GPU-Assisted Computation of Centroidal Voronoi Tessellation

Abstract

Abstract-Centroidal Voronoi tessellations (CVT) are widely used in computational science and engineering. The most commonly used method is Lloyd's method, and recently the L-BFGS method is shown to be faster than Lloyd's method for computing the CVT. However, these methods run on the CPU and are still too slow for many practical applications. We present techniques to implement these methods on the GPU for computing the CVT on 2D planes and on surfaces, and demonstrate significant speedup of these GPU-based methods over their CPU counterparts. For CVT computation on a surface, we use a geometry image stored in the GPU to represent the surface for computing the Voronoi diagram on the surface. In our implementation a new technique is proposed for parallel regional reduction on the GPU for evaluating integrals over Voronoi cells.

Robust and accurate optimal transportation map by self-adaptive sampling

“…The mesh adaption methods include centroidal Voronoi tessellation (CVT) (Du and Wang, 2005;Liu et al, 2009;Rong et al, 2011;Yan et al, 2011), interpolation error minimization (Kunert, 2002;Alauzet et al, 2006), optimal Delaunay triangulations (ODTs) (Chen and Xu, 2004), mesh refinement (Apel and Lube, 1996;Bossen and Heckbert, 1996;Löhner and Cebral, 2000;Berndt and Shashkov, 2003;Apel et al, 2004), monitor functions (Huang, 2006), mesh smoothing (Sirois, et al, 2006), and a branch of other methods (Bottasso, 2004;Dobrzynski and Frey, 2008). Many of these methods were generalized to produce anisotropic meshes by incorporating a metric tensor into the functional.…”

Meshless voronoi on the GPU

“…For the sake of completeness, we also mention that discrete pixel-based) Voronoi diagrams can be computed on the GPU in 2D, using the Z-Buffer to determine to which cell each pixel belongs [Hoff et al 1999], and related works devoted to compute distance fields both in 2D and 3D [Cuntz and Kolb 2007;Fischer and Gotsman 2006;Rong and Tan 2006;Schneider et al 2010;Sigg et al 2003;Sud et al 2006Sud et al , 2005Sud et al , 2004. A similar method is exploited to compute Centroidal Voronoi Tessellations [Fei et al 2014;Rong et al 2011] The approach that we propose is based on a completely different strategy: it is based on the observation that given a 3D search data These applications solely need the geometry of the Voronoi cells, and do not need a global mesh topology. Fig.…”

Anisotropic Sampling of Planar and Two-Manifold Domains for Texture Generation and Glyph Distribution

“…To guarantee interactivity, which is one of our most important requirements, we present a GPU implementation that computes the Voronoi cells in the fragment shader. In contrast to previous GPU implementations of generalized Voronoi diagrams in the planar domain [27] or centroidal Voronoi diagrams in the two-manifold domain [28], we do not need an explicit representation of the diagram, because we use the diagram for visualization purposes only. The implementation that we present in this paper does not require a surface parameterization.…”