On the Locality of Extracting a 2-Manifold in

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 extraction step 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.