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Journal of Scientific Computing

, Volume 63, Issue 1, pp 185–212 | Cite as

Fast Methods for Computing Centroidal Voronoi Tessellations

  • James C. Hateley
  • Huayi Wei
  • Long ChenEmail author
Article

Abstract

A Centroidal Voronoi tessellation (CVT) is a Voronoi tessellation in which the generators are the centroids for each Voronoi region. CVTs have many applications to computer graphics, image processing, data compression, mesh generation, and optimal quantization. Lloyd’s method, the most widely method used to generate CVTs, converges very slowly for larger scale problems. Recently quasi-Newton methods using the Hessian of the associated energy as a preconditioner are developed to speed up the rate of convergence. In this work a graph Laplacian preconditioner and a two-grid method are used to speed up quasi-Newton schemes. The proposed graph Laplacian is always symmetric, positive definite and easy to assemble, while the Hessian, in general, may not be positive definite nor easy to assemble. The two-grid method, in which an optimization method using a relaxed stopping criteria is applied on a coarse grid, and then the coarse grid is refined to generate a better initial guess in the fine grid, will further speed up the convergence and lower the energy. Numerical tests show that our preconditioned two-grid optimization methods converges fast and has nearly linear complexity.

Keywords

Centroidal Voronoi tessellation Lloyd’s method Numerical optimization Quasi-Newton methods 

Mathematics Subject Classification

62H30 65H10 65K10 65U05 68U10 

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Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  1. 1.IrvineUSA
  2. 2.Hunan Key Laboratory for Computation and Simulation in Science and Engineering, School of Mathematics and Computational ScienceXiangtan UinversityXiangtanPeople’s Republic of China
  3. 3.Department of MathematicsUniversity of California, IrvineIrvineUSA

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