Surface parameterization has been widely applied in many different fields, include surface registration, remeshing, resampling, and texture mapping, etc. In this paper, we propose a balance energy minimization (BEM) algorithm for the computation of disk-shaped balance parameterization of a simply connected open surface. Some numerical experiments shown in this thesis show that the efficiency compares with the existing state-of-the-art algorithm. In addition, applications of the BEM on the surface, e.g., remeshing and surface registration are presented thereafter. By using the BEM algorithm, choosing the optimal parameterization between angle- and area-preserving is easier.

[1] Pierre Alliez, Giuliana Ucelli, Craig Gotsman, and Marco Attene. Recent advances in remeshing of surfaces. Shape analysis and structuring, pages 53–82, 2008.
[2] Sigurd Angenent, Steven Haker, Allen Tannenbaum, and Ron Kikinis. On the laplace-beltrami operator and brain surface flattening. IEEE Transactions on Medical Imaging, 18(8):700–711, 1999.
[3] Michel A Audette, Frank P Ferrie, and Terry M Peters. An algorithmic overview of surface registration techniques for medical imaging. Medical image analysis, 4(3):201–217, 2000.
[4] Daniel Carroll, Eleanor Hankins, Emek Kose, and Ivan Sterling. A survey of the differential geometry of discrete curves. The Mathematical Intelligencer, 36(4):28–35, 2014.
[5] Keenan Crane. Discrete differential geometry: An applied introduction. Notices of the AMS, Communication, pages 1153–1159, 2018.
[6] Patrick Degener, Jan Meseth, and Reinhard Klein. An adaptable surface parameterization method. IMR, 3:201–213, 2003.
[7] Ayelet Dominitz and Allen Tannenbaum. Texture mapping via optimal mass transport. IEEE transactions on visualization and computer graphics, 16(3):419–433, 2009.
[8] Michael S Floater and Kai Hormann. Surface parameterization: a tutorial and survey. Advances in multiresolution for geometric modelling, pages 157–186, 2005.
[9] Gene H Golub and Charles F Van Loan. Matrix computations, volume 3. JHU press, 2013.
[10] Xianfeng Gu, Feng Luo, Jian Sun, and S-T Yau. Variational principles for minkowski type problems, discrete optimal transport, and discrete monge-ampere equations. arXiv preprint arXiv:1302.5472, 2013.
[11] Xianfeng David Gu and Shing-Tung Yau. Computational conformal geometry, volume 1. International Press Somerville, MA, 2008.
[12] Steven Haker, Sigurd Angenent, Allen Tannenbaum, Ron Kikinis, Guillermo Sapiro, and Michael Halle. Conformal surface parameterization for texture mapping. IEEE Transactions on Visualization and Computer Graphics, 6(2):181–189, 2000.
[13] Kai Hormann. Geometry Processing, pages 593–606. 01 2015.
[14] Kai Hormann and Günther Greiner. Mips: An efficient global parametrization method. Technical report, ERLANGENNUERNBERG UNIV (GERMANY) COMPUTER GRAPHICS GROUP, 2000.
[15] Wei-Qiang Huang, Xianfeng David Gu, Tsung-Ming Huang, SongSun Lin, Wen-Wei Lin, and Shing-Tung Yau. High performance computing for spherical conformal and riemann mappings. Geom. Imag. Comput, 1(2):223–258, 2014.
[16] Wei-Qiang Huang, Xianfeng David Gu, Wen-Wei Lin, and Shing-Tung Yau. A novel symmetric skew-hamiltonian isotropic lanczos algorithm for spectral conformal parameterizations. Journal of Scientific Computing, 61(3):558–583, 2014.
[17] Andrew Edie Johnson and Martial Hebert. Surface registration by matching oriented points. In Proceedings. International Conference on Recent Advances in 3-D Digital Imaging and Modeling (Cat. No. 97TB100134), pages 121–128. IEEE, 1997.
[18] Ligang Liu, Lei Zhang, Yin Xu, Craig Gotsman, and Steven J Gortler. A local/global approach to mesh parameterization. In Computer Graphics Forum, volume 27, pages 1495–1504. Wiley Online Library, 2008.
[19] Yong-Jin Liu, Chun-Xu Xu, Dian Fan, and Ying He. Efficient construction and simplification of delaunay meshes. ACM Transactions on Graphics (TOG), 34(6):1–13, 2015.
[20] Mark Meyer, Mathieu Desbrun, Peter Schröder, and Alan H Barr. Discrete differential-geometry operators for triangulated 2-manifolds. In Visualization and mathematics III, pages 35–57. Springer, 2003.
[21] Jason J Molitierno. Applications of combinatorial matrix theory to Laplacian matrices of graphs. CRC Press, 2016.
[22] Patrick Mullen, Yiying Tong, Pierre Alliez, and Mathieu Desbrun. Spectral conformal parameterization. In Computer Graphics Forum, volume 27, pages 1487–1494. Wiley Online Library, 2008.
[23] Saad Nadeem, Zhengyu Su, Wei Zeng, Arie Kaufman, and Xianfeng Gu. Spherical parameterization balancing angle and area distortions. IEEE transactions on visualization and computer graphics, 23(6):1663– 1676, 2016.
[24] Per-Olof Persson and Gilbert Strang. A simple mesh generator in matlab. SIAM review, 46(2):329–345, 2004.
[25] Alla Sheffer and Eric de Sturler. Parameterization of faceted surfaces for meshing using angle-based flattening. Engineering with computers, 17(3):326–337, 2001.
[26] Kehua Su, Li Cui, Kun Qian, Na Lei, Junwei Zhang, Min Zhang, and Xianfeng David Gu. Area-preserving mesh parameterization for polyannulus surfaces based on optimal mass transportation. Computer Aided Geometric Design, 46:76–91, 2016.
[27] Chunxue Wang, Zheng Liu, and Ligang Liu. As-rigid-as-possible spherical parametrization. Graphical models, 76(5):457–467, 2014.
[28] Zhao Wang, Zhong-xuan Luo, Jie-lin Zhang, and Emil Saucan. Arap++: an extension of the local/global approach to mesh parameterization. Frontiers of Information Technology & Electronic Engineering, 17(6):501–515, 2016.
[29] Shin Yoshizawa, Alexander Belyaev, and Hans-Peter Seidel. A fast and simple stretch-minimizing mesh parameterization. In Proceedings Shape Modeling Applications, 2004., pages 200–208. IEEE, 2004.
[30] Mei-Heng Yueh, Hsiao-Han Huang, Tiexiang Li, Wen-Wei Lin, and Shing-Tung Yau. Optimized surface parameterizations with applications to chinese virtual broadcasting. Electron. Trans. Numer. Anal., 53:383–405, 2020.
[31] Mei-Heng Yueh, Tiexiang Li, Wen-Wei Lin, and Shing-Tung Yau. A novel algorithm for volume-preserving parameterizations of 3- manifolds. SIAM Journal on Imaging Sciences, 12(2):1071–1098, 2019.
[32] Mei-Heng Yueh, Wen-Wei Lin, Chin-Tien Wu, and Shing-Tung Yau. An efficient energy minimization for conformal parameterizations. Journal of Scientific Computing, 73(1):203–227, 2017.
[33] Mei-Heng Yueh, Wen-Wei Lin, Chin-Tien Wu, and Shing-Tung Yau. A novel stretch energy minimization algorithm for equiareal parameterizations. Journal of Scientific Computing, 78(3):1353–1386, 2019.
[34] Hui Zhao, Xuan Li, Huabin Ge, Na Lei, Min Zhang, Xiaoling Wang, and Xianfeng Gu. Conformal mesh parameterization using discrete calabi flow. Computer Aided Geometric Design, 63:96–108, 2018.
[35] Xin Zhao, Zhengyu Su, Xianfeng David Gu, Arie Kaufman, Jian Sun, Jie Gao, and Feng Luo. Area-preservation mapping using optimal mass transport. IEEE Transactions on Visualization and Computer Graphics, 19(12):2838–2847, 2013.