Gauss-Helmert Model for Direct Solution to Absolute Orientation

Öz

Absolute orientation is an old and fundamental task in photogrammetry, computer vision and robotics. It involves obtaining the transformation of the stereo model developed from pairs of photographs from the image coordinate system to the object coordinate system. The traditional solution for absolute orientation is a numerical iterative least squares solution, which operates a linearisation process that requires a good approximation of the transformation parameters. In this study, we focus on the Gauss-Helmert model, which is used in three-dimensional similarity transformation for solving absolute orientation and provides a 'direct' solution. In order to test the performance of the proposed direct solution method, a coordinate data set with known properties in the literature is used for numerical implementation and analyses. The transformation parameters and accuracy values estimated by the conventional GMM method are compared with the results obtained by the direct solution GHM model. In general, for absolute orientation problems where the rotation angles are relatively small, the use of direct solution methods may be an appropriate choice. Numerical experiments have shown that the GHM model can produce more accurate transformation parameters with less time requirement.

Anahtar Kelimeler

• Absolute orientation
• least squares
• coordinate transformation
• Gauss-Helmert model

Mutlak Yöneltmenin Doğrudan Çözümü İçin Gauss-Helmert Modeli

Öz

Mutlak yöneltme, fotogrametri, bilgisayarlı görü (computer vision) ve robot biliminde eski ve temel görevlerden biridir. Fotoğraf çiftlerinden geliştirilen stereo modelin görüntü koordinat sisteminden nesne koordinat sistemine dönüşümünün elde edilmesini içerir. Mutlak yöneltmeye yönelik geleneksel çözüm, dönüşüm parametrelerinin iyi bir yaklaşımını gerektiren bir doğrusallaştırma sürecini işleten sayısal (nümerik) yinelemeli en küçük kareler çözümüdür. Bu çalışmada ise mutlak yöneltmenin çözümü için üç boyutlu benzerlik dönüşümünde kullanılan ve ‘doğrudan’ çözüm sağlayan Gauss-Helmert Modeli üzerinde durulmuştur. Önerilen doğrudan çözüm yönteminin performansını test etmek üzere; literatürde özellikleri bilinen bir koordinat veri seti sayısal uygulama ve analizler için kullanılmıştır. Geleneksel GMM yöntemiyle kestirilen dönüşüm parametreleri ve doğruluk değerleri, doğrudan çözüm sağlayan GHM modeliyle elde edilen sonuçlarla karşılaştırılmıştır. Genel olarak dönme açılarının nispeten küçük olduğu mutlak yöneltme problemlerinde doğrudan çözüm yöntemlerinin kullanılması uygun bir tercih olabilir. Örnek sayısal deneylerde GHM modelinin daha az zaman gereksinimiyle, daha duyarlı dönüşüm parametreleri üretebileceği gösterilmiştir.

Anahtar Kelimeler

• Mutlak yöneltme
• En küçük kareler
• Koordinat dönüşümü
• Gauss-Helmert modeli

Kaynakça

1. Akyılmaz, O., Acar, M., & Özlüdemir, M. (2007). Koordinat dönüşümünde En Küçük Kareler ve Toplam En Küçük Yöntemleri. Jeodezi ve Jeoinformasyon Dergisi,(97), 15-22.
2. Ansar, A., & Daniilidis, K. (2003). Linear pose estimation from points or lines. IEEE Transactions on Pattern Analysis and Machine Intelligence, 25(5), 578-589.
3. Arun, K. S., Huang, T. S., & Blostein, S. D. (1987). Least-squares fitting of two 3-D point sets. IEEE Transactions on pattern analysis and machine intelligence, (5), 698-700.
4. Chang, G. (2015). On least-squares solution to 3D similarity transformation problem under Gauss–Helmert model. Journal of Geodesy, 89(6), 573-576.
5. Chang, G., Xu, T., & Wang, Q. (2017). Error analysis of the 3D similarity coordinate transformation. GPS Solutions, 21, 963-971.
6. Eggert, D. W., Lorusso, A., & Fisher, R. B. (1997). Estimating 3-D rigid body transformations: a comparison of four major algorithms. Machine vision and applications, 9(5-6), 272-290.
7. Felus, Y. A., & Burtch, R. C. (2009). On symmetrical three-dimensional datum conversion. GPS solutions, 13, 65-74.
8. Fiore, P. D. (2001). Efficient linear solution of exterior orientation. IEEE Transactions on Pattern Analysis and Machine Intelligence, 23(2), 140-148.

9. Ghilani, C. D. (2017). Adjustment computations: spatial data analysis. John Wiley & Sons.
10. Grafarend, E. W., & Awange, J. L. (2003). Nonlinear analysis of the three-dimensional datum transformation [conformal group ℂ 7 (3)]. Journal of Geodesy, 77, 66-76.
11. Grussenmeyer, P., & Khalil, O. A. (2002). Solutions for exterior orientation in photogrammetry: a review. The photogrammetric record, 17(100), 615-634.
12. Guo, K., Ye, H., Zhao, Z., & Gu, J. (2021). An efficient closed form solution to the absolute orientation problem for camera with unknown focal length. Sensors, 21(19), 6480.
13. Horn, B. K. (1987). Closed-form solution of absolute orientation using unit quaternions. Josa a, 4(4), 629-642.
14. Horn, B. K., Hilden, H. M., & Negahdaripour, S. (1988). Closed-form solution of absolute orientation using orthonormal matrices. Josa a, 5(7), 1127-1135.
15. Jiang, G., Wang, J., & Zhang, R. (2007). A close-form solution of absolute orientation using unit quaternions. Journal of Zhengzhou Institute of Surveying and Mapping, 24(3), 193-195.
16. Kraus, K. (1997). Photogrammetry: Advanced methods and applications. Dümmler.
17. Kraus, K. (2007). Photogrammetry: geometry from images and laser scans (1). Walter de Gruyter.
18. Kumar, R., & Hanson, A. R. (1994). Robust methods for estimating pose and a sensitivity analysis. CVGIP: Image understanding, 60(3), 313-342.
19. Kurt, O. (2018). An integrated solution for reducing ill-conditioning and testing the results in non-linear 3D similarity transformations. Inverse problems in science and engineering, 26(5), 708-727.
20. Leick, A., Rapoport, L., & Tatarnikov, D. (2015). GPS satellite surveying. John Wiley & Sons.
21. Lepetit, V., Moreno-Noguer, F., & Fua, P. (2009). EP n P: An accurate O (n) solution to the P n P problem. International journal of computer vision, 81, 155-166.
22. Li, J., Hu, Q., Zhong, R., & Ai, M. (2017). Exterior orientation revisited: A robust method based on lq-norm. Photogrammetric Engineering & Remote Sensing, 83(1), 47-56.
23. Mahboub, V. (2012). On weighted total least-squares for geodetic transformations. Journal of geodesy, 86(5), 359-367.
24. Neitzel, F. (2010). Generalization of total least-squares on example of unweighted and weighted 2D similarity transformation. Journal of geodesy, 84, 751-762.
25. Schaffrin, B. (2006). A note on constrained total least-squares estimation. Linear algebra and its applications, 417(1), 245-258.
26. Schaffrin, B. (2006). A note on constrained total least-squares estimation. Linear algebra and its applications, 417(1), 245-258.
27. Schaffrin, B., & Felus, Y. A. (2008). On the multivariate total least-squares approach to empirical coordinate transformations. Three algorithms. Journal of Geodesy, 82, 373-383.
28. Teunissen, P. J. (1985). The geometry of geodetic inverse linear mapping and non-linear adjustment. Netherlands Geodetic Commission, 8(1).
29. Teunissen, P. J. G. (1985). Generalized inverses, adjustment, the datum problem and S-transformations. Optimization of Geodetic Networks.
30. Xu, P., Liu, J., & Shi, C. (2012). Total least squares adjustment in partial errors-in-variables models: algorithm and statistical analysis. Journal of geodesy, 86, 661-675.
31. Yan, L., Wan, J., Sun, Y., Fan, S., Yan, Y., & Chen, R. (2016). A novel absolute orientation method using local similarities representation. ISPRS International Journal of Geo-Information, 5(8), 135.
32. Yang, L., Sheng, Y., & Wang, B. (2016). 3D reconstruction of building facade with fused data of terrestrial LiDAR data and optical image. Optik, 127(4), 2165-2168.
33. Zeng, H., & Yi, Q. (2012). Simple and Efficient Direct Solution to Absolute Orientation. In Future Control and Automation: Proceedings of the 2nd International Conference on Future Control and Automation (ICFCA 2012), 2, 19-25. Springer Berlin Heidelberg.
34. Zhou, L., & Kaess, M. (2019, November). An efficient and accurate algorithm for the perspecitve-n-point problem. In 2019 IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS) (6245-6252). IEEE.