A General Framework for Trajectory Triangulation

  • Jeremy Yirmeyahu Kaminski
  • Mina Teicher
Article

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. Two cases are considered: (i) the motion is captured in the images by tracking the moving point itself, (ii) the tangents of the motion only are extracted from the images.

The first case is based on a new representation (to computer vision) of curves (trajectories) where a curve is represented by a family of hypersurfaces in the projective space ℙ5. The second case is handled by considering the dual curve of the curve generated by the trajectory.

In both cases these representations of curves allow: (i) the triangulation of the trajectory of a moving point from non-synchronized sequences, (ii) the recovery of more standard representation of the whole trajectory, (iii) the computations of the set of positions of the moving point at each time instant an image was made.

Furthermore, theoretical considerations lead to a general theorem stipulating how many independent constraints a camera provides on the motion of the point. This number of constraint is a function of the camera motion.

On the computation front, in both cases the triangulation leads to equations where the unknowns appear linearly. Therefore the problem reduces to estimate a high-dimensional parameter in presence of heteroscedastic noise. Several method are tested.

structure from motion trajectory triangulation mathematical methods in 3D reconstruction 

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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, Vol. 22, No. 4, pp. 348–357, 2000.Google Scholar
  2. 2.
    M. Barnabei, A. Brini, and G.C. Rota, “On the exterior calculus of invariant theory,” Journal of Algebra, Vol. 96, pp. 120–160, 1985.Google Scholar
  3. 3.
    B. Buchberger and F.Winkler, Gröbner Bases and Applications, Cambridge University Press, 1998.Google Scholar
  4. 4.
    D. Cox, J. Little, and D. O'Shea, Ideals, Varieties and Algorithms, 2nd., Springer-Verlag, 1996.Google Scholar
  5. 5.
    W. Chojnacki, M. Brooks, A. van den Hengel, and D. Gawley, “On the fitting of surfaces to data with covariances,” TIPAMI, Vol. 22, 2000.Google Scholar
  6. 6.
    J. Costeira and T. Kanade, “A multibody factorization method for independent moving objects.” International Journal of Computer Vision, Kluwer, Vol. 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, June 2000, pp. 891–906.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 2001.Google Scholar
  17. 17.
    J.Y. Kaminski and M. Teicher, “General trajectory triangulation,” in Proceedings of European Conference on Computer Vision, June 2002.Google Scholar
  18. 18.
    S. Lang, Algebra, 3rd edn., Springer-Verlag, 2002.Google Scholar
  19. 19.
    R.A. Manning and C.R. Dyer, “Interpolating view and scene motion by dynamic view morphing,” in Proceedings of IEEE Conference on Computer Vision and Pattern Recognition, June 1999, pp. 388–394.Google Scholar
  20. 20.
    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
  21. 21.
    D. Segal and A. Shashua, “3D reconstruction from tangent-ofsight measurements of a moving object seen from a moving camera,” in Proceedings of European Conference on Computer Vision, June 2000, pp. 507–521.Google Scholar
  22. 22.
    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, June 2000, pp. 507–521.Google Scholar
  23. 23.
    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
  24. 24.
    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

© Kluwer Academic Publishers 2004

Authors and Affiliations

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

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