Bootstrapping Errors-in-Variables Models

  • Bogdan Matei
  • Peter Meer
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 1883)


The bootstrap is a numerical technique, with solid theoretical foundations, to obtain statistical measures about the quality of an estimate by using only the available data. Performance assessment through bootstrap provides the same or better accuracy than the traditional error propagation approach, most often without requiring complex analytical derivations. In many computer vision tasks a regression problem in which the measurement errors are point dependent has to be solved. Such regression problems are called heteroscedastic and appear in the linearization of quadratic forms in ellipse fitting and epipolar geometry, in camera calibration, or in 3D rigid motion estimation. The performance of these complex vision tasks is difficult to evaluate analytically, therefore we propose in this paper the use of bootstrap. The technique is illustrated for 3D rigid motion and fundamental matrix estimation. Experiments with real and synthetic data show the validity of bootstrap as an evaluation tool and the importance of taking the heteroscedasticity into account.


Fundamental Matrix Rigid Motion Rotation Error Total Little Square Epipolar Line 
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 3D point sets”, IEEE Transactions on Pattern Analysis and Machine Intelligence. vol. 9, pp. 698–700, 1987.Google Scholar
  2. 2.
    S.D. Blostein and T.S. Huang, “Error analysis in stereo determination of 3D point positions”, IEEE Transactions on Pattern Analysis and Machine Intelligence, vol. 9, pp. 752–765, 1987.CrossRefGoogle Scholar
  3. 3.
    G. Csurka, C. Zeller, Z. Zhang and O. Faugeras, “Characterizing the Uncertainty of the Fundamental Matrix”, Computer Vision and Image Understanding, Vol. 68, pp. 18–36, 1997.CrossRefGoogle Scholar
  4. 4.
    A.C. Davison and D.V. Hinkley, Bootstrap Methods and their Application, Cambridge University Press, 1998.Google Scholar
  5. 5.
    B. Efron and R.J. Tibshirani, An Introduction to the Bootstrap, Chapman&Hall, 1993.Google Scholar
  6. 6.
    D.W. Eggert, A. Lorusso and R.B. Fisher, “Estimating 3-D rigid body transformations: A comparison of four major algorithms”, Machine Vision and Applications, Vol. 9, pp. 272–290, 1997.CrossRefGoogle Scholar
  7. 7.
    O. Faugeras, Three-dimensional Computer Vision. A Geometric Viewpoint, MIT Press, 1993.Google Scholar
  8. 8.
    S. Yi, R.H. Haralick and L. Shapiro, “Error propagation in machine vision”, Machine Vision and Applications, vol. 7, pp. 93–114, 1994.CrossRefGoogle Scholar
  9. 9.
    R.I. Hartley, “In Defense of the 8-Point Algorithm”, Proceedings of the 5th International Conference on Computer Vision, Cambridge (MA), pp. 1064–1070, 1995.Google Scholar
  10. 10.
    R. I. Hartley, “Triangulation”, Computer Vision and Image Understanding, vol. 68, pp. 146–157, 1997.CrossRefGoogle Scholar
  11. 11.
    B.K.P. Horn, H.M. Hilden and S. Negahdaripour, “Closed-form solution of absolute orientation using orthonormal matrices”, J. Opt. Soc. Am. vol. 5, pp. 1127–1135, 1988.MathSciNetGoogle Scholar
  12. 12.
    K. Kanatani, Geometric Computation for Machine Vision, Oxford Science Publications, 1993.Google Scholar
  13. 13.
    K. Kanatani, “Analysis of 3-D rotation fitting”, IEEE Transactions on Pattern Analysis and Machine Intelligence, vol. 16, No. 5, pp. 543–549, 1994.CrossRefGoogle Scholar
  14. 14.
    K. Kanatani, Statistical Optimization for Geometric Computation: Theory and Practice, Elsevier, 1996.Google Scholar
  15. 15.
    Y. Leedan and P. Meer, “Estimation with bilinear constraints in computer vision”, Proceedings of the 5th International Conference on Computer Vision, Bombay, India, pp. 733–738, 1998.Google Scholar
  16. 16.
    B. Matei and P. Meer, “Optimal rigid motion estimation and performance evaluation with bootstrap”, Proceedings of the Computer Vision and Pattern Recognition 99, Fort Collins Co., vol 1, pp. 339–345, 1999.Google Scholar
  17. 17.
    N. Ohta and K. Kanatani, “Optimal estimation of three-dimensional rotation and reliability evaluation”, Computer Vision-ECCV 98’, H. Burkhardt, B. Neumann Eds., Lecture Notes in Computer Science, Springer, pp. 175–187, 1998.CrossRefGoogle Scholar
  18. 18.
    T. Kato, A Short Introduction to Perturbation Theory for Linear Operators, Springer-Verlag, 1982.Google Scholar
  19. 19.
    H.C. Longuet-Higgins, “A Computer Algorithm for Reconstructing a Scene from Two Projections”, Nature, Vol. 293, pp. 133–135, 1981.CrossRefGoogle Scholar
  20. 20.
    X. Pennec and J.P. Thirion, “A framework for uncertainty and validation of 3-D registration methods based on points and frames”, International Journal on Computer Vision, vol. 25, pp. 203–229, 1997.CrossRefGoogle Scholar
  21. 21.
    H. Sahibi and A. Basu, “Analysis of error in depth perception with vergence and spatially varying sensing”, Computer Vision and Image Understanding vol. 63, pp. 447–461, 1996.CrossRefGoogle Scholar
  22. 22.
    L.L. Scharf, Statistical Signal Processing, Addison-Wesley, 1990.Google Scholar
  23. 23.
    E. Trucco and A. Verri, Introductory Techniques for 3-D Computer Vision, Prentice Hall, 1998.Google Scholar
  24. 24.
    S. Umeyama, “Least-squares estimation of transformation parameters between two point patterns”, IEEE Transactions on Pattern Analysis and Machine Intelligence, vol. 13, pp. 376–380, 1991.CrossRefGoogle Scholar
  25. 25.
    Z. Zhang, R. Deriche, O. Faugeras, Q.-T. Luong, “A robust technique for matching two uncalibrated images through the recovery of unknown epipolar geometry”, Artificial Intelligence, vol. 78, 87–119, 1995.CrossRefGoogle Scholar
  26. 26.
    Z. Zhang, “On the Optimization Criteria Used in Two View Motion Analysis”, IEEE Transactions on Pattern Analysis and Machine Intelligence, vol. 20, pp. 717–729, 1998.CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2000

Authors and Affiliations

  • Bogdan Matei
    • 1
  • Peter Meer
    • 1
  1. 1.Electrical and Computer Engineering DepartmentRutgers UniversityPiscatawayUSA

Personalised recommendations