Abstract

The bispectrum is the Fourier transform of the triple correlation, sometimes also referred to as the triple-product integral. The influence of photon noise on the bispectrum of an image-intensity distribution is discussed. As an example, the astronomical speckle-masking method is considered. Speckle masking is a method to overcome image degradation that is due to the turbulent atmosphere. It is shown theoretically that bispectral analysis in speckle masking should yield true, diffraction-limited images in all those cases in which the speckle-interferometry process has been successful in reconstructing the object autocorrelation.

© 1985 Optical Society of America

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References

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  1. J. W. Goodman, J. F. Belsher, “Photon limited images and their restoration,”ARPA Order No. 2646, (Rome Air Development Center, New York, 1976).
  2. J. W. Goodman, J. F. Belsher, “Precompensation and post-compensation of photon limited images,”ARPA Order No. 2646, (Rome Air Development Center, New York, 1976); “Photon limitations in imaging and image restoration,” (Rome Air Development Center, New York, 1977).
  3. A. Labeyrie, “Attainment of diffraction-limited resolution in large telescopes by Fourier analysing speckle patterns in star images,” Astron. Astrophys. 6, 85 (1970).
  4. J. C. Dainty, A. H. Greenaway, “Estimation of spatial power spectra in speckle interferometry,”J. Opt. Soc. Am. 64, 786 (1979).
    [Crossref]
  5. G. P. Weigelt, “Modified speckle interferometry: speckle masking,” Opt. Commun. 21, 55 (1977).
    [Crossref]
  6. G. P. Weigelt, B. Wirnitzer, “Image reconstruction by the speckle masking method,” Opt. Lett. 8, 389 (1983).
    [Crossref] [PubMed]
  7. A. W. Lohmann, G. P. Weigelt, B. Wirnitzer, “Speckle masking in astronomy—triple correlation theory and applications,” Appl. Opt. 22, 4028 (1983).
    [Crossref] [PubMed]
  8. H. Gamo, “Triple correlator of photo electric fluctuations as a spectroscopic tool,” J. Appl. Phys. 34, 875 (1963); “Phase determination of coherence functions by the intensity interferometer,” in Symposium on Electromagnetic Theory and Antennas (Pergamon, New York, 1963), pp. 801–810.
    [Crossref]
  9. T. Sato, S. Wadaka, J. Yamamoto, J. Ishij, “Imaging system using an intensity triple correlator,” Appl. Opt. 17, 2047 (1978).
    [Crossref] [PubMed]
  10. T. Sato, K. Sasaki, “Bispectral holography,”J. Acoust. Soc. Am. 62, 404 (1977).
    [Crossref]
  11. H. Bartelt, A. W. Lohmann, B. Wirnitzer, “Phase and amplitude recovery from bispectra,” Appl. Opt. 23, 3121 (1984).
    [Crossref] [PubMed]
  12. B. Wirnitzer, “Measurement of ultrashort laser pulses,” Opt. Commun. 48, 225 (1983).
    [Crossref]
  13. A. W. Lohmann, B. Wirnitzer, “Triple correlations,” Proc. IEEE 22, 889 (1984).
    [Crossref]
  14. M. G. Miller, “Noise considerations in stellar speckle interferometry,”J. Opt. Soc. Am. 67, 1176 (1977).
    [Crossref]
  15. J. C. Dainty, “The transfer function, signal-to-noise-ratio and limiting magnitude in stellar speckle interferometry,” Mon. Not. R. Astron. Soc. 169, 631 (1974).
  16. F. Roddier, “Signal-to-noise ratio in speckle interferometry,” presented at Meeting on Imaging in Astronomy, Boston, Mass., 1975.

1984 (2)

1983 (3)

1979 (1)

J. C. Dainty, A. H. Greenaway, “Estimation of spatial power spectra in speckle interferometry,”J. Opt. Soc. Am. 64, 786 (1979).
[Crossref]

1978 (1)

1977 (3)

T. Sato, K. Sasaki, “Bispectral holography,”J. Acoust. Soc. Am. 62, 404 (1977).
[Crossref]

G. P. Weigelt, “Modified speckle interferometry: speckle masking,” Opt. Commun. 21, 55 (1977).
[Crossref]

M. G. Miller, “Noise considerations in stellar speckle interferometry,”J. Opt. Soc. Am. 67, 1176 (1977).
[Crossref]

1974 (1)

J. C. Dainty, “The transfer function, signal-to-noise-ratio and limiting magnitude in stellar speckle interferometry,” Mon. Not. R. Astron. Soc. 169, 631 (1974).

1970 (1)

A. Labeyrie, “Attainment of diffraction-limited resolution in large telescopes by Fourier analysing speckle patterns in star images,” Astron. Astrophys. 6, 85 (1970).

1963 (1)

H. Gamo, “Triple correlator of photo electric fluctuations as a spectroscopic tool,” J. Appl. Phys. 34, 875 (1963); “Phase determination of coherence functions by the intensity interferometer,” in Symposium on Electromagnetic Theory and Antennas (Pergamon, New York, 1963), pp. 801–810.
[Crossref]

Bartelt, H.

Belsher, J. F.

J. W. Goodman, J. F. Belsher, “Photon limited images and their restoration,”ARPA Order No. 2646, (Rome Air Development Center, New York, 1976).

J. W. Goodman, J. F. Belsher, “Precompensation and post-compensation of photon limited images,”ARPA Order No. 2646, (Rome Air Development Center, New York, 1976); “Photon limitations in imaging and image restoration,” (Rome Air Development Center, New York, 1977).

Dainty, J. C.

J. C. Dainty, A. H. Greenaway, “Estimation of spatial power spectra in speckle interferometry,”J. Opt. Soc. Am. 64, 786 (1979).
[Crossref]

J. C. Dainty, “The transfer function, signal-to-noise-ratio and limiting magnitude in stellar speckle interferometry,” Mon. Not. R. Astron. Soc. 169, 631 (1974).

Gamo, H.

H. Gamo, “Triple correlator of photo electric fluctuations as a spectroscopic tool,” J. Appl. Phys. 34, 875 (1963); “Phase determination of coherence functions by the intensity interferometer,” in Symposium on Electromagnetic Theory and Antennas (Pergamon, New York, 1963), pp. 801–810.
[Crossref]

Goodman, J. W.

J. W. Goodman, J. F. Belsher, “Precompensation and post-compensation of photon limited images,”ARPA Order No. 2646, (Rome Air Development Center, New York, 1976); “Photon limitations in imaging and image restoration,” (Rome Air Development Center, New York, 1977).

J. W. Goodman, J. F. Belsher, “Photon limited images and their restoration,”ARPA Order No. 2646, (Rome Air Development Center, New York, 1976).

Greenaway, A. H.

J. C. Dainty, A. H. Greenaway, “Estimation of spatial power spectra in speckle interferometry,”J. Opt. Soc. Am. 64, 786 (1979).
[Crossref]

Ishij, J.

Labeyrie, A.

A. Labeyrie, “Attainment of diffraction-limited resolution in large telescopes by Fourier analysing speckle patterns in star images,” Astron. Astrophys. 6, 85 (1970).

Lohmann, A. W.

Miller, M. G.

Roddier, F.

F. Roddier, “Signal-to-noise ratio in speckle interferometry,” presented at Meeting on Imaging in Astronomy, Boston, Mass., 1975.

Sasaki, K.

T. Sato, K. Sasaki, “Bispectral holography,”J. Acoust. Soc. Am. 62, 404 (1977).
[Crossref]

Sato, T.

Wadaka, S.

Weigelt, G. P.

Wirnitzer, B.

Yamamoto, J.

Appl. Opt. (3)

Astron. Astrophys. (1)

A. Labeyrie, “Attainment of diffraction-limited resolution in large telescopes by Fourier analysing speckle patterns in star images,” Astron. Astrophys. 6, 85 (1970).

J. Acoust. Soc. Am. (1)

T. Sato, K. Sasaki, “Bispectral holography,”J. Acoust. Soc. Am. 62, 404 (1977).
[Crossref]

J. Appl. Phys. (1)

H. Gamo, “Triple correlator of photo electric fluctuations as a spectroscopic tool,” J. Appl. Phys. 34, 875 (1963); “Phase determination of coherence functions by the intensity interferometer,” in Symposium on Electromagnetic Theory and Antennas (Pergamon, New York, 1963), pp. 801–810.
[Crossref]

J. Opt. Soc. Am. (2)

J. C. Dainty, A. H. Greenaway, “Estimation of spatial power spectra in speckle interferometry,”J. Opt. Soc. Am. 64, 786 (1979).
[Crossref]

M. G. Miller, “Noise considerations in stellar speckle interferometry,”J. Opt. Soc. Am. 67, 1176 (1977).
[Crossref]

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

J. C. Dainty, “The transfer function, signal-to-noise-ratio and limiting magnitude in stellar speckle interferometry,” Mon. Not. R. Astron. Soc. 169, 631 (1974).

Opt. Commun. (2)

B. Wirnitzer, “Measurement of ultrashort laser pulses,” Opt. Commun. 48, 225 (1983).
[Crossref]

G. P. Weigelt, “Modified speckle interferometry: speckle masking,” Opt. Commun. 21, 55 (1977).
[Crossref]

Opt. Lett. (1)

Proc. IEEE (1)

A. W. Lohmann, B. Wirnitzer, “Triple correlations,” Proc. IEEE 22, 889 (1984).
[Crossref]

Other (3)

F. Roddier, “Signal-to-noise ratio in speckle interferometry,” presented at Meeting on Imaging in Astronomy, Boston, Mass., 1975.

J. W. Goodman, J. F. Belsher, “Photon limited images and their restoration,”ARPA Order No. 2646, (Rome Air Development Center, New York, 1976).

J. W. Goodman, J. F. Belsher, “Precompensation and post-compensation of photon limited images,”ARPA Order No. 2646, (Rome Air Development Center, New York, 1976); “Photon limitations in imaging and image restoration,” (Rome Air Development Center, New York, 1977).

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Figures (3)

Fig. 1
Fig. 1

One-dimensional example for a triple correlation and a bispectrum. (a) Double pulse, (b) its triple correlation, (c) its bispectrum.

Fig. 2
Fig. 2

Illustration of the influence of photon noise. (a) Expected value of the bispectrum of the pulse in Fig. 1(a) in the case of photon-limited raw data, (b) Fourier transform of the bispectrum in (a). We assume that the raw data contained five photons, on average.

Fig. 3
Fig. 3

SNR for various speckle methods: (a) SNR in the bispectrum, (b) in the power spectrum, (c) in the Fourier phase recovered from the bispectrum of Z frames of data. We assume that the speckle interferograms contain 2500 speckles on average and that õ(u) = 1.

Equations (52)

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I ( 3 ) ( x , x ) = I ( x ) I ( x + x ) I ( x + x ) d x ,
I ˜ ( 3 ) ( u , v ) = I ( 3 ) ( x , x ) exp [ - 2 π i ( ux + vx ) ] d x d x = I ˜ ( u ) I ˜ ( v ) I ˜ ( - u - v ) ,
I ˜ ( 3 ) ( u , v = 0 ) = I ˜ ( u ) I ˜ ( 0 ) I ˜ ( - u ) = const . I ˜ ( u ) 2 ,
const . = I ˜ ( 0 ) .
D ( x ) = j = 1 N δ ( x - x j ) ,
D ˜ ( u ) = j = 1 N δ ( x - x j ) exp ( - 2 π i ux ) d x = j = 1 N exp ( - 2 π i u x j ) .
D ˜ ( 3 ) ( u , v ) = D ˜ ( u ) D ˜ ( v ) D ˜ ( - u - v ) = j , k , l exp { - 2 π i [ u ( x j - x l ) + v ( x k - x l ) ] } .
p ( x j ) = c I ( x j ) ,
E j k l [ D ˜ ( 3 ) ( u , v ) ] = j , k , l E j k l ( exp { - 2 π i [ u ( x j - x l ) + v ( x k - x l ) ] } ) .
j = k = l = 1 N E j k l ( exp { - 2 π i [ u ( x j - x l ) + v ( x k - x l ) ] } ) = N .
j k = l = 1 N E j k l ( exp { - 2 π i [ u ( x j - x l ) + v ( x k - x k ) ] } ) = N ( N - 1 ) j k E j k ( exp { - 2 π i [ u ( x j - x k ) ] } ) = N ( N - 1 ) c 2 I ˜ ( u ) 2 ,
p ( x j , x k ) = p ( x j ) p ( x k ) = c 2 I ( x j ) I ( x k ) .
E j k l [ D ˜ ( 3 ) ( u , v ) ] = N + c 2 N ( N - 1 ) [ I ˜ ( u ) 2 + I ˜ ( v ) 2 + I ˜ ( - u - v ) 2 ] + c 3 N ( N - 1 ) ( N - 2 ) I ˜ ( 3 ) ( u , v ) .
E N [ N ( N - 1 ) ( N - r + 1 ) ] = N ¯ I r ,
E j k l , N [ D ˜ ( 3 ) ( u , v ) ] = N ¯ I + N ¯ I 2 [ i ˜ ( u ) 2 + i ˜ ( v ) 2 + i ˜ ( - u - v ) 2 ] + N ¯ I 3 i ˜ ( 3 ) ( u , v ) ,
E [ D ˜ ( 3 ) ( u , v ) ] = N ¯ + N ¯ 2 [ i ˜ ( u ) 2 + i ˜ ( v ) 2 + i ˜ ( - u - v ) 2 ] + N ¯ 3 i ˜ ( 3 ) ( u , v ) ,
E [ D ˜ ( u ) 2 ] = N ¯ + N ¯ 2 i ˜ ( u ) 2 .
Q ˜ ( 3 ) ( u , v ) = D ˜ ( 3 ) ( u , v ) - [ D ˜ ( u ) 2 + D ˜ ( v ) 2 + D ˜ ( - u - v ) 2 - 2 N ] .
E [ Q ˜ ( 3 ) ( u , v ) ] = N ¯ 3 i ˜ ( 3 ) ( u , v ) ,
I n ( x ) = O ( x ) * P n ( x ) ,
I ˜ n ( u ) 2 = O ˜ ( u ) 2 P ˜ n ( u ) 2 .
I ˜ n ( 3 ) ( u , v ) = O ˜ ( 3 ) ( u , v ) P ˜ n ( 3 ) ( u , v ) .
phase { I ˜ n ( 3 ) ( u , v ) } = phase { O ˜ ( 3 ) ( u , v ) } ,
Q ˜ n ( 3 ) ( u , v ) = D ˜ n ( 3 ) ( u , v ) - [ D ˜ n ( u ) 2 + D ˜ n ( v ) 2 + D ˜ n ( - u - v ) 2 - 2 N ]
E [ Q ˜ n ( 3 ) ( u , v ) ] = N ¯ 3 i ˜ n ( 3 ) ( u , v ) ,
( SNR BI ) Z = E [ Q ˜ n ( 3 ) ] σ Q / Z = N ¯ 3 i ˜ n ( 3 ) ( u , v ) σ Q / Z ,
( SNR BI ) Z Z             for n ¯ 1.
( SNR BI ) Z Z n ¯ 3 / 2 b ˜ ( 3 ) ( u , v ) o ˜ ( 3 ) ( u , v )             for n ¯ 1 ,
( SNR POW ) Z Z for n ¯ 1 , ( SNR POW ) Z Z n ¯ b ˜ ( u ) o ˜ ( u ) 2 for n ¯ 1 ,
Δ ϕ BI < arctan σ Q / Z I ˜ n ( 3 ) = arctan 1 / ( SNR BI ) Z 1 / ( SNR BI ) Z ,
arctan 1 / ( SNR BI ) Z 1 / ( SNR BI ) Z .
Δ ϕ PH 2 n ¯ s - 1 / 2 1 / ( SNR BI ) Z .
SNR PHASE = π / 2 Δ ϕ PH .
( SNR PHASE ) Z Z × ½ × n ¯ n 1 / 2             for n ¯ 1 , ( SNR PHASE ) Z Z × ½ × N ¯ 1 / 2 × n ¯ × b ˜ ( u ) × o ˜ ( u )             for n ¯ 1 ,
E [ Q ˜ n ( 3 ) ( u , v ) ] = N ¯ 3 i ˜ n ( 3 ) ( u , v ) .
σ Q 2 = E [ Q ˜ n ( 3 ) 2 ] - E [ Q ˜ n ( 3 ) ] 2 = E { D ˜ n ( 3 ) ( u , v ) 2 + D ˜ n ( u ) 4 + D ˜ n ( v ) 4 + D ˜ n ( - u - v ) 4 + 2 [ D ˜ n ( u ) D ˜ n ( v ) 2 + D ˜ n ( u ) D ˜ n ( - u - v ) 2 + D ˜ n ( v ) D ˜ n ( - u - v ) 2 ] + 4 N 2 - 4 N [ D ˜ n ( u ) 2 + D ˜ n ( v ) 2 + D ˜ n ( - u - v ) 2 ] - [ D ˜ n ( 3 ) ( u , v ) + D ˜ n ( 3 ) * ( u , v ) ] [ D ˜ n ( u ) 2 + D ˜ n ( v ) 2 + D ˜ n ( - u - v ) 2 - 2 N ] } - E [ Q n ( 3 ) ] 2 .
σ Q 2 = j , k , l , m , n , o E ( exp { - 2 π i [ u ( x j - x l - x m + x o ) + v ( x k - x l - x n + x o ) ] } ) ( a ) + j , k , l , m E ( exp { - 2 π i [ u ( x j - x k - x l + x m ] } ) ( b ) + j , k , l , m E ( exp { - 2 π i [ v ( x j - x k - x l + x m ] } ) ( c ) + j , k , l , m E ( exp { - 2 π i [ ( - u - v ) ( x j - x k - x l + x m ) ] } ) ( d ) + 2 j , k , l , m E ( exp { - 2 π i [ u ( x j - x k ) + v ( x l - x m ) ] } ) ( e ) + 2 j , k , l , m E ( exp { - 2 π i [ u ( x j - x k - x l + x m ) + v ( - x l + x m ) ] } ) ( f ) + 2 j , k , l , m E ( exp { - 2 π i [ u ( - x j + x k ) + v ( - x 1 + x m - x j + x k ] } ) ( g ) + 4 N 2 ( h ) - 4 j , k E ( exp { - 2 π i [ u ( x j - x k ) ] } ) ( i ) - 4 N j , k E ( exp { - 2 π i [ v ( x j - x k ) ] } ) ( j ) - 4 N j , k E ( exp { - 2 π i [ ( - u - v ) ( x j - x k ) ] } ) ( k ) - j , k , l , m , n E ( exp { - 2 π i [ u ( x j - x l + x m - x n ) + v ( x k - k l ) ] } ) ( l ) - j , k , l , m , n E ( exp { - 2 π i [ u ( x j - x l ) + v ( x k - x l + x m - x n ) ] } ) ( m ) - j , k , l , m , n E ( exp { - 2 π i [ u ( x j - x l + x m - x n ) + v ( x k - x l + x m - x n ) ] } ) ( n ) + 2 N j , k , l E ( exp { - 2 π i [ u ( x j - x l ) + v ( x k - x l ) ] } ) ( o ) + c . c . of terms ( l ) - ( o ) ( p ) - E ( Q ˜ n ( 3 ) 2 .
j k l m u 0 ,
N ¯ I + 4 N ¯ I 2 + N ¯ I 3 + + N ¯ I 6 i ˜ n ( 3 ) ( u , v ) 2 .
3 ( N ¯ I + 2 N ¯ I 2 ) + , ( b ) + ( c ) + ( d ) 3 ( 2 N ¯ I + 2 N ¯ I 2 ) + , ( e ) + ( f ) + ( g ) 4 N ¯ I 2 + 4 N ¯ I , ( h ) - 3 ( 4 N ¯ I 2 + 4 N ¯ i ) + , ( i ) + ( g ) + ( k ) - 3 ( N ¯ I + 2 N ¯ I 2 ) + , ( l ) + ( m ) + ( n ) + 2 N ¯ I 2 + 2 N ¯ r + , ( o ) - 3 ( N ¯ I + 2 N ¯ I 2 ) + 2 N ¯ I 2 + 2 N ¯ I + , ( p )
N ¯ I 3 + + N ¯ I 6 i ˜ n ( 3 ) ( u , v ) 2             ( a ) - ( p ) .
N ¯ 3 + N ¯ 6 i ˜ n ( 3 ) ( u , v ) 2             ( a ) - ( p ) .
E j , k , l , N [ Q ˜ n ( 3 ) ] = N ¯ I 3 i ˜ n ( 3 ) ( u , v ) = N ¯ I 3 i ˜ n ( u ) i ˜ n ( v ) i ˜ n ( - u - v ) .
E [ Q ˜ n ( 3 ) ] 2 0             for u , v 0 , u v .
σ Q 2 = N ¯ 3 + + N ¯ 6 i ˜ n ( 3 ) ( u , v ) 2 .
( SNR BI ) Z = N ¯ 3 i ˜ n ( 3 ) ( u , v ) ( N ¯ 3 + + N ¯ 6 i ˜ n ( 3 ) ( u , v ) 2 ) 1 / 2 Z .
i ˜ n ( 3 ) ( u , v ) = p ˜ n ( 3 ) ( u , v ) o ˜ ( 3 ) ( u , v ) .
p ˜ n ( 3 ) ( u , v ) n ¯ s 3 / 2 b ˜ ( 3 ) ( u , v ) u , v r 0 / λ f ,             n ¯ s 1 ,
p ˜ n ( 3 ) ( u , v ) 2 n ¯ s - 3 t ˜ ( 3 ) ( u , v ) 2 , u , v r 0 / λ f ,             n ¯ s 1 ,
( SNR BI ) Z = n ¯ s - 3 / 2 b ˜ ( 3 ) ( u , v ) o ˜ ( u , v ) ( N ¯ 3 + n ¯ s - 3 t ˜ ( 3 ) ( u , v ) 2 o ˜ ( u , v ) 2 ) 1 / 2 Z .
( SNR BI ) Z Z ,             n ¯ ,
( SNR BI ) Z n ¯ 3 / 2 b ˜ ( 3 ) ( u , v ) o ˜ ( 3 ) ( u , v ) Z ,             n ¯ 1 ,

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