How much geometry it takes to reconstruct a 2-manifold in R ³

Abstract

Known algorithms for reconstructing a 2-manifold from a point sample in R 3 are naturally based on decisions/predicates that take the geometry of the point sample into account. Facing the always present problem of round-off errors that easily compromise the exactness of those predicate decisions, an exact and robust implementation of these algorithms is far from being trivial and typically requires the employment of advanced datatypes for exact arithmetic as provided by libraries like CORE, LEDA or GMP. In this paper we present a new reconstruction algorithm, one of whose main novelties is to throw away geometry information early on in the reconstruction process and to mainly operate combinatorially on a graph structure. As such it is less susceptible to robustness problems due to round-off errors and also benefits from not requiring expensive exact arithmetic by faster running times. A more theoretical view on our algorithm including correctness proofs under suitable sampling conditions can be found in a companion paper [3].

BallMerge: High‐quality Fast Surface Reconstruction via Voronoi Balls

“…A faster computation is enabled via Ball‐pivoting [BMR*99], where the surface is reconstructed by growing a seed by adding additional triangles. Local tangent‐plane projection [Boi84], while handling local topological inconsistencies between tangent planes to correctly approximate the DT, is a popular acceleration that inspired others [GKS00; DFKM08; FR02; AGJ02; Kós01]. Attene et al [AS00] extend the reconstruction to objects with genus > 0.…”

Surface reconstruction by computing restricted Voronoi cells in parallel

“…Dumitriu et al [38] used this framework to design a surface reconstruction algorithm with correctness guarantees. Although the algorithm is quite complicated, they have produced an implementation [39]. Amenta and Kil [40] employed similar techniques and designed a parallel algorithm operating on the GPU.…”