The Community for Technology Leaders
RSS Icon
Subscribe
Issue No.11 - Nov. (2012 vol.18)
pp: 1880-1890
Lin Lu , Sch. of Comput. Sci. & Technol., Shandong Univ., Jinan, China
Feng Sun , Dept. of Comput. Sci., Univ. of Hong Kong, Hong Kong, China
Hao Pan , Dept. of Comput. Sci., Univ. of Hong Kong, Hong Kong, China
Wenping Wang , Dept. of Comput. Sci., Univ. of Hong Kong, Hong Kong, China
ABSTRACT
Centroidal Voronoi Tessellation (CVT) is a widely used geometric structure in applications including mesh generation, vector quantization and image processing. Global optimization of the CVT function is important in these applications. With numerical evidences, we show that the CVT function is highly nonconvex and has many local minima and therefore the global optimization of the CVT function is nontrivial. We apply the method of Monte Carlo with Minimization (MCM) to optimizing the CVT function globally and demonstrate its efficacy in producing much improved results compared with two other global optimization methods.
INDEX TERMS
vector quantisation, computational geometry, mesh generation, Monte Carlo methods, optimisation, MCM, global optimization, centroidal Voronoi tessellation, Monte Carlo approach, CVT function, geometric structure, mesh generation, vector quantization, image processing, local minima, Monte Carlo with minimization, Monte Carlo methods, Minimization, Density functional theory, Vectors, Mesh generation, Optimization methods, Monte Carlo with minimization, Centroidal Voronoi tessellation, global optimization
CITATION
Lin Lu, Feng Sun, Hao Pan, Wenping Wang, "Global Optimization of Centroidal Voronoi Tessellation with Monte Carlo Approach", IEEE Transactions on Visualization & Computer Graphics, vol.18, no. 11, pp. 1880-1890, Nov. 2012, doi:10.1109/TVCG.2012.28
REFERENCES
[1] S.P. Lloyd, "Least Squares Quantization in PCM," IEEE Trans. Information Theory, vol. TIT-28, no. 2, pp. 129-136, Mar. 1982.
[2] J.B. Macqueen, "Some Methods for Classification and Analysis of Multivariate Observations," Proc. Fifth Berkeley Symp. Math., Statistics, and Probability, pp. 281-297, 1967.
[3] Y. Liu, W. Wang, B. Lévy, F. Sun, D.-M. Yan, L. Lu, and C. Yang, "On Centroidal Voronoi Tessellation—Energy Smoothness and Fast Computation," ACM Trans. Graphic, vol. 28, no. 4, pp. 1-17, 2009.
[4] Q. Du, V. Faber, and M. Gunzburger, "Centroidal Voronoi Tessellations: Applications and Algorithms," SIAM Rev., vol. 41, no. 4, pp. 637-676, http://link.aip.org/link/?SIR/41/6371, 1999.
[5] Q. Du, M. Gunzburger, and L. Ju, "Advances in Studies and Applications of Centroidal Voronoi Tessellations," Numerical Math.: Theory, Methods and Applications, vol. 3, no. 2, pp. 119-142, 2010.
[6] A. Gersho, "Asymptotically Optimal Block Quantization," IEEE Trans. Information Theory, vol. TIT-25, no. 4, pp. 373-380, July 1979.
[7] Z. Li and H.A. Scheraga, "Monte Carlo-Minimization Approach to the Multiple-Minima Problem in Protein Folding," Proc. Nat'l Academy of Sciences of USA, vol. 84, no. 19, pp. 6611-6615, Oct. 1987.
[8] G.F. Tòth, "A Stability Criterion to the Moment Theorem," Studia Scientiarum Mathematicarum Hungarica, vol. 34, pp. 209-224, 2001.
[9] Q. Du and D. Wang, "The Optimal Centroidal Voronoi Tessellations and the Gersho's Conjecture in the Three-Dimensional Space," Computers and Math. with Applications, vol. 49, nos. 9/10, pp. 1355-1373, 2005.
[10] R. Gray and D. Neuhoff, "Quantization," IEEE Trans. Information Theory, vol. 44, no. 6, pp. 2325-2383, Oct. 1998.
[11] E. Hendrix and B.G. Tóth, Introduction to Nonlinear and Global Optimization. Springer, 2010.
[12] N. Metropolis and S. Ulam, "The Monte Carlo method," J. Am. Statistical Assoc., vol. 44, no. 247, pp. 335-341, 1949.
[13] J. Kennedy and R. Eberhart, "Particle Swarm Optimization," Proc. IEEE Int'l Conf. Neural Networks, pp. 1942-1948, 1995.
[14] Handbook of Evolutionary Computation, T. Baeck, D.B. Fogel, and Z. Michalewicz, eds. IOP Publishing Ltd., 1997.
[15] A. Nayeem, J. Vila, and H.A. Scheraga, "A Comparative Study of the Simulated-Annealing and Monte Carlo-with-Minimization Approaches to the Minimum-Energy Structures of Polypeptides: (met)-Enkephalin," J. Computational Chemistry, vol. 12, no. 5, pp. 594-605, 1991.
[16] S.B. Ozkan and H. Meirovitch, "Conformational Search of Peptides and Proteins: Monte Carlo Minimization with an Adaptive Bias Method Applied to the Heptapeptide Deltorphin," J. Computational Chemistry, vol. 25, no. 4, pp. 565-572, 2004.
[17] N. Metropolis, A.W. Rosenbluth, M.N. Rosenbluth, A.H. Teller, and E. Teller, "Equation of State Calculations by Fast Computing Machines," J. Chemical Physics, vol. 21, pp. 1087-1092, http://dx.doi.org/10.10631.1699114, June 1953.
[18] S. Kirkpatrick, C.D. Gelatt, and M.P. Vecchi, "Optimization by Simulated Annealing," Science, vol. 220, no. 4598, pp. 671-680, http://citeseerx.ist.psu.edu/viewdocsummary?doi=10.1.1. 18.4175 , 1983.
[19] R. Horst, P.M. Pardalos, and N.V. Thoai, "Nonconvex Optimization and Its Applications," Introduction to Global Optimization, second ed., vol. 48, Kluwer Academic Publishers, http://www.ams.orgmathscinet-getitem?mr=1799654 , 2000.
[20] P. Pardalos and E. Romeijn, Handbook of Global Optimization, vol. 2, Kluwer Academic Publishers, 2002.
[21] S. Kirkpatrick, "Optimization by Simulated Annealing: Quantitative Studies," J. Statistical Physics, vol. 34, no. 5, pp. 975-986, http://www.springerlink.com/contentR8316332T1U15773 , 1984.
[22] T. Weise, Global Optimization Algorithms—Theory and Application, second ed., http:/www.it-weise.de/, June 2009.
[23] J.A. Snyman and L.P. Fatti, "A Multi-Start Global Minimization Algorithm with Dynamic Search Trajectories," J. Optimization Theory and Applications, vol. 54, pp. 121-141, http://dx.doi.org/10.1007BF00940408, 1987, doi: 10.1007/BF00940408.
[24] R.W. Eglese, "Simulated Annealing: A Tool for Operational Research," European J. Operational Research, vol. 46, no. 3, pp. 271-281, http://ideas.repec.org/a/eee/ejoresv46y1990i3p271-281. html , June 1990.
[25] Q. Du and M. Gunzburger, "Grid Generation and Optimization Based on Centroidal Voronoi Tessellations," Applied Math. Computation, vol. 133, nos. 2/3, pp. 591-607, 2002.
[26] Q. Du, M.D. Gunzburger, and L. Ju, "Constrained Centroidal Voronoi Tessellations for Surfaces," SIAM J. Scientific Computing, vol. 24, pp. 1488-1506, http://portal.acm.orgcitation. cfm?id=767408.767429 , May 2002.
[27] D.-M. Yan, B. Lévy, Y. Liu, F. Sun, and W. Wang, "Isotropic Remeshing with Fast and Exact Computation of Restricted Voronoi Diagram," Computer Graphics Forum, vol. 28, no. 5, pp. 1445-1454, 2009.
45 ms
(Ver 2.0)

Marketing Automation Platform Marketing Automation Tool