Designing Shapes and Handling Noisy Data with Weighted Least Squares-Based Subdivision Schemes

This article explores the application of the weighted least squares(WLS)method in the development of subdivision schemes. These schemes offer a valuable means of approximating data points, whether they belong to linear or non-linear functions. Moreover, the schemes demonstrate effective handling of noisy data with outliers during the fitting process. Additionally, they prove to be highly effective in generating engineering shapes. In comparison to the traditional least squares (LS) approximation, WLS provides a more flexible approach. The resulting subdivision schemes produced by WLS are capable of generating smooth curves and surfaces without explicit function representations. In this work, generalized WLS-based methods for the creation of stationary and non-stationary subdivision schemes are presented. Furthermore, it includes a comparison between LS and WLS-based schemes, highlighting that the schemes produced by the LS method are a special case of those generated by WLS. By shedding light on the advantages of WLS over LS, this article contributes to the understanding and utilization of WLS-based subdivision schemes in various contexts.