Abstract
Computing centroidal Voronoi tessellations (CVT) has many applications in computer graphics. The existing methods, such as the Lloyd algorithm and the quasi-Newton solver, are efficient and easy to implement; however, they compute only the local optimal solutions due to the highly non-linear nature of the CVT energy. This paper presents a novel method, called manifold differential evolution (MDE), for computing globally optimal geodesic CVT energy on triangle meshes. Formulating the mutation operator using discrete geodesics, MDE naturally extends the powerful differential evolution framework from Euclidean spaces to manifold domains. Under mild assumptions, we show that MDE has a provable probabilistic convergence to the global optimum. Experiments on a wide range of 3D models show that MDE consistently out-performs the existing methods by producing results with lower energy. Thanks to its intrinsic and global nature, MDE is insensitive to initialization and mesh tessellation. Moreover, it is able to handle multiply-connected Voronoi cells, which are challenging to the existing geodesic CVT methods.
Supplemental Material
Available for Download
Supplemental file.
- Alliez, P., de Verdière, É. C., Devillers, O., and Isenburg, M. 2003. Isotropic surface remeshing. In Shape Modeling International, 49--58. Google ScholarDigital Library
- Alliez, P., de Verdière, É. C., Devillers, O., and Isenburg, M. 2005. Centroidal voronoi diagrams for isotropic surface remeshing. Graphical Models 67, 3, 204--231. Google ScholarDigital Library
- Berntsen, J., and Espelid, T. O. 1992. Algorithm 706: Dcutri: An algorithm for adaptive cubature over a collection of triangles. ACM Trans. Math. Softw. 18, 3, 329--342. Google ScholarDigital Library
- Boender, C. G. E., and Romeijn, H. E. 1995. Stochastic methods. In Handbook of Global Optimization, Kluwer Academic Publishers, 829--869.Google Scholar
- Cheng, P., Miao, C., Liu, Y.-J., Tu, C., and He, Y. 2016. Solving the initial value problem of discrete geodesics. Computer-Aided Design 70, 144--152. Google ScholarDigital Library
- Crane, K., Weischedel, C., and Wardetzky, M. 2013. Geodesics in heat: A new approach to computing distance based on heat flow. ACM Trans. Graph. 32, 5, 152:1--152:11. Google ScholarDigital Library
- Das, S., and Suganthan, P. N. 2011. Differential evolution: A survey of the state-of-the-art. IEEE Transactions on Evolutionary Computation 15, 1, 4--31. Google ScholarDigital Library
- Das, S., Mullick, S. S., and Suganthan, P. N. 2016. Recent advances in differential evolution - an updated survey. Swarm and Evolutionary Computation 27, 1--30.Google ScholarCross Ref
- Du, Q., Faber, V., and Gunzburger, M. 1999. Centroidal Voronoi tessellations: Applications and algorithms. SIAM Review 41, 4, 637--676. Google ScholarDigital Library
- Du, Q., Gunzburger, M. D., and Ju, L. 2003. Constrained centroidal Voronoi tessellations for surfaces. SIAM Journal on Scientific Computing 24, 5, 1488--1506. Google ScholarDigital Library
- Engelbrecht, A. P. 2006. Fundamentals of Computational Swarm Intelligence. John Wiley & Sons. Google ScholarDigital Library
- Ghosh, S., Das, S., Vasilakos, A. V., and Suresh, K. 2012. On convergence of differential evolution over a class of continuous functions with unique global optimum. IEEE Transactions on Systems, Man, and Cybernetics, Part B: Cybernetics 42, 1, 107--124. Google ScholarDigital Library
- Horst, R., Pardalos, P. M., and Thoai, N. V. 2000. Introduction to Global Optimization. 2nd edition, Kluwer Academic Publishers.Google Scholar
- Kimmel, R., and Sethian, J. 1998. Computing geodesic paths on manifolds. Proceedings of National Academy of Sciences 95, 8431--8435.Google ScholarCross Ref
- Korte, B., and Vygen, J. 2006. Combinatorial Optimization: Theory and Algorithms. 3rd edition, Springer-Verlag.Google ScholarDigital Library
- Leung, Y.-S., Wang, X., He, Y., Liu, Y.-J., and Wang, C. C. L. 2015. A unified framework for isotropic meshing based on narrow-banded euclidean distance transformation. Computational Visual Media 1, 3, 239--251.Google ScholarCross Ref
- Lévy, B., and Liu, Y. 2010. Lp centroidal voronoi tessellation and its applications. ACM Trans. Graph. 29, 4, 119:1--119:11. Google ScholarDigital Library
- 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, 101:1--101:17. Google ScholarDigital Library
- Liu, Y.-J., Chen, Z., and Tang, K. 2011. Construction of iso-contours, bisectors, and Voronoi diagrams on triangulated surfaces. IEEE Transactions on Pattern Analysis and Machine Intelligence 33, 8, 1502--1517. Google ScholarDigital Library
- Liu, Y.-J. 2015. Semi-continuity of skeletons in 2-manifold and discrete voronoi approximation. IEEE Transactions on Pattern Analysis and Machine Intelligence 37, 9, 1938--1944.Google ScholarCross Ref
- Lloyd, S. 1982. Least squares quantization in PCM. IEEE Transactions on Information Theory 28, 2, 129 -- 137. Google ScholarDigital Library
- Lu, L., Sun, F., Pan, H., and Wang, W. 2012. Global optimization of centroidal Voronoi tessellation with monte carlo approach. IEEE Transactions on Visualization and Computer Graphics 18, 11, 1880--1890. Google ScholarDigital Library
- Mitchell, J. S., Mount, D. M., and Papadimitriou, C. H. 1987. The discrete geodesic problem. SIAM Journal on Computing 16, 4, 647--668. Google ScholarDigital Library
- Papoulis, A. 1991. Probability, Random Variables and Stochastic Processes. Third Edition, McGraw-Hill, Inc.Google Scholar
- Price, K., Storn, R. M., and Lampinen, J. A. 2005. Differential Evolution: A Practical Approach to Global Optimization. Springer. Google ScholarDigital Library
- Qu, B., Liang, J., Wang, Z., Chen, Q., and Suganthan, P. 2016. Novel benchmark functions for continuous multimodal optimization with comparative results. Swarm and Evolutionary Computation 26, 23--34.Google ScholarCross Ref
- Rong, G., Jin, M., Shuai, L., and Guo, X. 2011. Centroidal voronoi tessellation in universal covering space of manifold surfaces. Comput. Aided Geom. Des. 28, 8, 475--496. Google ScholarDigital Library
- Rong, G., Liu, Y., Wang, W., Yin, X., Gu, X., and Guo, X. 2011. GPU-assisted computation of centroidal Voronoi tessellation. IEEE Transactions on Visualization and Computer Graphics 17, 3, 345--356. Google ScholarDigital Library
- Vesterstrom, J., and Thomsen, R. 2004. A comparative study of differential evolution, particle swarm optimization, and evolutionary algorithms on numerical benchmark problems. In Proc. 6th Congress on Evolutionary Computation, 1980--1987.Google Scholar
- Wandzurat, S., and Xiao, H. 2003. Symmetric quadrature rules on a triangle. Computers & Mathematics with Applications 45, 12, 1829--1840.Google ScholarCross Ref
- Wang, X., Ying, X., Liu, Y.-J., Xin, S.-Q., Wang, W., Gu, X., Mueller-Wittig, W., and He, Y. 2015. Intrinsic computation of centroidal Voronoi tessellation (cvt) on meshes. Computer-Aided Design 58, 51--61. Google ScholarDigital Library
- Xu, C., Wang, T. Y., Liu, Y., Liu, L., and He, Y. 2015. Fast wavefront propagation (FWP) for computing exact geodesic distances on meshes. IEEE Trans. Vis. Comput. Graph. 21, 7, 822--834.Google ScholarDigital Library
- Yan, D., and Wonka, P. 2016. Non-obtuse remeshing with centroidal Voronoi tessellation. IEEE Trans. Vis. Comput. Graph. 22, 9, 2136--2144.Google ScholarCross Ref
- Yan, D.-M., Lévy, B., Liu, Y., Sun, F., and Wang, W. 2009. Isotropic remeshing with fast and exact computation of restricted Voronoi diagram. Comput. Graph. Forum 28, 5, 1445--1454. Google ScholarDigital Library
- Yan, D., Bao, G., Zhang, X., and Wonka, P. 2014. Low-resolution remeshing using the localized restricted voronoi diagram. IEEE Trans. Vis. Comput. Graph. 20, 10, 1418--1427.Google ScholarCross Ref
- Ying, X., Wang, X., and He, Y. 2013. Saddle vertex graph (svg): A novel solution to the discrete geodesic problem. ACM Trans. Graph. 32, 6, 170:1--170:12. Google ScholarDigital Library
- Ying, X., Xin, S., and He, Y. 2014. Parallel Chen-Han (PCH) algorithm for discrete geodesics. ACM Trans. Graph. 33, 1, 9:1--9:11. Google ScholarDigital Library
- Zaharie, D. 2001. On the explorative power of differential evolution algorithms. In Proc. 3rd Int. Workshop Symbolic and Numerical Algorithms on Scientific Computing.Google Scholar
Index Terms
- Manifold differential evolution (MDE): a global optimization method for geodesic centroidal voronoi tessellations on meshes
Recommendations
Delaunay mesh simplification with differential evolution
Delaunay meshes (DM) are a special type of manifold triangle meshes --- where the local Delaunay condition holds everywhere --- and find important applications in digital geometry processing. This paper addresses the general DM simplification problem: ...
Constructing Intrinsic Delaunay Triangulations from the Dual of Geodesic Voronoi Diagrams
Intrinsic Delaunay triangulation (IDT) naturally generalizes Delaunay triangulation from R2 to curved surfaces. Due to many favorable properties, the IDT whose vertex set includes all mesh vertices is of particular interest in polygonal mesh processing. ...
Lp Centroidal Voronoi Tessellation and its applications
This paper introduces Lp-Centroidal Voronoi Tessellation (Lp-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