Abstract
The 3D similarity transformation models, e.g. Bursa model is usually applied in geodesy and photogrammetry. In general, they are suitable in small angle 3D transformation. However, a lot of large 3D transformations need to be performed. This contribution describes a 3D similarity transformation model suitable for any angle rotation, where the nine elements in the rotation matrix are used to replace the three rotation angles as unknown parameters. In the coordinate transformation model, the Errors-In-Variables (EIV) model will be adjusted according to the theory of Least Squares (LS) method within the nonlinear Gauss–Helmert (GH) model. At the end of the contribution, case studies are investigated to demonstrate the coordinate transformation method proposed in this paper. The results show that using the linearized iterative GH model the correct solution can be obtained and this mixed model can be applied no matter whether the variance covariance matrices are full or diagonal.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Akyilmaz O (2007) Total least squares solution of coordinate transformation. Surv Rev 39(303):68–80
Felus YA, Burtch RC (2009) On symmetrical three-dimensional datum conversion. GPS Solut 13(1):65–74
Felus YA, Schaffrin B (2005) Performing similarity transformations using the error-in-variables model. In: ASPRS 2005 annual conference, Baltimore, March, pp 7–11
Golub HG, Van Loan FC (1980) An analysis of the total least squares problem. SIAM J Numer Anal 17(6):883–893
Leick A (2004) GPS satellite surveying, 3rd edn. Wiley, Hoboken
Lu J, Chen Y, Zheng B (2008) Research study on three-dimensional datum transformation using total least squares. J Geod Geodyn 28(5):77–81
Neitzel F (2010) Generalization of total least-squares on example of unweighted and weighted 2D similarity transformation. J Geod 84(12):751–762
Pope AJ (1972) Some pitfalls to be avoided in the iterative adjustment of nonlinear problems. In: Proceedings of the 38th annual meeting of the American society of photogrammetry, Washington, pp 449–477
Schaffrin B (2006) A note on constrained total least-squares estimation. Linear Algebra Appl 417:245–258
Schaffrin B, Felus YA (2008) On the multivariate total least-squares approach to empirical coordinate transformation: three algorithms. J Geod 82(6):373–383
Schaffrin B, Felus YA (2009) An algorithmic approach to the total least-squares problem with linear and quadratic constraints. Stud Geophys Geod 53:1–16
Schaffrin B, Wieser A (2008) On weighted total least-squares adjustment for linear regression. J Geod 82(7):415–421
Schaffrin B, Neitzel F, Uzum S (2009) Empirical similarity transformation via TLS-adjustment: exact solution vs. Cadzow’s approximation. In: International geomatics forum, Qingdao, pp 28–30
Van Huffel S, Vandewalle J (1991) The total least squares problem. Computational aspects and analysis. Front Appl Math 9:1–87 [SIAM, Philadelphia]
Acknowledgement
Financial support: National Natural Science Foundation of China, Grant No. 41074017.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2015 Springer International Publishing Switzerland
About this paper
Cite this paper
Lu, J., Chen, Y., Fang, X., Zheng, B. (2015). Performing 3D Similarity Transformation Using the Weighted Total Least-Squares Method. In: Kutterer, H., Seitz, F., Alkhatib, H., Schmidt, M. (eds) The 1st International Workshop on the Quality of Geodetic Observation and Monitoring Systems (QuGOMS'11). International Association of Geodesy Symposia, vol 140. Springer, Cham. https://doi.org/10.1007/978-3-319-10828-5_11
Download citation
DOI: https://doi.org/10.1007/978-3-319-10828-5_11
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-10827-8
Online ISBN: 978-3-319-10828-5
eBook Packages: Earth and Environmental ScienceEarth and Environmental Science (R0)