Advances in Studies and Applications of Centroidal Voronoi Tessellations

Abstract

Abstract. Centroidal Voronoi tessellations (CVTs) have become a useful tool in many applications ranging from geometric modeling, image and data analysis, and numerical partial differential equations, to problems in physics, astrophysics, chemistry, and biology. In this paper, we briefly review the CVT concept and a few of its generalizations and well-known properties. We then present an overview of recent advances in both mathematical and computational studies and in practical applications of CVTs. Whenever possible, we point out some outstanding issues that still need investigating.

Centroidal Voronoi tessellation in universal covering space of manifold surfaces

“…We only review previous work closely related to geometric modeling. For other applications, please refer to [Du et al, 1999[Du et al, , 2010Rong et al, 2011] and the references therein.…”

“…The centroidal Voronoi tessellation (CVT) is a special Voronoi diagram where every site s i coincides with the centroid c i of its Voronoi cell [Du et al, 1999[Du et al, , 2010. To define the CVT in Euclidean, spherical, and hyperbolic spaces, we have to first define the centroid in these spaces.…”

Improved initialisation for centroidal Voronoi tessellation and optimal Delaunay triangulation

“…Minimising F(X) ensures that each subregion Ω i , and thus each site x i , represents approximately the same subvolume of Ω in the uniform density case. The properties and computation of CVT have been well studied [1,13] and several algorithms, including Lloyd's method [9], the Lloyd-Newton method [14] and the Quasi-Newton method [8] have been proposed for computing a CVT for a given region. We briefly review these methods for the reader's benefit.…”

Efficient implementation of Galerkin meshfree methods for large-scale problems with an emphasis on maximum entropy approximants

“…1, Appendix A). The proper choice of the locality parameter is problem dependent and not easy in general, which has motivated a systematic studies for general meshfree methods [38] and for LME approximants [25] in particular. In LME approximations, the locality parameter is an aspect ratio parameter , which allows us to smoothly move from linear finite elements shape functions ( > 4.0) to more spread out approximation schemes (e.g., = 0.6), as illustrated in Fig.…”