Grouped progressive iterative approximation method of data fitting

Zheng Guo,Zhang Li,Zhang Shijie,Du Zhuangping,Liu Yi,Tan Jieqing(School of Mathematics, Hefei University of Technology, Hefei 230009, China;School of Computer and Information, Hefei University of Technology, Hefei 230009, China)

Abstract
Objective The progressive iterative approximation (PIA) method has a wide range of applications for solving interpolation and fitting problems in computer-aided design. The PIA format presents an intuitive way of data fitting by adjusting the control points iteratively. This method format generates a sequence of curves/surfaces with fine precision. According to the PIA property, the limit of the curve/surface sequence interpolates the initial data points if the blending basis is normalized and totally positive and its corresponding collocation matrix is nonsingular. To increase the flexibility of the PIA method in the large-scale fitting of data points, a new PIA method, which is based on grouping, is proposed in this work.Method First, the initial data points to be fitted are divided into several groups. Second, by applying the PIA or LSPIA method on these grouped data points separately, we can obtain a sequence of curves/surfaces with the PIA property for each group of data. Then, by adjusting the control points on the boundary according to the continuity conditions, a blending algorithm is implemented on these separate curves/surfaces. Thus, we finally acquire one whole curve/surface piece and ensure its continuity. Moreover, by grouping the data points, we can reduce the computation and improve the iteration efficiency.Result We use the PIA, LSPIA, and proposed methods on a single data set to fit data points. The grouped PIA method is convergent because we use the PIA or LSPIA method to fit each group of data points. Compared with the fitting error of the PIA method, that of the grouped PIA method decreases according to the number of groups we divide. That is, the more groups we divide, the more the fitting error is reduced. Compared with the fitting error of the LSPIA method, that of the grouped PIA method is lower by approximately one half.Conclusion This study mainly aims to combine the grouping idea with the PIA method and proposes a grouped PIA method of fitting the data set. This method not only brings enhanced flexibility to the large-scale fitting of data points but also delivers a higher convergence rate and fewer errors than the PIA and LSPIA methods do. Numerous numerical examples are presented to show the effectiveness of this technique.
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