Abstract

Images obtained by an aberration-free system are digitally defocused with progressive values of defocusing. Wiener filtering restoration is applied to the defocused images, and the defocusing range for effective restoration is shown. The stability of the filters for restoring images with spatially variant defocusing is also discussed.

© 1983 Optical Society of America

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References

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  1. W. K. Pratt, Digital Image Processing (Wiley-Interscience, New York, 1978).
  2. G. Haüsler, Opt. Commun. 6, 38 (1972).
    [CrossRef]
  3. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968).
  4. H. H. Hopkins, M. J. Yzuel, Opt. Acta 17, 157 (1970).
    [CrossRef]

1972 (1)

G. Haüsler, Opt. Commun. 6, 38 (1972).
[CrossRef]

1970 (1)

H. H. Hopkins, M. J. Yzuel, Opt. Acta 17, 157 (1970).
[CrossRef]

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968).

Haüsler, G.

G. Haüsler, Opt. Commun. 6, 38 (1972).
[CrossRef]

Hopkins, H. H.

H. H. Hopkins, M. J. Yzuel, Opt. Acta 17, 157 (1970).
[CrossRef]

Pratt, W. K.

W. K. Pratt, Digital Image Processing (Wiley-Interscience, New York, 1978).

Yzuel, M. J.

H. H. Hopkins, M. J. Yzuel, Opt. Acta 17, 157 (1970).
[CrossRef]

Opt. Acta (1)

H. H. Hopkins, M. J. Yzuel, Opt. Acta 17, 157 (1970).
[CrossRef]

Opt. Commun. (1)

G. Haüsler, Opt. Commun. 6, 38 (1972).
[CrossRef]

Other (2)

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968).

W. K. Pratt, Digital Image Processing (Wiley-Interscience, New York, 1978).

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Figures (13)

Fig. 1
Fig. 1

Defocused MTFs: (a) case (1), δ0W20 = 0 (—), 0.6λ (---), λ (…), and 2λ (-.-.); (b) case (2), δ0W20 = 4λ (—), 8λ (---), 12λ (…), and 20λ (-.-.).

Fig. 2
Fig. 2

Defocused MTFs for case (1): W20 = 0.6λ and 1.0λ.

Fig. 3
Fig. 3

Original USC girl picture at 256 × 256 pixels and its Fourier transform in logarithmic scale.

Fig. 4
Fig. 4

Top, from left to right: images blurred for case (1) by defocusing values δ0W20 = 0.6λ, λ, and 2λ. Bottom, from left to right images restored from the above images by Wiener filters, with A = 0.0001.

Fig. 5
Fig. 5

Wiener filters (a) and product of defocused MTFs and Wiener filters (b) for defocusing values δ0W20 = 0.6λ and 2λ.

Fig. 6
Fig. 6

Top, from left to right: images blurred for case (2) by defocusing values δ0W20 = 4λ, 8λ, and 12λ. Bottom, from left to right: images restored from the above images by Wiener filters, with A = 0.0001.

Fig. 7
Fig. 7

Top, from left to right: images blurred for case (2) by defocusing values δ0W20 = 20λ, 32λ, and 46λ. Bottom, from left to right: images restored from the above images by Wiener filters with A = 0.0001.

Fig. 8
Fig. 8

Wiener filters for case (2): δ0W20 = 4λ (—), 8λ (---), 12λ (…), and 20λ (-.-.).

Fig. 9
Fig. 9

From left to right and top to bottom: image blurred by defocusing value δ0W20 = 20λ and images restored with values A = 0.01, 0.001, 0.0001, and 0.00001, respectively.

Fig. 10
Fig. 10

Logarithm of the Fourier transform magnitude of the restored image for δ0W20 = 20λ with A = 0.0001.

Fig. 11
Fig. 11

Image blurred by blocks (a), by values δ0W20 = λ (top left), 2λ (top right), 3λ (bottom left), and 4λ (bottom right). Image restored by blocks with Wiener filters (b), and images restored by applying to the four blocks the same filter corresponding to defocusing values 2λ (c), 3λ (d), and 4λ(e), with A = 0.001.

Fig. 12
Fig. 12

From left to right and top to bottom: original image, image blurred by blocks with values δ0W20 = 2λ, 3λ, 4λ, and 5λ, image restored by blocks, and whole image restored by filters corresponding to defocusing values 2λ, 3λ, and 3.5λ, respectively, with A = 0.001.

Fig. 13
Fig. 13

From left to right and top to bottom: original image, image restored by the filter corresponding to 3λ defocusing with A = 0.001, image restored by the same filter with A = 0.0001, image restored by the 4λ filter with A = 0.01, and image restored by the 4λ filter with A = 0.001.

Equations (3)

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g ( x , y ) = f ( x , y ) h ( x x , y y ) dxdy ,
G ( u , υ ) = F ( u , υ ) H ( u , υ ) ,
H * ( u , υ ) | H ( u , υ ) | 2 + ϕ n ( u , υ ) / ϕ f ( u , υ ) ,

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