SUBMITTED TO IEEE TVCG 1 Optimal Surface parameterization Using Inverse Curvature Map

Abstract — Mesh parameterization is a fundamental technique in computer graphics. The major goals during mesh param-eterization are to minimize both the angle distortion and the area distortion. Angle distortion can be eliminated by use of conformal mapping, in principle. 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 formula to represent area distortion by curvature change in the discrete conformal mapping, giving a dynamic Poisson equation. On a mesh, the vertex curvature is related to edge lengths by the curvature map. Our result shows the map is invertible, i.e. the edge lengths can be uniquely determined from prescribed curvatures under discrete conformal mappings. Furthermore, we give the explicit Jacobi matrix of the inverse curvature map. Secondly, we formulate the task of computing conformal param-eterizations 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; it has a natural physical interpretation. This result proves the common belief that optimal parameteri-zation with least area distortion has a unique solution and can be achieved by free boundary conformal mapping. Major theoretical results are deduced using symbolic computa-tion, for which proofs are given as Maple source code. Practical algorithms are presented for optimal parameterization based on the inverse curvature map. Comparisons are conducted with existing methods and using different energies. Novel pa-rameterization applications are also introduced. The theoretical framework of the inverse curvature map can be applied to further study discrete conformal mappings