Abstract

The majority of computer vision applications assumes that the camera adheres to the pinhole camera model. However, most optical systems will introduce undesirable effects. By far, the most evident of these effects is radial lensing, which is particularly noticeable in fish-eye camera systems, where the effect is relatively extreme. Several authors have developed models of fish-eye lenses that can be used to describe the fish-eye displacement. Our aim is to evaluate the accuracy of several of these models. Thus, we present a method by which the lens curve of a fish-eye camera can be extracted using well-founded assumptions and perspective methods. Several of the models from the literature are examined against this empirically derived curve.

© 2010 Optical Society of America

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References

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  1. D. C. Brown, “Decentering distortion of lenses,” Photograph. Eng. 32, 444–462 (1966).
  2. R. Tsai, “A versatile camera calibration technique for high-accuracy 3D machine vision metrology using off-the-shelf TV cameras and lenses,” IEEE Trans. Robot. Automat. 3, 323–344 (1987).
    [CrossRef]
  3. Z. Zhang, “A flexible new technique for camera calibration,” IEEE Trans. Pattern Anal. Mach. Intell. 22, 1330–1334 (2000).
    [CrossRef]
  4. A. Basu and S. Licardie, “Alternative models for fish-eye lenses,” Pattern Recogn. Lett. 16, 433–441 (1995).
    [CrossRef]
  5. F. Devernay and O. Faugeras, “Straight lines have to be straight: automatic calibration and removal of distortion from scenes of structured environments,” Mach. Vis. Appl. 13, 14–24 (2001).
    [CrossRef]
  6. C. Bräuer-Burchardt and K. Voss, “A new algorithm to correct fish-eye- and strong wide-angle-lens-distortion from single images,” in Proceedings of the IEEE International Conference on Image Processing (IEEE, 2001), pp. 225–228.
  7. A. W. Fitzgibbon, “Simultaneous linear estimation of multiple view geometry and lens distortion,” in Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition (IEEE, 2001), pp. 125–132.
  8. D. Schneider, E. Schwalbe, and H.-G. Maas, “Validation of geometric models for fisheye lenses,” ISPRS J. Photogramm. Remote Sens. 64, 259–266 (2009).
    [CrossRef]
  9. C. Hughes, R. McFeely, P. Denny, M. Glavin, and E. Jones, “Equidistant (fθ) fish-eye perspective with application in distortion centre estimation,” Image Vis. Comput. 28, 538–551(2010).
    [CrossRef]
  10. R. I. Hartley and S. B. Kang, “Parameter-free radial distortion correction with center of distortion estimation,” IEEE Trans. Pattern Anal. Mach. Intell. 29, 1309–1321 (2007).
    [CrossRef] [PubMed]
  11. K. V. Asari, “Design of an efficient VLSI architecture for non-linear spatial warping of wide-angle camera images,” J. Syst. Architect. 50, 743–755 (2004).
    [CrossRef]
  12. J. Weng, P. Cohen, and M. Herniou, “Camera calibration with distortion models and accuracy evaluation,” IEEE Trans. Pattern Anal. Mach. Intell. 14, 965–980 (1992).
    [CrossRef]
  13. G. P. Stein, “Internal camera calibration using rotation and geometric shapes,” M.S. thesis (MIT, 1993).
  14. R. I. Hartley and A. Zisserman, Multiple View Geometry in Computer Vision, 2nd ed. (Cambridge U. Press, 2004).
    [CrossRef]
  15. G. Xu, J. Terai, and H.-Y. Shum, “A linear algorithm for camera self-calibration, motion and structure recovery for multi-planar scenes from two perspective images,” in Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition (IEEE, 2000), pp. 474–479.
  16. K. Miyamoto, “Fish eye lens,” J. Opt. Soc. Am. 54, 1060–1061 (1964).
    [CrossRef]
  17. S. Shah and J. K. Aggarwal, “Intrinsic parameter calibration procedure for a (high-distortion) fish-eye lens camera with distortion model and accuracy estimation,” Pattern Recogn. 29, 1775–1788 (1996).
    [CrossRef]
  18. B. B. Jähne, Digital Image Processing, 5th ed. (Springer-Verlag, 2002), Chap. 12.
  19. Z. Wang, W. Wu, X. Xu, and D. Xue, “Recognition and location of the internal corners of planar checkerboard calibration pattern image,” Appl. Math. Comput. 185, 894–906 (2007).
    [CrossRef]
  20. D. W. Marquardt, “An algorithm for least-squares estimation of nonlinear parameters,” SIAM J. Appl. Math. 11, 431–441(1963).
    [CrossRef]
  21. W. S. Cleveland and S. J. Devlin, “Locally weighted regression: an approach to regression analysis by local fitting,” J. Am. Stat. Assoc. 83, 596–610 (1988).
    [CrossRef]
  22. D. G. Lowe, “Object recognition from local scale-invariant features,” Proceedings of the IEEE International Conference on Computer Vision (IEEE, 1999), pp. 1150–1157.
    [CrossRef]

2010 (1)

C. Hughes, R. McFeely, P. Denny, M. Glavin, and E. Jones, “Equidistant (fθ) fish-eye perspective with application in distortion centre estimation,” Image Vis. Comput. 28, 538–551(2010).
[CrossRef]

2009 (1)

D. Schneider, E. Schwalbe, and H.-G. Maas, “Validation of geometric models for fisheye lenses,” ISPRS J. Photogramm. Remote Sens. 64, 259–266 (2009).
[CrossRef]

2007 (2)

Z. Wang, W. Wu, X. Xu, and D. Xue, “Recognition and location of the internal corners of planar checkerboard calibration pattern image,” Appl. Math. Comput. 185, 894–906 (2007).
[CrossRef]

R. I. Hartley and S. B. Kang, “Parameter-free radial distortion correction with center of distortion estimation,” IEEE Trans. Pattern Anal. Mach. Intell. 29, 1309–1321 (2007).
[CrossRef] [PubMed]

2004 (1)

K. V. Asari, “Design of an efficient VLSI architecture for non-linear spatial warping of wide-angle camera images,” J. Syst. Architect. 50, 743–755 (2004).
[CrossRef]

2001 (1)

F. Devernay and O. Faugeras, “Straight lines have to be straight: automatic calibration and removal of distortion from scenes of structured environments,” Mach. Vis. Appl. 13, 14–24 (2001).
[CrossRef]

2000 (1)

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

1996 (1)

S. Shah and J. K. Aggarwal, “Intrinsic parameter calibration procedure for a (high-distortion) fish-eye lens camera with distortion model and accuracy estimation,” Pattern Recogn. 29, 1775–1788 (1996).
[CrossRef]

1995 (1)

A. Basu and S. Licardie, “Alternative models for fish-eye lenses,” Pattern Recogn. Lett. 16, 433–441 (1995).
[CrossRef]

1992 (1)

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

1988 (1)

W. S. Cleveland and S. J. Devlin, “Locally weighted regression: an approach to regression analysis by local fitting,” J. Am. Stat. Assoc. 83, 596–610 (1988).
[CrossRef]

1987 (1)

R. Tsai, “A versatile camera calibration technique for high-accuracy 3D machine vision metrology using off-the-shelf TV cameras and lenses,” IEEE Trans. Robot. Automat. 3, 323–344 (1987).
[CrossRef]

1966 (1)

D. C. Brown, “Decentering distortion of lenses,” Photograph. Eng. 32, 444–462 (1966).

1964 (1)

1963 (1)

D. W. Marquardt, “An algorithm for least-squares estimation of nonlinear parameters,” SIAM J. Appl. Math. 11, 431–441(1963).
[CrossRef]

Aggarwal, J. K.

S. Shah and J. K. Aggarwal, “Intrinsic parameter calibration procedure for a (high-distortion) fish-eye lens camera with distortion model and accuracy estimation,” Pattern Recogn. 29, 1775–1788 (1996).
[CrossRef]

Asari, K. V.

K. V. Asari, “Design of an efficient VLSI architecture for non-linear spatial warping of wide-angle camera images,” J. Syst. Architect. 50, 743–755 (2004).
[CrossRef]

Basu, A.

A. Basu and S. Licardie, “Alternative models for fish-eye lenses,” Pattern Recogn. Lett. 16, 433–441 (1995).
[CrossRef]

Bräuer-Burchardt, C.

C. Bräuer-Burchardt and K. Voss, “A new algorithm to correct fish-eye- and strong wide-angle-lens-distortion from single images,” in Proceedings of the IEEE International Conference on Image Processing (IEEE, 2001), pp. 225–228.

Brown, D. C.

D. C. Brown, “Decentering distortion of lenses,” Photograph. Eng. 32, 444–462 (1966).

Cleveland, W. S.

W. S. Cleveland and S. J. Devlin, “Locally weighted regression: an approach to regression analysis by local fitting,” J. Am. Stat. Assoc. 83, 596–610 (1988).
[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, 965–980 (1992).
[CrossRef]

Denny, P.

C. Hughes, R. McFeely, P. Denny, M. Glavin, and E. Jones, “Equidistant (fθ) fish-eye perspective with application in distortion centre estimation,” Image Vis. Comput. 28, 538–551(2010).
[CrossRef]

Devernay, F.

F. Devernay and O. Faugeras, “Straight lines have to be straight: automatic calibration and removal of distortion from scenes of structured environments,” Mach. Vis. Appl. 13, 14–24 (2001).
[CrossRef]

Devlin, S. J.

W. S. Cleveland and S. J. Devlin, “Locally weighted regression: an approach to regression analysis by local fitting,” J. Am. Stat. Assoc. 83, 596–610 (1988).
[CrossRef]

Faugeras, O.

F. Devernay and O. Faugeras, “Straight lines have to be straight: automatic calibration and removal of distortion from scenes of structured environments,” Mach. Vis. Appl. 13, 14–24 (2001).
[CrossRef]

Fitzgibbon, A. W.

A. W. Fitzgibbon, “Simultaneous linear estimation of multiple view geometry and lens distortion,” in Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition (IEEE, 2001), pp. 125–132.

Glavin, M.

C. Hughes, R. McFeely, P. Denny, M. Glavin, and E. Jones, “Equidistant (fθ) fish-eye perspective with application in distortion centre estimation,” Image Vis. Comput. 28, 538–551(2010).
[CrossRef]

Hartley, R. I.

R. I. Hartley and S. B. Kang, “Parameter-free radial distortion correction with center of distortion estimation,” IEEE Trans. Pattern Anal. Mach. Intell. 29, 1309–1321 (2007).
[CrossRef] [PubMed]

R. I. Hartley and A. Zisserman, Multiple View Geometry in Computer Vision, 2nd ed. (Cambridge U. Press, 2004).
[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, 965–980 (1992).
[CrossRef]

Hughes, C.

C. Hughes, R. McFeely, P. Denny, M. Glavin, and E. Jones, “Equidistant (fθ) fish-eye perspective with application in distortion centre estimation,” Image Vis. Comput. 28, 538–551(2010).
[CrossRef]

Jähne, B. B.

B. B. Jähne, Digital Image Processing, 5th ed. (Springer-Verlag, 2002), Chap. 12.

Jones, E.

C. Hughes, R. McFeely, P. Denny, M. Glavin, and E. Jones, “Equidistant (fθ) fish-eye perspective with application in distortion centre estimation,” Image Vis. Comput. 28, 538–551(2010).
[CrossRef]

Kang, S. B.

R. I. Hartley and S. B. Kang, “Parameter-free radial distortion correction with center of distortion estimation,” IEEE Trans. Pattern Anal. Mach. Intell. 29, 1309–1321 (2007).
[CrossRef] [PubMed]

Licardie, S.

A. Basu and S. Licardie, “Alternative models for fish-eye lenses,” Pattern Recogn. Lett. 16, 433–441 (1995).
[CrossRef]

Lowe, D. G.

D. G. Lowe, “Object recognition from local scale-invariant features,” Proceedings of the IEEE International Conference on Computer Vision (IEEE, 1999), pp. 1150–1157.
[CrossRef]

Maas, H.-G.

D. Schneider, E. Schwalbe, and H.-G. Maas, “Validation of geometric models for fisheye lenses,” ISPRS J. Photogramm. Remote Sens. 64, 259–266 (2009).
[CrossRef]

Marquardt, D. W.

D. W. Marquardt, “An algorithm for least-squares estimation of nonlinear parameters,” SIAM J. Appl. Math. 11, 431–441(1963).
[CrossRef]

McFeely, R.

C. Hughes, R. McFeely, P. Denny, M. Glavin, and E. Jones, “Equidistant (fθ) fish-eye perspective with application in distortion centre estimation,” Image Vis. Comput. 28, 538–551(2010).
[CrossRef]

Miyamoto, K.

Schneider, D.

D. Schneider, E. Schwalbe, and H.-G. Maas, “Validation of geometric models for fisheye lenses,” ISPRS J. Photogramm. Remote Sens. 64, 259–266 (2009).
[CrossRef]

Schwalbe, E.

D. Schneider, E. Schwalbe, and H.-G. Maas, “Validation of geometric models for fisheye lenses,” ISPRS J. Photogramm. Remote Sens. 64, 259–266 (2009).
[CrossRef]

Shah, S.

S. Shah and J. K. Aggarwal, “Intrinsic parameter calibration procedure for a (high-distortion) fish-eye lens camera with distortion model and accuracy estimation,” Pattern Recogn. 29, 1775–1788 (1996).
[CrossRef]

Shum, H.-Y.

G. Xu, J. Terai, and H.-Y. Shum, “A linear algorithm for camera self-calibration, motion and structure recovery for multi-planar scenes from two perspective images,” in Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition (IEEE, 2000), pp. 474–479.

Stein, G. P.

G. P. Stein, “Internal camera calibration using rotation and geometric shapes,” M.S. thesis (MIT, 1993).

Terai, J.

G. Xu, J. Terai, and H.-Y. Shum, “A linear algorithm for camera self-calibration, motion and structure recovery for multi-planar scenes from two perspective images,” in Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition (IEEE, 2000), pp. 474–479.

Tsai, R.

R. Tsai, “A versatile camera calibration technique for high-accuracy 3D machine vision metrology using off-the-shelf TV cameras and lenses,” IEEE Trans. Robot. Automat. 3, 323–344 (1987).
[CrossRef]

Voss, K.

C. Bräuer-Burchardt and K. Voss, “A new algorithm to correct fish-eye- and strong wide-angle-lens-distortion from single images,” in Proceedings of the IEEE International Conference on Image Processing (IEEE, 2001), pp. 225–228.

Wang, Z.

Z. Wang, W. Wu, X. Xu, and D. Xue, “Recognition and location of the internal corners of planar checkerboard calibration pattern image,” Appl. Math. Comput. 185, 894–906 (2007).
[CrossRef]

Weng, J.

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

Wu, W.

Z. Wang, W. Wu, X. Xu, and D. Xue, “Recognition and location of the internal corners of planar checkerboard calibration pattern image,” Appl. Math. Comput. 185, 894–906 (2007).
[CrossRef]

Xu, G.

G. Xu, J. Terai, and H.-Y. Shum, “A linear algorithm for camera self-calibration, motion and structure recovery for multi-planar scenes from two perspective images,” in Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition (IEEE, 2000), pp. 474–479.

Xu, X.

Z. Wang, W. Wu, X. Xu, and D. Xue, “Recognition and location of the internal corners of planar checkerboard calibration pattern image,” Appl. Math. Comput. 185, 894–906 (2007).
[CrossRef]

Xue, D.

Z. Wang, W. Wu, X. Xu, and D. Xue, “Recognition and location of the internal corners of planar checkerboard calibration pattern image,” Appl. Math. Comput. 185, 894–906 (2007).
[CrossRef]

Zhang, Z.

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

Zisserman, A.

R. I. Hartley and A. Zisserman, Multiple View Geometry in Computer Vision, 2nd ed. (Cambridge U. Press, 2004).
[CrossRef]

Appl. Math. Comput. (1)

Z. Wang, W. Wu, X. Xu, and D. Xue, “Recognition and location of the internal corners of planar checkerboard calibration pattern image,” Appl. Math. Comput. 185, 894–906 (2007).
[CrossRef]

IEEE Trans. Pattern Anal. Mach. Intell. (3)

Z. Zhang, “A flexible new technique for camera calibration,” IEEE Trans. Pattern Anal. Mach. Intell. 22, 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, 965–980 (1992).
[CrossRef]

R. I. Hartley and S. B. Kang, “Parameter-free radial distortion correction with center of distortion estimation,” IEEE Trans. Pattern Anal. Mach. Intell. 29, 1309–1321 (2007).
[CrossRef] [PubMed]

IEEE Trans. Robot. Automat. (1)

R. Tsai, “A versatile camera calibration technique for high-accuracy 3D machine vision metrology using off-the-shelf TV cameras and lenses,” IEEE Trans. Robot. Automat. 3, 323–344 (1987).
[CrossRef]

Image Vis. Comput. (1)

C. Hughes, R. McFeely, P. Denny, M. Glavin, and E. Jones, “Equidistant (fθ) fish-eye perspective with application in distortion centre estimation,” Image Vis. Comput. 28, 538–551(2010).
[CrossRef]

ISPRS J. Photogramm. Remote Sens. (1)

D. Schneider, E. Schwalbe, and H.-G. Maas, “Validation of geometric models for fisheye lenses,” ISPRS J. Photogramm. Remote Sens. 64, 259–266 (2009).
[CrossRef]

J. Am. Stat. Assoc. (1)

W. S. Cleveland and S. J. Devlin, “Locally weighted regression: an approach to regression analysis by local fitting,” J. Am. Stat. Assoc. 83, 596–610 (1988).
[CrossRef]

J. Opt. Soc. Am. (1)

J. Syst. Architect. (1)

K. V. Asari, “Design of an efficient VLSI architecture for non-linear spatial warping of wide-angle camera images,” J. Syst. Architect. 50, 743–755 (2004).
[CrossRef]

Mach. Vis. Appl. (1)

F. Devernay and O. Faugeras, “Straight lines have to be straight: automatic calibration and removal of distortion from scenes of structured environments,” Mach. Vis. Appl. 13, 14–24 (2001).
[CrossRef]

Pattern Recogn. (1)

S. Shah and J. K. Aggarwal, “Intrinsic parameter calibration procedure for a (high-distortion) fish-eye lens camera with distortion model and accuracy estimation,” Pattern Recogn. 29, 1775–1788 (1996).
[CrossRef]

Pattern Recogn. Lett. (1)

A. Basu and S. Licardie, “Alternative models for fish-eye lenses,” Pattern Recogn. Lett. 16, 433–441 (1995).
[CrossRef]

Photograph. Eng. (1)

D. C. Brown, “Decentering distortion of lenses,” Photograph. Eng. 32, 444–462 (1966).

SIAM J. Appl. Math. (1)

D. W. Marquardt, “An algorithm for least-squares estimation of nonlinear parameters,” SIAM J. Appl. Math. 11, 431–441(1963).
[CrossRef]

Other (7)

D. G. Lowe, “Object recognition from local scale-invariant features,” Proceedings of the IEEE International Conference on Computer Vision (IEEE, 1999), pp. 1150–1157.
[CrossRef]

C. Bräuer-Burchardt and K. Voss, “A new algorithm to correct fish-eye- and strong wide-angle-lens-distortion from single images,” in Proceedings of the IEEE International Conference on Image Processing (IEEE, 2001), pp. 225–228.

A. W. Fitzgibbon, “Simultaneous linear estimation of multiple view geometry and lens distortion,” in Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition (IEEE, 2001), pp. 125–132.

B. B. Jähne, Digital Image Processing, 5th ed. (Springer-Verlag, 2002), Chap. 12.

G. P. Stein, “Internal camera calibration using rotation and geometric shapes,” M.S. thesis (MIT, 1993).

R. I. Hartley and A. Zisserman, Multiple View Geometry in Computer Vision, 2nd ed. (Cambridge U. Press, 2004).
[CrossRef]

G. Xu, J. Terai, and H.-Y. Shum, “A linear algorithm for camera self-calibration, motion and structure recovery for multi-planar scenes from two perspective images,” in Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition (IEEE, 2000), pp. 474–479.

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Figures (7)

Fig. 1
Fig. 1

Rectilinear projection representation.

Fig. 2
Fig. 2

Fish-eye projection function representations, showing the projection of the point P to the projection sphere and then the reprojection of the point on the projection sphere to the image plane: (a) equidistant, (b) equisolid, (c) orthographic, and (d) stereographic.

Fig. 3
Fig. 3

Point of minimum distance between the distortion center and the distorted line lies on the line perpendicular to the undistorted line through the distortion center.

Fig. 4
Fig. 4

Two-point perspective: (a) shows how to find the vanishing points, horizon line, and ground line (which is parallel to the horizon line) and (b) shows how parallel lines in 3D space converge at a single point in perspective and cross the ground line at equal distances (marked as d in the figure).

Fig. 5
Fig. 5

Curves in undistorted space overlaid on the corresponding curves in distorted space for the (a) horizontal lines, (b) vertical lines, and (c) both sets of lines. (d) Shows the corners in the distorted space connected to the corners in the undistorted space and (e) shows the extracted points with locally weighted scatterplot smoothing (LOESS) applied.

Fig. 6
Fig. 6

Residuals after the fitting of each of the models to camera 1: (a) equidistant, equisolid, orthographic, and stereographic projection functions, (b) PFET, FET, and FOV models, and (c) all of the models with the additional radial distortion parameters included. In (c), the projection functions with the additional parameters are almost coincident. Note the change in scales between the graphs.

Fig. 7
Fig. 7

Residuals after the fitting of each of the models to camera 2: (a) equidistant, equisolid, orthographic, and stereographic projection functions, (b) PFET, FET, and FOV models, and (c) all of the models with the additional radial distortion parameters included. In (c), the projection functions with the additional parameters are almost coincident. Note the change in scales between the graphs.

Tables (5)

Tables Icon

Table 1 List of Tested Cameras

Tables Icon

Table 2 RMSE of the Functions Fitted to the Distortion Curves Extracted from Each of the Cameras a

Tables Icon

Table 3 RMSE of Fits to Camera 2 for PFET of Various Orders a

Tables Icon

Table 4 RMSE of Fits to Camera 5 for Equidistant Model with Additional Radial Distortion Parameters of Various Orders

Tables Icon

Table 5 Maximum Error of Various Models Fitted to Various Cameras, in Terms of Pixels (Assuming an Image Sensor Pixel Sample Resolution of 640 × 480 )

Equations (24)

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

r u = f tan ( θ ) ,
r d = f θ .
r d = f arctan ( r u f ) .
r u = f tan ( r d f ) .
r d = 2 f sin ( θ 2 ) .
r d = 2 f sin ( arctan ( r u / f ) 2 ) .
r u = f tan ( 2 arcsin ( r d 2 f ) ) .
r d = f sin ( θ ) .
r d = r u ( 1 + r u 2 f 2 ) 1 / 2 .
r u = r d ( 1 r d 2 f 2 ) 1 / 2 .
r d = 2 f tan ( θ 2 ) .
r d = 2 f tan ( arctan ( r u f ) 2 ) .
r u = f tan ( 2 arctan ( r d 2 f ) ) .
r d = r u 1 r u 2 4 f 2 .
r d = r u 1 λ r u 2 .
r d = n = 1 κ n r u n = κ 1 r u + κ 2 r u 2 + + κ n r u n + .
r d = s ln ( 1 + λ r u ) ,
r u = exp ( r d / s ) 1 λ .
r d = 1 ω arctan ( 2 r u tan ( ω 2 ) ) .
r u = tan ( r d ω ) 2 tan ( ω 2 ) ,
Δ r d = A 1 r u 3 + A 2 r u 5 + A 3 r u 7 ,
r d = f arctan ( r u f ) + Δ r d .
ξ = i = 1 n | Δ s i | ,
Δ s = { m s , | m s | < π / 2 π ( m s ) , | m s | > π / 2 .

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