1 Neiman Yu.M.
2 Sugaipova L.S.

Moscow State University of Geodesy and Cartography (MIIGAiK)

Determination of coordinate systems’ parameters transformation has always been and remains the core of surveyors’ practical and theoretical works. However, in most cases, the issue is interpreted under the assumption that the angular rotation of the coordinate axes and the similarity parameter are very small, which enables a significant simplifying of the algorithm. In this paper the strict theory of determining the 7 and 9 parameters of coordinate systems transformation within the framework of nonlinear method of least squares with limitations is considered. Strict algorithm is described and numerical experiments are performed. The connection of the coordinate systems transformation theory with the general one of mathematical statistics, known under the name of Procrust analysis, is specified. However, in general, artificial neural network theory is recommended for conversion of one coordinate system to another. This generally eliminates the need for a prior determination of the transformation’s appropriate parameters, inevitably associated with hypotheses for the transformation model, but quickly and accurately solves the main problem.
The article was prepared according to the state task FSBI Center of Geodesy, Cartography and SDI for 2022 in the framework of Research GEOTECH.
1.   Demmel' Dzh. Vychislitel'naya lineinaya algebra. Per. s angl. pod red. Kh. D. Ikramova. Moskva: Mir, 2001, 430 p.
2.   Kallan R. Osnovnye kontseptsii neironnykh setei. Per. s angl. A. G. Sivaka. Moskva: Vil'yamc, 2001, 287 p.
3.   Piskunov N. S. Differentsial'noe i integral'noe ischisleniya: Ucheb. dlya vtuzov. Moskva: Integral-Press, 2002, 2 Vol. 1, 416 p.
4.   Akca D. (2003) Generalized Procrustes analysis and its applications in photogrammetry. Zuerich: Institute of Geodesy and Photogrammetry. ETH-Hoenggerberg. 23 p. DOI: 10.3929/ethz-a-004656648.
5.   Awange J. L., Bae K.-H., Claessens S. J. (2008) Procrustean solution of the 9-parameter transformation problem. Earth Planets Space, no. 60, pp. 529–537.
6.   Crosilla F. (2003) Procrustes analysis and geodetic sciences. pp. 69–78. DOI: 10.1007/978-3-662-05296-9_29.
7.   Crosilla F., Maset E., Fusiello A. (2017) Procrustean photogrammetry: From exterior orientation to bundle adjustment. New advanced GNSS and 3D spatial techniques. pp. 157–165. DOI: 10.1007/978-3-319-56218-6_12.
8.   Hossainali M. M., Becker M., Groten E. (2011) Procrustean statistical inference of deformations. Journal of Geodetic Science, no. 1 (2), pp. 170–180. DOI: 10.2478/v10156-010-0020-5.
9.   Palancz B., Zaletnyik P., Awange J. L., Heck B. (2010) Extension of the ABC-Procrustes algorithm for 3D affine coordinate transformation. Earth Planets and Space, no. 62 (11), pp. 857–862.
10.   Paun C. D., Oniga V. E., Dragomir P. I. (2017) Three-dimensional transformation of coordinates systems using nonlinear analysis – Procrusters algorithm. International journal of engineering sciences and research technology, no. 6 (2), pp. 355–363. DOI: 10.5281/zenodo.291839.
11.   Schonemann P. H., Carroll R. M. (1970) Fitting one matrix to another under choice of a central dilation and a rigid motion. Psychometrika, no. 35 (2), pp. 245–255.
12.   Zeng H., Chang G., He H., Li K. (2020) WTLS iterative algorithm of 3D similarity coordinate transformation based on Gibbs vectors. Earth Planets and Space, no. 72 (53), pp. 1–12.
Neiman Yu.M., 
Sugaipova L.S., 
(2022) On the coordinate systems transformation. Geodesy and cartography = Geodezia i Kartografia, 83(9), pp. 21-29. (In Russian). DOI: 10.22389/0016-7126-2022-987-9-21-29