Abstract

An analytical method to calibrate a camera’s extrinsic parameters and effective focal lengths is presented. We used a single image with four coplanar control lines. Unique solutions of the camera’s extrinsic parameters and the two effective focal lengths are derived linearly and analytically. If the effective focal lengths are known, unique solutions of the extrinsic parameters can be derived more easily. Our method can easily be implemented and has better precision than the analogous method by use of control points. The results of simulated experiments are presented.

© 2004 Optical Society of America

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References

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  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. Rob. Autom. 3, 323–344 (1987).
    [CrossRef]
  2. J. Weng, P. Cohen, M. Herniou, “Camera calibration with distortion models and accuracy evaluation,” IEEE Trans. Pattern Anal. Mach. Intell. 14, 965–980 (1992).
    [CrossRef]
  3. Z. Zhang, “A flexible new technique for camera calibration,” Technical Report MSR-TR-98-71 (Microsoft Corporation, Redmond, Wash., December1998).
  4. M. A. Fishler, R. C. Bolles, “Random sample consensus: a paradigm for model fitting with applications to image analysis and automated cartography,” Commun. ACM 24, 381–395 (1981).
    [CrossRef]
  5. M. A. Penna, “Determining camera parameters from the perspective projection of a quadrilateral,” Pattern Recogn. 24, 533–541 (1991).
    [CrossRef]
  6. M. A. Abidi, T. Chandra, “A new efficient and direct solution for pose estimation using quadrangular targets: algorithm and evaluation,” IEEE Trans. Pattern Anal. Mach. Intell. 17, 534–538 (1995).
    [CrossRef]
  7. Z. Hu, C. Lei, F. Wu, “A short note on P4P problem,” Acta Automat. Sin. 27, 770–776 (2001).
  8. M. S. Scholl, “Optical processing for semiautonomous terminal navigation and docking,” Appl. Opt. 32, 5049–5055 (1993).
    [CrossRef] [PubMed]
  9. M. S. Scholl, Y. W. Wang, G. P. Padilla, “Design of a high-resolution telescope for an imaging sensor to characterize a (Martian) landing site,” Opt. Eng. 34, 3222–3228 (1995).
    [CrossRef]
  10. M. S. Scholl, G. P. Padilla, “Push-broom reconnaissance camera with time expansion for a (Martian) landing-site certification,” Opt. Eng. 36, 566–573 (1997).
    [CrossRef]

2001 (1)

Z. Hu, C. Lei, F. Wu, “A short note on P4P problem,” Acta Automat. Sin. 27, 770–776 (2001).

1997 (1)

M. S. Scholl, G. P. Padilla, “Push-broom reconnaissance camera with time expansion for a (Martian) landing-site certification,” Opt. Eng. 36, 566–573 (1997).
[CrossRef]

1995 (2)

M. S. Scholl, Y. W. Wang, G. P. Padilla, “Design of a high-resolution telescope for an imaging sensor to characterize a (Martian) landing site,” Opt. Eng. 34, 3222–3228 (1995).
[CrossRef]

M. A. Abidi, T. Chandra, “A new efficient and direct solution for pose estimation using quadrangular targets: algorithm and evaluation,” IEEE Trans. Pattern Anal. Mach. Intell. 17, 534–538 (1995).
[CrossRef]

1993 (1)

1992 (1)

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

1991 (1)

M. A. Penna, “Determining camera parameters from the perspective projection of a quadrilateral,” Pattern Recogn. 24, 533–541 (1991).
[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. Rob. Autom. 3, 323–344 (1987).
[CrossRef]

1981 (1)

M. A. Fishler, R. C. Bolles, “Random sample consensus: a paradigm for model fitting with applications to image analysis and automated cartography,” Commun. ACM 24, 381–395 (1981).
[CrossRef]

Abidi, M. A.

M. A. Abidi, T. Chandra, “A new efficient and direct solution for pose estimation using quadrangular targets: algorithm and evaluation,” IEEE Trans. Pattern Anal. Mach. Intell. 17, 534–538 (1995).
[CrossRef]

Bolles, R. C.

M. A. Fishler, R. C. Bolles, “Random sample consensus: a paradigm for model fitting with applications to image analysis and automated cartography,” Commun. ACM 24, 381–395 (1981).
[CrossRef]

Chandra, T.

M. A. Abidi, T. Chandra, “A new efficient and direct solution for pose estimation using quadrangular targets: algorithm and evaluation,” IEEE Trans. Pattern Anal. Mach. Intell. 17, 534–538 (1995).
[CrossRef]

Cohen, P.

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

Fishler, M. A.

M. A. Fishler, R. C. Bolles, “Random sample consensus: a paradigm for model fitting with applications to image analysis and automated cartography,” Commun. ACM 24, 381–395 (1981).
[CrossRef]

Herniou, M.

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

Hu, Z.

Z. Hu, C. Lei, F. Wu, “A short note on P4P problem,” Acta Automat. Sin. 27, 770–776 (2001).

Lei, C.

Z. Hu, C. Lei, F. Wu, “A short note on P4P problem,” Acta Automat. Sin. 27, 770–776 (2001).

Padilla, G. P.

M. S. Scholl, G. P. Padilla, “Push-broom reconnaissance camera with time expansion for a (Martian) landing-site certification,” Opt. Eng. 36, 566–573 (1997).
[CrossRef]

M. S. Scholl, Y. W. Wang, G. P. Padilla, “Design of a high-resolution telescope for an imaging sensor to characterize a (Martian) landing site,” Opt. Eng. 34, 3222–3228 (1995).
[CrossRef]

Penna, M. A.

M. A. Penna, “Determining camera parameters from the perspective projection of a quadrilateral,” Pattern Recogn. 24, 533–541 (1991).
[CrossRef]

Scholl, M. S.

M. S. Scholl, G. P. Padilla, “Push-broom reconnaissance camera with time expansion for a (Martian) landing-site certification,” Opt. Eng. 36, 566–573 (1997).
[CrossRef]

M. S. Scholl, Y. W. Wang, G. P. Padilla, “Design of a high-resolution telescope for an imaging sensor to characterize a (Martian) landing site,” Opt. Eng. 34, 3222–3228 (1995).
[CrossRef]

M. S. Scholl, “Optical processing for semiautonomous terminal navigation and docking,” Appl. Opt. 32, 5049–5055 (1993).
[CrossRef] [PubMed]

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

Wang, Y. W.

M. S. Scholl, Y. W. Wang, G. P. Padilla, “Design of a high-resolution telescope for an imaging sensor to characterize a (Martian) landing site,” Opt. Eng. 34, 3222–3228 (1995).
[CrossRef]

Weng, J.

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

Wu, F.

Z. Hu, C. Lei, F. Wu, “A short note on P4P problem,” Acta Automat. Sin. 27, 770–776 (2001).

Zhang, Z.

Z. Zhang, “A flexible new technique for camera calibration,” Technical Report MSR-TR-98-71 (Microsoft Corporation, Redmond, Wash., December1998).

Acta Automat. Sin. (1)

Z. Hu, C. Lei, F. Wu, “A short note on P4P problem,” Acta Automat. Sin. 27, 770–776 (2001).

Appl. Opt. (1)

Commun. ACM (1)

M. A. Fishler, R. C. Bolles, “Random sample consensus: a paradigm for model fitting with applications to image analysis and automated cartography,” Commun. ACM 24, 381–395 (1981).
[CrossRef]

IEEE J. Rob. Autom. (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. Rob. Autom. 3, 323–344 (1987).
[CrossRef]

IEEE Trans. Pattern Anal. Mach. Intell. (2)

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

M. A. Abidi, T. Chandra, “A new efficient and direct solution for pose estimation using quadrangular targets: algorithm and evaluation,” IEEE Trans. Pattern Anal. Mach. Intell. 17, 534–538 (1995).
[CrossRef]

Opt. Eng. (2)

M. S. Scholl, Y. W. Wang, G. P. Padilla, “Design of a high-resolution telescope for an imaging sensor to characterize a (Martian) landing site,” Opt. Eng. 34, 3222–3228 (1995).
[CrossRef]

M. S. Scholl, G. P. Padilla, “Push-broom reconnaissance camera with time expansion for a (Martian) landing-site certification,” Opt. Eng. 36, 566–573 (1997).
[CrossRef]

Pattern Recogn. (1)

M. A. Penna, “Determining camera parameters from the perspective projection of a quadrilateral,” Pattern Recogn. 24, 533–541 (1991).
[CrossRef]

Other (1)

Z. Zhang, “A flexible new technique for camera calibration,” Technical Report MSR-TR-98-71 (Microsoft Corporation, Redmond, Wash., December1998).

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Figures (7)

Fig. 1
Fig. 1

Pinhole camera model: O-XYZ, world coordinate system; S, optical center; ∑, image plane; I-uv, image coordinate system; P, a world point; p, its projection.

Fig. 2
Fig. 2

Calibration system of the simulated experiments: Π, control plane; L i and P i , control lines and control points, where i = 1, 2, 3, 4; O-XYZ, world coordinate system; S, optical center; ∑, image plane; I-uv, image coordinate system.

Fig. 3
Fig. 3

Results for t X and t Y obtained by computer simulations: δ is the rms of the extracting image points and σ tX tY ) is the rms of t X (t Y ).

Fig. 4
Fig. 4

Results for t Z obtained by computer simulations: δ is the rms of the extracting image points and σ tZ is the rms of t Z .

Fig. 5
Fig. 5

Results for ω and ϕ obtained by computer simulations: δ is the rms of the extracting image points and σωϕ) is the rms of ω(ϕ).

Fig. 6
Fig. 6

Results for κ obtained by computer simulations: δ is the rms of the extracting image points and σκ is the rms of κ.

Fig. 7
Fig. 7

Results for f u and f v obtained by computer simulations: δ is the rms of the extracting image points and σ fu fv ) is the rms of f u (f v ).

Tables (1)

Tables Icon

Table 1 Parameters of the Simulated Camera

Equations (18)

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

λuv1=fu0u000fvv000010 · r11r12r13tXr21r22r23tYr31r32r33tZ0001 · XYZ1,
λuv1=fu0u00fvv0001 · r11r12tXr21r22tYr31r32tZ · XY1.
H=h11h12h13h21h22h23h31h32h33=fur11+u0r31fur12+u0r32futX+u0tZfvr21+v0r31fvr22+v0r32fvtY+v0tZr31r32tZ.
λu v 1T=H · X Y 1T.
u=h11X+h12Y+h13/h31X+h32Y+h33,
v=h21X+h22Y+h23/h31X+h32Y+h33.
Y=ah11+bh31-h21-ah12-bh32+h22 X+ah13+bh33-h23-ah12-bh32+h22.
ah11+aAh12-h21-Ah22+bh31+bAh32=0,
aBh12+ah13-Bh22-h23+bBh32+bh33=0.
Mh11 h12 h13 h21 h22 h23 h31 h32T=h330 -b1 0 -b2 0 -b3 0 -b4T,
M=a1a1A10-1-A10b1b1A10a1B1a10-B1-10b1B1a2a2A20-1-A20b2b2A20a2B2a20-B2-10b2B2a3a3A30-1-A30b3b3A30a3B3a30-B3-10b3B3a4a4A40-1-A40b4b4A40a4B4a40-B4-10b4B4.
M · G=B,
M=a1a1A10-1-A10b1b1A10a1B1a10-B1-10b1B1a2a2A20-1-A20b2b2A20a2B2a20-B2-10b2B2ananAn0-1-An0bnbnAn0anBnan0-Bn-10bnBn2n×8,
G=MTM-1MTB.
fu0u0000000000fu0u0000000000fu0u00fvv00000000000fvv00000000000fvv0001000000000001000000000001 · r11r21r31r12r22r32tXtYtZ =ηg11ηg12ηg13ηg21ηg22ηg23ηg31ηg32η.
10u000000000010u000000000010u001v000000000001v000000000001v0001000000000001000000000001 · fur11fvr21r31fur12fvr22r32futXfvtYtZ =ηg11ηg12ηg13ηg21ηg22ηg23ηg31ηg32η.
k12k22-1k42k52-1k1k4k2k50 · 1/fu21/fv21/η2=-k32-k62-k3k6.
X1Y11000-u1X1-u1Y1000X1Y11-v1X1-v1Y1X2Y21000-u2X2-u2Y2000X2Y21-v2X2-v2Y2X3Y31000-u3X3-u3Y3000X3Y31-v3X3-v3Y3X4Y41000-u4X4-u4Y4000X4Y41-v4X4-v4Y4 · g11g12g13g21g22g23g31g32 =u1v1u2v2u3v3u4v4,

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