Summary
Centroidal Voronoi tessellations can be constructed using iterative improvement methods such as Lloyd’s method. Using iterative improvement methods implies that the convergence speed and the quality of the results depend on the initialization methods. In this chapter, we briefly describe how to construct centroidal Voronoi tessellations on surface meshes and propose efficient initialization methods. The proposed methods try to make initial tessellations mimic the properties of the centroidal Voronoi tessellations. We compare our methods with other initialization methods: random sampling and farthest point sampling. The experimental results show that our methods have the faster convergence speed than farthest point sampling and outperform random sampling.
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References
Alliez, P., Colin de Verdière, É., Devillers, O., Isenburg, M.: Isotropic surface remeshing. In: Proceedings of Shape Modeling International, pp. 49–58 (2003)
Cohen-Steiner, D., Alliez, P., Desbrun, M.: Variational shape approximation. Proceedings of SIGGRAPH of ACM Transactions on Graphics 23(3), 905–914 (2004)
Du, Q., Faber, V., Gunzburger, M.: Centroidal Voronoi tessellations: applications and algorithms. SIAM Review 41(4), 637–676 (1999)
Du, Q., Gunzburger, M.D., Ju, L.: Constrained centroidal Voronoi tessellations for surfaces. SIAM Journal on Scientific Computing 24(5), 1488–1506 (2003)
Eldar, Y., Lindenbaum, M., Porat, M., Zeevi, Y.Y.: The farthest point strategy for progressive image sampling. IEEE Transactions on Image Processing 6(9), 1305–1315 (1997)
Equitz, W.H.: A new vector quantization clustering algorithm. IEEE Transactions on Acoustics, Speech, and Signal Processing 37(10), 1568–1575 (1989)
Frey, P.J., Borouchaki, H.: Geometric surface mesh optimization. Computing and Visualization in Science 1(3), 113–121 (1998)
Garland, M., Heckbert, P.S.: Surface simplification using quadric error metrics. In: Proceedings of SIGGRAPH, pp. 209–216 (1997)
Garland, M., Willmott, A., Heckbert, P.S.: Hierarchical face clustering on polygonal surfaces. In: Proceedings of Symposium on Interactive 3D Graphics, pp. 49–58 (2001)
Gersho, A., Gray, R.M.: Vector Quantization and Signal Compression. Kluwer Academic Publishers, Dordrecht (1992)
Hoppe, H.: Progressive meshes. In: Proceedings of SIGGRAPH, pp. 99–108 (1996)
Hoppe, H., DeRose, T., Duchamp, T., McDonald, J., Stuetzle, W.: Mesh optimization. In: Proceedings of SIGGRAPH, pp. 19–26 (1993)
Iri, M., Murota, K., Ohya, T.: A fast Voronoi-diagram algorithm with applications to geographical optimization problems. In: Proceedings of the 11th IFIP Conference on System Modelling and Optimization, pp. 273–288 (1984)
Katsavounidis, I., Kuo, C.-C.J., Zhang, Z.: A new initialization technique for generalized Lloyd iteration. IEEE Signal Processing Letters 1(10), 144–146 (1994)
Kim, J., Lee, S.: Transitive mesh space of a progressive mesh. IEEE Transactions on Visualization and Computer Graphics 9(4), 463–480 (2003)
Kurita, T.: An efficient clustering algorithm for region merging. IEICE Trans. on Information and Systems E78-D (12), 1546–1551 (2003)
Lloyd, S.P.: Least squares quantization in PCM. IEEE Transactions on Information Theory 28(2), 129–137 (1982)
MacQueen, J.: Some methods for classification and analysis of multivariate observations. In: Proceedings of the fifth Berkeley symposium on mathematical statistics and probability, pp. 281–297 (1967)
Peyré, G., Cohen, L.: Surface segmentation using geodesic centroidal tesselation. In: Proceedings of 3D Data Processing, Visualization, and Transmission, pp. 995–1002 (2004)
Southern, R., Marais, P., Blake, E.: Generic memoryless polygonal simplification. In: Proceedings of AFRIGRAPH, pp. 7–15 (2001)
Surazhsky, V., Alliez, P., Gotsman, C.: Isotropic remeshing of surfaces: a local parameterization approach. In: Proceedings of 12th International Meshing Roundtable, pp. 215–224 (2003)
Valette, S., Chassery, J.-M.: Approximated centroidal Voronoi diagrams for uniform polygonal mesh coarsening. In: Proceedings of Eurographics, pp. 381–389 (2004)
Ward, J.H.: Hierarchical grouping to optimize an objection function. Journal of the American Statistical Association 58, 236–244 (1963)
Wu, J., Kobbelt, L.: Fast mesh decimation by multiple-choice techniques. In: Proceedings of Vision, Modeling and Visualization, pp. 241–248 (2002)
Wu, J., Kobbelt, L.: Structure recovery via hybrid variational surface approximation. In: Proceedings of Eurographics, pp. 277–284 (2005)
Xia, J.C., Varshney, A.: Dynamic view-dependent simplification for polygonal models. In: Proceedings of Visualization, pp. 327–334 (1996)
Yan, D.-M., Liu, Y., Wang, W.: Quadric surface extraction by variational shape approximation. In: Proceedings of Geometric Modeling and Processing, pp. 73–86 (2006)
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Moriguchi, M., Sugihara, K. (2009). Constructing Centroidal Voronoi Tessellations on Surface Meshes. In: Gavrilova, M.L. (eds) Generalized Voronoi Diagram: A Geometry-Based Approach to Computational Intelligence. Studies in Computational Intelligence, vol 158. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-85126-4_10
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DOI: https://doi.org/10.1007/978-3-540-85126-4_10
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