Subdivision Schemes Based on Weighted Local Polynomial Regression: A New Technique for the Convergence Analysis

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 new families of binary univariate linear subdivision schemes using weighted local polynomial regression which are suitable for noisy data. We study their properties, such as convergence, monotonicity, polynomial reproduction and approximation and denoising capabilities. We establish conditions that connect the weight function with two essential properties: noise elimination and approximation capability. We propose a multi-objective optimization problem for improving these two capabilities and conclude that only some weight functions are Pareto optimal. For the convergence study, we develop some new theoretical results. Finally, some examples are presented to confirm the proven properties.