Abstract

Two novel camera calibration methods are compared with traditional pinhole calibration: one new method uses an analytic geometrical version of Snell’s law (denoted as the Snell model); the other uses 6×6 matrix-based paraxial ray-tracing (referred to as the paraxial model). Pinhole model uses a perspective projection approximation to give a single lumped result for the multiple optical elements in a camera system. It is mathematically simple, but suffers from accuracy limitations since it does not consider the lens system. The Snell model is mathematically the most complex but potentially has the highest levels of accuracy for the widest range of conditions. The paraxial model has the merit of offering analytical equations for calibration.

© 2007 Optical Society of America

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References

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  1. M. Kawakita, K. Iizuka, R. Iwama, K. Takizawa, H. Kikuchi, and F. Sato, “Gain-modulated axi-vision camera (high speed high-accuracy depth-mapping camera),” Opt. Express 12,5336–5344 (2004).
    [Crossref] [PubMed]
  2. Y.I. Abdel-Aziz and H. M. Karara, “Direct linear transformation from comparator coordinates into object space coordinates in close-range photogrammetry,” Proceedings of the ASP Symposium on Close-Range Photogrammetry,1–18 (1971).
  3. H. Hatze, “High-precision three-dimensional photogrammetric calibration and object space reconstruction using a modified DLT-approach,“ J. Biomech. 21,533–538 (1988).
    [Crossref] [PubMed]
  4. R. K. Lenz and R. Y. Tsai, “Techniques for calibration of the scale factor and image center for high accuracy 3-D machine vision metrology,” IEEE Trans. Pattern Anal. Mach. Intell., 10, 713–720 (1988).
    [Crossref]
  5. F. Y. Wang, “An effecient coordinate frame calibration method for 3-D measurement by multiple camera systems,” IEEE Trans. Syst. Man Cybern., 35, 453–464 (2005).
    [Crossref]
  6. F. Y. Wang, “A simple and analytical procedure for calibrating extrinsic camewra parameters,” IEEE Trans. Rob. Autom., 20, 121–124 (2004).
    [Crossref]
  7. R. Klette, K. SchlUns, and A. Koschan, Computer Vision: Three-Dimension Data from Images, (Springer-Verlag Singapore, Pte. Ltd., 44, (1998).
  8. P. D. Lin and C. K. Sung, “Camera calibration based on Snell’s law,” J. Dyn. Syst. Meas. Control-Trans. ASME, 128, 548–557 (2006).
    [Crossref]
  9. P. D. Lin and C. K. Sung, “Matrix-based paraxial skew ray-tracing in 3D systems with non-coplanar optical axis,” Optik, 117, 329–340 (2006).
    [Crossref]
  10. R. P. Paul, Robot manipulators-mathematics, programming and control, (MIT press, Cambridge, Mass., 10–15,1982).

2006 (2)

P. D. Lin and C. K. Sung, “Camera calibration based on Snell’s law,” J. Dyn. Syst. Meas. Control-Trans. ASME, 128, 548–557 (2006).
[Crossref]

P. D. Lin and C. K. Sung, “Matrix-based paraxial skew ray-tracing in 3D systems with non-coplanar optical axis,” Optik, 117, 329–340 (2006).
[Crossref]

2005 (1)

F. Y. Wang, “An effecient coordinate frame calibration method for 3-D measurement by multiple camera systems,” IEEE Trans. Syst. Man Cybern., 35, 453–464 (2005).
[Crossref]

2004 (2)

1988 (2)

H. Hatze, “High-precision three-dimensional photogrammetric calibration and object space reconstruction using a modified DLT-approach,“ J. Biomech. 21,533–538 (1988).
[Crossref] [PubMed]

R. K. Lenz and R. Y. Tsai, “Techniques for calibration of the scale factor and image center for high accuracy 3-D machine vision metrology,” IEEE Trans. Pattern Anal. Mach. Intell., 10, 713–720 (1988).
[Crossref]

Abdel-Aziz, Y.I.

Y.I. Abdel-Aziz and H. M. Karara, “Direct linear transformation from comparator coordinates into object space coordinates in close-range photogrammetry,” Proceedings of the ASP Symposium on Close-Range Photogrammetry,1–18 (1971).

Hatze, H.

H. Hatze, “High-precision three-dimensional photogrammetric calibration and object space reconstruction using a modified DLT-approach,“ J. Biomech. 21,533–538 (1988).
[Crossref] [PubMed]

Iizuka, K.

Iwama, R.

Karara, H. M.

Y.I. Abdel-Aziz and H. M. Karara, “Direct linear transformation from comparator coordinates into object space coordinates in close-range photogrammetry,” Proceedings of the ASP Symposium on Close-Range Photogrammetry,1–18 (1971).

Kawakita, M.

Kikuchi, H.

Klette, R.

R. Klette, K. SchlUns, and A. Koschan, Computer Vision: Three-Dimension Data from Images, (Springer-Verlag Singapore, Pte. Ltd., 44, (1998).

Koschan, A.

R. Klette, K. SchlUns, and A. Koschan, Computer Vision: Three-Dimension Data from Images, (Springer-Verlag Singapore, Pte. Ltd., 44, (1998).

Lenz, R. K.

R. K. Lenz and R. Y. Tsai, “Techniques for calibration of the scale factor and image center for high accuracy 3-D machine vision metrology,” IEEE Trans. Pattern Anal. Mach. Intell., 10, 713–720 (1988).
[Crossref]

Lin, P. D.

P. D. Lin and C. K. Sung, “Camera calibration based on Snell’s law,” J. Dyn. Syst. Meas. Control-Trans. ASME, 128, 548–557 (2006).
[Crossref]

P. D. Lin and C. K. Sung, “Matrix-based paraxial skew ray-tracing in 3D systems with non-coplanar optical axis,” Optik, 117, 329–340 (2006).
[Crossref]

Paul, R. P.

R. P. Paul, Robot manipulators-mathematics, programming and control, (MIT press, Cambridge, Mass., 10–15,1982).

Sato, F.

SchlUns, K.

R. Klette, K. SchlUns, and A. Koschan, Computer Vision: Three-Dimension Data from Images, (Springer-Verlag Singapore, Pte. Ltd., 44, (1998).

Sung, C. K.

P. D. Lin and C. K. Sung, “Matrix-based paraxial skew ray-tracing in 3D systems with non-coplanar optical axis,” Optik, 117, 329–340 (2006).
[Crossref]

P. D. Lin and C. K. Sung, “Camera calibration based on Snell’s law,” J. Dyn. Syst. Meas. Control-Trans. ASME, 128, 548–557 (2006).
[Crossref]

Takizawa, K.

Tsai, R. Y.

R. K. Lenz and R. Y. Tsai, “Techniques for calibration of the scale factor and image center for high accuracy 3-D machine vision metrology,” IEEE Trans. Pattern Anal. Mach. Intell., 10, 713–720 (1988).
[Crossref]

Wang, F. Y.

F. Y. Wang, “An effecient coordinate frame calibration method for 3-D measurement by multiple camera systems,” IEEE Trans. Syst. Man Cybern., 35, 453–464 (2005).
[Crossref]

F. Y. Wang, “A simple and analytical procedure for calibrating extrinsic camewra parameters,” IEEE Trans. Rob. Autom., 20, 121–124 (2004).
[Crossref]

IEEE Trans. Pattern Anal. Mach. Intell. (1)

R. K. Lenz and R. Y. Tsai, “Techniques for calibration of the scale factor and image center for high accuracy 3-D machine vision metrology,” IEEE Trans. Pattern Anal. Mach. Intell., 10, 713–720 (1988).
[Crossref]

IEEE Trans. Rob. Autom. (1)

F. Y. Wang, “A simple and analytical procedure for calibrating extrinsic camewra parameters,” IEEE Trans. Rob. Autom., 20, 121–124 (2004).
[Crossref]

IEEE Trans. Syst. Man Cybern. (1)

F. Y. Wang, “An effecient coordinate frame calibration method for 3-D measurement by multiple camera systems,” IEEE Trans. Syst. Man Cybern., 35, 453–464 (2005).
[Crossref]

J. Biomech. (1)

H. Hatze, “High-precision three-dimensional photogrammetric calibration and object space reconstruction using a modified DLT-approach,“ J. Biomech. 21,533–538 (1988).
[Crossref] [PubMed]

J. Dyn. Syst. Meas. Control-Trans. ASME (1)

P. D. Lin and C. K. Sung, “Camera calibration based on Snell’s law,” J. Dyn. Syst. Meas. Control-Trans. ASME, 128, 548–557 (2006).
[Crossref]

Opt. Express (1)

Optik (1)

P. D. Lin and C. K. Sung, “Matrix-based paraxial skew ray-tracing in 3D systems with non-coplanar optical axis,” Optik, 117, 329–340 (2006).
[Crossref]

Other (3)

R. P. Paul, Robot manipulators-mathematics, programming and control, (MIT press, Cambridge, Mass., 10–15,1982).

R. Klette, K. SchlUns, and A. Koschan, Computer Vision: Three-Dimension Data from Images, (Springer-Verlag Singapore, Pte. Ltd., 44, (1998).

Y.I. Abdel-Aziz and H. M. Karara, “Direct linear transformation from comparator coordinates into object space coordinates in close-range photogrammetry,” Proceedings of the ASP Symposium on Close-Range Photogrammetry,1–18 (1971).

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Figures (5)

Fig. 1.
Fig. 1.

Pinhole model is an ideal model of camera.

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Fig. 2.
Fig. 2.

Schematic drawing of camera used in Snell model.

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Fig. 3
Fig. 3

Camera system used in paraxial model.

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Fig. 4.
Fig. 4.

The laser/plate assembly for quick determination of the source ray data.

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Fig. 5.
Fig. 5.

The camera calibration of paraxial model.

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Tables (3)

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Table 1. Calibration results from pinhole model and Snell model

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Table 2. Calibration results from paraxial model

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Table 3. Comparison of system performances of three models

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Equations (24)

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P c 0 = [ P c 0 x P c 0 y P c 0 z 1 ] T = A c w P w 0
A c w = Trans ( t x , t y , t z ) Rot ( z , ω z ) Rot ( y , ω y ) Rot ( x , ω x )
[ u v 1 ] = 1 P c 0 y [ f s u u 0 − fCΩ S u SΩ 0 0 v 0 f s v SΩ 0 0 1 0 0 ] A c w P w 0
P w i = [ P w i − lx + ℓ w i − lx λ i P w i − ly + ℓ w i − ly λ i P w i − lz + ℓ w i − lz λ i 0 ] T
ℓ w i = [ ℓ w ix ℓ w iy ℓ w iz 0 ] = [ − n w ix 1 − N i 2 + ( N i Cθ i ) 2 + N i ( ℓ w i − lx + n w i x Cθ i ) − n w iy 1 − N i 2 + ( N i Cθ i ) 2 + N i ( ℓ w i − ly + n w i y Cθ i ) − n w iz 1 − N i 2 + ( N i Cθ i ) 2 + N i ( ℓ w i − lz + n w i z Cθ i ) 0 ]
P n n = [ P n nx P n ny P n nz 1 ] T = A n w P w n
[ u v ] = [ u 0 v 0 ] + [ 1 S u − CΩ S u SΩ 0 1 S v SΩ ] [ P n nx P n nz ]
P c 0 = [ P c 0 x P c 0 y P c 0 z 1 ] T = A c w P w 0 ;
ℓ c 0 = [ ℓ c 0 x ℓ c 0 y ℓ c 0 z 0 ] T = A c w ℓ w 0 .
[ ∆ P c 0 ∆ ℓ c 0 ] T = [ P c 0 ℓ c 0 ] T − [ P̱ c 0 ℓ̱ c 0 ] T
= [ P c 0 x P c 0 y P c 0 z ℓ c 0 x ℓ c 0 y ℓ c 0 z ] T − [ 0 P c 0 y 0 0 1 0 ] T
= [ ( Cω y Cω z ) w P 0 x + ( Sω x Sω y Cω z − Cω x Cω z ) w P 0y + ( Cω x Sω y Cω z + Sω x Sω z ) w P 0z + t x 0 ( − Sω y ) w P 0 x + ( Sω x Cω y ) w P 0y + ( Cω x Cω y ) w P 0z + t z ( Cω y Cω z ) ℓ w 0 x + ( Sω x Sω y Cω z − Cω x Cω z ) w ℓ 0y + ( Cω x Sω y Cω z + Sω x Sω z ) w ℓ 0z ( Cω y Sω z ) w ℓ 0 x + ( Sω x Sω y Sω z + Cω x Cω z ) w ℓ 0y + ( Cω x Sω y Sω z + Sω x Cω z ) w ℓ 0z − 1 ( − Sω y ) w ℓ 0 x + ( Sω x Cω y ) w ℓ 0y + ( Cω x Cω y ) w ℓ 0z ]
[ Δ P n n Δ ℓ n n ] = [ Δ P c n Δ ℓ c n ] = ( T n ) ( M n − 1 T n − 1 M n − 2 T n − 2 ) ⋯ ( M 2 j T 2 j M 2 j − 1 T 2 j − 1 ) ⋯ ( M 2 T 2 M 1 T 1 ) [ Δ P c 0 Δ ℓ c 0 ]
[ u v ] = [ u 0 v 0 ] + [ 1 S u − CΩ S u SΩ 0 1 S v SΩ ] [ Δ P n nx Δ P n nz ]
Δ P 3 3 x = [ 1 + R 1 λ 3 ( 1 − N 2 ) + R 2 ( N 1 − 1 ) ( q 1 + e 2 N 2 ) + q 1 e 2 ( N 1 − 1 ) ( 1 − N 2 ) R 1 R 2 ] Δ P c 0x
+ { λ 1 + λ 2 [ R 1 e 2 ( 1 − N 2 ) + R 2 ( N 1 − 1 ) ( q 1 + e 2 N 2 ) + q 1 e 2 ( N 1 − 1 ) ( 1 − N 2 ) R 1 R 2 ]
+ N 1 [ q 1 + e 2 N 2 + q 1 e 1 ( 1 − N 2 ) R 2 ] } Δ ℓ c 0x
Δ P 3 3 z = [ 1 + R 1 λ 3 ( 1 − N 2 ) + R 2 ( N 1 − 1 ) ( q 1 + e 2 N 2 ) + q 1 e 2 ( N 1 − 1 ) ( 1 − N 2 ) R 1 R 2 ] Δ P c 0z
+ { λ 1 + λ 2 [ R 1 e 2 ( 1 − N 2 ) + R 2 ( N 1 − 1 ) ( q 1 + e 2 N 2 ) + q 1 e 2 ( N 1 − 1 ) ( 1 − N 2 ) R 1 R 2 ]
+ N 1 [ q 1 + e 2 N 2 + q 1 e 1 ( 1 − N 2 ) R 2 ] } Δ ℓ c 0z
Rot ( x , ω x ) = [ 1 0 0 0 0 C ω x − S ω x 0 0 S ω x C ω x 0 0 0 0 1 ]
Rot ( y , ω y ) = [ C ω y 0 Sω y 0 0 1 0 0 − Sω y 0 C ω y 0 0 0 0 1 ]
Rot ( z , ω z ) = [ Cω z − Sω z 0 0 Sω z Cω z 0 0 0 0 1 0 0 0 0 1 ]
Trans ( t x , t y , t z ) = [ 1 0 0 t x 0 1 0 t y 0 0 1 t z 0 0 0 1 ]

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