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Gauss–Seidel progressive iterative approximation (GS-PIA) for subdivision surface interpolation

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Abstract

We propose Gauss–Seidel progressive iterative approximation (GS-PIA) for subdivision surface interpolation by combining the Gauss–Seidel iterative method for linear systems and progressive iterative approximation (PIA) for free-form curve and surface interpolation. We address the details of GS-PIA for Loop and Catmull–Clark surface interpolation and prove that they are convergent. In addition, GS-PIA may also be applied to surface interpolation for other stationary approximating subdivision schemes with explicit limit position formula/masks. GS-PIA inherits many good properties of PIA, such as having intuitive geometric meaning and being easy to implement. Compared with some other existing interpolation methods by approximating subdivision schemes, GS-PIA has three main advantages. First, it has a faster convergence rate than PIA and weighted progressive iterative approximation (W-PIA). Second, GS-PIA does not need to compute optimal weights while W-PIA does. Third, GS-PIA does not modify the mesh topology but some methods with fairness measures do. Numerical examples for Loop and Catmull–Clark subdivision surface interpolation illustrated in this paper show the efficiency and effectiveness of GS-PIA.

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Acknowledgements

This work has been supported by the NSFC (61872121, 61761136010), Natural Science Foundation of Zhejiang Province (LQ17A010009) and Research Grants Council of Hong Kong (GRF Grant No. CityU 11206917).

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Authors and Affiliations

  1. School of Sciences, Hangzhou Dianzi University, Hangzhou, 310018, China

    Zhihao Wang, Yajuan Li, Jianzhen Liu & Chongyang Deng

  2. Department of Mechanical Engineering, City University of Hong Kong, Hong Kong, China

    Weiyin Ma

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  1. Zhihao Wang
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  2. Yajuan Li
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  3. Jianzhen Liu
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Correspondence to Chongyang Deng.

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Wang, Z., Li, Y., Liu, J. et al. Gauss–Seidel progressive iterative approximation (GS-PIA) for subdivision surface interpolation. Vis Comput 39, 139–148 (2023). https://doi.org/10.1007/s00371-021-02318-9

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