension for digital objects (that is, for their associated cubical complexes) with complexity \(\mathcal {O}(m^3 + \beta _q \cdot n^3)\), where m is the size of the bounding box of the object, n is the size of the object and \(\beta _q\) is the rank of its qth homology group. Our heuristic makes use of the thickness-breadth balls, a tool for visualizing and locating holes in digital objects.

We evaluate our algorithm with a data set of 3D digital objects and compare it with an adaptation of the best current algorithm for computing the minimum radius homology basis by Dey, Li and Wang [10].

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Authors and Affiliations

  1. Aix Marseille Univ, CNRS, LIS, Marseille, France

    Aldo Gonzalez-Lorenzo, Alexandra Bac & Jean-Luc Mari

Authors
  1. Aldo Gonzalez-Lorenzo
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  2. Alexandra Bac
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  3. Jean-Luc Mari
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Correspondence to Aldo Gonzalez-Lorenzo .

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Editors and Affiliations

  1. University of Strasbourg, Strasbourg, France

    Étienne Baudrier

  2. University of Strasbourg, Strasbourg, France

    Benoît Naegel

  3. University of Strasbourg, Strasbourg, France

    Adrien Krähenbühl

  4. University of Strasbourg, Strasbourg, France

    Mohamed Tajine

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Gonzalez-Lorenzo, A., Bac, A., Mari, JL. (2022). A Heuristic for Short Homology Basis of Digital Objects. In: Baudrier, É., Naegel, B., Krähenbühl, A., Tajine, M. (eds) Discrete Geometry and Mathematical Morphology. DGMM 2022. Lecture Notes in Computer Science, vol 13493. Springer, Cham. https://doi.org/10.1007/978-3-031-19897-7_6

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  • DOI: https://doi.org/10.1007/978-3-031-19897-7_6

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-031-19896-0

  • Online ISBN: 978-3-031-19897-7

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