On the Locality of Extracting a 2-Manifold in

Authors:
Daniel Dumitriu;Stefan Funke;Martin Kutz;Nikola Milosavljević
Affiliations:
Max-Planck-Institut für Informatik, Saarbrücken, Germany 66123;Ernst-Moritz-Arndt-Universität, Greifswald, Germany 17487;Stanford University, Stanford, USA CA 94305;Stanford University, Stanford, USA CA 94305
Venue:
SWAT '08 Proceedings of the 11th Scandinavian workshop on Algorithm Theory
Year:
2008

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Abstract

Algorithms for reconstructing a 2-manifold from a point sample in based on Voronoi-filtering like CRUST[1] or CoCone[2] still require --- after identifying a set of candidate triangles --- a so-called manifold extractionstep which identifies a subset of the candidate triangles to form the final reconstruction surface. Non-locality of the latter step is caused by so-called slivers--- configurations of four almost cocircular points having an empty circumsphere with center close to the manifold surface.We prove that under a certain mild condition --- local uniformity --- which typically holds in practice but can also be enforced theoretically, one can compute a reconstruction using an algorithm whose decisions about the adjacencies of a point only depend on nearby points.While the theoretical proof requires an extremely high sampling density, our prototype implementation, described in a companion paper [3], preforms well on typical sample sets. Due to its local mode of computation, it might be particularly suited for parallel computing or external memory scenarios.