Weighted local progressive-iterative approximation property for triangular Bézier surfaces

Progressive-iterative approximation (abbr. PIA) is an important and intuitive method for fitting and interpolating scattered data points. The triangular Bernstein basis with uniformly distributed parameters has the PIA property. For the sake of more flexibility, this paper presents a local progressive-iterative approximation (abbr. LPIA) format, which allows only a chosen subset of the initial control points to adjust and shows that the LPIA format is convergent for triangular Bézier surface of degree n≤17\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n \le 17$$\end{document} with uniform parameters. Furthermore, in order to accelerate the convergence rate, we develop a weighted LPIA format for triangular Bézier surfaces and prove that the weighted LPIA format has a faster convergence rate than the LPIA format when an optimal value of the weight is chosen. Finally, some numerical examples are presented to show the effectiveness of the LPIA method and the fast convergence of the weighted LPIA method.