Abstract

Asymptotic expressions are derived for the two-dimensional incoherent optical transfer function (OTF) of an optical system with defocus and spherical aberration. The two-dimensional stationary phase method is used to evaluate the aberrated OTF at large and moderately large defocus and spherical aberration. For small aberrations, the OTF is approximated by a power series in the aberration coefficients. An accurate approximation (in elementary functions) to the OTF is obtained for a defocused optical system with a circular pupil. We experimentally demonstrate the validity of the OTF approximations in sharp-focus image restoration from two defocused images. A digital focusing method is presented.

© 2010 Optical Society of America

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References

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  1. M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge University Press, 1999).
  2. J. W. Goodman, Introduction to Fourier Optics (Roberts, 2005).
  3. V. N. Mahajan, Optical Imaging and Aberrations (SPIE, 2001), Vol. 2.
    [CrossRef]
  4. R. C. Puetter, T. R. Gosnell, and A. Yahil, “Digital image reconstruction: deblurring and denoising,” Annu. Rev. Astron. Astrophys. 43, 139–194 (2005).
    [CrossRef]
  5. A. E. Savakis and H. J. Trussell, “On the accuracy of PSF representation in image restoration,” IEEE Trans. Image Process. 2, 252–259 (1993).
    [CrossRef] [PubMed]
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    [CrossRef]
  7. P. A. Stokseth, “Properties of a defocused optical system,” J. Opt. Soc. Am. 59, 1314–1321 (1969).
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  8. A. Pentland, S. Scherock, T. Darrell, and B. Girod, “Simple range cameras based on focal error,” J. Opt. Soc. Am. A 11, 2925–2934 (1994).
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  10. M. V. Fedoruk, Saddle Point Method (Nauka, 1977), pp. 92–161.
  11. R. Wong, Asymptotic Approximations of Integrals (Academic, 1989), pp. 477–515.
  12. A. B. Samokhin, A. N. Simonov, and M. C. Rombach, “Optical system invariant to second-order aberrations,” J. Opt. Soc. Am. A 26, 977–984 (2009).
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  15. W. J. Smith, Modern Optical Engineering, 3rd ed. (McGraw-Hill, 2000), pp. 348–355.
  16. A. N. Simonov and M. C. Rombach, “Sharp-focus image restoration from defocused images,” Opt. Lett. 34, 2111–2113 (2009).
    [CrossRef] [PubMed]
  17. F. R. Gantmacher, The Theory of Matrices (Chelsea, 1959), Vol. 1, pp. 294–317.
  18. D. Kaminski, “Exponentially improved stationary phase approximations for double integrals,” Methods Appl. Anal. 1, 44–56 (1994).
  19. M. Frigo and S. G. Johnson, “FFTW-3.2,” http://www.fftw.org.
  20. B. R. Frieden, “Image enhancement and restoration,” in Picture Processing and Digital Imaging, T.S.Huang, ed. (Springer-Verlag, 1979), pp. 179–249.

2009

2005

J. W. Goodman, Introduction to Fourier Optics (Roberts, 2005).

R. C. Puetter, T. R. Gosnell, and A. Yahil, “Digital image reconstruction: deblurring and denoising,” Annu. Rev. Astron. Astrophys. 43, 139–194 (2005).
[CrossRef]

2001

V. N. Mahajan, Optical Imaging and Aberrations (SPIE, 2001), Vol. 2.
[CrossRef]

2000

W. J. Smith, Modern Optical Engineering, 3rd ed. (McGraw-Hill, 2000), pp. 348–355.

1999

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge University Press, 1999).

1995

1994

A. Pentland, S. Scherock, T. Darrell, and B. Girod, “Simple range cameras based on focal error,” J. Opt. Soc. Am. A 11, 2925–2934 (1994).
[CrossRef]

D. Kaminski, “Exponentially improved stationary phase approximations for double integrals,” Methods Appl. Anal. 1, 44–56 (1994).

1993

A. E. Savakis and H. J. Trussell, “On the accuracy of PSF representation in image restoration,” IEEE Trans. Image Process. 2, 252–259 (1993).
[CrossRef] [PubMed]

1989

R. Wong, Asymptotic Approximations of Integrals (Academic, 1989), pp. 477–515.

M. Fedoruk, Asymptotic Methods in Analysis. Analysis. I, Vol. 13 of Encyclopaedia of Mathematical Sciences (Springer, 1989).

1979

B. R. Frieden, “Image enhancement and restoration,” in Picture Processing and Digital Imaging, T.S.Huang, ed. (Springer-Verlag, 1979), pp. 179–249.

1977

M. V. Fedoruk, Saddle Point Method (Nauka, 1977), pp. 92–161.

1969

1959

F. R. Gantmacher, The Theory of Matrices (Chelsea, 1959), Vol. 1, pp. 294–317.

1955

H. H. Hopkins, “The frequency response of a defocused optical system,” Proc. R. Soc. London, Ser. A 231, 91–103 (1955).
[CrossRef]

1952

Born, M.

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge University Press, 1999).

Cathey, W. T.

Darrell, T.

Dowski, E. R.

Fedoruk, M.

M. Fedoruk, Asymptotic Methods in Analysis. Analysis. I, Vol. 13 of Encyclopaedia of Mathematical Sciences (Springer, 1989).

Fedoruk, M. V.

M. V. Fedoruk, Saddle Point Method (Nauka, 1977), pp. 92–161.

Frieden, B. R.

B. R. Frieden, “Image enhancement and restoration,” in Picture Processing and Digital Imaging, T.S.Huang, ed. (Springer-Verlag, 1979), pp. 179–249.

Frigo, M.

M. Frigo and S. G. Johnson, “FFTW-3.2,” http://www.fftw.org.

Gantmacher, F. R.

F. R. Gantmacher, The Theory of Matrices (Chelsea, 1959), Vol. 1, pp. 294–317.

Girod, B.

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics (Roberts, 2005).

Gosnell, T. R.

R. C. Puetter, T. R. Gosnell, and A. Yahil, “Digital image reconstruction: deblurring and denoising,” Annu. Rev. Astron. Astrophys. 43, 139–194 (2005).
[CrossRef]

Hopkins, H. H.

H. H. Hopkins, “The frequency response of a defocused optical system,” Proc. R. Soc. London, Ser. A 231, 91–103 (1955).
[CrossRef]

Johnson, S. G.

M. Frigo and S. G. Johnson, “FFTW-3.2,” http://www.fftw.org.

Kaminski, D.

D. Kaminski, “Exponentially improved stationary phase approximations for double integrals,” Methods Appl. Anal. 1, 44–56 (1994).

Mahajan, V. N.

V. N. Mahajan, Optical Imaging and Aberrations (SPIE, 2001), Vol. 2.
[CrossRef]

Pentland, A.

Puetter, R. C.

R. C. Puetter, T. R. Gosnell, and A. Yahil, “Digital image reconstruction: deblurring and denoising,” Annu. Rev. Astron. Astrophys. 43, 139–194 (2005).
[CrossRef]

Rombach, M. C.

Samokhin, A. B.

Savakis, A. E.

A. E. Savakis and H. J. Trussell, “On the accuracy of PSF representation in image restoration,” IEEE Trans. Image Process. 2, 252–259 (1993).
[CrossRef] [PubMed]

Scherock, S.

Simonov, A. N.

Smith, W. J.

W. J. Smith, Modern Optical Engineering, 3rd ed. (McGraw-Hill, 2000), pp. 348–355.

Stokseth, P. A.

Trussell, H. J.

A. E. Savakis and H. J. Trussell, “On the accuracy of PSF representation in image restoration,” IEEE Trans. Image Process. 2, 252–259 (1993).
[CrossRef] [PubMed]

Wolf, E.

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge University Press, 1999).

E. Wolf, “On a new aberration function of optical instruments,” J. Opt. Soc. Am. 42, 547–552 (1952).
[CrossRef]

Wong, R.

R. Wong, Asymptotic Approximations of Integrals (Academic, 1989), pp. 477–515.

Yahil, A.

R. C. Puetter, T. R. Gosnell, and A. Yahil, “Digital image reconstruction: deblurring and denoising,” Annu. Rev. Astron. Astrophys. 43, 139–194 (2005).
[CrossRef]

Annu. Rev. Astron. Astrophys.

R. C. Puetter, T. R. Gosnell, and A. Yahil, “Digital image reconstruction: deblurring and denoising,” Annu. Rev. Astron. Astrophys. 43, 139–194 (2005).
[CrossRef]

Appl. Opt.

IEEE Trans. Image Process.

A. E. Savakis and H. J. Trussell, “On the accuracy of PSF representation in image restoration,” IEEE Trans. Image Process. 2, 252–259 (1993).
[CrossRef] [PubMed]

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

Methods Appl. Anal.

D. Kaminski, “Exponentially improved stationary phase approximations for double integrals,” Methods Appl. Anal. 1, 44–56 (1994).

Opt. Lett.

Proc. R. Soc. London, Ser. A

H. H. Hopkins, “The frequency response of a defocused optical system,” Proc. R. Soc. London, Ser. A 231, 91–103 (1955).
[CrossRef]

Other

M. Fedoruk, Asymptotic Methods in Analysis. Analysis. I, Vol. 13 of Encyclopaedia of Mathematical Sciences (Springer, 1989).

M. V. Fedoruk, Saddle Point Method (Nauka, 1977), pp. 92–161.

R. Wong, Asymptotic Approximations of Integrals (Academic, 1989), pp. 477–515.

W. J. Smith, Modern Optical Engineering, 3rd ed. (McGraw-Hill, 2000), pp. 348–355.

F. R. Gantmacher, The Theory of Matrices (Chelsea, 1959), Vol. 1, pp. 294–317.

M. Frigo and S. G. Johnson, “FFTW-3.2,” http://www.fftw.org.

B. R. Frieden, “Image enhancement and restoration,” in Picture Processing and Digital Imaging, T.S.Huang, ed. (Springer-Verlag, 1979), pp. 179–249.

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge University Press, 1999).

J. W. Goodman, Introduction to Fourier Optics (Roberts, 2005).

V. N. Mahajan, Optical Imaging and Aberrations (SPIE, 2001), Vol. 2.
[CrossRef]

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Figures (6)

Fig. 1
Fig. 1

Geometry of the image side of the optical system.

Fig. 2
Fig. 2

Calculation of the area of overlap for a circular pupil.

Fig. 3
Fig. 3

Approximation of the MTF at large defocus. Solid lines depict the MTF curves obtained by direct numerical simulations with Eq. (46); scatter plots represent the MTFs calculated using the approximate expressions (39) (open squares) and (23) (open triangles). The RMS errors ζ are calculated in the frequency range ( 0.2 / φ ) ω 2 .

Fig. 4
Fig. 4

Approximation of the MTF at small defocus. Solid lines depict the MTF curves obtained by direct numerical simulations with Eq. (46); scatter plots represent the MTFs calculated using the approximate expression (40). The RMS errors ζ are calculated in the frequency range 0 ω 0.2 .

Fig. 5
Fig. 5

Determination of the applicability ranges of the approximate expressions (44, 48). Panel 1 depicts the defocused OTF at φ = 20 ; panel 2 shows the calculated dependency ω 0 = ω 0 ( φ ) .

Fig. 6
Fig. 6

Reconstruction of images: 1 and 2, defocused images captured by the camera; 3 and 4, images reconstructed (separately) using the phase-diversity algorithm [16]; 5 and 6, images reconstructed by Wiener deconvolution.

Equations (67)

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P ( x , y ) = p ( x , y ) exp [ i Φ ( x , y ) ] ,
p ( x , y ) = { 1 , ( x , y ) Ω , 0 , ( x , y ) Ω , }
H ( ω x , ω y ) = 1 S P ( x + ω x 2 , y + ω y 2 ) P ( x ω x 2 , y ω y 2 ) d x d y ,
φ = π d 2 4 λ ( 1 z i 1 z a ) ,
Φ 4 ( x , y ) ψ ( x 2 + y 2 ) 2 ,
Φ ( x , y ) = φ ( x 2 + y 2 ) + ψ ( x 2 + y 2 ) 2 .
Φ ( x , y ) = φ r 2 ( 1 γ r 2 ) ,
H ( ω x , ω y ) = 1 S D exp [ i Φ ( x + ω x 2 , y + ω y 2 ) i Φ ( x ω x 2 , y ω y 2 ) ] d x d y ,
( x , y ) ( x , y ) = | cos   θ sin   θ sin   θ cos   θ | = 1 ,
H ( ω x , ω y ) H ( ω ) = 1 π D exp { i φ ω y [ 2 γ ω 2 4 γ ( x 2 + y 2 ) ] } d x d y ,
y = ω 2 + 1 x 2 ,     ( x , y ) C 1 ,
y = ω 2 1 x 2 ,     ( x , y ) C 2 .
F ( x , y , ω ) = ω y [ 2 γ ω 2 4 γ ( x 2 + y 2 ) ] .
H ( ω ) 2 φ n σ n | Δ ( x n , y n ) | 1 / 2 exp [ i φ F ( x n , y n , ω ) + i π 4 δ ( x n , y n ) ] + O ( φ 2 ) ,
F x ( x n , y n ) 8 γ ω x n y n = 0 ,
F y ( x n , y n ) 2 ω γ ω 3 4 γ ω x n 2 12 γ ω y n 2 = 0.
M ( x n , y n ) = [ 2 F x 2 ( x n , y n ) 2 F x y ( x n , y n ) 2 F y x ( x n , y n ) 2 F y 2 ( x n , y n ) ] ,
Δ ( x n , y n ) = det   M ( x n , y n ) = [ 2 F x 2 2 F y 2 2 F x y 2 F y x ] ( x n , y n ) .
σ n = { 1 , ( x n , y n ) D 0 , ( x n , y n ) D . }
x 1 = 0 ,     y 1 = 2 γ ω 2 12 γ ,
x 2 = 0 ,     y 2 = 2 γ ω 2 12 γ ,
x 3 = 2 γ ω 2 4 γ ,     y 3 = 0 ,
x 4 = 2 γ ω 2 4 γ ,     y 4 = 0.
2 ω 2 6 ω + 6 1 γ .
Δ ( x 1 , y 1 ) = Δ ( x 2 , y 2 ) = Δ ( x 3 , y 3 ) = Δ ( x 4 , y 4 ) = 16 γ ( 2 γ ω 2 ) ω 2 .
| 2 F x 2 ( x n , y n ) λ n 2 F x y ( x n , y n ) 2 F y x ( x n , y n ) 2 F y 2 ( x n , y n ) λ n | | 8 γ ω y n λ n 8 γ ω x n 8 γ ω x n 24 γ ω y n λ n | = 0 ,
δ ( x 1 , y 1 ) = 2 ,
δ ( x 2 , y 2 ) = 2 ,
δ ( x 3 , y 3 ) = 0 ,
δ ( x 4 , y 4 ) = 0.
H ( ω ) { sin ( φ F 1 ) φ ω | γ ( 2 γ ω 2 ) | , 0 < γ < 1 2 1 + sin ( φ F 1 ) φ ω | γ ( 2 γ ω 2 ) | , 1 2 γ 1 , }
F 1 F ( x 1 , y 1 , ω ) = ω 27 γ ( 2 γ ω 2 ) 3 / 2 .
H ( ω ) = 1 π D exp ( i φ 2 ω y ) exp { i ψ ω y [ ω 2 + 4 ( x 2 + y 2 ) ] } d x d y 1 π D exp ( i φ 2 ω y ) { 1 + i ψ ω y [ ω 2 + 4 ( x 2 + y 2 ) ] } d x d y .
H 1 ( ω ) = 1 π D exp ( i φ 2 ω y ) d x d y .
H 1 ( ω ) = i 2 π φ ω C 1 exp ( i φ 2 ω y ) d x + i 2 π φ ω C 2 exp ( i φ 2 ω y ) d x .
H 1 ( ω ) = 2 π φ ω 0 1 ω 2 / 4 sin [ i φ ( ω 2 + 2 ω 1 x 2 ) ] d x ,
H 1 ( ω ) = 1 π φ ω Im { 1 ω 2 / 4 1 ω 2 / 4 exp [ i φ ( ω 2 + 2 ω 1 x 2 ) ] d x } Im { H ̃ 1 ( ω ) } .
u ( x ) = ω 2 + 2 ω 1 x 2
H ̃ 1 ( ω ) exp [ i φ ω ( 2 ω ) i π / 4 ] π φ 3 ω 3 | k = 0 ( i ) k k ! ( 4 φ ω ) k ( x ) 2 k exp [ i φ ω ( 2 1 x 2 2 + x 2 ) ] | x = 0 .
H ̃ 1 ( ω ) exp [ i φ ω ( 2 ω ) i π / 4 ] π φ 3 ω 3 { 1 + i 3 16 φ ω + O ( φ 2 ) } .
H 1 ( ω ) 1 π φ 3 ω 3 { sin ( φ ω ( 2 ω ) π / 4 ) + 3 16 φ ω cos ( φ ω ( 2 ω ) π / 4 ) } + O ( φ 7 / 2 ) .
H 2 ( ω ) = i ψ ω π D y [ ω 2 + 4 ( x 2 + y 2 ) ] exp ( i φ 2 ω y ) d x d y .
H 2 ( ω ) = C 1 q ( x , y ) exp ( i φ 2 ω y ) d x + C 2 q ( x , y ) exp ( i φ 2 ω y ) d x ,
q ( x , y ) = i ψ ω π exp ( i φ 2 ω y ) y [ ω 2 + 4 ( x 2 + y 2 ) ] exp ( i φ 2 ω y ) d y .
H 2 ( ω ) ψ ω 3 φ 4 π φ ω { h 1   cos ( φ ω ( 2 ω ) π / 4 ) + h 2   sin ( φ ω ( 2 ω ) π / 4 ) + O ( φ 1 ) } ,
h 1 = φ ω ( 2 ω ) h 0 ,
h 2 = 6 3 h 0 φ 2 ω 4 ,
h 0 = 3 + 2 φ 2 ω 3 2 φ 2 ω 2 φ 2 ω 4 .
H ( ω ) = H 1 ( ω ) + H 2 ( ω ) 1 π φ 3 ω 3 { [ 1 ψ h 2 ω 2 φ 3 ] sin ( φ ω ( 2 ω ) π / 4 ) + [ 3 16 φ ω ψ h 1 ω 2 φ 3 ] × cos ( φ ω ( 2 ω ) π / 4 ) } .
H ( ω ) 1 π D { 1 1 2 ( φ 2 ω y + ψ ω y [ ω 2 + 4 ( x 2 + y 2 ) ] ) 2 } d x d y H 0 ( ω ) + H a ( ω ) ,
H 0 ( ω ) 1 π D d x d y = 2 π { arccos ( ω 2 ) ω 2 1 ω 2 4 }
H a ( ω ) = 1 2 π D ( φ 2 ω y + ψ ω y [ ω 2 + 4 ( x 2 + y 2 ) ] ) 2 d x d y = ω 2 π ( g 20 φ 2 + g 11 φ ψ + g 02 ψ 2 ) ,
g 20 = ω 24 ( ω 2 + 26 ) 4 ω 2 ( 1 + ω 2 ) arccos ( ω 2 ) ,
g 11 = ω 18 ( ω 4 + 50 ω 2 + 84 ) 4 ω 2 ( 2 ω 4 + 8 ω 2 + 8 / 3 ) arccos ( ω 2 ) ,
g 02 = ω 360 ( 7 ω 6 + 584 ω 4 + 2502 ω 2 + 1740 ) 4 ω 2 ( ω 6 + 8 ω 4 + 34 ω 2 / 3 + 2 ) arccos ( ω 2 ) .
H ( ω ) H 0 ( ω ) + φ 2 ω 2 24 π { ω ( ω 2 + 26 ) 4 ω 2 24 ( 1 + ω 2 ) arccos ( ω 2 ) } .
H ( ω ) 1 2 ω π + φ 2 ω 2 2 + O ( ω 3 ) ,
H num ( ω ) = 2 π S F ̂ 1 { | F ̂ [ P ( x , y ) ] | 2 } ,
ζ 2 = 2 ω b 2 ω a 2 ω a ω b [ H ( ω ) H num ( ω ) ] 2 ω d ω ,
H ( ω , φ ) sin ( φ ω ( 2 ω ) π / 4 ) π φ 3 ω 3 .
H ( ω , φ ) sin ( φ ω ( 2 ω ) + π / 4 ) π | φ | 3 ω 3 .
| H num ( ω 0 , φ ) H Eq. 44 ( ω 0 , φ ) | = | H num ( ω 0 , φ ) H Eq. 48 ( ω 0 , φ ) | ,
ω 0 = c + b / ( φ + a )
H ( ω , φ ) { ω ω 0 ( φ ) , H Eq. 44 ( ω , φ ) , ω > ω 0 ( φ ) , H Eq. 48 ( ω , φ ) . }
Q ( ω x , ω y ) = B 0 ( ω x , ω y ) B 0 ( ω x , ω y ) 2 + α G η ( ω x , ω y ) / G I 0 ( ω x , ω y ) ,
B 0 ( ω x , ω y ) Δ φ { ( H ( ω x , ω y , φ ) φ ) 2 2 H ( ω x , ω y , φ ) φ 2 H ( ω x , ω y , φ ) }
μ 2 = [ I 0 ( x , y ) I 0 ( x , y ) ] 2 d x d y = α G η ( ω x , ω y ) | B 0 ( ω x , ω y ) | 2 + α G η ( ω x , ω y ) / G I 0 ( ω x , ω y ) d ω x d ω y ,

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