Issue No. 05 - September/October (2008 vol. 14)
DOI Bookmark: http://doi.ieeecomputersociety.org/10.1109/TVCG.2008.54
Yong-Liang Yang , Tsinghua University, Beijing
Junho Kim , State University of New York at Stony Brook, Stony Brook and Dong-Eui University, Busan
Feng Luo , Rutgers University, Piscataway
Shi-Min Hu , Tsinghua University, Beijing
Xianfeng Gu , State University of New York at Stony Brook, Stony Brook
Mesh parameterization is a fundamental technique in computer graphics. Our paper focuses on solving the problem of finding the best discrete conformal mapping that also minimizes area distortion. Firstly, we deduce an exact analytical differential formula to represent area distortion by curvature change in the discrete conformal mapping, giving a dynamic Poisson equation. Our result shows the curvature map is invertible. Furthermore, we give the explicit Jacobi matrix of the inverse curvature map. Secondly, we formulate the task of computing conformal parameterizations with least area distortions as a constrained nonlinear optimization problem in curvature space. We deduce explicit conditions for the optima. Thirdly, we give an energy form to measure the area distortions, and show it has a unique global minimum. We use this to design an efficient algorithm, called free boundary curvature diffusion, which is guaranteed to converge to the global minimum. This result proves the common belief that optimal parameterization with least area distortion has a unique solution and can be achieved by free boundary conformal mapping. Major theoretical results and practical algorithms are presented for optimal parameterization based on the inverse curvature map. Comparisons are conducted with existing methods and using different energies. Novel parameterization applications are also introduced.
Computational Geometry and Object Modeling, Optimization, Partial Differential Equations, Discrete Mathematics
S. Hu, X. Gu, F. Luo, J. Kim and Y. Yang, "Optimal Surface Parameterization Using Inverse Curvature Map," in IEEE Transactions on Visualization & Computer Graphics, vol. 14, no. , pp. 1054-1066, 2008.