Abstract

We present a simple method to determine the relative distortion of axially symmetric lens systems. This method uses graphs to determine every parametric value instead of nonlinear minimization computation and is composed of an LCD screen to display a square grid pattern of pixel-wide spots and a set of analyzing processes for the spots in the image. The two Cartesian components of the spot locations are processed by a two-step linear least-square fitting to third-order polynomials. The graphs for the coefficients enable us to determine the amount of decentering of the camera lens axis with respect to the center of the image array and the tip/tilt of the screen, which in turn gives the relative distortion coefficient. We present experimental results to demonstrate the utility of the method by comparing our results with the corresponding values determined by open source software available online.

© 2012 Optical Society of America

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References

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  1. A. E. Conrady, “Decentered lens-systems,” Mon. Not. R. Astron. Soc. 79, 384–390 (1919).
  2. D. Brown, “Close range camera calibration,” Photogramm. Eng. 8, 855–866 (1971).
  3. “Camera calibration toolbox for Matlab,” http://www.vision.caltech.edu/bouguetj/calib_doc/index.html#system .
  4. 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]
  5. C. Ricolfe-Viala and A. Sanchez-Salmeron, “Lens distortion models evaluation,” Appl. Opt. 49, 5914–5928 (2010).
    [CrossRef]
  6. D. B. Gennery, “Generalized camera calibration including fish-eye lenses,” Int. J. Comput. Vis. 68, 239–266 (2006).
    [CrossRef]
  7. Z. Zhang, “Flexible camera calibration by viewing a plane from unknown orientations,” Proceedings of the Seventh IEEE International Conference on Computer Vision (IEEE, 1999), pp. 666–673.
  8. W. T. Wellford, Aberrations of Optical Systems (Adam Hilger, 1986), p. 126.
  9. O. Faugeras, Three-Dimensional Computer Vision: A Geometric Viewpoint (MIT, 1993).
  10. The MathWorks, Inc., http://www.mathworks.com/ .

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]

2006 (1)

D. B. Gennery, “Generalized camera calibration including fish-eye lenses,” Int. J. Comput. Vis. 68, 239–266 (2006).
[CrossRef]

1971 (1)

D. Brown, “Close range camera calibration,” Photogramm. Eng. 8, 855–866 (1971).

1919 (1)

A. E. Conrady, “Decentered lens-systems,” Mon. Not. R. Astron. Soc. 79, 384–390 (1919).

Brown, D.

D. Brown, “Close range camera calibration,” Photogramm. Eng. 8, 855–866 (1971).

Conrady, A. E.

A. E. Conrady, “Decentered lens-systems,” Mon. Not. R. Astron. Soc. 79, 384–390 (1919).

Faugeras, O.

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

Gennery, D. B.

D. B. Gennery, “Generalized camera calibration including fish-eye lenses,” Int. J. Comput. Vis. 68, 239–266 (2006).
[CrossRef]

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]

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]

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]

Wellford, W. T.

W. T. Wellford, Aberrations of Optical Systems (Adam Hilger, 1986), p. 126.

Zhang, Z.

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

Appl. Opt. (1)

Int. J. Comput. Vis. (1)

D. B. Gennery, “Generalized camera calibration including fish-eye lenses,” Int. J. Comput. Vis. 68, 239–266 (2006).
[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]

Mon. Not. R. Astron. Soc. (1)

A. E. Conrady, “Decentered lens-systems,” Mon. Not. R. Astron. Soc. 79, 384–390 (1919).

Photogramm. Eng. (1)

D. Brown, “Close range camera calibration,” Photogramm. Eng. 8, 855–866 (1971).

Other (5)

“Camera calibration toolbox for Matlab,” http://www.vision.caltech.edu/bouguetj/calib_doc/index.html#system .

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

W. T. Wellford, Aberrations of Optical Systems (Adam Hilger, 1986), p. 126.

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

The MathWorks, Inc., http://www.mathworks.com/ .

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Figures (4)

Fig. 1.
Fig. 1.

Setup for measuring the relative distortion of a camera lens with an LCD screen.

Fig. 2.
Fig. 2.

(a) Second-order coefficients a2T as a function of spot number in horizontal lines (blue) and vertical lines (red). (b) Zeroth-order a0T coefficients as a function of spot number in horizontal lines (blue) and vertical lines (red), after the linear term and the constant term have been removed.

Fig. 3.
Fig. 3.

Theoretical distorted spot locations, denoted by lines of (a) top row, (b) bottom row, (c) left end column, and (d) right end column, are compared with the experimental data shown in symbols. Note that the spot location scales in (a) and (b) are ascending from top to bottom to represent the actual spots in the image. A 90 deg clockwise rotation of (c) and (d) renders them to be compared with the actual spots in the image.

Fig. 4.
Fig. 4.

(a) Measured extrinsic parameters and (b) reprojection errors calculated by the software [3] for the camera lens studied in the current study. Please refer to [3] for more details on these figures.

Tables (3)

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Table 1. Set of Specifications for the Setup Used in This Work

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Table 2. Measured Relative Distortion

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Table 3. Calibration Results for the Camera Lens Studied in the Current Study Determined by the Software [3]a

Equations (7)

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

rrr=K1r2,
x(i,j)=[1+K1Δ2(i2+j2)]Δiy(i,j)=[1+K1Δ2(i2+j2)]Δj},
x(i,j)=[1+K1Δ2(i2+j2)]Δi+xoy(i,j)=[1+K1Δ2(i2+j2)]Δj+yo},
y(i,jo)=Δ[1+K1Δ2(i2+jo2)]jo+yo,
y(i,jo)=a2T(jo)i2+a0T(jo),
a2T(jo)=K1Δ3joa0T(jo)=K1Δ3jo3+Δjo+yo}.
x(i,j)=Δ(iΔdqθi2)y(i,j)=Δ(jΔdqϕj2)},

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