Abstract

Visual measurement technology has been widely used in fields such as industrial production or measurement and monitoring. Camera calibration is a very important link of visual measurement, because it directly determines the accuracy and precision of visual measurement. It often does not need extremely high-precision measurement but simple, rapid, effective measurement in many engineering surveys. Therefore, the camera self-calibration technique is really needed. The advantages of camera self-calibration are that it does not need any calibration target or complex mechanical structure that is used to control the camera’s motion. In this paper, we propose an efficient camera self-calibration method based on the abstract quadric that is simple for calculation and has good robustness.

© 2013 Optical Society of America

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References

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  1. W. Faig, “Calibration of close- range photogrammetric systems: mathematical formulation,” Photogramm. Eng. Remote Sens. 41, 1479–1486 (1975).
  2. R. Shapiro, “Direct linear transformation method for three-dimensional cinematography,” Research Q. 49, 197–205 (1978).
  3. R. Y. Tsai, “An efficient and accurate camera calibration technique for 3D machine vision,” in Proceedings of IEEE Computer Society Conference on Computer Vision and Pattern Recognition, CVPR’86 (IEEE, 1986), pp. 364–374.
  4. Z. Zhang, “Flexible camera calibration by viewing a plane from unknown orientations,” in Proceedings of 7th International Conference on Computer Vision (IEEE, 1999), pp. 666–673.
  5. Z. Zhang, “A flexible new technique for camera calibration,” IEEE Trans. Pattern Anal. Mach. Intel. 22, 1330–1334 (2000).
    [CrossRef]
  6. Z. Zhang, “Camera calibration with one-dimensional objects,” IEEE Trans. Pattern Anal. Mach. Intel. 26, 892–899 (2004).
    [CrossRef]
  7. F. C. Wu, Z. Y. Hu, and H. J. Zhu, “Camera calibration with moving one-dimensional objects,” Pattern Recogn. 38, 755–765(2005).
    [CrossRef]
  8. O. D. Faugeras, Q. T. Luong, and S. J. Maybank, “Camera self-calibration: Theory and experiments,” in Proceedings of the Second European Conference on Computer Vision, ECCV’92 (Springer, 1992), pp. 321–334.
  9. S. Maybank and O. Faugeras, “A theory of self-calibration of moving camera,” Int. J. Comput. Vis. 8, 123–151 (1992).
    [CrossRef]
  10. T. Moons, L. Van Gool, and M. Proesmans, “Affine re-construction from perspective image pairs with a relative object-camera translation in between,” IEEE Trans. Pattern Anal. Mach. Intel. 18, 77–83 (1996).
    [CrossRef]
  11. B. Triggs, “Autocalibration and absolute quadric,” in Proceedings of IEEE Computer Society Conference on Computer Vision and Pattern Recognition, CVPR’1997 (IEEE, 1997), pp. 604–614.
  12. S. D. Ma, “A self-calibration technique for active vision system,” IEEE Trans. Robot. Autom. 12, 114–120 (1996).
    [CrossRef]
  13. R. I. Hartley, “Euclidean reconstruction from un-calibrated views,” in Applications of Invariance in Computer Science, Second Joint European-U.S. Workshop Proceedings (1994), pp. 235–256.
    [CrossRef]
  14. M. Pollefeys, R. Koch, and L. Van Gool, “Self-calibration and metric reconstruction inspite of varying and unknown instrinsic camera parameters,” Int. J. Comput. Vis. 32, 7–25 (1999).
    [CrossRef]
  15. P. Sturm and B. Triggs, “A factorization based algorithm for multi-image projective structure and motion,” in 4th European Conference on Computer Vision, ECCV’96 (Springer, 1996), pp. 709–720.
  16. T. Ueshiba and F. Tomita, “A factorization method for multiple perspective Views via iterative depth estimation,” Systems and Computers in Japan 31, 87–95 (2000).
    [CrossRef]
  17. R. Hartley and A. Zisserman, Multiple View Geometry in Computer Vision (Cambridge University, 2000), pp. 585–587.

2005

F. C. Wu, Z. Y. Hu, and H. J. Zhu, “Camera calibration with moving one-dimensional objects,” Pattern Recogn. 38, 755–765(2005).
[CrossRef]

2004

Z. Zhang, “Camera calibration with one-dimensional objects,” IEEE Trans. Pattern Anal. Mach. Intel. 26, 892–899 (2004).
[CrossRef]

2000

Z. Zhang, “A flexible new technique for camera calibration,” IEEE Trans. Pattern Anal. Mach. Intel. 22, 1330–1334 (2000).
[CrossRef]

T. Ueshiba and F. Tomita, “A factorization method for multiple perspective Views via iterative depth estimation,” Systems and Computers in Japan 31, 87–95 (2000).
[CrossRef]

1999

M. Pollefeys, R. Koch, and L. Van Gool, “Self-calibration and metric reconstruction inspite of varying and unknown instrinsic camera parameters,” Int. J. Comput. Vis. 32, 7–25 (1999).
[CrossRef]

1996

T. Moons, L. Van Gool, and M. Proesmans, “Affine re-construction from perspective image pairs with a relative object-camera translation in between,” IEEE Trans. Pattern Anal. Mach. Intel. 18, 77–83 (1996).
[CrossRef]

S. D. Ma, “A self-calibration technique for active vision system,” IEEE Trans. Robot. Autom. 12, 114–120 (1996).
[CrossRef]

1992

S. Maybank and O. Faugeras, “A theory of self-calibration of moving camera,” Int. J. Comput. Vis. 8, 123–151 (1992).
[CrossRef]

1978

R. Shapiro, “Direct linear transformation method for three-dimensional cinematography,” Research Q. 49, 197–205 (1978).

1975

W. Faig, “Calibration of close- range photogrammetric systems: mathematical formulation,” Photogramm. Eng. Remote Sens. 41, 1479–1486 (1975).

Faig, W.

W. Faig, “Calibration of close- range photogrammetric systems: mathematical formulation,” Photogramm. Eng. Remote Sens. 41, 1479–1486 (1975).

Faugeras, O.

S. Maybank and O. Faugeras, “A theory of self-calibration of moving camera,” Int. J. Comput. Vis. 8, 123–151 (1992).
[CrossRef]

Faugeras, O. D.

O. D. Faugeras, Q. T. Luong, and S. J. Maybank, “Camera self-calibration: Theory and experiments,” in Proceedings of the Second European Conference on Computer Vision, ECCV’92 (Springer, 1992), pp. 321–334.

Hartley, R.

R. Hartley and A. Zisserman, Multiple View Geometry in Computer Vision (Cambridge University, 2000), pp. 585–587.

Hartley, R. I.

R. I. Hartley, “Euclidean reconstruction from un-calibrated views,” in Applications of Invariance in Computer Science, Second Joint European-U.S. Workshop Proceedings (1994), pp. 235–256.
[CrossRef]

Hu, Z. Y.

F. C. Wu, Z. Y. Hu, and H. J. Zhu, “Camera calibration with moving one-dimensional objects,” Pattern Recogn. 38, 755–765(2005).
[CrossRef]

Koch, R.

M. Pollefeys, R. Koch, and L. Van Gool, “Self-calibration and metric reconstruction inspite of varying and unknown instrinsic camera parameters,” Int. J. Comput. Vis. 32, 7–25 (1999).
[CrossRef]

Luong, Q. T.

O. D. Faugeras, Q. T. Luong, and S. J. Maybank, “Camera self-calibration: Theory and experiments,” in Proceedings of the Second European Conference on Computer Vision, ECCV’92 (Springer, 1992), pp. 321–334.

Ma, S. D.

S. D. Ma, “A self-calibration technique for active vision system,” IEEE Trans. Robot. Autom. 12, 114–120 (1996).
[CrossRef]

Maybank, S.

S. Maybank and O. Faugeras, “A theory of self-calibration of moving camera,” Int. J. Comput. Vis. 8, 123–151 (1992).
[CrossRef]

Maybank, S. J.

O. D. Faugeras, Q. T. Luong, and S. J. Maybank, “Camera self-calibration: Theory and experiments,” in Proceedings of the Second European Conference on Computer Vision, ECCV’92 (Springer, 1992), pp. 321–334.

Moons, T.

T. Moons, L. Van Gool, and M. Proesmans, “Affine re-construction from perspective image pairs with a relative object-camera translation in between,” IEEE Trans. Pattern Anal. Mach. Intel. 18, 77–83 (1996).
[CrossRef]

Pollefeys, M.

M. Pollefeys, R. Koch, and L. Van Gool, “Self-calibration and metric reconstruction inspite of varying and unknown instrinsic camera parameters,” Int. J. Comput. Vis. 32, 7–25 (1999).
[CrossRef]

Proesmans, M.

T. Moons, L. Van Gool, and M. Proesmans, “Affine re-construction from perspective image pairs with a relative object-camera translation in between,” IEEE Trans. Pattern Anal. Mach. Intel. 18, 77–83 (1996).
[CrossRef]

Shapiro, R.

R. Shapiro, “Direct linear transformation method for three-dimensional cinematography,” Research Q. 49, 197–205 (1978).

Sturm, P.

P. Sturm and B. Triggs, “A factorization based algorithm for multi-image projective structure and motion,” in 4th European Conference on Computer Vision, ECCV’96 (Springer, 1996), pp. 709–720.

Tomita, F.

T. Ueshiba and F. Tomita, “A factorization method for multiple perspective Views via iterative depth estimation,” Systems and Computers in Japan 31, 87–95 (2000).
[CrossRef]

Triggs, B.

P. Sturm and B. Triggs, “A factorization based algorithm for multi-image projective structure and motion,” in 4th European Conference on Computer Vision, ECCV’96 (Springer, 1996), pp. 709–720.

B. Triggs, “Autocalibration and absolute quadric,” in Proceedings of IEEE Computer Society Conference on Computer Vision and Pattern Recognition, CVPR’1997 (IEEE, 1997), pp. 604–614.

Tsai, R. Y.

R. Y. Tsai, “An efficient and accurate camera calibration technique for 3D machine vision,” in Proceedings of IEEE Computer Society Conference on Computer Vision and Pattern Recognition, CVPR’86 (IEEE, 1986), pp. 364–374.

Ueshiba, T.

T. Ueshiba and F. Tomita, “A factorization method for multiple perspective Views via iterative depth estimation,” Systems and Computers in Japan 31, 87–95 (2000).
[CrossRef]

Van Gool, L.

M. Pollefeys, R. Koch, and L. Van Gool, “Self-calibration and metric reconstruction inspite of varying and unknown instrinsic camera parameters,” Int. J. Comput. Vis. 32, 7–25 (1999).
[CrossRef]

T. Moons, L. Van Gool, and M. Proesmans, “Affine re-construction from perspective image pairs with a relative object-camera translation in between,” IEEE Trans. Pattern Anal. Mach. Intel. 18, 77–83 (1996).
[CrossRef]

Wu, F. C.

F. C. Wu, Z. Y. Hu, and H. J. Zhu, “Camera calibration with moving one-dimensional objects,” Pattern Recogn. 38, 755–765(2005).
[CrossRef]

Zhang, Z.

Z. Zhang, “Camera calibration with one-dimensional objects,” IEEE Trans. Pattern Anal. Mach. Intel. 26, 892–899 (2004).
[CrossRef]

Z. Zhang, “A flexible new technique for camera calibration,” IEEE Trans. Pattern Anal. Mach. Intel. 22, 1330–1334 (2000).
[CrossRef]

Z. Zhang, “Flexible camera calibration by viewing a plane from unknown orientations,” in Proceedings of 7th International Conference on Computer Vision (IEEE, 1999), pp. 666–673.

Zhu, H. J.

F. C. Wu, Z. Y. Hu, and H. J. Zhu, “Camera calibration with moving one-dimensional objects,” Pattern Recogn. 38, 755–765(2005).
[CrossRef]

Zisserman, A.

R. Hartley and A. Zisserman, Multiple View Geometry in Computer Vision (Cambridge University, 2000), pp. 585–587.

IEEE Trans. Pattern Anal. Mach. Intel.

Z. Zhang, “A flexible new technique for camera calibration,” IEEE Trans. Pattern Anal. Mach. Intel. 22, 1330–1334 (2000).
[CrossRef]

Z. Zhang, “Camera calibration with one-dimensional objects,” IEEE Trans. Pattern Anal. Mach. Intel. 26, 892–899 (2004).
[CrossRef]

T. Moons, L. Van Gool, and M. Proesmans, “Affine re-construction from perspective image pairs with a relative object-camera translation in between,” IEEE Trans. Pattern Anal. Mach. Intel. 18, 77–83 (1996).
[CrossRef]

IEEE Trans. Robot. Autom.

S. D. Ma, “A self-calibration technique for active vision system,” IEEE Trans. Robot. Autom. 12, 114–120 (1996).
[CrossRef]

Int. J. Comput. Vis.

M. Pollefeys, R. Koch, and L. Van Gool, “Self-calibration and metric reconstruction inspite of varying and unknown instrinsic camera parameters,” Int. J. Comput. Vis. 32, 7–25 (1999).
[CrossRef]

S. Maybank and O. Faugeras, “A theory of self-calibration of moving camera,” Int. J. Comput. Vis. 8, 123–151 (1992).
[CrossRef]

Pattern Recogn.

F. C. Wu, Z. Y. Hu, and H. J. Zhu, “Camera calibration with moving one-dimensional objects,” Pattern Recogn. 38, 755–765(2005).
[CrossRef]

Photogramm. Eng. Remote Sens.

W. Faig, “Calibration of close- range photogrammetric systems: mathematical formulation,” Photogramm. Eng. Remote Sens. 41, 1479–1486 (1975).

Research Q.

R. Shapiro, “Direct linear transformation method for three-dimensional cinematography,” Research Q. 49, 197–205 (1978).

Systems and Computers in Japan

T. Ueshiba and F. Tomita, “A factorization method for multiple perspective Views via iterative depth estimation,” Systems and Computers in Japan 31, 87–95 (2000).
[CrossRef]

Other

R. Hartley and A. Zisserman, Multiple View Geometry in Computer Vision (Cambridge University, 2000), pp. 585–587.

P. Sturm and B. Triggs, “A factorization based algorithm for multi-image projective structure and motion,” in 4th European Conference on Computer Vision, ECCV’96 (Springer, 1996), pp. 709–720.

R. I. Hartley, “Euclidean reconstruction from un-calibrated views,” in Applications of Invariance in Computer Science, Second Joint European-U.S. Workshop Proceedings (1994), pp. 235–256.
[CrossRef]

B. Triggs, “Autocalibration and absolute quadric,” in Proceedings of IEEE Computer Society Conference on Computer Vision and Pattern Recognition, CVPR’1997 (IEEE, 1997), pp. 604–614.

R. Y. Tsai, “An efficient and accurate camera calibration technique for 3D machine vision,” in Proceedings of IEEE Computer Society Conference on Computer Vision and Pattern Recognition, CVPR’86 (IEEE, 1986), pp. 364–374.

Z. Zhang, “Flexible camera calibration by viewing a plane from unknown orientations,” in Proceedings of 7th International Conference on Computer Vision (IEEE, 1999), pp. 666–673.

O. D. Faugeras, Q. T. Luong, and S. J. Maybank, “Camera self-calibration: Theory and experiments,” in Proceedings of the Second European Conference on Computer Vision, ECCV’92 (Springer, 1992), pp. 321–334.

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Figures (8)

Fig. 1.
Fig. 1.

Camera imaging model.

Fig. 2.
Fig. 2.

Relative error of focal length fu.

Fig. 3.
Fig. 3.

Relative error of focal length fv.

Fig. 4.
Fig. 4.

Relative error of principal point coordinates u0.

Fig. 5.
Fig. 5.

Relative error of principal point coordinates v0.

Fig. 6.
Fig. 6.

Relative errors of the camera’s internal parameters in different degrees of random noises from 0.01 pixels to 0.1 pixels.

Fig. 7.
Fig. 7.

Relative errors of the camera’s internal parameters in different degrees of random noises from 0.1 pixels to 1 pixel.

Fig. 8.
Fig. 8.

Relative error of different self-calibration methods.

Tables (1)

Tables Icon

Table 1. Error Analysis of the Camera’s Internal Parameters

Equations (22)

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

λx=PX,
P=K[R|t],
K=[fusu00fvv0001],
Q*e=[I3×300T0].
PeQ*ePeT=KRRTKT=KKT=ω*.
Pe=PpH,
PpHQ*eHTPpT=PpQ*pPpT=KKT=ω*,
Tk=[10ut01vt001].
λx=λTkx=TkPpXp=PpXp,
ω*=KKT=[fu2+u02+s2u0v0+sfvu0u0v0+sfvfv2+v02v0u0v01].
ω*=KKT=[fu2+s2sfv0sfvfv20001].
(PpiQ*pPpiT)13=(PpiQ*pPpiT)23=0(i=1,2,3,,m).
[ P 11 P 31 P 11 P 31 + P 11 P 32 P 13 P 31 + P 11 P 33 P 14 P 31 + P 11 P 34 P 12 P 32 P 13 P 32 + P 12 P 33 P 14 P 32 + P 12 P 34 P 13 P 33 P 14 P 33 + P 13 P 34 P 14 P 34 P 21 P 31 P 21 P 31 + P 21 P 32 P 23 P 31 + P 21 P 33 P 24 P 31 + P 21 P 34 P 22 P 32 P 23 P 32 + P 22 P 33 P 24 P 32 + P 22 P 34 P 23 P 33 P 24 P 33 + P 23 P 34 P 24 P 34 ] [ Q 11 Q 12 Q 13 Q 14 Q 22 Q 23 Q 24 Q 33 Q 34 Q 44 ] = A q = 0 ,
ω*=KKT=[fu2000fv20001].
(PpQ*pPpT)12=(PpQ*pPpT)13=(PpQ*PPpT)23=0.
[ P 11 P 21 P 11 P 21 + P 11 P 22 P 13 P 21 + P 11 P 23 P 14 P 21 + P 11 P 34 P 12 P 22 P 13 P 22 + P 12 P 23 P 14 P 22 + P 12 P 24 P 13 P 23 P 14 P 23 + P 13 P 24 P 14 P 24 P 11 P 31 P 11 P 31 + P 11 P 32 P 13 P 31 + P 11 P 33 P 14 P 31 + P 11 P 34 P 12 P 32 P 13 P 32 + P 12 P 33 P 14 P 32 + P 12 P 34 P 13 P 33 P 14 P 33 + P 13 P 34 P 14 P 34 P 21 P 31 P 21 P 31 + P 21 P 32 P 23 P 31 + P 21 P 33 P 24 P 31 + P 21 P 34 P 22 P 32 P 23 P 32 + P 22 P 33 P 24 P 32 + P 22 P 34 P 23 P 33 P 24 P 33 + P 23 P 34 P 24 P 34 ] [ Q 11 Q 12 Q 13 Q 14 Q 22 Q 23 Q 24 Q 33 Q 34 Q 44 ] = A q = 0 ,
ω*i=KiKiT=PpiQ*pPpiT(i=1,2,3,,m).
PpiQ*pPpiT=PpjQ*pPpjT.
{(PpiQ*pPpiT)11/(PpiQ*pPpiT)33=(PpjQ*pPpjT)11/(PpjQ*pPpjT)33(PpiQ*pPpiT)12/(PpiQ*pPpiT)33=(PpjQ*pPpjT)12/(PpjQ*pPpjT)33(PpiQ*pPpiT)13/(PpiQ*pPpiT)33=(PpjQ*pPpjT)13/(PpjQ*pPpjT)33(PpiQ*pPpiT)22/(PpiQ*pPpiT)33=(PpjQ*pPpjT)22/(PpjQ*pPpjT)33(PpiQ*pPpiT)23/(PpiQ*pPpiT)33=(PpjQ*pPpjT)23/(PpjQ*pPpjT)33,
K0=[fu0u00fvv0001].
K=[10145010240101451024001].
Ks=[10144.99986.5e-101023.9978010144.99981023.9989001].

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