Abstract

Nonindustrial low-cost cameras have the advantages of cheap and simple structure, but have the disadvantages of low resolution and large image noise. When the existing camera calibration methods are used to calibrate nonindustrial low-cost cameras, high-accuracy calibration cannot be obtained. A high-accuracy calibration method using a high-accuracy planar target is introduced in this study to solve this problem. First, the initial values and the uncertainties of all image feature points are determined by the multiscale image analysis method. Then, the image disturbance factor is added to each target image feature point. In addition, the image projection error is established as the minimum objective function according to the homography matrix between the target plane and the image plane. Thus, the optimal coordinates of all image feature points are obtained by the nonlinear optimization method. Finally, the calibration of the intrinsic and extrinsic parameters of the camera will be achieved by using Zhang’s method according to the image feature points obtained from the previous step. Simulative and real experiments have been conducted to evaluate the performance of the proposed method, and results show that the calibration accuracy of the proposed method is at least three times that of Zhang’s method.

© 2016 Optical Society of America

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References

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    [Crossref]
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    [Crossref]
  7. J. S. Kim, P. Gurdjos, and I. S. Kweon, “Geometric and algebraic constraints of projected concentric circles and their applications to camera calibration,” IEEE Trans. Pattern Anal. Mach. Intell. 27(4), 637–642 (2005).
    [Crossref] [PubMed]
  8. Y. H. Wu, X. J. Li, F. C. Wu, and Z. Hu, “Coplanar circles, quasi-affine invariance and calibration,” Image Vis. Comput. 24(4), 319–326 (2006).
    [Crossref]
  9. D. Douxchamps and K. Chihara, “High-accuracy and robust localization of large control markers for geometric camera calibration,” IEEE Trans. Pattern Anal. Mach. Intell. 31(2), 376–383 (2009).
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    [Crossref] [PubMed]
  11. Y. Hong, G. Ren, and E. Liu, “Non-iterative method for camera calibration,” Opt. Express 23(18), 23992–24003 (2015).
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    [Crossref] [PubMed]
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    [Crossref] [PubMed]
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    [Crossref]
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    [Crossref]
  17. H. Zhang, K. Y. Wong, and G. Zhang, “Camera calibration from images of spheres,” IEEE Trans. Pattern Anal. Mach. Intell. 29(3), 499–502 (2007).
    [Crossref] [PubMed]
  18. X. Ying and H. Zha, “Geometric interpretations of the relation between the image of the absolute conic and sphere images,” IEEE Trans. Pattern Anal. Mach. Intell. 28(12), 2031–2036 (2006).
    [Crossref] [PubMed]
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    [Crossref]
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    [Crossref]
  21. A. Albarelli, E. Rodola, and A. Torsello, “Robust camera calibration using inaccurate target,” inProceedings of The British Maching Vision Conference (2010), pp. 1 - 10.
  22. H. Strobl and G. Hirzinger, “More accurate camera and hand-eye calibration with unknown grid pattern dimensions,” inProceedings of IEEE Conference on Robotics and Automation (IEEE, 2008), pp. 1398 - 1405.
    [Crossref]
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    [Crossref] [PubMed]
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    [Crossref]
  25. A. Datta, J. S. Kim, and T. Kanade, “Accurate camera calibration using iterative refinement of control points,” inProceedings of IEEE Conference on Computer Vision Workshops (IEEE, 2009), pp. 1201 - 1208.
    [Crossref]
  26. J. Y. Bouguet, “The MATLAB open source calibration toolbox,” http://www.vision.caltech.edu/bouguetj/calib _doc/ .
  27. J. MORE, The Levenberg-Marquardt Algorithm, Implementation and Theory (Numerical Analysis, 1977).

2015 (2)

Z. Liu, F. J. Li, X. J. Li, and G. J. Zhang, “A novel and accurate calibration method for cameras with large field of view using combined small targets,” Measurement 64(3), 1–16 (2015).

Y. Hong, G. Ren, and E. Liu, “Non-iterative method for camera calibration,” Opt. Express 23(18), 23992–24003 (2015).
[Crossref] [PubMed]

2014 (1)

S. Shirmohammadi and A. Ferrero, “Camera as the instrument: the rising trend of vision based measurement,” IEEE Trans. Instrum. Meas. 17(3), 41–47 (2014).
[Crossref]

2013 (2)

2012 (1)

2011 (1)

2009 (1)

D. Douxchamps and K. Chihara, “High-accuracy and robust localization of large control markers for geometric camera calibration,” IEEE Trans. Pattern Anal. Mach. Intell. 31(2), 376–383 (2009).
[Crossref] [PubMed]

2007 (2)

F. Qi, Q. Li, Y. Luo, and D. Hu, “Camera calibration with one-dimensional objects moving under gravity,” Pattern Recognit. 40(1), 343–345 (2007).
[Crossref]

H. Zhang, K. Y. Wong, and G. Zhang, “Camera calibration from images of spheres,” IEEE Trans. Pattern Anal. Mach. Intell. 29(3), 499–502 (2007).
[Crossref] [PubMed]

2006 (2)

X. Ying and H. Zha, “Geometric interpretations of the relation between the image of the absolute conic and sphere images,” IEEE Trans. Pattern Anal. Mach. Intell. 28(12), 2031–2036 (2006).
[Crossref] [PubMed]

Y. H. Wu, X. J. Li, F. C. Wu, and Z. Hu, “Coplanar circles, quasi-affine invariance and calibration,” Image Vis. Comput. 24(4), 319–326 (2006).
[Crossref]

2005 (2)

J. S. Kim, P. Gurdjos, and I. S. Kweon, “Geometric and algebraic constraints of projected concentric circles and their applications to camera calibration,” IEEE Trans. Pattern Anal. Mach. Intell. 27(4), 637–642 (2005).
[Crossref] [PubMed]

F. C. Wu, Z. Y. Hu, and H. J. Zhu, “Camera calibration with moving one-dimensional objects,” Pattern Recognit. 38(5), 755–765 (2005).
[Crossref]

2004 (1)

Z. Zhang, “Camera calibration with one-dimensional objects,” IEEE Trans. Pattern Anal. Mach. Intell. 26(7), 892–899 (2004).
[Crossref] [PubMed]

2003 (2)

E. N. Malamas, E. G. M. Petrakis, M. Zervakis, L. Petit, and J. D. Legat, “A survey on industrial vision systems, applications and tools,” Image Vis. Comput. 21(2), 171–188 (2003).
[Crossref]

K. K. Wong, R. S. P. Mendonca, and R. Cipolla, “Camera calibration from surfaces of revolution,” IEEE Trans. Pattern Anal. Mach. Intell. 25(2), 147–161 (2003).
[Crossref]

2000 (2)

J. Heikkila, “Geometric camera calibration using circular control points,” IEEE Trans. Pattern Anal. Mach. Intell. 22(10), 1066–1077 (2000).
[Crossref]

Z. Y. Zhang, “A flexible new technique for camera calibration,” IEEE Trans. Pattern Anal. Mach. Intell. 22(11), 1330–1334 (2000).
[Crossref]

1987 (1)

R. Y. Tsai, “A versatile camera calibration technique for high-accuracy 3D machine vision metrology using off-the-shelf TV camera and lenses,” IEEE J. Robot. Autom. 3(4), 323–344 (1987).
[Crossref]

Albarelli, A.

A. Albarelli, E. Rodola, and A. Torsello, “Robust camera calibration using inaccurate target,” inProceedings of The British Maching Vision Conference (2010), pp. 1 - 10.

Asundi, A.

Chihara, K.

D. Douxchamps and K. Chihara, “High-accuracy and robust localization of large control markers for geometric camera calibration,” IEEE Trans. Pattern Anal. Mach. Intell. 31(2), 376–383 (2009).
[Crossref] [PubMed]

Cipolla, R.

K. K. Wong, R. S. P. Mendonca, and R. Cipolla, “Camera calibration from surfaces of revolution,” IEEE Trans. Pattern Anal. Mach. Intell. 25(2), 147–161 (2003).
[Crossref]

Datta, A.

A. Datta, J. S. Kim, and T. Kanade, “Accurate camera calibration using iterative refinement of control points,” inProceedings of IEEE Conference on Computer Vision Workshops (IEEE, 2009), pp. 1201 - 1208.
[Crossref]

Douxchamps, D.

D. Douxchamps and K. Chihara, “High-accuracy and robust localization of large control markers for geometric camera calibration,” IEEE Trans. Pattern Anal. Mach. Intell. 31(2), 376–383 (2009).
[Crossref] [PubMed]

Ferrero, A.

S. Shirmohammadi and A. Ferrero, “Camera as the instrument: the rising trend of vision based measurement,” IEEE Trans. Instrum. Meas. 17(3), 41–47 (2014).
[Crossref]

Gurdjos, P.

J. S. Kim, P. Gurdjos, and I. S. Kweon, “Geometric and algebraic constraints of projected concentric circles and their applications to camera calibration,” IEEE Trans. Pattern Anal. Mach. Intell. 27(4), 637–642 (2005).
[Crossref] [PubMed]

Hasegawa, Y.

K. Nakano, M. Okutomi, and Y. Hasegawa, “Camera calibration with precise extraction of feature points using projective transformation,” inProceedings of IEEE Conference on Robotics and Automation (IEEE, 2002), pp. 2532 - 2538.
[Crossref]

Heikkila, J.

J. Heikkila, “Geometric camera calibration using circular control points,” IEEE Trans. Pattern Anal. Mach. Intell. 22(10), 1066–1077 (2000).
[Crossref]

Hirzinger, G.

H. Strobl and G. Hirzinger, “More accurate camera and hand-eye calibration with unknown grid pattern dimensions,” inProceedings of IEEE Conference on Robotics and Automation (IEEE, 2008), pp. 1398 - 1405.
[Crossref]

Hong, Y.

Hu, D.

F. Qi, Q. Li, Y. Luo, and D. Hu, “Camera calibration with one-dimensional objects moving under gravity,” Pattern Recognit. 40(1), 343–345 (2007).
[Crossref]

Hu, Z.

Y. H. Wu, X. J. Li, F. C. Wu, and Z. Hu, “Coplanar circles, quasi-affine invariance and calibration,” Image Vis. Comput. 24(4), 319–326 (2006).
[Crossref]

Hu, Z. Y.

F. C. Wu, Z. Y. Hu, and H. J. Zhu, “Camera calibration with moving one-dimensional objects,” Pattern Recognit. 38(5), 755–765 (2005).
[Crossref]

Huang, L.

Kanade, T.

A. Datta, J. S. Kim, and T. Kanade, “Accurate camera calibration using iterative refinement of control points,” inProceedings of IEEE Conference on Computer Vision Workshops (IEEE, 2009), pp. 1201 - 1208.
[Crossref]

Kim, J. H.

Kim, J. S.

J. S. Kim, P. Gurdjos, and I. S. Kweon, “Geometric and algebraic constraints of projected concentric circles and their applications to camera calibration,” IEEE Trans. Pattern Anal. Mach. Intell. 27(4), 637–642 (2005).
[Crossref] [PubMed]

A. Datta, J. S. Kim, and T. Kanade, “Accurate camera calibration using iterative refinement of control points,” inProceedings of IEEE Conference on Computer Vision Workshops (IEEE, 2009), pp. 1201 - 1208.
[Crossref]

Koo, B. K.

Kweon, I. S.

J. S. Kim, P. Gurdjos, and I. S. Kweon, “Geometric and algebraic constraints of projected concentric circles and their applications to camera calibration,” IEEE Trans. Pattern Anal. Mach. Intell. 27(4), 637–642 (2005).
[Crossref] [PubMed]

Legat, J. D.

E. N. Malamas, E. G. M. Petrakis, M. Zervakis, L. Petit, and J. D. Legat, “A survey on industrial vision systems, applications and tools,” Image Vis. Comput. 21(2), 171–188 (2003).
[Crossref]

Li, F. J.

Z. Liu, F. J. Li, X. J. Li, and G. J. Zhang, “A novel and accurate calibration method for cameras with large field of view using combined small targets,” Measurement 64(3), 1–16 (2015).

Li, Q.

F. Qi, Q. Li, Y. Luo, and D. Hu, “Camera calibration with one-dimensional objects moving under gravity,” Pattern Recognit. 40(1), 343–345 (2007).
[Crossref]

Li, X. J.

Z. Liu, F. J. Li, X. J. Li, and G. J. Zhang, “A novel and accurate calibration method for cameras with large field of view using combined small targets,” Measurement 64(3), 1–16 (2015).

Y. H. Wu, X. J. Li, F. C. Wu, and Z. Hu, “Coplanar circles, quasi-affine invariance and calibration,” Image Vis. Comput. 24(4), 319–326 (2006).
[Crossref]

Liu, E.

Liu, Z.

Z. Liu, F. J. Li, X. J. Li, and G. J. Zhang, “A novel and accurate calibration method for cameras with large field of view using combined small targets,” Measurement 64(3), 1–16 (2015).

Luo, Y.

F. Qi, Q. Li, Y. Luo, and D. Hu, “Camera calibration with one-dimensional objects moving under gravity,” Pattern Recognit. 40(1), 343–345 (2007).
[Crossref]

Malamas, E. N.

E. N. Malamas, E. G. M. Petrakis, M. Zervakis, L. Petit, and J. D. Legat, “A survey on industrial vision systems, applications and tools,” Image Vis. Comput. 21(2), 171–188 (2003).
[Crossref]

Mendonca, R. S. P.

K. K. Wong, R. S. P. Mendonca, and R. Cipolla, “Camera calibration from surfaces of revolution,” IEEE Trans. Pattern Anal. Mach. Intell. 25(2), 147–161 (2003).
[Crossref]

Nakano, K.

K. Nakano, M. Okutomi, and Y. Hasegawa, “Camera calibration with precise extraction of feature points using projective transformation,” inProceedings of IEEE Conference on Robotics and Automation (IEEE, 2002), pp. 2532 - 2538.
[Crossref]

Okutomi, M.

K. Nakano, M. Okutomi, and Y. Hasegawa, “Camera calibration with precise extraction of feature points using projective transformation,” inProceedings of IEEE Conference on Robotics and Automation (IEEE, 2002), pp. 2532 - 2538.
[Crossref]

Petit, L.

E. N. Malamas, E. G. M. Petrakis, M. Zervakis, L. Petit, and J. D. Legat, “A survey on industrial vision systems, applications and tools,” Image Vis. Comput. 21(2), 171–188 (2003).
[Crossref]

Petrakis, E. G. M.

E. N. Malamas, E. G. M. Petrakis, M. Zervakis, L. Petit, and J. D. Legat, “A survey on industrial vision systems, applications and tools,” Image Vis. Comput. 21(2), 171–188 (2003).
[Crossref]

Qi, F.

F. Qi, Q. Li, Y. Luo, and D. Hu, “Camera calibration with one-dimensional objects moving under gravity,” Pattern Recognit. 40(1), 343–345 (2007).
[Crossref]

Ren, G.

Ricolfe-Viala, C.

Rodola, E.

A. Albarelli, E. Rodola, and A. Torsello, “Robust camera calibration using inaccurate target,” inProceedings of The British Maching Vision Conference (2010), pp. 1 - 10.

Sanchez-Salmeron, A. J.

Shirmohammadi, S.

S. Shirmohammadi and A. Ferrero, “Camera as the instrument: the rising trend of vision based measurement,” IEEE Trans. Instrum. Meas. 17(3), 41–47 (2014).
[Crossref]

Strobl, H.

H. Strobl and G. Hirzinger, “More accurate camera and hand-eye calibration with unknown grid pattern dimensions,” inProceedings of IEEE Conference on Robotics and Automation (IEEE, 2008), pp. 1398 - 1405.
[Crossref]

Torsello, A.

A. Albarelli, E. Rodola, and A. Torsello, “Robust camera calibration using inaccurate target,” inProceedings of The British Maching Vision Conference (2010), pp. 1 - 10.

Tsai, R. Y.

R. Y. Tsai, “A versatile camera calibration technique for high-accuracy 3D machine vision metrology using off-the-shelf TV camera and lenses,” IEEE J. Robot. Autom. 3(4), 323–344 (1987).
[Crossref]

Wong, K. K.

K. K. Wong, R. S. P. Mendonca, and R. Cipolla, “Camera calibration from surfaces of revolution,” IEEE Trans. Pattern Anal. Mach. Intell. 25(2), 147–161 (2003).
[Crossref]

Wong, K. Y.

H. Zhang, K. Y. Wong, and G. Zhang, “Camera calibration from images of spheres,” IEEE Trans. Pattern Anal. Mach. Intell. 29(3), 499–502 (2007).
[Crossref] [PubMed]

Wu, F. C.

Y. H. Wu, X. J. Li, F. C. Wu, and Z. Hu, “Coplanar circles, quasi-affine invariance and calibration,” Image Vis. Comput. 24(4), 319–326 (2006).
[Crossref]

F. C. Wu, Z. Y. Hu, and H. J. Zhu, “Camera calibration with moving one-dimensional objects,” Pattern Recognit. 38(5), 755–765 (2005).
[Crossref]

Wu, Y. H.

Y. H. Wu, X. J. Li, F. C. Wu, and Z. Hu, “Coplanar circles, quasi-affine invariance and calibration,” Image Vis. Comput. 24(4), 319–326 (2006).
[Crossref]

Ying, X.

X. Ying and H. Zha, “Geometric interpretations of the relation between the image of the absolute conic and sphere images,” IEEE Trans. Pattern Anal. Mach. Intell. 28(12), 2031–2036 (2006).
[Crossref] [PubMed]

Zervakis, M.

E. N. Malamas, E. G. M. Petrakis, M. Zervakis, L. Petit, and J. D. Legat, “A survey on industrial vision systems, applications and tools,” Image Vis. Comput. 21(2), 171–188 (2003).
[Crossref]

Zha, H.

X. Ying and H. Zha, “Geometric interpretations of the relation between the image of the absolute conic and sphere images,” IEEE Trans. Pattern Anal. Mach. Intell. 28(12), 2031–2036 (2006).
[Crossref] [PubMed]

Zhang, G.

H. Zhang, K. Y. Wong, and G. Zhang, “Camera calibration from images of spheres,” IEEE Trans. Pattern Anal. Mach. Intell. 29(3), 499–502 (2007).
[Crossref] [PubMed]

Zhang, G. J.

Z. Liu, F. J. Li, X. J. Li, and G. J. Zhang, “A novel and accurate calibration method for cameras with large field of view using combined small targets,” Measurement 64(3), 1–16 (2015).

Zhang, H.

H. Zhang, K. Y. Wong, and G. Zhang, “Camera calibration from images of spheres,” IEEE Trans. Pattern Anal. Mach. Intell. 29(3), 499–502 (2007).
[Crossref] [PubMed]

Zhang, Q.

Zhang, Z.

Z. Zhang, “Camera calibration with one-dimensional objects,” IEEE Trans. Pattern Anal. Mach. Intell. 26(7), 892–899 (2004).
[Crossref] [PubMed]

Zhang, Z. Y.

Z. Y. Zhang, “A flexible new technique for camera calibration,” IEEE Trans. Pattern Anal. Mach. Intell. 22(11), 1330–1334 (2000).
[Crossref]

Zhu, H. J.

F. C. Wu, Z. Y. Hu, and H. J. Zhu, “Camera calibration with moving one-dimensional objects,” Pattern Recognit. 38(5), 755–765 (2005).
[Crossref]

Appl. Opt. (1)

IEEE J. Robot. Autom. (1)

R. Y. Tsai, “A versatile camera calibration technique for high-accuracy 3D machine vision metrology using off-the-shelf TV camera and lenses,” IEEE J. Robot. Autom. 3(4), 323–344 (1987).
[Crossref]

IEEE Trans. Instrum. Meas. (1)

S. Shirmohammadi and A. Ferrero, “Camera as the instrument: the rising trend of vision based measurement,” IEEE Trans. Instrum. Meas. 17(3), 41–47 (2014).
[Crossref]

IEEE Trans. Pattern Anal. Mach. Intell. (8)

D. Douxchamps and K. Chihara, “High-accuracy and robust localization of large control markers for geometric camera calibration,” IEEE Trans. Pattern Anal. Mach. Intell. 31(2), 376–383 (2009).
[Crossref] [PubMed]

J. Heikkila, “Geometric camera calibration using circular control points,” IEEE Trans. Pattern Anal. Mach. Intell. 22(10), 1066–1077 (2000).
[Crossref]

Z. Y. Zhang, “A flexible new technique for camera calibration,” IEEE Trans. Pattern Anal. Mach. Intell. 22(11), 1330–1334 (2000).
[Crossref]

J. S. Kim, P. Gurdjos, and I. S. Kweon, “Geometric and algebraic constraints of projected concentric circles and their applications to camera calibration,” IEEE Trans. Pattern Anal. Mach. Intell. 27(4), 637–642 (2005).
[Crossref] [PubMed]

H. Zhang, K. Y. Wong, and G. Zhang, “Camera calibration from images of spheres,” IEEE Trans. Pattern Anal. Mach. Intell. 29(3), 499–502 (2007).
[Crossref] [PubMed]

X. Ying and H. Zha, “Geometric interpretations of the relation between the image of the absolute conic and sphere images,” IEEE Trans. Pattern Anal. Mach. Intell. 28(12), 2031–2036 (2006).
[Crossref] [PubMed]

K. K. Wong, R. S. P. Mendonca, and R. Cipolla, “Camera calibration from surfaces of revolution,” IEEE Trans. Pattern Anal. Mach. Intell. 25(2), 147–161 (2003).
[Crossref]

Z. Zhang, “Camera calibration with one-dimensional objects,” IEEE Trans. Pattern Anal. Mach. Intell. 26(7), 892–899 (2004).
[Crossref] [PubMed]

Image Vis. Comput. (2)

Y. H. Wu, X. J. Li, F. C. Wu, and Z. Hu, “Coplanar circles, quasi-affine invariance and calibration,” Image Vis. Comput. 24(4), 319–326 (2006).
[Crossref]

E. N. Malamas, E. G. M. Petrakis, M. Zervakis, L. Petit, and J. D. Legat, “A survey on industrial vision systems, applications and tools,” Image Vis. Comput. 21(2), 171–188 (2003).
[Crossref]

Measurement (1)

Z. Liu, F. J. Li, X. J. Li, and G. J. Zhang, “A novel and accurate calibration method for cameras with large field of view using combined small targets,” Measurement 64(3), 1–16 (2015).

Opt. Express (3)

Opt. Lett. (1)

Pattern Recognit. (2)

F. C. Wu, Z. Y. Hu, and H. J. Zhu, “Camera calibration with moving one-dimensional objects,” Pattern Recognit. 38(5), 755–765 (2005).
[Crossref]

F. Qi, Q. Li, Y. Luo, and D. Hu, “Camera calibration with one-dimensional objects moving under gravity,” Pattern Recognit. 40(1), 343–345 (2007).
[Crossref]

Other (7)

C. Colombo, D. Comanducci, and A. D. Bimbo, “Camera calibration with two arbitrary coaxial circles,” in Proceedings of European Conference on Computer Vision 1. (Springer Verlag, 2006), pp. 265 - 276.
[Crossref]

A. Albarelli, E. Rodola, and A. Torsello, “Robust camera calibration using inaccurate target,” inProceedings of The British Maching Vision Conference (2010), pp. 1 - 10.

H. Strobl and G. Hirzinger, “More accurate camera and hand-eye calibration with unknown grid pattern dimensions,” inProceedings of IEEE Conference on Robotics and Automation (IEEE, 2008), pp. 1398 - 1405.
[Crossref]

K. Nakano, M. Okutomi, and Y. Hasegawa, “Camera calibration with precise extraction of feature points using projective transformation,” inProceedings of IEEE Conference on Robotics and Automation (IEEE, 2002), pp. 2532 - 2538.
[Crossref]

A. Datta, J. S. Kim, and T. Kanade, “Accurate camera calibration using iterative refinement of control points,” inProceedings of IEEE Conference on Computer Vision Workshops (IEEE, 2009), pp. 1201 - 1208.
[Crossref]

J. Y. Bouguet, “The MATLAB open source calibration toolbox,” http://www.vision.caltech.edu/bouguetj/calib _doc/ .

J. MORE, The Levenberg-Marquardt Algorithm, Implementation and Theory (Numerical Analysis, 1977).

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Figures (15)

Fig. 1
Fig. 1 Sketch of the initial coordinates and the uncertainties of the image feature points.
Fig. 2
Fig. 2 Uncertainties of all target image feature point.
Fig. 3
Fig. 3 Homography relationship between the target plane and the image plane.
Fig. 4
Fig. 4 The transformation of the image feature point coordinates with the image disturbance factor.
Fig. 5
Fig. 5 RMSEs of the intrinsic parameters based on Zhang’s method and the proposed method.
Fig. 6
Fig. 6 Reprojection errors of the target points. (a) Reprojection errors in the u direction; (b) Reprojection errors in the v direction.
Fig. 7
Fig. 7 Reprojection errors of the target feature points based on two calibration methods. (a) Reprojection errors based on Zhang’s method; (b) Reprojection errors based on the proposed method; (c) The statistical form of the reprojection errors based on Zhang’s method; (d) The statistical form of the reprojection errors based on the proposed method.
Fig. 8
Fig. 8 Back-projection errors based on two calibration methods. (a) Back-projection errors based on Zhang’s method; (b) The statistical form of the back-projection errors based on Zhang’s method; (c) 3D coordinate errors of the target feature points based on Zhang’s calibration results; (d) Back-projection errors based on the proposed method; (e) The statistical form of the back-projection errors based on the proposed method; (f) 3D coordinate errors of the target feature points based on the proposed method calibration results.
Fig. 9
Fig. 9 Physical map of the calibration setup.
Fig. 10
Fig. 10 Images used for calibration.
Fig. 11
Fig. 11 Disturbance factors in the real image.
Fig. 12
Fig. 12 Reprojection errors of the target feature points based on the two methods. (a) Reprojection errors distribution based on Zhang’s method; (b) Reprojection errors distribution based on the proposed method; (c) The arrow form of the reprojection errors based on Zhang’s method; (d) The arrow form of the reprojection errors based on the proposed method; (e) The statistical form of the reprojection errors based on Zhang’s method; (f) The statistical form of the reprojection errors based on the proposed method.
Fig. 13
Fig. 13 Back-projection errors of the image feature points based on the two methods. (a) Back-projection errors distribution based on Zhang’s method; (b) Back-projection errors distribution based on the proposed method; (c) The arrow form of the back-projection errors based on Zhang’s method; (d) The arrow form of the back-projection errors based on the proposed method; (e) The statistical form of the back-projection errors based on Zhang’s method; (f) The statistical form of the back-projection errors based on the proposed method.
Fig. 14
Fig. 14 3D display form of the back-projection errors of the image feature points based on the two methods. (a) The back-projection errors of different feature points obtained by Zhang’s method; (b) The back-projection errors of different feature points obtained by the proposed method.
Fig. 15
Fig. 15 Statistical distributions of the distance deviation of the feature point pairs based on two calibration results.

Tables (2)

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Table 1 Comparative results of the intrinsic parameters

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Table 2 Comparative results of the uncertainties of the intrinsic parameters

Equations (11)

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ρp=K[ 1 0 0 0 0 1 0 0 0 0 1 0 ][ R t 0 1 ]q
u ˜ =u+(u u 0 )( k 1 r 2 + k 2 r 4 ) v ˜ =v+(v v 0 )( k 1 r 2 + k 2 r 4 )
s[ u ij v ij 1 ]= H i [ x j y j 1 ]
s p ^ ij =s[ u ij +Δ u ij v ij +Δ v ij 1 ]= H i [ x j y j 1 ]
e 1 (a)=min( i=1 M j=1 N Dist( p ^ ij , H i q ij ) + i=1 M j=1 N Dist( p ij , H i q ij )+ i=1 M j=1 N Dist( q j , q ^ ij ) )
e 2 (a)=min(( i=1 M j=1 N q j i=1 M j=1 N q ^ ij ))
E(a)= e 1 (a)+ e 2 (a)
{ a x n U ax a x a x +n U ax a y n U ay a y a y +n U ay γn U γ γγ+n U γ u 0 n U u0 u 0 u 0 +n U u0 v 0 n U v0 v 0 v 0 +n U v0 k 1 n U k1 k 1 k 1 +n U k1 k 2 n U k2 k 2 k 2 +n U k2 n U Δ u ij Δ u ij n U Δ u ij n U Δ v ij Δ v ij n U Δ v ij
s H i =s[ h 1i h 2i h 3i ]=K[ r 1i r 2i t i ]
{ h 1i T K T K 1 h 2i T =0 h 1i T K T K 1 h 1i T = h 2i T K T K 1 h 2i T
e 1 (a)=min( i=1 M j=1 N Dist( p ^ ij , H i q ij ) )

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