On the limited memory BFGS method for large scale optimization
Mathematical Programming: Series A and B
Line search algorithms with guaranteed sufficient decrease
ACM Transactions on Mathematical Software (TOMS)
The quickhull algorithm for convex hulls
ACM Transactions on Mathematical Software (TOMS)
Incomplete Cholesky Factorizations with Limited Memory
SIAM Journal on Scientific Computing
Delaunay triangulations and Voronoi diagrams for Riemannian manifolds
Proceedings of the sixteenth annual symposium on Computational geometry
A Revised Modified Cholesky Factorization Algorithm
SIAM Journal on Optimization
Grid generation and optimization based on centroidal Voronoi tessellations
Applied Mathematics and Computation
Constrained Centroidal Voronoi Tessellations for Surfaces
SIAM Journal on Scientific Computing
Geodesic Voronoi Diagrams on Parametric Surfaces
CGI '97 Proceedings of the 1997 Conference on Computer Graphics International
An adaptive numerical cubature algorithm for simplices
ACM Transactions on Mathematical Software (TOMS)
Variational shape approximation
ACM SIGGRAPH 2004 Papers
Surface Segmentation Using Geodesic Centroidal Tesselation
3DPVT '04 Proceedings of the 3D Data Processing, Visualization, and Transmission, 2nd International Symposium
Centroidal Voronoi Tessellation Based Algorithms for Vector Fields Visualization and Segmentation
VIS '04 Proceedings of the conference on Visualization '04
Anisotropic Centroidal Voronoi Tessellations and Their Applications
SIAM Journal on Scientific Computing
Variational tetrahedral meshing
ACM SIGGRAPH 2005 Papers
Centroidal Voronoi diagrams for isotropic surface remeshing
Graphical Models - Special issue on SMI 2003
Convergence of the Lloyd Algorithm for Computing Centroidal Voronoi Tessellations
SIAM Journal on Numerical Analysis
The Effectiveness of Lloyd-Type Methods for the k-Means Problem
FOCS '06 Proceedings of the 47th Annual IEEE Symposium on Foundations of Computer Science
k-means++: the advantages of careful seeding
SODA '07 Proceedings of the eighteenth annual ACM-SIAM symposium on Discrete algorithms
Generic Remeshing of 3D Triangular Meshes with Metric-Dependent Discrete Voronoi Diagrams
IEEE Transactions on Visualization and Computer Graphics
Computers & Mathematics with Applications
Isotropic remeshing with fast and exact computation of Restricted Voronoi Diagram
SGP '09 Proceedings of the Symposium on Geometry Processing
Vector field analysis and visualization through variational clustering
EUROVIS'05 Proceedings of the Seventh Joint Eurographics / IEEE VGTC conference on Visualization
Packing circles and spheres on surfaces
ACM SIGGRAPH Asia 2009 papers
Isotropic remeshing with fast and exact computation of Restricted Voronoi Diagram
SGP '09 Proceedings of the Symposium on Geometry Processing
Lp Centroidal Voronoi Tessellation and its applications
ACM SIGGRAPH 2010 papers
Hyperbolic centroidal Voronoi tessellation
Proceedings of the 14th ACM Symposium on Solid and Physical Modeling
Regularization of B-spline objects
Computer Aided Geometric Design
Geodesic Methods in Computer Vision and Graphics
Foundations and Trends® in Computer Graphics and Vision
Sparse meshless models of complex deformable solids
ACM SIGGRAPH 2011 papers
HOT: Hodge-optimized triangulations
ACM SIGGRAPH 2011 papers
SMI 2011: Full Paper: Capacity-Constrained Delaunay Triangulation for point distributions
Computers and Graphics
The Geometric Stability of Voronoi Diagrams with Respect to Small Changes of the Sites
Proceedings of the twenty-seventh annual symposium on Computational geometry
Efficient and good Delaunay meshes from random points
Computer-Aided Design
Centroidal Voronoi tessellation in universal covering space of manifold surfaces
Computer Aided Geometric Design
Obtuse triangle suppression in anisotropic meshes
Computer Aided Geometric Design
PolyMesher: a general-purpose mesh generator for polygonal elements written in Matlab
Structural and Multidisciplinary Optimization
Efficient computation of 3d clipped voronoi diagram
GMP'10 Proceedings of the 6th international conference on Advances in Geometric Modeling and Processing
Robust modeling of constant mean curvature surfaces
ACM Transactions on Graphics (TOG) - SIGGRAPH 2012 Conference Proceedings
Applications of Geometry Processing: Blue noise sampling of surfaces
Computers and Graphics
Centroidal Voronoi Tessellation of Line Segments and Graphs
Computer Graphics Forum
Computing Voronoi Treemaps: Faster, Simpler, and Resolution-independent
Computer Graphics Forum
Blue noise through optimal transport
ACM Transactions on Graphics (TOG) - Proceedings of ACM SIGGRAPH Asia 2012
Generalized edge-weighted centroidal Voronoi tessellations for geometry processing
Computers & Mathematics with Applications
Graphics Interaction: 5-6-7 Meshes: Remeshing and analysis
Computers and Graphics
Efficient computation of clipped Voronoi diagram for mesh generation
Computer-Aided Design
Computing self-supporting surfaces by regular triangulation
ACM Transactions on Graphics (TOG) - SIGGRAPH 2013 Conference Proceedings
Particle-based anisotropic surface meshing
ACM Transactions on Graphics (TOG) - SIGGRAPH 2013 Conference Proceedings
Broadcast control of multi-agent systems
Automatica (Journal of IFAC)
Improving spatial coverage while preserving the blue noise of point sets
Computer-Aided Design
Special Section on CAD/Graphics 2013: Geometry-constrained crowd formation animation
Computers and Graphics
Approximating functions on a mesh with restricted Voronoï diagrams
SGP '13 Proceedings of the Eleventh Eurographics/ACMSIGGRAPH Symposium on Geometry Processing
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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.