On the Estimation of the Fundamental Matrix: A Convex Approach to Constrained Least-Squares

  • G. Chesi
  • A. Garulli
  • A. Vicino
  • R. Cipolla
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 1842)


In this paper we consider the problem of estimating the fundamental matrix from point correspondences. It is well known that the most accurate estimates of this matrix are obtained by criteria minimizing geometric errors when the data are affected by noise. It is also well known that these criteria amount to solving non-convex optimization problems and, hence, their solution is affected by the optimization starting point. Generally, the starting point is chosen as the fundamental matrix estimated by a linear criterion but this estimate can be very inaccurate and, therefore, inadequate to initialize methods with other error criteria.

Here we present a method for obtaining a more accurate estimate of the fundamental matrix with respect to the linear criterion. It consists of the minimization of the algebraic error taking into account the rank 2 constraint of the matrix. Our aim is twofold. First, we show how this non-convex optimization problem can be solved avoiding local minima using recently developed convexification techniques. Second, we show that the estimate of the fundamental matrix obtained using our method is more accurate than the one obtained from the linear criterion, where the rank constraint of the matrix is imposed after its computation by setting the smallest singular value to zero. This suggests that our estimate can be used to initialize non-linear criteria such as the distance to epipolar lines and the gradient criterion, in order to obtain a more accurate estimate of the fundamental matrix. As a measure of the accuracy, the obtained estimates of the epipolar geometry are compared in experiments with synthetic and real data.


Linear Matrix Inequality Geometric Error Fundamental Matrix Point Correspondence 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.


  1. 1.
    S. Boyd, L. El Ghaoui, E. Feron, and V. Balakrishnan: Linear Matrix Inequalities in System and Control Theory. SIAM, Philadelphia (Penn.), 1994.Google Scholar
  2. 2.
    G. Chesi, A. Tesi, A. Vicino, and R. Genesio: A convex approach to a class of minimum norm problems. Robustness in Identification and Control, pages 359–372, A. Garulli, A. Tesi, A. Vicino, Eds., Springer-Verlag, London (UK), 1999.CrossRefGoogle Scholar
  3. 3.
    G. Chesi, A. Tesi, A. Vicino, and R. Genesio: On convexification of some minimum distance problems. Proc. 5th European Control Conference, Karlsruhe (Germany), 1999.Google Scholar
  4. 4.
    R. Deriche, Z. Zhang, Q.-T. Luong, and O. Faugeras: Robust recovery of the epipolar geometry for an uncalibrated stereo rig. Proc. 3rd European Conference on Computer Vision, Stockholm (Sweden), 1994.Google Scholar
  5. 5.
    O. Faugeras: What can be seen in three dimensions with an uncalibrated stereo rig? Proc. 2nd European Conference on Computer Vision, pages 563–578, Santa Margherita Ligure (Italy), 1992.Google Scholar
  6. 6.
    O. Faugeras: Three-Dimensional Computer Vision: A Geometric Viewpoint. MIT Press, Cambridge (Mass), 1993.Google Scholar
  7. 7.
    R. Hartley: Estimation of relative camera positions for uncalibrated cameras. Proc. 2nd European Conference on Computer Vision, pages 579–587, Santa Margherita Ligure (Italy), 1992.Google Scholar
  8. 8.
    R. Hartley: In defence of the 8-point algorithm: Proc. International Conference on Computer Vision, pages 1064–1070, 1995.Google Scholar
  9. 9.
    R. Hartley: Minimizing algebraic error in geometric estimation problem. Proc. 6th International Conference on Computer Vision, pages 469–476, Bombay (India), 1998.Google Scholar
  10. 10.
    K. Kanatani: Geometric computation for machine vision. Oxford Univeristy Press, Oxford (UK), 1992.Google Scholar
  11. 11.
    D. Luenberger: Linear and Nonlinear Programming. Addison-Wesley, USA, 1984.zbMATHGoogle Scholar
  12. 12.
    Q.-T. Luong and O. Faugeras: The fundamental matrix: theory, algorithms, and stability analysis. The International Journal of Computer Vision, 17(1):43–76, 1996.CrossRefGoogle Scholar
  13. 13.
    Q.-T. Luong and O. Faugeras: Camera calibration, scene motion and structure recovery from point correspondences and fundamental matrices. The International Journal of Computer Vision, 22(3):261–289, 1997.CrossRefGoogle Scholar
  14. 14.
    S. Maybank: Properties of essential matrices. International Journal of Imaging Systems and Technology, 2:380–384, 1990.CrossRefGoogle Scholar
  15. 15.
    P. Mendonca and R. Cipolla: A simple techinique for self-calibration. Proc. IEEE Conference on Computer Vision and Pattern Recognition, pages 500–505, Fort Collins (Colorado), 1999.Google Scholar
  16. 16.
    Y. Nesterov and A. Nemirovsky: Interior Point Polynomial Methods in Convex Programming: Theory and Applications. SIAM, Philadelphia (Penn.), 1993.Google Scholar
  17. 17.
    P. Torr and D. Murray: The development and comparison of robust methods for estimating the fundamental matrix. International Journal of Computer Vision, 3(24):271–300, 1997.CrossRefGoogle Scholar
  18. 18.
    P. Torr and A. Zisserman: Performance characterizaton of fundamental matrix estimation under image degradation. Machine Vision and Applications, 9:321–333, 1997.CrossRefGoogle Scholar
  19. 19.
    P. Torr, A. Zisserman, and S. Maybank: Robust detection of degenerate configurations for the fundamental matrix. Computer Vision and Image Understanding, 71(3):312–333, 1998.CrossRefGoogle Scholar
  20. 20.
    Z. Zhang: Determining the epipolar geometry and its uncertainty-a review. The International Journal of Computer Vision, 27(2):161–195, 1998.CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2000

Authors and Affiliations

  • G. Chesi
    • 1
  • A. Garulli
    • 1
  • A. Vicino
    • 1
  • R. Cipolla
    • 2
  1. 1.Dipartimento di Ingegneria dell’InformazioneUniversita di SienaSienaItaly
  2. 2.Department of EngineeringUniversity of CambridgeCambridgeUK

Personalised recommendations