Families of non-linear subdivision schemes for scattered data fitting and their non-tensor product extensions

TL;DR

This paper introduces families of non-linear subdivision schemes based on polynomials up to degree three, designed for fitting scattered noisy data while preserving shape and avoiding overfitting, with extensions to bivariate cases.

Contribution

It presents novel non-linear subdivision schemes using dynamic re-weighted least squares, applicable to scattered data fitting with shape preservation and polynomial reproduction capabilities.

Findings

Non-linear schemes outperform linear ones in data fitting.

Schemes effectively preserve shape and avoid overfitting.

Extensions to bivariate schemes are successfully demonstrated.

Abstract

In this article, families of non-linear subdivision schemes are presented that are based on univariate polynomials up to degree three. Theses families of schemes are constructed by using dynamic iterative re-weighed least squares method. These schemes are suitable for fitting scattered data with noise and outliers. Although these schemes are non-interpolatory, but have the ability to preserve the shape of the initial polygon in case of non-noisy initial data. The numerical examples illustrate that the schemes constructed by non-linear polynomials give better performance than the schemes that are constructed by linear polynomials (Computer-Aided Design, 58, 189-199). Moreover, the numerical examples show that these schemes have the ability to reproduce polynomials and do not cause over and under fitting of the data. Furthermore, families of non-linear bivariate subdivision schemes are also presented that are based on linear and non-linear bivariate polynomials.