17/08/2020

Efficient bijective parameterizations

Jian-Ping Su, Chunyang Ye, Ligang Liu, Xiao-Ming Fu

Keywords: coarse shell meshes, triangle inequality-based barriers, bijective parameterizations

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Abstract: We propose a novel method to efficiently compute bijective parameterizations with low distortion on disk topology meshes. Our method relies on a second-order solver. To design an efficient solver, we develop two key techniques. First, we propose a coarse shell to substantially reduce the number of collision constraints that are used to guarantee overlap-free boundaries. During the optimization process, the shell ensures the Hessian matrix with a fixed nonzero structure and a low density, thereby significantly accelerating the optimization. The second is a triangle inequality-based barrier function that effectively ensures non-intersecting boundaries. Our barrier function is C∞ inside the locally supported region and its convex second-order approximation is able to be analytically obtained. Compared to state-of-the-art methods for optimizing bijective parameterizations, our method exhibits better scalability and is about six times faster. The performance of our bijective parameterization algorithm is comparable to state-of-the-art methods of locally flip-free parameterizations. A large number of experimental results have shown the capability and feasibility of our method.

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