三次均匀B样条扩展曲线的渐进迭代逼近法
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摘要
为了得到收敛速度更快的几何迭代法,提出带形状参数的三次均匀B样条扩展曲线的(加权)渐进迭代逼近法.首先基于三次均匀B样条扩展曲线提出(加权)渐进迭代逼近法的迭代格式;然后通过分析迭代矩阵的谱半径,探讨迭代法的最优形状参数及加权渐进迭代逼近法的最优权系数;最后指出双三次均匀B样条扩展曲面同样具有(加权)渐进迭代逼近性质.数值实例结果表明,所求的最优形状参数及权系数使得迭代法具有最快的收敛速度.
In order to obtain faster convergence rate, we proposed the(weighted) progressive-iterative approximation by the extension of the cubic uniform B-spline curves with shape parameters. Firstly, we presented the(weighted) progressive-iterative approximation expression, and then studied the optimal shape parameters and the optimal weight by analyzing the spectral radius of the iterative matrix of the progressive-iterative approximation. Finally, we also proposed the(weighted) progressive-iterative approximation by the extension of bicubic uniform B-spline surfaces, and pointed out that the initial surface also has the(weighted) progressive iteration approximation property. Numerical results show that the optimal shape parameters and the optimal weight make the iterative methods have the fastest convergence rate.
In order to obtain faster convergence rate, we proposed the(weighted) progressive-iterative approximation by the extension of the cubic uniform B-spline curves with shape parameters. Firstly, we presented the(weighted) progressive-iterative approximation expression, and then studied the optimal shape parameters and the optimal weight by analyzing the spectral radius of the iterative matrix of the progressive-iterative approximation. Finally, we also proposed the(weighted) progressive-iterative approximation by the extension of bicubic uniform B-spline surfaces, and pointed out that the initial surface also has the(weighted) progressive iteration approximation property. Numerical results show that the optimal shape parameters and the optimal weight make the iterative methods have the fastest convergence rate.
引文
[1]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):575-586
[2]Maekawa T,Matsumoto Y,Namiki K.Interpolation by geometric algorithm[J].Computer-Aided Design,2007,39(4):313-323
[3]Lin H W.The convergence of the geometric interpolation algorithm[J].Computer-Aided Design,2010,42(6):505-508
[4]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)
[5]Lin H W,Maekawa T,Deng C Y.Survey on geometric iterative methods and their applications[J].Computer-Aided Design,2017,95:40-51
[6]Delgado J,Pe?a J M.Progressive iterative approximation and bases with the fastest convergence rates[J].Computer Aided Geometric Design,2007,24(1):10-18
[7]Han Xuli,Liu Shengjun.An extension of the cubic uniform B-spline curve[J].Journal of Computer-Aided Design&Computer Graphics,2003,15(5):576-578(in Chinese)(韩旭里,刘圣军.三次均匀B样条曲线的扩展[J].计算机辅助设计与图形学学报,2003,15(5):576-578)
[8]Han X L.Piecewise quartic polynomial curves with a local shape parameter[J].Journal of Computational and Applied Mathematics,2006,195(1/2):34-45
[9]Xu Gang,Wang Guozhao.Extensions of uniform cubic B-spline curve with local shape parameters[J].Journal of Computer Research and Development,2007,44(6):1032-1037(in Chinese)(徐岗,汪国昭.带局部形状参数的三次均匀B样条曲线的扩展[J].计算机研究与发展,2007,44(6):1032-1037)
[10]Cao J,Wang G Z.Non-uniform B-spline curves with multiple shape parameters[J].Journal of Zhejiang University Science C,2011,12:800-808
[11]Cao J,Wang G Z.The structure of uniform B-spline curves with parameters[J].Progress in Natural Science,2008,18(3):303-308
[12]Lu L Z.Weighted progressive iterative approximation and convergence analysis[J].Computer Aided Geometric Design,2010,27(2):129-137
[13]Carnicer J M,Delgado J,Pe?a J M.Richardson method and totally nonnegative linear systems[J].Linear Algebra and its Applications,2010,433(11/12):2010-2017
[14]Yang Shengliang.Eigenvalue of tridiagonal matrix and its applications[J].Mathematics in Practice and Theory,2010,40(3):155-160(in Chinese)(杨胜良.三对角矩阵的特征值及其应用[J].数学的实践与认识,2010,40(3):155-160
[15]Marco A,Martínez J J.A fast and accurate algorithm for solving Bernstein-Vandermonde linear systems[J].Linear Algebra and its Applications,2012,422(2/3):616-628
[16]Lin Hongwei,Wang Guojin,Dong Chenshi.Constructing iterative non-uniform B-spline curve and surface to fit data points[J].Science in China:Series E,2003,33(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:E辑,2003,33(10):912-923)
[17]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
[2]Maekawa T,Matsumoto Y,Namiki K.Interpolation by geometric algorithm[J].Computer-Aided Design,2007,39(4):313-323
[3]Lin H W.The convergence of the geometric interpolation algorithm[J].Computer-Aided Design,2010,42(6):505-508
[4]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)
[5]Lin H W,Maekawa T,Deng C Y.Survey on geometric iterative methods and their applications[J].Computer-Aided Design,2017,95:40-51
[6]Delgado J,Pe?a J M.Progressive iterative approximation and bases with the fastest convergence rates[J].Computer Aided Geometric Design,2007,24(1):10-18
[7]Han Xuli,Liu Shengjun.An extension of the cubic uniform B-spline curve[J].Journal of Computer-Aided Design&Computer Graphics,2003,15(5):576-578(in Chinese)(韩旭里,刘圣军.三次均匀B样条曲线的扩展[J].计算机辅助设计与图形学学报,2003,15(5):576-578)
[8]Han X L.Piecewise quartic polynomial curves with a local shape parameter[J].Journal of Computational and Applied Mathematics,2006,195(1/2):34-45
[9]Xu Gang,Wang Guozhao.Extensions of uniform cubic B-spline curve with local shape parameters[J].Journal of Computer Research and Development,2007,44(6):1032-1037(in Chinese)(徐岗,汪国昭.带局部形状参数的三次均匀B样条曲线的扩展[J].计算机研究与发展,2007,44(6):1032-1037)
[10]Cao J,Wang G Z.Non-uniform B-spline curves with multiple shape parameters[J].Journal of Zhejiang University Science C,2011,12:800-808
[11]Cao J,Wang G Z.The structure of uniform B-spline curves with parameters[J].Progress in Natural Science,2008,18(3):303-308
[12]Lu L Z.Weighted progressive iterative approximation and convergence analysis[J].Computer Aided Geometric Design,2010,27(2):129-137
[13]Carnicer J M,Delgado J,Pe?a J M.Richardson method and totally nonnegative linear systems[J].Linear Algebra and its Applications,2010,433(11/12):2010-2017
[14]Yang Shengliang.Eigenvalue of tridiagonal matrix and its applications[J].Mathematics in Practice and Theory,2010,40(3):155-160(in Chinese)(杨胜良.三对角矩阵的特征值及其应用[J].数学的实践与认识,2010,40(3):155-160
[15]Marco A,Martínez J J.A fast and accurate algorithm for solving Bernstein-Vandermonde linear systems[J].Linear Algebra and its Applications,2012,422(2/3):616-628
[16]Lin Hongwei,Wang Guojin,Dong Chenshi.Constructing iterative non-uniform B-spline curve and surface to fit data points[J].Science in China:Series E,2003,33(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:E辑,2003,33(10):912-923)
[17]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