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

作者：Wang, Zhihao; Li, Yajuan; Liu, Jianzhen; Ma, Weiyin; Deng, Chongyang^*

来源：VISUAL COMPUTER, 2023, 39(1): 139-148.

DOI：10.1007/s00371-021-02318-9

• Gauss-Seidel iterative method
• Progressive iterative approximation
• Surface interpolation
• Loop subdivision surface
• Catmull-Clark subdivision surface

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.