arXiv subjects

Scott A. Mitchell

Publications and source records attributed to Scott A. Mitchell.

4 recordsLinked to original sources

arXiv2023

VoroCrust: Voronoi Meshing Without Clipping

Polyhedral meshes are increasingly becoming an attractive option with particular advantages over traditional meshes for certain applications. What has been missing is a robust polyhedral meshing algorithm that can handle broad classes of domains exhibiting arbitrarily curved boundaries and sharp features. In addition, the power of primal-dual mesh pairs, exemplified by Voronoi-Delaunay meshes, has been recognized as an important ingredient in numerous formulations. The VoroCrust algorithm is the first provably-correct algorithm for conforming polyhedral Voronoi meshing for non-convex and non-manifold domains with guarantees on the quality of both surface and volume elements. A robust refinement process estimates a suitable sizing field that enables the careful placement of Voronoi seeds across the surface circumventing the need for clipping and avoiding its many drawbacks. The algorithm has the flexibility of filling the interior by either structured or random samples, while preserving all sharp features in the output mesh. We demonstrate the capabilities of the algorithm on a variety of models and compare against state-of-the-art polyhedral meshing methods based on clipped Voronoi cells establishing the clear advantage of VoroCrust output.

cs.GR↗

arXiv2018

Spoke-Darts for High-Dimensional Blue-Noise Sampling

Blue noise sampling has proved useful for many graphics applications, but remains underexplored in high-dimensional spaces due to the difficulty of generating distributions and proving properties about them. We present a blue noise sampling method with good quality and performance across different dimensions. The method, spoke-dart sampling, shoots rays from prior samples and selects samples from these rays. It combines the advantages of two major high-dimensional sampling methods: the locality of advancing front with the dimensionality-reduction of hyperplanes, specifically line sampling. We prove that the output sampling is saturated with high probability, with bounds on distances between pairs of samples and between any domain point and its nearest sample. We demonstrate spoke-dart applications for approximate Delaunay graph construction, global optimization, and robotic motion planning. Both the blue-noise quality of the output distribution and the adaptability of the intermediate processes of our method are useful in these applications.

cs.GR↗

arXiv2018

Sampling Conditions for Conforming Voronoi Meshing by the VoroCrust Algorithm

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 $α$-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 $ε$-sample on the bounding surface, with a weak $σ$-sparsity condition, we work with the balls of radius $δ$ 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 $ε$, $σ$ and $δ$, 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.

cs.CG↗

arXiv2013

k-d Darts: Sampling by k-Dimensional Flat Searches

We formalize the notion of sampling a function using k-d darts. A k-d dart is a set of independent, mutually orthogonal, k-dimensional subspaces called k-d flats. Each dart has d choose k flats, aligned with the coordinate axes for efficiency. We show that k-d darts are useful for exploring a function's properties, such as estimating its integral, or finding an exemplar above a threshold. We describe a recipe for converting an algorithm from point sampling to k-d dart sampling, assuming the function can be evaluated along a k-d flat. We demonstrate that k-d darts are more efficient than point-wise samples in high dimensions, depending on the characteristics of the sampling domain: e.g. the subregion of interest has small volume and evaluating the function along a flat is not too expensive. We present three concrete applications using line darts (1-d darts): relaxed maximal Poisson-disk sampling, high-quality rasterization of depth-of-field blur, and estimation of the probability of failure from a response surface for uncertainty quantification. In these applications, line darts achieve the same fidelity output as point darts in less time. We also demonstrate the accuracy of higher dimensional darts for a volume estimation problem. For Poisson-disk sampling, we use significantly less memory, enabling the generation of larger point clouds in higher dimensions.

cs.GR↗