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Algebraic varieties in multiple view geometry

  • Anders Heyden
  • Kalle Åström
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 1065)

Abstract

In this paper we will investigate the different algebraic varieties and ideals that can be generated from multiple view geometry with uncalibrated cameras. The natural descriptor, Vn, is the image of \(\mathcal{P}^3\) in \(\mathcal{P}^2 \times \mathcal{P}^2 \times \cdot \cdot \cdot \times \mathcal{P}^2\) under n different projections. However, we will show that Vn is not a variety.

Another descriptor, the variety Vb, is generated by all bilinear forms between pairs of views and consists of all points in \(\mathcal{P}^2 \times \mathcal{P}^2 \times \cdot \cdot \cdot \times \mathcal{P}^2\) where all bilinear forms vanish. Yet another descriptor, the variety, Vt, is the variety generated by all trilinear forms between triplets of views. We will show that when n=3, Vt is a reducible variety with one component corresponding to Vb and another corresponding to the trifocal plane. In ideal theoretic terms this is called a primary decomposition. This settles the discussion on the connection between the bilinearities and the trilinearities.

Furthermore, we will show that when n=3, Vt is generated by the three bilinearities and one trilinearity and when n≥4, Vt is generated by the ( 2 n ) bilinearities. This shows that four images is the generic case in the algebraic setting, because Vt can be generated by just bilinearities.

Keywords

Bilinear Form Algebraic Variety Primary Decomposition Epipolar Line Natural Descriptor 
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.
    Faugeras, O., D., What can be seen in three dimensions with an uncalibrated stereo rig?, EGCV'92, Lecture notes in Computer Science, Vol 588. Ed. G. Sandini, Springer-Verlag, 1992, pp. 563–578.Google Scholar
  2. 2.
    Faugeras, O., D., Mourrain, B., On the geometry and algebra on the point and line correspondences between N images, Proc. ICCV'95, IEEE Computer Society Press, 1995, pp. 951–956.Google Scholar
  3. 3.
    Faugeras, O., D., Mourrain, B., About the correspondences of points between N images, Proc. IEEE Workshop on Representation of Visual Scenes, 1995.Google Scholar
  4. 4.
    Hartley, R., Hartley, I., Projective Reconstruction and Invariants from Multiple Images, IEEE Trans. Pattern Anal. Machine Intell., vol. 16, no. 10, pp. 1036–1041, 1994.Google Scholar
  5. 5.
    Hartley, A linear method for reconstruction from lines and points, Proc. ICCV'95, IEEE Computer Society Press, 1995, pp. 882–887.Google Scholar
  6. 6.
    Heyden, A., Reconstruction from Image Sequences by means of Relative Depths, Proc. ICCV'95, IEEE Computer Society Press, 1995, pp. 1058–1063, Also to appear in IJCV, International Journal of Computer Vision.Google Scholar
  7. 7.
    Heyden, A., Åström, K., A Canonical Framework for Sequences of Images, Proc. IEEE Workshop on Representation of Visual Scenes, 1995.Google Scholar
  8. 8.
    Heyden, A., Åström, K., Algebraic Properties of Multilinear Constraints, Technical Report, CODEN: LUFTD2/TFMA—96/7001—SE, Lund, Sweden, 1996.Google Scholar
  9. 9.
    Luong, Q.-T., Vieville, T., Canonic Representations for the Geometries of Multiple Projective Views, ECCV'94, Lecture notes in Computer Science, Vol 800. Ed. Jan-Olof Eklund, Springer-Verlag, 1994, pp. 589–599.Google Scholar
  10. 10.
    Schafarevich, I., R., Basic Algebraic Geometry I — Varieties in Projective Space, Springer Verlag, 1988.Google Scholar
  11. 11.
    Sharp, R., Y., Steps in Commutative Algebra, London Mathematical Society Texts, 1990.Google Scholar
  12. 12.
    Shashua, A., Trilinearity in Visual Recognition by Alignment, ECCV'94, Lecture notes in Computer Science, Vol 800. Ed. Jan-Olof Eklund, Springer-Verlag, 1994, pp. 479–484.Google Scholar
  13. 13.
    Shahsua, A., Werman, M., Trilinearity of Three Perspective Views and its Associated Tensor, Proc. ICCV'95, IEEE Computer Society Press, 1995, pp. 920–925.Google Scholar
  14. 14.
    Triggs, B., Matching Constraints and the Joint Image, Proc. ICCV '95, IEEE Computer Society Press, 1995, pp. 338–343.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1996

Authors and Affiliations

  • Anders Heyden
    • 1
  • Kalle Åström
    • 1
  1. 1.Dept of MathematicsLund UniversityLundSweden

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