GPU-based computation of discrete periodic centroidal Voronoi tessellation in hyperbolic space

Periodic centroidal Voronoi tessellation (CVT) in hyperbolic space provides a nice theoretical framework for computing the constrained CVT on high-genus (genus > 1) surfaces. This paper addresses two computational issues related to such hyperbolic CVT framework: (1) efficient reduction of unnecessary site copies in neighbor domains on the universal covering space, based on two special rules; (2) GPU-based parallel algorithms to compute a discrete version of the hyperbolic CVT. Our experiments show that with the dramatically reduced number of unnecessary site copies in neighbor domains and the GPU-based parallel algorithms, we significantly speed up the computation of CVT for high-genus surfaces. The proposed discrete hyperbolic CVT guarantees to converge and produces high-quality results.