Restricting Voronoi diagrams to meshes using corner validation

• Published in: Computer graphics forum Vol. 36; no. 5; pp. 81 - 91
• Main Authors: Sainlot, M., Nivoliers, V., Attali, D.
• Format: Journal Article
• Language: English
• Published: Oxford Blackwell Publishing Ltd 01.08.2017; Wiley
• Subjects: Categories and Subject Descriptors (according to ACM CCS); Computation; Computational Geometry; Computer Science; Decision trees; I.3.3 [Computer Graphics]: Picture/Image Generation—Line and curve generation; Triangles; Voronoi graphs
• ISSN: 0167-7055; 1467-8659; 1467-8659
• DOI: 10.1111/cgf.13247

Restricted Voronoi diagrams are a fundamental geometric structure used in many applications such as surface reconstruction from point sets or optimal transport. Given a set of sites V = {vk}nk=1 ⊂ ℝd and a mesh X with vertices in ℝd connected by triangles, the restricted Voronoi diagram partitions X by computing for each site the portion of X for which the site is the nearest. The restricted Voronoi diagram is the intersection between the regular Voronoi diagram and the mesh. Depending on the site distribution or the ambient space dimension computing the regular Voronoi diagram may not be feasible using classical algorithms. In this paper, we extend Lévy and Bonneel's approach [LB12] based on nearest neighbor queries. We show that their method is limited when the sites are not located on X. We propose a new algorithm for computing restricted Voronoi which reduces the number of sites considered for each triangle of the mesh and scales smoothly when the sites are far from the surface.