Optimal Voronoi Tessellations with Hessian-based Anisotropy

Max Budninskiy (1) , Beibei Liu (1) , Fernando de Goes (2) , Yiying Tong (3) , Pierre Alliez (4, 5) , Mathieu Desbrun (5, 1)

1 CS CALTECH - Computer Science Department
2 Pixar Animation Studios
3 Michigan State University [East Lansing]
4 COMUE UCA - COMUE Université Côte d'Azur (2015-2019)
5 TITANE - Geometric Modeling of 3D Environments

This paper presents a variational method to generate cell complexes with local anisotropy conforming to the Hessian of any given convex function and for any given local mesh density. Our formulation builds upon approximation theory to offer an anisotropic extension of Centroidal Voronoi Tessellations which can be seen as a dual form of Optimal Delaunay Triangulation. We thus refer to the resulting anisotropic polytopal meshes as Optimal Voronoi Tessel-lations. Our approach sharply contrasts with previous anisotropic versions of Voronoi diagrams as it employs first-type Bregman diagrams , a generalization of power diagrams where sites are augmented with not only a scalar-valued weight but also a vector-valued shift. As such, our OVT meshes contain only convex cells with straight edges, and admit an embedded dual triangulation that is combinatorially-regular. We show the effectiveness of our technique using off-the-shelf computational geometry libraries.

Mots clés: Voronoi; anisotropic

Domaines: Géométrie algorithmique [cs.CG]

Max Budninskiy, Beibei Liu, Fernando de Goes, Yiying Tong, Pierre Alliez, et al.. Optimal Voronoi Tessellations with Hessian-based Anisotropy. ACM Transactions on Graphics, 2016, Proceedings of SIGGRAPH Asia, pp.12. ⟨hal-01376243⟩