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Camera calibration based on paraxial raytracing

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Abstract

Traditional camera calibration is based on the pinhole model, which is an approximation algorithm using untrue geometrical assumptions and giving a single lumped result for the multiple optical elements in a camera. To provide an alternative method of camera calibration, we extend the traditional 2×2 matrix-based paraxial raytracing method to 6×6 in order to trace paraxial rays by using the first-order Taylor series expansion of Snell’s laws. Then we establish the geometric relationship between images and objects. Compared with the Snell’s Law camera calibration model of our previous work, the paraxial model offers explicit analytical sensitivity analysis for the mathematical manipulation of problematical conditions. Compared with the existing pinhole model, the proposed method, in addition to five intrinsic and six extrinsic parameters, gives the position parameters of each optical element of the camera system.

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Abbreviations

(xyz)w :

world coordinate frame

(xyz)0 :

camera coordinate frame

j :

number to label optical elements, j=1,2,…,k

i :

number to label boundary surface, i=1,2,…,n (=2k−1)

(xyz) i :

coordinate frame embedded in the ith boundary surface

r i :

ith boundary surface with unit normal n i

P i :

incidence point at r i

ℓ i :

refracted unit directional vector of a light ray at r i

j A i :

pose matrix of frame (xyz) i with respect to frame (xyz) j

ξ i :

the refractive index of medium i

N i =ξ i−1/ξ i :

the relative refractive index across the boundary surface r i

E ex=[t x t y t z ω x ω y ω z ]T :

vector of 6 extrinsic parameters

q j :

thickness of element j (j=1,2,…,k)

e j :

position parameter of element j (j=1,2,…,k)

E in=[u 0 v 0 1/s u 1/s v Ω]T :

vector of 5 intrinsic parameters

E total=[E ex e 2 e 3 ⋅⋅⋅ e k E in]T :

vector of 10+k parameters of a camera

References

  1. P.D. Lin, T.T. Liao, J. Dyn. Syst. Meas. Control-Trans. ASME 126, 102–114 (2004)

    Article  Google Scholar 

  2. D. Murray, A. Basu, Trans. Pattern Anal. Mach. Intell. IEEE 16, 449–459 (1994)

    Article  Google Scholar 

  3. C.J. Taylor, D.J. Kriegman, Trans. Robot. Autom. IEEE 14, 417–426 (1998)

    Article  Google Scholar 

  4. Li. Wenjing, S.T. Marcelo, G. Ulf, Opt. Express 14, 12887–12901 (2006)

    Article  Google Scholar 

  5. B. Baumann, E. Götzinger, M. Pircher, C.K. Hitzenberger, Opt. Express 15, 1054–1063 (2007)

    Article  ADS  Google Scholar 

  6. Y.I. Abdel-Aziz, H.M. Karara, in Proc. Symp. Close-Range Photo (Am. Soc. Photogr., Falls Church, 1971), pp. 1–18

    Google Scholar 

  7. H. Hatze, J. Biomech. 21, 533–538 (1988)

    Article  Google Scholar 

  8. R. Lenz, R. Tsai, Trans. Pattern Anal. Mach. Intell. IEEE PAMI 10(5), 713–720 (1988)

    Article  Google Scholar 

  9. J. Weng, P. Cohen, M. Herniou, Trans. Pattern Anal. Mach. Intell. IEEE 14, 965–980 (1992)

    Article  Google Scholar 

  10. L. Ma, Y.Q. Chen, K.L. Morre, in Proceedings of the IEEE International Symposium on Intelligence Control, 2003, pp. 799–804

  11. Z. Zhang, in IEEE International Conference on Computer Vision, 1999, pp. 666–673

  12. F.Y. Wang, IEEE Trans. Syst. Man Cybern. 35, 453–464 (2005)

    Article  Google Scholar 

  13. F.Y. Wang, Trans. Robot. Autom. IEEE 20, 121–124 (2004)

    Article  Google Scholar 

  14. P.D. Lin, C.K. Sung, Opt. Express 15, 3012–3022 (2007)

    Article  ADS  Google Scholar 

  15. P.D. Lin, C.K. Sung, OPTIK 117, 329–340 (2006)

    ADS  Google Scholar 

  16. R.P. Paul, Robot Manipulators. Mathematics, Programming and Control (MIT Press, Cambridge, 1982)

    Google Scholar 

  17. P.D. Lin, C.K. Sung, J. Dyn. Syst. Meas. Control-Trans. ASME 128, 548–557 (2006)

    Article  Google Scholar 

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Authors and Affiliations

  1. Department of Electrical Engineering, Republic of China Military Academy, Kaohsiung, Taiwan, 83059

    C.-H. Lu & C.-K. Sung

  2. Department of Mechanical Engineering, National Cheng Kung University, Tainan, Taiwan, 70101

    P. D. Lin

Authors
  1. C.-H. Lu
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  2. P. D. Lin
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  3. C.-K. Sung
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Correspondence to C.-H. Lu.

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Lu, CH., Lin, P.D. & Sung, CK. Camera calibration based on paraxial raytracing. Appl. Phys. B 94, 307–317 (2009). https://doi.org/10.1007/s00340-008-3311-0

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  • DOI: https://doi.org/10.1007/s00340-008-3311-0

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