Discrete Surfaces and Frontier Orders
 Xavier Daragon,
 Michel Couprie,
 Gilles Bertrand
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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 nsurfaces 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 nsurface is an npseudomanifold, and that any ndimensional combinatorial manifold is an nsurface. Moreover, given a subset of an nsurface (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 nmanifold, 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 Cubeslike algorithms can be designed using the framework of partially ordered sets.
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 Title
 Discrete Surfaces and Frontier Orders
 Journal

Journal of Mathematical Imaging and Vision
Volume 23, Issue 3 , pp 379399
 Cover Date
 20051101
 DOI
 10.1007/s1085100520294
 Print ISSN
 09249907
 Online ISSN
 15737683
 Publisher
 Kluwer Academic Publishers
 Additional Links
 Topics
 Keywords

 discrete topology
 discrete surfaces
 partially ordered sets
 simplicial complexes
 frontier orders
 Industry Sectors
 Authors

 Xavier Daragon ^{(1)} ^{(2)}
 Michel Couprie ^{(1)} ^{(2)}
 Gilles Bertrand ^{(1)} ^{(2)}
 Author Affiliations

 1. École Supérieure d’Ingénieurs en Électrotechnique et Électronique, Laboratoire A2 SI, 2, boulevard Blaise Pascal, Cité DESCARTES, BP 99, 93162, Noisy le Grand CEDEX, France
 2. IGM, Unité Mixte de Recherche CNRSUMLVESIEE UMR 8049, France