research-article

L_p Centroidal Voronoi Tessellation and its applications

Authors:

Yang LiuAuthors Info & Claims

ACM Transactions on Graphics (TOG), Volume 29, Issue 4

Article No.: 119, Pages 1 - 11

https://doi.org/10.1145/1778765.1778856

Published: 26 July 2010 Publication History

Abstract

This paper introduces L_p-Centroidal Voronoi Tessellation (L_p-CVT), a generalization of CVT that minimizes a higher-order moment of the coordinates on the Voronoi cells. This generalization allows for aligning the axes of the Voronoi cells with a predefined background tensor field (anisotropy). L_p-CVT is computed by a quasi-Newton optimization framework, based on closed-form derivations of the objective function and its gradient. The derivations are given for both surface meshing (Ω is a triangulated mesh with per-facet anisotropy) and volume meshing (Ω is the interior of a closed triangulated mesh with a 3D anisotropy field). Applications to anisotropic, quad-dominant surface remeshing and to hexdominant volume meshing are presented. Unlike previous work, L_p-CVT captures sharp features and intersections without requiring any pre-tagging.

Supplementary Material

JPG File (tp080-10.jpg)

Download
17.15 KB

Supplemental material. (119.zip)

LpCVT.zip contains data, sourcecode, videos and supplemental examples.

Download
109.91 MB

MP4 File (tp080-10.mp4)

Download
98.75 MB

References

[1]

Alliez, P., Cohen-Steiner, D., Devillers, O., Lévy, B., and Desbrun, M. 2003. Anisotropic polygonal remeshing. ACM Transactions on Graphics 22, 3, 485--493.

Digital Library

[2]

Alliez, P., Cohen-Steiner, D., Yvinec, M., and Desbrun, M. 2005. Variational tetrahedral meshing. ACM Transactions on Graphics 24, 3, 617--625.

Digital Library

[3]

Boissonnat, J.-D., and Oudot, S. 2006. Provably good sampling and meshing of Lipschitz surfaces. In SCG '06: Proceedings of the twenty-second annual symposium on Computational geometry, 337--346.

Digital Library

[4]

Bommes, D., Zimmer, H., and Kobbelt, L. 2009. Mixedinteger quadrangulation. ACM Transactions on Graphics 28, 3, 1--10.

Digital Library

[5]

Busaryev, O., Dey, T. K., and Levine, J. A. 2009. Repairing and meshing imperfect shapes with Delaunay refinement. In SPM '09: 2009 SIAM/ACM Joint Conference on Geometric and Physical Modeling, 25--33.

Digital Library

[6]

Chen, L., and Xu, J. 2004. Optimal Delaunay triangulations. Journal of Computational Mathematics 22, 2, 299--308.

[7]

Cohen-Steiner, D., Alliez, P., and Desbrun, M. 2004. Variational shape approximation. ACM Transactions on Graphics 23, 905--914.

Digital Library

[8]

Daniels-II, J., Silva, C. T., and Cohen, E. 2009. Localized quadrilateral coarsening. Computer Graphics Forum 28, 1437--1444.

Digital Library

[9]

Dey, T. K., and Levine, J. A. 2008. Delaunay meshing of piecewise smooth complexes without expensive predicates. Algorithms 2, 1327--1349. OSU-CISRC-7/08-TR40.

[10]

Dong, S., Bremer, P.-T., Garland, M., Pascucci, V., and Hart, J. C. 2006. Spectral surface quadrangulation. ACM Transactions on Graphics 25, 1057--1066.

Digital Library

[11]

Du, Q., and Emelianenko, M. 2006. Acceleration schemes for computing centroidal Voronoi tessellations. Numerical Linear Algebra with Applications 13, 2--3, 173--192.

[12]

Du, Q., and Wang, D. 2003. Tetrahedral mesh generation and optimization based on centroidal Voronoi tessellations. International Journal for Numerical Methods in Engineering 56, 1355--1373.

[13]

Du, Q., and Wang, D. 2005. Anisotropic centroidal Voronoi tessellations and their applications. SIAM Journal on Scientific Computing 26, 3, 737--761.

Digital Library

[14]

Du, Q., Faber, V., and Gunzburger, M. 1999. Centroidal Voronoi tessellations: applications and algorithms. SIAM Review 41, 4, 637--676.

Digital Library

[15]

Edelsbrunner, H., and Shah, N. R. 1997. Triangulating topological spaces. International Journal of Computational Geometry & Applications 7, 4, 365--378.

[16]

Huang, J., Zhang, M., Ma, J., Liu, X., Kobbelt, L., and Bao, H. 2008. Spectral quadrangulation with orientation and alignment control. ACM Transactions on Graphics 27.

Digital Library

[17]

Iri, M., Murota, K., and Ohya, T. 1984. A fast Voronoidiagram algorithm with applications to geographical optimization problems. In Proc. IFIP, 273--288.

[18]

Itoh, T., and Shimada, K. 2001. Automatic conversion of triangular meshes into quadrilateral meshes with directionality. International Journal of CAD/CAM.

[19]

Kalberer, F., Nieser, M., and Polthier, K. 2007. Quadcover --- surface parameterization using branched coverings. Computer Graphics Forum 26, 375--384.

[20]

Labelle, F., and Shewchuk, J. R. 2003. Anisotropic Voronoi diagrams and guaranteed-quality anisotropic mesh generation. In SCG '03: Proceedings of the nineteenth annual symposium on Computational geometry, 191--200.

Digital Library

[21]

Lai, Y.-K., Kobbelt, L., and Hu, S.-M. 2008. An incremental approach to feature aligned quad dominant remeshing. In ACM SPM conference proceedings.

Digital Library

[22]

Lasserre, J., and Avrachenkov, K. 2001. Integration on a simplex: The multi-dimensional version of ∫_a ^b x ^p dx. American Mathematics Monthly 108, 151--154.

[23]

Liu, Y., Wang, W., Lévy, B., Sun, F., Yan, D.-M., Lu, L., and Yang, C. 2009. On centroidal Voronoi tessellation---energy smoothness and fast computation. ACM Transactions on Graphics 28, 4, 1--17.

Digital Library

[24]

Lloyd, S. P. 1982. Least squares quantization in PCM. IEEE Transactions on Information Theory 28, 2, 129--137.

Digital Library

[25]

Maréchal, L. 2009. Advances in octree-based all-hexahedral mesh generation: handling sharp features. In 18th International Meshing Roundtable conference proceedings.

[26]

Martin, T., Cohen, E., and Kirby, M. 2008. Volumetric parameterization and trivariate B-spline fitting using harmonic functions. In SPM '08: Proceedings of the 2008 ACM symposium on Solid and physical modeling, 269--280.

Digital Library

[27]

Meshkat, S., and Talmor, D. 2000. Generating a mixed mesh of hexahedra, pentahedra and tetrahedra from an underlying tetrahedral mesh. International Journal for Numerical Methods in Engineering 49, 17--30.

[28]

Minka, T. 1997. Old and new matrix algebra useful for statistics. Tech. rep., MIT Media Lab. revised 12/00.

[29]

Mount, D. M., and Arya, S. 1997. ANN: A library for approximate nearest neighbor searching. In Proceedings CGC Workshop on Computational Geometry, 33--40.

[30]

Owen, S. J., and Saigal, S. 2000. H-Morph: An indirect approach to advancing front hex meshing. International Journal for Numerical Methods in Engineering 49, 289--312.

[31]

Ray, N., Li, W. C., Lévy, B., Sheffer, A., and Alliez, P. 2006. Periodic global parameterization. ACM Transactions on Graphics 25, 4, 1460--1485.

Digital Library

[32]

Shepherd, J. F., and Johnson, C. R. 2008. Hexahedral mesh generation constraints. Engineering with Computers 24, 3, 195--213.

Digital Library

[33]

Shepherd, J., Mitchell, S. A., Knupp, P., and White, D. 2000. Methods for multisweep automation. In 9th Intl. Meshing Roundtable conference proceedings.

[34]

Staten, M. L., Owen, S. J., and Blacker, T. D. 2005. Unconstrained paving and plastering: A new idea for all hexahedral mesh generation. In International Meshing Roundtable conference proceedings.

[35]

Steiner, D., and Fischer, A. 2005. Planar parameterization for closed manifold genus-g meshes using any type of positive weights. Journal of Computational Methods in Information Science and Engineering 5, 2.

[36]

Tchon, K.-F., and Camarero, R. 2006. Quad-dominant mesh adaptation using specialized simplicial optimization. In 15th International Meshing Roundtable conference proceedings.

[37]

Tong, Y., Alliez, P., Cohen-Steiner, D., and Desbrun, M. 2006. Designing quadrangulations with discrete harmonic forms. In SGP '06: Proceedings of the fourth Eurographics symposium on Geometry processing, 201--210.

Digital Library

[38]

Tournois, J., Wormser, C., Alliez, P., and Desbrun, M. 2009. Interleaving delaunay refinement and optimization for practical isotropic tetrahedron mesh generation. ACM Transactions on Graphics 28, 75:1--75:9.

Digital Library

[39]

Valette, S., Chassery, J.-M., and Prost, R. 2008. Generic remeshing of 3D triangular meshes with metric-dependent discrete Voronoi diagrams. IEEE Transactions on Visualization and Computer Graphics 14, 2, 369--381.

Digital Library

[40]

Vyas, V., and Shimada, K. 2009. Tensor-guided hex-dominant mesh generation with targeted all-hex regions. In 18th International Meshing Roundtable conference proceedings.

[41]

Yamakawa, S., and Shimada, K. 2003. Anisotropic tetrahedral meshing via bubble packing and advancing front. International Journal for Numerical Methods in Engineering 57.

[42]

Yamakawa, S., and Shimada, K. 2003. Fully-automated hexdominant mesh generation with directionality control via packing rectangular solid cells. International Journal for Numerical Methods in Engineering 57, 2099--2129.

[43]

Yan, D.-M., Lévy, B., Liu, Y., Sun, F., and Wang, W. 2009. Isotropic remeshing with fast and exact computation of restricted Voronoi diagram. Computer Graphics Forum 28, 5, 1445--1454.

Digital Library

Cited By

Xu RLiu LWang NChen SXin SGuo XZhong ZKomura TWang WTu C(2024)CWF: Consolidating Weak Features in High-quality Mesh SimplificationACM Transactions on Graphics10.1145/365815943:4(1-14)Online publication date: 19-Jul-2024
https://dl.acm.org/doi/10.1145/3658159
Hu JWang SHe YLuo ZLei NLiu L(2024)A Parametric Design Method for Engraving Patterns on Thin ShellsIEEE Transactions on Visualization and Computer Graphics10.1109/TVCG.2023.324050330:7(3719-3730)Online publication date: 1-Jul-2024
https://dl.acm.org/doi/10.1109/TVCG.2023.3240503
Zhang WYue YPan HChen ZWang CPfister HWang W(2024)Marching Windows: Scalable Mesh Generation for Volumetric Data With Multiple MaterialsIEEE Transactions on Visualization and Computer Graphics10.1109/TVCG.2022.322552630:3(1728-1742)Online publication date: Mar-2024
https://doi.org/10.1109/TVCG.2022.3225526
Show More Cited By

Index Terms

L_p Centroidal Voronoi Tessellation and its applications
1. Computing methodologies
  1. Computer graphics
    1. Shape modeling
2. Theory of computation
  1. Randomness, geometry and discrete structures

Recommendations

On centroidal voronoi tessellation—energy smoothness and fast computation

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 ...
L_p Centroidal Voronoi Tessellation and its applications
SIGGRAPH '10: ACM SIGGRAPH 2010 papers

This paper introduces L_p-Centroidal Voronoi Tessellation (L_p-CVT), a generalization of CVT that minimizes a higher-order moment of the coordinates on the Voronoi cells. This generalization allows for aligning the axes of the Voronoi cells with a ...
Hyperbolic centroidal Voronoi tessellation
SPM '10: Proceedings of the 14th ACM Symposium on Solid and Physical Modeling

The centroidal Voronoi tessellation (CVT) has found versatile applications in geometric modeling, computer graphics, and visualization. In this paper, we extend the concept of the CVT from Euclidean space to hyperbolic space. A novel hyperbolic CVT ...

Comments

Information & Contributors

Information

Published In

cover image ACM Transactions on Graphics

ACM Transactions on Graphics Volume 29, Issue 4

July 2010

942 pages

ISSN:0730-0301

EISSN:1557-7368

DOI:10.1145/1778765

Issue’s Table of Contents

Copyright © 2010 ACM.

Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. Copyrights for components of this work owned by others than ACM must be honored. Abstracting with credit is permitted. To copy otherwise, or republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee. Request permissions from [email protected]

Publisher

Association for Computing Machinery

New York, NY, United States

Publication History

Published: 26 July 2010

Published in TOG Volume 29, Issue 4

Permissions

Request permissions for this article.

Request Permissions

Check for updates

Author Tags

Qualifiers

Research-article

Funding Sources

European Research Council

Contributors

Other Metrics

View Article Metrics

Bibliometrics & Citations

Bibliometrics

Article Metrics

166
Total Citations
View Citations
3,382
Total Downloads

Downloads (Last 12 months)128
Downloads (Last 6 weeks)25

Reflects downloads up to 28 Dec 2024

Other Metrics

View Author Metrics

Citations

Cited By

Xu RLiu LWang NChen SXin SGuo XZhong ZKomura TWang WTu C(2024)CWF: Consolidating Weak Features in High-quality Mesh SimplificationACM Transactions on Graphics10.1145/365815943:4(1-14)Online publication date: 19-Jul-2024
https://dl.acm.org/doi/10.1145/3658159
Hu JWang SHe YLuo ZLei NLiu L(2024)A Parametric Design Method for Engraving Patterns on Thin ShellsIEEE Transactions on Visualization and Computer Graphics10.1109/TVCG.2023.324050330:7(3719-3730)Online publication date: 1-Jul-2024
https://dl.acm.org/doi/10.1109/TVCG.2023.3240503
Zhang WYue YPan HChen ZWang CPfister HWang W(2024)Marching Windows: Scalable Mesh Generation for Volumetric Data With Multiple MaterialsIEEE Transactions on Visualization and Computer Graphics10.1109/TVCG.2022.322552630:3(1728-1742)Online publication date: Mar-2024
https://doi.org/10.1109/TVCG.2022.3225526
Prouvost LBelme AFuster D(2024)A metric-based adaptive mesh refinement criterion under constrain for solving elliptic problems on quad/octree gridsJournal of Computational Physics10.1016/j.jcp.2024.112941506(112941)Online publication date: Jul-2024
https://doi.org/10.1016/j.jcp.2024.112941
Després B(2024)Lagrangian Voronoï meshes and particle dynamics with shocksComputer Methods in Applied Mechanics and Engineering10.1016/j.cma.2023.116427418(116427)Online publication date: Jan-2024
https://doi.org/10.1016/j.cma.2023.116427
Dai JWang YYan D(2024)Feature-preserving shrink wrapping with adaptive alphaComputer Aided Geometric Design10.1016/j.cagd.2024.102321111(102321)Online publication date: Jul-2024
https://doi.org/10.1016/j.cagd.2024.102321
Dai YSu JFu X(2024)Anisotropic triangular meshing using metric-adapted embeddingsComputer Aided Geometric Design10.1016/j.cagd.2024.102314111(102314)Online publication date: Jul-2024
https://doi.org/10.1016/j.cagd.2024.102314
Wang SYang BBao HHuang J(2024)Alternating size field optimizing and parameterization domain CAD model remeshingComputer Aided Geometric Design10.1016/j.cagd.2024.102294111(102294)Online publication date: Jul-2024
https://doi.org/10.1016/j.cagd.2024.102294
Weng YCao JChen Z(2024)Global optimization of optimal Delaunay triangulation with modified whale optimization algorithmEngineering with Computers10.1007/s00366-023-01928-240:4(2595-2616)Online publication date: 1-Aug-2024
https://dl.acm.org/doi/10.1007/s00366-023-01928-2
Gao JXiao ZXu CShen SXu G(2024)Quadrilateral surface mesh generation with improved quality by combination of trianglesInternational Journal for Numerical Methods in Engineering10.1002/nme.7539125:16Online publication date: 27-May-2024
https://doi.org/10.1002/nme.7539
Show More Cited By

View Options

Login options

Check if you have access through your login credentials or your institution to get full access on this article.

Full Access

Get this Article

View options

PDF

View or Download as a PDF file.

eReader

View online with eReader.

Media

Figures

Other

Tables

View Issue’s Table of Contents