Abstract

We present a noniterative algorithm for sharp-focus image restoration from multiple defocused images having known relative defocus values between the image planes but unknown absolute defocus values with respect to the in-focus plane. Starting from an arbitrary value of defocus relative to the in-focus plane and using the optical transfer function, the algorithm evaluates the absolute defocus and retrieves the in-focus image of an incoherently illuminated planar object. Experiments with a circular-aperture imaging system confirm the efficiency and robustness of the algorithm. A reconstruction time of 100ms is attained with two defocused 512×512  pixel images (captured by a 10 bit camera) on a 2.16GHz laptop.

© 2009 Optical Society of America

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References

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  1. R. C. Puetter, T. R. Gosnell, and A. Yahil, Annu. Rev. Astron. Astrophys. 43, 139 (2005).
    [CrossRef]
  2. J. W. Goodman, Introduction to Fourier Optics (Roberts & Co., 2005).
  3. A. W. Lohmann, D. Mendlovic, and Z. Zalevsky, Appl. Opt. 36, 7204 (1997).
    [CrossRef]
  4. I. Raveh, D. Mendlovic, and Z. Zalevsky, Opt. Eng. 38, 1620 (1999).
    [CrossRef]
  5. H. S. Wilf, Generating Functionology (Academic, 1994).
  6. H. H. Hopkins, Proc. R. Soc. London, Ser. A 231, 91 (1955).
    [CrossRef]
  7. M. Frigo and S. G. Johnson, http://www.fftw.org.
  8. T. E. Gureyev and K. A. Nugent, J. Opt. Soc. Am. A 13, 1670 (1996).
    [CrossRef]
  9. D. Paganin and K. A. Nugent, Phys. Rev. Lett. 80, 2586 (1998).
    [CrossRef]

2005

R. C. Puetter, T. R. Gosnell, and A. Yahil, Annu. Rev. Astron. Astrophys. 43, 139 (2005).
[CrossRef]

1999

I. Raveh, D. Mendlovic, and Z. Zalevsky, Opt. Eng. 38, 1620 (1999).
[CrossRef]

1998

D. Paganin and K. A. Nugent, Phys. Rev. Lett. 80, 2586 (1998).
[CrossRef]

1997

1996

1955

H. H. Hopkins, Proc. R. Soc. London, Ser. A 231, 91 (1955).
[CrossRef]

Frigo, M.

M. Frigo and S. G. Johnson, http://www.fftw.org.

Goodman, J. W.

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

Gosnell, T. R.

R. C. Puetter, T. R. Gosnell, and A. Yahil, Annu. Rev. Astron. Astrophys. 43, 139 (2005).
[CrossRef]

Gureyev, T. E.

Hopkins, H. H.

H. H. Hopkins, Proc. R. Soc. London, Ser. A 231, 91 (1955).
[CrossRef]

Johnson, S. G.

M. Frigo and S. G. Johnson, http://www.fftw.org.

Lohmann, A. W.

Mendlovic, D.

I. Raveh, D. Mendlovic, and Z. Zalevsky, Opt. Eng. 38, 1620 (1999).
[CrossRef]

A. W. Lohmann, D. Mendlovic, and Z. Zalevsky, Appl. Opt. 36, 7204 (1997).
[CrossRef]

Nugent, K. A.

D. Paganin and K. A. Nugent, Phys. Rev. Lett. 80, 2586 (1998).
[CrossRef]

T. E. Gureyev and K. A. Nugent, J. Opt. Soc. Am. A 13, 1670 (1996).
[CrossRef]

Paganin, D.

D. Paganin and K. A. Nugent, Phys. Rev. Lett. 80, 2586 (1998).
[CrossRef]

Puetter, R. C.

R. C. Puetter, T. R. Gosnell, and A. Yahil, Annu. Rev. Astron. Astrophys. 43, 139 (2005).
[CrossRef]

Raveh, I.

I. Raveh, D. Mendlovic, and Z. Zalevsky, Opt. Eng. 38, 1620 (1999).
[CrossRef]

Wilf, H. S.

H. S. Wilf, Generating Functionology (Academic, 1994).

Yahil, A.

R. C. Puetter, T. R. Gosnell, and A. Yahil, Annu. Rev. Astron. Astrophys. 43, 139 (2005).
[CrossRef]

Zalevsky, Z.

I. Raveh, D. Mendlovic, and Z. Zalevsky, Opt. Eng. 38, 1620 (1999).
[CrossRef]

A. W. Lohmann, D. Mendlovic, and Z. Zalevsky, Appl. Opt. 36, 7204 (1997).
[CrossRef]

Annu. Rev. Astron. Astrophys.

R. C. Puetter, T. R. Gosnell, and A. Yahil, Annu. Rev. Astron. Astrophys. 43, 139 (2005).
[CrossRef]

Appl. Opt.

J. Opt. Soc. Am. A

Opt. Eng.

I. Raveh, D. Mendlovic, and Z. Zalevsky, Opt. Eng. 38, 1620 (1999).
[CrossRef]

Phys. Rev. Lett.

D. Paganin and K. A. Nugent, Phys. Rev. Lett. 80, 2586 (1998).
[CrossRef]

Proc. R. Soc. London, Ser. A

H. H. Hopkins, Proc. R. Soc. London, Ser. A 231, 91 (1955).
[CrossRef]

Other

M. Frigo and S. G. Johnson, http://www.fftw.org.

H. S. Wilf, Generating Functionology (Academic, 1994).

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

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Figures (2)

Fig. 1
Fig. 1

Experimental arrangement: 1, object; 2, LED; 3, imaging lens; 4, circular diaphragm; 5, CCD camera; 6, motorized translation stage.

Fig. 2
Fig. 2

Reconstruction of a sharp-focus image: 1 and 2, defocused images captured by the camera; 3, reconstructed image; 4, in-focus image.

Equations (12)

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H ( ω x , ω y , ϕ ) = 1 Ω P 0 ( x + ω x 2 , y + ω y 2 ) P 0 ( x ω x 2 ,
y ( ω y 2 ) exp [ i 2 ϕ ( ω x x + ω y y ) ] d x d y ,
ϕ n = π D 2 4 λ ( 1 z 0 1 z n ) ,
G ( ω x , ω y , δ ϕ ) = n = 1 N A n H ( ω x , ω y , ϕ 1 ( 0 ) + Δ ϕ n + δ ϕ ) = p = 0 B p ( ω x , ω y ) δ ϕ p ,
B p ( ω x , ω y ) = 1 p ! ( p ) G ( ω x , ω y , δ ϕ ) δ ϕ ( p ) .
B p ( ω x , ω y ) = 1 p ! n = 1 N A n ( p ) H ( ω x , ω y , ϕ 1 ( 0 ) + Δ ϕ n ) ϕ ( p ) .
G ( ω x , ω y , δ ϕ ) = B 0 ( ω x , ω y ) + O ( δ ϕ M + 1 ) .
I n ( ω x , ω y ) = H ( ω x , ω y , ϕ 1 ( 0 ) + Δ ϕ n + δ ϕ ) I 0 ( ω x , ω y ) + η n ( ω x , ω y ) ,
n = 1 N A n I n ( ω x , ω y ) = { B 0 ( ω x , ω y ) + O ( δ ϕ M + 1 ) } I 0 ( ω x , ω y ) + n = 1 N A n η n .
I 0 ( ω x , ω y ) = B 0 * ( ω x , ω y ) B 0 ( ω x , ω y ) 2 + ɛ n = 1 N A n I n ( ω x , ω y ) ,
G ( ω x , ω y , δ ϕ ) = B 0 ( ω x , ω y ) + B 1 ( ω x , ω y ) δ ϕ + O ( δ ϕ M + 1 ) .
δ ϕ 1 B 1 ( ω x , ω y ) { B 0 ( ω x , ω y ) n = 1 N A n I n ( ω x , ω y ) n = 1 N A n I n ( ω x , ω y ) B 0 ( ω x , ω y ) } .

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