Abstract

We define self-cross correlation as the cross correlation between an image and a truncated part of it. A new method for finding tight object-support bounds by use of the self-cross-correlation information is addressed. These bounds are used in the generalized-projection method to reconstruct photon-limited atmosphere-degraded images. The iterative Fourier-transform algorithm and the generalized-projection method, with use of the autocorrelation function only, have not been preferred to the Knox–Thompson and the triple-correlation methods for phase-retrieval problems, because the use of the autocorrelation function only usually does not give tight object-support bounds. The tight-object-support-bounds constraint is crucial for the uniqueness of phase retrieval by generalized projections. We find that the information contained in the average self cross correlation gives the required tight object-support bounds. The advantage of our method over the triple-correlation method lies in the speed of computation.

© 1994 Optical Society of America

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References

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  1. A. Labeyrrie, “Attainment of diffraction limited resolution in large telescopes by Fourier speckle patterns in stars’ images,” Astron. Astrophys. 6, 85–87 (1970).
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    [CrossRef]
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    [CrossRef] [PubMed]
  5. A. Levi, H. Stark, “Restoration from phase and magnitude by generalized projections,” in Image Recovery: Theory and Application, H. Stark, ed. (Academic, New York, 1987), pp. 277–319.
  6. M. Nieto-Versperinas, R. Navarro, F. J. Fuentes, “Performance of a simulated annealing algorithm for phase retrieval,” J. Opt. Soc. Am. A 5, 30–38 (1988).
    [CrossRef]
  7. M. J. Perez, M. Nieto-Vesperinas, “Phase retrieval of photon-limited stellar images from information of the power spectrum only,” J. Opt. Soc. Am. A 8, 908–917 (1991).
    [CrossRef]
  8. J. H. Seldin, J. R. Fienup, “Numerical investigation of the uniqueness of phase retrieval,” J. Opt. Soc. Am. A 7, 412–427 (1990).
    [CrossRef]
  9. J. C. Dainty, J. R. Fienup, “Phase retrieval and image reconstruction for astronomy,” in Image Recovery: Theory and Application, H. Stark, ed. (Academic, New York, 1987), pp. 231–275.
  10. K. T. Knox, B. J. Thompson, “Recovery of images from atmospherically degraded short-exposure photographs,” Astron. J. 193, L45–L48 (1974).
    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef]
  15. H. Takajo, T. Takahashi, “Least-squares phase recovery from the bispectrum phase: an algorithm for a two-dimensional object,” J. Opt. Soc. Am. A 8, 1038–1047 (1991).
    [CrossRef]
  16. J. Meng, G. J. M. Aitken, E. K. Hege, J. S. Morgan, “Triple-correlation subplane reconstruction of photon-address stellar images,” J. Opt. Soc. Am. A 7, 1243–1250 (1990).
    [CrossRef]
  17. J. Sebag, J. Arnaud, G. Lelievre, J. L. Nieto, E. L. Coarer, “High-resolution imaging using pupil segmentation,” J. Opt. Soc. Am. A 7, 1237–1242 (1990).
    [CrossRef]
  18. X. Shi, R. K. Ward, “Restoration of images degraded by atmospheric turbulence and detection noise,” J. Opt. Soc. Am. A 9, 364–370 (1992).
    [CrossRef]
  19. T. R. Crimmins, J. R. Fienup, B. J. Thelen, “Improved bounds on object support from autocorrelation support and application to phase retrieval,” J. Opt. Soc. Am. A 7, 3–13 (1990).
    [CrossRef]
  20. B. J. Brames, “Efficient method of support reduction,” Opt. Commun. 64, 333–337 (1990).
    [CrossRef]
  21. J. R. Fienup, T. R. Crimmins, W. Holsztynski, “Reconstruction of the support of an object from the support of its autocorrelation,” J. Opt. Soc. Am. 72, 610–624 (1982).
    [CrossRef]
  22. E. K. Hege, “Notes on noise calibration of speckle imagery,” in Diffraction-Limited Imaging with Very Large Telescopes, D. M. Alloin, J. M. Mariotti, eds. (Kluwer, Norwell, Mass., 1989), pp. 113–124.
    [CrossRef]
  23. J. W. Goodman, Statistical Optics (Wiley, New York, 1985), Chap. 9.
  24. B. K. P. Horn, Robot Vision (MIT Press, Cambridge, Mass., 1986), Chap. 8.
  25. A. V. Oppenheim, J. S. Lim, “The importance of phase in signals,” Proc. IEEE 69, 529–541 (1981).
    [CrossRef]
  26. R. H. T. Bates, F. W. Cady, “Toward true imaging by wideband speckle interferometry,” Opt. Commun. 32, 365 (1980).
    [CrossRef]
  27. M. J. Northcott, G. R. Ayers, J. C. Dainty, “Algorithms for image reconstruction from photon-limited data using the triple correlation,” J. Opt. Soc. Am. A 5, 986–992 (1988).
    [CrossRef]

1992 (1)

1991 (2)

1990 (6)

1988 (2)

1983 (2)

1982 (2)

1981 (1)

A. V. Oppenheim, J. S. Lim, “The importance of phase in signals,” Proc. IEEE 69, 529–541 (1981).
[CrossRef]

1980 (1)

R. H. T. Bates, F. W. Cady, “Toward true imaging by wideband speckle interferometry,” Opt. Commun. 32, 365 (1980).
[CrossRef]

1979 (1)

J. R. Fienup, “Space object imaging through the turbulent atmosphere,” Opt. Eng. 18, 529–534 (1979).
[CrossRef]

1978 (1)

1977 (1)

1974 (1)

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

1970 (1)

A. Labeyrrie, “Attainment of diffraction limited resolution in large telescopes by Fourier speckle patterns in stars’ images,” Astron. Astrophys. 6, 85–87 (1970).

Aitken, G. J. M.

Arnaud, J.

Ayers, G. R.

Bates, R. H. T.

R. H. T. Bates, F. W. Cady, “Toward true imaging by wideband speckle interferometry,” Opt. Commun. 32, 365 (1980).
[CrossRef]

Brames, B. J.

B. J. Brames, “Efficient method of support reduction,” Opt. Commun. 64, 333–337 (1990).
[CrossRef]

Cady, F. W.

R. H. T. Bates, F. W. Cady, “Toward true imaging by wideband speckle interferometry,” Opt. Commun. 32, 365 (1980).
[CrossRef]

Coarer, E. L.

Crimmins, T. R.

Dainty, J. C.

M. J. Northcott, G. R. Ayers, J. C. Dainty, “Algorithms for image reconstruction from photon-limited data using the triple correlation,” J. Opt. Soc. Am. A 5, 986–992 (1988).
[CrossRef]

J. C. Dainty, J. R. Fienup, “Phase retrieval and image reconstruction for astronomy,” in Image Recovery: Theory and Application, H. Stark, ed. (Academic, New York, 1987), pp. 231–275.

Fienup, J. R.

Fuentes, F. J.

Goodman, J. W.

J. W. Goodman, Statistical Optics (Wiley, New York, 1985), Chap. 9.

Hege, E. K.

J. Meng, G. J. M. Aitken, E. K. Hege, J. S. Morgan, “Triple-correlation subplane reconstruction of photon-address stellar images,” J. Opt. Soc. Am. A 7, 1243–1250 (1990).
[CrossRef]

E. K. Hege, “Notes on noise calibration of speckle imagery,” in Diffraction-Limited Imaging with Very Large Telescopes, D. M. Alloin, J. M. Mariotti, eds. (Kluwer, Norwell, Mass., 1989), pp. 113–124.
[CrossRef]

Holsztynski, W.

Horn, B. K. P.

B. K. P. Horn, Robot Vision (MIT Press, Cambridge, Mass., 1986), Chap. 8.

Hudgin, R. H.

Knox, K. T.

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

Labeyrrie, A.

A. Labeyrrie, “Attainment of diffraction limited resolution in large telescopes by Fourier speckle patterns in stars’ images,” Astron. Astrophys. 6, 85–87 (1970).

Lelievre, G.

Levi, A.

A. Levi, H. Stark, “Restoration from phase and magnitude by generalized projections,” in Image Recovery: Theory and Application, H. Stark, ed. (Academic, New York, 1987), pp. 277–319.

Lim, J. S.

A. V. Oppenheim, J. S. Lim, “The importance of phase in signals,” Proc. IEEE 69, 529–541 (1981).
[CrossRef]

Lohmann, A. W.

Meng, J.

Morgan, J. S.

Navarro, R.

Nieto, J. L.

Nieto-Versperinas, M.

Nieto-Vesperinas, M.

Northcott, M. J.

Oppenheim, A. V.

A. V. Oppenheim, J. S. Lim, “The importance of phase in signals,” Proc. IEEE 69, 529–541 (1981).
[CrossRef]

Perez, M. J.

Sebag, J.

Seldin, J. H.

Shi, X.

Stark, H.

A. Levi, H. Stark, “Restoration from phase and magnitude by generalized projections,” in Image Recovery: Theory and Application, H. Stark, ed. (Academic, New York, 1987), pp. 277–319.

Takahashi, T.

Takajo, H.

Thelen, B. J.

Thompson, B. J.

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

Ward, R. K.

Weigelt, G.

Weigelt, G. P.

Wirnitzer, B.

Appl. Opt. (2)

Astron. Astrophys. (1)

A. Labeyrrie, “Attainment of diffraction limited resolution in large telescopes by Fourier speckle patterns in stars’ images,” Astron. Astrophys. 6, 85–87 (1970).

Astron. J. (1)

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

J. Opt. Soc. Am. (2)

J. Opt. Soc. Am. A (10)

H. Takajo, T. Takahashi, “Suppression of the influence of noise in least-squares phase estimation from the phase difference,” J. Opt. Soc. Am. A 7, 1153–1162 (1990).
[CrossRef]

H. Takajo, T. Takahashi, “Least-squares phase recovery from the bispectrum phase: an algorithm for a two-dimensional object,” J. Opt. Soc. Am. A 8, 1038–1047 (1991).
[CrossRef]

J. Meng, G. J. M. Aitken, E. K. Hege, J. S. Morgan, “Triple-correlation subplane reconstruction of photon-address stellar images,” J. Opt. Soc. Am. A 7, 1243–1250 (1990).
[CrossRef]

J. Sebag, J. Arnaud, G. Lelievre, J. L. Nieto, E. L. Coarer, “High-resolution imaging using pupil segmentation,” J. Opt. Soc. Am. A 7, 1237–1242 (1990).
[CrossRef]

X. Shi, R. K. Ward, “Restoration of images degraded by atmospheric turbulence and detection noise,” J. Opt. Soc. Am. A 9, 364–370 (1992).
[CrossRef]

T. R. Crimmins, J. R. Fienup, B. J. Thelen, “Improved bounds on object support from autocorrelation support and application to phase retrieval,” J. Opt. Soc. Am. A 7, 3–13 (1990).
[CrossRef]

M. Nieto-Versperinas, R. Navarro, F. J. Fuentes, “Performance of a simulated annealing algorithm for phase retrieval,” J. Opt. Soc. Am. A 5, 30–38 (1988).
[CrossRef]

M. J. Perez, M. Nieto-Vesperinas, “Phase retrieval of photon-limited stellar images from information of the power spectrum only,” J. Opt. Soc. Am. A 8, 908–917 (1991).
[CrossRef]

J. H. Seldin, J. R. Fienup, “Numerical investigation of the uniqueness of phase retrieval,” J. Opt. Soc. Am. A 7, 412–427 (1990).
[CrossRef]

M. J. Northcott, G. R. Ayers, J. C. Dainty, “Algorithms for image reconstruction from photon-limited data using the triple correlation,” J. Opt. Soc. Am. A 5, 986–992 (1988).
[CrossRef]

Opt. Commun. (2)

R. H. T. Bates, F. W. Cady, “Toward true imaging by wideband speckle interferometry,” Opt. Commun. 32, 365 (1980).
[CrossRef]

B. J. Brames, “Efficient method of support reduction,” Opt. Commun. 64, 333–337 (1990).
[CrossRef]

Opt. Eng. (1)

J. R. Fienup, “Space object imaging through the turbulent atmosphere,” Opt. Eng. 18, 529–534 (1979).
[CrossRef]

Opt. Lett. (2)

Proc. IEEE (1)

A. V. Oppenheim, J. S. Lim, “The importance of phase in signals,” Proc. IEEE 69, 529–541 (1981).
[CrossRef]

Other (5)

E. K. Hege, “Notes on noise calibration of speckle imagery,” in Diffraction-Limited Imaging with Very Large Telescopes, D. M. Alloin, J. M. Mariotti, eds. (Kluwer, Norwell, Mass., 1989), pp. 113–124.
[CrossRef]

J. W. Goodman, Statistical Optics (Wiley, New York, 1985), Chap. 9.

B. K. P. Horn, Robot Vision (MIT Press, Cambridge, Mass., 1986), Chap. 8.

A. Levi, H. Stark, “Restoration from phase and magnitude by generalized projections,” in Image Recovery: Theory and Application, H. Stark, ed. (Academic, New York, 1987), pp. 277–319.

J. C. Dainty, J. R. Fienup, “Phase retrieval and image reconstruction for astronomy,” in Image Recovery: Theory and Application, H. Stark, ed. (Academic, New York, 1987), pp. 231–275.

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Figures (9)

Fig. 1
Fig. 1

(a) Test object, (b) phasor image of the test object shown in (a), (c) typical short-exposure PSF, (d) phasor image of the average self-cross correlation 〈pg〉, (e) three-dimensional plot of the average cross correlation 〈hp〉, (e′) cross section of (e), (f) cross section of the derivatives |2hp/∂x∂y| of 〈hp〉 in (e), (g) phasor image of the average cross correlation 〈hp〉, (h) three-dimensional plot of the phasor image shown in (g).

Fig. 2
Fig. 2

(a) First computer-simulated object, (b) second computer-simulated object.

Fig. 3
Fig. 3

(a) Phasor image of the object shown in Fig. 2(a), (b) phasor image of the object shown in Fig. 2(b).

Fig. 4
Fig. 4

(a) Long exposure image corresponding to the object shown in Fig. 2(a), (b) subimage truncated from (a).

Fig. 5
Fig. 5

(a) Long-exposure image corresponding to the object shown in Fig. 2(b), (b) subimage truncated from (a).

Fig. 6
Fig. 6

(a) Average of five phasor images of the self cross spectra corresponding to the image shown in Fig. 4(a), (b) average of five phasor images of the self cross spectra corresponding to the image sown in Fig. 4(b).

Fig. 7
Fig. 7

(a) Scaled and thresholded average phasor image shown in Fig. 6(a), (b) scaled and thresholded average phasor image shown in Fig. 6(b).

Fig. 8
Fig. 8

(a) Reconstructed image corresponding to the object shown in Fig. 2(a), (b) reconstructed image corresponding to the object shown in Fig. 2(b).

Fig. 9
Fig. 9

(a) Long exposure image of the triple star ADS 11344, (b) reconstructed image of the triple star ADS 11344.

Equations (34)

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r f 1 f 2 = f 1 f 2 = f 1 ( r + r 1 ) f 2 ( r 1 ) d r 1 ,
R f 1 f 2 = F 1 ( ω x , ω y ) F 2 * ( ω x , ω y ) ,
f 1 ( x , y ) 0 ( x , y ) S , f 2 ( x , y ) = { 0 f 1 ( x , y ) ( x , y ) R R ,
p t ( x , y ) = g t ( x , y ) cir ( r ) = { 0 ( x , y ) g t ( x , y ) ( x , y ) S ,
cir ( r ) = cir ( x x 0 , y y 0 ) = 1 r = [ ( x x 0 ) 2 + ( y y 0 ) 2 ] 1 / 2 d .
g ( x , y ) = m = 1 K g δ ( x x m , y y m ) + n g ( x , y ) ( x , y ) S ,
p ( x , y ) = m = 1 K p δ ( x x m , y y m ) + n p ( x , y ) ( x , y ) ,
g ( x , y ) = h ( x , y ) f g ( x , y ) + n photon [ x , y , g 0 ( x , y ) ] + n g ( x , y ) ( x , y ) S ,
g 0 ( x , y ) = E [ g ( x , y ) | h ( x , y ) ] = h ( x , y ) * f g ( x , y ) ( x , y ) S .
p 0 ( x , y ) = E [ p ( x , y ) | h ( x , y ) ] ( x , y ) .
G ( ω x , ω y ) = k = 1 K g exp [ j ( ω x x k + ω y y k ) ] + N g ( ω x , ω y ) ,
P ( ω x , ω y ) = k = 1 K p exp [ j ( ω x x k + ω y y k ) ] + N p ( ω x , ω y ) .
g ( x , y ) = h ( x , y ) f g ( x , y ) + n g ( x , y ) = g 0 ( x , y ) + n g ( x , y ) ( x , y ) S ,
p ( x , y ) = p 0 ( x , y ) + n p ( x , y ) ( x , y ) ,
G P * = k = 1 K g l = 1 K p exp { j [ ω x ( x k x l ) + ω y ( y k y l ) ] } .
E [ G P * | H , K g ] = K p + K g K p K p K g K p G 0 P 0 * = K p + k g 1 K g G 0 P 0 * K p + K g 1 K g H F g P 0 * ,
( E [ G P * | H , K g ] K p ) K g K g 1 = G 0 P 0 * = H F g P 0 * .
( G P * K p ) K g K g 1 = G 0 P 0 * + N g p = H F g P 0 * + N g p ,
( G P * K p ) K g K g 1 = G 0 P 0 * + N g p = F g H P 0 * + N g p ,
G P * = F H P * = F H P * .
G 0 P 0 * K g K g 1 [ G ( ω x , ω y ) P * ( ω x , ω y ) K p = T { K g K g 1 [ g ( x , y ) p ( x , y ) K p δ ( x , y ) ] } = T { K g K g 1 k = 1 K g l = 1 , k l K p δ ( x k x l x , y k y l y } ,
b im ( x , y ) = I T ( G 0 P 0 * | G 0 P 0 * | ) .
b im e ( x , y ) = 1 J I = 1 J I T ( G 0 P 0 I * | G 0 P 0 I * | ) .
b im e ( x , y ) = { 1 b im e ( x , y ) > thres 0 b im e ( x , y ) thres ,
R g g ( ω x , ω y ) = | H ( ω x , ω y ) | 2 F g ( ω x , ω y ) 2 T { K g K g 1 [ g ( x , y ) g ( x , y ) K g δ ( x , y ) ] } = T { K g K g 1 k = 1 K g l = 1 , k l K g δ ( x k x l x , y k y l y } .
| F g | = [ R g g ( ω x , ω y ) | H ( ω x , ω y ) | 2 + c ( K g ¯ + K h ¯ ) A ( ω x , ω y ) ] 1 / 2 ,
F g ( ω ) = | F g ( ω ) | exp [ j ϕ f g ( ω ) ] ,
1 = { f g ( x , y ) : { f g ( x , y ) 0 ( x , y ) f g ( x , y ) = 0 ( x , y ) } ,
2 = { f g ( x , y ) | F g ( ω ) | exp [ j ϕ f g ( ω ) ] : | F g ( ω ) | = | F ̂ g ( ω x , ω y ) | } ,
P 1 f g ( i ) ( x , y ) = { f g ( i + 1 ) ( x , y ) 0 ( x , y ) R 0 ( x , y ) R ,
P 2 f g ( i ) ( x , y ) | F ̂ g ( ω ) | exp [ j ϕ f g ( i ) ( ω ) ] .
f g ( i + 1 ) ( x , y ) = T 2 T 1 f g ( i ) ( x , y ) i = 0 , 1 , 2 , ,
T 1 = 1 + λ 1 ( P 1 1 ) , T 2 = 1 + λ 2 ( P 2 1 ) .
h ( x , y , t ) = α exp [ j φ ( x , y , t ) ] exp [ j φ ( x , y , t ) ] ,

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