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Invariants of 6 points from 3 uncalibrated images

  • Long Quan
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 801)

Abstract

There are three projective invariants of a set of six points in general position in space. It is well known that these invariants cannot be recovered from one image, however an invariant relationship does exist between space invariants and image invariants. This invariant relationship will first be derived for a single image. Then this invariant relationship is used to derive the space invariants, when multiple images are available.

This paper establishes that the minimum number of images for computing these invariants is three, and invariants from three images can have as many as three solutions. Algorithms are presented for computing these invariants in closed form. The accuracy and stability with respect to image noise, selection of the triplets of images and distance between viewing positions are studied both through real and simulated images.

Application of these invariants is also presented, this extends the results of projective reconstruction of Faugeras [6] and Hartley et al. [10] and the method of epipolar geometry determination of Sturm [18] for two uncalibrated images to the case of three uncalibrated images.

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References

  1. 1.
    E.B. Barrett, M.H. Brill, N.N. Haag, and P.M. Payton. Invariant linear methods in photogrammetry and model-matching. In J. Mundy and A. Zisserman, editors, Geometric Invariance in Computer Vision, pages 277–292. The MIT press, 1992.Google Scholar
  2. 2.
    B. Boufama, R. Mohr, and F. Veillon. Euclidian constraints for uncalibrated reconstruction. In Proceedings of the 4th International Conference on Computer Vision, Berlin, Germany, pages 466–470, May 1993.Google Scholar
  3. 3.
    J.B. Burns, R. Weiss, and E.M. Riseman. View variation of point set and line segment features. In Proceedings of DARPA Image Understanding Workshop, Pittsburgh, Pennsylvania, USA, pages 650–659, 1990.Google Scholar
  4. 4.
    A.B. Coble. Algebraic Geometry and Theta Functions. American Mathematical Society, 1961.Google Scholar
  5. 5.
    J. Dixmier. Quelques aspects de la théorie des invariants. Gazette des Mathématiciens, 43:39–64, January 1990.Google Scholar
  6. 6.
    O. Faugeras. What can be seen in three dimensions with an uncalibrated stereo rig? In G. Sandini, editor, Proceedings of the 2nd European Conference on Computer Vision, Santa Margherita Ligure, Italy, pages 563–578. Springer-Verlag, May 1992.Google Scholar
  7. 7.
    O.D. Faugeras, Q.T. Luong, and S.J. Maybank. Camera Self-Calibration: Theory and Experiments. In G. Sandini, editor, Proceedings of the 2nd European Conference on Computer Vision, Santa Margherita Ligure, Italy, pages 321–334. Springer-Verlag, May 1992.Google Scholar
  8. 8.
    P. Gros and L. Quan. 3D projective invariants from two images. In Geometric Methods in Computer Vision II, SPIE's 199S International Symposium on Optical Instrumentation and Applied Science, pages 75–86, July 1993.Google Scholar
  9. 9.
    R. Hartley. Invariants of Points Seen in Multiple Images. Technical report, G.E. CRD, Schenectady, 1992.Google Scholar
  10. 10.
    R. Hartley, R. Gupta, and T. Chang. Stereo from uncalibrated cameras. In Proceedings of the Conference on Computer Vision and Pattern Recognition, Urbana-Champaign, Illinois, USA, pages 761–764, 1992.Google Scholar
  11. 11.
    J.J. Koenderink and A. J. van Doom. Affine structure from motion. Technical report, Utrecht University, Utrecht, The Netherlands, October 1989.Google Scholar
  12. 12.
    R. Mohr, F. Veillon, and L. Quan. Relative 3D reconstruction using multiple uncalibrated images. In Proceedings of IEEE Conference on Computer Vision and Pattern Recognition, New York, USA, pages 543–548, June 1993.Google Scholar
  13. 13.
    J.L. Mundy and A. Zisserman, editors. Geometric Invariance in Computer Vision. MIT Press, Cambridge, Massachusetts, USA, 1992.Google Scholar
  14. 14.
    L. Quan. Invariants of 6 Points from 3 Uncalibrated Images. Rapport Technique RT 101 IMAG 19 LIFIA, LIFIA-IMAG, Grenoble, October 1993.Google Scholar
  15. 15.
    L. Quan and R. Mohr. Affine shape representation from motion through reference points. Journal of Mathematical Imaging and Vision, 1:145–151, 1992. Also in IEEE Workshop on Visual Motion, New Jersey, pages 249–254, 1991.Google Scholar
  16. 16.
    J.G. Semple and G.T. Kneebone. Algebraic Projective Geometry. Oxford Science Publication, 1952.Google Scholar
  17. 17.
    G. Sparr. An algebraic/analytic method for reconstruction from image correspondance. In Proceedings of the 7th Scandinavian Conference on Image Analysis, Aalborg, Denmark, pages 274–281, 1991.Google Scholar
  18. 18.
    R. Sturm. Das problem der projektivität und seine anwendung auf die flächen zweiten grades. Math. Ann., 1:533–574, 1869.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1994

Authors and Affiliations

  • Long Quan
    • 1
  1. 1.LIFIA - CNRS - INRIAGrenobleFrance

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