Efficient Two-Dimensional Randomized Progressive Iterative Approximation for Large-Scale B-Spline Fitting

Abstract

The randomized progressive iterative approximation (RPIA) is a local and approximate geometric iteration method designed for large-scale data fitting. At each iteration, RPIA updates only the control points indexed by a specific set, leaving the others unchanged. In this work, we introduce a two-dimensional RPIA (D2RPIA) for fitting B-spline curves and surfaces. Unlike RPIA, D2RPIA updates the control points with an adaptive step-size, which is determined by imposing a constraint on the new control points. This adaptive step-size allows D2RPIA to achieve the current optimal result, thereby enhancing the convergence rate compared to RPIA. We prove that D2RPIA converges linearly in the mean square to the least-squares solution. Several numerical studies are presented to validate our theoretical results.