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.