加速的B样条曲线曲面拟合最小二乘渐进迭代逼近  

Accelerated least squares progressive-iterative approximation for B-spline curve and surface fittings

作　　者：刘成志 吴念慈 李军成 LIU Chengzhi;WU Nianci;LI Juncheng(College of Mathematics and Finance,Hunan University of Humanities,Science and Technology,Loudi 417000,Hunan Province,China;School of Mathematics and Statistics,South-Central Minzu University,Wuhan 430074,China)

机构地区：[1]湖南人文科技学院数学与金融学院,湖南娄底417000 [2]中南民族大学数学与统计学院,湖北武汉430074

出　　处：《浙江大学学报(理学版)》2025年第3期346-356,370,共12页Journal of Zhejiang University(Science Edition)

基　　金：国家自然科学基金资助项目(12101225,12201651);湖南省自然科学基金资助项目(2023JJ50080);中南民族大学中央高校基本科研业务费专项资金资助项目(CZQ23004)。

摘　　要：最小二乘渐进迭代逼近(least squares progressive-iterative approximation,LSPIA)算法可近似地生成拟合给定数据点集的曲线或曲面。标准的LSPIA是用Landweber迭代格式计算控制顶点的,收敛速度相对较慢。为此,基于切比雪夫半迭代格式,提出了一种加速的LSPIA算法(简记为CLSPIA)。根据切比雪夫多项式外推形式,通过拟合曲线或曲面控制顶点的历史信息以及自适应步长参数选取策略更新控制顶点。收敛性分析表明,采用三次B样条曲线曲面拟合CLSPIA算法较传统LSPIA算法具有更快的收敛速度。数值实例进一步验证了理论结果正确,也证实了CLSPIA算法是可行和有效的。The least squares progressive-iterative approximation(LSPIA)method can approximate curves or surfaces to fit given data point sets.The standard LSPIA method employs the Landweber iterative format to calculate the control points,but it converges relatively slowly.In this paper,an accelerated LSPIA method is proposed based on the Chebyshev semi-iterative scheme.The main idea is to use the extrapolation form of Chebyshev polynomials,taking into account the historical information of the control points of the fitting curve or surface,as well as an adaptive step size parameter selection strategy to update the control points(denoted by CLSPIA).The convergence analysis indicates that the CLSPIA method for cubic B-spline curve and surface fitting has a faster convergence rate than the standard LSPIA method.Numerical examples further validate the theoretical results and demonstrate that the CLSPIA method is feasible and effective.

关 键 词：三次B样条曲线曲面 最小二乘拟合 渐进迭代逼近法 切比雪夫多项式 

分 类 号：TP391[自动化与计算机技术—计算机应用技术]