Subdivision schemes based on weighted local polynomial regression. A new technique for the convergence analysis

TL;DR

This paper introduces new subdivision schemes based on weighted local polynomial regression, offering improved convergence, denoising, and approximation capabilities for data with noise in CAD applications.

Contribution

The paper develops novel binary univariate subdivision schemes using weighted local polynomial regression and provides new theoretical results on their convergence and properties.

Findings

Schemes demonstrate convergence and polynomial reproduction.

Effective denoising capabilities shown through examples.

Enhanced approximation properties compared to traditional methods.

Abstract

The generation of curves and surfaces from given data is a well-known problem in Computer-Aided Design that can be approached using subdivision schemes. They are powerful tools that allow obtaining new data from the initial one by means of simple calculations. However, in some applications, the collected data are given with noise and most of schemes are not adequate to process them. In this paper, we present some new families of binary univariate linear subdivision schemes using weighted local polynomial regression. We study their properties, such as convergence, monotonicity, polynomial reproduction and approximation and denoising capabilities. For the convergence study, we develop some new theoretical results. Finally, some examples are presented to confirm the proven properties.