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Critical Configurations for Projective Reconstruction from Multiple Views

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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.

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References

  • 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.

    Article  MATH  MathSciNet  Google Scholar 

  • Buchanan, T. 1988. The twisted cubic and camera calibration. Computer Vision, Graphics and Image Processing, 42:130–132.

    Article  MATH  MathSciNet  Google Scholar 

  • 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.

  • Evelyn, C.J.A., Money-Coutts, G.B., and Tyrrell, J.A. 1974. The Seven Circles Theorem and Other New Theorems. Stacey International: London.

    MATH  Google Scholar 

  • Hartley, R. 2000. Ambiguous configurations for 3-view projective reconstruction. In European Conf. Computer Vision, Dublin, Ireland, Vol. I, pp. 922–935.

  • 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.

  • 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.

  • Hartley, R.I. 1994. Projective reconstruction and invariants from multiple images. IEEE Transactions on Pattern Analysis and Machine Intelligence, 16:1036–1041.

    Article  Google Scholar 

  • Hartley, R. I., and Zisserman, A. 2003. Multiple View Geometry in Computer Vision (2nd edn.), Cambridge University Press.

  • Kahl, F. 2001. Geometry and Critical Configurations of Multiple Views. PhD thesis, Sweden, Lund Institute of Technology.

  • 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.

  • 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.

  • Krames, J. 1940. Zur Ermittlung eines Objectes aus zwei Perspectiven (Ein Beitrag zur Theorie der gefährlichen Örter). Monatsh. Math. Phys., 49:327–354.

    Article  Google Scholar 

  • Maybank, S. 1993. Theory of Reconstruction from Image Motion. Heidelberg: Berlin: Springer-Verlag: New York.

    MATH  Google Scholar 

  • Maybank, S. 1995. The critical line congruences for reconstruction from three images. Applicable Algebra in Engineering, Communication and Computing, 6(2), pp. 89–113.

  • Maybank, S., and Shashua, A. 1998. Ambiguity in reconstruction from images of six points. In Int. Conf. Computer Vision, Mumbai, India, pp. 703–708.

  • 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.

  • Semple, J. G., and Kneebone, G. T. 1979. Algebraic Projective Geometry. Oxford University Press.

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Authors and Affiliations

  1. Australian National University, and National ICT Australia, Australia

    Richard Hartley

  2. Australian National University, and Centre for Mathematical Sciences, Lund University, Sweden

    Fredrik Kahl

Authors
  1. Richard Hartley
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  2. Fredrik Kahl
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National ICT Australia is funded by the Australian Government's Department of Communications, Information Technology, and the Arts and the Australian Research Council through Backing Australia's Ability and the ICT Research Centre of Excellence programs.

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Hartley, R., Kahl, F. Critical Configurations for Projective Reconstruction from Multiple Views. Int J Comput Vision 71, 5–47 (2007). https://doi.org/10.1007/s11263-005-4796-1

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  • DOI: https://doi.org/10.1007/s11263-005-4796-1

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