Progressive Iterative Approximation of Non-Uniform Cubic B-Spline Curves and Surfaces via Successive Over-Relaxation Iteration

Abstract

Geometric iteration (GI) is one of the most efficient curve- or surface-fitting techniques in recent years, which is famous for its remarkable geometric significance. In essence, GI can be thought of as the sum of iterative methods for solving systems of linear equations, such as progressive iterative approximation (PIA) which relies on the theory of Richardson iteration. Thus, when the curve- or surface-fitting error is at a desired level, we want to have as few iterations as possible to improve efficiency when dealing with large data sets. Based on the idea of successive over-relaxation (SOR) iteration, we formulate a faster PIA curve and surface interpolation scheme using classical non-uniform cubic B-splines, named SOR-PIA. The genetic algorithm is utilized to estimate the best approximate relaxation factor of SOR-PIA. Similar to standard PIA, SOR-PIA can also be regarded as a process in which the control points move in one direction, but it can greatly reduce the number of iterations in the iterative process with the same fitting accuracy. By comparing with the standard PIA and WPIA algorithms, the effectiveness of the SOR-PIA iterative interpolation algorithm can be verified.