Abstract

The linear camera equipped with cylindrical lenses has prominent advantages in high-precision coordinate measurement and dynamic position-tracking. However, the serious distortion of the cylindrical lenses limits the application of this camera. To overcome this obstacle, a precise two-step calibration method is developed. In the first step, a radial basis function-based (RBF-based) mapping technique is employed to recover the projection mapping of the imaging system by interpolating the correspondence between incident rays and image points. For an object point in 3D space, the plane passing through the object point in camera coordinate frame can be calculated accurately by this technique. The second step is the calibration of extrinsic parameters, which realizes the coordinate transformation from the camera coordinate frame to world coordinate frame. The proposed method has three aspects of advantage. Firstly, this method (black box calibration) is still effective even if the distortion is high and asymmetric. Secondly, the coupling between extrinsic parameters and other parameters, which is normally occurred and may lead to the failure of calibration, is avoided because this method simplifies the pinhole model and only extrinsic parameters are concerned in the simplified model. Thirdly, the nonlinear optimization, which is widely used to refine camera parameters, is better conditioned since fewer parameters are needed and more accurate initial iteration value is estimated. Both simulative and real experiments have been carried out and good results have been obtained.

© 2015 Optical Society of America

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References

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2013 (1)

2012 (2)

2011 (4)

2010 (3)

2009 (2)

N. Gonçalves and H. Araújo, “Estimating parameters of noncentral catadioptric systems using bundle adjustment,” Comput. Vis. Image Underst. 113(1), 11–28 (2009).
[Crossref]

W. Cuypers, N. Van Gestel, A. Voet, J.-P. Kruth, J. Mingneau, and P. Bleys, “Optical measurement techniques for mobile and large-scale dimensional metrology,” Opt. Lasers Eng. 47(3), 292–300 (2009).
[Crossref]

2008 (2)

2005 (1)

M. D. Grossberg and S. K. Nayar, “The raxel imaging model and ray-based calibration,” Int. J. Comput. Vis. 61(2), 119–137 (2005).
[Crossref]

2004 (1)

B. Fornberg and G. Wright, “Stable computation of multiquadric interpolants for all values of the shape parameter,” Comput. Math. Appl. 48(5), 853–867 (2004).
[Crossref]

2000 (2)

E. J. Kansa and Y. C. Hon, “Circumventing the ill-conditioning problem with multiquadric radial basis functions: applications to elliptic partial differential equations,” Comput. Math. Appl. 39(7), 123–137 (2000).
[Crossref]

Z. Zhang, “A flexible new technique for camera calibration,” IEEE Trans. Pattern Anal. Mach. Intell. 22(11), 1330–1334 (2000).
[Crossref]

1999 (1)

M. A. Golberg, C. S. Chen, and H. Bowman, “Some recent results and proposals for the use of radial basis functions in the BEM,” Eng. Anal. Bound. Elem. 23(4), 285–296 (1999).
[Crossref]

1992 (2)

F. Gazzani, “Performance of a 7-parameter DLT method for the calibration of stereo photogrammetric systems using 1-D transducers,” J. Biomed. Eng. 14(6), 476–482 (1992).
[Crossref] [PubMed]

J. Weng, P. Cohen, and M. Herniou, “Camera calibration with distortion models and accuracy evaluation,” IEEE Trans. Pattern Anal. Mach. Intell. 14(10), 965–980 (1992).
[Crossref]

1990 (1)

E. J. Kansa, “Multiquadrics—a scattered data approximation scheme with applications to computational fluid-dynamics—I surface approximations and partial derivative estimates,” Comput. Math. Appl. 19(8), 127–145 (1990).
[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. Robot. Autom. 3(4), 323–344 (1987).
[Crossref]

1983 (1)

V. Macellari, “CoSTEL: a computer peripheral remote sensing device for 3-dimensional monitoring of human motion,” Med. Biol. Eng. Comput. 21(3), 311–318 (1983).
[Crossref] [PubMed]

1982 (1)

R. Franke, “Scattered data interpolation: Tests of some methods,” Math. Comput. 38(157), 181–200 (1982).

Ai, L.

Araújo, H.

N. Gonçalves and H. Araújo, “Estimating parameters of noncentral catadioptric systems using bundle adjustment,” Comput. Vis. Image Underst. 113(1), 11–28 (2009).
[Crossref]

Bauer, A.

Bergmann, R. B.

Bleys, P.

W. Cuypers, N. Van Gestel, A. Voet, J.-P. Kruth, J. Mingneau, and P. Bleys, “Optical measurement techniques for mobile and large-scale dimensional metrology,” Opt. Lasers Eng. 47(3), 292–300 (2009).
[Crossref]

Bothe, T.

Bowman, H.

M. A. Golberg, C. S. Chen, and H. Bowman, “Some recent results and proposals for the use of radial basis functions in the BEM,” Eng. Anal. Bound. Elem. 23(4), 285–296 (1999).
[Crossref]

Cakmakci, O.

Chen, C. S.

M. A. Golberg, C. S. Chen, and H. Bowman, “Some recent results and proposals for the use of radial basis functions in the BEM,” Eng. Anal. Bound. Elem. 23(4), 285–296 (1999).
[Crossref]

Cohen, P.

J. Weng, P. Cohen, and M. Herniou, “Camera calibration with distortion models and accuracy evaluation,” IEEE Trans. Pattern Anal. Mach. Intell. 14(10), 965–980 (1992).
[Crossref]

Colkesen, I.

Cuypers, W.

W. Cuypers, N. Van Gestel, A. Voet, J.-P. Kruth, J. Mingneau, and P. Bleys, “Optical measurement techniques for mobile and large-scale dimensional metrology,” Opt. Lasers Eng. 47(3), 292–300 (2009).
[Crossref]

Dai, X.

Ding, Z.

Draréni, J.

J. Draréni, S. Roy, and P. Sturm, “Plane-based calibration for linear cameras,” Int. J. Comput. Vis. 91(2), 146–156 (2011).
[Crossref]

Ergun, B.

Fornberg, B.

B. Fornberg and G. Wright, “Stable computation of multiquadric interpolants for all values of the shape parameter,” Comput. Math. Appl. 48(5), 853–867 (2004).
[Crossref]

Foroosh, H.

Franke, R.

R. Franke, “Scattered data interpolation: Tests of some methods,” Math. Comput. 38(157), 181–200 (1982).

Gazzani, F.

F. Gazzani, “Performance of a 7-parameter DLT method for the calibration of stereo photogrammetric systems using 1-D transducers,” J. Biomed. Eng. 14(6), 476–482 (1992).
[Crossref] [PubMed]

Golberg, M. A.

M. A. Golberg, C. S. Chen, and H. Bowman, “Some recent results and proposals for the use of radial basis functions in the BEM,” Eng. Anal. Bound. Elem. 23(4), 285–296 (1999).
[Crossref]

Gonçalves, N.

N. Gonçalves and H. Araújo, “Estimating parameters of noncentral catadioptric systems using bundle adjustment,” Comput. Vis. Image Underst. 113(1), 11–28 (2009).
[Crossref]

Gong, X.

Grossberg, M. D.

M. D. Grossberg and S. K. Nayar, “The raxel imaging model and ray-based calibration,” Int. J. Comput. Vis. 61(2), 119–137 (2005).
[Crossref]

Herniou, M.

J. Weng, P. Cohen, and M. Herniou, “Camera calibration with distortion models and accuracy evaluation,” IEEE Trans. Pattern Anal. Mach. Intell. 14(10), 965–980 (1992).
[Crossref]

Hon, Y. C.

E. J. Kansa and Y. C. Hon, “Circumventing the ill-conditioning problem with multiquadric radial basis functions: applications to elliptic partial differential equations,” Comput. Math. Appl. 39(7), 123–137 (2000).
[Crossref]

Jüptner, W. P.

Kansa, E. J.

E. J. Kansa and Y. C. Hon, “Circumventing the ill-conditioning problem with multiquadric radial basis functions: applications to elliptic partial differential equations,” Comput. Math. Appl. 39(7), 123–137 (2000).
[Crossref]

E. J. Kansa, “Multiquadrics—a scattered data approximation scheme with applications to computational fluid-dynamics—I surface approximations and partial derivative estimates,” Comput. Math. Appl. 19(8), 127–145 (1990).
[Crossref]

Kavzoglu, T.

Kruth, J.-P.

W. Cuypers, N. Van Gestel, A. Voet, J.-P. Kruth, J. Mingneau, and P. Bleys, “Optical measurement techniques for mobile and large-scale dimensional metrology,” Opt. Lasers Eng. 47(3), 292–300 (2009).
[Crossref]

Li, W.

Li, Y.

Li, Y. J.

Luma, C. A.

C. A. Luma and M. Mazo, “Calibration of line-scan cameras,” IEEE Trans. on Inst. Meas. 59(8), 2185–2190 (2010).

Macellari, V.

V. Macellari, “CoSTEL: a computer peripheral remote sensing device for 3-dimensional monitoring of human motion,” Med. Biol. Eng. Comput. 21(3), 311–318 (1983).
[Crossref] [PubMed]

Martinez-Berti, E.

Mazo, M.

C. A. Luma and M. Mazo, “Calibration of line-scan cameras,” IEEE Trans. on Inst. Meas. 59(8), 2185–2190 (2010).

Mingneau, J.

W. Cuypers, N. Van Gestel, A. Voet, J.-P. Kruth, J. Mingneau, and P. Bleys, “Optical measurement techniques for mobile and large-scale dimensional metrology,” Opt. Lasers Eng. 47(3), 292–300 (2009).
[Crossref]

Nayar, S. K.

M. D. Grossberg and S. K. Nayar, “The raxel imaging model and ray-based calibration,” Int. J. Comput. Vis. 61(2), 119–137 (2005).
[Crossref]

Parkins, K.

Ricolfe-Viala, C.

Rodriguez, F.

Rolland, J.

Rolland, J. P.

Roy, S.

J. Draréni, S. Roy, and P. Sturm, “Plane-based calibration for linear cameras,” Int. J. Comput. Vis. 91(2), 146–156 (2011).
[Crossref]

Sahin, C.

Sanchez-Salmeron, A. J.

Schulte, M.

Sturm, P.

J. Draréni, S. Roy, and P. Sturm, “Plane-based calibration for linear cameras,” Int. J. Comput. Vis. 91(2), 146–156 (2011).
[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 J. Robot. Autom. 3(4), 323–344 (1987).
[Crossref]

Van Gestel, N.

W. Cuypers, N. Van Gestel, A. Voet, J.-P. Kruth, J. Mingneau, and P. Bleys, “Optical measurement techniques for mobile and large-scale dimensional metrology,” Opt. Lasers Eng. 47(3), 292–300 (2009).
[Crossref]

Vo, S.

Voet, A.

W. Cuypers, N. Van Gestel, A. Voet, J.-P. Kruth, J. Mingneau, and P. Bleys, “Optical measurement techniques for mobile and large-scale dimensional metrology,” Opt. Lasers Eng. 47(3), 292–300 (2009).
[Crossref]

von Kopylow, C.

Wang, G.

Weng, J.

J. Weng, P. Cohen, and M. Herniou, “Camera calibration with distortion models and accuracy evaluation,” IEEE Trans. Pattern Anal. Mach. Intell. 14(10), 965–980 (1992).
[Crossref]

Wright, G.

B. Fornberg and G. Wright, “Stable computation of multiquadric interpolants for all values of the shape parameter,” Comput. Math. Appl. 48(5), 853–867 (2004).
[Crossref]

Xiang, Z.

Yuan, F.

Zhang, Z.

Z. Zhang, “A flexible new technique for camera calibration,” IEEE Trans. Pattern Anal. Mach. Intell. 22(11), 1330–1334 (2000).
[Crossref]

Zhou, H. C.

Appl. Opt. (3)

Chin. Opt. Lett. (1)

Comput. Math. Appl. (3)

E. J. Kansa, “Multiquadrics—a scattered data approximation scheme with applications to computational fluid-dynamics—I surface approximations and partial derivative estimates,” Comput. Math. Appl. 19(8), 127–145 (1990).
[Crossref]

E. J. Kansa and Y. C. Hon, “Circumventing the ill-conditioning problem with multiquadric radial basis functions: applications to elliptic partial differential equations,” Comput. Math. Appl. 39(7), 123–137 (2000).
[Crossref]

B. Fornberg and G. Wright, “Stable computation of multiquadric interpolants for all values of the shape parameter,” Comput. Math. Appl. 48(5), 853–867 (2004).
[Crossref]

Comput. Vis. Image Underst. (1)

N. Gonçalves and H. Araújo, “Estimating parameters of noncentral catadioptric systems using bundle adjustment,” Comput. Vis. Image Underst. 113(1), 11–28 (2009).
[Crossref]

Eng. Anal. Bound. Elem. (1)

M. A. Golberg, C. S. Chen, and H. Bowman, “Some recent results and proposals for the use of radial basis functions in the BEM,” Eng. Anal. Bound. Elem. 23(4), 285–296 (1999).
[Crossref]

IEEE J. Robot. 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. Robot. Autom. 3(4), 323–344 (1987).
[Crossref]

IEEE Trans. on Inst. Meas. (1)

C. A. Luma and M. Mazo, “Calibration of line-scan cameras,” IEEE Trans. on Inst. Meas. 59(8), 2185–2190 (2010).

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

Z. Zhang, “A flexible new technique for camera calibration,” IEEE Trans. Pattern Anal. Mach. Intell. 22(11), 1330–1334 (2000).
[Crossref]

J. Weng, P. Cohen, and M. Herniou, “Camera calibration with distortion models and accuracy evaluation,” IEEE Trans. Pattern Anal. Mach. Intell. 14(10), 965–980 (1992).
[Crossref]

Int. J. Comput. Vis. (2)

M. D. Grossberg and S. K. Nayar, “The raxel imaging model and ray-based calibration,” Int. J. Comput. Vis. 61(2), 119–137 (2005).
[Crossref]

J. Draréni, S. Roy, and P. Sturm, “Plane-based calibration for linear cameras,” Int. J. Comput. Vis. 91(2), 146–156 (2011).
[Crossref]

J. Biomed. Eng. (1)

F. Gazzani, “Performance of a 7-parameter DLT method for the calibration of stereo photogrammetric systems using 1-D transducers,” J. Biomed. Eng. 14(6), 476–482 (1992).
[Crossref] [PubMed]

J. Opt. Soc. Am. A (1)

Math. Comput. (1)

R. Franke, “Scattered data interpolation: Tests of some methods,” Math. Comput. 38(157), 181–200 (1982).

Med. Biol. Eng. Comput. (1)

V. Macellari, “CoSTEL: a computer peripheral remote sensing device for 3-dimensional monitoring of human motion,” Med. Biol. Eng. Comput. 21(3), 311–318 (1983).
[Crossref] [PubMed]

Opt. Express (3)

Opt. Lasers Eng. (1)

W. Cuypers, N. Van Gestel, A. Voet, J.-P. Kruth, J. Mingneau, and P. Bleys, “Optical measurement techniques for mobile and large-scale dimensional metrology,” Opt. Lasers Eng. 47(3), 292–300 (2009).
[Crossref]

Opt. Lett. (2)

Other (2)

P. Sturm and S. Ramalingam, “A generic concept for camera calibration,” in Computer Vision-ECCV 2004 (Springer, 2004), pp. 1-13.

Basler, “Basler vision technologies,” (2014). http://www.baslerweb.com/ .

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Figures (10)

Fig. 1
Fig. 1 Optical layout of the camera lens with a 40 × 40 deg FOV and f-number/19. The focal length is 38mm. The position of the aperture stop (AS) is indicated above. (a) YZ view, (b) XZ view.
Fig. 2
Fig. 2 (a) Projection mapping model of the LCEWCL. (b) Lens distortion distribution.
Fig. 3
Fig. 3 Schematic of setting up the correspondence ( α i , β i ) u i for i th sample point.
Fig. 4
Fig. 4 Schematic of 3D coordinate measurement based on multi-plane constraint.
Fig. 5
Fig. 5 (a) Set of sample points and test points corresponding to the red points and blue points, respectively. Not all points are drawn for clarity; (b) Angle mapping error for the test points.
Fig. 6
Fig. 6 The error of RBF-based mapping with respect to different configurations by computer simulation: (a) N sample and (b) χ .
Fig. 7
Fig. 7 Experimental platform for setting up the correspondence. (1) linear camera, (2) laser tracker, (3) light spot, (4) 1D rotation stage, (5) 6D adjusting stage, (6) motion stage, (7) SMR.
Fig. 8
Fig. 8 The error of RBF-based mapping with respect to different configurations by real data: (a) N sample and (b) χ .
Fig. 9
Fig. 9 Photo of 3D coordinate measurement system. Measurement results are compared with the laser tracker. (1) camera 1, (2) camera 2, (3) camera 3, (4) light spot, (5) laser tracker.
Fig. 10
Fig. 10 3D coordinate error distribution for 60 measurement points using different calibration methods: 7DLT method and the proposed method. (a)-(c): the 3D coordinate measurement error along X, Y and Z directions, respectively.

Tables (1)

Tables Icon

Table 1 Summary of statistic results for 3D coordinate measurement error using different calibration methods. Coordinate discrepancies are shown as maximum values.

Equations (22)

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

z(x)= i=1 n w i ϕ i (x)+ j=1 m c j p j (x)= A T (x)W+ P T (x)C,
W T =[ w 1 , w 2 ,, w n ], C T =[ c 1 , c 2 ,, c m ], A T (x)=[ ϕ 1 (x), ϕ 2 (x),, ϕ n (x)], P T (x)=[ p 1 (x), p 2 (x),, p m (x)].
ϕ i (x)= ϕ i ( r i )= ϕ i (x,y),
r i = [ (x x i ) 2 + (y y i ) 2 ] 1/2 .
P T (x)=[1,x,y, x 2 ,xy, y 2 ,].
f k =z( x k , y k )= i=1 n w i ϕ i ( x k , y k )+ j=1 m c j p j ( x k , y k ),k=1,2,,n.
i=1 n w i p j ( x i , y i )=0,j=1,2,,m.
[ A P P T 0 ][ W C ]=[ Z 0 ],
Z= [ f 1 , f 2 , f 3 , f n ] T ,
A=[ ϕ 1 ( x 1 , y 1 ) ϕ 2 ( x 1 , y 1 ) ϕ n ( x 1 , y 1 ) ϕ 1 ( x 2 , y 2 ) ϕ 2 ( x 2 , y 2 ) ϕ n ( x 2 , y 2 ) ϕ 1 ( x n , y n ) ϕ 2 ( x n , y n ) ϕ n ( x n , y n ) ],
P=[ p 1 ( x 1 , y 1 ) p 2 ( x 1 , y 1 ) p m ( x 1 , y 1 ) p 1 ( x 2 , y 2 ) p 2 ( x 2 , y 2 ) p m ( x 2 , y 2 ) p 1 ( x n , y n ) p 2 ( x n , y n ) p m ( x n , y n ) ].
[ W C ]= [ A P P T 0 ] 1 [ Z 0 ].
α(x)= i=1 N sample w i ϕ i (x)+ j=1 3 c j p j (x)= A T (x)W+ P T (x)C,x=(u,β)
ϕ i (x)= ( r 2 + χ 2 ) 1/2 = [ (u u i ) 2 +(β β i ) 2 + χ 2 ] 1/2 ,
[ x c1 y c1 z c1 ]=[ R t ][ X Y Z 1 ]=[ r 11 r 12 r 13 t 1 r 21 r 22 r 23 t 2 r 31 r 32 r 33 t 3 ][ X Y Z 1 ],
tanα= x c1 / z c1 .
tanα= r 11 X+ r 12 Y+ r 13 Z+ t 1 r 31 X+ r 32 Y+ r 33 Z+ t 3 ,
F i = X i r 11 + Y i r 12 + Z i r 13 + t 1 tan α i X i r 31 tan α i Y i r 32 tan α i Z i r 33 tan α i t 3 =0.
{ f 1 = r 11 2 + r 12 2 + r 13 2 1=0 f 2 = r 31 2 + r 32 2 + r 33 2 1=0 f 3 = r 11 r 31 + r 12 r 32 + r 13 r 33 =0 .
E= i=1 n F i 2 +M j=1 3 f j 2 ,
AX=0,
A=[ X 1 Y 1 Z 1 1 tan α 1 X 1 tan α 1 Y 1 tan α 1 Z 1 tan α 1 X 2 Y 2 Z 2 1 tan α 2 X 2 tan α 2 Y 2 tan α 2 Z 2 tan α 2 X n Y n Z n 1 tan α n X n tan α n Y n tan α n Z n tan α n ]

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