What can two images tell us about a third one?

  • Olivier Faugeras
  • Luc Robert
Calibration and Multiple Views
Part of the Lecture Notes in Computer Science book series (LNCS, volume 800)

Abstract

This paper discusses the problem of predicting image features in an image from image features in two other images and the epipolar geometry between the three images. We adopt the most general camera model of perpective projection and show that a point can be predicted in the third image as a bilinear function of its images in the first two cameras, that the tangents to three corresponding curves are related by a trilinear function, and that the curvature of a curve in the third image is a linear function of the curvatures at the corresponding points in the other two images. We thus answer completely the following question: given two views of an object, what would a third view look like? We show that in the special case of orthographic projection our results for points reduce to those of Ullman and Basri [19]. We demonstrate on synthetic as well as on real data the applicability of our theory.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Eamon B. Barrett, Michael H. Brill, Nils N. Haag, and Paul M. Payton. Invariant Linear Methods in Photogrammetry and Model-Matching. In Joseph L. Mundy and Andrew Zimmerman, editors, Geometric Invariance in Computer Vision, chapter 14. MIT Press, 1992.Google Scholar
  2. 2.
    Ronen Basri. On the Uniqueness of Correspondence under Orthographic and Perspective Projections. In Proc. of the Image Understanding Workshop, pages 875–884, 1993.Google Scholar
  3. 3.
    R. Deriche. Using Canny's Criteria to Derive an Optimal Edge Detector Recursively Implemented. In The International Journal of Computer Vision, volume 2, pages 15–20, April 1987.Google Scholar
  4. 4.
    O.D. Faugeras and L. Robert. What Can Two Images Tell us About a Third One ? Rapport de recherche 2018, INRIA, Projet Robotique et Vision, September 1993.Google Scholar
  5. 5.
    Olivier D. Faugeras. What can be seen in three dimensions with an uncalibrated stereo rig. In Giulio Sandini, editor, Proceedings of the 2nd European Conference on Computer Vision, pages 563–578. Springer-Verlag, Lecture Notes in Computer Science 588, May 1992.Google Scholar
  6. 6.
    Olivier D. Faugeras. Three-Dimensional Computer Vision: a Geometric Viewpoint. MIT Press, 1993.Google Scholar
  7. 7.
    Olivier D. Faugeras, Tuan Luong, and Steven Maybank. Camera self-calibration: theory and experiments. In Giulio Sandini, editor, Proceedings of the 2nd European Conference on Computer Vision, pages 321–334. Springer-Verlag, Lecture Notes in Computer Science 588, 1992.Google Scholar
  8. 8.
    Richard Hartley, Rajiv Gupta, and Tom Chang. Stereo from Uncalibrated Cameras. In Proceedings of CVPR92, Champaign, Illinois, pages 761–764, June 1992.Google Scholar
  9. 9.
    Kenichi Kanatani. Computational Projective Geometry. CVGIP: Image Understanding, 54(3):333–348, November 1991.Google Scholar
  10. 10.
    Quang-Tuan Luong, Rachid Deriche, Olivier Faugeras, and Théodore Papadopoulo. On Determining the Fundamental Matrix: Analysis of Different Methods and Experimental Results. Technical Report 1894, INRIA, 1993.Google Scholar
  11. 11.
    R. Mohr and E. Arbogast. It can be done without camera calibration. Pattern Recognition Letters, 12:39–43, 1990.Google Scholar
  12. 12.
    Roger Mohr, Luce Morin, and Enrico Grosso. Relative positioning with poorly calibrated cameras. In J.L. Mundy and A. Zisserman, editors, Proceedings of DARPA-ESPRIT Workshop on Applications of Invariance in Computer Vision, pages 7–46, 1991.Google Scholar
  13. 13.
    Joseph L. Mundy and Andrew Zimmerman, editors. Geometric Invariance in Computer Vision. MIT Press, 1992.Google Scholar
  14. 14.
    M. Pietikainen and D. Harwood. Progress in trinocular stereo. In Proceedings NATO Advanced Workshop on Real-time Object and Environment Measurement and classification, Maratea, Italy, August 31–September 3 1987.Google Scholar
  15. 15.
    L. Robert. Perception Stéréoscopigue de Courbes et de Surfaces Tridimensionnelles. Applications à la Robotique Mobile. PhD thesis, Ecole Polytechnique, Mars 1993.Google Scholar
  16. 16.
    J.G. Semple and G.T. Kneebone. Algebraic Projective Geometry. Oxford: Clarendon Press, 1952. Reprinted 1979.Google Scholar
  17. 17.
    A. Shashua. On Geometric and Algebraic Aspects of 3D Affine and Projective Structures from Perspective 2D Views. Technical Report A.I. Memo No. 1405, MIT, July 1993.Google Scholar
  18. 18.
    A. Shashua. Projective Depth: A Geometric Invariant for 3D Reconstruction From Two Perspective Orthographic Views and For Visual Recognition. In Proc. Fourth International Conference on Computer Vision, pages 583–590, 1993.Google Scholar
  19. 19.
    Shimon Ullman and Ronen Basri. Recognition by Linear Combinations of Models. IEEE Transactions on Pattern Analysis and Machine Intelligence, 13(10):992–1006, 1991.Google Scholar

Copyright information

© Springer-Verlag 1994

Authors and Affiliations

  • Olivier Faugeras
    • 1
  • Luc Robert
    • 1
  1. 1.INRIASophia-AntipolisFrance

Personalised recommendations