Fast disk conformal parameterization of simply-connected open surfaces. (English) Zbl 1329.65050
Summary: Surface parameterizations have been widely used in computer graphics and geometry processing. In particular, as simply-connected open surfaces are conformally equivalent to the unit disk, it is desirable to compute the disk conformal parameterizations of the surfaces. In this paper, we propose a novel algorithm for the conformal parameterization of a simply-connected open surface onto the unit disk, which significantly speeds up the computation, enhances the conformality and stability, and guarantees the bijectivity. The conformality distortions at the inner region and on the boundary are corrected by two steps, with the aid of an iterative scheme using quasi-conformal theories. Experimental results demonstrate the effectiveness of our proposed method.
MSC:
| 65D17 | Computer-aided design (modeling of curves and surfaces) |
| 53A30 | Conformal differential geometry (MSC2010) |
Keywords:
disk conformal parameterization; simply-connected open surface; Beltrami differential; conformal geometry; quasi-conformal theory; numerical examples; graphical examples; computer graphics; geometry processing; algorithmSoftware:
ABF++References:
| [1] | Choi, P.T., Lam, K.C., Lui, L.M.: FLASH: Fast landmark aligned spherical harmonic parameterization for genus-0 closed brain surfaces. SIAM J. Imaging Sci. 8(1), 67-94 (2015) · Zbl 1316.65028 |
| [2] | Gu, X., Wang, Y., Chan, T.F., Thompson, P., Yau, S.T.: Genus zero surface conformal mapping and its application to brain surface mapping. IEEE Trans. Med. Imag. 23, 949-958 (2004) · doi:10.1109/TMI.2004.831226 |
| [3] | Lui, L.M., Wang, Y., Chan, T.F., Thompson, P.M.: Landmark constrained genus zero surface conformal mapping and its application to brain mapping research. Appl. Numer. Math. 57, 847-858 (2007) · Zbl 1111.92007 · doi:10.1016/j.apnum.2006.07.031 |
| [4] | Lui, L.M., Wong, T.W., Gu, X.F., Thompson, P.M., Chan, T.F., Yau, S.T.: Shape-based diffeomorphic registration on hippocampal surfaces using Beltrami holomorphic flow, medical image computing and computer assisted intervention (MICCAI). Part II, LNCS 6362, 323-330 (2010) |
| [5] | Krichever, I., Mineev-Weinstein, M., Wiegmannd, P., Zabrodin, A.: Laplacian growth and Whitham equations of soliton theory. Phys. D Nonlinear Phenom. 198(12), 1-28 (2004) · Zbl 1078.76025 · doi:10.1016/j.physd.2004.06.003 |
| [6] | Krichever, I., Marshakov, A., Zabrodin, A.: Integrable structure of the Dirichlet boundary problem in multiply-connected domains. Commun. Math. Phys. 259(1), 1-44 (2005) · Zbl 1091.37019 · doi:10.1007/s00220-005-1387-5 |
| [7] | Lui, L.M., Gu, X., Chan, T.F., Yau, S.-T.: Variational method on Riemann surfaces using conformal parameterization and its applications to image processin. J. Methods Appl. Anal. 15(4), 513-538 (2008) · Zbl 1185.68807 |
| [8] | Floater, M.: Parameterization and smooth approximation of surface triangulations. Comput. Aided Geom. Des. 14(3), 231-250 (1997) · Zbl 0906.68162 · doi:10.1016/S0167-8396(96)00031-3 |
| [9] | Hormann, K., Greiner, G.: MIPS: An efficient global parametrization method. In: Curve and Surface Design, 153-162 (2000) |
| [10] | Sheffer, A., de Sturler, E.: Surface parameterization for meshing by triangulation flattening. In: Proceedings of 9th International Meshing Roundtable, pp. 161-172 (2000) · Zbl 1092.14514 |
| [11] | Sheffer, A., Lévy, B., Mogilnitsky, M., Bogomyakov, A.: ABF++: fast and robust angle based flattening. ACM Trans. Gr. 24(2), 311-330 (2005) · doi:10.1145/1061347.1061354 |
| [12] | Desbrun, M., Meyer, M., Alliez, P.: Intrinsic parameterizations of surface meshes. Comput. Gr. Forum 21, 209-218 (2002) · doi:10.1111/1467-8659.00580 |
| [13] | Lévy, B., Petitjean, S., Ray, N., Maillot, J.: Least squares conformal maps for automatic texture atlas generation. In: ACM SIGGRAPH 2002, 362-371 (2002) |
| [14] | Floater, M.: Mean value coordinates. Comput. Aided Geom. Des. 20(1), 19-27 (2003) · Zbl 1069.65553 |
| [15] | Kharevych, L., Springborn, B., Schröder, P.: Discrete conformal mappings via circle patterns. ACM Trans. Gr. 25(2), 412-438 (2006) · doi:10.1145/1138450.1138461 |
| [16] | Mullen, P., Tong, Y., Alliez, P., Desbrun, M.: Spectral conformal parameterization. Comput. Gr. Forum 27(5), 1487-1494 (2008) |
| [17] | Jin, M., Wang, Y., Gu, X., Yau, S.T.: Optimal global conformal surface parameterization for visualization. Commun. Inform. Syst. 4(2), 117-134 (2005) · Zbl 1092.14515 |
| [18] | Jin, M., Kim, J., Luo, F., Gu, X.: Discrete surface Ricci flow. IEEE Trans. Vis. Comput. Gr. 14(5), 1030-1043 (2008) · doi:10.1109/TVCG.2008.57 |
| [19] | Yang, Y.L., Guo, R., Luo, F., Hu, S.M., Gu, X.F.: Generalized discrete Ricci flow. Comput. Gr. Forum 28(7), 2005-2014 (2009) · doi:10.1111/j.1467-8659.2009.01579.x |
| [20] | Luo, F.: Combinatorial Yamabe flow on surfaces. Commun. Contemp. Math. 6, 765-780 (2004) · Zbl 1075.53063 · doi:10.1142/S0219199704001501 |
| [21] | Gu, X., Yau, S.T.: Computing conformal structures of surfaces. Commun. Inform. Syst. 2(2), 121-146 (2002) · Zbl 1092.14514 · doi:10.4310/CIS.2002.v2.n2.a2 |
| [22] | Sheffer, A., Praun, E., Rose, K.: Mesh parameterization methods and their applications. Found. Trends Comput Gr. Vis. 2(2), 105-171 (2006) · Zbl 1112.68130 · doi:10.1561/0600000011 |
| [23] | Hormann, K., Lévy, B., Sheffer, A.: Mesh parameterization: theory and practice. In: SIGGRAPH 2007 Course Notes 2, 1-122 (2007) |
| [24] | Floater, M., Hormann, K.: Parameterization of Triangulations and Unorganized Points. Tutorials on Multiresolution in Geometric Modelling 287-316 (2002) · Zbl 1004.65029 |
| [25] | Floater, M., Hormann, K.: Surface parameterization: a tutorial and survey. In: Advances in Multiresolution for Geometric Modelling 157-186 (2005) · Zbl 1065.65030 |
| [26] | do Carmo, M.: Differential Geometry of Curves and Surfaces. Prentice Hall, Englewood Cliffs (1976) · Zbl 0326.53001 |
| [27] | Zou, G., Hu, J., Gu, X., Hua, J.: Authalic parameterization of general surfaces using lie advection. IEEE Trans. Vis. Comput. Gr. 17(12), 2005-2014 (2011) · Zbl 1219.90081 · doi:10.1109/TVCG.2011.171 |
| [28] | Zhao, X., Su, Z., David Gu, X., Kaufman, A., Sun, J., Gao, J., Luo, F.: Area-preservation mapping using optimal mass transport. IEEE Trans. Vis. Comput. Gr. 19(12), 2838-2847 (2013) · Zbl 1091.37019 |
| [29] | Schoen, R., Yau, S.: Lectures on Harmonic Maps. Harvard University, International Press, Cambridge, MA (1997) · Zbl 0886.53004 |
| [30] | Pinkall, U., Polthier, K.: Computing discrete minimal surfaces and their conjugates. Exp. Math. 2(1), 15-36 (1993) · Zbl 0799.53008 · doi:10.1080/10586458.1993.10504266 |
| [31] | Eck, M., DeRose, T., Duchamp, T., Hoppe, H., Lounsbery, M., Stuetzle, W.: Multiresolution analysis of arbitrary meshes. In: ACM SIGGRAPH 95, 173-182 (1995) |
| [32] | Angenent, S., Haker, S., Tannenbaum, A., Kikinis, R.: Conformal Geometry and Brain Flattening, Medical Image Computing and Computer-Assisted Intervention, 271-278 (1999) |
| [33] | Haker, S., Angenent, S., Tannenbaum, A., Kikinis, R., Sapiro, G., Halle, M.: Conformal surface parameterization for texture mapping. IEEE Trans. Vis. Comput. Gr. 6(2), 181-189 (2000) · doi:10.1109/2945.856998 |
| [34] | Hurdal, M.K., Stephenson, K.: Cortical cartography using the discrete conformal approach of circle packings. Neuroimage 23, S119-S128 (2004) · doi:10.1016/j.neuroimage.2004.07.018 |
| [35] | Lai, R., Wen, Z., Yin, W., Gu, X., Lui, L.M.: Folding-free global conformal mapping for genus-0 surfaces by Harmonic energy minimization. J. Sci. Comput. 58(3), 705-725 (2013) · Zbl 1298.30007 |
| [36] | Gu, X., Yau, ST.: Global conformal surface parameterization. In: ACM Symposium on Geometry Processing, pp. 127-137 (2003) |
| [37] | David Gu, Xianfeng, Yau, Shing-Tung: Computational Conformal Geometry. Advanced Lectures in Mathematics (ALM), vol. 3. International Press, Somerville (2008) · Zbl 1144.65008 |
| [38] | Schoen, R., Yau, S.T.: Lectures on Differential Geometry. International Press, Somerville (1994) · Zbl 0830.53001 |
| [39] | Fletcher, A., Markovic, V.: Quasiconformal Maps and Teichmüller Theory. Oxford University Press, Oxford (2006) · Zbl 1103.30002 |
| [40] | Gardiner, F., Lakic, N.: Quasiconformal Teichmüller Theory. American Mathematics Society, Providence (2000) · Zbl 0949.30002 |
| [41] | Lui, L.M., Lam, K.C., Wong, T.W., Gu, X.: Texture map and video compression using Beltrami representation. SIAM J. Imag. Sci. 6(4), 1880-1902 (2013) · Zbl 1281.65036 · doi:10.1137/120866129 |
| [42] | Lui, L.M., Wong, T.W., Zeng, W., Gu, X.F., Thompson, P.M., Chan, T.F., Yau, S.T.: Optimization of surface registrations using Beltrami holomorphic flow. J. Sci. Comput. 50(3), 557-585 (2012) · Zbl 1244.65096 · doi:10.1007/s10915-011-9506-2 |
| [43] | http://shapes.aim-at-shape.net/ · Zbl 1244.65096 |
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