Abstract

An algebraic derivation of the Kruppa equations is presented. This derivation is much simpler and thus easier to understand than the conventional derivation based on projective geometry. Using the newly derived Kruppa equations, we further propose a new algorithm for determining a camera’s intrinsic parameters. Experimental results and discussions are given for both synthetic and real images.

© 1999 Optical Society of America

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References

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  1. R. Y. Tsai, “Synopsis of recent progress on camera calibration for 3D machine vision,” in The Robotics Review, O. Khatib, J. J. Craig, T. Lozano-Pirez, eds. (MIT Press, Cambridge, Mass., 1989), pp. 147–159.
  2. O. D. Faugeras, Three-Dimensional Computer Vision: A Geometric Viewpoint (MIT Press, Cambridge, Mass., 1993).
  3. Z. Zhang, “A flexible new technique for camera calibration,” (Microsoft Research, Redmond, Wash.1998).
  4. S. J. Maybank, O. D. Faugeras, “A theory of self-calibration of a moving camera,” Int. J. Comput. Vision 8, 123–152 (1992).
    [CrossRef]
  5. C. Zeller, O. Faugeras, “Camera self-calibration from video sequences: the Kruppa equations revisited,” (Institut National de Recherche en Informatique et d’Automatique, Sophia-Antipolis, France, 1996).
  6. Q.-T. Luong, O. Faugeras, “Self-calibration of a moving camera from point correspondences and fundamental matrices,” Int. J. Comput. Vision 22, 261–289 (1997).
    [CrossRef]
  7. R. Hartley, “Kruppa’s equations derived from the fundamental matrix,” IEEE Trans. Pattern. Anal. Mach. Intell. 19, 133–135 (1997).
    [CrossRef]
  8. S. Bougnoux, “From projective to Euclidean space under any practical situation, a critism of self-calibration,” in Proceedings of the Sixth International Conference on Computer Vision (Narosa, New Delhi, India, 1998), pp. 790–796.
  9. M. Pollefeys, R. Koch, L. van Gool, “Self-calibration and metric reconstruction in spite of varying and unknown internal camera parameters,” in Proceedings of the Sixth International Conference on Computer Vision, (Narosa, New Delhi, India, 1998), pp. 90–95.
  10. B. Trigg, “Autocalibration and the absolute quadric,” in Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition (IEEE, New York, 1997), pp. 609–614.
  11. A. Heyden, X. Astrom, “Euclidean reconstruction from image sequences with varying and unknown focal length and principal point,” in Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition (IEEE, New York, 1997), pp. 438–443.
  12. G. Xu, Z. Zhang, Epipolar Geometry in Stereo, Motion and Object Recognition: A Unified Approach (Kluwer Academic, Dordrecht, The Netherlands, 1996).
  13. Z. Zhang, “Determining the epipolar geometry and its uncertainty: a review,” Int. J. Comput. Vision 27, 161–195 (1998).
    [CrossRef]
  14. H. C. Longuet-Higgins, “A computer algorithm for reconstructing a scene from two projections,” Nature 293, 133–135 (1981).
    [CrossRef]
  15. Z. Zhang, R. Deriche, O. Faugeras, Q.-T. Luong, “A robust technique for matching two uncalibrated images through the recovery of the unknown epipolar geometry,” Artif. Intel. 78, 87–119 (1995).
    [CrossRef]
  16. R. Hartley, “In defense of the eight-point algorithm,” IEEE Trans. Pattern. Anal. Mach. Intell. 19, 580–593 (1997).
    [CrossRef]
  17. W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes in C (Cambridge U. Press, Cambridge, UK, 1988).
  18. R. Hartley, “Estimation of relative camera positions for uncalibrated cameras,” in Proceedings of the Second European Conference on Computer Vision (Springer-Verlag, Berlin, 1992), pp. 579–588.
  19. Z. Zhang, “Motion and structure from two perspective views: from essential parameters to Euclidean motion through the fundamental matrix,” J. Opt. Soc. Am. A 14, 2938–2950 (1997).
    [CrossRef]
  20. K. Kanatani, Geometric Computation for Machine Vision (Oxford Sci. Publ., Oxford, UK, 1993).
  21. K. Kanatani, “Optimal estimation of fundamental matrices and its reliability,” http://www.ail.cs.gunmau.ac.jp/labo/paper/fundamatrix.ps.gz (Department of Computer Science, Gunma University, Kiryu, Japan, 1998; in Japanese).

1998

Z. Zhang, “Determining the epipolar geometry and its uncertainty: a review,” Int. J. Comput. Vision 27, 161–195 (1998).
[CrossRef]

1997

Q.-T. Luong, O. Faugeras, “Self-calibration of a moving camera from point correspondences and fundamental matrices,” Int. J. Comput. Vision 22, 261–289 (1997).
[CrossRef]

R. Hartley, “Kruppa’s equations derived from the fundamental matrix,” IEEE Trans. Pattern. Anal. Mach. Intell. 19, 133–135 (1997).
[CrossRef]

R. Hartley, “In defense of the eight-point algorithm,” IEEE Trans. Pattern. Anal. Mach. Intell. 19, 580–593 (1997).
[CrossRef]

Z. Zhang, “Motion and structure from two perspective views: from essential parameters to Euclidean motion through the fundamental matrix,” J. Opt. Soc. Am. A 14, 2938–2950 (1997).
[CrossRef]

1995

Z. Zhang, R. Deriche, O. Faugeras, Q.-T. Luong, “A robust technique for matching two uncalibrated images through the recovery of the unknown epipolar geometry,” Artif. Intel. 78, 87–119 (1995).
[CrossRef]

1992

S. J. Maybank, O. D. Faugeras, “A theory of self-calibration of a moving camera,” Int. J. Comput. Vision 8, 123–152 (1992).
[CrossRef]

1981

H. C. Longuet-Higgins, “A computer algorithm for reconstructing a scene from two projections,” Nature 293, 133–135 (1981).
[CrossRef]

Astrom, X.

A. Heyden, X. Astrom, “Euclidean reconstruction from image sequences with varying and unknown focal length and principal point,” in Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition (IEEE, New York, 1997), pp. 438–443.

Bougnoux, S.

S. Bougnoux, “From projective to Euclidean space under any practical situation, a critism of self-calibration,” in Proceedings of the Sixth International Conference on Computer Vision (Narosa, New Delhi, India, 1998), pp. 790–796.

Deriche, R.

Z. Zhang, R. Deriche, O. Faugeras, Q.-T. Luong, “A robust technique for matching two uncalibrated images through the recovery of the unknown epipolar geometry,” Artif. Intel. 78, 87–119 (1995).
[CrossRef]

Faugeras, O.

Q.-T. Luong, O. Faugeras, “Self-calibration of a moving camera from point correspondences and fundamental matrices,” Int. J. Comput. Vision 22, 261–289 (1997).
[CrossRef]

Z. Zhang, R. Deriche, O. Faugeras, Q.-T. Luong, “A robust technique for matching two uncalibrated images through the recovery of the unknown epipolar geometry,” Artif. Intel. 78, 87–119 (1995).
[CrossRef]

C. Zeller, O. Faugeras, “Camera self-calibration from video sequences: the Kruppa equations revisited,” (Institut National de Recherche en Informatique et d’Automatique, Sophia-Antipolis, France, 1996).

Faugeras, O. D.

S. J. Maybank, O. D. Faugeras, “A theory of self-calibration of a moving camera,” Int. J. Comput. Vision 8, 123–152 (1992).
[CrossRef]

O. D. Faugeras, Three-Dimensional Computer Vision: A Geometric Viewpoint (MIT Press, Cambridge, Mass., 1993).

Flannery, B. P.

W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes in C (Cambridge U. Press, Cambridge, UK, 1988).

Hartley, R.

R. Hartley, “In defense of the eight-point algorithm,” IEEE Trans. Pattern. Anal. Mach. Intell. 19, 580–593 (1997).
[CrossRef]

R. Hartley, “Kruppa’s equations derived from the fundamental matrix,” IEEE Trans. Pattern. Anal. Mach. Intell. 19, 133–135 (1997).
[CrossRef]

R. Hartley, “Estimation of relative camera positions for uncalibrated cameras,” in Proceedings of the Second European Conference on Computer Vision (Springer-Verlag, Berlin, 1992), pp. 579–588.

Heyden, A.

A. Heyden, X. Astrom, “Euclidean reconstruction from image sequences with varying and unknown focal length and principal point,” in Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition (IEEE, New York, 1997), pp. 438–443.

Kanatani, K.

K. Kanatani, Geometric Computation for Machine Vision (Oxford Sci. Publ., Oxford, UK, 1993).

K. Kanatani, “Optimal estimation of fundamental matrices and its reliability,” http://www.ail.cs.gunmau.ac.jp/labo/paper/fundamatrix.ps.gz (Department of Computer Science, Gunma University, Kiryu, Japan, 1998; in Japanese).

Koch, R.

M. Pollefeys, R. Koch, L. van Gool, “Self-calibration and metric reconstruction in spite of varying and unknown internal camera parameters,” in Proceedings of the Sixth International Conference on Computer Vision, (Narosa, New Delhi, India, 1998), pp. 90–95.

Longuet-Higgins, H. C.

H. C. Longuet-Higgins, “A computer algorithm for reconstructing a scene from two projections,” Nature 293, 133–135 (1981).
[CrossRef]

Luong, Q.-T.

Q.-T. Luong, O. Faugeras, “Self-calibration of a moving camera from point correspondences and fundamental matrices,” Int. J. Comput. Vision 22, 261–289 (1997).
[CrossRef]

Z. Zhang, R. Deriche, O. Faugeras, Q.-T. Luong, “A robust technique for matching two uncalibrated images through the recovery of the unknown epipolar geometry,” Artif. Intel. 78, 87–119 (1995).
[CrossRef]

Maybank, S. J.

S. J. Maybank, O. D. Faugeras, “A theory of self-calibration of a moving camera,” Int. J. Comput. Vision 8, 123–152 (1992).
[CrossRef]

Pollefeys, M.

M. Pollefeys, R. Koch, L. van Gool, “Self-calibration and metric reconstruction in spite of varying and unknown internal camera parameters,” in Proceedings of the Sixth International Conference on Computer Vision, (Narosa, New Delhi, India, 1998), pp. 90–95.

Press, W. H.

W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes in C (Cambridge U. Press, Cambridge, UK, 1988).

Teukolsky, S. A.

W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes in C (Cambridge U. Press, Cambridge, UK, 1988).

Trigg, B.

B. Trigg, “Autocalibration and the absolute quadric,” in Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition (IEEE, New York, 1997), pp. 609–614.

Tsai, R. Y.

R. Y. Tsai, “Synopsis of recent progress on camera calibration for 3D machine vision,” in The Robotics Review, O. Khatib, J. J. Craig, T. Lozano-Pirez, eds. (MIT Press, Cambridge, Mass., 1989), pp. 147–159.

van Gool, L.

M. Pollefeys, R. Koch, L. van Gool, “Self-calibration and metric reconstruction in spite of varying and unknown internal camera parameters,” in Proceedings of the Sixth International Conference on Computer Vision, (Narosa, New Delhi, India, 1998), pp. 90–95.

Vetterling, W. T.

W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes in C (Cambridge U. Press, Cambridge, UK, 1988).

Xu, G.

G. Xu, Z. Zhang, Epipolar Geometry in Stereo, Motion and Object Recognition: A Unified Approach (Kluwer Academic, Dordrecht, The Netherlands, 1996).

Zeller, C.

C. Zeller, O. Faugeras, “Camera self-calibration from video sequences: the Kruppa equations revisited,” (Institut National de Recherche en Informatique et d’Automatique, Sophia-Antipolis, France, 1996).

Zhang, Z.

Z. Zhang, “Determining the epipolar geometry and its uncertainty: a review,” Int. J. Comput. Vision 27, 161–195 (1998).
[CrossRef]

Z. Zhang, “Motion and structure from two perspective views: from essential parameters to Euclidean motion through the fundamental matrix,” J. Opt. Soc. Am. A 14, 2938–2950 (1997).
[CrossRef]

Z. Zhang, R. Deriche, O. Faugeras, Q.-T. Luong, “A robust technique for matching two uncalibrated images through the recovery of the unknown epipolar geometry,” Artif. Intel. 78, 87–119 (1995).
[CrossRef]

G. Xu, Z. Zhang, Epipolar Geometry in Stereo, Motion and Object Recognition: A Unified Approach (Kluwer Academic, Dordrecht, The Netherlands, 1996).

Z. Zhang, “A flexible new technique for camera calibration,” (Microsoft Research, Redmond, Wash.1998).

Artif. Intel.

Z. Zhang, R. Deriche, O. Faugeras, Q.-T. Luong, “A robust technique for matching two uncalibrated images through the recovery of the unknown epipolar geometry,” Artif. Intel. 78, 87–119 (1995).
[CrossRef]

IEEE Trans. Pattern. Anal. Mach. Intell.

R. Hartley, “In defense of the eight-point algorithm,” IEEE Trans. Pattern. Anal. Mach. Intell. 19, 580–593 (1997).
[CrossRef]

R. Hartley, “Kruppa’s equations derived from the fundamental matrix,” IEEE Trans. Pattern. Anal. Mach. Intell. 19, 133–135 (1997).
[CrossRef]

Int. J. Comput. Vision

S. J. Maybank, O. D. Faugeras, “A theory of self-calibration of a moving camera,” Int. J. Comput. Vision 8, 123–152 (1992).
[CrossRef]

Z. Zhang, “Determining the epipolar geometry and its uncertainty: a review,” Int. J. Comput. Vision 27, 161–195 (1998).
[CrossRef]

Q.-T. Luong, O. Faugeras, “Self-calibration of a moving camera from point correspondences and fundamental matrices,” Int. J. Comput. Vision 22, 261–289 (1997).
[CrossRef]

J. Opt. Soc. Am. A

Nature

H. C. Longuet-Higgins, “A computer algorithm for reconstructing a scene from two projections,” Nature 293, 133–135 (1981).
[CrossRef]

Other

K. Kanatani, Geometric Computation for Machine Vision (Oxford Sci. Publ., Oxford, UK, 1993).

K. Kanatani, “Optimal estimation of fundamental matrices and its reliability,” http://www.ail.cs.gunmau.ac.jp/labo/paper/fundamatrix.ps.gz (Department of Computer Science, Gunma University, Kiryu, Japan, 1998; in Japanese).

W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes in C (Cambridge U. Press, Cambridge, UK, 1988).

R. Hartley, “Estimation of relative camera positions for uncalibrated cameras,” in Proceedings of the Second European Conference on Computer Vision (Springer-Verlag, Berlin, 1992), pp. 579–588.

C. Zeller, O. Faugeras, “Camera self-calibration from video sequences: the Kruppa equations revisited,” (Institut National de Recherche en Informatique et d’Automatique, Sophia-Antipolis, France, 1996).

R. Y. Tsai, “Synopsis of recent progress on camera calibration for 3D machine vision,” in The Robotics Review, O. Khatib, J. J. Craig, T. Lozano-Pirez, eds. (MIT Press, Cambridge, Mass., 1989), pp. 147–159.

O. D. Faugeras, Three-Dimensional Computer Vision: A Geometric Viewpoint (MIT Press, Cambridge, Mass., 1993).

Z. Zhang, “A flexible new technique for camera calibration,” (Microsoft Research, Redmond, Wash.1998).

S. Bougnoux, “From projective to Euclidean space under any practical situation, a critism of self-calibration,” in Proceedings of the Sixth International Conference on Computer Vision (Narosa, New Delhi, India, 1998), pp. 790–796.

M. Pollefeys, R. Koch, L. van Gool, “Self-calibration and metric reconstruction in spite of varying and unknown internal camera parameters,” in Proceedings of the Sixth International Conference on Computer Vision, (Narosa, New Delhi, India, 1998), pp. 90–95.

B. Trigg, “Autocalibration and the absolute quadric,” in Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition (IEEE, New York, 1997), pp. 609–614.

A. Heyden, X. Astrom, “Euclidean reconstruction from image sequences with varying and unknown focal length and principal point,” in Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition (IEEE, New York, 1997), pp. 438–443.

G. Xu, Z. Zhang, Epipolar Geometry in Stereo, Motion and Object Recognition: A Unified Approach (Kluwer Academic, Dordrecht, The Netherlands, 1996).

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Figures (6)

Fig. 1
Fig. 1

Change of error in αu with the standard deviation of the Gaussian noise.

Fig. 2
Fig. 2

Change of error in αv with the standard deviation of the Gaussian noise.

Fig. 3
Fig. 3

Change of error in u0 with the standard deviation of the Gaussian noise.

Fig. 4
Fig. 4

Change of error in v0 with the standard deviation of the Gaussian noise.

Fig. 5
Fig. 5

Three images of a room. The blue points were manually chosen and matched over the three images, and the red points are backprojections of the computed 3D points.

Fig. 6
Fig. 6

Recovered room shown by a VRML browser for a new viewpoint.

Tables (1)

Tables Icon

Table 1 Algorithm for Choosing Three Independent Equations

Equations (28)

Equations on this page are rendered with MathJax. Learn more.

A=αubu00αvv0001,
m˜=Ax˜,
m˜TFm˜=0,
x˜TEx˜=0,
E=[t]×R,
FTe=0,Fe=0.
ATFA=[t]×R
F=A-T[t]×RA-1.
A-TRT[t]×TA-1e=0,
t=sA-1e ,
FA=sA-T[A-1e]×R.
det(M)[M-1v]×=MT[v]×M.
[A-1e]×=1αuαv AT[e]×A.
FA=s[e]×AR,
FCFT=s[e]×C[e]×T,
FCFTe=s[e]×C[e]×Te=0,
(e × y)TC(e × y)=0,
(FTy)TC(FTy)=0,
yT[e]×C[e]×Ty=0,
yTFCFTy=0.
FCFT=s[e]×C[e]×T,
Gij-sHij=0,i, j=1, 2, 3.
(Gi1-sHi1)e1
+(Gi2-sHi2)e2+(Gi3-sHi3)e3=0,
i=1, 2, 3 .
E=k=13l=13 cT(glk-skhlk)(glk-skhlk)Tc,
sk=cTMkc2cTNkc,
E=k=13cTLkc-(cTMkc)24cTNkc,

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