Noniterative Approach for Solving the Indirect Problems of Linear Reference Frame Transformations
Publication: Journal of Surveying Engineering
Volume 136, Issue 4
Abstract
A linear transformation, including a similarity model and an affine model, plays an important role in reference frame transformation problems. It establishes geometric relations between different reference frames by adopting physically interpretable parameters, and allows one to integrate positional coordinates across various systems. In this study, an analytical noniterative approach for solving the indirect problem (i.e., parameter estimation) of a linear transformation model is proposed. It gives a closed-form solution for the transformation without any prerequisite on the magnitude of parameters. Consequently, neither good initial values nor iterations are required during the parameter estimation process. Through numerical examples, it is illustrated that the proposed approach is capable of giving accurate parameter estimates as a classic least-squares technique does, but with much higher computational efficiency. In a reference frame transformation problem of 1,000 control points, the proposed approach takes less than 0.1 s to produce an accurate solution, while an ordinary least-squares approach takes 10–17 s to finish the same analysis. With the above advantages, this approach would especially benefit the cases in which a real-time application or large network is involved in an analysis.
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Acknowledgments
The writer thanks Dr. Tomás Soler and the anonymous reviewers for their constructive comments, which significantly improved the quality of the original manuscript. The funding support by the National Science Council of Taiwan (under Contract No. NSCTNSC 96-2218-E-002-027) is also gratefully acknowledged.
References
Altamimi, Z., Sillard, P., and Boucher, C. (2002). “ITRF2000: A new release of the international terrestrial reference frame for earth science applications.” J. Geophys. Res., 107(B10), 2214.
Awange, J. L., Bae, K. B., and Claessens, S. (2008). “Procrustean solution of the 9-parameter transformation problem.” Earth, Planets Space, 60(6), 529–537.
Awange, J. L., and Grafarend, E. W. (2003). “Closed form solution of the overdetermined nonlinear 7 parameter datum transformation.” Allgemeine Vermessungs-Nachrichten, 110, 130–148.
Awange, J. L., and Grafarend, E. W. (2005). Solving algebraic computational problems in geodesy and geoinformatics, Springer, Berlin.
Badekas, J. (1969). “Investigations related to the establishment of a world geodetic system.” Rep. No. 124, Dept. of Geodetic Science, Ohio State Univ., Columbus, Ohio.
Burša, M. (1967). “On the possibility of determining the rotation elements of geodetic reference systems on the basis of satellite observations.” Stud. Geophys. Geod., 11(4), 390–396.
Chen, L. -C., Lo, C. Y., and Rau, J. Y. (2003). “Generation of digital orthophotos from IKONOS satellite images.” J. Surv. Eng., 129(2), 73–78.
Han, J. Y. (2006). “Time-variant transformations for modern terrestrial reference frames.” Ph.D. thesis, Purdue Univ., West Lafayette, Ind.
Han, J. Y., and van Gelder, B. H. W. (2006). “Stepwise parameter estimations for a time-variant similarity transformation.” J. Surv. Eng., 132(4), 141–148.
Han, J. Y., van Gelder, B. H. W., and Lin, S. L. (2010). “Rotation- and translation-free estimation of symmetric, rank-two tensors with a case study in LIDAR surveying.” J. Surv. Eng., 136(1), 23–28.
Han, J. Y., van Gelder, B. H. W., Soler, T., and Snay, R. A. (2008). “Geometric combination of multiple terrestrial network solutions.” J. Surv. Eng., 134(4), 126–131.
Jaw, J. J., and Chuang, T. Y. (2008). “Registration of ground-based LIDAR point clouds by means of 3D line features.” J. Chin. Inst. Eng., 31(6), 1031–1045.
Leick, A., and van Gelder, B. H. W. (1975). “On similarity transformations and geodetic network distortions based on Doppler satellite observations.” Rep. No. 235, Dept. of Geodetic Science, Ohio State Univ., Columbus, Ohio.
Mikhail, E. M., and Ackermann, F. (1976). Observations and least squares, IEP-A Dun-Donnelley Publisher, New York.
Mikhail, E. M., Bethel, J. S., and McGlone, J. C. (2001). Introduction to modern photogrammetry, Wiley, New York.
Molodenskii, M. S., Eremeev, V. F., and Yurkina, M. I. (1962). Methods for study of the external gravitational field and figure of the earth (transl. from Russian 1960), National Technical Information Service, Washington, D.C.
Rabbani, T., Dijkman, S., van den Heuvel, F., and Vosselman, G. (2007). “An integrated approach for modelling and global registration of point clouds.” ISPRS J. Photogramm. Remote Sens., 61(6), 355–370.
Soler, T. (1998). “A compendium of transformation formulas useful in GPS work.” J. Geodyn., 72(7–8), 482–490.
Soler, T., and Snay, R. A. (2004). “Transforming positions and velocities between the international terrestrial reference frame of 2000 and North American datum of 1983.” J. Surv. Eng., 130(2), 49–55.
Soler, T., and van Gelder, B. H. W. (1987). “On differential scale changes and the satellite Doppler system Z-shift.” Geophys. J. R. Astron. Soc., 91(3), 639–656.
van Gelder, B. H. W. (2003). “Geodesy.” The civil engineering handbook, W. F. Chen and J. Y. Richard Liew, eds., CRC, Boca Raton, Fla.
Vaniček, P., and Wells, D. E. (1974). “Positioning of horizontal geodetic datums.” Can. Surveyor, 28(5), 531–538.
von Hansen, W., Gross, H., and Thoennessen, U. (2008). “Line-based registration of terrestrial and airborne LIDAR data.” Int. Arch. Photogramm. Remote Sens. Spat. Inf. Sci., 37(B3a), 161–166.
Wang, J., Di, K., and Li, R. (2005). “Evaluation and improvement of geopositioning accuracy of IKONOS stereo imagery.” J. Surv. Eng., 131(2), 35–42.
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Published In
Journal of Surveying Engineering
Volume 136 • Issue 4 • November 2010
Pages: 150 - 156
Copyright
© 2010 ASCE.
History
Received: Apr 19, 2009
Accepted: Dec 30, 2009
Published online: Jan 5, 2010
Published in print: Nov 2010
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