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Integrating Local Affine into Global Projective Images in the Joint Image Space

  • P. Anandan
  • Shai Avidan
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 1842)

Abstract

The fundamental matrix defines a nonlinear 3D variety in the joint image space of multiple projective (or “uncalibrated perspective”) images. We show that, in the case of two images, this variety is a 4D cone whose vertex is the joint epipole (namely the 4D point obtained by stacking the two epipoles in the two images). Affine (or “para-perspective”) projection approximates this nonlinear variety with a linear subspace, both in two views and in multiple views. We also show that the tangent to the projective joint image at any point on that image is obtained by using local affine projection approximations around the corresponding 3D point. We use these observations to develop a new approach for recovering multiview geometry by integrating multiple local affine joint images into the global projective joint image. Given multiple projective images, the tangents to the projective joint image are computed using local affine approximations for multiple image patches. The affine parameters from different patches are combined to obtain the epipolar geometry of pairs of projective images. We describe two algorithms for this purpose, including one that directly recovers the image epipoles without recovering the fundamental matrix as an intermediate step.

Keywords

Fundamental Matrix Match Point Epipolar Line Epipolar Geometry Tangent Hyperplane 
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.

References

  1. 1.
    R. Basri Paraperspective Equiv Affine In International Journal of Computer Vision, Vol. 19, pp. 169–179, 1996CrossRefGoogle Scholar
  2. 2.
    Rikard Berthilsson, Anders Heyden, Gunnar Sparr: Recursive Structure and Motion from Image Sequences using Shape and Depth Spaces In Computer Vision and Pattern Recognition, Puerto Rico, 1997.Google Scholar
  3. 3.
    Chen, Q. and Medioni, G. Efficient Iterative Solutions to M-View Projective Reconstruction Problem, In Computer Vision and Pattern Recognition, Puerto Rico, 1999.Google Scholar
  4. 4.
    S. Christy and R. Horaud Euclidean shape and motion from multiple perspective views by affine iterations In IEEE Transactions on Pattern Analysis and Machine Intelligence PAMI, 18(10):1098–1104, November 1996.Google Scholar
  5. 5.
    O.D. Faugeras. What can be seen in three dimensions with an uncalibrated stereo rig? In Proceedings of the European Conference on Computer Vision, pages 563–578, Santa Margherita Ligure, Italy, June 1992.Google Scholar
  6. 6.
    Hanna, K. J. and Okamoto, N.E. Combining Stereo And Motion Analysis For Direct Estimation Of Scene Structure In Proceedings of the International Conference on Computer Vision, pages 357–365, 1993.Google Scholar
  7. 7.
    J.M. Lawn and R. Cipolla. Robust egomotion estimation from affine motion-parallax In Proceedings of the European Conference on Computer Vision, pages I:205–210, 1994.Google Scholar
  8. 8.
    J.M. Lawn and R. Cipolla. Reliable extraction of camera motion using constraints on the epipole. In Proceedings of the European Conference on Computer Vision, pages II:161–173, 1996.Google Scholar
  9. 9.
    C. J. Poelman, T. Kanade A paraperspective factorization method for shape and motion recovery In Proceedings of the European Conference on Computer Vision, pages 97–108, 1994.Google Scholar
  10. 10.
    J.H. Rieger, D.T. Lawton Processing differential image motion In Journal of the Optical Society of America Vol. 2, Februay 1985.Google Scholar
  11. 11.
    Shapiro, L.S. Affine Analysis Of Image Sequences, Cambridge University Press, CambridgeGoogle Scholar
  12. 12.
    J.G. Semple, G.T. Kneebone Algebraic Projective Geometry, Oxford university pressGoogle Scholar
  13. 13.
    A. Shashua and S. Avidan. The rank4 constraint in multiple view geometry. In Proceedings of the European Conference on Computer Vision, Cambridge, UK, April 1996.Google Scholar
  14. 14.
    R. Szeliski and P. Torr Geometrically constrained structure from motion: Points on planes. In European Workshop on 3D Structure from Multiple Images of Large-Scale Environments (SMILE), pages 171–186, Freiburg, Germany, June 1998.Google Scholar
  15. 15.
    B. Triggs Matching constraints and the joint image In Proceedings of the International Conference on Computer Vision (ICCV), 1995.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2000

Authors and Affiliations

  • P. Anandan
    • 1
  • Shai Avidan
    • 1
  1. 1.Microsoft ResearchRedmondUSA

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