Abstract

This Letter proposes a generalized unified model (GUM) for the calibration of noncentral catadioptric cameras. Releasing the constraint on the projection center and the orientation of the imaging plane that the traditional unified projection model has, the new model is able to well compensate the misalignment between the mirror and the camera. Being a compact and approximate central model, the GUM inherits the flexibility and simplicity from the unified model while maintaining accuracy even under severe misalignment. The calibration algorithm to compute the describing parameters of the model is also given. With the GUM, the calibration of central or noncentral systems could be treated with equal simplicity (or complexity). Experiments on both synthetic data and real images proved our success.

© 2013 Optical Society of America

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References

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  1. C. Geyer and K. Daniilidis, in 6th European Conference on Computer Vision (Springer, 2000), pp. 445–461.
  2. C. Mei and P. Rives, in International Conference on Robotics and Automation (IEEE, 2007), pp. 3945–3950.
  3. M. Grossberg and S. Nayar, Int. J. Comput. Vis. 61, 119 (2005).
    [CrossRef]
  4. N. Goncalves and H. Araújo, Comput. Vis. Image Underst. 113, 11 (2009).
    [CrossRef]
  5. Z. Xiang, B. Sun, and X. Dai, Sensors, 12, 7299 (2012).
    [CrossRef]

2012 (1)

Z. Xiang, B. Sun, and X. Dai, Sensors, 12, 7299 (2012).
[CrossRef]

2009 (1)

N. Goncalves and H. Araújo, Comput. Vis. Image Underst. 113, 11 (2009).
[CrossRef]

2005 (1)

M. Grossberg and S. Nayar, Int. J. Comput. Vis. 61, 119 (2005).
[CrossRef]

Araújo, H.

N. Goncalves and H. Araújo, Comput. Vis. Image Underst. 113, 11 (2009).
[CrossRef]

Dai, X.

Z. Xiang, B. Sun, and X. Dai, Sensors, 12, 7299 (2012).
[CrossRef]

Daniilidis, K.

C. Geyer and K. Daniilidis, in 6th European Conference on Computer Vision (Springer, 2000), pp. 445–461.

Geyer, C.

C. Geyer and K. Daniilidis, in 6th European Conference on Computer Vision (Springer, 2000), pp. 445–461.

Goncalves, N.

N. Goncalves and H. Araújo, Comput. Vis. Image Underst. 113, 11 (2009).
[CrossRef]

Grossberg, M.

M. Grossberg and S. Nayar, Int. J. Comput. Vis. 61, 119 (2005).
[CrossRef]

Mei, C.

C. Mei and P. Rives, in International Conference on Robotics and Automation (IEEE, 2007), pp. 3945–3950.

Nayar, S.

M. Grossberg and S. Nayar, Int. J. Comput. Vis. 61, 119 (2005).
[CrossRef]

Rives, P.

C. Mei and P. Rives, in International Conference on Robotics and Automation (IEEE, 2007), pp. 3945–3950.

Sun, B.

Z. Xiang, B. Sun, and X. Dai, Sensors, 12, 7299 (2012).
[CrossRef]

Xiang, Z.

Z. Xiang, B. Sun, and X. Dai, Sensors, 12, 7299 (2012).
[CrossRef]

Comput. Vis. Image Underst. (1)

N. Goncalves and H. Araújo, Comput. Vis. Image Underst. 113, 11 (2009).
[CrossRef]

Int. J. Comput. Vis. (1)

M. Grossberg and S. Nayar, Int. J. Comput. Vis. 61, 119 (2005).
[CrossRef]

Sensors (1)

Z. Xiang, B. Sun, and X. Dai, Sensors, 12, 7299 (2012).
[CrossRef]

Other (2)

C. Geyer and K. Daniilidis, in 6th European Conference on Computer Vision (Springer, 2000), pp. 445–461.

C. Mei and P. Rives, in International Conference on Robotics and Automation (IEEE, 2007), pp. 3945–3950.

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Figures (4)

Fig. 1.
Fig. 1.

Proposed generalized unified model (GUM).

Fig. 2.
Fig. 2.

Reprojection error after calibration under different misaligned configurations. (a) Lateral rotation and (b) lateral translation.

Fig. 3.
Fig. 3.

Calibration results with the hyperbolic mirror. (a) The position change of the projection center with respect to the rotation and (b) the reprojection error and the covariance with respect to the noise levels.

Fig. 4.
Fig. 4.

Calibration of the real system. (a) An example of the calibration image and (b) the reprojection error and the covariance under different levels of rotation misalignments.

Tables (2)

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Table 1. Standard Configuration of the Cameras

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Table 2. Computed Angle Errors

Equations (7)

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(Xw)Fm=Rm(Xw)w+Tm,
(Xs)Fm=(Xw)Fm|(Xw)Fm|=(xs,ys,zs)T.
mu=(xs+ξ1zs+ξ3,ys+ξ2zs+ξ3,1)T=h((Xs)Fm).
p=Kmu=[γ1γ1αu00γ2v0001]mu=k(mu),
L(ρ)=1+k1ρ2+k2ρ4,
F(V)=12i=1nj=1mi[G(V,gij)eij]2.
(Xs)Fm=(xs,ys,zs)T=h1(mu)=s[xy1212(x2+y2)],

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