Abstract

A four-step phase-restoration procedure that provides faithful reconstructions of images consisting of faint detail superimposed upon bright, broad backgrounds is presented. Such images tend to cause existing phase-retrieval algorithms considerable difficulty. The two middle steps consist of the simplest of our recently reported phaserestoration schemes and of Fienup’s algorithm. The first and crucial step is to subtract a fraction of the central lobe of the given intensity of the Fourier transform of the image. The image is then reconstructed in two parts, which are combined in the final step. Encouraging computational results are presented.

© 1983 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. Y. M. Bruck and L. G. Sodin, “On the ambiguity of the image reconstruction problem,” Opt. Commun. 30, 304–308 (1979).
    [CrossRef]
  2. W. Lawton, “A numerical algorithm for 2-D wavefront reconstruction from intensity measurements in a single plane,” Proc. Soc. Photo-Opt. Instrum. Eng. 231, 94–98 (1980).
  3. R. H. T. Bates, “Fourier phase problems are uniquely solvable in more than one dimension: I: underlying theory,” Optik 61, 247–262 (1982).
  4. C. van Schooneveld, ed., Image Formation from Coherence Functions in Astronomy (Reidel, Dordrecht, The Netherlands, 1979), Part V.
    [CrossRef]
  5. J. R. Fienup, “Reconstruction of an object from the modulus of its Fourier transform,” Opt. Lett. 3, 27–29 (1978);“Space object imaging through the turbulent atmosphere,” Opt. Eng. 18, 529–534 (1979);“Phase retrieval algorithms: a comparison,” Appl. Opt. 21, 2758–2769 (1982).
    [CrossRef] [PubMed]
  6. W. R. Fright and R. H. T. Bates, “Fourier phase problems are uniquely solvable in more than one dimension: III: computational examples for two dimensions,” Optik (to be published).
  7. K. L. Garden and R. H. T. Bates, “Fourier phase problems are uniquely solvable in more than one dimension: II: one-dimensional considerations,” Optik (to be published).
  8. R. N. Bracewell, The Fourier Transform and Its Applications (McGraw-Hill, New York, 1965).
  9. K. T. Knox and B. J. Thompson, “Recovery of images from atmospherically-degraded short-exposure photographs,” Astrophys. J. 193, L45–L48 (1974).
    [CrossRef]
  10. R. H. T. Bates and W. R. Fright, “Towards imaging with a speckle-interferometric optical synthesis telescope,” Mon. Not. R. Astron. Soc. 198, 1017–1031 (1982).

1982 (2)

R. H. T. Bates, “Fourier phase problems are uniquely solvable in more than one dimension: I: underlying theory,” Optik 61, 247–262 (1982).

R. H. T. Bates and W. R. Fright, “Towards imaging with a speckle-interferometric optical synthesis telescope,” Mon. Not. R. Astron. Soc. 198, 1017–1031 (1982).

1980 (1)

W. Lawton, “A numerical algorithm for 2-D wavefront reconstruction from intensity measurements in a single plane,” Proc. Soc. Photo-Opt. Instrum. Eng. 231, 94–98 (1980).

1979 (1)

Y. M. Bruck and L. G. Sodin, “On the ambiguity of the image reconstruction problem,” Opt. Commun. 30, 304–308 (1979).
[CrossRef]

1978 (1)

1974 (1)

K. T. Knox and B. J. Thompson, “Recovery of images from atmospherically-degraded short-exposure photographs,” Astrophys. J. 193, L45–L48 (1974).
[CrossRef]

Bates, R. H. T.

R. H. T. Bates, “Fourier phase problems are uniquely solvable in more than one dimension: I: underlying theory,” Optik 61, 247–262 (1982).

R. H. T. Bates and W. R. Fright, “Towards imaging with a speckle-interferometric optical synthesis telescope,” Mon. Not. R. Astron. Soc. 198, 1017–1031 (1982).

W. R. Fright and R. H. T. Bates, “Fourier phase problems are uniquely solvable in more than one dimension: III: computational examples for two dimensions,” Optik (to be published).

K. L. Garden and R. H. T. Bates, “Fourier phase problems are uniquely solvable in more than one dimension: II: one-dimensional considerations,” Optik (to be published).

Bracewell, R. N.

R. N. Bracewell, The Fourier Transform and Its Applications (McGraw-Hill, New York, 1965).

Bruck, Y. M.

Y. M. Bruck and L. G. Sodin, “On the ambiguity of the image reconstruction problem,” Opt. Commun. 30, 304–308 (1979).
[CrossRef]

Fienup, J. R.

Fright, W. R.

R. H. T. Bates and W. R. Fright, “Towards imaging with a speckle-interferometric optical synthesis telescope,” Mon. Not. R. Astron. Soc. 198, 1017–1031 (1982).

W. R. Fright and R. H. T. Bates, “Fourier phase problems are uniquely solvable in more than one dimension: III: computational examples for two dimensions,” Optik (to be published).

Garden, K. L.

K. L. Garden and R. H. T. Bates, “Fourier phase problems are uniquely solvable in more than one dimension: II: one-dimensional considerations,” Optik (to be published).

Knox, K. T.

K. T. Knox and B. J. Thompson, “Recovery of images from atmospherically-degraded short-exposure photographs,” Astrophys. J. 193, L45–L48 (1974).
[CrossRef]

Lawton, W.

W. Lawton, “A numerical algorithm for 2-D wavefront reconstruction from intensity measurements in a single plane,” Proc. Soc. Photo-Opt. Instrum. Eng. 231, 94–98 (1980).

Sodin, L. G.

Y. M. Bruck and L. G. Sodin, “On the ambiguity of the image reconstruction problem,” Opt. Commun. 30, 304–308 (1979).
[CrossRef]

Thompson, B. J.

K. T. Knox and B. J. Thompson, “Recovery of images from atmospherically-degraded short-exposure photographs,” Astrophys. J. 193, L45–L48 (1974).
[CrossRef]

Astrophys. J. (1)

K. T. Knox and B. J. Thompson, “Recovery of images from atmospherically-degraded short-exposure photographs,” Astrophys. J. 193, L45–L48 (1974).
[CrossRef]

Mon. Not. R. Astron. Soc. (1)

R. H. T. Bates and W. R. Fright, “Towards imaging with a speckle-interferometric optical synthesis telescope,” Mon. Not. R. Astron. Soc. 198, 1017–1031 (1982).

Opt. Commun. (1)

Y. M. Bruck and L. G. Sodin, “On the ambiguity of the image reconstruction problem,” Opt. Commun. 30, 304–308 (1979).
[CrossRef]

Opt. Lett. (1)

Optik (1)

R. H. T. Bates, “Fourier phase problems are uniquely solvable in more than one dimension: I: underlying theory,” Optik 61, 247–262 (1982).

Proc. Soc. Photo-Opt. Instrum. Eng. (1)

W. Lawton, “A numerical algorithm for 2-D wavefront reconstruction from intensity measurements in a single plane,” Proc. Soc. Photo-Opt. Instrum. Eng. 231, 94–98 (1980).

Other (4)

C. van Schooneveld, ed., Image Formation from Coherence Functions in Astronomy (Reidel, Dordrecht, The Netherlands, 1979), Part V.
[CrossRef]

W. R. Fright and R. H. T. Bates, “Fourier phase problems are uniquely solvable in more than one dimension: III: computational examples for two dimensions,” Optik (to be published).

K. L. Garden and R. H. T. Bates, “Fourier phase problems are uniquely solvable in more than one dimension: II: one-dimensional considerations,” Optik (to be published).

R. N. Bracewell, The Fourier Transform and Its Applications (McGraw-Hill, New York, 1965).

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (8)

Fig. 1
Fig. 1

Image space. The smaller and larger squares are what are referred to, respectively, in the text as the inner and outer squares.

Fig. 2
Fig. 2

Fourier space. The filled circles and crosses represent, respectively, actual and in-between sample points. To accord with Eq. (3) and the discussion in Section 3.B, there must be a total of (2M + 1)2 actual and 4M2 in-between sample points.

Fig. 3
Fig. 3

Magnitude of the visibility of the combined image for ξ = 0.1 (refer to Fig. 6). The solid jagged line is the closed curve C (it is jagged because the image is displayed as a square array of discrete pixels). The magnitude within C has been multiplied by 0.03 in order to be able to display the structure of |F(u, υ)| simultaneously both inside and outside C. The central peak of |F(u, υ)| is in fact about 50 times as large as any other peak.

Fig. 4
Fig. 4

High-contrast image: (a) f(x, y); (b) fe (x, y) from Fienup, with N = 50 and with a pseudorandom initial Φ(u, υ); (c) fe (x, y) from crude phase estimation with γ = 0.5; (d) fe(x, y) from Fienup, with N = 10 and with an initial Φ(u, υ) generated by crude phase estimation.

Fig. 5
Fig. 5

High-contrast image. fe(x, y) from crude phase estimation: (a) γ = 0.45, (b) γ = 0.52. fe(x, y) from Fienup, with N = 10 and with an initial Φ(u, υ) generated by crude phase estimation: (c) γ = 0.45, (d) γ = 0.52.

Fig. 6
Fig. 6

The two parts of the combined image: (a) the fog, (b) the detail. Both parts are shown here with their maximum magnitudes the same.

Fig. 7
Fig. 7

Low-contrast image (i.e., combined image with ξ = 0.1). fe(x, y) after detail abstraction: (a) from Fienup, N =50 and with a pseudorandom initial Φ(u, υ); (b) from crude phase estimation with γ = 0.5. fr(x, y) after preliminary defogging, with k = 0.05: phase estimation with γ = 0.5; (d) from Fienup, with N = 10 and with an initial Φ(u, υ) generated by crude phase estimation.

Fig. 8
Fig. 8

fr(x, y) for low-contrast image (ξ = 0.1) after preliminary defogging. From crude phase estimation with γ = 0.5: (a) k = 0.03, (b) k = 0.06. From Fienup, with N = 10 and with an initial Φ(u, υ) generated by crude phase estimation, with γ = 0.5: (c) k = 0.03, (d) k = 0.06.

Equations (17)

Equations on this page are rendered with MathJax. Learn more.

F ( u , υ ) = | F ( u , υ ) | exp { i phase [ F ( u , υ ) ] } f ( x , y ) ,
| F ( u , υ ) | exp [ i Φ ( u , υ ) ] = F e ( u , υ ) f e ( x , y ) ,
f ( x , y ) = 0 , | x | , | y | , > 1 / 2 α , = α 2 m , n = M M F m , n exp [ i 2 π α ( m x + n y ) ] , | x | , | y | , < 1 / 2 α ,
f ( x , y ) 0 , | x | , | y | , > 1 / 2 β .
| F ( u , υ ) | 2 f f ( x , y ) ,
F e ( u , υ ) = m , n = M M F e m , n samp ( u / α m ) samp ( υ / α n )
| F e m , n | = | F m , n |
samp ( u ) = samp ( u ) = 0 , | u | 1 .
samp ( 0 ) = 0 , samp ( 1 / 2 ) = γ ,
F e ( u , υ ) = F s ( u , υ ) + F r ( u , υ ) , F s ( u , υ ) = 0 outside C , | F r ( u , υ ) | = k | F ( u , υ ) | inside C ,
| F s ( u , υ ) | ( 1 k ) | F ( u , υ ) | inside C
F s ( u , υ ) f s ( x , y ) , F r ( u , υ ) f r ( x , y ) .
cos ( θ l + 1 θ l ) = ( γ 2 B l A l A l + 1 ) / 2 a l a l + 1 .
F e e ( u , υ ) f e e ( x , y )
Φ ( u , υ ) = phase [ F e e ( u , υ ) ] .
| F e e ( u , υ ) | = | F ( u , υ ) |
f e ( x , y ) = f s ( x , y ) + f r ( x , y ) .