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Journal of Geodetic Science

Editor-in-Chief: Eshagh, Mehdi

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Closed-form and iterative weighted least squares solutions of Helmert transformation parameters

L.E. Sjöberg
  • Division of Geodesy and Geoinformatics, Royal Institute of Technology (KTH), SE 10044, Stockholm, Sweden
  • Other articles by this author:
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Published Online: 2013-04-30 | DOI: https://doi.org/10.2478/jogs-2013-0002


The Helmert transformation is the most common transformation between different geodetic systems. In 2-D, in contrast to higher dimensions, it is a well-known procedure how to determine the 4 transformation parameters in a closed form. Here we derive the closed-form weighted least squares solution in m-dimensional space for an arbitrary number (≥ m) of coordinate set-ups in two related systems. The solution employs singular value decomposition (SVD) for the rotation matrix, while the translation vector and scale parameters are obtained in simpler ways. To avoid the SVD routine, we also present an iterative approach to solve for the rotation matrix. The paper is completed with a test procedure for detecting outlying coordinate pairs.

Keywords: Helmert transformation; Procrustes problem; registration; singular value decomposition

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About the article

Published Online: 2013-04-30

Published in Print: 2013-03-01

Citation Information: Journal of Geodetic Science, Volume 3, Issue 1, Pages 7–11, ISSN (Print) 2081-9943, DOI: https://doi.org/10.2478/jogs-2013-0002.

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