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Simplicialization of digital volumes in 26-adjacency: Application to topological analysis

  • Representation, Processing, Analysis, and Understanding of Images
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Abstract

In this paper, we introduce a simple and original algorithm to compute a three-dimensional simplicial complex topologically equivalent to a 3D digital object V, according to the 26-adjacency. The use of this adjacency generates issues like auto-intersecting triangles that unnecessarily increase the dimensionality of the associated simplicial complex. To avoid these problems, we present an approach based on a modified Delaunay tetrahedralization of the digital object, that preserves its topological characteristics. Considering the resulting complex as an input in algebraic-topological format (fixing a ground ring for the coefficients), we develop propositions regardless of the adjacency considered. These potential applications are related to topological analysis like thinning, homology computation, topological characterization and control. Moreover, our technique is susceptible to be extended to higher dimensions.

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Authors and Affiliations

  1. Information and System Science Laboratory (LSIS), Computer Graphics Dept. (“Image and Models” Team), University of Marseille II, ESIL, Campus de Luminy, case 925, 13288, Marseille cedex 9, France

    J. -L. Mari

  2. Dpto. de Matemática Aplicada I, Escuela Técnica Superior de Ingeniería Informatica, Universidad de Sevilla, Avda. Reina Mercedes, s/n., 41012, Sevilla, Spain

    P. Real

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  1. J. -L. Mari
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  2. P. Real
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Correspondence to J. -L. Mari.

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Jean-Luc Mari received his PhD degree in 2002. He has been an Associate Professor since 2003 in the Department of Computer Science at the Faculté des Sciences de Luminy (University of Marseilles). He is also a member of the Information and System Science Laboratory (LSIS), in the team “Image and Models” (Computer Graphics group). His research interests include geometrical modeling, model representation, implicit and subdivision surfaces, meshes, multiresolution, skeleton based objects and reconstruction.

Pedro Real received his PhD degree in 1993. He has been an Associate Professor since 1995 in the Department of Applied Mathematics I at Higher Technical School of Computer Engineering (University of Seville, Spain). He is the main responsible of the andalusian research group “Computational Topology and Applied Mathematics.” His research interests include computational algebraic topology, topological analysis of digital images, algebraic pattern recognition and computational algebra.

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Mari, J.L., Real, P. Simplicialization of digital volumes in 26-adjacency: Application to topological analysis. Pattern Recognit. Image Anal. 19, 231–238 (2009). https://doi.org/10.1134/S1054661809020035

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