摘要
Not many methods for parameterization guarantee bijectivity or local injectivity, which is essential for foldover- free mappings. Stretch-minimizing parameterization which is widely used for surface parameterization, provides foldover-free mappings and is capable of trading off between angle and area distortions. We extend its usage to volumetric parameterization in this paper by deriving a 3D version of stretch-distortion energy and incorporating fixed boundary conditions. Our energy definition includes a naturM barrier term which effectively prevents elements from collapsing and folding over. It saves the effort in other methods of formulating additional energy or constrains to ensure the local injectivity. We propose to minimize the overall energy integrated over the whole mesh with a relaxation-enhanced solver and optimize the energy globally. This is different from the conventional approach of surface parameterization where mesh nodes are optimized individually. Compared with other volumetric parameterizations, method, being foldover-free and offering a good trade-off between our approach bears the advantages of stretch-minimizing angle and volume distortions.
Not many methods for parameterization guarantee bijectivity or local injectivity, which is essential for foldover- free mappings. Stretch-minimizing parameterization which is widely used for surface parameterization, provides foldover-free mappings and is capable of trading off between angle and area distortions. We extend its usage to volumetric parameterization in this paper by deriving a 3D version of stretch-distortion energy and incorporating fixed boundary conditions. Our energy definition includes a naturM barrier term which effectively prevents elements from collapsing and folding over. It saves the effort in other methods of formulating additional energy or constrains to ensure the local injectivity. We propose to minimize the overall energy integrated over the whole mesh with a relaxation-enhanced solver and optimize the energy globally. This is different from the conventional approach of surface parameterization where mesh nodes are optimized individually. Compared with other volumetric parameterizations, method, being foldover-free and offering a good trade-off between our approach bears the advantages of stretch-minimizing angle and volume distortions.
基金
This work was supported by the National Natural Science Foundation of China under Grant No. 61170141, the National High Technology Research and Development 863 Program of China under Grant No. 2013AA013903, the People Programme （Marie Curie Ac- tions） of the European Union＇s Seventh Framework Programme FP7/2007-2013/ under REA Grant Agreement n^° [612627]-＂AniNex＂, and the Zhejiang Provincal Natural Science Foundation of China under Grant No. LY13F020036. Acknowledgement We would thank all anonymous reviewers for their helpful suggestions and Xin Li and Noam Aigerman for providing data and codes on their homepagcs.
