Partially error-affected point-wise weighted closed-form solution to the similarity transformation and its variants

Marcin Ligas 1
  • 1 AGH University of Science and Technology, Faculty of Mining Surveying and Environmental Engineering, Department of Integrated Geodesy and Cartography, Krakow, Poland
Marcin Ligas
  • Corresponding author
  • 49811AGH University of Science and Technology, Faculty of Mining Surveying and Environmental Engineering, Department of Integrated Geodesy and Cartography, Krakow, Poland
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The paper presents a closed-form solution to the point-wise weighted similarity transformation and its variants in the least squares framework under two estimation scenarios.

In the first scenario a target system is subject to random errors whilst in the second one a source system is considered to be erroneous. These transformation models will be named asymmetric in contrast to the symmetrical one solved under the errors-in-variables model where both systems are contaminated by random errors. The entire derivation is based on Procrustes Analysis. The formulas presented herein hold for both 2D and 3D transformations without any modification. The solution uses a polar decomposition to recover the rotation matrix.

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