Abstract

We present the correction of distortion for a novel type of an all-reflective zoom objective. The all- reflective unobscured optical-power zoom (OPZ) objective with four mirrors has been previously designed and presented. The magnification of the OPZ can be varied by changing the curvatures of the first and the last mirror, which results in a zoom factor of 3. However, the objective exhibits significant distortion. For the unobscured design principle, we present the basic distortion model with its different types of distortion. Based on simulation data of the objective design, we optimized the parameters of the model and verified that model by applying it to images taken with the objective.

© 2010 Optical Society of America

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References

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  1. K. Seidl, J. Knobbe, and H. Grüger, “Design of an all-reflective unobscured optical-power zoom objective,” Appl. Opt. 48, 4097–4107 (2009).
    [CrossRef] [PubMed]
  2. F. C. Wippermann, P. Schreiber, A. Bräuer, and P. Craen, “Bifocal liquid lens zoom objective for mobile phone applications,” Proc. SPIE 6501, 650109 (2007).
    [CrossRef]
  3. S. Kuiper, B. H. W. Hendriks, J. F. Suijver, S. Deladi, and I. Helwegen, “Zoom camera based on liquid lenses,” Proc. SPIE 6466, 64660F (2007).
    [CrossRef]
  4. W. Greger, T. Hösel, T. Fellner, A. Schoth, C. Mueller, J. Wilde, and H. Reinicke, “Low-cost deformable mirror for laser focusing,” Proc. SPIE 6374, 63740F (2006).
    [CrossRef]
  5. 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]
  6. A. E. Conrady, “Decentered lens-systems,” Mon. Notes R. Astron. Soc. 79, 384–390 (1919).
  7. D. C. Brown, “Decentering distortion of lenses,” Photogramm. Eng. Remote Sens. 32, 444–462 (1966).
  8. J. Wang, F. Shi, J. Zhang, and Y. Liu, “A new calibration model of camera lens distortion,” Pattern Recog. 41, 607–615(2008).
    [CrossRef]
  9. T. A. Clarke and J. G. Fryer, “The development of camera calibration methods and models,” Photogramm. Rec. 16, 51–66 (1998).
    [CrossRef]
  10. Z. Zhang, “A flexible new technique for camera calibration,” IEEE Trans. Pattern Anal. Mach. Intell. 22, 1330–1334(2000).
    [CrossRef]
  11. F. Devernay and O. Faugeras, “Straight lines have to be straight,” Mach. Vision Appl. 13, 14–24 (2001).
    [CrossRef]
  12. J. P. de Villiers, “Correction of radially asymmetric lens distortion with a closed form solution and inverse function,” Ph.D. dissertation (University of Pretoria, 2007).
  13. J. Perš and S. Kovačič, “Nonparametric, model-based radial lens distortion correction using tilted camera assumption,” in Proceedings of the Computer Vision Winter Workshop (IEEE, 2002), Vol. 1, pp. 286–295.
  14. C.Slama, ed., Manual of Photogrammetry, 4th ed. (American Society of Photogrammetry, 1980).
  15. H. Gross, H. Zügge, M. Peschka, and F. Blechinger, Handbook of Optical Systems—Volume 3: Aberration Theory and Correction of Optical Systems (Wiley, 2007).
  16. C. S. Fraser and S. Al-Ajlouni, “Zoom-dependent camera calibration in digital close range photogrammetry,” Photogramm. Eng. Remote Sens. 72, 1017–1026 (2006).
  17. K. T. Gribbon and D. G. Bailey, “A novel approach to real-time bilinear interpolation,” in Proceedings of the Second IEEE International Workshop on Electronic Design, Test and Applications (IEEE, 2004).
    [CrossRef]
  18. H. Gross, Handbook of Optical Systems—Volume 1: Fundamentals of Technical Optics (Wiley, 2008).
  19. H. Haferkorn, Optik—Physikalisch-technische Grundlagen und Anwendungen, 3rd ed. (Wiley, 1994).

2009

2008

J. Wang, F. Shi, J. Zhang, and Y. Liu, “A new calibration model of camera lens distortion,” Pattern Recog. 41, 607–615(2008).
[CrossRef]

H. Gross, Handbook of Optical Systems—Volume 1: Fundamentals of Technical Optics (Wiley, 2008).

2007

J. P. de Villiers, “Correction of radially asymmetric lens distortion with a closed form solution and inverse function,” Ph.D. dissertation (University of Pretoria, 2007).

H. Gross, H. Zügge, M. Peschka, and F. Blechinger, Handbook of Optical Systems—Volume 3: Aberration Theory and Correction of Optical Systems (Wiley, 2007).

F. C. Wippermann, P. Schreiber, A. Bräuer, and P. Craen, “Bifocal liquid lens zoom objective for mobile phone applications,” Proc. SPIE 6501, 650109 (2007).
[CrossRef]

S. Kuiper, B. H. W. Hendriks, J. F. Suijver, S. Deladi, and I. Helwegen, “Zoom camera based on liquid lenses,” Proc. SPIE 6466, 64660F (2007).
[CrossRef]

2006

W. Greger, T. Hösel, T. Fellner, A. Schoth, C. Mueller, J. Wilde, and H. Reinicke, “Low-cost deformable mirror for laser focusing,” Proc. SPIE 6374, 63740F (2006).
[CrossRef]

C. S. Fraser and S. Al-Ajlouni, “Zoom-dependent camera calibration in digital close range photogrammetry,” Photogramm. Eng. Remote Sens. 72, 1017–1026 (2006).

2004

K. T. Gribbon and D. G. Bailey, “A novel approach to real-time bilinear interpolation,” in Proceedings of the Second IEEE International Workshop on Electronic Design, Test and Applications (IEEE, 2004).
[CrossRef]

2002

J. Perš and S. Kovačič, “Nonparametric, model-based radial lens distortion correction using tilted camera assumption,” in Proceedings of the Computer Vision Winter Workshop (IEEE, 2002), Vol. 1, pp. 286–295.

2001

F. Devernay and O. Faugeras, “Straight lines have to be straight,” Mach. Vision Appl. 13, 14–24 (2001).
[CrossRef]

2000

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

1998

T. A. Clarke and J. G. Fryer, “The development of camera calibration methods and models,” Photogramm. Rec. 16, 51–66 (1998).
[CrossRef]

1994

H. Haferkorn, Optik—Physikalisch-technische Grundlagen und Anwendungen, 3rd ed. (Wiley, 1994).

1992

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]

1980

C.Slama, ed., Manual of Photogrammetry, 4th ed. (American Society of Photogrammetry, 1980).

1966

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

1919

A. E. Conrady, “Decentered lens-systems,” Mon. Notes R. Astron. Soc. 79, 384–390 (1919).

Al-Ajlouni, S.

C. S. Fraser and S. Al-Ajlouni, “Zoom-dependent camera calibration in digital close range photogrammetry,” Photogramm. Eng. Remote Sens. 72, 1017–1026 (2006).

Bailey, D. G.

K. T. Gribbon and D. G. Bailey, “A novel approach to real-time bilinear interpolation,” in Proceedings of the Second IEEE International Workshop on Electronic Design, Test and Applications (IEEE, 2004).
[CrossRef]

Blechinger, F.

H. Gross, H. Zügge, M. Peschka, and F. Blechinger, Handbook of Optical Systems—Volume 3: Aberration Theory and Correction of Optical Systems (Wiley, 2007).

Bräuer, A.

F. C. Wippermann, P. Schreiber, A. Bräuer, and P. Craen, “Bifocal liquid lens zoom objective for mobile phone applications,” Proc. SPIE 6501, 650109 (2007).
[CrossRef]

Brown, D. C.

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

Clarke, T. A.

T. A. Clarke and J. G. Fryer, “The development of camera calibration methods and models,” Photogramm. Rec. 16, 51–66 (1998).
[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]

Conrady, A. E.

A. E. Conrady, “Decentered lens-systems,” Mon. Notes R. Astron. Soc. 79, 384–390 (1919).

Craen, P.

F. C. Wippermann, P. Schreiber, A. Bräuer, and P. Craen, “Bifocal liquid lens zoom objective for mobile phone applications,” Proc. SPIE 6501, 650109 (2007).
[CrossRef]

de Villiers, J. P.

J. P. de Villiers, “Correction of radially asymmetric lens distortion with a closed form solution and inverse function,” Ph.D. dissertation (University of Pretoria, 2007).

Deladi, S.

S. Kuiper, B. H. W. Hendriks, J. F. Suijver, S. Deladi, and I. Helwegen, “Zoom camera based on liquid lenses,” Proc. SPIE 6466, 64660F (2007).
[CrossRef]

Devernay, F.

F. Devernay and O. Faugeras, “Straight lines have to be straight,” Mach. Vision Appl. 13, 14–24 (2001).
[CrossRef]

Faugeras, O.

F. Devernay and O. Faugeras, “Straight lines have to be straight,” Mach. Vision Appl. 13, 14–24 (2001).
[CrossRef]

Fellner, T.

W. Greger, T. Hösel, T. Fellner, A. Schoth, C. Mueller, J. Wilde, and H. Reinicke, “Low-cost deformable mirror for laser focusing,” Proc. SPIE 6374, 63740F (2006).
[CrossRef]

Fraser, C. S.

C. S. Fraser and S. Al-Ajlouni, “Zoom-dependent camera calibration in digital close range photogrammetry,” Photogramm. Eng. Remote Sens. 72, 1017–1026 (2006).

Fryer, J. G.

T. A. Clarke and J. G. Fryer, “The development of camera calibration methods and models,” Photogramm. Rec. 16, 51–66 (1998).
[CrossRef]

Greger, W.

W. Greger, T. Hösel, T. Fellner, A. Schoth, C. Mueller, J. Wilde, and H. Reinicke, “Low-cost deformable mirror for laser focusing,” Proc. SPIE 6374, 63740F (2006).
[CrossRef]

Gribbon, K. T.

K. T. Gribbon and D. G. Bailey, “A novel approach to real-time bilinear interpolation,” in Proceedings of the Second IEEE International Workshop on Electronic Design, Test and Applications (IEEE, 2004).
[CrossRef]

Gross, H.

H. Gross, Handbook of Optical Systems—Volume 1: Fundamentals of Technical Optics (Wiley, 2008).

H. Gross, H. Zügge, M. Peschka, and F. Blechinger, Handbook of Optical Systems—Volume 3: Aberration Theory and Correction of Optical Systems (Wiley, 2007).

Grüger, H.

Haferkorn, H.

H. Haferkorn, Optik—Physikalisch-technische Grundlagen und Anwendungen, 3rd ed. (Wiley, 1994).

Helwegen, I.

S. Kuiper, B. H. W. Hendriks, J. F. Suijver, S. Deladi, and I. Helwegen, “Zoom camera based on liquid lenses,” Proc. SPIE 6466, 64660F (2007).
[CrossRef]

Hendriks, B. H. W.

S. Kuiper, B. H. W. Hendriks, J. F. Suijver, S. Deladi, and I. Helwegen, “Zoom camera based on liquid lenses,” Proc. SPIE 6466, 64660F (2007).
[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]

Hösel, T.

W. Greger, T. Hösel, T. Fellner, A. Schoth, C. Mueller, J. Wilde, and H. Reinicke, “Low-cost deformable mirror for laser focusing,” Proc. SPIE 6374, 63740F (2006).
[CrossRef]

Knobbe, J.

Kovacic, S.

J. Perš and S. Kovačič, “Nonparametric, model-based radial lens distortion correction using tilted camera assumption,” in Proceedings of the Computer Vision Winter Workshop (IEEE, 2002), Vol. 1, pp. 286–295.

Kuiper, S.

S. Kuiper, B. H. W. Hendriks, J. F. Suijver, S. Deladi, and I. Helwegen, “Zoom camera based on liquid lenses,” Proc. SPIE 6466, 64660F (2007).
[CrossRef]

Liu, Y.

J. Wang, F. Shi, J. Zhang, and Y. Liu, “A new calibration model of camera lens distortion,” Pattern Recog. 41, 607–615(2008).
[CrossRef]

Mueller, C.

W. Greger, T. Hösel, T. Fellner, A. Schoth, C. Mueller, J. Wilde, and H. Reinicke, “Low-cost deformable mirror for laser focusing,” Proc. SPIE 6374, 63740F (2006).
[CrossRef]

Perš, J.

J. Perš and S. Kovačič, “Nonparametric, model-based radial lens distortion correction using tilted camera assumption,” in Proceedings of the Computer Vision Winter Workshop (IEEE, 2002), Vol. 1, pp. 286–295.

Peschka, M.

H. Gross, H. Zügge, M. Peschka, and F. Blechinger, Handbook of Optical Systems—Volume 3: Aberration Theory and Correction of Optical Systems (Wiley, 2007).

Reinicke, H.

W. Greger, T. Hösel, T. Fellner, A. Schoth, C. Mueller, J. Wilde, and H. Reinicke, “Low-cost deformable mirror for laser focusing,” Proc. SPIE 6374, 63740F (2006).
[CrossRef]

Schoth, A.

W. Greger, T. Hösel, T. Fellner, A. Schoth, C. Mueller, J. Wilde, and H. Reinicke, “Low-cost deformable mirror for laser focusing,” Proc. SPIE 6374, 63740F (2006).
[CrossRef]

Schreiber, P.

F. C. Wippermann, P. Schreiber, A. Bräuer, and P. Craen, “Bifocal liquid lens zoom objective for mobile phone applications,” Proc. SPIE 6501, 650109 (2007).
[CrossRef]

Seidl, K.

Shi, F.

J. Wang, F. Shi, J. Zhang, and Y. Liu, “A new calibration model of camera lens distortion,” Pattern Recog. 41, 607–615(2008).
[CrossRef]

Suijver, J. F.

S. Kuiper, B. H. W. Hendriks, J. F. Suijver, S. Deladi, and I. Helwegen, “Zoom camera based on liquid lenses,” Proc. SPIE 6466, 64660F (2007).
[CrossRef]

Wang, J.

J. Wang, F. Shi, J. Zhang, and Y. Liu, “A new calibration model of camera lens distortion,” Pattern Recog. 41, 607–615(2008).
[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]

Wilde, J.

W. Greger, T. Hösel, T. Fellner, A. Schoth, C. Mueller, J. Wilde, and H. Reinicke, “Low-cost deformable mirror for laser focusing,” Proc. SPIE 6374, 63740F (2006).
[CrossRef]

Wippermann, F. C.

F. C. Wippermann, P. Schreiber, A. Bräuer, and P. Craen, “Bifocal liquid lens zoom objective for mobile phone applications,” Proc. SPIE 6501, 650109 (2007).
[CrossRef]

Zhang, J.

J. Wang, F. Shi, J. Zhang, and Y. Liu, “A new calibration model of camera lens distortion,” Pattern Recog. 41, 607–615(2008).
[CrossRef]

Zhang, Z.

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

Zügge, H.

H. Gross, H. Zügge, M. Peschka, and F. Blechinger, Handbook of Optical Systems—Volume 3: Aberration Theory and Correction of Optical Systems (Wiley, 2007).

Appl. Opt.

IEEE Trans. Pattern Anal. Mach. Intell.

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]

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

Mach. Vision Appl.

F. Devernay and O. Faugeras, “Straight lines have to be straight,” Mach. Vision Appl. 13, 14–24 (2001).
[CrossRef]

Mon. Notes R. Astron. Soc.

A. E. Conrady, “Decentered lens-systems,” Mon. Notes R. Astron. Soc. 79, 384–390 (1919).

Pattern Recog.

J. Wang, F. Shi, J. Zhang, and Y. Liu, “A new calibration model of camera lens distortion,” Pattern Recog. 41, 607–615(2008).
[CrossRef]

Photogramm. Eng. Remote Sens.

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

C. S. Fraser and S. Al-Ajlouni, “Zoom-dependent camera calibration in digital close range photogrammetry,” Photogramm. Eng. Remote Sens. 72, 1017–1026 (2006).

Photogramm. Rec.

T. A. Clarke and J. G. Fryer, “The development of camera calibration methods and models,” Photogramm. Rec. 16, 51–66 (1998).
[CrossRef]

Proc. SPIE

F. C. Wippermann, P. Schreiber, A. Bräuer, and P. Craen, “Bifocal liquid lens zoom objective for mobile phone applications,” Proc. SPIE 6501, 650109 (2007).
[CrossRef]

S. Kuiper, B. H. W. Hendriks, J. F. Suijver, S. Deladi, and I. Helwegen, “Zoom camera based on liquid lenses,” Proc. SPIE 6466, 64660F (2007).
[CrossRef]

W. Greger, T. Hösel, T. Fellner, A. Schoth, C. Mueller, J. Wilde, and H. Reinicke, “Low-cost deformable mirror for laser focusing,” Proc. SPIE 6374, 63740F (2006).
[CrossRef]

Other

K. T. Gribbon and D. G. Bailey, “A novel approach to real-time bilinear interpolation,” in Proceedings of the Second IEEE International Workshop on Electronic Design, Test and Applications (IEEE, 2004).
[CrossRef]

H. Gross, Handbook of Optical Systems—Volume 1: Fundamentals of Technical Optics (Wiley, 2008).

H. Haferkorn, Optik—Physikalisch-technische Grundlagen und Anwendungen, 3rd ed. (Wiley, 1994).

J. P. de Villiers, “Correction of radially asymmetric lens distortion with a closed form solution and inverse function,” Ph.D. dissertation (University of Pretoria, 2007).

J. Perš and S. Kovačič, “Nonparametric, model-based radial lens distortion correction using tilted camera assumption,” in Proceedings of the Computer Vision Winter Workshop (IEEE, 2002), Vol. 1, pp. 286–295.

C.Slama, ed., Manual of Photogrammetry, 4th ed. (American Society of Photogrammetry, 1980).

H. Gross, H. Zügge, M. Peschka, and F. Blechinger, Handbook of Optical Systems—Volume 3: Aberration Theory and Correction of Optical Systems (Wiley, 2007).

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Figures (11)

Fig. 1
Fig. 1

Final design of a Schiefspiegler OPZ objective: (a) f = 5.2 mm , (b) f = 7.8 mm , and (c) f = 15.6 mm ; 1, first deformable mirror (DM); 2, off-axis biconic concave mirror; 3, biconic convex mirror (acts as an aperture stop); 4, second DM; 5, cover glass; and 6, image sensor.

Fig. 2
Fig. 2

(a) Sectional 3D-CAD-model and (b) optomechanical setup.

Fig. 3
Fig. 3

Distorted images taken with OPZ setups: (a) f = 5.2 mm , (b) f = 7.8 mm , and (c) f = 15.6 mm .

Fig. 4
Fig. 4

Distortion of symmetrical optical systems: (a) barrel distortion and (b) pincushion distortion.

Fig. 5
Fig. 5

(a) Keystone distortion, (b) bowing distortion, (c) combination of stretching and shortening distortion, and (d) anamorphotic distortion.

Fig. 6
Fig. 6

Simulated distortion of the OPZ system: (a) f = 5.2 mm , (b) f = 7.8 mm , and (c) f = 15.6 mm .

Fig. 7
Fig. 7

Optimized distortion parameters with the according regressions: (a) R 1 ( f ) , (b) R 2 ( f ) , A 1 ( f ) , A 2 ( f ) , and (c) B ( f ) , K ( f ) , S ( f ) .

Fig. 8
Fig. 8

Images corrected for distortion: (a) f = 5.2 mm , (b) f = 7.8 mm , and (c) f = 15.6 mm .

Fig. 9
Fig. 9

Jagged-edge artifacts: (a) detail from Fig. 3a without distortion correction, (b) after distortion correction based on nearest neighbor interpolation, and (c) after distortion with bilinear interpolation.

Fig. 10
Fig. 10

Interpolation principles: (a) nearest neighbor, (b) bilinear interpolation, and (c) definition of the horizontal and vertical deviations of the calculated position to the center of the nearest neighbor pixel.

Fig. 11
Fig. 11

Change in relative illumination in aperture values (AV): (a) due to distortion, (b) due to natural vignetting, and (c) both effects superposed.

Tables (3)

Tables Icon

Table 1 Optimized Distortion Parameters and Residual Deviation for Wide-Angle Zoom Position ( f = 5.17 mm )

Tables Icon

Table 2 Optimized Distortion Parameters and Residual Deviation

Tables Icon

Table 3 Residual Root Mean Square and Maximum Distances before Correction and after Correction with Final Distortion Model Based on 101 × 101 Simulated Points

Equations (19)

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

x = x + Δ x , y = y + Δ y ,
Δ x radial = x r 2 ( R 1 + R 2 r 2 + ) , Δ y radial = y r 2 ( R 1 + R 2 r 2 + ) ,
Δ x decentering = [ sin ϕ 0 ( r 2 + 2 x 2 ) + 2 cos ϕ 0 x y ] ( D 1 + D 2 r 2 + ) , Δ y decentering = [ 2 sin ϕ 0 x y + cos ϕ 0 ( r 2 + 2 y 2 ) ] ( D 1 + D 2 r 2 + ) .
Δ x decentering = 2 x y ( D 1 + D 2 r 2 ) , Δ y decentering = ( r 2 + 2 y 2 ) ( D 1 + D 2 r 2 ) .
Δ x keystone = 2 x y ( K 1 + K 2 r 2 ) .
Δ y bowing = r 2 ( B 1 + B 2 r 2 ) .
Δ y stretching = 2 y 2 ( S 1 + S 2 r 2 ) .
Δ x anamorphotism = A 1 x , Δ y anamorphotism = 3 A 1 y .
Δ x anamorphotism = A 1 x + A 2 x 3 , Δ y anamorphotism = 3 A 1 y + A 3 y 3 .
x = x + x r 2 ( R 1 + R 2 r 2 ) + 2 x y ( K 1 + K 2 r 2 ) + A 1 x + A 2 x 3 , y = y + y r 2 ( R 1 + R 2 r 2 ) + r 2 ( B 1 + B 2 r 2 ) + 2 y 2 ( S 1 + S 2 r 2 ) + 3 A 1 y + A 3 y 3 .
x = x + x r 2 ( R 1 + R 2 r 2 ) + 2 K x y ( R 1 + R 2 r 2 ) + A 1 x + A 2 x 3 , y = y + y r 2 ( R 1 + R 2 r 2 ) B r 2 ( R 1 + R 2 r 2 ) + 2 S y 2 ( R 1 + R 2 r 2 ) + 3 A 1 y A 2 y 3 .
R 1 ( f ) = R 10 + R 11 ( R 12 + R 13 f ) R 14 .
x ( f ) = x + x r 2 [ R 1 ( f ) + R 2 ( f ) r 2 ] + 2 K ( f ) x y [ R 1 ( f ) + R 2 ( f ) r 2 ] + A 1 ( f ) x + A 2 ( f ) x 3 , y ( f ) = y + y r 2 [ R 1 ( f ) + R 2 ( f ) r 2 ] B ( f ) r 2 [ R 1 ( f ) + R 2 ( f ) r 2 ] + 2 S ( f ) y 2 [ R 1 ( f ) + R 2 ( f ) r 2 ] + 3 A 1 ( f ) y A 2 ( f ) y 3 ,
R 1 ( f ) = R 10 + R 11 ( R 12 + R 13 f ) R 14 , R 2 ( f ) = R 20 + R 21 ( R 22 + R 23 f ) R 24 , A 1 ( f ) = A 10 + A 11 ( A 12 + A 13 f ) A 14 , A 2 ( f ) = A 20 + A 21 ( A 22 + A 23 f ) A 24 , B ( f ) = B 0 + B 1 f , K ( f ) = K 0 + K 1 f , S ( f ) = S 0 + S 1 f .
( R G B ) undistorted = ( 1 h ) ( 1 v ) ( R G B ) A + ( 1 h ) v ( R G B ) B + h ( 1 v ) ( R G B ) C + h v ( R G B ) D .
E w natural = E 0 cos 4 w .
E 0 = π Ω 0 L sin 2 u β 2 .
d ( w ) = β ( w ) β 0 1.
E w OPZ = E 0 cos 4 w ( d ( w ) + 1 ) 2 .

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