Abstract

We show that by using an iterative, digital restoration algorithm (Wiener or Kalman), it is possible to improve substantially the defocused optical transfer function of a previously apodized optical system. Consequently, high focal depth can be achieved by the use of an apodizer at the recording step, and a posteriori step of digital restoration. Computer-simulated images exhibit the focal depth achieved.

© 1988 Optical Society of America

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References

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  1. J. E. Pearson, R. H. Freeman, H. C. Reynolds, “Adaptive Optical Techniques for Wave-Front Correction,” in Applied Optics and Optical Engineering, R. R. Shannon, J. C. Wyant, Eds. (Academic, New York, 1979), Chap. 8.
    [CrossRef]
  2. S. H. Lee, Optical Information Processing (Springer-Verlag, Berlin, 1981), Chaps. 1 and 2.
    [CrossRef]
  3. H. C. Andrews, B. R. Hunt, Digital Image Restoration (Prentice-Hall, Englewood Cliffs, NJ, 1977), Chaps. 4, 6, and 8.
  4. B. R. Hunt, “Image Restoration,” in Digital Image Processing Techniques, M. P. Ekstrom, Ed. (Academic, Orlando, FL, 1984), Chap. 2.
  5. W. T. Welford, “Use of Annular Apertures to Increase Focal Depth,” J. Opt. Soc. Am. 50, 749 (1960).
    [CrossRef]
  6. J. Ojeda-Castaneda, L. R. Berriel-Valdos, E. L. Montes, “Line-Spread Function Relatively Insensitive to Defocus,” Opt. Lett. 8, 458 (1983).
    [CrossRef] [PubMed]
  7. J. Ojeda-Castaneda, L. R. Berriel-Valdos, E. L. Montes, “Spatial Filter for Increasing the Depth of Focus,” Opt. Lett. 10, 520 (1985).
    [CrossRef] [PubMed]
  8. G. Indebetouw, H. X. Bai, “Imaging with Fresnel Zone Pupil Masks: Extended Depth of Field,” Appl. Opt. 23, 4299 (1984).
    [CrossRef] [PubMed]
  9. C. Varamit, G. Indebetouw, “Imaging Properties of Defocused Partitioned Pupils,” J. Opt. Soc. Am. A 2, 799 (1985).
    [CrossRef]
  10. J. P. Mills, B. J. Thompson, “Effect of Aberrations and Apodization on the Performance of Coherent Optical Systems. I. The Amplitude Impulse Response,” J. Opt. Soc. Am. A 3, 694 (1986).
    [CrossRef]
  11. G. Hausler, “A Method to Increase the Depth of Focus by Two Step Image Processing,” Opt. Commun. 6, 38 (1972).
    [CrossRef]
  12. R. J. Pieper, A. Korpel, “Image Processing for Extended Depth of Field,” Appl. Opt. 22, 1449 (1983).
    [CrossRef] [PubMed]
  13. A. Erhardt, G. Zinser, D. Komitowski, J. Bille, “Reconstructing 3-D Light-Microscopic Images by Digital Image Processing,” Appl. Opt. 24, 194 (1985).
    [CrossRef] [PubMed]
  14. S. A. Sugimoto, Y. Ichioka, “Digital Composition of Images with Increased Depth of Focus Considering Depth Information,” Appl. Opt. 24, 2076 (1985).
    [CrossRef] [PubMed]
  15. J. Maeda, “Image Restoration by an Iterative Damped Least-Squares Method with Non-Negativity Constraint,” Appl. Opt. 24, 751 (1985).
    [CrossRef] [PubMed]
  16. J. Maeda, “Image Restoration by an Iterative Approximate Wiener Filtering Method,” in Conference Digest, ICO-13, Sapporo ’84 (1984), p. 218.
  17. R. Ramos-López, A. Noyola, L. R. Berriel-Valdos, “Restauración iteractiva de imágenes por medio del filtro de Kalman,” Memorias de la VII Reunión de Ingeniería Electrónica ELECTRO ’85 (1985), p. 282.

1986 (1)

1985 (6)

1984 (1)

1983 (2)

1972 (1)

G. Hausler, “A Method to Increase the Depth of Focus by Two Step Image Processing,” Opt. Commun. 6, 38 (1972).
[CrossRef]

1960 (1)

Andrews, H. C.

H. C. Andrews, B. R. Hunt, Digital Image Restoration (Prentice-Hall, Englewood Cliffs, NJ, 1977), Chaps. 4, 6, and 8.

Bai, H. X.

Berriel-Valdos, L. R.

J. Ojeda-Castaneda, L. R. Berriel-Valdos, E. L. Montes, “Spatial Filter for Increasing the Depth of Focus,” Opt. Lett. 10, 520 (1985).
[CrossRef] [PubMed]

R. Ramos-López, A. Noyola, L. R. Berriel-Valdos, “Restauración iteractiva de imágenes por medio del filtro de Kalman,” Memorias de la VII Reunión de Ingeniería Electrónica ELECTRO ’85 (1985), p. 282.

J. Ojeda-Castaneda, L. R. Berriel-Valdos, E. L. Montes, “Line-Spread Function Relatively Insensitive to Defocus,” Opt. Lett. 8, 458 (1983).
[CrossRef] [PubMed]

Bille, J.

Erhardt, A.

Freeman, R. H.

J. E. Pearson, R. H. Freeman, H. C. Reynolds, “Adaptive Optical Techniques for Wave-Front Correction,” in Applied Optics and Optical Engineering, R. R. Shannon, J. C. Wyant, Eds. (Academic, New York, 1979), Chap. 8.
[CrossRef]

Hausler, G.

G. Hausler, “A Method to Increase the Depth of Focus by Two Step Image Processing,” Opt. Commun. 6, 38 (1972).
[CrossRef]

Hunt, B. R.

H. C. Andrews, B. R. Hunt, Digital Image Restoration (Prentice-Hall, Englewood Cliffs, NJ, 1977), Chaps. 4, 6, and 8.

B. R. Hunt, “Image Restoration,” in Digital Image Processing Techniques, M. P. Ekstrom, Ed. (Academic, Orlando, FL, 1984), Chap. 2.

Ichioka, Y.

Indebetouw, G.

Komitowski, D.

Korpel, A.

Lee, S. H.

S. H. Lee, Optical Information Processing (Springer-Verlag, Berlin, 1981), Chaps. 1 and 2.
[CrossRef]

Maeda, J.

J. Maeda, “Image Restoration by an Iterative Damped Least-Squares Method with Non-Negativity Constraint,” Appl. Opt. 24, 751 (1985).
[CrossRef] [PubMed]

J. Maeda, “Image Restoration by an Iterative Approximate Wiener Filtering Method,” in Conference Digest, ICO-13, Sapporo ’84 (1984), p. 218.

Mills, J. P.

Montes, E. L.

Noyola, A.

R. Ramos-López, A. Noyola, L. R. Berriel-Valdos, “Restauración iteractiva de imágenes por medio del filtro de Kalman,” Memorias de la VII Reunión de Ingeniería Electrónica ELECTRO ’85 (1985), p. 282.

Ojeda-Castaneda, J.

Pearson, J. E.

J. E. Pearson, R. H. Freeman, H. C. Reynolds, “Adaptive Optical Techniques for Wave-Front Correction,” in Applied Optics and Optical Engineering, R. R. Shannon, J. C. Wyant, Eds. (Academic, New York, 1979), Chap. 8.
[CrossRef]

Pieper, R. J.

Ramos-López, R.

R. Ramos-López, A. Noyola, L. R. Berriel-Valdos, “Restauración iteractiva de imágenes por medio del filtro de Kalman,” Memorias de la VII Reunión de Ingeniería Electrónica ELECTRO ’85 (1985), p. 282.

Reynolds, H. C.

J. E. Pearson, R. H. Freeman, H. C. Reynolds, “Adaptive Optical Techniques for Wave-Front Correction,” in Applied Optics and Optical Engineering, R. R. Shannon, J. C. Wyant, Eds. (Academic, New York, 1979), Chap. 8.
[CrossRef]

Sugimoto, S. A.

Thompson, B. J.

Varamit, C.

Welford, W. T.

Zinser, G.

Appl. Opt. (5)

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (2)

Memorias de la VII Reunión de Ingeniería Electrónica ELECTRO ’85 (1)

R. Ramos-López, A. Noyola, L. R. Berriel-Valdos, “Restauración iteractiva de imágenes por medio del filtro de Kalman,” Memorias de la VII Reunión de Ingeniería Electrónica ELECTRO ’85 (1985), p. 282.

Opt. Commun. (1)

G. Hausler, “A Method to Increase the Depth of Focus by Two Step Image Processing,” Opt. Commun. 6, 38 (1972).
[CrossRef]

Opt. Lett. (2)

Other (5)

J. Maeda, “Image Restoration by an Iterative Approximate Wiener Filtering Method,” in Conference Digest, ICO-13, Sapporo ’84 (1984), p. 218.

J. E. Pearson, R. H. Freeman, H. C. Reynolds, “Adaptive Optical Techniques for Wave-Front Correction,” in Applied Optics and Optical Engineering, R. R. Shannon, J. C. Wyant, Eds. (Academic, New York, 1979), Chap. 8.
[CrossRef]

S. H. Lee, Optical Information Processing (Springer-Verlag, Berlin, 1981), Chaps. 1 and 2.
[CrossRef]

H. C. Andrews, B. R. Hunt, Digital Image Restoration (Prentice-Hall, Englewood Cliffs, NJ, 1977), Chaps. 4, 6, and 8.

B. R. Hunt, “Image Restoration,” in Digital Image Processing Techniques, M. P. Ekstrom, Ed. (Academic, Orlando, FL, 1984), Chap. 2.

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Figures (8)

Fig. 1
Fig. 1

Optical transfer functions (OTFs) for the in-focus OTF for a clear aperture (broken line) and the out-of-focus OTF, W20 = λ, for a clear aperture (solid line).

Fig. 2
Fig. 2

Same as Fig. 1, but for the apodized optical system.

Fig. 3
Fig. 3

Restored OTFs: solid lines, obtained using Wiener’s algorithm; dashed line, the in-focus OTF for a clear aperture.

Fig. 4
Fig. 4

Same as Fig. 3, but using Kalman’s algorithm.

Fig. 5
Fig. 5

Same as Fig. 3, but for the apodized optical system.

Fig. 6
Fig. 6

Same as Fig. 4, but when using the apodizer.

Fig. 7
Fig. 7

Image of a spoke pattern: column A, images obtained with a clear aperture; column B, images obtained with the apodized pupil; along line (a), in-focus images; along line (b), the out-of-focus images, with W20 = λ.

Fig. 8
Fig. 8

Restored images of a spoke pattern: column A, those obtained with a clear aperture; column B, those obtained with the apodized pupil; along line (a) Wiener’s restoration scheme; along line (b) Kalman’s restoration scheme.

Equations (19)

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I = HO + N .
E [ O ^ ] = I - H O ^ 2 + ɛ Δ O ^ 2 .
O ^ ( k + 1 ) = O ^ ( k ) + [ H T H + ɛ ( k ) 1 ] - 1 H T [ I - H O ^ ( k ) ] .
O ˜ ( k + 1 ) = O ˜ ( k ) + W ( k ) [ I ˜ - H ˜ O ˜ ( k ) ] ,
W ( k ) = H ˜ * [ H ˜ 2 + ɛ ( k ) ] - 1 ;
O ( k ) = A O ( k - 1 ) .
O ^ ( k ) = A O ^ ( k - 1 ) + G ( k ) I .
P ( k ) = E [ O ( k ) - O ^ ( k ) ] 2 .
E { [ O ( k ) - O ^ ( k ) ] O ^ ( k - 1 ) } = 0 ,
E { [ O ^ ( k ) - O ( k ) ] I } = 0.
O ^ ( k ) = A O ^ ( k - 1 ) + G ( k ) [ I - H A O ^ ( k - 1 ) ] ,
G ( k ) = A P ( k - 1 ) H T [ H P ( k - 1 ) H T - R ( k ) ] - 1
R ( k ) = E [ NN T ] ,
R ( k ) = σ 2 1 ;
P ( k ) = P ( k - 1 ) - G ( k ) HP ( k - 1 ) .
P ( O ) = 256 1 ,
A = 1.
G ˜ ( k ) = P ˜ ( k - 1 ) H ˜ * / [ P ˜ ( k - 1 ) H ˜ 2 + R ˜ ( k ) ] ,
O ˜ ( k ) = O ˜ ( k - 1 ) + G ˜ ( k ) [ I ˜ - H ˜ O ˜ ( k - 1 ) ] ,

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