Optimal Surface Parameterization Using Inverse Curvature Map

Yang, Y. L., Kim, J., Luo, F., Hu, Shi-Min and Gu, X. 2008. Optimal Surface Parameterization Using Inverse Curvature Map. IEEE Transactions on Visualization and Computer Graphics 14 (5) , pp. 1054-1066. 10.1109/TVCG.2008.54

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Abstract

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.

Item Type:	Article
Date Type:	Publication
Status:	Published
Schools:	Computer Science & Informatics
Subjects:	Q Science > QA Mathematics > QA75 Electronic computers. Computer science
Uncontrolled Keywords:	Computational Geometry and Object Modeling, Discrete Mathematics, Optimization, Partial Differential Equations
Publisher:	Institute of Electrical and Electronics Engineers
ISSN:	1077-2626
Last Modified:	19 Mar 2016 23:06
URI:	https://orca.cardiff.ac.uk/id/eprint/38666

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