通用EIV平差模型的线性化估计算法

曾文宪, 刘泽邦, 方兴, 李玉兵

曾文宪, 刘泽邦, 方兴, 李玉兵. 通用EIV平差模型的线性化估计算法[J]. 武汉大学学报 ( 信息科学版). DOI: 10.13203/j.whugis20200243
引用本文: 曾文宪, 刘泽邦, 方兴, 李玉兵. 通用EIV平差模型的线性化估计算法[J]. 武汉大学学报 ( 信息科学版). DOI: 10.13203/j.whugis20200243
ZENG Wenxian, LIU Zebang, FANG Xing, LI Yubing. Linearization Estimation Algorithm for Universal EIV Adjustment Model[J]. Geomatics and Information Science of Wuhan University. DOI: 10.13203/j.whugis20200243
Citation: ZENG Wenxian, LIU Zebang, FANG Xing, LI Yubing. Linearization Estimation Algorithm for Universal EIV Adjustment Model[J]. Geomatics and Information Science of Wuhan University. DOI: 10.13203/j.whugis20200243

通用EIV平差模型的线性化估计算法

基金项目: 

国家自然科学基金(41674002,41774009),湖北省自然科学基金(2018CFB578)

详细信息
    作者简介:

    曾文宪(1975-),女,博士,主要从事测量数据处理理论与应用的研究。E-mail:wxzeng@sgg.whu.edu.cn

    通讯作者:

    刘泽邦(1998-),男,硕士。E-mail:lzb13142350585@163.com

  • 中图分类号: P207

Linearization Estimation Algorithm for Universal EIV Adjustment Model

Funds: 

The National Natural Science Foundation of China (Nos. 41674002,41774009), The Natural Science Foundation of Hubei province(No.2018CFB578)

  • 摘要: 通用EIV模型将EIV模型扩展到了最一般化的形式,其加权整体最小二乘算法同时顾及了观测向量、观测向量的系数矩阵和参数的系数矩阵中的随机误差。本文将非线性的通用EIV函数模型展开,并将二阶项纳入模型的常数项,从而将通用EIV模型表示为线性形式的高斯-赫尔默特模型,推导出了通用EIV模型的线性化整体最小二乘算法和近似精度估计公式。通过模拟数据和实例进行了评估和分析,该算法与通用EIV模型的WTLS算法估计结果一致,验证了该算法的正确性和可行性。当模型含大量估计量时,通用EIV模型的线性化估计算法显著提升了计算效率,收敛速度更快。
    Abstract: The universal EIV model extends the EIV model to the most general form, and the weighted total least squares (WTLS) algorithm is proposed to take into account the random errors in observation vector, observation vector coefficient matrix and parameter coefficient matrix. In this paper, the nonlinear universal EIV function model is expanded, and the second-order term is included into the constant term of the model, so the universal EIV model is represented as Gauss-Helmert model in linear form, and the Linearized total least squares algorithm and approximate precision estimation formula of the universal EIV model are derived. Through the simulation data and examples, this algorithm is consistent with the estimation results of the WTLS algorithm of the universal EIV model, which verifies the correctness and feasibility of this algorithm. When the model contains a large number of estimators, the linearized estimation algorithm of the universal EIV model significantly improves the computational efficiency and converges faster.
  • 图  1   1 000组实验数据的LTLS参数解与参数真值偏差统计图

    Figure  1.   Statistical Graph of Deviations Between Truth Values and Parameter Values Soluted by LTLS in 1 000 Experiments

    图  2   1 000组实验数据的WTLS参数解与参数真值偏差统计图

    Figure  2.   Statistical Graph of Deviations Between Truth Values and Parameter Values Soluted by WTLS in 1 000 Experiments

    图  3   摄影测量实例图

    Figure  3.   Diagram of the Photogrammetry Example in This Paper

    表  1   参数解及其方差估计值

    Table  1   Parameter Values and Mean Square Deviations

    算法 参数解 中误差
    $ \hat {\boldsymbol{X}} _1 $ $ \hat {\boldsymbol{X}} _2 $ $ σ _{\hat{\boldsymbol{X}}_1} $ $ σ_ {\hat{\boldsymbol{X}}_2} $
    LTLS算法 5.012 551 9.994 964 0.039 9 0.051 9
    WTLS算法 5.012 551 9.994 964 0.039 9 0.051 9
    下载: 导出CSV

    表  2   1000组实验的参数解均值和协因数阵

    Table  2   Average Parameter Values and Co⁃variance Matrix in 1 000 Experiments

    算法 参数解均值 协因数阵
    $ {\rm{avg}} ( \hat{\boldsymbol{X}}_1) $ $ {\rm{avg}} ( \hat{\boldsymbol{X}}_2) $ $ {\boldsymbol{Q }}( \hat{\boldsymbol{X}} ) $ $ \bar{ \boldsymbol{Q }}( \hat{\boldsymbol{X}} ) $
    LTLS算法 5.009 427 9.998 257 0.004 3 −0.005 1 0.004 2 −0.004 9
    −0.005 1 0.007 3 −0.004 9 0.007 2
    WTLS算法 5.009 427 9.998 257 0.004 3 −0.005 1 0.004 2 −0.004 9
    −0.005 1 0.007 3 −0.004 9 0.007 2
    下载: 导出CSV

    表  3   LTLS算法和WTLS算法计算效率的比较

    Table  3   Comparison of Computational Efficiency Between LTLS Algorithm and WTLS Algorithm

    待估量数量 NLTLS NWTLS tLTLS tWTLS 减少比例/%
    10 5.18 4.5 0.224 ms 0.191 ms
    100 5.04 6.36 0.681 ms 0.724 ms 5.9
    1 000 5 6.75 0.033 s 0.044 s 25.0
    10 000 5 7.5 2.681 s 3.909 s 31.4
    下载: 导出CSV

    表  4   距离观测值及其中误差

    Table  4   Distance Observations and Standard Deviations

    统计项 l1/mm l2/mm l3/mm l4/mm l5/mm l6/mm y1/m y2/m
    观测值 14.1 16.6 6.1 7.1 22.1 26.3 10.0 8.0
    中误差 0.10 0.10 0.10 0.10 0.10 0.10 0.05 0.05
    下载: 导出CSV

    表  5   P1和点P2的坐标估值/m

    Table  5   Coordinate Estimates of P1 and P2 /m

    算法 坐标估值
    $ \hat x_1 $ $ \hat x_2 $ $ \hat x_3 $ $ \hat x_4 $
    LTLS算法 6.995 056 5 49.715 632 6.981 465 5 41.968 315 9
    WTLS算法 6.995 056 5 49.715 632 6.981 465 5 41.968 315 9
    下载: 导出CSV

    表  6   距离观测值估值

    Table  6   Estimation of Distance Observations

    算法 $ \hat l_1 $/mm $ \hat l_2 $/mm $ \hat l_3 $/mm $ \hat l_4 $/mm $ \hat l_5 $5/mm $ \hat l_6 $/mm $ \hat y_1 $/m $ \hat y_2 $/m
    LTLS算法 14.070 1 16.635 1 6.032 4 7.178 4 22.137 7 26.256 7 9.994 1 8.006 8
    WTLS算法 14.070 1 16.635 1 6.032 4 7.178 4 22.137 7 26.256 7 9.994 1 8.006 8
    下载: 导出CSV
  • [1] The Group of Surveying Adjustment in the School of Geodesy and Geomatics.Wuhan University.Error Theory and Foundation of Surveying Adjustment[M].Wuhan:Wuhan University Press, 2003.63-98(武汉大学测绘学院测量平差学科组. 误差理论与测量平差基础[M]. 武汉:武汉大学出版社,2003. 63-98).
    [2]

    VAN HUFFEL S, VANDEWALLE J. The Total Least Squares Problem:Computational Aspects and Analysis[M]. Philadelphia:Society for Industrial and Applied Mathematics, 1991.

    [3]

    Adcock R J. Note on the Method of Least Squares[J]. The Analyst, 1877,4(6):183-184.

    [4]

    Golub G H, Loan C F V. An Analysis of the Total Least Squares Problem[M]//An analysis of the total least squares problem., 1980.

    [5]

    Van Huffel S, Vandewalle J. The Total Least Squares Problem:Computational Aspects and Analysis[M]. Philadelphia:SLAM,1991

    [6]

    Yunzhong Shen, Bofeng Li, Yi Chen. An iterative solution of weighted total least-squares adjustment[J]. Journal of Geodesy, 2011, 85(4):229-238.

    [7]

    FANG Xing. Weighted Total Least Squares:Necessary and Sufficient Conditions, Fixed and Random Parameters[J]. Journal of Geodesy, 2013, 87(8):733-749

    [8] Liu Jingnan, Zeng Wenxian, Xu Peiliang. Overview of Total Least Squares Methods[J]. Geomatics and Information Science of Wuhan University, 2013, 38(5):505-512. (刘经南, 曾文宪, 徐培亮. 整体最小二乘估计的研究进展[J]. 武汉大学学报·信息科学版, 2013, 38(5):505-512).
    [9]

    FANG Xing. A structured and constrained Total Least-Squares solution with cross-covariances [J]. Studia Geophysica Et Geodaetica, 2014, 58(1):1-16.

    [10]

    FANG Xing. On non-combinatorial weighted total least squares with inequality constraints[J]. Journal of Geodesy, 2014, 88(8):805-816.

    [11]

    FANG Xing. Weighted total least-squares with constraints:a universal formula for geodetic symmetrical transformations[J]. Journal of Geodesy, 2015, 89(5):459-469.

    [12]

    Zeng Wenxian,Liu Jinnan,Yao Yibin, On Partial Errors-in-variables Models with Inequality Constraints of Parameters and Variables, Journal of Geodesy, 2015.2.89(2):111~119.

    [13] XIE Jian, LONG Sichun, LI Li,et al. An Aggregate Function Method for Weighted Total Least Squares with Inequality Constraints[J]. Geomatics and Information Science of Wuhan University, 2018, 43(10):1526-1530(谢建, 龙四春, 李黎, 等. 不等式约束加权整体最小二乘的凝聚函数法[J]. 武汉大学学报·信息科学版, 2018, 43(10):1526-1530)
    [14] LI Sida, LIU Lintao, LIU Zhiping,et al. Robust Total Least Squares Method for Multivariable EIV Model[J]. Geomatics and Information Science of Wuhan University, 2019, 44(8):1241-1248(李思达, 柳林涛, 刘志平,等. 多变量稳健总体最小二乘平差方法[J]. 武汉大学学报·信息科学版, 2019, 44(8):1241-1248).
    [15]

    Fang Xing, Zeng Wenxian, Zhou Yongjun, et al. On the Total Least Median of Squares adjustment for the pattern recognition in point clouds[J]. Measurement, 2020:107794.

    [16]

    XU Peiliang, LIU Jingnan, SHI, Chuang. Total least squares adjustment in partial Errors-in-variables models:algorithm and statistical analysis. Journal of geodesy, 2012, 86(8):661-675.

    [17] ZENG Wenxian, FANG Xing, LIU Jingnan,et al. Weighted Total Least Squares of Universal EIV Adjustment Model. Acta Geodaetica et Cartographica Sinica,2016, 45(8):890-894(曾文宪, 方兴, 刘经南,等. 通用EIV平差模型及其加权整体最小二乘估计[J]. 测绘学报, 2016, 45(8):890-894).
    [18]

    Amiri-Simkooei A R, Mortazavi S, Asgari J. Weighted total least squares applied to mixed observation model[J]. Empire Survey Review, 2015, 48(349):278-286.

    [19] ZENG Wenxian, TAO Benzao. Nonlinear model of three-dimensional coordinate transformation. Geomatics and Information Science of Wuhan University.2003, 28(5):566-568(曾文宪, 陶本藻. 三维坐标转换的非线性模型[J]. 武汉大学学报(信息科学版), 2003, 28(5):566-568).
    [20] ZENG Wenxian. Effect of the random design matrix on adjustment of an EIV model and its reliability theory[D].WuHan:WuHan University. 2013(曾文宪. 系数矩阵误差对EIV模型平差结果的影响研究[D].武汉:武汉大学, 2013).
    [21]

    Leberl F. Observations and Least Squares:Edward M. Mikhail, with contributions by F. Ackermann. Dun-Donelly, New York, N.Y. 1976, 497 pp. hard cover, U.S[J]. Photogrammetria, 1978, 34(6):261-262.

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出版历程
  • 收稿日期:  2020-05-23
  • 网络出版日期:  2023-09-26

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