research-article

Optimal voronoi tessellations with hessian-based anisotropy

Authors:

Max Budninskiy,

Fernando de Goes,

Mathieu DesbrunAuthors Info & Claims

ACM Transactions on Graphics (TOG), Volume 35, Issue 6

Article No.: 242, Pages 1 - 12

https://doi.org/10.1145/2980179.2980245

Published: 05 December 2016 Publication History

Abstract

This paper presents a variational method to generate cell complexes with local anisotropy conforming to the Hessian of any given convex function and for any given local mesh density. Our formulation builds upon approximation theory to offer an anisotropic extension of Centroidal Voronoi Tessellations which can be seen as a dual form of Optimal Delaunay Triangulation. We thus refer to the resulting anisotropic polytopal meshes as Optimal Voronoi Tessellations. Our approach sharply contrasts with previous anisotropic versions of Voronoi diagrams as it employs first-type Bregman diagrams, a generalization of power diagrams where sites are augmented with not only a scalar-valued weight but also a vector-valued shift. As such, our OVT meshes contain only convex cells with straight edges, and admit an embedded dual triangulation that is combinatorially-regular. We show the effectiveness of our technique using off-the-shelf computational geometry libraries.

Supplementary Material

ZIP File (a242-budninskiy.zip)

Supplemental file.

Download
172.56 KB

References

[1]

Alliez, P., Cohen-Steiner, D., Yvinec, M., and Desbrun, M. 2005. Variational tetrahedral meshing. In ACM SIGGRAPH, vol. 24(3), 617--625.

Digital Library

[2]

App, A., and Reif, U. 2010. Piecewise linear orthogonal approximation. SIAM J. Num. Anal. 48, 3, 840--856.

Digital Library

[3]

Aurenhammer, F. 1987. A criterion for the affine equivalence of cell complexes in R<sup>d</sup> and convex polyhedra in R<sup>d+1</sup>. Disc. & Comput. Geom. 2, 1, 49--64.

Digital Library

[4]

Boissonnat, J.-D., Cohen-Steiner, D., and Yvinec, M. 2006. Comparison of algorithms for anisotropic meshing and adaptive refinement. Tech. Rep. ACS-TR-362603, INRIA.

[5]

Boissonnat, J.-D., Wormser, C., and Yvinec, M. 2008. Anisotropic diagrams: Labelle-Shewchuk approach revisited. Theor. Comput. Sci. 408, 2--3, 163--173.

Digital Library

[6]

Boissonnat, J.-D., Nielsen, F., and Nock, R. 2010. Bregman Voronoi diagrams. Disc. & Comput. Geom. 44, 2, 281--307.

Digital Library

[7]

Boissonnat, J.-D., Shi, K.-L., Tournois, J., and Yvinec, M. 2014. Anisotropic Delaunay meshes of surfaces. ACM Trans. Graph.

Digital Library

[8]

Bossen, Frank, J., and Heckbert, P. S. 1996. A pliant method for anisotropic mesh generation. In Int. Meshing Roundtable, 63--76.

[9]

CGAL. 2016. CGAL 4.8 User and Reference Manual. CGAL Editorial Board, http://www.cgal.org.

[10]

Chen, L., and Holst, M. J. 2011. Efficient mesh optimization schemes based on Optimal Delaunay Triangulations. Comput. Methods Appl. Mech. Engrg. 200, 967--984.

[11]

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

[12]

Chen, L., Sun, P., and Xu, J. 2007. Optimal anisotropic simplicial meshes for minimizing interpolation errors in L<sup>p</sup>-norm. Math. of Comput. 76, 257, 179--204.

[13]

Chen, Z., Wang, W., Lvy, B., Liu, L., and Sun, F. 2014. Revisiting Optimal Delaunay Triangulation for 3D graded mesh generation. SIAM J. Sci. Comput. 36, 3, 930--954.

[14]

Chen, L. 2004. Mesh smoothing schemes based on Optimal Delaunay Triangulations. In Int. Meshing Roundtable, 109--120.

[15]

Cheng, S.-W., Dey, T. K., Ramos, E. A., and Wenger, R. 2006. Anisotropic surface meshing. In Symp. Disc. Alg., 202--211.

Digital Library

[16]

Dassi, F., Si, H., Perotto, S., and Streckenbach, T. 2015. Anisotropic finite element mesh adaptation via higher dimensional embedding. Procedia Engineering 124, 265--277.

[17]

D'Azevedo, E. F., and Simpson, R. B. 1989. On optimal interpolation triangle incidences. SIAM J. Sci. Stat. Comput. 10, 6, 1063--1075.

Digital Library

[18]

de Goes, F., Breeden, K., Ostromoukhov, V., and Desbrun, M. 2012. Blue noise through optimal transport. ACM Trans. Graph. 31, 6, Art. 171.

Digital Library

[19]

de Goes, F., Alliez, P., Owhadi, H., and Desbrun, M. 2013. On the equilibrium of simplicial masonry structures. ACM Trans. Graph. 32, 4, Art. 93.

Digital Library

[20]

de Goes, F., Wallez, C., Huang, J., Pavlov, D., and Desbrun, M. 2015. Power particles: An incompressible fluid solver based on power diagrams. ACM Trans. Graph. 34, 4, Art. 50.

Digital Library

[21]

Du, Q., and Emelianenko, M. 2006. Acceleration schemes for computing centroidal Voronoi tessellations. Numer. Linear Algebr. 13, 2--3, 173--192.

[22]

Du, Q., and Wang, D. 2005. Anisotropic centroidal Voronoi tessellations and their applications. SIAM J. Sci. Comp. 26, 3, 737--761.

Digital Library

[23]

Du, Q., Faber, V., and Gunzburger, M. 1999. Centroidal Voronoi tesselations. SIAM Review 41, 4, 637--676.

Digital Library

[24]

Frey, P., and Alauzet, F. 2004. Anisotropic metrics for mesh adaptation. In Comput. Fluid Solid Mech., K. Bathe, Ed., 24--28.

[25]

Fu, X.-M., Liu, Y., Snyder, J., and Guo, B. 2014. Anisotropic simplicial meshing using local convex functions. ACM Trans. Graph. 33, 6, Art. 182.

Digital Library

[26]

Gao, Z., Yu, Z., and Holst, M. 2012. Quality tetrahedral mesh smoothing via boundary-optimized Delaunay triangulation. Comput. Aided Geom. Design 29, 9, 707--721.

Digital Library

[27]

George, P., Frey, P., and Alauzet, F. 2002. Automatic generation of 3D adapted meshes. World Congress on Comput. Mech.

[28]

Kuzmin, D. 2010. A Guide to Numerical Methods for Transport Equations. University of Erlangen-Nuremberg.

[29]

Labelle, F., and Shewchuk, J. R. 2003. Anisotropic Voronoi diagrams and guaranteed-quality anisotropic mesh generation. In Symp. Comp. Geom., 191--200.

Digital Library

[30]

Lee, C. 1991. Regular triangulations of convex polytopes. Applied Geom. & Disc. Math. 4, 443--456.

[31]

Lévy, B., and Bonneel, N. 2013. Variational anisotropic surface meshing with Voronoi parallel linear enumeration. In Int. Meshing Roundtable. 349--366.

[32]

Lévy, B., and Liu, Y. 2010. L<sub>p</sub> Centroidal Voronoi Tessellation and its applications. ACM Trans. Graph. 29, 4, Art. 119.

Digital Library

[33]

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 Trans. Graph. 28, 4, Art. 101.

Digital Library

[34]

Liu, Y., Pan, H., Snyder, J., Wang, W., and Guo, B. 2013. Computing self-supporting surfaces by regular triangulation. ACM Trans. Graph. 32, 4, Art. 92.

Digital Library

[35]

Loseille, A., and Alauzet, F. 2009. Optimal 3D highly anisotropic mesh adaptation based on the continuous mesh framework. In Int. Meshing Roundtable. 575--594.

[36]

Loseille, A., and Alauzet, F. 2011. Continuous mesh framework, Part I: Well-posed continuous interpolation error. SIAM J. Num. Anal. 49, 1, 38--60.

Digital Library

[37]

Marsden, J. E., and Tromba, A. 2003. Vector Calculus, 5th ed. W. H. Freeman.

[38]

Memari, P., Mullen, P., and Desbrun, M. 2012. Parametrization of generalized primal-dual triangulations. In Int. Meshing Roundtable. 237--253.

[39]

Mirebeau, J.-M., and Cohen, A. 2011. Greedy bisection generates optimally adapted triangulations. Math. Comp. 81, 811--837.

[40]

Mullen, P., Memari, P., de Goes, F., and Desbrun, M. 2011. HOT: Hodge-optimized triangulations. ACM Trans. Graph. 30, 4, Art. 103.

Digital Library

[41]

Nadler, E. 1986. Piecewise linear best L<sub>2</sub> approximation on triangulations. In Approx. Theory V, C. K. Chui, L. L. Schumaker, and J. D. Ward, Eds. Academic Press, 499--502.

[42]

Nielsen, F., and Nock, R. 2011. Skew Jensen-Bregman Voronoi diagrams. In Transactions on Computational Science XIV, Special Issue on Voronoi Diagrams and Delaunay Triangulation, M. L. Gavrilova, C. J. K. Tan, and M. A. Mostafavi, Eds. Springer, 102--128.

Digital Library

[43]

Richter, R., and Alexa, M. 2015. Mahalanobis centroidal Voronoi tessellations. Comp. & Graph. 46, 48--54.

Digital Library

[44]

Rycroft, C. H. 2009. Voro++: A three-dimensional Voronoi cell library in C++. Chaos: An Interdisciplinary Journal of Nonlinear Science 19, 4.

[45]

Shewchuk, J. 1998. Tetrahedral mesh generation by Delaunay refinement. In Symp. on Comp. Geometry, 86--95.

Digital Library

[46]

Sun, F., Choi, Y.-K., Wang, W., Yan, D.-M., Liu, Y., and Lévy, B. 2011. Obtuse triangle suppression in anisotropic meshes. Comput. Aided Geom. Design 28, 9, 537--548.

Digital Library

[47]

Tournois, J., Wormser, C., Alliez, P., and Desbrun, M. 2009. Interleaving Delaunay refinement and optimization for practical isotropic tetrahedron mesh generation. ACM Trans. Graph. 28, 3, Art. 75.

Digital Library

[48]

Valette, S., Chassery, J.-M., and Prost, R. 2008. Generic remeshing of 3D triangular meshes with metric-dependent discrete Voronoi diagrams. IEEE Trans. Vis. Comp. Graph. 14, 2, 369--381.

Digital Library

[49]

Yan, D.-M., Wang, W., Lévy, B., and Liu, Y. 2013. Efficient computation of clipped Voronoi diagram. Computer-Aided Design 45, 4, 843 -- 852.

Digital Library

[50]

Zhong, Z., Guo, X., Wang, W., Lévy, B., Sun, F., Liu, Y., and Mao, W. 2013. Particle-based anisotropic surface meshing. ACM Trans. Graph. 32, 4, Art. 99.

Digital Library

Cited By

Dai YSu JFu X(2024)Anisotropic triangular meshing using metric-adapted embeddingsComputer Aided Geometric Design10.1016/j.cagd.2024.102314(102314)Online publication date: Apr-2024
https://doi.org/10.1016/j.cagd.2024.102314
Birgin ELaurain AMenezes T(2023)Sensitivity analysis and tailored design of minimization diagramsMathematics of Computation10.1090/mcom/383992:344(2715-2768)Online publication date: 12-May-2023
https://doi.org/10.1090/mcom/3839
Xiao YCao JXu SChen Z(2023)Meshless power diagramsComputers & Graphics10.1016/j.cag.2023.06.014114(247-256)Online publication date: Aug-2023
https://doi.org/10.1016/j.cag.2023.06.014
Show More Cited By

Index Terms

Optimal voronoi tessellations with hessian-based anisotropy
1. Computing methodologies
  1. Computer graphics
    1. Shape modeling
2. Mathematics of computing
  1. Mathematical analysis
    1. Numerical analysis
      1. Mesh generation

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

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 ...
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 ...

Comments

Information & Contributors

Information

Published In

cover image ACM Transactions on Graphics

ACM Transactions on Graphics Volume 35, Issue 6

November 2016

1045 pages

ISSN:0730-0301

EISSN:1557-7368

DOI:10.1145/2980179

Issue’s Table of Contents

Copyright © 2016 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 the author(s) 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: 05 December 2016

Published in TOG Volume 35, Issue 6

Permissions

Request permissions for this article.

Request Permissions

Check for updates

Author Tags

Qualifiers

Research-article

Contributors

Other Metrics

View Article Metrics

Bibliometrics & Citations

Bibliometrics

Article Metrics

17
Total Citations
View Citations
397
Total Downloads

Downloads (Last 12 months)54
Downloads (Last 6 weeks)6

Reflects downloads up to 05 Jan 2025

Other Metrics

View Author Metrics

Citations

Cited By

Dai YSu JFu X(2024)Anisotropic triangular meshing using metric-adapted embeddingsComputer Aided Geometric Design10.1016/j.cagd.2024.102314(102314)Online publication date: Apr-2024
https://doi.org/10.1016/j.cagd.2024.102314
Birgin ELaurain AMenezes T(2023)Sensitivity analysis and tailored design of minimization diagramsMathematics of Computation10.1090/mcom/383992:344(2715-2768)Online publication date: 12-May-2023
https://doi.org/10.1090/mcom/3839
Xiao YCao JXu SChen Z(2023)Meshless power diagramsComputers & Graphics10.1016/j.cag.2023.06.014114(247-256)Online publication date: Aug-2023
https://doi.org/10.1016/j.cag.2023.06.014
Khan DPlopski AFujimoto YKanbara MJabeen GZhang YZhang XKato H(2022)Surface Remeshing: A Systematic Literature Review of Methods and Research DirectionsIEEE Transactions on Visualization and Computer Graphics10.1109/TVCG.2020.301664528:3(1680-1713)Online publication date: 1-Mar-2022
https://doi.org/10.1109/TVCG.2020.3016645
Abdelkader ABajaj CEbeida MMahmoud AMitchell SOwens JRushdi A(2020)VoroCrustACM Transactions on Graphics10.1145/333768039:3(1-16)Online publication date: 13-May-2020
https://dl.acm.org/doi/10.1145/3337680
Preiner RBoubekeur TWimmer M(2019)Gaussian-product subdivision surfacesACM Transactions on Graphics10.1145/3306346.332302638:4(1-11)Online publication date: 12-Jul-2019
https://dl.acm.org/doi/10.1145/3306346.3323026
Xu QYan DLi WYang Y(2019)Anisotropic Surface Remeshing without Obtuse AnglesComputer Graphics Forum10.1111/cgf.1387738:7(755-763)Online publication date: 14-Nov-2019
https://doi.org/10.1111/cgf.13877
Garanzha VKudryavtseva LTsvetkova V(2019)Structured Orthogonal Near-Boundary Voronoi Mesh Layers for Planar DomainsNumerical Geometry, Grid Generation and Scientific Computing10.1007/978-3-030-23436-2_2(25-44)Online publication date: 11-Oct-2019
https://doi.org/10.1007/978-3-030-23436-2_2
Portenier THu QSzabó ABigdeli SFavaro PZwicker M(2018)FaceshopACM Transactions on Graphics10.1145/3197517.320139337:4(1-13)Online publication date: 30-Jul-2018
https://dl.acm.org/doi/10.1145/3197517.3201393
Fei YBatty CGrinspun EZheng C(2018)A multi-scale model for simulating liquid-fabric interactionsACM Transactions on Graphics10.1145/3197517.320139237:4(1-16)Online publication date: 30-Jul-2018
https://dl.acm.org/doi/10.1145/3197517.3201392
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