Abstract

Given the rapid convergence characteristic of the stochastic parallel gradient descent (SPGD) algorithm, this study proposes a method that applies the algorithm to a two-step camera calibration method to resolve the frequent iteration and long calibration time deficiencies that exist under the traditional two-step camera calibration method, thereby achieving rapid calibration. The method first uses image coordinates obtained with subpixel positioning technology as initial values of control variables, in addition to positive disturbances produced on a two-dimensional plane, then uses two-step theory to calculate the average value of aberrations. Based on the same rationale, negative disturbances are then produced and the average value of the aberrations is calculated. Finally if, after assessing whether to continue with further iterations based on the difference in these values, continued iterations confirm new control variables based on the SPGD algorithm iteration formula, a new cycle is started until the results satisfy requirements. Theoretical analysis and experimental results show that the proposed rapid calibration method using the SPGD algorithm in the two-step camera calibration method is 3–4 times faster than the traditional two-step calibration method, and that it has significant potential value for use in certain time-constrained projects.

© 2012 Optical Society of America

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  1. W. Qi, F. Li, and L. Zhengzhong, “Review on camera calibration,” presented at the Chinese Control and Design Conference, Shengyang, China, 2010.
  2. G. Xu and N. Sugimoto, “A linear algorithm for motion from three weak perspective images using Euler angles,” IEEE Trans. Pattern Anal. Machine Intell. 20, 54–57 (1998).
  3. W. Faig, “Calibration of close-range photogrammetry systems: mathematical formulation,” Photogramm. Eng. Remote Sens. 41, 1479–1486 (1975).
  4. Y. I. Abdel-Aziz and H. M. Karara, “Direct linear transformation into object space coordinates in close-range photogrammetry,” in Proceedings of Symposium on Close-Range Photogrammetry (University of Illinois, 1971), pp. 1–18.
  5. K. W. Wong, “Mathematical foundation and digital analysis in close-range photogrammetry,” Photogram. Eng. Remote Sens. 41, 1355–1373 (1975).
  6. H. M. Karara, Handbook of Close-Range Photogrammetry (American Society of Photogrammetry, 1979).
  7. R. Y. Tsai, “A versatile camera calibration technique for high accuracy 3D machine vision metrology using off-the-shelf TV cameras and lenses,” IEEE J. Robot. Automon. 3, 323–344 (1987).
  8. J. Weng, P. Cohen, and M. Herniou, “Camera calibration with distortion models and accuracy evaluation,” IEEE Trans. Pattern Anal. Machine Intell. 14, 965–980 (1992).
    [CrossRef]
  9. R. I. Hartley, “Self-calibration of stationary camera,” Int. J. Comput. Vis. 22, 5–23 (1997).
    [CrossRef]
  10. S. D. Ma, “A self-calibration technique for active vision systems,” IEEE Trans. Robot Autom. 12, 114–120 (1996).
    [CrossRef]
  11. R. I. Hartley, “Estimation of relative camera positions for uncalibrated cameras,” in Proceedings of the ECCV92 (Springer, 1992), pp. 379–387.
  12. R. I. Hartley, “In defence of the 8-point algorithm,” in Proceedings of 5th International Conference on Computer Vision (IEEE, 1995), pp. 1064–1070.
  13. S. Maybank and O. Faugeras, “A theory of self-calibration of a moving camera,” Int. J. Comput. Vis. 8, 123–151 (1992).
    [CrossRef]
  14. R. I. Hartley, “Euclidean reconstruction and invariants from multiple images,” IEEE Trans. Pattern Anal. Machine Intell. 16, 1036–1041 (1994).
    [CrossRef]
  15. B. Triggs, “Autocalibration and the absolute quadric,” in Proceedings of International Conference on Pattern Recognition (IEEE, 1997), pp. 609–614.
  16. M. Pollefeys, L. Van Gool, and M. Oosterlinck, “The modulus constraint: a new constraint for self-calibration,” in Proceedings of International Conference on Pattern Recognition (IEEE, 1996), pp. 349–353.
  17. A. Heyden and K. Astrom, “Euclidean reconstruction from image sequences with varying and unknown focal length and principal point,” in Proceedings of Conference on Computer Vision and Pattern Recognition (IEEE, 1997), pp. 438–443.
  18. M. Pollefeys, R. Koch, and G. L. Van, “Self-calibration and metric reconstruction in spite of varying and unknown internal camera parameters,” in Proceedings of the 6th International Conference on Computer Vision (IEEE, 1998), pp. 90–95.
  19. J. C. Spall, Introduction to Stochastic Search and Optimization: Estimation, Simulation, and Control (Wiley, 2003).
  20. J. C. Spall, “Multivariate stochastic approximation using a simultaneous perturbation gradient approximation,” IEEE Trans. Automat. Contr. 37, 332–341 (1992).
  21. G. A. Cauwenberghs, “Fast stochastic error-descent algorithm for supervised learning and optimization,” in Advances in Neural Information Processing Systems (Morgan Kaufman, 1993), pp. 244–251.
  22. M. A. Vorontsov, G. W. Carhart, and J. C. Ricklin, “Adaptive phase-distortion correction based on parallel gradient-descent optimization,” Opt. Lett. 22, 907–909 (1997).
    [CrossRef]
  23. G. W. Carhart, J. C. Ricklin, V. P. Sivokon, and M. A. Vorontsov, “Parallel perturbation gradient descent algorithm for adaptive wavefront correction,” Proc. SPIE 3126, 221–227 (1997).
    [CrossRef]
  24. V. I. Polejaev and M. A. Vorontsov, “Adaptive active imaging system based on radiation focusing for extended targets,” Proc. SPIE 3126, 216–220 (1997).
    [CrossRef]
  25. T. Weyrauch, M. A. Vorontsov, J. W. Gowens, and T. G. Bifano, “Fiber coupling with adaptive optics for free-space optical communication,” Proc SPIE 4489, 177–184 (2002).
    [CrossRef]
  26. M. H. Cohen, M. A. Vorontsov, G. W. Carhart, and G. Cauwenberghs, “Adaptive wavefront correction: a hybrid VLSI/optical system implementing parallel stochastic gradient descent,” Proc. SPIE 3866, 176–182 (1999).
    [CrossRef]
  27. T. Weyrauch, M. A. Vorontsov, T. G. Bifano, J. A. Hammer, M. Cohen, and G. Cauwenberghs, “Microscale adaptive optics: wave-front control with a μ-mirror array and a VLSI stochastic gradient descent controller,” Appl. Opt. 40, 4243–4253 (2001).
    [CrossRef]
  28. L. Liu and M. A. Vorontsov, “Phase-locking of tiled fiber array using spgd feedback controller,” Proc. SPIE 5895, 58950P (2005).
    [CrossRef]
  29. A. Mathieu and M. A. Vorontsov, “Imaging with an array of adaptive subapertures,” Opt. Lett. 33, 10–12 (2008).
    [CrossRef]
  30. T. Weyrauch and M. A. Vorontsov, “Atmospheric compensation with a speckle beacon in strong scintillation conditions: directed energy and laser communication applications,” Appl. Opt. 44, 6388–6400 (2005).
    [CrossRef]
  31. R. Hartley and A. Zisserman, Multiple View Geometry in Computer Vision (Cambridge University, 2000).
  32. Z. Zhang, “A flexible new technique for camera calibration,” Microsoft Corporation, Technical Report, MSR-TR-98-71 (Microsoft, 1998).
  33. D. D. Udrea, P. J. Bryanston-Cross, W. K. Lee, and M. Funes-Gallanzi, “Two sub-pixel processing algorithms for high accuracy particle centre estimate on in low seeding density particle image velocimetry,” Opt. Laser Technol. 28, 389–396 (1996).
    [CrossRef]

2008 (1)

2005 (2)

2002 (1)

T. Weyrauch, M. A. Vorontsov, J. W. Gowens, and T. G. Bifano, “Fiber coupling with adaptive optics for free-space optical communication,” Proc SPIE 4489, 177–184 (2002).
[CrossRef]

2001 (1)

1999 (1)

M. H. Cohen, M. A. Vorontsov, G. W. Carhart, and G. Cauwenberghs, “Adaptive wavefront correction: a hybrid VLSI/optical system implementing parallel stochastic gradient descent,” Proc. SPIE 3866, 176–182 (1999).
[CrossRef]

1998 (1)

G. Xu and N. Sugimoto, “A linear algorithm for motion from three weak perspective images using Euler angles,” IEEE Trans. Pattern Anal. Machine Intell. 20, 54–57 (1998).

1997 (4)

R. I. Hartley, “Self-calibration of stationary camera,” Int. J. Comput. Vis. 22, 5–23 (1997).
[CrossRef]

M. A. Vorontsov, G. W. Carhart, and J. C. Ricklin, “Adaptive phase-distortion correction based on parallel gradient-descent optimization,” Opt. Lett. 22, 907–909 (1997).
[CrossRef]

G. W. Carhart, J. C. Ricklin, V. P. Sivokon, and M. A. Vorontsov, “Parallel perturbation gradient descent algorithm for adaptive wavefront correction,” Proc. SPIE 3126, 221–227 (1997).
[CrossRef]

V. I. Polejaev and M. A. Vorontsov, “Adaptive active imaging system based on radiation focusing for extended targets,” Proc. SPIE 3126, 216–220 (1997).
[CrossRef]

1996 (2)

D. D. Udrea, P. J. Bryanston-Cross, W. K. Lee, and M. Funes-Gallanzi, “Two sub-pixel processing algorithms for high accuracy particle centre estimate on in low seeding density particle image velocimetry,” Opt. Laser Technol. 28, 389–396 (1996).
[CrossRef]

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

1994 (1)

R. I. Hartley, “Euclidean reconstruction and invariants from multiple images,” IEEE Trans. Pattern Anal. Machine Intell. 16, 1036–1041 (1994).
[CrossRef]

1992 (3)

J. C. Spall, “Multivariate stochastic approximation using a simultaneous perturbation gradient approximation,” IEEE Trans. Automat. Contr. 37, 332–341 (1992).

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

J. Weng, P. Cohen, and M. Herniou, “Camera calibration with distortion models and accuracy evaluation,” IEEE Trans. Pattern Anal. Machine Intell. 14, 965–980 (1992).
[CrossRef]

1987 (1)

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

1975 (2)

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

K. W. Wong, “Mathematical foundation and digital analysis in close-range photogrammetry,” Photogram. Eng. Remote Sens. 41, 1355–1373 (1975).

Abdel-Aziz, Y. I.

Y. I. Abdel-Aziz and H. M. Karara, “Direct linear transformation into object space coordinates in close-range photogrammetry,” in Proceedings of Symposium on Close-Range Photogrammetry (University of Illinois, 1971), pp. 1–18.

Astrom, K.

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

Bifano, T. G.

Bryanston-Cross, P. J.

D. D. Udrea, P. J. Bryanston-Cross, W. K. Lee, and M. Funes-Gallanzi, “Two sub-pixel processing algorithms for high accuracy particle centre estimate on in low seeding density particle image velocimetry,” Opt. Laser Technol. 28, 389–396 (1996).
[CrossRef]

Carhart, G. W.

M. H. Cohen, M. A. Vorontsov, G. W. Carhart, and G. Cauwenberghs, “Adaptive wavefront correction: a hybrid VLSI/optical system implementing parallel stochastic gradient descent,” Proc. SPIE 3866, 176–182 (1999).
[CrossRef]

G. W. Carhart, J. C. Ricklin, V. P. Sivokon, and M. A. Vorontsov, “Parallel perturbation gradient descent algorithm for adaptive wavefront correction,” Proc. SPIE 3126, 221–227 (1997).
[CrossRef]

M. A. Vorontsov, G. W. Carhart, and J. C. Ricklin, “Adaptive phase-distortion correction based on parallel gradient-descent optimization,” Opt. Lett. 22, 907–909 (1997).
[CrossRef]

Cauwenberghs, G.

T. Weyrauch, M. A. Vorontsov, T. G. Bifano, J. A. Hammer, M. Cohen, and G. Cauwenberghs, “Microscale adaptive optics: wave-front control with a μ-mirror array and a VLSI stochastic gradient descent controller,” Appl. Opt. 40, 4243–4253 (2001).
[CrossRef]

M. H. Cohen, M. A. Vorontsov, G. W. Carhart, and G. Cauwenberghs, “Adaptive wavefront correction: a hybrid VLSI/optical system implementing parallel stochastic gradient descent,” Proc. SPIE 3866, 176–182 (1999).
[CrossRef]

Cauwenberghs, G. A.

G. A. Cauwenberghs, “Fast stochastic error-descent algorithm for supervised learning and optimization,” in Advances in Neural Information Processing Systems (Morgan Kaufman, 1993), pp. 244–251.

Cohen, M.

Cohen, M. H.

M. H. Cohen, M. A. Vorontsov, G. W. Carhart, and G. Cauwenberghs, “Adaptive wavefront correction: a hybrid VLSI/optical system implementing parallel stochastic gradient descent,” Proc. SPIE 3866, 176–182 (1999).
[CrossRef]

Cohen, P.

J. Weng, P. Cohen, and M. Herniou, “Camera calibration with distortion models and accuracy evaluation,” IEEE Trans. Pattern Anal. Machine Intell. 14, 965–980 (1992).
[CrossRef]

Faig, W.

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

Faugeras, O.

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

Funes-Gallanzi, M.

D. D. Udrea, P. J. Bryanston-Cross, W. K. Lee, and M. Funes-Gallanzi, “Two sub-pixel processing algorithms for high accuracy particle centre estimate on in low seeding density particle image velocimetry,” Opt. Laser Technol. 28, 389–396 (1996).
[CrossRef]

Gowens, J. W.

T. Weyrauch, M. A. Vorontsov, J. W. Gowens, and T. G. Bifano, “Fiber coupling with adaptive optics for free-space optical communication,” Proc SPIE 4489, 177–184 (2002).
[CrossRef]

Hammer, J. A.

Hartley, R.

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

Hartley, R. I.

R. I. Hartley, “Self-calibration of stationary camera,” Int. J. Comput. Vis. 22, 5–23 (1997).
[CrossRef]

R. I. Hartley, “Euclidean reconstruction and invariants from multiple images,” IEEE Trans. Pattern Anal. Machine Intell. 16, 1036–1041 (1994).
[CrossRef]

R. I. Hartley, “Estimation of relative camera positions for uncalibrated cameras,” in Proceedings of the ECCV92 (Springer, 1992), pp. 379–387.

R. I. Hartley, “In defence of the 8-point algorithm,” in Proceedings of 5th International Conference on Computer Vision (IEEE, 1995), pp. 1064–1070.

Herniou, M.

J. Weng, P. Cohen, and M. Herniou, “Camera calibration with distortion models and accuracy evaluation,” IEEE Trans. Pattern Anal. Machine Intell. 14, 965–980 (1992).
[CrossRef]

Heyden, A.

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

Karara, H. M.

H. M. Karara, Handbook of Close-Range Photogrammetry (American Society of Photogrammetry, 1979).

Y. I. Abdel-Aziz and H. M. Karara, “Direct linear transformation into object space coordinates in close-range photogrammetry,” in Proceedings of Symposium on Close-Range Photogrammetry (University of Illinois, 1971), pp. 1–18.

Koch, R.

M. Pollefeys, R. Koch, and G. L. Van, “Self-calibration and metric reconstruction in spite of varying and unknown internal camera parameters,” in Proceedings of the 6th International Conference on Computer Vision (IEEE, 1998), pp. 90–95.

Lee, W. K.

D. D. Udrea, P. J. Bryanston-Cross, W. K. Lee, and M. Funes-Gallanzi, “Two sub-pixel processing algorithms for high accuracy particle centre estimate on in low seeding density particle image velocimetry,” Opt. Laser Technol. 28, 389–396 (1996).
[CrossRef]

Li, F.

W. Qi, F. Li, and L. Zhengzhong, “Review on camera calibration,” presented at the Chinese Control and Design Conference, Shengyang, China, 2010.

Liu, L.

L. Liu and M. A. Vorontsov, “Phase-locking of tiled fiber array using spgd feedback controller,” Proc. SPIE 5895, 58950P (2005).
[CrossRef]

Ma, S. D.

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

Mathieu, A.

Maybank, S.

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

Oosterlinck, M.

M. Pollefeys, L. Van Gool, and M. Oosterlinck, “The modulus constraint: a new constraint for self-calibration,” in Proceedings of International Conference on Pattern Recognition (IEEE, 1996), pp. 349–353.

Polejaev, V. I.

V. I. Polejaev and M. A. Vorontsov, “Adaptive active imaging system based on radiation focusing for extended targets,” Proc. SPIE 3126, 216–220 (1997).
[CrossRef]

Pollefeys, M.

M. Pollefeys, R. Koch, and G. L. Van, “Self-calibration and metric reconstruction in spite of varying and unknown internal camera parameters,” in Proceedings of the 6th International Conference on Computer Vision (IEEE, 1998), pp. 90–95.

M. Pollefeys, L. Van Gool, and M. Oosterlinck, “The modulus constraint: a new constraint for self-calibration,” in Proceedings of International Conference on Pattern Recognition (IEEE, 1996), pp. 349–353.

Qi, W.

W. Qi, F. Li, and L. Zhengzhong, “Review on camera calibration,” presented at the Chinese Control and Design Conference, Shengyang, China, 2010.

Ricklin, J. C.

M. A. Vorontsov, G. W. Carhart, and J. C. Ricklin, “Adaptive phase-distortion correction based on parallel gradient-descent optimization,” Opt. Lett. 22, 907–909 (1997).
[CrossRef]

G. W. Carhart, J. C. Ricklin, V. P. Sivokon, and M. A. Vorontsov, “Parallel perturbation gradient descent algorithm for adaptive wavefront correction,” Proc. SPIE 3126, 221–227 (1997).
[CrossRef]

Sivokon, V. P.

G. W. Carhart, J. C. Ricklin, V. P. Sivokon, and M. A. Vorontsov, “Parallel perturbation gradient descent algorithm for adaptive wavefront correction,” Proc. SPIE 3126, 221–227 (1997).
[CrossRef]

Spall, J. C.

J. C. Spall, “Multivariate stochastic approximation using a simultaneous perturbation gradient approximation,” IEEE Trans. Automat. Contr. 37, 332–341 (1992).

J. C. Spall, Introduction to Stochastic Search and Optimization: Estimation, Simulation, and Control (Wiley, 2003).

Sugimoto, N.

G. Xu and N. Sugimoto, “A linear algorithm for motion from three weak perspective images using Euler angles,” IEEE Trans. Pattern Anal. Machine Intell. 20, 54–57 (1998).

Triggs, B.

B. Triggs, “Autocalibration and the absolute quadric,” in Proceedings of International Conference on Pattern Recognition (IEEE, 1997), pp. 609–614.

Tsai, R. Y.

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

Udrea, D. D.

D. D. Udrea, P. J. Bryanston-Cross, W. K. Lee, and M. Funes-Gallanzi, “Two sub-pixel processing algorithms for high accuracy particle centre estimate on in low seeding density particle image velocimetry,” Opt. Laser Technol. 28, 389–396 (1996).
[CrossRef]

Van, G. L.

M. Pollefeys, R. Koch, and G. L. Van, “Self-calibration and metric reconstruction in spite of varying and unknown internal camera parameters,” in Proceedings of the 6th International Conference on Computer Vision (IEEE, 1998), pp. 90–95.

Van Gool, L.

M. Pollefeys, L. Van Gool, and M. Oosterlinck, “The modulus constraint: a new constraint for self-calibration,” in Proceedings of International Conference on Pattern Recognition (IEEE, 1996), pp. 349–353.

Vorontsov, M. A.

A. Mathieu and M. A. Vorontsov, “Imaging with an array of adaptive subapertures,” Opt. Lett. 33, 10–12 (2008).
[CrossRef]

L. Liu and M. A. Vorontsov, “Phase-locking of tiled fiber array using spgd feedback controller,” Proc. SPIE 5895, 58950P (2005).
[CrossRef]

T. Weyrauch and M. A. Vorontsov, “Atmospheric compensation with a speckle beacon in strong scintillation conditions: directed energy and laser communication applications,” Appl. Opt. 44, 6388–6400 (2005).
[CrossRef]

T. Weyrauch, M. A. Vorontsov, J. W. Gowens, and T. G. Bifano, “Fiber coupling with adaptive optics for free-space optical communication,” Proc SPIE 4489, 177–184 (2002).
[CrossRef]

T. Weyrauch, M. A. Vorontsov, T. G. Bifano, J. A. Hammer, M. Cohen, and G. Cauwenberghs, “Microscale adaptive optics: wave-front control with a μ-mirror array and a VLSI stochastic gradient descent controller,” Appl. Opt. 40, 4243–4253 (2001).
[CrossRef]

M. H. Cohen, M. A. Vorontsov, G. W. Carhart, and G. Cauwenberghs, “Adaptive wavefront correction: a hybrid VLSI/optical system implementing parallel stochastic gradient descent,” Proc. SPIE 3866, 176–182 (1999).
[CrossRef]

V. I. Polejaev and M. A. Vorontsov, “Adaptive active imaging system based on radiation focusing for extended targets,” Proc. SPIE 3126, 216–220 (1997).
[CrossRef]

G. W. Carhart, J. C. Ricklin, V. P. Sivokon, and M. A. Vorontsov, “Parallel perturbation gradient descent algorithm for adaptive wavefront correction,” Proc. SPIE 3126, 221–227 (1997).
[CrossRef]

M. A. Vorontsov, G. W. Carhart, and J. C. Ricklin, “Adaptive phase-distortion correction based on parallel gradient-descent optimization,” Opt. Lett. 22, 907–909 (1997).
[CrossRef]

Weng, J.

J. Weng, P. Cohen, and M. Herniou, “Camera calibration with distortion models and accuracy evaluation,” IEEE Trans. Pattern Anal. Machine Intell. 14, 965–980 (1992).
[CrossRef]

Weyrauch, T.

Wong, K. W.

K. W. Wong, “Mathematical foundation and digital analysis in close-range photogrammetry,” Photogram. Eng. Remote Sens. 41, 1355–1373 (1975).

Xu, G.

G. Xu and N. Sugimoto, “A linear algorithm for motion from three weak perspective images using Euler angles,” IEEE Trans. Pattern Anal. Machine Intell. 20, 54–57 (1998).

Zhang, Z.

Z. Zhang, “A flexible new technique for camera calibration,” Microsoft Corporation, Technical Report, MSR-TR-98-71 (Microsoft, 1998).

Zhengzhong, L.

W. Qi, F. Li, and L. Zhengzhong, “Review on camera calibration,” presented at the Chinese Control and Design Conference, Shengyang, China, 2010.

Zisserman, A.

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

Appl. Opt. (2)

IEEE J. Robot. Automon. (1)

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

IEEE Trans. Automat. Contr. (1)

J. C. Spall, “Multivariate stochastic approximation using a simultaneous perturbation gradient approximation,” IEEE Trans. Automat. Contr. 37, 332–341 (1992).

IEEE Trans. Pattern Anal. Machine Intell. (3)

R. I. Hartley, “Euclidean reconstruction and invariants from multiple images,” IEEE Trans. Pattern Anal. Machine Intell. 16, 1036–1041 (1994).
[CrossRef]

J. Weng, P. Cohen, and M. Herniou, “Camera calibration with distortion models and accuracy evaluation,” IEEE Trans. Pattern Anal. Machine Intell. 14, 965–980 (1992).
[CrossRef]

G. Xu and N. Sugimoto, “A linear algorithm for motion from three weak perspective images using Euler angles,” IEEE Trans. Pattern Anal. Machine Intell. 20, 54–57 (1998).

IEEE Trans. Robot Autom. (1)

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

Int. J. Comput. Vis. (2)

R. I. Hartley, “Self-calibration of stationary camera,” Int. J. Comput. Vis. 22, 5–23 (1997).
[CrossRef]

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

Opt. Laser Technol. (1)

D. D. Udrea, P. J. Bryanston-Cross, W. K. Lee, and M. Funes-Gallanzi, “Two sub-pixel processing algorithms for high accuracy particle centre estimate on in low seeding density particle image velocimetry,” Opt. Laser Technol. 28, 389–396 (1996).
[CrossRef]

Opt. Lett. (2)

Photogram. Eng. Remote Sens. (1)

K. W. Wong, “Mathematical foundation and digital analysis in close-range photogrammetry,” Photogram. Eng. Remote Sens. 41, 1355–1373 (1975).

Photogramm. Eng. Remote Sens. (1)

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

Proc SPIE (1)

T. Weyrauch, M. A. Vorontsov, J. W. Gowens, and T. G. Bifano, “Fiber coupling with adaptive optics for free-space optical communication,” Proc SPIE 4489, 177–184 (2002).
[CrossRef]

Proc. SPIE (4)

M. H. Cohen, M. A. Vorontsov, G. W. Carhart, and G. Cauwenberghs, “Adaptive wavefront correction: a hybrid VLSI/optical system implementing parallel stochastic gradient descent,” Proc. SPIE 3866, 176–182 (1999).
[CrossRef]

G. W. Carhart, J. C. Ricklin, V. P. Sivokon, and M. A. Vorontsov, “Parallel perturbation gradient descent algorithm for adaptive wavefront correction,” Proc. SPIE 3126, 221–227 (1997).
[CrossRef]

V. I. Polejaev and M. A. Vorontsov, “Adaptive active imaging system based on radiation focusing for extended targets,” Proc. SPIE 3126, 216–220 (1997).
[CrossRef]

L. Liu and M. A. Vorontsov, “Phase-locking of tiled fiber array using spgd feedback controller,” Proc. SPIE 5895, 58950P (2005).
[CrossRef]

Other (13)

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

Z. Zhang, “A flexible new technique for camera calibration,” Microsoft Corporation, Technical Report, MSR-TR-98-71 (Microsoft, 1998).

Y. I. Abdel-Aziz and H. M. Karara, “Direct linear transformation into object space coordinates in close-range photogrammetry,” in Proceedings of Symposium on Close-Range Photogrammetry (University of Illinois, 1971), pp. 1–18.

H. M. Karara, Handbook of Close-Range Photogrammetry (American Society of Photogrammetry, 1979).

R. I. Hartley, “Estimation of relative camera positions for uncalibrated cameras,” in Proceedings of the ECCV92 (Springer, 1992), pp. 379–387.

R. I. Hartley, “In defence of the 8-point algorithm,” in Proceedings of 5th International Conference on Computer Vision (IEEE, 1995), pp. 1064–1070.

W. Qi, F. Li, and L. Zhengzhong, “Review on camera calibration,” presented at the Chinese Control and Design Conference, Shengyang, China, 2010.

G. A. Cauwenberghs, “Fast stochastic error-descent algorithm for supervised learning and optimization,” in Advances in Neural Information Processing Systems (Morgan Kaufman, 1993), pp. 244–251.

B. Triggs, “Autocalibration and the absolute quadric,” in Proceedings of International Conference on Pattern Recognition (IEEE, 1997), pp. 609–614.

M. Pollefeys, L. Van Gool, and M. Oosterlinck, “The modulus constraint: a new constraint for self-calibration,” in Proceedings of International Conference on Pattern Recognition (IEEE, 1996), pp. 349–353.

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

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Figures (5)

Fig. 1.
Fig. 1.

Image of the full coordinate system relationship in camera imaging.

Fig. 2.
Fig. 2.

Solution procedure based on rear surface control points under the two-step camera calibration method.

Fig. 3.
Fig. 3.

SPGD algorithm procedure.

Fig. 4.
Fig. 4.

Calibration procedure for the two-step camera calibration method based on the SPGD algorithm.

Fig. 5.
Fig. 5.

Schematic diagram of camera calibration positioning arrangement.

Tables (8)

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Table 1. Internal and External Parameters and Aberration Coefficients Between Camera and Marked Object

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Table 2. Global Coordinates of Marked Points and Image Point Coefficients

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Table 3. Calibration Parameters (Simulated) Obtained When Using the SPGD Algorithm Two-Step Method

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Table 4. Calibration Parameters Under the Traditional Two-Step Method (Simulated)

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Table 5. Global Coordinates of Marked Points

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Table 6. Image Point Image Coordinates

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Table 7. Calibration Parameters Obtained Using the SPGD Algorithm Two-Step Method (Experiment)

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Table 8. Calibration Parameters Obtained Using the Traditional Two-Step Method (Experiment)

Equations (17)

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{Xm0+Ym1+Zm2+m3xXm8xYm9xZm10xm11=0Xm4+Ym5+Zm6+m7yXm8yYm9yZm10ym11=0,
{x˜=m0X+m1Y+m2Z+m3m8X+m9Y+m10Z+m11y˜=m4X+m5Y+m6Z+m7m8X+m9Y+m10Z+m11.
{xd(xd2+yd2)k0+(xd2+yd2)k1+xd2k3+xdydk4=δxyd(xd2+yd2)k0+(xd2+yd2)k2+xdydk3+yd2k4=δy.
{xd=(x˜Cx)/Fxyd=(y˜Cy)/Fy,
δJ=J(u1+δu1,,uj+δuj,,uN+δuN)J(u1,,uj,,uN).
δJ=j=1NJujδuj+12j,iN2Jujuiδujδui+
δJδul=Jul(δul)2+ψl,
ψl=jlNJujδujδul+12j,iN2Jujuiδujδuiδul+.
δui=0,δuiδuj=σ2δij,
δJδul=Jul(δul)2+ψl,
ψl=jlNJujδujδul+12j,iN2Jujuiδujδuiδul+=O(σ4).
uj(n+1)=uj(n)γnJuj|uj=uj(n),
uj(n+1)=uj(n)μδJ(n)δuj(n).
ΔJ=J(u(n+1))J(u(n))i=1NJui(μδJδui)=μi=1NJui[Jui(δui)2+ψi]μi=1N(Juiδui)2μi,jiNJuiJujδuiδuj+,
ΔJ=μσ2i=1N(Jui)2+O(μσ4).
{m0·m2=Fxro·r2+Cxr2·r2=Cxm1·m2=Fyr1·r2+Cyr2·r2=Cy|m0×m2|=Fx|r0×r2|+Cx|r2×r2|=Fx|m1×m2|=Fy|r1×r2|+Cy|r2×r2|=Fy.
{x=xγ(S¯+S¯)δxy=yγ(S¯+S¯)δy.

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