On Computing Metric Upgrades of Projective Reconstructions under the Rectangular Pixel Assumption

  • Jean Ponce
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 2018)


This paper shows how to upgrade the projective reconstruction of a scene to a metric one in the case where the only assumption made about the cameras observing that scene is that they have rectangular pixels (zero-skew cameras). The proposed approach is based on a simple characterization of zero-skew projection matrices in terms of line geometry, and it handles zero-skew cameras with arbitrary or known aspect ratios in a unified framework. The metric upgrade computation is decomposed into a sequence of linear operations, including linear least-squares parameter estimation and eigenvalue-based symmetric matrix factorization, followed by an optional non-linear least-squares refinement step. A few classes of critical motions for which a unique solution cannot be found are spelled out. A MATLAB implementation has been constructed and preliminary experiments with real data are presented.


Projective Transformation Optical Center Quadratic Constraint Projection Matrice World Coordinate System 
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  1. 1.
    B. Boufama, R. Mohr, and F. Veillon. Euclidian constraints for uncalibrated reconstruction. In Proc. Int. Conf. Comp. Vision, pages 466–470, Berlin, Germany, May 1993.Google Scholar
  2. 2.
    J. Costeira and T. Kanade. A multi-body factorization method for motion analysis. Int. J. of Comp. Vision, 29(3):159–180, September 1998.CrossRefGoogle Scholar
  3. 3.
    O. Faugeras and B. Mourrain. On the geometry and algebra of the point and line correspondences between n images. Technical Report 2665, INRIA Sophia-Antipolis, 1995.Google Scholar
  4. 4.
    O. Faugeras and T. Papadopoulo. Gaussman-Caylay algebra for modeling systems of cameras and the algebraic equations of the manifold of trifocal tensors. Technical Report 3225, INRIA Sophia-Antipolis, 1997.Google Scholar
  5. 5.
    O.D. Faugeras. What can be seen in three dimensions with an uncalibrated stereo rig? In G. Sandini, editor, Proc. European Conf. Comp. Vision, volume 588 of Lecture Notes in Computer Science, pages 563–578, Santa Margherita, Italy, 1992. Springer-Verlag.Google Scholar
  6. 6.
    O.D. Faugeras. Three-Dimensional Computer Vision. MIT Press, 1993.Google Scholar
  7. 7.
    O.D. Faugeras. Stratification of 3D vision: projective, affine and metric representations. J. Opt. Soc. Am. A, 12(3):465–484, March 1995.Google Scholar
  8. 8.
    O.D. Faugeras, Q.-T. Luong, and S.J. Maybank. Camera self-calibration: theory and experiments. In G. Sandini, editor, Proc. European Conf. Comp. Vision, volume 588 of Lecture Notes in Computer Science, pages 321–334, Santa Margherita, Italy, 1992. Springer-Verlag.Google Scholar
  9. 9.
    A. Fitzgibbon and A. Zisserman. Automatic 3D model acquisition and generation of new images from video sequences. In European Signal Processing Conference, pages 311–326, Rhodes, Greece, 1998.Google Scholar
  10. 10.
    P.E. Gill and W. Murray. Newton-type methods for unconstrained and linearly constrained optimization. Math. Programming, 28:311–350, 1974.CrossRefMathSciNetGoogle Scholar
  11. 11.
    C. Harris and M. Stephens. A combined edge and corner detector. In 4th Alvey Vision Conference, pages 189–192, Manchester, UK, 1988.Google Scholar
  12. 12.
    R. Hartley. Lines and points in three views and the trifocal tensor. Int. J. of Comp. Vision, 22(2):125–140, March 1997.CrossRefGoogle Scholar
  13. 13.
    R.I. Hartley. An algorithm for self calibration from several views. In Proc. IEEE Conf. Comp. Vision Patt. Recog., pages 908–912, Seattle,WA, June 1994.Google Scholar
  14. 14.
    R.I. Hartley. In defence of the 8-point algorithm. In Proc. Int. Conf. Comp. Vision, pages 1064–1070, Boston, MA, 1995.Google Scholar
  15. 15.
    R.I. Hartley, R. Gupta, and T. Chang. Stereo from uncalibrated cameras. In Proc. IEEE Conf. Comp. Vision Patt. Recog., pages 761–764, Champaign, IL, 1992.Google Scholar
  16. 16.
    A. Heyden. Geometry and algebra of multiple projective transformations. PhD thesis, Lund University, Sweden, 1995.Google Scholar
  17. 17.
    A. Heyden and K. Åström Minimal conditions on intrinsic parameters for Euclidean reconstruction. In Asian Conference on Computer Vision, Hong Kong, 1998.Google Scholar
  18. 18.
    A. Heyden and K. Åström. Flexible calibration: minimal cases for auto-calibration. In Proc. Int. Conf. Comp. Vision, pages 350–355, Kerkyra, Greece, September 1999.Google Scholar
  19. 19.
    J.J. Koenderink and A.J. Van Doorn. Affine structure from motion. J. Opt. Soc. Am. A, 8:377–385, 1990.Google Scholar
  20. 20.
    S.J. Maybank and O.D. Faugeras. A theory of self-calibration of a moving camera. Int. J. of Comp. Vision, 8(2):123–151, 1992.CrossRefGoogle Scholar
  21. 21.
    R. Mohr, L. Quan, F. Veillon, and B. Boufama. Relative 3D reconstruction using multiple uncalibrated images. Technical Report RT 84-IMAG 12-LIFIA, LIFIA-IRIMAG, June 1992.Google Scholar
  22. 22.
    C.J. Poelman and T. Kanade. A paraperspective factorization method for shape and motion recovery. IEEE Trans. Patt. Anal. Mach. Intell., 19(3):206–218, March 1997.CrossRefGoogle Scholar
  23. 23.
    M. Pollefeys. Self-calibration and metric 3D reconstruction from uncalibrated image sequences. PhD thesis, Katholieke Universiteit Leuven, 1999.Google Scholar
  24. 24.
    M. Pollefeys, R. Koch, and L. Van Gool. Self-calibration and metric reconstruction in spite of varying and unknown internal camera parameters. Int. J. of Comp. Vision, 32(1):7–26, August 1999.CrossRefGoogle Scholar
  25. 25.
    L. Quan. Self-calibration of an affine cameras from multiple views. Int. J. of Comp. Vision, 19:93–110, 1996.CrossRefGoogle Scholar
  26. 26.
    R.B. Schnabel and E. Eskow. A new modified Cholesky factorization. SIAM J. Sci. Comput., 11:1136–1158, 1990.zbMATHCrossRefMathSciNetGoogle Scholar
  27. 27.
    A. Shashua. Projective depth: a geometric invariant for 3D reconstruction from two perspective/ orthographic views and for visual recognition. In Proc. Int. Conf. Comp. Vision, pages 583–590, Berlin, Germany, 1993.Google Scholar
  28. 28.
    A. Shashua. Trilinearity in visual recognition by alignment. In J.-O. Eklundh, editor, Proc. European Conf. Comp. Vision, volume 800 of Lecture Notes in Computer Science, pages 479–484. Springer-Verlag, 1994.Google Scholar
  29. 29.
    P. Sturm. Critical motion sequences for monocular self-calibration and uncalibrated Euclidean reconstruction. In Proc. IEEE Conf. Comp. Vision Patt. Recog., pages 1100–1105, San Juan, Puerto Rico, June 1997.Google Scholar
  30. 30.
    C. Tomasi and T. Kanade. Shape and motion from image streams under orthography: a factorization method. Int. J. of Comp. Vision, 9(2):137–154, 1992.CrossRefGoogle Scholar
  31. 31.
    P.H.S. Torr, A.W. Fitzgibbon, and A. Zisserman. The problem of degeneracy in structure and motion estimation from uncalibrated motion sequences. Int. J. of Comp. Vision, 32(1):27–44, August 1999.CrossRefGoogle Scholar
  32. 32.
    R.Y. Tsai and T.S. Huang. Uniqueness and estimation of 3D motion parameters of rigid bodies with curved surfaces. IEEE Trans. Patt. Anal. Mach. Intell., 6:13–27, 1984.CrossRefGoogle Scholar
  33. 33.
    S. Ullman. The Interpretation of Visual Motion. The MIT Press, Cambridge, MA, 1979.Google Scholar
  34. 34.
    D. Weinshall and C. Tomasi. Linear and incremental acquisition of invariant shape models from image sequences. IEEE Trans. Patt. Anal. Mach. Intell., 17(5), May 1995.Google Scholar
  35. 35.
    Z. Zhang, R. Deriche, O.D. Faugeras, and Q.-T. Luong. A robust technique for matching two uncalibrated images through the recovery of the unknown epipolar geometry. Artificial Intelligence Journal, 78:87–119, October 1995.CrossRefGoogle Scholar


  1. 1.
    T. Papadopoulo and M. Lourakis. Estimating the jacobian of the singular value decomposition: Theory and applications. In Proc. European Conference on Computer Vision, pages 554–570, 2000.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2001

Authors and Affiliations

  • Jean Ponce
    • 1
  1. 1.Dept. of Computer Science and Beckman InstituteUniversity of Illinois at Urbana-ChampaignUSA

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