International Journal of Computer Vision

, Volume 22, Issue 2, pp 125–140 | Cite as

Lines and Points in Three Views and the Trifocal Tensor

  • Richard I. Hartley
Article

Abstract

This paper discusses the basic role of the trifocal tensor in scene reconstruction from three views. This 3× 3× 3 tensor plays a role in the analysis of scenes from three views analogous to the role played by the fundamental matrix in the two-view case. In particular, the trifocal tensor may be computed by a linear algorithm from a set of 13 line correspondences in three views. It is further shown in this paper, that the trifocal tensor is essentially identical to a set of coefficients introduced by Shashua to effect point transfer in the three view case. This observation means that the 13-line algorithm may be extended to allow for the computation of the trifocal tensor given any mixture of sufficiently many line and point correspondences. From the trifocal tensor the camera matrices of the images may be computed, and the scene may be reconstructed. For unrelated uncalibrated cameras, this reconstruction will be unique up to projectivity. Thus, projective reconstruction of a set of lines and points may be carried out linearly from three views.

trifocal tensor projective reconstruction trilinear relation structure from motion 

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References

  1. Armstrong, M., Zisserman, A., and Hartley, R.I. 1996. Self-calibration from image triplets. In Computer Vision-ECCV'96, LCNS-Series, Springer Verlag, Vol 1064, pp. 3-16.Google Scholar
  2. Atkinson, K.E. 1989. An Introduction to Numerical Analysis, 2nd edition. John Wiley and Sons: New York.Google Scholar
  3. Beardsley, P.A., Zisserman, A., and Murray, D.W. 1994. Sequential update of projective and affine structure from motion. Report OUEL 2012/94, Oxford University (to appear in IJCV).Google Scholar
  4. Faugeras, O.D. 1992. What can be seen in three dimensions with an uncalibrated stereo rig? In Computer Vision-ECCV'92, LNCS Series, Springer-Verlag, Vol. 588, pp. 563-578.Google Scholar
  5. Faugeras, O. and Mourrain, B. 1995. On the geometry and algebra of the point and line correspondences between N images. In Proc. International Conference on Computer Vision, pp. 951-956.Google Scholar
  6. Golub, G.H. and Van Loan, C.F. 1989. Matrix Computations, 2nd Ed. The Johns Hopkins University Press: Baltimore, London.Google Scholar
  7. Hartley, R.I. 1993. Camera calibration using line correspondences. In Proc. DARPA Image Understanding Workshop, pp. 361- 366.Google Scholar
  8. Hartley, R.I. 1994a. Projective reconstruction and invariants from multiple images. IEEE Trans. on Pattern Analysis and Machine Intelligence, 16:1036-1041.Google Scholar
  9. Hartley, R.I. 1994b. Projective reconstruction from line correspondences. In Proc. IEEE Conf. on Computer Vision and Pattern Recognition, pp. 903-907.Google Scholar
  10. Hartley, R.I. 1995. In defence of the 8-point algorithm. In Proc. International Conference on Computer Vision, pp. 1064-1070.Google Scholar
  11. Hartley, R.I. and Sturm, P. 1994. Triangulation. In Proc. ARPA Image Understanding Workshop, pp. 957-966. Also in Proc. Computer Analysis of Images and Patterns, Prague, Sept. 1995, LNCS-Series, Springer Verlag, Vol. 970, pp. 190-197.Google Scholar
  12. Hartley, R., Gupta, R., and Chang, T. 1992. Stereo from uncalibrated cameras. In Proc. IEEE Conf. on Computer Vision and Pattern Recognition, pp. 761-764.Google Scholar
  13. Heyden, A. 1995a. Geometry and algebra of multiple projective transformations. Ph.D. thesis, Department of Mathematics, Lund University, Sweden.Google Scholar
  14. Heyden, A. 1995b. Reconstruction from image sequences by means of relative depth. In Proc. International Conference on Computer Vision, pp. 1058-1063.Google Scholar
  15. Heyden, A. 1995c. Reconstruction from multiple images using kinetic depths. Technical Report, ISRN LUFTD2/TFMA-95/7003-SE, Department of Mathematics, Lund University.Google Scholar
  16. Horn, B.K.P. 1990. Relative orientation. International Journal of Computer Vision, 4:59-78.Google Scholar
  17. Horn, B.K.P. 1991. Relative orientation revisited. Journal of the Optical Society of America, A, 8(10):1630-1638.Google Scholar
  18. Longuet-Higgins, H.C. 1981. A computer algorithm for reconstructing a scene from two projections. Nature, 293:133-135.Google Scholar
  19. Press, W.H., Flannery, B.P., Teukolsky, S.A., and Vetterling, W.T. 1988. Numerical Recipes in C: The Art of Scientific Computing. Cambridge University Press.Google Scholar
  20. Shashua, A. 1994. Trilinearity in visual recognition by alignment. In Computer Vision-ECCV'94, LNCS-Series, Vol. 800, Springer-Verlag, Vol. I, pp. 479-484.Google Scholar
  21. Shashua, A. 1995. Algebraic functions for recognition. IEEE Trans.on Pattern Analysis and Machine Intelligence, 17(8):779-789.Google Scholar
  22. Shashua, A. and Werman, M. 1995. Trilinearity of three perspective views and its associated tensor. In Proc. International Conference on Computer Vision, pp. 920-925.Google Scholar
  23. Shashua, A. and Anandan, P. 1996. Trilinear constraints revisited: Generalized trilinear constraints and the tensor brightness constraint. In Proc. ARPA Image Understanding Workshop, Palm Springs, pp. 815-820.Google Scholar
  24. Spetsakis, M.E. 1992. A linear algorithm for point and line-based structure from motion. CVGIP: Image Understanding, 56(2):230- 241.Google Scholar
  25. Spetsakis, M.E. and Aloimonos, J. 1990a. Structure from motion using line correspondences. International Journal of Computer Vision, 4(3):171-183.Google Scholar
  26. Spetsakis, M.E. and Aloimonos, J. 1990b. A unified theory of structure from motion. In DARPA IU Proceedings, pp. 271- 283.Google Scholar
  27. Triggs, B. 1995a. The geometry of projective reconstruction I: Matching constraints and the joint image. Unpublished report.Google Scholar
  28. Triggs, B. 1995b. Matching constraints and the joint image. In Proc. International Conference on Computer Vision, pp. 338- 343.Google Scholar
  29. Vieville, T. and Luong, Q.T. 1993. Motion of points and lines in the uncalibrated case. Report RR-2054, INRIA.Google Scholar
  30. Weng, J., Huang, T.S., and Ahuja, N. 1992. Motion and structure from line correspondences: Closed-form solution, uniqueness and optimization. IEEE Trans. on Pattern Analysis and Machine Intelligence, 14(3):318-336.Google Scholar
  31. Werman, M. and Shashua, A. 1995. The study of 3D-from-2D using elimination. In Proc. International Conference on Computer Vision, pp. 473-479.Google Scholar

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© Kluwer Academic Publishers 1997

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  • Richard I. Hartley

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