Abstract

Camera calibration is a two-step process where first a linear algebraic approximation is followed by a nonlinear minimization. The nonlinear minimization adjusts the pin-hole and lens distortion models to the calibrating data. Since both models are coupled, nonlinear minimization can converge to a local solution easily. Moreover, nonlinear minimization is poorly conditioned since parameters with different effects in the minimization function are calculated simultaneously (some are in pixels, some in world coordinates, and some are lens distortion parameters). A local solution is adapted to parameters, which minimize the function easily, and the remaining parameters are just adapted to this solution. We propose a calibration method where traditional calibration steps are inverted. First, a nonlinear minimization is done, and after, camera parameters are computed in a linear step. Using projective geometry constraints in a nonlinear minimization process, detected point locations in the images are corrected. The pin-hole and lens distortion models are computed separately with corrected point locations. The proposed method avoids the coupling between both models. Also, the condition of nonlinear minimization increases since points coordinates are computed alone.

© 2012 Optical Society of America

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References

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  1. S. Shih, Y. Hung, and W. Lin, “When should we consider lens distortion in camera calibration,” Pattern Recogn. 28, 447–461(1995).
    [CrossRef]
  2. D. C. Brown, “Close-range camera calibration,” Photogramm. Eng. Remote Sensing 37, 855–866 (1971).
  3. A. Basu and S. Licardie, “Alternative models for fish-eye lenses,” Pattern Recogn. Lett. 16, 433–441 (1995).
    [CrossRef]
  4. S. S. Beauchemin and R. Bajcsy, “Modelling and removing radial and tangential distortions in spherical lenses,” in Multi-Image Analysis, Vol.  2032 of Lecture Notes in Computer Science (Springer, 2001), pp. 1–21.
  5. F. Devernay and O. Faugeras, “Straight lines have to be straight,” Mach. Vis. Appl. 13, 14–24 (2001).
    [CrossRef]
  6. C. McGlone, E. Mikhail, and J. Bethel, Manual of Photogrammetry, 5th ed. (American Society of Photogrammetry and Remote Sensing, 2004).
  7. L. Ma, Y. Q. Chen, and K. L. Moore, “Flexible camera calibration using a new analytical radial undistortion formula with application to mobile robot localization,” in IEEE International Symposium on Intelligent Control (IEEE, 2003), pp. 799–804.
  8. J. Mallon and P. F. Whelan, “Precise radial un-distortion of images,” in Proceedings of the 17th International Conference on Pattern Recognition (IEEE, 2004), pp. 18–21.
  9. P. Sturm and S. Ramalingam, “A generic concept for camera calibration,” in Proceedings of the 5th European Conference on Computer Vision (Springer, 2004), pp. 1–13.
  10. J. Wang, F. Shi, J. Zhang, and Y. Liu, “A new calibration model of camera lens distortion,” Pattern Recogn. 41, 607–615 (2008).
    [CrossRef]
  11. 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]
  12. Z. Zhang, “A flexible new technique for camera calibration,” IEEE Trans. Pattern Anal. Machine Intell. 22, 1330–1334(2000).
    [CrossRef]
  13. A. Wang, T. Qiu, and L. Shao, “A simple method of radial distortion correction with centre of distortion estimation,” J. Math. Imaging Vis. 35, 165–172 (2009).
    [CrossRef]
  14. R. Hartley and S. Kang, “Paremeter-free radial distortion correction with centre of distortion estimation,” in Proceedings of the Tenth IEEE International Conference on Computer Vision (IEEE, 2005), pp. 1834–1841.
  15. A. Fitzgibbon, “Simultaneous linear estimation of multiple view geometry and lens distortion,” in IEEE International Conference on Computer Vision and Pattern Recognition (IEEE, 2001), pp. I-125–I-132.
  16. D. Claus and A. Fitzgibbon, “A rational function lens distortion model for general cameras,” in Proceedings of the IEEE International Conference on Computer Vision and Pattern Recognition (IEEE, 2005), pp. 213–219.
  17. Z. Zhang, “Camera calibration with one-dimensional objects,” IEEE Trans. Pattern Anal. Machine Intell. 26, 892–899 (2004).
    [CrossRef]
  18. B. Caprile and V. Torre, “Using vanishing points for camera calibration,” Int. J. Comput. Vis. 4, 127–140 (1990).
    [CrossRef]
  19. R. Hartley, “An algorithm for self calibration from several views,” in Proceedings of the Conference on Computer Vision and Pattern Recognition (IEEE, 1994), pp. 908–912.
  20. B. Triggs, “Autocalibration and the absolute quadric,” in Proceedings of the Conference on Computer Vision and Pattern Recognition (IEEE, 1997), pp. 609–614.
  21. J. Guillemaunt, A. Aguado, and J. Illingworth, “Using points at infinity for decoupling in camera calibration, ” IEEE Trans. Pattern Anal. Machine Intell. 27, 265–270 (2005).
    [CrossRef]
  22. W. Chojnacki, M. Brooks, A. Hengel, and D. Gawley, “Revisiting Hartley’s normalized eight-point algorithm,” IEEE Trans. Pattern Anal. Machine Intell. 25, 1172–1177(2003).
    [CrossRef]
  23. R. Swaminathan and S. Nayar, “Non metric calibration of wide-angle lenses and polycameras, ” IEEE Trans. Pattern Anal. Machine Intell. 22, 1172–178 (2000).
    [CrossRef]
  24. M. Ahmed and A. Farag, “Non-metric calibration of camera lens distortion: differential methods and robust estimation,” IEEE Trans. Image Process. 14, 1215–1230(2005).
    [CrossRef]
  25. S. Becker and V. Bove, “Semiautomatic 3D-model extraction from uncalibrated 2D-camera views, ” Proc. SPIE 2410, 447–461 (1881).
    [CrossRef]
  26. M. Penna, “Camera calibration: a quick and easy way to detection of scale factor,” IEEE Trans. Pattern Anal. Machine Intell. 13, 1240–1245 (1991).
    [CrossRef]
  27. Z. Zhang, “On the epipolar geometry between two images with lens distortion,” in Proceedings of International Conference on Computer Vision and Pattern Recognition (IEEE, 1996), Vol.  1, pp. 407–411.
  28. J. Bouguet and P. Perona, “Closed-form camera calibration in dual-space geometry,” in Proceedings of the European Conference on Computer Vision (ECCV98) (Springer, 1998).
  29. C. Ricolfe-Viala and A. Sanchez-Salmeron, “Lens distortion models evaluation,” Appl. Opt. 49, 5914–5928 (2010).
    [CrossRef]

2010 (1)

2009 (1)

A. Wang, T. Qiu, and L. Shao, “A simple method of radial distortion correction with centre of distortion estimation,” J. Math. Imaging Vis. 35, 165–172 (2009).
[CrossRef]

2008 (1)

J. Wang, F. Shi, J. Zhang, and Y. Liu, “A new calibration model of camera lens distortion,” Pattern Recogn. 41, 607–615 (2008).
[CrossRef]

2005 (2)

J. Guillemaunt, A. Aguado, and J. Illingworth, “Using points at infinity for decoupling in camera calibration, ” IEEE Trans. Pattern Anal. Machine Intell. 27, 265–270 (2005).
[CrossRef]

M. Ahmed and A. Farag, “Non-metric calibration of camera lens distortion: differential methods and robust estimation,” IEEE Trans. Image Process. 14, 1215–1230(2005).
[CrossRef]

2004 (1)

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

2003 (1)

W. Chojnacki, M. Brooks, A. Hengel, and D. Gawley, “Revisiting Hartley’s normalized eight-point algorithm,” IEEE Trans. Pattern Anal. Machine Intell. 25, 1172–1177(2003).
[CrossRef]

2001 (1)

F. Devernay and O. Faugeras, “Straight lines have to be straight,” Mach. Vis. Appl. 13, 14–24 (2001).
[CrossRef]

2000 (2)

R. Swaminathan and S. Nayar, “Non metric calibration of wide-angle lenses and polycameras, ” IEEE Trans. Pattern Anal. Machine Intell. 22, 1172–178 (2000).
[CrossRef]

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

1995 (2)

A. Basu and S. Licardie, “Alternative models for fish-eye lenses,” Pattern Recogn. Lett. 16, 433–441 (1995).
[CrossRef]

S. Shih, Y. Hung, and W. Lin, “When should we consider lens distortion in camera calibration,” Pattern Recogn. 28, 447–461(1995).
[CrossRef]

1992 (1)

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]

1991 (1)

M. Penna, “Camera calibration: a quick and easy way to detection of scale factor,” IEEE Trans. Pattern Anal. Machine Intell. 13, 1240–1245 (1991).
[CrossRef]

1990 (1)

B. Caprile and V. Torre, “Using vanishing points for camera calibration,” Int. J. Comput. Vis. 4, 127–140 (1990).
[CrossRef]

1971 (1)

D. C. Brown, “Close-range camera calibration,” Photogramm. Eng. Remote Sensing 37, 855–866 (1971).

1881 (1)

S. Becker and V. Bove, “Semiautomatic 3D-model extraction from uncalibrated 2D-camera views, ” Proc. SPIE 2410, 447–461 (1881).
[CrossRef]

Aguado, A.

J. Guillemaunt, A. Aguado, and J. Illingworth, “Using points at infinity for decoupling in camera calibration, ” IEEE Trans. Pattern Anal. Machine Intell. 27, 265–270 (2005).
[CrossRef]

Ahmed, M.

M. Ahmed and A. Farag, “Non-metric calibration of camera lens distortion: differential methods and robust estimation,” IEEE Trans. Image Process. 14, 1215–1230(2005).
[CrossRef]

Bajcsy, R.

S. S. Beauchemin and R. Bajcsy, “Modelling and removing radial and tangential distortions in spherical lenses,” in Multi-Image Analysis, Vol.  2032 of Lecture Notes in Computer Science (Springer, 2001), pp. 1–21.

Basu, A.

A. Basu and S. Licardie, “Alternative models for fish-eye lenses,” Pattern Recogn. Lett. 16, 433–441 (1995).
[CrossRef]

Beauchemin, S. S.

S. S. Beauchemin and R. Bajcsy, “Modelling and removing radial and tangential distortions in spherical lenses,” in Multi-Image Analysis, Vol.  2032 of Lecture Notes in Computer Science (Springer, 2001), pp. 1–21.

Becker, S.

S. Becker and V. Bove, “Semiautomatic 3D-model extraction from uncalibrated 2D-camera views, ” Proc. SPIE 2410, 447–461 (1881).
[CrossRef]

Bethel, J.

C. McGlone, E. Mikhail, and J. Bethel, Manual of Photogrammetry, 5th ed. (American Society of Photogrammetry and Remote Sensing, 2004).

Bouguet, J.

J. Bouguet and P. Perona, “Closed-form camera calibration in dual-space geometry,” in Proceedings of the European Conference on Computer Vision (ECCV98) (Springer, 1998).

Bove, V.

S. Becker and V. Bove, “Semiautomatic 3D-model extraction from uncalibrated 2D-camera views, ” Proc. SPIE 2410, 447–461 (1881).
[CrossRef]

Brooks, M.

W. Chojnacki, M. Brooks, A. Hengel, and D. Gawley, “Revisiting Hartley’s normalized eight-point algorithm,” IEEE Trans. Pattern Anal. Machine Intell. 25, 1172–1177(2003).
[CrossRef]

Brown, D. C.

D. C. Brown, “Close-range camera calibration,” Photogramm. Eng. Remote Sensing 37, 855–866 (1971).

Caprile, B.

B. Caprile and V. Torre, “Using vanishing points for camera calibration,” Int. J. Comput. Vis. 4, 127–140 (1990).
[CrossRef]

Chen, Y. Q.

L. Ma, Y. Q. Chen, and K. L. Moore, “Flexible camera calibration using a new analytical radial undistortion formula with application to mobile robot localization,” in IEEE International Symposium on Intelligent Control (IEEE, 2003), pp. 799–804.

Chojnacki, W.

W. Chojnacki, M. Brooks, A. Hengel, and D. Gawley, “Revisiting Hartley’s normalized eight-point algorithm,” IEEE Trans. Pattern Anal. Machine Intell. 25, 1172–1177(2003).
[CrossRef]

Claus, D.

D. Claus and A. Fitzgibbon, “A rational function lens distortion model for general cameras,” in Proceedings of the IEEE International Conference on Computer Vision and Pattern Recognition (IEEE, 2005), pp. 213–219.

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]

Devernay, F.

F. Devernay and O. Faugeras, “Straight lines have to be straight,” Mach. Vis. Appl. 13, 14–24 (2001).
[CrossRef]

Farag, A.

M. Ahmed and A. Farag, “Non-metric calibration of camera lens distortion: differential methods and robust estimation,” IEEE Trans. Image Process. 14, 1215–1230(2005).
[CrossRef]

Faugeras, O.

F. Devernay and O. Faugeras, “Straight lines have to be straight,” Mach. Vis. Appl. 13, 14–24 (2001).
[CrossRef]

Fitzgibbon, A.

D. Claus and A. Fitzgibbon, “A rational function lens distortion model for general cameras,” in Proceedings of the IEEE International Conference on Computer Vision and Pattern Recognition (IEEE, 2005), pp. 213–219.

A. Fitzgibbon, “Simultaneous linear estimation of multiple view geometry and lens distortion,” in IEEE International Conference on Computer Vision and Pattern Recognition (IEEE, 2001), pp. I-125–I-132.

Gawley, D.

W. Chojnacki, M. Brooks, A. Hengel, and D. Gawley, “Revisiting Hartley’s normalized eight-point algorithm,” IEEE Trans. Pattern Anal. Machine Intell. 25, 1172–1177(2003).
[CrossRef]

Guillemaunt, J.

J. Guillemaunt, A. Aguado, and J. Illingworth, “Using points at infinity for decoupling in camera calibration, ” IEEE Trans. Pattern Anal. Machine Intell. 27, 265–270 (2005).
[CrossRef]

Hartley, R.

R. Hartley and S. Kang, “Paremeter-free radial distortion correction with centre of distortion estimation,” in Proceedings of the Tenth IEEE International Conference on Computer Vision (IEEE, 2005), pp. 1834–1841.

R. Hartley, “An algorithm for self calibration from several views,” in Proceedings of the Conference on Computer Vision and Pattern Recognition (IEEE, 1994), pp. 908–912.

Hengel, A.

W. Chojnacki, M. Brooks, A. Hengel, and D. Gawley, “Revisiting Hartley’s normalized eight-point algorithm,” IEEE Trans. Pattern Anal. Machine Intell. 25, 1172–1177(2003).
[CrossRef]

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]

Hung, Y.

S. Shih, Y. Hung, and W. Lin, “When should we consider lens distortion in camera calibration,” Pattern Recogn. 28, 447–461(1995).
[CrossRef]

Illingworth, J.

J. Guillemaunt, A. Aguado, and J. Illingworth, “Using points at infinity for decoupling in camera calibration, ” IEEE Trans. Pattern Anal. Machine Intell. 27, 265–270 (2005).
[CrossRef]

Kang, S.

R. Hartley and S. Kang, “Paremeter-free radial distortion correction with centre of distortion estimation,” in Proceedings of the Tenth IEEE International Conference on Computer Vision (IEEE, 2005), pp. 1834–1841.

Licardie, S.

A. Basu and S. Licardie, “Alternative models for fish-eye lenses,” Pattern Recogn. Lett. 16, 433–441 (1995).
[CrossRef]

Lin, W.

S. Shih, Y. Hung, and W. Lin, “When should we consider lens distortion in camera calibration,” Pattern Recogn. 28, 447–461(1995).
[CrossRef]

Liu, Y.

J. Wang, F. Shi, J. Zhang, and Y. Liu, “A new calibration model of camera lens distortion,” Pattern Recogn. 41, 607–615 (2008).
[CrossRef]

Ma, L.

L. Ma, Y. Q. Chen, and K. L. Moore, “Flexible camera calibration using a new analytical radial undistortion formula with application to mobile robot localization,” in IEEE International Symposium on Intelligent Control (IEEE, 2003), pp. 799–804.

Mallon, J.

J. Mallon and P. F. Whelan, “Precise radial un-distortion of images,” in Proceedings of the 17th International Conference on Pattern Recognition (IEEE, 2004), pp. 18–21.

McGlone, C.

C. McGlone, E. Mikhail, and J. Bethel, Manual of Photogrammetry, 5th ed. (American Society of Photogrammetry and Remote Sensing, 2004).

Mikhail, E.

C. McGlone, E. Mikhail, and J. Bethel, Manual of Photogrammetry, 5th ed. (American Society of Photogrammetry and Remote Sensing, 2004).

Moore, K. L.

L. Ma, Y. Q. Chen, and K. L. Moore, “Flexible camera calibration using a new analytical radial undistortion formula with application to mobile robot localization,” in IEEE International Symposium on Intelligent Control (IEEE, 2003), pp. 799–804.

Nayar, S.

R. Swaminathan and S. Nayar, “Non metric calibration of wide-angle lenses and polycameras, ” IEEE Trans. Pattern Anal. Machine Intell. 22, 1172–178 (2000).
[CrossRef]

Penna, M.

M. Penna, “Camera calibration: a quick and easy way to detection of scale factor,” IEEE Trans. Pattern Anal. Machine Intell. 13, 1240–1245 (1991).
[CrossRef]

Perona, P.

J. Bouguet and P. Perona, “Closed-form camera calibration in dual-space geometry,” in Proceedings of the European Conference on Computer Vision (ECCV98) (Springer, 1998).

Qiu, T.

A. Wang, T. Qiu, and L. Shao, “A simple method of radial distortion correction with centre of distortion estimation,” J. Math. Imaging Vis. 35, 165–172 (2009).
[CrossRef]

Ramalingam, S.

P. Sturm and S. Ramalingam, “A generic concept for camera calibration,” in Proceedings of the 5th European Conference on Computer Vision (Springer, 2004), pp. 1–13.

Ricolfe-Viala, C.

Sanchez-Salmeron, A.

Shao, L.

A. Wang, T. Qiu, and L. Shao, “A simple method of radial distortion correction with centre of distortion estimation,” J. Math. Imaging Vis. 35, 165–172 (2009).
[CrossRef]

Shi, F.

J. Wang, F. Shi, J. Zhang, and Y. Liu, “A new calibration model of camera lens distortion,” Pattern Recogn. 41, 607–615 (2008).
[CrossRef]

Shih, S.

S. Shih, Y. Hung, and W. Lin, “When should we consider lens distortion in camera calibration,” Pattern Recogn. 28, 447–461(1995).
[CrossRef]

Sturm, P.

P. Sturm and S. Ramalingam, “A generic concept for camera calibration,” in Proceedings of the 5th European Conference on Computer Vision (Springer, 2004), pp. 1–13.

Swaminathan, R.

R. Swaminathan and S. Nayar, “Non metric calibration of wide-angle lenses and polycameras, ” IEEE Trans. Pattern Anal. Machine Intell. 22, 1172–178 (2000).
[CrossRef]

Torre, V.

B. Caprile and V. Torre, “Using vanishing points for camera calibration,” Int. J. Comput. Vis. 4, 127–140 (1990).
[CrossRef]

Triggs, B.

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

Wang, A.

A. Wang, T. Qiu, and L. Shao, “A simple method of radial distortion correction with centre of distortion estimation,” J. Math. Imaging Vis. 35, 165–172 (2009).
[CrossRef]

Wang, J.

J. Wang, F. Shi, J. Zhang, and Y. Liu, “A new calibration model of camera lens distortion,” Pattern Recogn. 41, 607–615 (2008).
[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]

Whelan, P. F.

J. Mallon and P. F. Whelan, “Precise radial un-distortion of images,” in Proceedings of the 17th International Conference on Pattern Recognition (IEEE, 2004), pp. 18–21.

Zhang, J.

J. Wang, F. Shi, J. Zhang, and Y. Liu, “A new calibration model of camera lens distortion,” Pattern Recogn. 41, 607–615 (2008).
[CrossRef]

Zhang, Z.

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

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

Z. Zhang, “On the epipolar geometry between two images with lens distortion,” in Proceedings of International Conference on Computer Vision and Pattern Recognition (IEEE, 1996), Vol.  1, pp. 407–411.

Appl. Opt. (1)

IEEE Trans. Image Process. (1)

M. Ahmed and A. Farag, “Non-metric calibration of camera lens distortion: differential methods and robust estimation,” IEEE Trans. Image Process. 14, 1215–1230(2005).
[CrossRef]

IEEE Trans. Pattern Anal. Machine Intell. (7)

J. Guillemaunt, A. Aguado, and J. Illingworth, “Using points at infinity for decoupling in camera calibration, ” IEEE Trans. Pattern Anal. Machine Intell. 27, 265–270 (2005).
[CrossRef]

W. Chojnacki, M. Brooks, A. Hengel, and D. Gawley, “Revisiting Hartley’s normalized eight-point algorithm,” IEEE Trans. Pattern Anal. Machine Intell. 25, 1172–1177(2003).
[CrossRef]

R. Swaminathan and S. Nayar, “Non metric calibration of wide-angle lenses and polycameras, ” IEEE Trans. Pattern Anal. Machine Intell. 22, 1172–178 (2000).
[CrossRef]

M. Penna, “Camera calibration: a quick and easy way to detection of scale factor,” IEEE Trans. Pattern Anal. Machine Intell. 13, 1240–1245 (1991).
[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]

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

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

Int. J. Comput. Vis. (1)

B. Caprile and V. Torre, “Using vanishing points for camera calibration,” Int. J. Comput. Vis. 4, 127–140 (1990).
[CrossRef]

J. Math. Imaging Vis. (1)

A. Wang, T. Qiu, and L. Shao, “A simple method of radial distortion correction with centre of distortion estimation,” J. Math. Imaging Vis. 35, 165–172 (2009).
[CrossRef]

Mach. Vis. Appl. (1)

F. Devernay and O. Faugeras, “Straight lines have to be straight,” Mach. Vis. Appl. 13, 14–24 (2001).
[CrossRef]

Pattern Recogn. (2)

S. Shih, Y. Hung, and W. Lin, “When should we consider lens distortion in camera calibration,” Pattern Recogn. 28, 447–461(1995).
[CrossRef]

J. Wang, F. Shi, J. Zhang, and Y. Liu, “A new calibration model of camera lens distortion,” Pattern Recogn. 41, 607–615 (2008).
[CrossRef]

Pattern Recogn. Lett. (1)

A. Basu and S. Licardie, “Alternative models for fish-eye lenses,” Pattern Recogn. Lett. 16, 433–441 (1995).
[CrossRef]

Photogramm. Eng. Remote Sensing (1)

D. C. Brown, “Close-range camera calibration,” Photogramm. Eng. Remote Sensing 37, 855–866 (1971).

Proc. SPIE (1)

S. Becker and V. Bove, “Semiautomatic 3D-model extraction from uncalibrated 2D-camera views, ” Proc. SPIE 2410, 447–461 (1881).
[CrossRef]

Other (12)

Z. Zhang, “On the epipolar geometry between two images with lens distortion,” in Proceedings of International Conference on Computer Vision and Pattern Recognition (IEEE, 1996), Vol.  1, pp. 407–411.

J. Bouguet and P. Perona, “Closed-form camera calibration in dual-space geometry,” in Proceedings of the European Conference on Computer Vision (ECCV98) (Springer, 1998).

S. S. Beauchemin and R. Bajcsy, “Modelling and removing radial and tangential distortions in spherical lenses,” in Multi-Image Analysis, Vol.  2032 of Lecture Notes in Computer Science (Springer, 2001), pp. 1–21.

C. McGlone, E. Mikhail, and J. Bethel, Manual of Photogrammetry, 5th ed. (American Society of Photogrammetry and Remote Sensing, 2004).

L. Ma, Y. Q. Chen, and K. L. Moore, “Flexible camera calibration using a new analytical radial undistortion formula with application to mobile robot localization,” in IEEE International Symposium on Intelligent Control (IEEE, 2003), pp. 799–804.

J. Mallon and P. F. Whelan, “Precise radial un-distortion of images,” in Proceedings of the 17th International Conference on Pattern Recognition (IEEE, 2004), pp. 18–21.

P. Sturm and S. Ramalingam, “A generic concept for camera calibration,” in Proceedings of the 5th European Conference on Computer Vision (Springer, 2004), pp. 1–13.

R. Hartley and S. Kang, “Paremeter-free radial distortion correction with centre of distortion estimation,” in Proceedings of the Tenth IEEE International Conference on Computer Vision (IEEE, 2005), pp. 1834–1841.

A. Fitzgibbon, “Simultaneous linear estimation of multiple view geometry and lens distortion,” in IEEE International Conference on Computer Vision and Pattern Recognition (IEEE, 2001), pp. I-125–I-132.

D. Claus and A. Fitzgibbon, “A rational function lens distortion model for general cameras,” in Proceedings of the IEEE International Conference on Computer Vision and Pattern Recognition (IEEE, 2005), pp. 213–219.

R. Hartley, “An algorithm for self calibration from several views,” in Proceedings of the Conference on Computer Vision and Pattern Recognition (IEEE, 1994), pp. 908–912.

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

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Figures (5)

Fig. 1
Fig. 1

Diagram of the proposed camera calibration process. Points detected in the captured images q d i , j are corrected to obtain the precise position of points in the image q o i , j . Precise position is computed taking into account perspective projective constraints. With detected points q d i , j and corrected points q o i , j , lens distortion model is calibrated. With corrected points q o i , j and a point from the calibration template p i , the pin-hole model is calibrated.

Fig. 2
Fig. 2

Perspective projective constraints used for correcting points detected in the image. Cross ratio guarantees that parallel lines remain parallel under perspective projection. Points are corrected to belong to straight lines. All parallel lines meet in a vanishing point in an image. All vanishing points form the horizon line.

Fig. 3
Fig. 3

Results with real data. (a)–(c)  640 × 480 captured images with an Axis 212 PTZ with 2.7 mm lens mounted, which gives 85 ° field of view. (d)–(f) Images corrected with the “easyCalib” method proposed by Zhang [12]. (g)–(i) Images corrected with the proposed method in this paper. (j)–(l) Differences between point locations in images if several methods are used to calibrate the camera. The proposed method obtains accurate results since perspective projection rules are taken into account in the calibration process.

Fig. 4
Fig. 4

Result with simulated data varying the noise level. Zh, Zhang method; Pr, proposed method. The reprojection error is smaller with the proposed method because the rational function lens distortion model can represent lens distortion accurately. The Zhang method uses the radial, tangential, and prism distortion model, which does not work properly with high lens distortion. With the proposed calibration method, the pin-hole and lens distortion models are computed separately, and reprojection error remains when data changes. Also, with the proposed method, focal length error does not depend on noise since they are always set in the center of the image.

Fig. 5
Fig. 5

Result with simulated data changing the number of images. Results are quite similar to the variation of noise. The translation vector has significant different values in both methods. Reprojection error has small values, and therefore the mapping from template points to distorted image points is correct for calibrating data. The reprojection error gives both results as valid parameters a priori.

Tables (1)

Tables Icon

Table 1 Camera Parameters Computed with Both Calibration Methods

Equations (31)

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

[ c , d , b j ] = f ( p i , q d i , j ) .
q i , j = g ( p i , c , d , b j ) .
j i q d i , j g ( p i , f ( p i , q d i , j ) ) 0.
J CR = l = 1 n k = 1 m 3 ( CR ( q k , q k + 1 , l , q k + 2 , l , q k + 3 , l ) CR ( p 1 , p 2 , p 3 , p 4 ) ) 2 .
[ u 1 , l v 1 , l w 1 , l u 2 , l v 2 , l w 2 , l u m , l v m , l w m , l ] · [ a l b l c l ] = [ 0 0 0 0 ] ,
J ST = l = 1 n k = 1 m d ( q k , l , s l ) = l = 1 n k = 1 m | a l · u k , l + b l · v k , l + c l · w k , l | a l 2 + b l 2 .
[ a 1 b 1 c 1 a 2 b 2 c 2 a t b t c t ] · [ u v p v v p w v p ] = [ 0 0 0 0 ] .
B · q v p = 0.
J VP = i = 1 4 l = 1 t d ( q v p i , s l ) = i = 1 4 l = 1 t | a l · u v p i + b l · v v p i + c l · w v p i | a l 2 + b l 2 .
[ q v p T 1 q v p T 2 q v p T 3 q v p T 4 ] · [ a h b h c h ] = [ 0 0 0 0 ] .
J HL = i = 1 4 d ( q v p i , s h ) = i = 1 4 | a h · u v p i + b h · v v p i + c h · w v p i | a h 2 + b h 2 .
K = [ 1 / f x 0 0 0 1 / f y 0 0 0 1 ] .
u v p 1 · u v p 2 f x 2 + v v p 1 · v v p 2 f y 2 + w v p 1 · w v p 2 = 0 , u v p 3 · u v p 4 f x 2 + v v p 3 · v v p 4 f y 2 + w v p 3 · w v p 4 = 0 ,
[ u v p 1 · u v p 2 v v p 1 · v v p 2 u v p 3 · u v p 4 v v p 3 · v v p 4 ] · [ 1 / f x 2 1 / f y 2 ] = [ w v p 1 · w v p 2 w v p 3 · w v p 4 ] .
f = [ 1 / f x 2 1 / f y 2 ] , D = [ u v p 1 , 1 · u v p 2 , 1 v v p 1 , 1 · v v p 2 , 1 u v p 3 , 1 · u v p 4 , 1 v v p 3 , 1 · v v p 4 , 1 u v p 1 , s · u v p 2 , s v v p 1 , s · v v p 2 , s u v p 3 , s · u v p 4 , s v v p 3 , s · v v p 4 , s ] , E = [ w v p 1 , 1 · w v p 2 , 1 w v p 3 , 1 · w v p 4 , 1 w v p 1 , s · w v p 2 , s w v p 3 , s · w v p 4 , s ] .
J FL = G T · G ,
J = J CR + J ST + J VP + J HL + J FL .
q 0 i , j = K · [ R j t j ] · p i = K · [ r 1 j r 2 j r 3 j t j ] · p i ,
K = [ α u β u 0 0 α v v 0 0 0 1 ] .
H j = K · [ r 1 j r 2 j t j ] .
[ x i y i 1 0 0 0 u 0 i , j · x i u 0 i , j · y i u 0 i , j 0 0 0 x i y i 1 v 0 i , j · x i v 0 i , j · y i v 0 i , j ] · h j = 0 ,
r 1 j = λ j · K 1 · h 1 j , r 2 j = λ j · K 1 · h 2 j , r 3 j = r 1 j × r 2 j , t j = λ j · K 1 · h 3 j ,
d ( u d , v d ) = [ a 11 · u d 2 + a 12 · u d · v d + a 13 · v d 2 + a 14 · u d + a 15 · v d + a 16 a 21 · u d 2 + a 22 · u d · v d + a 23 · v d 2 + a 24 · u d + a 25 · v d + a 26 a 31 · u d 2 + a 32 · u d · v d + a 33 · v d 2 + a 34 · u d + a 35 · v d + a 36 ] .
d ( u d , v d ) = A · x ( u d , v d ) ,
x ( u d , v d ) = [ u d 2 u d · v d v d 2 u d v d 1 ] T .
q o = ( u o , v o ) = ( a 1 T · x ( u d , v d ) a 3 T · x ( u d , v d ) , a 2 T · x ( u d , v d ) a 3 T · x ( u d , v d ) ) ,
a 3 T · x ( u i , d , v i , d ) · u i , 0 = a 1 T · x ( u i , d , v i , d ) a 3 T · x ( u i , d , v i , d ) · v i , 0 = a 2 T · x ( u i , d , v i , d ) ,
[ x ( u i , d , v i , d ) T 0 u i , 0 · x ( u i , d , v i , d ) T 0 x ( u i , d , v i , d ) T v i , 0 · x ( u i , d , v i , d ) T ] · [ a 1 a 2 a 3 ] = 0.
J RT = i = 1 n · m ( u i , 0 a 1 T · x ( u d , v d ) a 3 T · x ( u d , v d ) + v i , 0 a 2 T · x ( u i , d , v i , d ) a 3 T · x ( u i , d , v i , d ) ) .
e = 1 n · m j i q d i , j K · [ R j t j ] · p i 0.
u d = u o + Δ u o · ( k 1 · r o 2 + k 2 · r o 4 ) + p 1 ( 3 Δ u o 2 + Δ v o 2 ) + 2 p 2 · Δ u o · Δ v o , v d = v o + Δ v o · ( k 1 · r o 2 + k 2 · r o 4 ) + 2 p 1 · Δ u o · Δ v o + p 2 ( Δ u o 2 + 3 Δ v o 2 ) ,

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