Optimal estimation of three-dimensional rotation and reliability evaluation

  • Naoya Ohta
  • Kenichi Kanatani
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 1406)


We discuss optimal rotation estimation from two sets of 3-D points in the presence of anisotropic and inhomogeneous noise. We first present a theoretical accuracy bound and then give a method that attains that bound, which can be viewed as describing the reliability of the solution. We also show that an efficient computational scheme can be obtained by using quaternions and applying renormalization. Using real stereo images for 3-D reconstruction, we demonstrate that our method is superior to the least-squares method and confirm the theoretical predictions of our theory by applying the bootstrap procedure.


Stereo Image Stereo Vision Moment Matrix Theoretical Accuracy True Rotation 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    K. S. Arun, T. S. Huang and S. D. Blostein, Least-squares fitting of two 3-D point sets, IEEE Trans. Patt. Anal. Mach. Intell., 9-5 (1987), 698–700.Google Scholar
  2. 2.
    B. Efron and R. J. Tibshirani, An Introduction to Bootstrap, Chapman-Hall, New York, 1993.zbMATHGoogle Scholar
  3. 3.
    B. K. P. Horn, Closed-form solution of absolute orientation using unit quaternions, J. Opt. Soc. Am., A4 (1987), 629–642.CrossRefGoogle Scholar
  4. 4.
    B. K. P. Horn, H. M. Hilden and S. Negahdaripour, Closed-form solution of absolute orientation using orthonormal matrices, J. Opt. Soc. Am., A5 (1988), 1127–1135.MathSciNetGoogle Scholar
  5. 5.
    D. Goryn and S. Hein, On the estimation of rigid body rotation from noisy data, IEEE Trans. Patt. Anal. Mach. Intell., 17-12 (1995), 1219–1220.CrossRefGoogle Scholar
  6. 6.
    K. Kanatani, Group-Theoretical Methods in Image Understanding, Springer, Berlin, 1990zbMATHGoogle Scholar
  7. 7.
    K. Kanatani, Geometric Computation for Machine Vision, Oxford University Press, Oxford, 1993.zbMATHGoogle Scholar
  8. 8.
    K. Kanatani, Analysis of 3-D rotation fitting, IEEE Trans. Patt. Anal. Mach. Intell., 16-5 (1994), 543–5490.CrossRefGoogle Scholar
  9. 9.
    K. Kanatani, Statistical Optimization for Geometric Computation: Theory and Practice, Elsevier, Amsterdam, 1996.zbMATHGoogle Scholar
  10. 10.
    Y. Kanazawa and K. Kanatani, Reliability of 3-D reconstruction by stereo vision, IEICE Trans. Inf. & Syst., E78-D-10 (1995), 1301–1306.Google Scholar
  11. 11.
    Y. Kanazawa and K. Kanatani, Reliability of fitting a plane to range data, IEICE Trans. Inf. & Syst., E78-D-12 (1995), 1630–1635.Google Scholar
  12. 12.
    J. Oliensis, Rigorous bounds for two-frame structure from motion, Proc. 4th European Conf. Computer Vision, April 1996, Cambridge, Vol. 2, pp. 184–195.Google Scholar
  13. 13.
    I. Shimizu and K. Deguchi, A method to register multiple range images from unknown viewing directions, Proc. MVA'96, Nov. 1996, Tokyo, pp. 406–409.Google Scholar
  14. 14.
    S. Umeyama, Least-squares estimation of transformation parameters between two point sets, IEEE Trans. Patt. Anal. Mach. Intell., 13–4 (1991), 379–380.Google Scholar
  15. 15.
    J. Weng, T. S. Huang and N. Ahuja, Motion and Structure from Image Sequences, Springer, Berlin, 1993.zbMATHGoogle Scholar
  16. 16.
    Z. Zhang and O. Faugeras, 3D Dynamic Scene Analysis, Springer, Berlin, 1992.zbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1998

Authors and Affiliations

  • Naoya Ohta
    • 1
  • Kenichi Kanatani
    • 1
  1. 1.Department of Computer ScienceGunma UniversityGunmaJapan

Personalised recommendations