Abstract

An iterative Fourier-transform-based deconvolution method for resolution enhancement is presented. This method makes use of the a priori information that the data are real and positive. The method is robust in the presence of noise and is efficient especially for large data sets, since the fast Fourier transform can be employed.

© 2009 Optical Society of America

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References

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  1. Y. Biraud, “A new approach for increasing the resolving power by data processing,” Astron. Astrophys. 1, 124-127 (1969).
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    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef]
  5. P. A. Jansson, Deconvolution of Images and Spectra (Academic, 1997).
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    [CrossRef]
  7. P. J. Tadrous, BiaQIm image processing software, Version 21.01, 2008, URL http://www.bialith.com.
  8. S. J. Howard, “Method for continuing Fourier spectra given by the fast Fourier transform,” J. Opt. Soc. Am. 71, 95-98 (1981).

1981 (2)

1972 (2)

1970 (1)

1969 (1)

Y. Biraud, “A new approach for increasing the resolving power by data processing,” Astron. Astrophys. 1, 124-127 (1969).

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Figures (3)

Fig. 1
Fig. 1

Upper trace, unprocessed data; lower trace, data after inverse filtering. The traces are not plotted to the same scale and have been offset for clarity.

Fig. 2
Fig. 2

Deconvolved data after 1 (top), 10 (middle), and 100 (bottom) iterations. The traces are not plotted to the same scale and have been offset for clarity.

Fig. 3
Fig. 3

Upper trace, data deconvolved by the method presented here from the 0.08 mm slit width data. Middle trace, 0.01 mm slit width data. Bottom trace, 0.08 mm slit width data deconvolved by the Landweber method with a nonnegativity constraint. The traces are not plotted to the same scale and have been offset for clarity.

Equations (21)

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D ̃ ( v ) = N 1 τ = 0 N 1 D ( τ ) exp ( i 2 π ( v N ) τ ) ,
R ̃ ( v ) = N 1 τ = 0 N 1 R ( τ ) exp ( i 2 π ( v N ) τ ) ,
C ̃ ( v ) = D ̃ ( v ) R ̃ ( v ) ,
C ( τ ) = v = 0 N 1 C ̃ ( v ) exp ( i 2 π ( τ N ) v ) .
A ̃ ( v ) = C ̃ ( v ) R ̃ ( v ) if R ( v ) M ,
A ̃ ( v ) = 0 if R ( v ) < M.
A ( τ ) = v = 0 N 1 A ̃ ( v ) exp ( i 2 π ( τ N ) v ) .
B ( τ ) = v = 0 N 1 B ̃ ( v ) exp ( i 2 π ( τ N ) v ).
A ̃ ( v ) B ̃ ( v ) = 0.
B ̃ ( v ) = N 1 τ = 0 N 1 B ( τ ) exp ( i 2 π ( v N ) τ ) .
H ( y ) = 0 y < 0 , H ( y ) = 1 2 y = 0 , H ( y ) = 1 y > 0 ,
τ = 0 N 1 [ H ( { A ( τ ) + B ( τ ) } ) ] [ A ( τ ) + B ( τ ) ] 2 .
H ( y ) = lim K [ 1 + exp ( K y ) ] 1 .
τ = 0 N 1 [ 1 + exp ( K { A ( τ ) + B ( τ ) } ) ] 1 [ A ( τ ) + B ( τ ) ] 2 .
B ̃ ( v ) τ = 0 N 1 [ 1 + exp ( K { A ( τ ) + B ( τ ) } ) ] 1 [ A ( τ ) + B ( τ ) ] 2 = 0 .
τ = 0 N 1 2 [ A ( τ ) + B ( τ ) ] [ 1 + exp ( K { A ( τ ) + B ( τ ) } ) ] 1 [ exp ( i 2 π v τ N ) ] [ A ( τ ) + B ( τ ) ] 2 [ 1 + exp ( K { A ( τ ) + B ( τ ) } ) ] 2 [ K exp ( K { A ( τ ) + B ( τ ) } ) ] [ exp ( i 2 π v τ N ) ] = 0 .
τ = 0 N 1 [ A ( τ ) + B ( τ ) ] exp ( i 2 π v τ N ) = 0 ,
[ A ( τ ) + B ( τ ) ] = [ A ( τ ) + B ( τ ) ] + + [ A ( τ ) + B ( τ ) ] .
N 1 τ = 0 N 1 exp ( i 2 π v m τ N ) [ A ( τ ) + B ( τ ) ] = N 1 τ = 0 N 1 exp ( i 2 π v m τ N ) [ A ( τ ) + B ( τ ) ] + .
N 1 τ = 0 N 1 exp ( i 2 π v m τ N ) B ( τ ) = N 1 τ = 0 N 1 exp ( i 2 π v m τ N ) [ A ( τ ) + B ( τ ) ] + .
B ̃ ( v ) = N 1 τ = 0 N 1 exp ( i 2 π v τ N ) [ A ( τ ) + B ( τ ) ] + .

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