International Journal of Computer Vision

, Volume 26, Issue 1, pp 41–61 | Cite as


  • Richard I. Hartley


It is known that a set of points in three-dimensions is determined up to projectivity from two views with uncalibrated cameras. It is shown in this paper that this result may be improved by distinguishing between points in front of and behind the camera. Any point that lies in an image must lie in front of the camera producing that image. Using this idea, it is shown that the scene is determined from two views up to a more restricted class of mappings known as quasi-affine transformations, which are precisely those projectivities that preserve the convex hull of an object of interest. An invariant of quasi-affine transformation known as the chiral sequence of a set of points is defined and it is shown how the chiral sequence may be computed using two uncalibrated views. As demonstrated theoretically and by experiment the chiral sequence may distinguish between sets of points that are projectively equivalent. These results lead to necessary and sufficient conditions for a set of corresponding pixels in two images to be realizable as the images of a set of points in three dimensions.Using similar methods, a necessary and sufficient condition is given for the orientation of a set of points to be determined by two views. If the perspective centres are not separated from the point set by a plane, then the orientation of the set of points is determined from two views.

chirality projective invariant quasi-affine reconstruction projective reconstruction oriented projective geometry 


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  1. Faugeras, O.D. 1992. What can be seen in three dimensions with an uncalibrated stereo rig? Computer Vision-ECCV' 92, LNCS-Series Vol. 588,Springer-Verlag, pp. 563-578.Google Scholar
  2. Hartley, R. 1992. Invariants of points seen in multiple images. Unpublished report.Google Scholar
  3. Hartley, R.I. 1993a. Invariants of lines in space. In Proc. DARPA Image Understanding Workshop, pp. 737-744.Google Scholar
  4. Hartley, R.I. 1993b. Euclidean reconstruction from uncalibrated views. In Proc. of the Second Europe-US Workshop on Invariance, Ponta Delgada, Azores, pp. 187-202.Google Scholar
  5. Hartley, R.I. and Kawauchi, A. 1979. Polynomials of amphicheiral knots. Math. Ann., 243:63-70.Google Scholar
  6. Hartley, R., Gupta, R. and Chang, T. 1992. Stereo from uncalibrated cameras. In Proc. IEEE Conf. on Computer Vision and Pattern Recognition, pp. 761-764.Google Scholar
  7. Longuet-Higgins, H.C. 1981. A computer algorithm for reconstructing a scene from two projections. Nature, 293:133-135.Google Scholar
  8. Mohr, R., Veillon, F., and Quan, L. 1993. Relative 3D reconstruction using multiple uncalibrated images. In Proc. IEEE Conf. on Computer Vision and Pattern Recognition, pp. 543-548.Google Scholar
  9. Morin, L. 1993. Quelques Contributions des Invariants Projectifs À la Vision par Ordinateur. Ph.D. Thesis, Institut National Polytechnique de Grenoble.Google Scholar
  10. Morin, L., Brand, P., and Mohr, R., 1995. Indexing with projective invariants. In Proceedings of the Syntactical and Structural Pattern Recognition Workshop, Nahariya, Israel. World Scientific Pub.Google Scholar
  11. Press, W.H., Flannery, B.P., Teukolsky, S.A., and Vetterling, W.T. 1988. Numerical Recipes in C: The Art of Scientific Computing, CambridgeUniversity Press.Google Scholar
  12. Robert, L. and Faugeras, O.D. 1993. Relative 3D positioning and 3D convex hull computation from a weakly calibrated stereo pair. In Proc. International Conference on Computer Vision, pp. 540-544.Google Scholar
  13. Rothwell, C.A., Zisserman, A., Forsyth, D.A., and Mundy, J.L. 1992. Canonical frames for planar object recognition. In Computer Vision-ECCV' 92, LNCS-Series Vol. 588, Springer-Verlag, pp. 757-772.Google Scholar
  14. Sparr, G., 1992. Depth computations from polyhedral images. In Computer Vision-ECCV' 92, LNCS-Series Vol. 588, Springer-Verlag, pp. 378-386.Google Scholar
  15. Sutherland, I.E. 1980. Sketchpad: A man-machine graphical communications system. Technical Report 296, MIT Lincoln Laboratories, 1963. Also published by Garland Publishing Inc., New York.Google Scholar
  16. Wolfram, S. 1988. Mathematica: A System for Doing Mathematics by Computer. Addison-Wesley: Redwood City, California.Google Scholar

Copyright information

© Kluwer Academic Publishers 1998

Authors and Affiliations

  • Richard I. Hartley
    • 1
  1. 1.GE–Corporate Research and DevelopmentSchenectady

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