Abstract

Virtually all imaging devices introduce some amount of geometric lens distortion. A technique is presented for blindly removing these distortions in the absence of any calibration information or explicit knowledge of the imaging device. The basic approach exploits the fact that lens distortion introduces specific higher-order correlations in the frequency domain. These correlations can be detected by using tools from polyspectral analysis. The amount of distortion is then estimated by minimizing these correlations.

© 2001 Optical Society of America

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References

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  1. W. Faig, “Calibration of close-range photogrammetric systems: mathematical formulation,” Photogramm. Eng. Remote Sens. 41, 1479–1486 (1975).
  2. R. Y. Tsai, “A versatile camera calibration technique for high-accuracy 3D machine vision metrology using off-the-shelf TV cameras and lenses,” IEEE Trans. Rob. Autom. RA-3, 323–344 (1987).
    [CrossRef]
  3. J. Weng, “Camera calibration with distortion models and accuracy evaluation,” IEEE Trans. Pattern Anal. Mach. Intell. 14, 965–979 (1992).
    [CrossRef]
  4. F. Devernay, O. Faugeras, “Automatic calibration and removal of distortion from scenes of structured environments,” in Investigative and Trial Image Processing. L. I. Rudin, S. K. Bramble, eds., Proc. SPIE2567, 62–72 (1995).
  5. R. Swaminatha, S. K. Nayar, “Non-metric calibration of wide-angle lenses and polycameras,” in IEEE Conference on Computer Vision and Pattern Recognition (IEEE Computer Society Press, Los Alamitos, Calif., 1999), pp. 413–419.
  6. H. Farid, “Blind inverse gamma correction,” IEEE Trans. Image Process (to be published).
  7. I. S. Gradshteyn, I. M. Ryzhik, Table of Integrals, Series, and Products (Academic, San Diego, Calif., 1994).
  8. J. M. Mendel, “Tutorial on higher order statistics (spectra) in signal processing and system theory: theoretical results and some applications,” Proc. IEEE 79, 278–305 (1996).
    [CrossRef]
  9. Y. C. Kim, E. J. Powers, “Digital bispectral analysis and its applications to nonlinear wave interactions,” IEEE Trans. Plasma Sci. PS-7, 120–131 (1979).
    [CrossRef]

1996

J. M. Mendel, “Tutorial on higher order statistics (spectra) in signal processing and system theory: theoretical results and some applications,” Proc. IEEE 79, 278–305 (1996).
[CrossRef]

1992

J. Weng, “Camera calibration with distortion models and accuracy evaluation,” IEEE Trans. Pattern Anal. Mach. Intell. 14, 965–979 (1992).
[CrossRef]

1987

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

1979

Y. C. Kim, E. J. Powers, “Digital bispectral analysis and its applications to nonlinear wave interactions,” IEEE Trans. Plasma Sci. PS-7, 120–131 (1979).
[CrossRef]

1975

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

Devernay, F.

F. Devernay, O. Faugeras, “Automatic calibration and removal of distortion from scenes of structured environments,” in Investigative and Trial Image Processing. L. I. Rudin, S. K. Bramble, eds., Proc. SPIE2567, 62–72 (1995).

Faig, W.

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

Farid, H.

H. Farid, “Blind inverse gamma correction,” IEEE Trans. Image Process (to be published).

Faugeras, O.

F. Devernay, O. Faugeras, “Automatic calibration and removal of distortion from scenes of structured environments,” in Investigative and Trial Image Processing. L. I. Rudin, S. K. Bramble, eds., Proc. SPIE2567, 62–72 (1995).

Gradshteyn, I. S.

I. S. Gradshteyn, I. M. Ryzhik, Table of Integrals, Series, and Products (Academic, San Diego, Calif., 1994).

Kim, Y. C.

Y. C. Kim, E. J. Powers, “Digital bispectral analysis and its applications to nonlinear wave interactions,” IEEE Trans. Plasma Sci. PS-7, 120–131 (1979).
[CrossRef]

Mendel, J. M.

J. M. Mendel, “Tutorial on higher order statistics (spectra) in signal processing and system theory: theoretical results and some applications,” Proc. IEEE 79, 278–305 (1996).
[CrossRef]

Nayar, S. K.

R. Swaminatha, S. K. Nayar, “Non-metric calibration of wide-angle lenses and polycameras,” in IEEE Conference on Computer Vision and Pattern Recognition (IEEE Computer Society Press, Los Alamitos, Calif., 1999), pp. 413–419.

Powers, E. J.

Y. C. Kim, E. J. Powers, “Digital bispectral analysis and its applications to nonlinear wave interactions,” IEEE Trans. Plasma Sci. PS-7, 120–131 (1979).
[CrossRef]

Ryzhik, I. M.

I. S. Gradshteyn, I. M. Ryzhik, Table of Integrals, Series, and Products (Academic, San Diego, Calif., 1994).

Swaminatha, R.

R. Swaminatha, S. K. Nayar, “Non-metric calibration of wide-angle lenses and polycameras,” in IEEE Conference on Computer Vision and Pattern Recognition (IEEE Computer Society Press, Los Alamitos, Calif., 1999), pp. 413–419.

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 Trans. Rob. Autom. RA-3, 323–344 (1987).
[CrossRef]

Weng, J.

J. Weng, “Camera calibration with distortion models and accuracy evaluation,” IEEE Trans. Pattern Anal. Mach. Intell. 14, 965–979 (1992).
[CrossRef]

IEEE Trans. Pattern Anal. Mach. Intell.

J. Weng, “Camera calibration with distortion models and accuracy evaluation,” IEEE Trans. Pattern Anal. Mach. Intell. 14, 965–979 (1992).
[CrossRef]

IEEE Trans. Plasma Sci.

Y. C. Kim, E. J. Powers, “Digital bispectral analysis and its applications to nonlinear wave interactions,” IEEE Trans. Plasma Sci. PS-7, 120–131 (1979).
[CrossRef]

IEEE Trans. Rob. Autom.

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

Photogramm. Eng. Remote Sens.

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

Proc. IEEE

J. M. Mendel, “Tutorial on higher order statistics (spectra) in signal processing and system theory: theoretical results and some applications,” Proc. IEEE 79, 278–305 (1996).
[CrossRef]

Other

F. Devernay, O. Faugeras, “Automatic calibration and removal of distortion from scenes of structured environments,” in Investigative and Trial Image Processing. L. I. Rudin, S. K. Bramble, eds., Proc. SPIE2567, 62–72 (1995).

R. Swaminatha, S. K. Nayar, “Non-metric calibration of wide-angle lenses and polycameras,” in IEEE Conference on Computer Vision and Pattern Recognition (IEEE Computer Society Press, Los Alamitos, Calif., 1999), pp. 413–419.

H. Farid, “Blind inverse gamma correction,” IEEE Trans. Image Process (to be published).

I. S. Gradshteyn, I. M. Ryzhik, Table of Integrals, Series, and Products (Academic, San Diego, Calif., 1994).

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Figures (9)

Fig. 1
Fig. 1

One-parameter radially symmetric lens distortion [Eqs. (1)].

Fig. 2
Fig. 2

Top: power spectrum and bicoherence for a signal with random amplitudes and phases. Bottom: same signal with one frequency, ω3, whose amplitude and phase are correlated to ω1 and ω2. The horizontal axis of the bicoherence corresponds to ω1, and the vertical to ω2. The origin is in the center, and the axis range is [-π, π].

Fig. 3
Fig. 3

Shown in the left column is a fractal signal, the log of its normalized power spectrum, and its bicoherence. Shown in the right column is a distorted version of the signal. While the distortion leaves the power spectrum largely unchanged, there is a significant increase in the average bispectral response.

Fig. 4
Fig. 4

Bicoherence computed for a range of lens distortion (κ). The bicoherence is minimal when κ=0, i.e., no distortion.

Fig. 5
Fig. 5

Synthetic images with no distortion (center), negative distortion (left), and positive distortion (right).

Fig. 6
Fig. 6

Blindly estimated distortion parameters. Each data point corresponds to the average from ten synthetic images. See also Table 1.

Fig. 7
Fig. 7

Top: small low-grade camera and a calibration target used to manually calibrate the lens distortion. Bottom: image of the calibration target before (left) and after (right) calibration.

Fig. 8
Fig. 8

Several distorted images (left) and the results of blindly estimating and removing the lens distortion (right).

Fig. 9
Fig. 9

Several distorted images (left) and the results of blindly estimating and removing the lens distortion (right).

Tables (1)

Tables Icon

Table 1 Blindly Estimated Distortion Parameters (Mean, Standard Deviation (s.d.), and Minimum and Maximum Values) Averaged over Ten Independent Synthetic Images a

Equations (22)

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

x˜=x(1+κr2),y˜=y(1+κr2),
fu(x)=a cos(bx).
fd(x)=a cos(bx2).
Fd(ω)=- fd(x)exp(-iωx)dx
=20a cos(bx2)cos(ωx)dx.
Fd(ω)=2aπ2bcosω22b+sinω22b.
Fu(ω)=1,|ω|=b0,|ω|b,
F(ω)=k=-f(k)exp(-iωk).
P(ω)=E{F(ω)F*(ω)},
B(ω1, ω2)=E{F(ω1)F(ω2)F*(ω1+ω2)}.
Bˆ(ω1, ω2)=1Nk=1NFk(ω1)Fk(ω2)Fk*(ω1+ω2),
b2(ω1, ω2)=|B(ω2, ω2)|2E{|F(ω2)F(ω2)|2}E{|F(ω1+ω2)|2}.
bˆ(ω1, ω2)=1NkFk(ω1)Fk(ω2)Fk*(ω1+ω2)[1Nk|Fk(ω1)Fk(ω2)|21Nk|Fk(ω1+ω2)|2]1/2.
1N2ω1=-N/2N/2ω2=-N/2N/2 bˆ2πω1N,2πω2N.
fd(x)=fu(x(1+κx2)),
x˜=x(1+κr2),y˜=y(1+κr2),
r=x2+y2,θ=tan-1(y/x).
r˜=x˜2+y˜2,θ˜=tan-1(y˜/x˜).
r˜=r(1+κr2),θ˜=tan-1(y/x).
fu(x, y)=n=1Nansin{ωn[cos(θn)x+sin(θn)y]+ϕn}.
fd(x, y)=n=1Nansin{ωn[cos(θn)x˜+sin(θn)y˜]+ϕn},
κ=-1.5784κ3-0.7752κ2+1.6621κ-0.0089.

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