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General Trajectory Triangulation

  • Jeremy Yirmeyahu Kaminski
  • Mina Teicher
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 2351)

Abstract

The multiple view geometry of static scenes is now well understood. Recently attention was turned to dynamic scenes where scene points may move while the cameras move. The triangulation of linear trajectories is now well handled. The case of quadratic trajectories also received some attention.

We present a complete generalization and address the Problem of general trajectory triangulation of moving points from non-synchronized cameras. Our method is based on a particular representation of curves (trajectories) where a curve is represented by a family of hypersurfaces in the projective space ℙ5. This representation is linear, even for highly non-linear trajectories. We show how this representation allows the recovery of the trajectory of a moving point from non-synchronized sequences. We show how this representation can be converted into a more standard representation. We also show how one can extract directly from this representation the positions of the moving point at each time instant an image was made. Experiments on synthetic data and on real images demonstrate the feasibility of our approach.

Keywords

Structure from motion 

References

  1. 1.
    S. Avidan and A. Shashua, Trajectory triangulation: 3D reconstruction of moving points from a monocular image sequence. IEEE Transactions on Pattern Analysis and Machine Intelligence, 22(4):348–357, 2000.CrossRefGoogle Scholar
  2. 2.
    M. Barnabei, A. Brini and G.C. Rota, On the exterior calculus of invariant theory. Journal of Algebra, 96, 120–160(1985)zbMATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    B. Buchberger and F. Winkler, Gröbner Bases and Applications. Cambridge University Press, 1998.Google Scholar
  4. 4.
    W. Chojnacki, M. Brooks, A. van den Hengel and D. Gawley, On the Fitting of Surfaces to Data with Covariances. PAMI, vol. 22, Nov. 2000.Google Scholar
  5. 5.
    D. Cox, J. Little and D. O’Shea Ideals, Varieties and Algorithms, 2nd. Springer-Verlag, 1996.Google Scholar
  6. 6.
    A Multibody Factorization Method for Independent Moving Objects. International Journal Of Computer Vision, Kluwer, Vo. 29, Sep. 1998.Google Scholar
  7. 7.
    D. Eisenbud, Commutative Algebra with a view toward algebraic geometry. Springer-Verlag, 1995.Google Scholar
  8. 8.
    D. Eisenbud and J. Harris, The Geometry of Schemes. Springer-Verlag, 2000.Google Scholar
  9. 9.
    O.D. Faugeras and Q.T. Luong, The Geometry Of Multiple Images. MIT Press, 2001.Google Scholar
  10. 10.
    W. Fulton, Algebraic Curves: An Introduction to Algebraic Geometry. Addison-Wesley Publishing Company: The Advanced Book Program.Google Scholar
  11. 11.
    A.W. Fitzgibbon and A. Zisserman, Multibody Structure and Motion: 3D Reconstruction of Independently Moving Objects. In Proceedings of European Conference on Computer Vision, pages 891–906, June 2000.Google Scholar
  12. 12.
    M. Han and T. Kanade, Reconstruction of a Scene with Multiple Linearly Moving Points. In Proceedings of IEEE Conference on Computer Vision and Pattern recognition, June 2000.Google Scholar
  13. 13.
    J. Harris, Algebraic Geometry, a first course. Springer-Verlag, 1992.Google Scholar
  14. 14.
    R.I. Hartley and A. Zisserman, Multiple View Geometry in computer vision. Cambridge University Press, 2000.Google Scholar
  15. 15.
    R. Hartshorne, Algebraic Geometry. Springer-Verlag, 1977.Google Scholar
  16. 16.
    J.Y. Kaminski, M. Fryers, A. Shashua and M. Teicher, Multiple View Geometry of Non-planar Algebraic Curves. In Proceedings of International Conference on Computer Vision, July 2001Google Scholar
  17. 17.
    R.A. Manning C.R. Dyer, Interpolating view and scene motion by dynamic view morphing. In Proceedings of IEEE Conference on Computer Vision and Pattern recognition, pages 388–394, June 1999.Google Scholar
  18. 18.
    B. Matei and P. Meer, A General Method for Errors-in-variables Problems in Computer Vision. In Proceedings of IEEE Conference on Computer Vision and Pattern recognition, 2000.Google Scholar
  19. 19.
    D. Segal and A. Shashua, 3D Reconstruction from Tangent-of-Sight Measurements of a Moving Object Seen from a Moving Camera. In Proceedings of European Conference on Computer Vision, pages 507–521, June 2000.Google Scholar
  20. 20.
    A. Shashua and L. Wolf, Homography Tensors: On Algebraic Entities That Represent Three Views of Static or Moving Points. In Proceedings of European Conference on Computer Vision, pages 507–521, June 2000.Google Scholar
  21. 21.
    Y. Wexler and A. Shashua, On the synthesis of dynamic scenes from reference view. In Proceedings of IEEE Conference on Computer Vision and Pattern recognition, June 2000.Google Scholar
  22. 22.
    L. Wolf and A. Shashua, On Projection Matrices ℙk → ℙ2, k = 3,..., 6, and their Applications in Computer Vision. In Proceedings of IEEE International Conference on Computer Vision, July 2001.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2002

Authors and Affiliations

  • Jeremy Yirmeyahu Kaminski
    • 1
  • Mina Teicher
    • 1
  1. 1.Department of Mathematics and StatisticsBar-Ilan UniversityRamat-GanIsrael

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