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A new characterization of the trifocal tensor

  • Théodore Papadopoulo
  • Olivier Faugeras
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 1406)

Abstract

This paper deals with the problem of characterizing and parametrizing the manifold of trifocal tensors that describe the geometry of three views like the fundamental matrix characterizes the geometry of two. The paper contains two new results. First a new, simpler, set of algebraic constraints that characterizes the set of trifocal tensors is presented. Second, we give a new parametrization of the trifocal tensor based upon those constraints which is also simpler than previously known parametrizations. Some preliminary experimental results of the use of these constraints and parametrization to estimate the trifocal tensor from image correspondences are presented.

Keywords

Fundamental Matrix Projection Matrice Epipolar Line Epipolar Geometry Algebraic Constraint 
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.

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References

  1. [AS96]
    S. Avidan and A. Shashua. Tensorial transfer: Representation of n > 3 views of 3d scenes. In Proceedings of the ARPA Image Understanding Workshop. darpa, morgan-kaufmann, February 1996.Google Scholar
  2. [Car94]
    Stefan Carlsson. Multiple image invariance using the double algebra. In Joseph L. Mundy, Andrew Zissermann, and David Forsyth, editors, Applications of Invariance in Computer Vision, volume 825 of Lecture Notes in Computer Science, pages 145–164. Springer-Verlag, 1994.Google Scholar
  3. [DZLF94]
    R. Deriche, Z. Zhang, Q.-T. Luong, and O. Faugeras. Robust recovery of the epipolar geometry for an uncalibrated stereo rig. In Eklundh [Ekl94], pages 567–576, Vol. 1.Google Scholar
  4. [Ekl94]
    J-O. Eklundh, editor. volume 800–801 of Lecture Notes in Computer Science, Stockholm, Sweden, May 1994. Springer-Verlag.Google Scholar
  5. [FM95]
    Olivier Faugeras and Bernard Mourrain. On the geometry and algebra of the point and line correspondences between n images. In Proceedings of the 5th International Conference on Computer Vision [icc95], pages 951–956.Google Scholar
  6. [FP97]
    Olivier Faugeras and Théodore Papadopoulo. A nonlinear method for estimating the projective geometry of three views. RR 3221, INRIA, July 1997.Google Scholar
  7. [FP98]
    Olivier Faugeras and Théodore Papadopoulo. A nonlinear method for estimating the projective geometry of three views. In Proceedings of the 6th International Conference on Computer Vision, pages 477–484, Bombay, India, January 1998. IEEE Computer Society Press.Google Scholar
  8. [Har94]
    Richard Hartley. Lines and points in three views-an integrated approach. In Proceedings of the ARPA Image Understanding Workshop. Defense Advanced Research Projects Agency, Morgan Kaufmann Publishers, Inc., 1994.Google Scholar
  9. [Har95]
    R.I. Hartley. In defence of the 8-point algorithm. In Proceedings of the 5th International Conference on Computer Vision [icc95], pages 1064–1070.Google Scholar
  10. [Har97]
    Richard I. Hartley. Lines and points in three views and the trifocal tensor. The International Journal of Computer Vision, 22(2): 125–140, March 1997.CrossRefGoogle Scholar
  11. [Hey95]
    Anders Heyden. Reconstruction from image sequences by means of relative depths. In Proceedings of the 5th International Conference on Computer Vision [icc95], pages 1058–1063.Google Scholar
  12. [icc95]
    Boston, MA, June 1995. IEEE Computer Society Press.Google Scholar
  13. [LF96]
    Quang-Tuan Luong and Olivier D. Faugeras. The fundamental matrix: Theory, algorithms and stability analysis. The International Journal of Computer Vision, 1(17):43–76, January 1996.CrossRefGoogle Scholar
  14. [LV94]
    Q.-T. Luong and T. Viéville. Canonic representations for the geometries of multiple projective views. In Eklundh [Ekl94], pages 589–599.Google Scholar
  15. [SA90]
    Minas E. Spetsakis and Y. Aloimonos. A unified theory of structure from motion. In Proc. DARPA IU Workshop, pages 271–283, 1990.Google Scholar
  16. [Sha94]
    Amnon Shashua. Trilinearity in visual recognition by alignment. In Eklundh [Ekl94], pages 479–484.Google Scholar
  17. [Sha95]
    Amnon Shashua. Algebraic functions for recognition. IEEE Transactions on Pattern Analysis and Machine Intelligence, 17(8):779–789, 1995.CrossRefGoogle Scholar
  18. [SW95]
    A. Shashua and M. Werman. On the trilinear tensor of three perspective views and its underlying geometry. In Proceedings of the 5th International Conference on Computer Vision [icc95].Google Scholar
  19. [TZ97a]
    P.H.S. Torr and A. Zisserman. Robust parameterization and computation of the trifocal tensor. Image and Vision Computing, 15:591–605, 1997.CrossRefGoogle Scholar
  20. [TZ97b]
    P.H.S. Torr and A. Zissermann. Performance characterization of fundamental matrix estimation under image degradation. Machine Vision and Applications, 9:321–333, 1997.CrossRefGoogle Scholar
  21. [ZDFL95]
    Z. Zhang, R. Deriche, O. Faugeras, and Q.-T. Luong. A robust technique for matching two uncalibrated images through the recovery of the unknown epipolar geometry. Artificial Intelligence Journal, 78:87–119, October 1995.CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1998

Authors and Affiliations

  • Théodore Papadopoulo
    • 1
  • Olivier Faugeras
    • 1
  1. 1.INRIA Sophia AntipolisSophia-Antipolis CedexFrance

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