Sampling Conditions for Conforming Voronoi Meshing by the VoroCrust Algorithm

TL;DR

This paper introduces VoroCrust, a provably-correct algorithm for decomposing smooth bounded volumes into conforming Voronoi cells, enabling accurate surface reconstruction and volume meshing with convex, fat Voronoi cells.

Contribution

VoroCrust is the first algorithm to produce conforming Voronoi meshes for smooth surfaces with provable correctness, unweighted cells, and enhanced flexibility over previous methods.

Findings

Produces surface reconstructions isotopic to the original surface.

Generates convex, fat Voronoi cells for the volume mesh.

Outperforms clipping-based methods in cell quality and flexibility.

Abstract

We study the problem of decomposing a volume bounded by a smooth surface into a collection of Voronoi cells. Unlike the dual problem of conforming Delaunay meshing, a principled solution to this problem for generic smooth surfaces remained elusive. VoroCrust leverages ideas from $\alpha$-shapes and the power crust algorithm to produce unweighted Voronoi cells conforming to the surface, yielding the first provably-correct algorithm for this problem. Given an $\epsilon$-sample on the bounding surface, with a weak $\sigma$-sparsity condition, we work with the balls of radius $\delta$ times the local feature size centered at each sample. The corners of this union of balls are the Voronoi sites, on both sides of the surface. The facets common to cells on opposite sides reconstruct the surface. For appropriate values of $\epsilon$, $\sigma$ and $\delta$, we prove that the surface reconstruction is isotopic to the bounding surface. With the surface protected, the enclosed volume can be further decomposed into an isotopic volume mesh of fat Voronoi cells by generating a bounded number of sites in its interior. Compared to state-of-the-art methods based on clipping, VoroCrust cells are full Voronoi cells, with convexity and fatness guarantees. Compared to the power crust algorithm, VoroCrust cells are not filtered, are unweighted, and offer greater flexibility in meshing the enclosed volume by either structured grids or random samples.