# Weighted total least-squares with constraints: a universal formula for geodetic symmetrical transformations

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## 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.

## Keywords

Total least-squares Errors-in-variables model Nonlinear constraints Newton iteration Universal formula Statistical analysis Geodetic transformations## Notes

### 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).

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