Randomized progressive iterative approximation for B-spline curve and surface fittings

TL;DR

This paper introduces a randomized iterative method for fitting B-spline curves and surfaces, improving efficiency and convergence properties for large-scale data fitting tasks.

Contribution

It proposes a novel randomized progressive iterative approximation (RPIA) method that adjusts control points based on random criteria, with proven convergence to least-squares solutions.

Findings

RPIA converges in expectation to least-squares fits.

Numerical experiments confirm the effectiveness of RPIA.

RPIA offers efficiency benefits for large-scale data fitting.

Abstract

For large-scale data fitting, the least-squares progressive iterative approximation is a widely used method in many applied domains because of its intuitive geometric meaning and efficiency. In this work, we present a randomized progressive iterative approximation (RPIA) for the B-spline curve and surface fittings. In each iteration, RPIA locally adjusts the control points according to a random criterion of index selections. The difference for each control point is computed concerning the randomized block coordinate descent method. From geometric and algebraic aspects, the illustrations of RPIA are provided. We prove that RPIA constructs a series of fitting curves (resp., surfaces), whose limit curve (resp., surface) can converge in expectation to the least-squares fitting result of the given data points. Numerical experiments are given to confirm our results and show the benefits of RPIA.