%0 Journal Article %1 liu2009centroidal %A Liu, Yang %A Wang, Wenping %A Lévy, Bruno %A Sun, Feng %A Yan, Dong-Ming %A Lu, Lin %A Yang, Chenglei %C New York, NY, USA %D 2009 %I ACM %J ACM Trans. Graph. %K lloyd smoothness voronoi %N 4 %P 101:1--101:17 %R 10.1145/1559755.1559758 %T On Centroidal Voronoi Tessellation - Energy Smoothness and Fast Computation %U http://doi.acm.org/10.1145/1559755.1559758 %V 28 %X Centroidal Voronoi tessellation (CVT) is a particular type of Voronoi tessellation that has many applications in computational sciences and engineering, including computer graphics. The prevailing method for computing CVT is Lloyd's method, which has linear convergence and is inefficient in practice. We develop new efficient methods for CVT computation and demonstrate the fast convergence of these methods. Specifically, we show that the CVT energy function has 2nd order smoothness for convex domains with smooth density, as well as in most situations encountered in optimization. Due to the 2nd order smoothness, it is possible to minimize the CVT energy functions using Newton-like optimization methods and expect fast convergence. We propose a quasi-Newton method to compute CVT and demonstrate its faster convergence than Lloyd's method with various numerical examples. It is also significantly faster and more robust than the Lloyd-Newton method, a previous attempt to accelerate CVT. We also demonstrate surface remeshing as a possible application.

@article{liu2009centroidal, abstract = {Centroidal Voronoi tessellation (CVT) is a particular type of Voronoi tessellation that has many applications in computational sciences and engineering, including computer graphics. The prevailing method for computing CVT is Lloyd's method, which has linear convergence and is inefficient in practice. We develop new efficient methods for CVT computation and demonstrate the fast convergence of these methods. Specifically, we show that the CVT energy function has 2nd order smoothness for convex domains with smooth density, as well as in most situations encountered in optimization. Due to the 2nd order smoothness, it is possible to minimize the CVT energy functions using Newton-like optimization methods and expect fast convergence. We propose a quasi-Newton method to compute CVT and demonstrate its faster convergence than Lloyd's method with various numerical examples. It is also significantly faster and more robust than the Lloyd-Newton method, a previous attempt to accelerate CVT. We also demonstrate surface remeshing as a possible application.}, acmid = {1559758}, added-at = {2016-01-27T17:33:18.000+0100}, address = {New York, NY, USA}, articleno = {101}, author = {Liu, Yang and Wang, Wenping and L{\'e}vy, Bruno and Sun, Feng and Yan, Dong-Ming and Lu, Lin and Yang, Chenglei}, biburl = {https://puma.ub.uni-stuttgart.de/bibtex/27577cb562f4c2846811c36f44601dd2c/mariusoei}, description = {On centroidal voronoi tessellation—energy smoothness and fast computation}, doi = {10.1145/1559755.1559758}, interhash = {b43a71ef903965902d05839c24d2f2ae}, intrahash = {7577cb562f4c2846811c36f44601dd2c}, issn = {0730-0301}, issue_date = {August 2009}, journal = {ACM Trans. Graph.}, keywords = {lloyd smoothness voronoi}, month = sep, number = 4, numpages = {17}, pages = {101:1--101:17}, publisher = {ACM}, timestamp = {2016-03-10T22:23:23.000+0100}, title = {On Centroidal Voronoi Tessellation - Energy Smoothness and Fast Computation}, url = {http://doi.acm.org/10.1145/1559755.1559758}, volume = 28, year = 2009 }