圆曲线拟合的总体最小二乘算法分析
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摘要
针对圆曲线拟合参数求解的问题,该文在分析现有的圆曲线模型构建及其参数求解方法的基础上,以圆曲线的参数方程建立圆曲线拟合模型,采用总体最小二乘算法的求解模型参数。通过实例数据和模拟数据,对比分析现有以一般方程构建的圆曲线拟合模型的最小二乘算法、混合总体最小二算法以及根据协方差传播定率定权的加权总体最小二乘算法计算结果。实验结果表明,采用该文提出的总体最小二乘算法求解由参数方程构建的拟合模型,效率和准确度均有所提高。
Aiming at the problem of circular curve fitting,this paper in-depth analyzed the construction method for solving the parameters of the existing model of the circular curve and took total least squares to solve the parameters of circle fitting model established by its parametric equation.Comparing the existing least squares algorithm,mixed total least squares algorithm,weighted total least squares algorithm for circle fitting model established by its standard form,the examples results illustrated that the efficiency and feasibility of the algorithm proposed in this paper were better.
Aiming at the problem of circular curve fitting,this paper in-depth analyzed the construction method for solving the parameters of the existing model of the circular curve and took total least squares to solve the parameters of circle fitting model established by its parametric equation.Comparing the existing least squares algorithm,mixed total least squares algorithm,weighted total least squares algorithm for circle fitting model established by its standard form,the examples results illustrated that the efficiency and feasibility of the algorithm proposed in this paper were better.
引文
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[10]陈明晶,方源敏,陈杰.最小二乘法和迭代法圆曲线拟合[J].测绘科学,2016,41(1):194-197,202.(CHEN Mingjing,FANG Yuanmin,CHEN Jie.Fitting of circular curve based on least square method and iterative method[J].Science of Surveying and Mapping,2016,41(1):194-197,202.)
[11]GANDER W,GOLUB G H,STREBEL R.Leastsquares fitting of circles and ellipses[J].BIT Numerical Mathematics,1994,34(4):558-578.
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[13]SCHAFFRIN B,WIENER A.On weighted total least squares adjustment for linear regressiom[J].Journal of Geodesy,2008,82(7):415-421.
[14]FANG Xing.Weighted total least squares:necessary and sufficient conditions,fixed and random parameters[J].Journal of Geodesy,2013,87(8):733-749.
[15]许光煜,王乐洋.加权总体最小二乘平差方法的比较与分析[J].工程勘察,2014,42(6):67-72.(XU Guangyu,WANG Leyang.The comparison and analysis of three weighted total least-squares methods[J].Geotechnical Investigation&Surveying,2014,42(6):67-72.)
[16]同济大学数学系.高等数学[M].6版.上海:高等教育出版社,2010.(Department of Mathematics,Tongji University.Advanced mathematics[M].6th ed.Shanghai:Higher Education Press,2010.)
[2]李德仁,袁修孝.误差处理与可靠性理论[M].武汉:武汉大学出版社,2012.(LI Deren,YUAN Xiuxiao.Error processing and reliability theory[M].Wuhan:Wuhan University Press,2012.)
[3] HEARN D W,VIJAY J.Efficient algorithms for the(weighted)minimum circle problem[J].Operations Research,1982,30(4):777-795.
[4]马洪磊,刘成龙,宋韬,等.一种铁路既有线纵断面圆曲线拟合新方法[J].武汉大学学报(信息科学版),2016,41(12):1696-1700.(MA Honglei,LIU Chenglong,SONG Tao,et al.A new fitting method of profile circular curve of existing railway[J].Geomatics and Information Science of Wuhan University,2016,41(12):1696-1700.)
[5]杨佳萌,郭建锋.圆曲线拟合算法中的粗差探测与识别[J].测绘科学,2017,42(8):15-18.(YANG Jiameng,GUO Jianfeng.Detection and identification of outliers for circle fitting[J].Science of Surveying and Mapping,2017,42(8):15-18.)
[6] COOPE I D.Circle fitting by linear and nonlinear least squares[J].Journal of Optimization Theory and Applications,1993,76(2):381-388.
[7] DAVIS T G.Total least squares spiral curve fitting[J].Journal of Surveying Engineering,1999,125(4):159-176.
[8]袁豹,岳东杰,许义,等.加权总体最小二乘在曲面拟合中的应用[J].勘察科学技术,2013(3):47-50.(YUAN Bao,YUE Dongjie,XU Yi,et al.Application of weighted total least-squares in surface fitting[J].Site Investigation Science and Technology,2013(3):47-50.)
[9]武汉大学测绘学院测量平差学科组.误差处理与测量平差基础[M].武汉:武汉大学出版社,2009.(Department of Surveying and Mapping,Wuhan University.Error theory and foundation of surveying adjustment[M].Wuhan:Wuhan University Press,2009.)
[10]陈明晶,方源敏,陈杰.最小二乘法和迭代法圆曲线拟合[J].测绘科学,2016,41(1):194-197,202.(CHEN Mingjing,FANG Yuanmin,CHEN Jie.Fitting of circular curve based on least square method and iterative method[J].Science of Surveying and Mapping,2016,41(1):194-197,202.)
[11]GANDER W,GOLUB G H,STREBEL R.Leastsquares fitting of circles and ellipses[J].BIT Numerical Mathematics,1994,34(4):558-578.
[12]鲁铁定,宁津生.总体最小二乘平差理论及其应用[M].北京:中国科学技术出版社,2011.(LU Tieding,NING Jinsheng.Total least squares adjustment theory and its application[M].Beijing:China Science and Technology Press,2011.)
[13]SCHAFFRIN B,WIENER A.On weighted total least squares adjustment for linear regressiom[J].Journal of Geodesy,2008,82(7):415-421.
[14]FANG Xing.Weighted total least squares:necessary and sufficient conditions,fixed and random parameters[J].Journal of Geodesy,2013,87(8):733-749.
[15]许光煜,王乐洋.加权总体最小二乘平差方法的比较与分析[J].工程勘察,2014,42(6):67-72.(XU Guangyu,WANG Leyang.The comparison and analysis of three weighted total least-squares methods[J].Geotechnical Investigation&Surveying,2014,42(6):67-72.)
[16]同济大学数学系.高等数学[M].6版.上海:高等教育出版社,2010.(Department of Mathematics,Tongji University.Advanced mathematics[M].6th ed.Shanghai:Higher Education Press,2010.)