Singular Vector Methods for Fundamental Matrix Computation

  • Ferran Espuny
  • Pascal Monasse
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8333)

Abstract

The normalized eight-point algorithm is broadly used for the computation of the fundamental matrix between two images given a set of correspondences. However, it performs poorly for low-size datasets due to the way in which the rank-two constraint is imposed on the fundamental matrix. We propose two new algorithms to enforce the rank-two constraint on the fundamental matrix in closed form. The first one restricts the projection on the manifold of fundamental matrices along the most favorable direction with respect to algebraic error. Its complexity is akin to the classical seven point algorithm. The second algorithm relaxes the search to the best plane with respect to the algebraic error. The minimization of this error amounts to finding the intersection of two bivariate cubic polynomial curves. These methods are based on the minimization of the algebraic error and perform equally well for large datasets. However, we show through synthetic and real experiments that the proposed algorithms compare favorably with the normalized eight-point algorithm for low-size datasets.

Keywords

3D Reconstruction Fundamental Matrix Closed Form 

References

  1. 1.
    Aanæs, H., Dahl, A.L., Pedersen, K.S.: Interesting interest points. Int. J. Comput. Vis. 97, 18–35 (2012)CrossRefGoogle Scholar
  2. 2.
    Chesi, G., Garulli, A., Vicino, A., Cipolla, R.: Estimating the fundamental matrix via constrained least-squares: a convex approach. IEEE Trans. Pattern Anal. Mach. Intell. 24(3), 397–401 (2002)CrossRefGoogle Scholar
  3. 3.
    Chum, O., Matas, J., Kittler, J.: Locally optimized RANSAC. In: Michaelis, B., Krell, G. (eds.) DAGM 2003. LNCS, vol. 2781, pp. 236–243. Springer, Heidelberg (2003)CrossRefGoogle Scholar
  4. 4.
    Faugeras, O., Luong, Q.-T., Papadopoulou, T.: The Geometry of Multiple Images: The Laws That Govern the Formation of Images of A Scene and Some of Their Applications. MIT Press, Cambridge (2001) ISBN 0262062208Google Scholar
  5. 5.
    Fischler, M.A., Bolles, R.C.: Random sample consensus: A paradigm for model fitting with applications to image analysis and automated cartography. Comm. ACM 24(6), 381–395 (1981)CrossRefMathSciNetGoogle Scholar
  6. 6.
    Gluckman, J., Nayar, S.K.: Rectifying transformations that minimize resampling effects. In: Proc. CVPR (2001)Google Scholar
  7. 7.
    Hartley, R., Zisserman, A.: Multiple View Geometry in Computer Vision, 2nd edn. Cambridge University Press (2004) ISBN 0521540518Google Scholar
  8. 8.
    Hartley, R.I.: Theory and practice of projective rectification. Int. J. Comput. Vis. 35, 115–127 (1999)CrossRefGoogle Scholar
  9. 9.
    Hartley, R.I.: In defence of the 8-point algorithm. In: Proc. ICCV (1995)Google Scholar
  10. 10.
    Hartley, R.I.: Minimizing algebraic error in geometric estimation problems. In: Proc. ICCV (1998)Google Scholar
  11. 11.
    Kahl, F., Henrion, D.: Globally optimal estimates for geometric reconstruction problems. In: Proc. ICCV (2005)Google Scholar
  12. 12.
    Lowe, D.: Distinctive image features from scale-invariant keypoints. Int. J. Comput. Vis. 60, 91–110 (2004)CrossRefGoogle Scholar
  13. 13.
    Moisan, L., Stival, B.: A probabilistic criterion to detect rigid point matches between two images and estimate the fundamental matrix. Int. J. Comput. Vis. 57(3), 201–218 (2004)CrossRefGoogle Scholar
  14. 14.
    Schaffalitzky, F., Zisserman, A., Hartley, R.I., Torr, P.: A six point solution for structure and motion. In: Vernon, D. (ed.) ECCV 2000. LNCS, vol. 1842, pp. 632–648. Springer, Heidelberg (2000)CrossRefGoogle Scholar
  15. 15.
    Triggs, B., Mclauchlan, P., Hartley, R., Fitzgibbon, A.: Bundle adjustment – a modern synthesis. In: Proc. VA Workshop (1999)Google Scholar
  16. 16.
    Zhang, Z.: Determining the epipolar geometry and its uncertainty: a review. Int. J. Comput. Vis. 27(2), 161–195 (1998)CrossRefGoogle Scholar
  17. 17.
    Zheng, Y., Sugimoto, S., Okumoti, M.: A branch and contract algorithm for globally optimal fundamental matrix estimation. In: Proc. CVPR (2011)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  • Ferran Espuny
    • 1
  • Pascal Monasse
    • 2
  1. 1.School of Environmental SciencesUniversity of LiverpoolUK
  2. 2.LIGM (UMR CNRS 8049), Center for Visual Computing, ENPCUniversité Paris-EstMarne-la-ValléeFrance

Personalised recommendations