On a progressive and iterative approximation method with memory for least square fitting

TL;DR

This paper introduces MLSPIA, a new iterative method with memory for least squares fitting that guarantees convergence even with deficient data matrices, and demonstrates faster convergence than previous methods through theoretical analysis and examples.

Contribution

It proposes MLSPIA, a novel iterative fitting method with memory, providing convergence guarantees and improved convergence rate over existing approaches.

Findings

MLSPIA converges to the least squares fitting curve.

Theoretical convergence rate is faster than previous methods.

Examples confirm the improved convergence performance.

Abstract

In this paper, we present a progressive and iterative approximation method with memory for least square fitting(MLSPIA). It adjusts the control points and the weighted sums iteratively to construct a series of fitting curves (surfaces) with three weights. For any normalized totally positive basis even when the collocation matrix is of deficient column rank, we obtain a condition to guarantee that these curves (surfaces) converge to the least square fitting curve (surface) to the given data points. It is proved that the theoretical convergence rate of the method is faster than the one of the progressive and iterative approximation method for least square fitting (LSPIA) in [Deng C-Y, Lin H-W. Progressive and iterative approximation for least squares B-spline curve and surface fitting. Computer-Aided Design 2014;47:32-44] under the same assumption. Examples verify this phenomenon.