二维直角建筑物规则化的加权总体最小二乘平差方法
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摘要
针对基于遥感数据的二维建筑物的直角化问题,以建筑物边界点的坐标为观测值,以顾及边界正交限制条件的直线斜率和截距为参数,建立附有限制条件的变量误差(errors-in-variables, EIV)模型。考虑观测向量和设计矩阵相关的情况,给出了增广设计矩阵的协方差阵的计算方法,推导了附限制条件的通用加权总体最小二乘(weighted total least squares, WTLS)平差算法,以及近似精度评定算法和仅含二次型限制条件的WTLS平差方法。理论和算例分析表明,在建筑物重建问题中,附有限制条件的EIV模型比经典附有限制条件的Gauss-Helmert模型易于构建,所提的WTLS算法快速收敛速度快,对拓展WTLS平差方法的应用具有理论与实践意义。
An errors-in-variable(EIV) with arbitrary constraints is proposed for the purpose of buil-ding regularization from remote sensed data, in which the edge points are treated as measurements, the constrained slopes and intercepts of each edge are chosen as parameters. Assuming the measurement vector and the design matrix are mutually correlated, the scheme of calculating the dispersion matrix of augmented matrix is suggested. A generic constrained weighted total least squares(WTLS) algorithm is derived with an approximate accuracy assessment method, and the WTLS algorithm of a quadratic constrained EIV problem is given as a specific case. Theoretic analysis and data experiment demonstrate the advantages of an EIV model compared with a Gauss-Helmert model in building regularization problem, and the rapid convergence rate of proposed WTLS algorithm. It aims to promote WTLS adjustment methods, and to expand the applications of total least squares method in new surveying technology with a certain theoretical and practical significance.
An errors-in-variable(EIV) with arbitrary constraints is proposed for the purpose of buil-ding regularization from remote sensed data, in which the edge points are treated as measurements, the constrained slopes and intercepts of each edge are chosen as parameters. Assuming the measurement vector and the design matrix are mutually correlated, the scheme of calculating the dispersion matrix of augmented matrix is suggested. A generic constrained weighted total least squares(WTLS) algorithm is derived with an approximate accuracy assessment method, and the WTLS algorithm of a quadratic constrained EIV problem is given as a specific case. Theoretic analysis and data experiment demonstrate the advantages of an EIV model compared with a Gauss-Helmert model in building regularization problem, and the rapid convergence rate of proposed WTLS algorithm. It aims to promote WTLS adjustment methods, and to expand the applications of total least squares method in new surveying technology with a certain theoretical and practical significance.
引文
[1] Markovsky I, van Huffel S. Overview of Total Least Squares Methods[J]. Signal Process,2007, 87(10): 2 283-2 302
[2] Markovsky I, Rastello M, Premoli P, et al. The Element-Wise Weighted Total Least Squares Problem[J]. Computational Statistics & Data Analysis, 2006,50(1):181-209
[3] Schaffrin B, Wieser A. On Weighted Total Least-Squares Adjustment for Linear Regression[J]. Journal of Geodesy,2008, 82(7):415-421
[4] Shen Y Z, Li B F, Chen Y. An Iterative Solution of Weighted Total Least-Squares Adjustment[J]. Journal of Geodesy , 2011,85(4):229-238
[5] Mahboub V. On Weighted Total Least-Squares for Geodetic Transformations [J]. Journal of Geodesy, 2012,86(5):359-367
[6] Snow K. Topics in Total Least-Squares Adjustment Within the Errors-in-Variables Model: Singular Cofactor Matrices and Prior Information[D]. Ohio: Ohio State University, 2012
[7] Fang X. Weighted Total Least Squares: Necessary and Sufficient Conditions, Fixed and Random Parameters[J]. Journal of Geodesy,2013, 87(8):733-749
[8] Zhou Yongjun, Kou Xinjian, Zhu Jianjun. A Newton Algorithm for Weighted Total Least-Squares Solution to a Specific Errors-in-Variables Model with Correlated Measurements[J]. Stud Geophys Geod, 2014,58(3): 349-375
[9] van Huffel S, Lemmerling P. Total Least Squares and Errors-in-Variables Modeling: Analysis, Algorithms and Applications[M].Dordrecht: Kluwer Academic Publishers, 2002
[10] Schaffrin B.A Note on Constrained Total Least-Squares Estimation[J].Linear Algebra Appl,2006, 417(1):245-258
[11] Golub G H, Hansen P C, O’Leary D P. Tikhonov Regularization and Total Least Squares[J].SIAM J Matrix Anal Appl ,1999,21(1):185-194
[12] Beck A, Ben-Tal A.On the Solution of the Tikhonov Regularization of Total Least-Squares[J].SIAM J Optim, 2006, 17(1):98-118
[13] Schaffrin B, Felus A Y. An Algorithmic Approach to the Total Least-Squares Problem with Linear and Quadratic Constraints[J]. Stud Geophys Geod, 2009,53(1): 1-16
[14] Mahboub V, Sharifi M A. On Weighted Total Least-Squares with Linear and Quadratic Constraints[J]. J Geod, 2013,87(3): 279-286
[15] Zhang S L, Tong X H, Zhang K. A Solution to EIV Model with Inequality Constraints and Its Geodetic Applications[J]. J Geod, 2013, 87(1):23-28
[16] Zeng W, Liu J, Yao Y. On Partial Errors-in-Variables Models with Inequality Constraints of Parameters and Variables[J]. J Geod, 2015, 89(2):111-119
[17] Fang X. Weighted Total Least-Squares with Constraints: A Universal Formula for Geodetic Symmetrical Transformations[J]. J Geod, 2015, 89(2):1-11
[18] Tong Xiaohua, Deng Susu, Shi Wenzhong.A New Least Squares Adjustment Method for Map Conflation[J]. Geomatics and Information Science of Wuhan University, 2007, 32(7):621-625 (童小华, 邓愫愫, 史文中. 数字地图合并的平差原理与方法[J]. 武汉大学学报·信息科学版, 2007, 32(7):621-625)
[19] Sampath A, Shan J. Building Boundary Tracing and Regularization from Airborne LiDAR Point Clouds[J]. Photogrammetric Engineering & Remote Sensing, 2007, 73(7):805-812
[20] Xu P L, Liu J N, Shi C. Total Least Squares Adjustment in Partial Errors-in-Variables Models: Algorithm and Statistical Analysis[J]. J Geod, 2012, 86(8):661-675
[21] Nocedal J, Wright S. Numerical Optimization[M]. Berlin: Springer, 2006
[2] Markovsky I, Rastello M, Premoli P, et al. The Element-Wise Weighted Total Least Squares Problem[J]. Computational Statistics & Data Analysis, 2006,50(1):181-209
[3] Schaffrin B, Wieser A. On Weighted Total Least-Squares Adjustment for Linear Regression[J]. Journal of Geodesy,2008, 82(7):415-421
[4] Shen Y Z, Li B F, Chen Y. An Iterative Solution of Weighted Total Least-Squares Adjustment[J]. Journal of Geodesy , 2011,85(4):229-238
[5] Mahboub V. On Weighted Total Least-Squares for Geodetic Transformations [J]. Journal of Geodesy, 2012,86(5):359-367
[6] Snow K. Topics in Total Least-Squares Adjustment Within the Errors-in-Variables Model: Singular Cofactor Matrices and Prior Information[D]. Ohio: Ohio State University, 2012
[7] Fang X. Weighted Total Least Squares: Necessary and Sufficient Conditions, Fixed and Random Parameters[J]. Journal of Geodesy,2013, 87(8):733-749
[8] Zhou Yongjun, Kou Xinjian, Zhu Jianjun. A Newton Algorithm for Weighted Total Least-Squares Solution to a Specific Errors-in-Variables Model with Correlated Measurements[J]. Stud Geophys Geod, 2014,58(3): 349-375
[9] van Huffel S, Lemmerling P. Total Least Squares and Errors-in-Variables Modeling: Analysis, Algorithms and Applications[M].Dordrecht: Kluwer Academic Publishers, 2002
[10] Schaffrin B.A Note on Constrained Total Least-Squares Estimation[J].Linear Algebra Appl,2006, 417(1):245-258
[11] Golub G H, Hansen P C, O’Leary D P. Tikhonov Regularization and Total Least Squares[J].SIAM J Matrix Anal Appl ,1999,21(1):185-194
[12] Beck A, Ben-Tal A.On the Solution of the Tikhonov Regularization of Total Least-Squares[J].SIAM J Optim, 2006, 17(1):98-118
[13] Schaffrin B, Felus A Y. An Algorithmic Approach to the Total Least-Squares Problem with Linear and Quadratic Constraints[J]. Stud Geophys Geod, 2009,53(1): 1-16
[14] Mahboub V, Sharifi M A. On Weighted Total Least-Squares with Linear and Quadratic Constraints[J]. J Geod, 2013,87(3): 279-286
[15] Zhang S L, Tong X H, Zhang K. A Solution to EIV Model with Inequality Constraints and Its Geodetic Applications[J]. J Geod, 2013, 87(1):23-28
[16] Zeng W, Liu J, Yao Y. On Partial Errors-in-Variables Models with Inequality Constraints of Parameters and Variables[J]. J Geod, 2015, 89(2):111-119
[17] Fang X. Weighted Total Least-Squares with Constraints: A Universal Formula for Geodetic Symmetrical Transformations[J]. J Geod, 2015, 89(2):1-11
[18] Tong Xiaohua, Deng Susu, Shi Wenzhong.A New Least Squares Adjustment Method for Map Conflation[J]. Geomatics and Information Science of Wuhan University, 2007, 32(7):621-625 (童小华, 邓愫愫, 史文中. 数字地图合并的平差原理与方法[J]. 武汉大学学报·信息科学版, 2007, 32(7):621-625)
[19] Sampath A, Shan J. Building Boundary Tracing and Regularization from Airborne LiDAR Point Clouds[J]. Photogrammetric Engineering & Remote Sensing, 2007, 73(7):805-812
[20] Xu P L, Liu J N, Shi C. Total Least Squares Adjustment in Partial Errors-in-Variables Models: Algorithm and Statistical Analysis[J]. J Geod, 2012, 86(8):661-675
[21] Nocedal J, Wright S. Numerical Optimization[M]. Berlin: Springer, 2006