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

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.

Automated summary

We propose a Gauss-Seidel progressive iterative approximation (GS-PIA) method for subdivision surface interpolation. GS-PIA inherits the good properties of progressive iterative approximation (PIA) and has a faster convergence rate than PIA and weighted progressive iterative approximation (W-PIA), does not require computing optimal weights, and preserves the mesh topology.