Skip to main navigation menu Skip to main content Skip to site footer

Vigorous 3D Angular Resection Model Using Levenberg – Marquardt Method

Abstract

The resection in 3D space is a common problem in surveying engineering and photogrammetry based on observed distances, angles, and coordinates. This resection problem is nonlinear and comprises redundant observations which is normally solved using the least-squares method in an iterative approach. In this paper, we introduce a vigorous angular based resection method that converges to the global minimum even with very challenging starting values of the unknowns. The method is based on deriving oblique angles from the measured horizontal and vertical angles by solving spherical triangles. The derived oblique angles tightly connected the rays enclosed between the resection point and the reference points. Both techniques of the nonlinear least square adjustment either using the Gauss-Newton or Levenberg – Marquardt are applied in two 3D resection experiments. In both numerical methods, the results converged steadily to the global minimum using the proposed angular resection even with improper starting values. However, applying the Levenberg – Marquardt method proved to reach the global minimum solution in all the challenging situations and outperformed the Gauss-Newton method.

Keywords

3D Resection, Oblique Angle, Nonlinear Least Squares, Gauss-Newton, Levenberg, Marquardt

PDF

Author Biography

Yaseen T. Mustafa

Dept. of Environmental Science, Faculty of Science, University of Zakho, Kurdistan Region-Iraq;
Center of Applied Remote Sensing and GIS, University of Zakho, Kurdistan Region-Iraq


References

  1. S. Bektas, N. Karaahmetoglu, R. Yildirm, A. Uyar, U. K. Yildirim, and E. Pakel, "Three Point Resection in 3D With Distances," International Journal of Research -GRANTHAALAYAH, vol. 6, no. 2, pp. 2394-3629, 2018, doi: 10.5281/zenodo.1194735.
  2. S. J. Gordon and D. D. Lichti, "Terrestrial Laser Scanners with A Narrow Field of View: The Effect on 3D Resection Solutions," Survey Review, vol. 37, no. 292, pp. 448-468, 2004/04/01 2004, doi: 10.1179/sre.2004.37.292.448.
  3. F. A. Shepherd, Advanced Engineering Surveying: Problems and Solutions. Hodder Arnold 1982.
  4. B. Alsadik, "A Modified Method for Image Triangulation Using Inclined Angles," Int. Arch. Photogramm. Remote Sens. Spatial Inf. Sci., vol. XLI-B3, pp. 453-458, 2016, doi: 10.5194/isprs-archives-XLI-B3-453-2016.
  5. W. Schofield, Engineering surveying : theory and examination problems for students / third ed. London: Butterworths, 2007.
  6. B. Alsadik, "Robust Resection Model for Aligning the Mobile Mapping Systems Trajectories at Degraded and Denied Urban Environments," ISPRS Ann. Photogramm. Remote Sens. Spatial Inf. Sci., vol. V-1-2020, pp. 215-222, 2020, doi: 10.5194/isprs-annals-V-1-2020-215-2020.
  7. N. Borlin. "Nonlinear Optimization Least Squares Problems — The Gauss-Newton method." Umeå University. https://www8.cs.umu.se/kurser/5DA001/HT07/lectures/lsq-handouts (accessed March 15th, 2020).
  8. D. W. Marquardt, "An algorithm for least-squares estimation of nonlinear parameters," SIAM Journal on Applied Mathematics, vol. 11, no. 2, pp. 431-441, 1963.
  9. K. Levenberg, "A method for the solution of certain nonlinear problems in least squares," Quarterly of Applied Mathematics, no. 2, pp. 164–168, 1944.
  10. H. P. Gavin. "The Levenberg-Marquardt method for nonlinear least squares curve-fitting " http://people.duke.edu/~hpgavin/ce281/lm.pdf (accessed January 20th, 2019).
  11. COPASI. "Levenberg - Marquardt." http://copasi.org/Support/User_Manual/Methods/Optimization_Methods/Levenberg_-_Marquardt/ (accessed January 10th, 2019).
  12. B. Alsadik, Adjustment Models in 3D Geomatics and Computational Geophysics: With MATLAB Examples. Elsevier Science, 2019.
  13. B. Alsadik, "Robust Resection Model for Aligning the Mobile Mapping Systems Trajectories at Degraded and Denied Urban Environments," in ISPRS, Nice, France, 2020: ISPRS.

Metrics

Metrics Loading ...