Abstract
Although the analytical solutions for total least-squares with multiple linear and single quadratic constraints were developed quite recently in different geodetic publications, these methods are restricted in number and type of constraints, and currently their computational efficiency and applications are mostly unknown. In this contribution, it is shown how the weighted total least-squares (WTLS) problem with arbitrary applicable constraints can be solved based on a Newton type methodology. This iterative process with quadratic convergence is expanded upon to become a compact solution for the WTLS with or without constraints. This compact solution is then further interpreted as a universal formula for the symmetrical adjustment of the errors-in-variables model which represents affine, similarity and rigid transformations in two- and three-dimensional space. Furthermore, statistical analysis of the constrained WTLS including the first-order approximation of precision and the bias was investigated. In order to substantiate our proposed method’s applicability, it was used to solve the affine, similarity and rigid transformation problem in two- and three-dimensional cases, where the structure of the coefficient matrix and multiple constraints were taken into account simultaneously.
Similar content being viewed by others
References
Amiri-Simkooei A, Jazaeri S (2012) Weighted total least squares formulated by standard least squares theory. J Geod Sci 2(2):113–124
Amiri-Simkooei A (2013) Application of least squares variance component estimation to errors-in-variables models. J Geod 87(10–12):935–944
Amiri-Simkooei A, Jazaeri S (2013) Data-snooping procedure applied to errors-in-variables models. Studia Geophysica et Geodaetica 57(3):426–441
Amiri-Simkooei A, Zangeneh-Nejad F, Asgari J, Jazaeri S (2014) Estimation of straight line parameters with fully correlated coordinates. Measurement 48:378–386
Beck A, Ben-Tal A (2006) On the solution of the Tikhonov regularization of the total least squares. SIAM J Optim 17:98–118
Borre K, Strang G (2012) Algorithms for global positioning. Wellesley-Cambridge Press, Cambridge
Box MJ (1971) Bias in nonlinear estimation (with discussions). J Roy stat Soc B 33:171–201
Cai J, Grafarend E (2009) Systematical analysis of the transformation between Gauss–Krueger–Coordinate/DHDN and UTM-coordinate/ETRS89 in Baden–Württemberg with different estimation methods. In: Geodetic reference frames, International Association of Geodesy Symposia, vol 134, pp 205–211
Dowling EM, Degroat RD, Linebarger DA (1992) Total least-squares with linear constraints. In: Acoustics, speech, and signal processing (ICASSP-92), vol 5, pp 341–344. doi:10.1109/ICASSP.1992.226613
De Moor B (1990) Total linear least squares with inequality constraints. ESAT-SISTA Report 1990-02, Department of Electrical Engineering, Katholieke Universiteit Leuven, Belgium
Fang X (2011) Weighted total least squares solution for application in geodesy. Dissertation, Leibniz University Hanover, Nr 294
Fang X (2013) Weighted total least squares: necessary and sufficient conditions, fixed and random parameters. J Geod 87(8):733–749
Fang X (2014a) A structured and constrained total least-squares solution with cross-covariances. Studia Geophysica et Geodaetica 58(1):1–16
Fang X (2014b) On non-combinatorial weighted total least squares with inequality constraints. J Geod 88(8):805–816
Fang X (2014c) A total least squares solution for geodetic datum transformations. Acta Geodaetica et Geophysica 49(2):189–207
Felus F, Burtch R (2009) On symmetrical three-dimensional datum conversion. GPS Sol 13(1):65–74
Golub G, Van Loan C (1980) An analysis of the total least-squares problem. SIAM J Numer Anal 17(6):883–893
Golub GH, Hansen PC, O’Leary DP (1999) Tikhonov regularization and total least-squares. SIAM J Matrix Anal Appl 21:185–194
Grafarend E, Awange JL (2012) Applications of linear and nonlinear models. Fixed effects, random effects, and total least squares. Springer, Berlin
Jazaeri S, Amiri-Simkooei AR, Sharifi MA (2014) An iterative algorithm for weighted total least-squares adjustment. Surv Rev 46:16–27
Mahboub V (2012) On weighted total least-squares for geodetic transformation. J Geod 86(5):359–367
Mahboub V, Sharifi MA (2013) On weighted total least-squares with linear and quadratic constraints. J Geod 87(3):279–286
Mikhail EM, Gracie G (1981) Analysis and adjustment of survey measurements. Van Nostrand Reinhold Company, New York
Neitzel F (2010) Generalization of total least-squares on example of unweighted and weighted 2D similarity transformation. J Geod 84:751–762
Nocedal J, Wright S (2006) Numerical optimization. Springer, Berlin
Schaffrin B, Felus YA (2005) On total least-squares adjustment with constraints. In: Sansò F (ed) A window on the future of geodesy. International Association of Geodesy Symposia, vol 128. Springer, Berlin, pp 175–180
Schaffrin B (2006) A note on constrained total least-squares estimation. Linear Alg Appl 417:245–258
Schaffrin B, Felus Y (2008) On the multivariate total least-squares approach to empirical coordinate transformations. Three algorithms. J Geod 82:373–383
Schaffrin B, Wieser A (2008) On weighted total least-squares adjustment for linear regression. J Geod 82:415–421
Schaffrin B, Felus Y (2009) An algorithmic approach to the total least-squares problem with linear and quadratic constraints. Studia Geophysica et Geodaetica 53(1):1–16
Schaffrin B, Wieser A (2011) Total least-squares adjustment of condition equation. Studia Geophysica et Geodaetica 55(3):529–536
Schaffrin B, Neitzel F, Uzun S, Mahboub V (2012) Modifying Cadzow’s algorithm to generate the optimal TLS-solution for the structured EIV-model of a similarity transformation. J Geod Sci 2(2):98–106
Shen Y, Li B, Chen Y (2011) An iterative solution to weighted total least squares adjustment. J Geod 85(4):229–238. doi:10.1007/s00190-010-0431-1
Sima DM, van Huffel S, Golub GH (2004) Regularized total least-squares based on quadratic eigenvalue problem solver. Bit 44:793–812
Snow K (2012) Topics in total least-squares adjustment within the errors-in-variables model: singular cofactor matrices and priori information. PhD Dissertation, report No, 502, Geodetic Science Program, School of Earth Sciences, The Ohio State University, Columbus Ohio, USA
Teunissen PJG (1985) The geometry of geodetic inverse linear mapping and nonlinear adjustment. Netherlands Geodetic Commission, Publications on Geodesy, new series, vol. 8, no 1, pp 1–186
Teunissen PJG (1988) The non-linear 2D symmetric Helmert transformation: an exact non-linear least-squares solution. J Geod 62(1):1–15
Teunissen PJG (1989a) A note on the bias in the Symmetric Helmert Transformation, Festschrift Torben Krarup, pp 1–8
Teunissen PJG (1989b) First and second moments of nonlinear least-squares estimators. J Geod 63:253–262
Teunissen PJG (1990) Nonlinear least squares. Manuscripta Geodaetica 15(3):137–150
Van Huffel S, Vandewalle J (1991) The total least -squares problem. Computational aspects and analysis. Society for Industrial and Applied Mathematics, Philadelphia
Xu PL, Liu JN, Shi C (2012) Total least squares adjustment in partial errors-in-variables models: algorithm and statistical analysis. J Geod 86(8):661–675
Xu PL, Liu JN (2014) Variance components in errors-in-variables models: estimability, stability and bias analysis. J Geod 88(8):719–734
Zhang S, Tong X, Zhang K (2013) A solution to EIV model with inequality constraints and its geodetic applications. J Geod 87(1):23–28
Acknowledgments
I would like to thank the President at the Federal Agency for Cartography and Geodesy (BKG) in Germany, Prof. Kutterer, for his guidance. The author thanks three reviewers for their constructive comments. The second reviewer is particularly appreciated for his broad knowledge on nonlinear LS estimation and suggestions for the quality description, which have been fully implemented to improve the paper. This research was supported by the National Natural Science Foundation of China (41404005; 41474006) and the Fundamental Research Funds for the Central Universities (2042014kf053).
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Fang, X. Weighted total least-squares with constraints: a universal formula for geodetic symmetrical transformations. J Geod 89, 459–469 (2015). https://doi.org/10.1007/s00190-015-0790-8
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00190-015-0790-8