带多权值局部插值型的几何迭代法
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摘要
针对参数曲线/曲面重要细节特征处理的问题,提出带多个权值的、局部插值的几何迭代算法.首先将初始控制顶点分为插值点和非插值点2组,在调整不同的插值点时对调整向量赋予不同的权值,非插值点则不进行调整;然后给出算法的迭代矩阵并分析了收敛性;最后将文中算法推广到三维曲面情形.数值实例结果表明,通过适当地选取权值,该算法不仅可以灵活地处理曲线/曲面的局部细节特征,而且迭代产生的误差相对较小.
In order to deal with detail features of parameter curves and surfaces, one local interpolation type of geometric iterative approximation method with multiple weights is presented. Firstly, the initial data points are classified into two groups: interpolating points and noninterpolating points. For the interpolating points, different weights are distributed to the corresponding adjusting vectors. For the noninterpolating points, there is no adjustment. Then, the iteration matrix is given and the convergence is further analyzed. Finally, the method has been generalized to 3 D surfaces. The results of numerical examples show that, the method can deal with detail features flexibly and the iteration errors are relatively small.
In order to deal with detail features of parameter curves and surfaces, one local interpolation type of geometric iterative approximation method with multiple weights is presented. Firstly, the initial data points are classified into two groups: interpolating points and noninterpolating points. For the interpolating points, different weights are distributed to the corresponding adjusting vectors. For the noninterpolating points, there is no adjustment. Then, the iteration matrix is given and the convergence is further analyzed. Finally, the method has been generalized to 3 D surfaces. The results of numerical examples show that, the method can deal with detail features flexibly and the iteration errors are relatively small.
引文
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[3]Lin Hongwei,Wang Guojin,Dong Chenshi.Constructing iterative non-uniform B-spline curve and surface to fit data points[J].Science in China:Series F,2004,33(10):912-923(in Chinese)(蔺宏伟,王国瑾,董辰世.用迭代非均匀B-spline曲线(曲面)拟合给定点集[J].中国科学:F辑,2004,33(10):912-923
[4]Lin H W,Bao H J,Wang G J.Totally positive bases and progressive iteration approximation[J].Computers and Mathematics with Applications,2005,50(3/4):575-586
[5]Delgado J,Pena J M.Progressive iterative approximation and bases with the fastest convergence rates[J].Computer Aided Geometric Design,2007,24(1):10-18
[6]Delgado J,Pena J M.A comparison of different progressive iteration approximation methods[C]//Proceedings of the 7th International Conference on Mathematical Methods for Curves and Surfaces.Heidelberg:Springer,2008,5862:136-152
[7]Chen J,Wang G J.Progressive iterative approximation for triangular Bézier surfaces[J].Computer-Aided Design,2011,43(8):889-895
[8]Zhao Y,Lin H W.The PIA property of low degree non-uniform triangular B-B patches[C]//Proceedings of the 12th International Conference on Computer-Aided Design and Computer Graphics.Los Alamitos:IEEE Compute Society Press,2011:239-242
[9]Deng Shaohui,Wang Guozhao.Numerical analysis of the progressive iterative approximation method[J].Journal of ComputerAided Design&Computer Graphics,2012,24(7):879-884(in Chinese)(邓少辉,汪国昭.渐进迭代逼近方法的数值分析[J].计算机辅助设计与图形学学报,2012,24(7):879-884)
[10]Lin H W.Local progressive-iterative approximation format for blending curves and patches[J].Computer Aided Geometric Design,2010,27(4):322-339
[11]Lin H W.Adaptive data fitting by the progressive-iterative approximation[J].Computer Aided Geometric Design,2012,29(7):463-473
[12]Zhao Yu,Lin Hongwei.Real-time interactive modification of B-spline by PIA[J].Journal of Computer-Aided Design&Computer Graphics,2011,23(12):2013-2018(in Chinese)(赵宇,蔺宏伟.基于PIA的B-spline曲面实时交互修改方法[J].计算机辅助设计与图形学学报,2011,23(12):2013-2018)
[13]Lu L Z.Weighted progressive iteration approximation and convergence analysis[J].Computer Aided Geometric Design2010,27(2):129-137
[14]Chen Jie,Wang Guojin,Jin Congjian.Two kinds of generalized progressive iterative approximations[J].Acta Automatica Sinica,2012,38(1):135-139(in Chinese)(陈杰,王国瑾,金聪健.两类推广的渐近迭代逼近[J].自动化学报,2012,38(1):135-139)
[15]Lin H W,Zhang Z Y.An extended iterative format for the progressive-iteration approximation[J].Computers&Graphics,2011,35(5):967-975
[16]Deng C Y,Lin H W.Progressive and iterative approximation for least squares B-spline curve and surface fitting[J].Computer-Aided Design,2014,47:32-44
[17]Liu Xiaoyan,Deng Chongyang.Jacobi-PIA algorithm for nonuniform cubic B-spline curve interpolation[J].Journal of Computer-Aided Design&Computer Graphics,2015,27(3):485-491(in Chinese)(刘晓艳,邓重阳,非均匀三次B样条曲线插值的Jacobi-PIA算法[J].计算机辅助设计与图形学学报,2015,27(3):485-491)
[18]Lin Hongwei.Survey on geometric iterative methods with applications[J].Journal of Computer-Aided Design&Computer Graphics,2015,27(4):582-589(in Chinese)(蔺宏伟.几何迭代法及其应用综述[J].计算机辅助设计与图形学学报,2015,27(4):582-589)
[19]Lin H W,Maekawa T,Deng C Y.Survey on geometric iterative methods and their applications[J].Computer-Aided Design,2018,95:40-51
[2]de Boor C.How does Agee’s smoothing method work?[R].Washington D C:Army Research Office 79-3,1979:299-302
[3]Lin Hongwei,Wang Guojin,Dong Chenshi.Constructing iterative non-uniform B-spline curve and surface to fit data points[J].Science in China:Series F,2004,33(10):912-923(in Chinese)(蔺宏伟,王国瑾,董辰世.用迭代非均匀B-spline曲线(曲面)拟合给定点集[J].中国科学:F辑,2004,33(10):912-923
[4]Lin H W,Bao H J,Wang G J.Totally positive bases and progressive iteration approximation[J].Computers and Mathematics with Applications,2005,50(3/4):575-586
[5]Delgado J,Pena J M.Progressive iterative approximation and bases with the fastest convergence rates[J].Computer Aided Geometric Design,2007,24(1):10-18
[6]Delgado J,Pena J M.A comparison of different progressive iteration approximation methods[C]//Proceedings of the 7th International Conference on Mathematical Methods for Curves and Surfaces.Heidelberg:Springer,2008,5862:136-152
[7]Chen J,Wang G J.Progressive iterative approximation for triangular Bézier surfaces[J].Computer-Aided Design,2011,43(8):889-895
[8]Zhao Y,Lin H W.The PIA property of low degree non-uniform triangular B-B patches[C]//Proceedings of the 12th International Conference on Computer-Aided Design and Computer Graphics.Los Alamitos:IEEE Compute Society Press,2011:239-242
[9]Deng Shaohui,Wang Guozhao.Numerical analysis of the progressive iterative approximation method[J].Journal of ComputerAided Design&Computer Graphics,2012,24(7):879-884(in Chinese)(邓少辉,汪国昭.渐进迭代逼近方法的数值分析[J].计算机辅助设计与图形学学报,2012,24(7):879-884)
[10]Lin H W.Local progressive-iterative approximation format for blending curves and patches[J].Computer Aided Geometric Design,2010,27(4):322-339
[11]Lin H W.Adaptive data fitting by the progressive-iterative approximation[J].Computer Aided Geometric Design,2012,29(7):463-473
[12]Zhao Yu,Lin Hongwei.Real-time interactive modification of B-spline by PIA[J].Journal of Computer-Aided Design&Computer Graphics,2011,23(12):2013-2018(in Chinese)(赵宇,蔺宏伟.基于PIA的B-spline曲面实时交互修改方法[J].计算机辅助设计与图形学学报,2011,23(12):2013-2018)
[13]Lu L Z.Weighted progressive iteration approximation and convergence analysis[J].Computer Aided Geometric Design2010,27(2):129-137
[14]Chen Jie,Wang Guojin,Jin Congjian.Two kinds of generalized progressive iterative approximations[J].Acta Automatica Sinica,2012,38(1):135-139(in Chinese)(陈杰,王国瑾,金聪健.两类推广的渐近迭代逼近[J].自动化学报,2012,38(1):135-139)
[15]Lin H W,Zhang Z Y.An extended iterative format for the progressive-iteration approximation[J].Computers&Graphics,2011,35(5):967-975
[16]Deng C Y,Lin H W.Progressive and iterative approximation for least squares B-spline curve and surface fitting[J].Computer-Aided Design,2014,47:32-44
[17]Liu Xiaoyan,Deng Chongyang.Jacobi-PIA algorithm for nonuniform cubic B-spline curve interpolation[J].Journal of Computer-Aided Design&Computer Graphics,2015,27(3):485-491(in Chinese)(刘晓艳,邓重阳,非均匀三次B样条曲线插值的Jacobi-PIA算法[J].计算机辅助设计与图形学学报,2015,27(3):485-491)
[18]Lin Hongwei.Survey on geometric iterative methods with applications[J].Journal of Computer-Aided Design&Computer Graphics,2015,27(4):582-589(in Chinese)(蔺宏伟.几何迭代法及其应用综述[J].计算机辅助设计与图形学学报,2015,27(4):582-589)
[19]Lin H W,Maekawa T,Deng C Y.Survey on geometric iterative methods and their applications[J].Computer-Aided Design,2018,95:40-51