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1. 武汉大学 测绘学院,湖北 武汉 430079；
2. 武汉大学 卫星导航定位技术研究中心,湖北 武汉 430079

A General Total Least Squares Algorithm for Three-dimensional Coordinate Transformations
FANG Xing1, ZENG Wenxian1, LIU Jingnan1,2, YAO Yibin1
1. School of Geodesy and Geomatics,Wuhan University,Wuhan 430079,China;
2. Research Center of GNSS,Wuhan University,Wuhan 430079,China
First author: FANG Xing(1981—),male,PhD,majors in the theory and method of surveying data processing.E-mail：xfang@sgg.whu.edu.cn
Corresponding author：ZENG Wenxian E-mail：wxzeng@sgg.whu.edu.cn
Abstract: The model of three-dimensional coordinate transformation is a nonlinear errors-in-variables model. The methods proposed in published literatures are always restricted in practice for their special assumptions,such as size of rotation angles,structured design matrix and special weight matrix. A general weighted TLS algorithm of a three-dimensional coordinate transformation is investigated in this paper. The new algorithm can be applied in transformations with an arbitrary rotation angles and any applicable weights of the observations,as well as the structured design matrix or the design matrix with both random and fixed elements. The example given in this paper illustrates that this algorithm is general and suits to all kinds of three-dimensional coordinate transformations in practice.
Key words: total least-squares method     three-dimensional coordinate transformations     errors-in-variables model     nonlinear program

1 引 言

2 三维坐标转换的加权整体最小二乘算法

(1) 给出模型的初始解βc0，根据式(5)—式(6)计算模型的梯度g(β)。

(2) 迭代计算拟牛顿法中的海森(Hessian)矩阵的近似表达式量，直到前后两次参数估计之差Δβi小于设定的正微小量

(3) 根据以下Broyden-Fletcher-Goldfarb-Shanno (BFGS)算法公式，得到三维坐标转换模型的整体最小二乘估计结果

3 实 例

 点号 源坐标值 目标坐标值 X Y Z x y z 1 30 40 10 290 150 15 2 100 40 10 420 80 2 3 100 130 10 540 200 20 4 30 130 10 390 300 5

 参数 LS(等权) TLS(等权) TLS(加权) 1 54′45.356 1″ 54′45.356 1″ 30′51.466 6″ 2 -57′47.402 9″ -57′47.402 9″ -4°31′21.125 5″ 3 -35°49′30.616 6″ -35°49′30.616 6″ -33°32′19.511 1″ 2.082 975 4 2.121 636 2 2.176 126 9 Δ 196.970 86 193.016 96 188.977 14 Δ 118.588 95 117.402 74 101.517 20 Δ -14.935 29 -15.407 38 -33.380 08 1 283.77 236.89 319.004

4 结 论

 [1] FANG Xing. Weighted Total Least Squares Solution for Application in Geodesy[D]. Hanover: Leibniz University of Hanover, 2011. [2] HUFFEL S V, VANDEWALLE J.The Total Least-squares Problem: Computational Aspects and Analysis[M]. Philadelphia: Society for Industrial and Applied Mathematics, 1991. [3] FANG Xing. A Structured and Constrained Total Least-squares Solution with Cross-covariances[J]. Studia Geophysica et Geodaetica, 2014, 58 (1): 1-16. [4] BLEICH P, ILLNER M. Strenge Lsung der Rumlichen Koordinatentransformation Durch Iiterative Berechnung[J]. Allgemeine Vermessungs-Nachrichten,1989,96 (4):133-144. [5] POPE A. Some Pitfalls to be Avoided in the Iterative Adjustment of Nonlinear Problems[C]//Proceedings of the 38th Annual Meeting of American Society Photogrammetry. Washington DC: [s.n.], 1972: 449-473. [6] ACAR A, YLVDEMIR MT, AKYILMAZ O, et al. Deformation Analysis with Total Least Squares[J]. Natural Hazards and Earth System Sciences, 2006, 6: 663-669. [7] FELUS F, BURTCH R. On Symmetrical Three-dimensional Datum Conversion[J]. GPS Solutions, 2009, 13(1):65-74. [8] AKYILMAZ O. Solution of the Heteroscedastic Datum Transformation Problem[R]. Munich: International Association of Geodesy, 2012. [9] LU Jue, CHEN Yi, FANG Xing, et al. Performing 3-D Similarity Transformation Using the Weighted Total Least-squares Method[R]. Munich: International Association of Geodesy, 2012. [10] NEITZEL F. Generalization of Total Least-squares on Example of Unweighted and Weighted 2D Similarity Transformation[J]. Journal of Geodesy, 2010, 84(12):751-762. [11] LI Bofeng, SHEN Yunzhong, LI Weixiao. The Seamless Model for Three-dimensional Datum Transformation[J]. Science China: Earth Science, 2012, 55 (12): 2099-2108. [12] LI Bofeng, SHEN Yunzhong, ZHANG Xingfu, et al. Seamless Multivariate Affine Error-in-variables Transformation and Its Application to Map Rectification[J]. International Journal of Information Science, 2013, 27(8): 1572-1592. [13] CHEN Yi, LU Jue. Performing 3D Similarity Transformation by Robust Total Least Squares[J]. Acta Geodaetica et Cartographica Sinica, 2012, 41(5): 715-722. (陈义,陆珏.以三维坐标转换为例解算稳健总体最小二乘方法[J].测绘学报,2012,41(5):715-722.) [14] LU Jue, CHEN Yi, ZHENG Bo. Applying Total Least Squares to the Three-dimensional Datum Transformation[J]. Journal of Geodesy and Geodynamics, 2008, 28(5):77-81. (陆珏, 陈义, 郑波. 总体最小二乘方法在三维坐标转换中的应用[J]. 大地测量与地球动力学, 2008, 28(5): 77-81.) [15] XU Chaoqian, YAO Yibin, XIONG Siting, et al. 3D Rectangular Coordinate Transformation Adapted to Arbitrary Rotation Angle Based on Total Least-Squares Regression[J]. Journal of Geomatics, 2010, 35(5): 46-48. (许超钤,姚宜斌,熊思婷,等. 三维任意旋转角度坐标转换的整体最小二乘回归解法[J]. 测绘信息与工程,2010, 35(5):46-48.) [16] XU Peiliang, LIU Jingnan, SHI Chuang. Total Least Squares Adjustment in Partial Errors-in-variables Models: Algorithm and Statistical Analysis[J]. Journal of Geodesy, 2012, 86(8): 661-675. [17] FANG Xing. Weighted Total Least Squares: Necessary and Sufficient Conditions, Fixed and Random Parameters[J]. Journal of Geodesy, 2013, 87(8): 733-749. [18] GE Xuming, WU Jicang. Iterative Method of Weight Constraint Total Least Squares for Three Dimensional Datum Transfor-mation[J]. Science of Wuhan University:Geomatics and Information, 2012, 37(2): 178-182. (葛旭明,伍吉仓. 三维基准转换的约束加权混合整体最小二乘的迭代解法[J]. 武汉大学学报:信息科学版,2012, 37(2):178-182.) [19] NOCEDAL J, WRIGHT S. Numerical Optimization[M]. Berlin: Springer, 2006. [20] LENZMANN L, LENZMANN E. Zur Lsung des Nichtlinearen Gauss-Markov-Modells[J]. Zeitschrift für Geodsie, Geoinformation und Landmanagement, 2007, 132: 108-120. [21] KONG Jian, YAO Yibin, WU Han. Iterative Method for Total Least-squares[J]. Science of Wuhan University:Geomatics and Information, 2010, 35(6): 711-714. (孔建,姚宜斌,吴寒. 整体最小二乘的迭代解法[J]. 武汉大学学报:信息科学版,2010,35(6):711-714.) [22] SCHAFFRIN B, FELUS Y A. On the Multivariate Total Least-squares Approach to Empirical Coordinate Transformations: Three Algorithms[J]. Journal of Geodesy, 2008, 82(6):373-383. [23] CAI Jianqing, GRAFAREND EW. Systematical Analysis of the Transformation between Gauss-Krueger-coordinate/DHDN and UTM-coordinate/ETRS89 in Baden-Württemberg with Different Estimation Methods[C]//Geodetic Reference Frame International Association of Geodesy Symposia. Munich: International Association of Geodesy, 2009, 134:205-211.
http://dx.doi.org/10.13485/j.cnki.11-2089.2014.0193

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#### 文章信息

FANG Xing,ZENG Wenxian,LIU Jingnan,et al

A General Total Least Squares Algorithm for Three-dimensional Coordinate Transformations

Acta Geodaeticaet Cartographica Sinica,2014,43(11):1139-1143.
http://dx.doi.org/10.13485/j.cnki.11-2089.2014.0193