International Journal of Computer Vision

, Volume 71, Issue 1, pp 5–47

Critical Configurations for Projective Reconstruction from Multiple Views

  • Richard Hartley
  • Fredrik Kahl
Article

Abstract

This paper investigates a classical problem in computer vision: Given corresponding points in multiple images, when is there a unique projective reconstruction of the 3D geometry of the scene points and the camera positions? A set of points and cameras is said to be critical when there is more than one way of realizing the resulting image points. For two views, it has been known for almost a century that the critical configurations consist of points and camera lying on a ruled quadric surface. We give a classification of all possible critical configurations for any number of points in three images, and show that in most cases, the ambiguity extends to any number of cameras.

The underlying framework for deriving the critical sets is projective geometry. Using a generalization of Pascal's Theorem, we prove that any number of cameras and scene points on an elliptic quartic form a critical set. Another important class of critical configurations consists of cameras and points on rational quartics. The theoretical results are accompanied by many examples and illustrations.

Keywords:

projective geometry structure from motion, degeneracy critical sets, multiple view geometry geometry 3D reconstruction 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Åström, K., and Kahl, F. 2003. Ambiguous configurations for the lD structure and motion problem. Journal of Mathematical Imaging and Vision, 18(2):191–203.MATHMathSciNetCrossRefGoogle Scholar
  2. Buchanan, T. 1988. The twisted cubic and camera calibration. Computer Vision, Graphics and Image Processing, 42:130–132.MATHMathSciNetCrossRefGoogle Scholar
  3. Carlsson, S. 1995. Duality of reconstruction and positioning from projective views. In IEEE Workshop on Representation of Visual Scenes, Cambridge Ma, USA, pp. 85–92.Google Scholar
  4. Evelyn, C.J.A., Money-Coutts, G.B., and Tyrrell, J.A. 1974. The Seven Circles Theorem and Other New Theorems. Stacey International: London.MATHGoogle Scholar
  5. Hartley, R. 2000. Ambiguous configurations for 3-view projective reconstruction. In European Conf. Computer Vision, Dublin, Ireland, Vol. I, pp. 922–935.Google Scholar
  6. Hartley, R., and Debunne, G. 1998. Dualizing scene reconstruction algorithms. In 3D Structure from Multiple Images of Large-Scale Environments, European Workshop, SMILE, Freiburg, Germany, pp. 14–31.Google Scholar
  7. Hartley, R., and Kahl, F. 2003. A critical configuration for reconstruction from rectilinear motion. In Conf. Computer Vision and Pattern Recognition, Madison, USA, Vol. I, pp. 511–517.Google Scholar
  8. Hartley, R.I. 1994. Projective reconstruction and invariants from multiple images. IEEE Transactions on Pattern Analysis and Machine Intelligence, 16:1036–1041.CrossRefGoogle Scholar
  9. Hartley, R. I., and Zisserman, A. 2003. Multiple View Geometry in Computer Vision (2nd edn.), Cambridge University Press.Google Scholar
  10. Kahl, F. 2001. Geometry and Critical Configurations of Multiple Views. PhD thesis, Sweden, Lund Institute of Technology.Google Scholar
  11. Kahl, F., and Hartley, R. 2002. Critical curves and surfaces for Euclidean reconstruction. In European Conf. Computer Vision, Copenhagen, Denmark, Vol. II, pp. 447–462.Google Scholar
  12. Kahl, F., Hartley, R., and Åström, K. 2001. Critical configurations for N-view projective reconstruction. In Conf. Computer Vision and Pattern Recognition, Hawaii, USA, Vol. II, pp. 158–163.Google Scholar
  13. Krames, J. 1940. Zur Ermittlung eines Objectes aus zwei Perspectiven (Ein Beitrag zur Theorie der gefährlichen Örter). Monatsh. Math. Phys., 49:327–354.CrossRefGoogle Scholar
  14. Maybank, S. 1993. Theory of Reconstruction from Image Motion. Heidelberg: Berlin: Springer-Verlag: New York.MATHGoogle Scholar
  15. Maybank, S. 1995. The critical line congruences for reconstruction from three images. Applicable Algebra in Engineering, Communication and Computing, 6(2), pp. 89–113.Google Scholar
  16. Maybank, S., and Shashua, A. 1998. Ambiguity in reconstruction from images of six points. In Int. Conf. Computer Vision, Mumbai, India, pp. 703–708.Google Scholar
  17. Navab, N., and Faugeras, O. D. 1993. The critical sets of lines for camera displacement estimation: A mixed Euclidean-projective and constructive approach. In Int. Conf. Computer Vision, pp. 713–723.Google Scholar
  18. Semple, J. G., and Kneebone, G. T. 1979. Algebraic Projective Geometry. Oxford University Press.Google Scholar

Copyright information

© Springer Science + Business Media, LLC 2006

Authors and Affiliations

  • Richard Hartley
    • 1
  • Fredrik Kahl
    • 2
  1. 1.Australian National University, and National ICT AustraliaAustralia
  2. 2.Australian National University, and Centre for Mathematical Sciences, Lund UniversitySweden

Personalised recommendations