Article

Journal of Mathematical Imaging and Vision

, Volume 23, Issue 3, pp 379-399

Discrete Surfaces and Frontier Orders

  • Xavier DaragonAffiliated withÉcole Supérieure d’Ingénieurs en Électrotechnique et Électronique, Laboratoire A2 SI, 2, boulevard Blaise Pascal, Cité DESCARTES, BP 99IGM, Unité Mixte de Recherche CNRS-UMLV-ESIEE UMR 8049 Email author 
  • , Michel CouprieAffiliated withÉcole Supérieure d’Ingénieurs en Électrotechnique et Électronique, Laboratoire A2 SI, 2, boulevard Blaise Pascal, Cité DESCARTES, BP 99IGM, Unité Mixte de Recherche CNRS-UMLV-ESIEE UMR 8049
  • , Gilles BertrandAffiliated withÉcole Supérieure d’Ingénieurs en Électrotechnique et Électronique, Laboratoire A2 SI, 2, boulevard Blaise Pascal, Cité DESCARTES, BP 99IGM, Unité Mixte de Recherche CNRS-UMLV-ESIEE UMR 8049

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Abstract

Many applications require the extraction of an object boundary from a discrete image. In most cases, the result of such a process is expected to be, topologically, a surface, and this property might be required in subsequent operations. However, only through careful design can such a guarantee be provided. In the present article we will focus on partially ordered sets and the notion of n-surfaces introduced by Evako et al. to deal with this issue. Partially ordered sets are topological spaces that can represent the topology of a wide range of discrete spaces, including abstract simplicial complexes and regular grids. It will be proved in this article that (in the framework of simplicial complexes) any n-surface is an n-pseudomanifold, and that any n-dimensional combinatorial manifold is an n-surface. Moreover, given a subset of an n-surface (an object), we show how to build a partially ordered set called frontier order, which represents the boundary of this object. Similarly to the continuous case, where the boundary of an n-manifold, if not empty, is an (n−1)-manifold, we prove that the frontier order associated to an object is a union of disjoint (n−1)-surfaces. Thanks to this property, we show how topologically consistent Marching Cubes-like algorithms can be designed using the framework of partially ordered sets.

Keywords

discrete topology discrete surfaces partially ordered sets simplicial complexes frontier orders