Abstract：Focusing on the issue that non-zero genus surface parameterization has large deformation and high computational complexity, an improved global parameterization approach based on holomorphic 1-form was proposed, which starts from the gradient field and adapts easier and faster method to compute homology and cohomology group. Firstly, a simplified cut graph method was used to construct homology group to determine the topology. Secondly, cohomology group of the linear space formed by the gradient field was calculated by defining special harmonic function to figure out closed 1-form. Thirdly, homology group was diffused to harmonic 1-form through minimizing harmonic energy. Finally, holomorphic 1-form was computed by combining linearly harmonic 1-form and the parameterization was obtained by integrating holomorphic 1-form on the surface basic domain. Theoretical analysis of homology group and cohomology group shows that the parameterization is a global, border-free conformal mapping. Experimental results based on non-zero genus model show that, compared with the former global parameterization based on holomorphic 1-form, the proposed algorithm has better visual effect, smaller average error and higher operation efficiency.
[1] FLOATER M S. Parametrization and smooth approximation of surface triangulations [J]. Computer Aided Geometric Design, 1997, 14(3): 231-250. [2] ECK M, DEROSE T, DUCHAMP T, et al. Multiresolution analysis of arbitrary meshes [C]// SIGGRAPH '95: Proceedings of the 22nd Annual Conference on Computer Graphics and Interactive Techniques. New York: ACM, 1995: 173-182. [3] DESBRUN M, MEYER M, ALLIEZ P. Intrinsic parameterizations of surface meshes [J]. Computer Graphics Forum, 2002, 21(3): 209-218. [4] SANDER P V, SNYDER J, GORTLER S J, et al. Texture mapping progressive meshes [C]// SIGGRAPH '01: Proceedings of the 28th Annual Conference on Computer Graphics and Interactive Techniques. New York: ACM, 2001: 409-416. [5] JIN Y, QIAN G P, ZHAO J Y, et al. Stretch-minimizing volumetric parameterization [J]. Journal of Computer Science and Technology, 2015, 30(3): 553-564. [6] LÉVY B, MALLET J L. Non-distorted texture mapping for sheared triangulated meshes [C]// SIGGRAPH '98: Proceedings of the 25th Annual Conference on Computer Graphics and Interactive Techniques. New York:ACM, 1998: 343-352. [7] SHEFFER A, DE STURLER E. Parameterization of faceted surfaces for meshing using angle-based flattening [J]. Engineering with Computers, 2001, 17(3): 326-337. [8] HORMANN K, GREINER G. MIPS: an efficient global parametrization method [EB/OL]. [2015-11-23]. http://www.inf.usi.ch/hormann/papers/Hormann.2000.MAE.pdf. [9] GU X, YAU S T. Global conformal surface parameterization [C]// SGP '03: Proceedings of the 2003 Eurographics/ACM SIGGRAPH Symposium on Geometry Processing. Aire-la-Ville, Switzerland: Eurographics Association, 2003: 127-137. [10] RAY N, LI W C, LÉVY B, et al. Periodic global parameterization [J]. ACM Transactions on Graphics, 2006, 25(4): 1460-1485. [11] TONG Y, ALLIEZ P, COHEN-STEINER D, et al. Designing quadrangulations with discrete harmonic forms [C]// SGP '06: Proceedings of the 4th Eurographics Symposium on Geometry Processing. Aire-la-Ville, Switzerland: Eurographics Association, 2006: 201-210. [12] ZENG W, YIN X, ZHANG M, et al. Generalized Koebe's method for conformal mapping multiply connected domains [C]// SPM '09: Proceedings of the 2009 SIAM/ACM Joint Conference on Geometric and Physical Modeling. New York: ACM, 2009: 89-100. [13] ZENG W, LUO F, YAU S T, et al. Surface quasi-conformal mapping by solving Beltrami equations [M]// HANCOCK E R, MARTIN R R, SABIN M A. Mathematics of Surfaces XIII, LNCS 5654. Berlin: Springer, 2009: 391-408. [14] RODIN B, SULLIVAN D. The convergence of circle packings to the Riemann mapping [J]. Journal of Differential Geometry, 1987, 26(26): 349-360. [15] KHAREVYCH L, SPRINGBORN B, SCHRÖDER P. Discrete conformal mappings via circle patterns [J]. ACM Transactions on Graphics, 2006, 25(2): 412-438. [16] MYLES A, ZORIN D. Global parametrization by incremental flattening [J]. ACM Transactions on Graphics, 2012, 31(4): Article No. 109. [17] MYLES A, PIETRONI N, ZORIN D. Robust field-aligned global parametrization [J]. ACM Transactions on Graphics, 2014, 33(4): Article No. 135. [18] HOPPE H. View-dependent refinement of progressive meshes [C]// SIGGRAPH '97: Proceedings of the 24th Annual Conference on Computer Graphics and Interactive Techniques. New York: ACM, 1997: 189-198. [19] EPPSTEIN D. Dynamic generators of topologically embedded graphs [C]// SODA '03: Proceedings of the 14th Annual ACM-SIAM Symposium on Discrete Algorithms. Philadelphia, PA: SIAM, 2003: 599-608. [20] 胡国飞,方兴,彭群生.凸组合球面参数化[J].计算机辅助设计与图形学学报,2004,16(5):632-637.(HU G F, FANG X, PENG Q S. Convex combi nation spherical parameterization [J]. Journal of Computer-Aided Design and Computer Graphics, 2004, 16(5): 632-637.)
2016, Vol. 36 
