Critical points of the distance to an epsilon-sampling of a surface and flow-complex-based surface reconstruction

Abstract

The distance function to surfaces in three dimensions plays a key role in many geometric modeling applications such as medial axis approximations, surface reconstructions, offset computations, feature extractions and others. In most cases, the distance function induced by the surface is approximated by a discrete distance function induced by a discrete sample of the surface. The critical points of the distance function determine the topology of the set inducing the function. However, no earlier theoretical result has linked the critical points of the distance to a sampling of geometric structures to their topological properties. We provide this link by showing that the critical points of the distance function induced by a discrete sample of a surface either lie very close to the surface or near its medial axis and this closeness is quantified with the sampling density. Based on this result, we provide a new flow-complex-based surface reconstruction algorithm that, given a tight ε-sampling of a surface, approximates the surface geometrically, both in Hausdorff distance and normals, and captures its topology.

Medial Axis Approximation and Unstable Flow Complex

“…The proof is rather similar to the one given in [8] and is provided in the full-version of the paper.…”

“…Dey, et al [8] observed that if P is an ε-sample of the smooth boundary Σ of a shape S, then the critical points of the discrete distance function hP cannot reside everywhere in S. Rather they have to be either very close to Σ or very close to M . Theorem 1.…”

“…Theorem 1. [8] Let P be an ε-sample of a smooth surface Σ. Then for every critical point c of hP , either…”

“…Dey et al [8] show that the critical points of the distance function to an ε-sampling of the boundary of a shape are naturally separated; each such critical point is either very close to the surface, or oppositely, is very close to the medial axis of the shape. The two classes can be separated algorithmically provided that the sample is sufficiently dense.…”

Minimizing edge length to connect sparsely sampled unstructured point sets

“…While the flow complex is well established, and performs well in reconstructing surfaces with boundaries, its given guarantees to extract a closed manifold homeomorphic to the sampled surface ( [22], [23]) do not surpass previously given ones, or refer only to the weaker homotopy [24]. Besides in [20] it is mentioned that the computation of the flow complex is of (possibly much) higher order than the Delaunay complex.…”