Abstract
The problem of spherical parametrization is that of map** a genus-zero mesh onto a spherical surface. For a given mesh, different parametrizations can be obtained by different methods. And for a certain application, some parametrization results might behave better than others. In this paper, we will propose a method to parametrize a genus-zero mesh so that a surface fitting algorithm with PHT-splines can generate good result. Here the parametrization results are obtained by minimizing discrete harmonic energy subject to spherical constraints. Then some applications are given to illustrate the advantages of our results. Based on PHT-splines, parametric surfaces can be constructed efficiently and adaptively to fit genus-zero meshes after their spherical parametrization has been obtained.
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References
Deng, J., Chen, F., Feng, Y., 2006. Dimensions of spline spaces over T-meshes. Journal of Computational and Applied Mathematics, 194(2):267–283. [doi:10.1016/j.cam.2005.07.009]
Desbrun, M., Meyer, M., Alliez, P., 2002. Intrinsic parameterizations of surface meshes. Comput. Graph. Forum, 21(3):209–218. [doi:10.1111/1467-8659.00580]
Eck, M., DeRose, T., Duchamp, T., Hoppe, H., Lounsbery, M., Stuetzle, W., 1995. Multiresolution analysis of arbitrary meshes. Computer Graphics, 29:173–182.
Fletcher, R., 1987. Practical Methods of Optimization (Second Ed.). John Wiley & Sons Ltd.
Floater, M.S., 1997. Parametrization and smooth approximation of surface triangulations. Computer Aided Geometric Design, 14(3):231–250. [doi:10.1016/S0167-8396(96)00031-3]
Floater, M.S., 2003. Mean value coordinates. Computer Aided Geometric Design, 20(1):19–27. [doi:10.1016/S0167-8396(03)00002-5]
Floater, M.S., Hormann, K., 2002. Parameterization of Triangulations and Unorganized Points. In: Iske, A., Quak, E., Floater, M.S. (Eds.), Tutorials on Multiresolution in Geometric Modeling. Springer-Verlag, Heidelberg, p.287–316.
Floater, M.S., Hormann, K., 2005. Surface Parameterization: A Tutorial and Survey. In: Dodgson, N.A., Floater, M.S., Sabin, M.A. (Eds.), Advances in Multiresolution for Geometric Modelling. Springer-Verlag, Heidelberg, p.157–186. [doi:10.1007/3-540-26808-1_9]
Gotsman, C., Gu, X., Sheffer, A., 2003. Fundamentals of spherical parameterization for 3D meshes. ACM Trans. Graph., 22(3):358–363. [doi:10.1145/882262.882276]
Gu, X., Yau, S.T., 2003. Global conformal surface parameterization. In: Kobbelt, L., Schröder, P., Hoppe, H. (Eds.), Proceedings of the 2003 Eurographics Symposium on Geometry Processing. Eurographics Association, p.127–137.
Haker, S., Angenent, S., Tannenbaum, A., Kikinis, R., Sapiro, G., Halle, M., 2000. Conformal surface parameterization for texture map**. IEEE Trans. on Visualization and Computer Graphics, 6(2):181–189. [doi:10.1109/2945.856998]
Maillot, J., Yahia, H., Verroust, A., 1993. Interactive Texture Map**. Proceedings of the 20th Annual Conference on Computer Graphics and Interactive Techniques, SIGGRAPH’93, p.27–34. [doi:10.1145/166117.166120]
Praun, E., Hoppe, H., 2003. Spherical parametrization and remeshing. ACM Trans. Graph., 22(3):340–349. [doi:10.1145/8882262.882274]
Sander, P.V., Snyder, J., Gortler, S.J., Hoppe, H., 2001. Texture Map** Progressive Meshes. SIGGRAPH 2001, p.409–416.
Sheffer, A., Sturler, E.D., 2000. Surface Parameterization for Meshing by Triangulation Flattening. Proceedings of the 9th International Meshing Roundtable, Sandia National Laboratories, p.161–172.
Sheffer, A., Gotsman, C., Dyn, N., 2004. Robust spherical parameterization of triangular meshes. Computing, 72(1–2):185–193. [doi:10.1007/s00607-004-0056-9]
Tutte, W.T., 1963. How to Draw a Graph. Proceedings of London Mathematical Society, 13:743–768.
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Project supported by the Outstanding Youth Grant of Natural Science Foundation of China (No. 60225002), the National Basic Research Program (973) of China (No. 2004CB318000), the National Natural Science Foundation of China (Nos. 60533060 and 60473132)
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Li, Y., Yang, Zw. & Deng, Js. Spherical parametrization of genus-zero meshes by minimizing discrete harmonic energy. J. Zhejiang Univ. - Sci. A 7, 1589–1595 (2006). https://doi.org/10.1631/jzus.2006.A1589
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DOI: https://doi.org/10.1631/jzus.2006.A1589