Abstract

We present a novel linear algorithm with which to calibrate stereo cameras from two perpendicular planes. Stereo cameras are two cameras aligned in a special configuration with coplanar image planes and parallel axes that are increasingly more widely used in computer vision tasks. Our objective is to present a more practical and simplified linear algorithm for these special configuration cameras, as traditional linear algorithms usually require too-strong constraints either on three-dimensional scenes or on the camera’s motion. We developed the proposed algorithm from a new constraint by exploiting the orthogonality of two planes. The algorithm has much weaker constraints on three-dimensional scenes because two perpendicular planes are commonly found in daily life. We tested the algorithm with synthetic data and real image data. Experimental results show that it is both accurate and practical.

© 2005 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 Trans. Rob. Autom. 3, 323–344 (1987).
    [CrossRef]
  2. S. J. Maybank, O. D. Faugeras, “A theory of self-calibration of a moving camera,” Int. J. Comput. Vision 18, 123–151 (1992).
    [CrossRef]
  3. R. Hartley, “Self-calibration from multiple views with a rotating camera,” Vols. 800 and 801 of Lecture Notes in Computer Science (Springer-Verlag, 1994), pp. 471–478.
    [CrossRef]
  4. S. D. Ma, “A self-calibration technique for active vision systems,” IEEE Trans. Rob. Autom. 12, 114–120 (1996).
    [CrossRef]
  5. B. Caprile, V. Torre, “Using vanishing points for camera calibration,” Int. J. Comput. Vision 12, 127–140 (1990).
    [CrossRef]
  6. R. Cipolla, T. W. Drummond, D. Robertson, “Camera calibration from vanishing points in images of architectural scenes,” presented at the British Machine Vision Conference, Nottingham, UK, September 1999.
  7. Z. Zhang, “Flexible new technique for camera calibration,” IEEE Trans. Pattern Anal. Mach. Intell. 23, 604–616 (2001).
  8. K. Y. K. Wong, P. R. S. Mendonca, R. Cipolla, “Camera calibration from surface of revolution,” IEEE Trans. Pattern Anal. Mach. Intell. 25, 147–161 (2003).
    [CrossRef]
  9. H. Li, F. C. Wu, Z. Y. Hu, “New linear camera self-calibration technique,” Chin. J. Comput. 23, 1121–1129 (2000).
  10. R. Hartley, A. Zisserman, Multiple View Geometry in Computer Vision (Cambridge U. Press, 2000).

2003

K. Y. K. Wong, P. R. S. Mendonca, R. Cipolla, “Camera calibration from surface of revolution,” IEEE Trans. Pattern Anal. Mach. Intell. 25, 147–161 (2003).
[CrossRef]

2001

Z. Zhang, “Flexible new technique for camera calibration,” IEEE Trans. Pattern Anal. Mach. Intell. 23, 604–616 (2001).

2000

H. Li, F. C. Wu, Z. Y. Hu, “New linear camera self-calibration technique,” Chin. J. Comput. 23, 1121–1129 (2000).

1996

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

1992

S. J. Maybank, O. D. Faugeras, “A theory of self-calibration of a moving camera,” Int. J. Comput. Vision 18, 123–151 (1992).
[CrossRef]

1990

B. Caprile, V. Torre, “Using vanishing points for camera calibration,” Int. J. Comput. Vision 12, 127–140 (1990).
[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. 3, 323–344 (1987).
[CrossRef]

Caprile, B.

B. Caprile, V. Torre, “Using vanishing points for camera calibration,” Int. J. Comput. Vision 12, 127–140 (1990).
[CrossRef]

Cipolla, R.

K. Y. K. Wong, P. R. S. Mendonca, R. Cipolla, “Camera calibration from surface of revolution,” IEEE Trans. Pattern Anal. Mach. Intell. 25, 147–161 (2003).
[CrossRef]

R. Cipolla, T. W. Drummond, D. Robertson, “Camera calibration from vanishing points in images of architectural scenes,” presented at the British Machine Vision Conference, Nottingham, UK, September 1999.

Drummond, T. W.

R. Cipolla, T. W. Drummond, D. Robertson, “Camera calibration from vanishing points in images of architectural scenes,” presented at the British Machine Vision Conference, Nottingham, UK, September 1999.

Faugeras, O. D.

S. J. Maybank, O. D. Faugeras, “A theory of self-calibration of a moving camera,” Int. J. Comput. Vision 18, 123–151 (1992).
[CrossRef]

Hartley, R.

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

R. Hartley, “Self-calibration from multiple views with a rotating camera,” Vols. 800 and 801 of Lecture Notes in Computer Science (Springer-Verlag, 1994), pp. 471–478.
[CrossRef]

Hu, Z. Y.

H. Li, F. C. Wu, Z. Y. Hu, “New linear camera self-calibration technique,” Chin. J. Comput. 23, 1121–1129 (2000).

Li, H.

H. Li, F. C. Wu, Z. Y. Hu, “New linear camera self-calibration technique,” Chin. J. Comput. 23, 1121–1129 (2000).

Ma, S. D.

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

Maybank, S. J.

S. J. Maybank, O. D. Faugeras, “A theory of self-calibration of a moving camera,” Int. J. Comput. Vision 18, 123–151 (1992).
[CrossRef]

Mendonca, P. R. S.

K. Y. K. Wong, P. R. S. Mendonca, R. Cipolla, “Camera calibration from surface of revolution,” IEEE Trans. Pattern Anal. Mach. Intell. 25, 147–161 (2003).
[CrossRef]

Robertson, D.

R. Cipolla, T. W. Drummond, D. Robertson, “Camera calibration from vanishing points in images of architectural scenes,” presented at the British Machine Vision Conference, Nottingham, UK, September 1999.

Torre, V.

B. Caprile, V. Torre, “Using vanishing points for camera calibration,” Int. J. Comput. Vision 12, 127–140 (1990).
[CrossRef]

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. 3, 323–344 (1987).
[CrossRef]

Wong, K. Y. K.

K. Y. K. Wong, P. R. S. Mendonca, R. Cipolla, “Camera calibration from surface of revolution,” IEEE Trans. Pattern Anal. Mach. Intell. 25, 147–161 (2003).
[CrossRef]

Wu, F. C.

H. Li, F. C. Wu, Z. Y. Hu, “New linear camera self-calibration technique,” Chin. J. Comput. 23, 1121–1129 (2000).

Zhang, Z.

Z. Zhang, “Flexible new technique for camera calibration,” IEEE Trans. Pattern Anal. Mach. Intell. 23, 604–616 (2001).

Zisserman, A.

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

Chin. J. Comput.

H. Li, F. C. Wu, Z. Y. Hu, “New linear camera self-calibration technique,” Chin. J. Comput. 23, 1121–1129 (2000).

IEEE Trans. Pattern Anal. Mach. Intell.

Z. Zhang, “Flexible new technique for camera calibration,” IEEE Trans. Pattern Anal. Mach. Intell. 23, 604–616 (2001).

K. Y. K. Wong, P. R. S. Mendonca, R. Cipolla, “Camera calibration from surface of revolution,” IEEE Trans. Pattern Anal. Mach. Intell. 25, 147–161 (2003).
[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. 3, 323–344 (1987).
[CrossRef]

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

Int. J. Comput. Vision

B. Caprile, V. Torre, “Using vanishing points for camera calibration,” Int. J. Comput. Vision 12, 127–140 (1990).
[CrossRef]

S. J. Maybank, O. D. Faugeras, “A theory of self-calibration of a moving camera,” Int. J. Comput. Vision 18, 123–151 (1992).
[CrossRef]

Other

R. Hartley, “Self-calibration from multiple views with a rotating camera,” Vols. 800 and 801 of Lecture Notes in Computer Science (Springer-Verlag, 1994), pp. 471–478.
[CrossRef]

R. Cipolla, T. W. Drummond, D. Robertson, “Camera calibration from vanishing points in images of architectural scenes,” presented at the British Machine Vision Conference, Nottingham, UK, September 1999.

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

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Figures (2)

Fig. 1
Fig. 1

(a), (b) Two stereo image pairs taken with stereo cameras viewing two perpendicular sides of a paper box from two different directions.

Fig. 2
Fig. 2

Three stereo image pairs taken by stereo cameras at different orientations: a side of the desktop and a side of the computer were used as two perpendicular planes.

Tables (2)

Tables Icon

Table 1 Calibration Results with Several Noise Levelsa

Tables Icon

Table 2 Calibration Results with a Small Rotation Angle about the y Axis

Equations (23)

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α [ u v 1 ] T = P [ X Y Z 1 ] T ,
P = K [ R t ] ,
K = [ f u ς u 0 0 f u v 0 0 0 1 ] = [ f ς u 0 0 a f v 0 0 0 1 ] .
F 12 = K - T [ t ] x R K - 1 ,
F 12 = K - T [ t ] x K - 1 .
β H = K B K - 1 + K t N T d K - 1 = H + K t N T d K - 1 ,
F 21 = F 12 T = - K - T R T [ t ] x K - 1 .
S = F 21 H = - 1 β K - T R T [ t ] x K - 1 ( K R K - 1 + K t N T d K - 1 ) = - 1 β K - T R T [ t ] x R K - 1
S T = 1 β K - 1 K - T R T [ t ] x R K - 1 = - S .
D = λ H - H = β H - H = K t N T d K - 1 .
D = γ 1 e N T K - 1 ,
F 21 D = λ F 21 H - F 21 H = - λ ( F 21 H ) T + ( F 21 H ) T = - ( F 21 D ) T .
D = [ D ( 1 ) , α 1 D ( 1 ) , α 2 D ( 1 ) ] ,
F 21 ( 1 ) D ( 1 ) = 0 , α 1 F 21 ( 2 ) D ( 1 ) = 0 ;             F 21 ( 3 ) D ( 1 ) = 0 ,
D = γ 2 e [ 1 , α 1 , α 2 ] .
Δ D = ( λ H - H ) - ( λ H - H ) = ( λ - λ ) H .
Δ D = γ 1 e N T K - 1 - γ 2 e [ 1 , α 1 , α 2 ] = e ( γ 1 N T K - 1 - γ 2 [ 1 , α 1 , α 2 ] ) .
H = [ h 1 h 2 h 3 h 4 h 5 h 6 h 7 h 8 h 9 ] ,
D = λ [ h 1 h 2 h 3 h 4 h 5 h 6 h 7 h 8 h 9 ] - [ 1 0 0 0 1 0 0 0 1 ] = [ λ h 1 - 1 λ h 2 λ h 3 λ h 4 λ h 5 - 1 λ h 6 λ h 7 λ h 8 λ h 9 - 1 ] .
n 1 N 1 T K - 1 ,             n 2 N 2 T K - 1 ,
N 1 T n 1 K ,             N 2 T n 2 K .
N 1 T , N 2 T = n 1 K , n 2 K = 0 ,
n 1 K K T n 2 T = n 1 C n 2 T = 0 ,

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