# MathSciDoc: An Archive for Mathematician ∫

#### Computational Geometrymathscidoc:1608.09005

Computer-Aided Design, 58, 51, 2015
Centroidal Voronoi tessellation (CVT) is a special type of Voronoi diagram such that the generating point of each Voronoi cell is also its center of mass. The CVT has broad applications in computer graphics, such as meshing, stippling, sampling, etc. The existing methods for computing CVTs on meshes either require a global parameterization or compute it in the restricted sense (that is, intersecting a 3D CVT with the surface). Therefore, these approaches often fail on models with complicated geometry and/or topology. This paper presents two intrinsic algorithms for computing CVT on triangle meshes. The first algorithm adopts the Lloyd framework, which iteratively moves the generator of each geodesic Voronoi diagram to its mass center. Based on the discrete exponential map, our method can efficiently compute the Riemannian center and the center of mass for any geodesic Voronoi diagram. The second algorithm uses the L-BFGS method to accelerate the intrinsic CVT computation. Thanks to the intrinsic feature, our methods are independent of the embedding space, and work well for models with arbitrary topology and complicated geometry, where the existing extrinsic approaches often fail. The promising experimental results show the advantages of our method.
Voronoi diagram, Centroidal Voronoi tessellation (CVT), Discrete geodesics, Exponential map, Riemannian center, The L-BFGS method
@inproceedings{xiaoning2015intrinsic,
title={Intrinsic computation of centroidal Voronoi tessellation (CVT) on meshes},
author={Xiaoning Wang, Xiang Ying, Yong-Jin Liu, Shi-Qing Xin, Wenping Wang, Xianfeng Gu, Wolfgang Mueller-Wittig, and Ying He},
url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20160819143748489963260},
booktitle={Computer-Aided Design},
volume={58},
pages={51},
year={2015},
}
Xiaoning Wang, Xiang Ying, Yong-Jin Liu, Shi-Qing Xin, Wenping Wang, Xianfeng Gu, Wolfgang Mueller-Wittig, and Ying He. Intrinsic computation of centroidal Voronoi tessellation (CVT) on meshes. 2015. Vol. 58. In Computer-Aided Design. pp.51. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20160819143748489963260.