融合正交几何信息的非线性等式约束整体最小二乘平差及迭代算法

doi:10.11947/j.AGCS.2020.20190112

测绘学报 ›› 2020, Vol. 49 ›› Issue (7): 816-823.doi: 10.11947/j.AGCS.2020.20190112

融合正交几何信息的非线性等式约束整体最小二乘平差及迭代算法

胡川¹, 方兴², 赵立都¹

1. 重庆交通大学土木工程学院, 重庆 400074;
2. 武汉大学测绘学院, 湖北武汉 430079

收稿日期:2019-04-20 修回日期:2019-12-05 发布日期:2020-07-14
作者简介:胡川(1983-),男,博士,副教授,研究方向为测量数据处理理论与方法。E-mail:hucch@cqjtu.edu.cn
基金资助:
国家自然科学基金（41774009）；重庆市基础科学与前沿技术研究（一般）项目（cstc2017jcyjAX0102）；重庆市教委科学技术研究（KJ1705132）；中央高校自主科研学科交叉项目（2042018kf0230）；2016、2017年重庆交通大学高层次人才科研启动项目（16JDKJC-A025；17JDKJC-A027）

Nonlinear equality constrained total least squares adjustment combined with orthogonal geometry information and its iterative algorithm

HU Chuan¹, FANG Xing², ZHAO Lidu¹

1. School of Civil Engineering, Chongqing Jiaotong University, Chongqing 400074, China;
2. School of Geodesy and Geomatics, Wuhan University, Wuhan 430079, Chinat

Received:2019-04-20 Revised:2019-12-05 Published:2020-07-14
Supported by:
The National Natural Science Foundation of China (No. 41774009);The Natural Science Foundation of Chongqing(No. cstc2017jcyjAX0102);Scientific and Technological Research Program of Chongqing Municipal Education Commission (No. KJ1705132);The Fundamental Research Funds for the Central Universities(No. 2042018kf0230);The Research Foundation for Talented Scholars of Chongqing Jiaotong University(Nos. 16JDKJC-A025;17JDKJC-A027)

摘要/Abstract

摘要： 正交距离最小二乘和加权整体最小二乘是解自变量含误差拟合问题的两种独立准则。加权整体最小二乘与正交距离最小二乘不同，它不考虑测量点与拟合点之间的连线垂直于拟合对象的几何信息，不能确保测量点到拟合对象的距离的平方和为极小值。针对该问题，将正交几何信息作为约束条件融入加权整体最小二乘，提出一种约束方程带有误差改正数的非线性等式约束整体最小二乘平差法。首先，把加权整体最小二乘平差的函数式看作是非线性方程，连同正交几何约束方程一并线性化，得到线性的平差函数方程；然后，采用拉格朗日乘数法推导其参数估计及精度评定公式，并给出迭代计算算法；最后，以平面直线拟合为例，对新方法和计算算法进行验证。试验结果表明：①新的方法和算法具有可行性；②与加权最小二乘和加权整体最小二乘相比，新方法计算的测量点到拟合直线的垂直距离平方和最小；③新方法计算的测量点到拟合直线的距离与测量点到拟合点的距离相等。

关键词: 非线性, 等式约束, 加权整体最小二乘, 正交距离, 直线拟合

Abstract: For the problem of fitting the independent variables with error, orthogonal distance least squares and weighted total least squares two independent approaches can be employed to solve it. However, the weighted total least square is unlike orthogonal distance least squares. It does not taking into account the orthogonal geometry information, namely, the line segment consisted of observation point and fitted point is vertical to fitted object. Aimed to solve this problem, a nonlinear equality constrained total least squares adjustment model is proposed, in which total least squares is combined with orthogonal geometry information that has been transformed into a nonlinear equality constraints function with unknown corrected errors. After the function model and the nonlinear equality constraints function are linearized, the Lagrange multiplier method is introduced to derive the calculation formula for estimating parameter and assessing accuracy. Two iterative algorithms are given correspondingly. The suggested method and the designed algorithm are tested by the example of fitting a straight-line. The results show that, ①the proposed model and the iterative algorithm are feasible; ②compared with weighted least squares and weighted total least squares, the sum square of orthogonal distance from the measured point to the fitted line calculated by the new method is the smallest value; ③this orthogonal distance is equal to the distance from the measured point to the corrected point calculated by the suggested method, and the other two methods are not like this.

Key words: nonlinear, equality constraints, total least squares, orthogonal distance, straight-line fit

中图分类号:

P221

胡川, 方兴, 赵立都. 融合正交几何信息的非线性等式约束整体最小二乘平差及迭代算法[J]. 测绘学报, 2020, 49(7): 816-823.

HU Chuan, FANG Xing, ZHAO Lidu. Nonlinear equality constrained total least squares adjustment combined with orthogonal geometry information and its iterative algorithm[J]. Acta Geodaetica et Cartographica Sinica, 2020, 49(7): 816-823.

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融合正交几何信息的非线性等式约束整体最小二乘平差及迭代算法

Nonlinear equality constrained total least squares adjustment combined with orthogonal geometry information and its iterative algorithm

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