Advertisement

What Do Four Points in Two Calibrated Images Tell Us about the Epipoles?

  • David Nistér
  • Frederik Schaffalitzky
Part of the Lecture Notes in Computer Science book series (LNCS, volume 3022)

Abstract

Suppose that two perspective views of four world points are given, that the intrinsic parameters are known, but the camera poses and the world point positions are not. We prove that the epipole in each view is then constrained to lie on a curve of degree ten. We give the equation for the curve and establish many of the curve’s properties. For example, we show that the curve has four branches through each of the image points and that it has four additional points on each conic of the pencil of conics through the four image points. We show how to compute the four curve points on each conic in closed form. We show that orientation constraints allow only parts of the curve and find that there are impossible configurations of four corresponding point pairs. We give a novel algorithm that solves for the essential matrix given three corresponding points and one epipole. We then use the theory to describe a solution, using a 1-parameter search, to the notoriously difficult problem of solving for the pose of three views given four corresponding points.

Keywords

Image Point Geometric Construction Point Pair Point Correspondence Curve Branch 
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.: Three-Dimensional Computer Vision: a Geometric Viewpoint. MIT Press, Cambridge (1993) ISBN 0-262-06158-9Google Scholar
  2. 2.
    Fischler, M., Bolles, R.: Random Sample Consensus: a Paradigm for Model Fitting with Application to Image Analysis and Automated Cartography. Commun. Assoc. Comp. Mach. 24, 381–395 (1981)MathSciNetGoogle Scholar
  3. 3.
    Haralick, R., Lee, C., Ottenberg, K., Nölle, M.: Review and Analysis of Solutions of the Three Point Perspective Pose Estimation Problem. International Journal of Computer Vision 13(3), 331–356 (1994)CrossRefGoogle Scholar
  4. 4.
    Hartley, R., Zisserman, A.: Multiple View Geometry in Computer Vision. Cambridge University Press, Cambridge (2000) ISBN 0-521-62304-9zbMATHGoogle Scholar
  5. 5.
    Holt, R., Netravali, A.: Uniqueness of Solutions to Three PerspectiveViews of Four Points. IEEE Transactions on Pattern Analysis and Machine Intelligence 17(3) (March 1995)Google Scholar
  6. 6.
    Kirwan, F.: Complex Algebraic Curves. Cambridge University Press, Cambridge (1995) ISBN 0-521-42353-8Google Scholar
  7. 7.
    Maybank, S.: Theory of Reconstruction from Image Motion. Springer, Heidelberg (1993) ISBN 3-540- 55537-4zbMATHGoogle Scholar
  8. 8.
    Nistér, D.: An Efficient Solution to the Five-Point Relative Pose Problem. IEEE Conference on Computer Vision and Pattern Recognition 2, 195–202 (2003)Google Scholar
  9. 9.
    Nistér, D.: PreemptiveRANSACfor Live Structure and Motion Estimation. In: IEEE International Conference on Computer Vision, pp. 199–206 (2003)Google Scholar
  10. 10.
    Quan, L., Triggs, B., Mourrain, B., Ameller, A.: Uniqueness of Minimal Euclidean Reconstruction from 4 Points (2003) (unpublished)Google Scholar
  11. 11.
    Quan, L.: Invariants of Six Points and Projective Reconstruction from Three Uncalibrated Images. IEEE Transactions on Pattern Analysis and Machine Intelligence 1(17), 34–46 (1995)CrossRefGoogle Scholar
  12. 12.
    Schaffalitzky, F., Zisserman, A., Hartley, R., Torr, P.: A Six Point Solution for Structure and Motion. In: Vernon, D. (ed.) ECCV 2000. LNCS, vol. 1842, pp. 632–648. Springer, Heidelberg (2000)CrossRefGoogle Scholar
  13. 13.
    Semple, J., Kneebone, G.: Algebraic Projective Geometry. Oxford University Press, Oxford (1952) ISBN 0-19-8503636zbMATHGoogle Scholar
  14. 14.
    Werner, T.: Constraints on Five Points in Two Images. IEEE Conference on Computer Vision and Pattern Recognition 2, 203–208 (2003)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2004

Authors and Affiliations

  • David Nistér
    • 1
  • Frederik Schaffalitzky
    • 2
  1. 1.Sarnoff CorporationPrincetonUSA
  2. 2.Australian National University 

Personalised recommendations