Abstract

New methods are described for forming intensity estimates of randomly moving objects from quantum-limited data. These methods assume a preprocessed data set consisting of photon differences. Image estimates are formed with the use of the method of maximum likelihood. Algorithms for recovering an image from a measurement of its autocorrelation or triple correlation are derived by considering the limit as the photoconversion rate tends toward infinity.

© 1991 Optical Society of America

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References

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  1. D. L. Snyder, T. J. Schulz, “High-resolution imaging at low-light levels through weak turbulence,” J. Opt. Soc. Am. A 7, 1251–1265 (1990).
    [CrossRef]
  2. L. C. de Freitas, M. J. Northcott, B. J. Brames, J. C. Dainty, “Object reconstruction from photon-limited centroided data of randomly translating images,” in Digital Image Recovery and Synthesis, P. S. Idell, ed., Proc. Soc. Photo-Opt. Instrum. Eng.828, 62–72 (1987).
    [CrossRef]
  3. J. C. Dainty, M. J. Northcott, “Imaging a randomly translating object at low light levels using the triple correlation,” Opt. Commun. 58, 11–14 (1986).
    [CrossRef]
  4. J. R. Fienup, “Reconstruction of an object from the modulus of its Fourier transform,” Opt. Lett. 3, 27–29 (1978).
    [CrossRef] [PubMed]
  5. J. R. Fienup, “Reconstruction and synthesis applications of an iterative algorithm,” in Transformations in Optical Signal Processing, W. T. Rhodes, J. R. Fienup, B. E. A. Saleh, eds., Proc. Soc. Photo-Opt. Instrum. Eng.373, 147–160 (1981).
    [CrossRef]
  6. J. R. Fienup, “Phase retrieval algorithms: a comparison,” Appl. Opt. 21, 2758–2769 (1982).
    [CrossRef] [PubMed]
  7. J. R. Fienup, C. C. Wackerman, “Phase-retrieval stagnation problems and solutions,” J. Opt. Soc. Am. A 3, 1897–1907 (1986).
    [CrossRef]
  8. C. C. Wackerman, A. E. Yagle, “Use of Fourier domain real-plane zeros in phase retrieval,” in Signal Recovery and Synthesis III, Vol. 15 of 1989 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1989), pp. 128–131.
  9. A. W. Lohmann, G. Weigelt, B. Wirnitzer, “Speckle masking in astronomy: triple correlation theory and applications,” Appl. Opt. 22, 4028–4037 (1983).
    [CrossRef] [PubMed]
  10. G. R. Ayers, M. J. Northcott, J. C. Dainty, “Knox–Thompson and triple-correlation imaging through atmospheric turbulence,” J. Opt. Soc. Am. A 5, 963–985 (1988).
    [CrossRef]
  11. A. W. Lohmann, B. Wirnitzer, “Triple correlations,” Proc. IEEE 72, 889–901 (1984).
    [CrossRef]
  12. T. Matsuoka, T. J. Ulrych, “Phase estimation using the bispectrum,” Proc. IEEE 72, 1403–1411 (1984).
    [CrossRef]
  13. J. C. Marron, P. P. Sanchez, R. C. Sullivan, “Unwrapping algorithm for least-squares phase recovery from the modulo 2πbispectrum phase,” J. Opt. Soc. Am. A 7, 14–20 (1990).
    [CrossRef]
  14. 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]
  15. J. C. Dainty, A. H. Greenaway, “Estimation of spatial power spectra in speckle interferometry,”J. Opt. Soc. Am. 69, 786–790 (1979).
    [CrossRef]
  16. D. L. Snyder, Random Point Processes (Wiley, New York, 1975).
  17. A. P. Dempster, N. M. Laird, D. B. Rubin, “Maximum likelihood from incomplete data via the EM algorithm,”J. R. Statist. Soc. Ser. B 39, 1–38 (1977).
  18. These data were provided to us by C. Wackerman of the Environmental Research Institute of Michigan, Ann Arbor, Mich.
  19. The computations for the 128 × 128 image were performed at a rate of 30 iterations per minute using the AMT DAP 500, a 32 × 32 mesh-connected array of processors manufactured by Active Memory Technology Inc., Irvine, Calif.

1990 (2)

1988 (2)

1986 (2)

J. C. Dainty, M. J. Northcott, “Imaging a randomly translating object at low light levels using the triple correlation,” Opt. Commun. 58, 11–14 (1986).
[CrossRef]

J. R. Fienup, C. C. Wackerman, “Phase-retrieval stagnation problems and solutions,” J. Opt. Soc. Am. A 3, 1897–1907 (1986).
[CrossRef]

1984 (2)

A. W. Lohmann, B. Wirnitzer, “Triple correlations,” Proc. IEEE 72, 889–901 (1984).
[CrossRef]

T. Matsuoka, T. J. Ulrych, “Phase estimation using the bispectrum,” Proc. IEEE 72, 1403–1411 (1984).
[CrossRef]

1983 (1)

1982 (1)

1979 (1)

1978 (1)

1977 (1)

A. P. Dempster, N. M. Laird, D. B. Rubin, “Maximum likelihood from incomplete data via the EM algorithm,”J. R. Statist. Soc. Ser. B 39, 1–38 (1977).

Ayers, G. R.

Brames, B. J.

L. C. de Freitas, M. J. Northcott, B. J. Brames, J. C. Dainty, “Object reconstruction from photon-limited centroided data of randomly translating images,” in Digital Image Recovery and Synthesis, P. S. Idell, ed., Proc. Soc. Photo-Opt. Instrum. Eng.828, 62–72 (1987).
[CrossRef]

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]

G. R. Ayers, M. J. Northcott, J. C. Dainty, “Knox–Thompson and triple-correlation imaging through atmospheric turbulence,” J. Opt. Soc. Am. A 5, 963–985 (1988).
[CrossRef]

J. C. Dainty, M. J. Northcott, “Imaging a randomly translating object at low light levels using the triple correlation,” Opt. Commun. 58, 11–14 (1986).
[CrossRef]

J. C. Dainty, A. H. Greenaway, “Estimation of spatial power spectra in speckle interferometry,”J. Opt. Soc. Am. 69, 786–790 (1979).
[CrossRef]

L. C. de Freitas, M. J. Northcott, B. J. Brames, J. C. Dainty, “Object reconstruction from photon-limited centroided data of randomly translating images,” in Digital Image Recovery and Synthesis, P. S. Idell, ed., Proc. Soc. Photo-Opt. Instrum. Eng.828, 62–72 (1987).
[CrossRef]

de Freitas, L. C.

L. C. de Freitas, M. J. Northcott, B. J. Brames, J. C. Dainty, “Object reconstruction from photon-limited centroided data of randomly translating images,” in Digital Image Recovery and Synthesis, P. S. Idell, ed., Proc. Soc. Photo-Opt. Instrum. Eng.828, 62–72 (1987).
[CrossRef]

Dempster, A. P.

A. P. Dempster, N. M. Laird, D. B. Rubin, “Maximum likelihood from incomplete data via the EM algorithm,”J. R. Statist. Soc. Ser. B 39, 1–38 (1977).

Fienup, J. R.

Greenaway, A. H.

Laird, N. M.

A. P. Dempster, N. M. Laird, D. B. Rubin, “Maximum likelihood from incomplete data via the EM algorithm,”J. R. Statist. Soc. Ser. B 39, 1–38 (1977).

Lohmann, A. W.

Marron, J. C.

Matsuoka, T.

T. Matsuoka, T. J. Ulrych, “Phase estimation using the bispectrum,” Proc. IEEE 72, 1403–1411 (1984).
[CrossRef]

Northcott, M. J.

G. R. Ayers, M. J. Northcott, J. C. Dainty, “Knox–Thompson and triple-correlation imaging through atmospheric turbulence,” J. Opt. Soc. Am. A 5, 963–985 (1988).
[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]

J. C. Dainty, M. J. Northcott, “Imaging a randomly translating object at low light levels using the triple correlation,” Opt. Commun. 58, 11–14 (1986).
[CrossRef]

L. C. de Freitas, M. J. Northcott, B. J. Brames, J. C. Dainty, “Object reconstruction from photon-limited centroided data of randomly translating images,” in Digital Image Recovery and Synthesis, P. S. Idell, ed., Proc. Soc. Photo-Opt. Instrum. Eng.828, 62–72 (1987).
[CrossRef]

Rubin, D. B.

A. P. Dempster, N. M. Laird, D. B. Rubin, “Maximum likelihood from incomplete data via the EM algorithm,”J. R. Statist. Soc. Ser. B 39, 1–38 (1977).

Sanchez, P. P.

Schulz, T. J.

Snyder, D. L.

Sullivan, R. C.

Ulrych, T. J.

T. Matsuoka, T. J. Ulrych, “Phase estimation using the bispectrum,” Proc. IEEE 72, 1403–1411 (1984).
[CrossRef]

Wackerman, C. C.

J. R. Fienup, C. C. Wackerman, “Phase-retrieval stagnation problems and solutions,” J. Opt. Soc. Am. A 3, 1897–1907 (1986).
[CrossRef]

C. C. Wackerman, A. E. Yagle, “Use of Fourier domain real-plane zeros in phase retrieval,” in Signal Recovery and Synthesis III, Vol. 15 of 1989 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1989), pp. 128–131.

Weigelt, G.

Wirnitzer, B.

Yagle, A. E.

C. C. Wackerman, A. E. Yagle, “Use of Fourier domain real-plane zeros in phase retrieval,” in Signal Recovery and Synthesis III, Vol. 15 of 1989 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1989), pp. 128–131.

Appl. Opt. (2)

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (5)

J. R. Statist. Soc. Ser. B (1)

A. P. Dempster, N. M. Laird, D. B. Rubin, “Maximum likelihood from incomplete data via the EM algorithm,”J. R. Statist. Soc. Ser. B 39, 1–38 (1977).

Opt. Commun. (1)

J. C. Dainty, M. J. Northcott, “Imaging a randomly translating object at low light levels using the triple correlation,” Opt. Commun. 58, 11–14 (1986).
[CrossRef]

Opt. Lett. (1)

Proc. IEEE (2)

A. W. Lohmann, B. Wirnitzer, “Triple correlations,” Proc. IEEE 72, 889–901 (1984).
[CrossRef]

T. Matsuoka, T. J. Ulrych, “Phase estimation using the bispectrum,” Proc. IEEE 72, 1403–1411 (1984).
[CrossRef]

Other (6)

J. R. Fienup, “Reconstruction and synthesis applications of an iterative algorithm,” in Transformations in Optical Signal Processing, W. T. Rhodes, J. R. Fienup, B. E. A. Saleh, eds., Proc. Soc. Photo-Opt. Instrum. Eng.373, 147–160 (1981).
[CrossRef]

L. C. de Freitas, M. J. Northcott, B. J. Brames, J. C. Dainty, “Object reconstruction from photon-limited centroided data of randomly translating images,” in Digital Image Recovery and Synthesis, P. S. Idell, ed., Proc. Soc. Photo-Opt. Instrum. Eng.828, 62–72 (1987).
[CrossRef]

These data were provided to us by C. Wackerman of the Environmental Research Institute of Michigan, Ann Arbor, Mich.

The computations for the 128 × 128 image were performed at a rate of 30 iterations per minute using the AMT DAP 500, a 32 × 32 mesh-connected array of processors manufactured by Active Memory Technology Inc., Irvine, Calif.

D. L. Snyder, Random Point Processes (Wiley, New York, 1975).

C. C. Wackerman, A. E. Yagle, “Use of Fourier domain real-plane zeros in phase retrieval,” in Signal Recovery and Synthesis III, Vol. 15 of 1989 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1989), pp. 128–131.

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Figures (2)

Fig. 1
Fig. 1

Image reconstruction from Fourier magnitude: (A) original object, (B) Fourier magnitude of the original object, (C) autocorrelation function of the original object, (D) reconstructed image after 1000 iterations of the iterative algorithm described by Eq. (38).

Fig. 2
Fig. 2

Results of the iterative reconstruction procedure for different numbers of iterations: (A) 0, (B) 50, (C) 100, (D) 500.

Equations (40)

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μ ( u ) = R h ( u - x ) λ ( x ) d x ,
μ ( u , t ) = μ [ u - m ( t ) ] ,
Pr [ N ( U × T ) = N ] = ( T M 0 ) N exp ( - T M 0 ) / N ! ,
T T d t ,
M 0 U μ ( u ) d u .
u i = z i + m ( t i ) ,
m ( t ) = [ m ( t 1 ) m ( t N ) ] T .
m ( t ) = m k ,             t T k , k = 1 , , K ,
u k i = z k i + m k ,             k = 1 , , K , i = 1 , , N k ,
y k i = u k i - u k N k , = z k i - z k N k ,             k = 1 , , K ,             i = 1 , , N k - 1 ,
Pr [ N k = N ] = ( T k M 0 ) N exp ( - T k M 0 ) / N ! ,
p z k i N k ( z N ) = M 0 - 1 μ ( z ) ,
p y k i N k ( y / N ) = M 0 - 2 μ ( z ) μ ( z + y ) d z
p y k N k ( y / N ) = M 0 - N μ ( z ) i = 1 N - 1 μ ( z + y i ) d z ,
y k [ y k 1 y k N k - 1 ] T .
R μ ( N ) ( y ) μ ( z ) i = 1 N - 1 μ ( z + y i ) d z
p y k N k ( y / N ) = M 0 - N R μ ( N ) ( y ) .
L ( μ ) = ln k = 1 K p y k N k ( y / N ) Pr [ N k = N ] = - T u ( z ) d z + k = 1 K ln R μ ( N k ) ( y k ) ,
1 T k = 1 K i = 1 N k { j = 1 j i N k - 1 μ ^ ( z - y k i + y k j ) / [ R μ ^ ( N k ) ( y k ) ] } = 1 ,             μ ^ ( z ) > 0 ,
L c d ( u ) = - T ( u ) d u + k = 1 K i = 1 N k ln μ ( z k i ) ,
Q [ μ μ ^ ( r ) ] = E [ L c d ( μ ) { y k , N k } , μ ^ ( r ) ]
Q [ μ μ ^ ( r ) ] = - T μ ( u ) d u + k = 1 K i = 1 N k p ^ z k i y k , N k ( r ) ( z y k , N k ) × ln μ ( z ) d z ,
μ ( u ) = 1 T k = 1 K i = 1 N k p ^ z k i y k , N k ( r ) ( u y k , N k ) ,
p ^ z k i y k , N k ( r ) ( u y k , N k ) = Π j = 1 N k u ^ ( r ) ( u - y k i + y k j ) R μ ^ ( r ) ( N k ) ( y k )
μ ^ ( + 1 ) ( u ) = 1 T k = 1 K i = 1 N k Π j = 1 N k u ^ ( r ) ( u - y k i + y k j ) R μ ^ ( r ) ( N k ) ( y k ) .
p T n ( τ n = M K ) = ( τ M 0 ) ( M K - 1 ) ( M K - 1 ) ! M 0 exp ( - τ M 0 ) ,             τ 0.
p y k n ( y n = M K ) = M 0 - M R μ ( M ) ( y ) .
L ( μ ) = - T μ ( u ) d u + k = 1 K ln R μ ( M ) ( y k ) ,
1 T k = 1 K i = 1 M { j = 1 j i M - 1 μ ^ ( z - y k i + y k j ) / [ R μ ^ ( M ) ( y k ) ] } = 1 ,             μ ^ ( z ) > 0 ,
μ ^ ( r + 1 ) ( u ) = 1 T k = 1 K i = 1 M Π j = 1 M μ ^ ( r ) ( u + y k j - y k i ) R μ ^ ( r ) ( M ) ( y k ) ,
μ ^ ( r + 1 ) ( u ) = μ ^ ( r ) ( u ) 1 T k = 1 K μ ^ ( r ) ( u + y k ) + μ ^ ( r ) ( u - y k ) R μ ^ ( r ) ( 2 ) ( y k ) ,
μ ^ ( r + 1 ) ( u ) = μ ^ ( r ) ( u ) 1 T k = 1 K f ( u , y k 1 , y k 2 ; μ ^ ( r ) ) R μ ^ ( r ) ( 3 ) ( y k 1 , y k 2 ) ,
f ( u , y k 1 , y k 2 ; μ ^ ( r ) ) = μ ^ ( r ) ( u - y k 1 ) μ ^ ( r ) ( u + [ y k 2 - y k 1 ] ) + μ ^ ( r ) ( u - y k 2 ) μ ^ ( r ) ( u + [ y k 1 - y k 2 ] ) + μ ^ ( r ) ( u + y k 1 ) μ ^ ( r ) ( u + y k 2 )
μ ^ ( r + 1 ) ( u ) = μ ^ ( r ) ( u ) 1 T μ ^ ( r ) ( u + y ) + μ ^ ( r ) ( u - y ) R μ ^ ( r ) ( y ) M ( d y ) ,
M N ( d y ) = n = 1 N M n ( d y ) ,
μ ^ ( r + 1 ) ( u ) = μ ^ ( r ) ( u ) μ ^ ( r ) ( u + y ) + μ ^ ( r ) ( u - y ) R μ ^ ( r ) ( y ) [ M N ( d y ) T N ] .
μ ^ ( r + 1 ) ( u ) = μ ^ ( r ) ( u ) 1 2 M 0 μ ^ ( r ) ( u + y ) + μ ^ ( r ) ( u - y ) R μ ^ ( r ) ( y ) R μ ( y ) d y .
μ ^ ( r + 1 ) ( u ) = μ ^ ( r ) ( u ) 1 M 0 μ ^ ( r ) ( u + y ) R μ ^ ( y ) R μ ( y ) d y .
μ ^ ( r + 1 ) ( u ) = 1 M 0 μ ^ ( r ) ( u ) W ( r ) ( u ) ,
μ ^ ( r + 1 ) ( u ) = μ ^ ( r ) ( u ) 1 M 0 2 μ ^ ( r ) ( u + y ) μ ^ ( r ) ( u + z ) R μ ^ ( r ) ( 3 ) ( y , z ) × R μ ( 3 ) ( y , z ) d y d z .

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