Properties of the Catadioptric Fundamental Matrix

  • Christopher Geyer
  • Kostas Daniilidis
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 2351)


The geometry of two uncalibrated views obtained with a parabolic catadioptric device is the subject of this paper. We introduce the notion of circle space, a natural representation of line images, and the set of incidence preserving transformations on this circle space which happens to equal the Lorentz group. In this space, there is a bilinear constraint on transformed image coordinates in two parabolic catadioptric views involving what we call the catadioptric fundamental matrix. We prove that the angle between corresponding epipolar curves is preserved and that the transformed image of the absolute conic is in the kernel of that matrix, thus enabling a Euclidean reconstruction from two views. We establish the necessary and sufficient conditions for a matrix to be a catadioptric fundamental matrix.


Image Point Singular Vector Polar Plane Stereographic Projection Line Image 
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.
    R. Benosman and S.B. Kang. Panoramic Vision. Springer-Verlag, 2000.Google Scholar
  2. 2.
    O. Faugeras, Q.-T. Luong, and T. Papadopoulo. The Geometry of Multiple Images: The Laws That Govern the Formation of Multiple Images of a Scene and Some of Their Applications. MIT Press, 2001.Google Scholar
  3. 3.
    C. Geyer and K. Daniilidis. Catadioptric projective geometry. International Journal of Computer Vision, 43:223–243, 2001.CrossRefGoogle Scholar
  4. 4.
    C. Geyer and K. Daniilidis. Structure and motion from uncalibrated catadioptric views. In IEEE Conf. Computer Vision and Pattern Recognition, Hawaii, Dec. 11–13, 2001.Google Scholar
  5. 5.
    J. Gluckman and S.K. Nayar. Ego-motion and omnidirectional cameras. In Proc. Int. Conf. on Computer Vision, pages 999–1005, Bombay, India, Jan. 3–5, 1998.Google Scholar
  6. 6.
    R. Hartley and A. Zisserman. Multiple View Geometry. Cambridge Univ. Press, 2000.Google Scholar
  7. 7.
    V. Heine. Group Theory in Quantum Mechanics. Pergamon Press, Oxford, 1960.zbMATHGoogle Scholar
  8. 8.
    S.B. Kang. Catadioptric self-calibration. In IEEE Conf. Computer Vision and Pattern Recognition, pages I-201–207, Hilton Head Island, SC, June 13–15, 2000.Google Scholar
  9. 9.
    Y. Ma, K. Huang, R. Vidal, J. Kosecka, and S. Sastry. Rank conditions of the multiple view matrix. Technical Report UILU-ENG 01-2214 (DC-220), University of Illinois at Urbana-Champaign, CSL-Technical Report, June 2001.Google Scholar
  10. 10.
    S. Nayar. Catadioptric omnidirectional camera. In IEEE Conf. Computer Vision and Pattern Recognition, pages 482–488, Puerto Rico, June 17–19, 1997.Google Scholar
  11. 11.
    T. Needham. Visual Complex Analysis. Clarendon Press, Oxford, 1997.zbMATHGoogle Scholar
  12. 12.
    D. Pedoe. Geometry: A comprehensive course. Dover Publications, New York, NY, 1970.Google Scholar
  13. 13.
    J. Semple and G. Kneebone. Algebraic Projective Geometr. Oxford University Press, 1979.Google Scholar
  14. 14.
    T. Svoboda, T. Pajdla, and V. Hlavac. Epipolar geometry for panoramic cameras. In Proc. 5th European Conference on Computer Vision, pages 218–231, 1998.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2002 2002

Authors and Affiliations

  • Christopher Geyer
    • 1
  • Kostas Daniilidis
    • 1
  1. 1.GRASP LaboratoryUniversity of PennsylvaniaPhiladelphiaUSA

Personalised recommendations