Abstract

The properties of the phase-gradient algorithm for retrieving the Fourier phase from stellar speckle images are examined with respect to random image motion, centroiding, photon noise, and stray photons. The phase gradient is found to be immune to the degradation from image motion and distortions at low photon levels that occur in the Knox–Thompson process. The signal-to-noise ratios of the two methods are found to be comparable. Bias terms that are due to stray photons caused by background light or detector noise are derived.

© 1989 Optical Society of America

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References

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  1. F. Roddier, “The effects of atmospheric turbulence in optical astronomy,” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1981), Vol. 19, pp. 281–376.
    [CrossRef]
  2. A. Labeyrie, “Attainment of diffraction-limited resolution in large telescopes by Fourier analysing the speckle patterns in star images,” Astron. Astrophys. 6, 85–87 (1970).
  3. K. T. Knox, “Image retrieval from astronomical speckle patterns,”J. Opt. Soc. Am. 66, 1236–1239 (1976).
    [CrossRef]
  4. G. J. M. Aitken, R. Johnson, R. Houtman, “Phase-gradient stellar image reconstruction,” Opt. Commun. 56, 379–383 (1986).
    [CrossRef]
  5. G. J. M. Aitken, R. Johnson, “Phase-gradient reconstruction from photon-limited stellar speckle images,” Appl. Opt. 26, 4246–4249 (1987).
    [CrossRef] [PubMed]
  6. 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]
  7. S. E. Ebstein, “Speckle imaging of active galactic nuclei: NGC 1068 and NGC 4151,” doctoral dissertation (Harvard University, Cambridge, Mass., 1987).
  8. J. W. Goodman, J. F. Belsher, “Fundamental limitations in linear invariant restoration of atmospherically degraded images,” in Imaging through the Atmosphere, J. C. Wyant, ed., Proc. Soc. Photo-Opt. Instrum. Eng.75, 141–154 (1976).
    [CrossRef]
  9. A. Papoulis, Probability, Random Variables and Stochastic Processes (McGraw-Hill, New York, 1965).

1988 (1)

1987 (1)

1986 (1)

G. J. M. Aitken, R. Johnson, R. Houtman, “Phase-gradient stellar image reconstruction,” Opt. Commun. 56, 379–383 (1986).
[CrossRef]

1976 (1)

1970 (1)

A. Labeyrie, “Attainment of diffraction-limited resolution in large telescopes by Fourier analysing the speckle patterns in star images,” Astron. Astrophys. 6, 85–87 (1970).

Aitken, G. J. M.

G. J. M. Aitken, R. Johnson, “Phase-gradient reconstruction from photon-limited stellar speckle images,” Appl. Opt. 26, 4246–4249 (1987).
[CrossRef] [PubMed]

G. J. M. Aitken, R. Johnson, R. Houtman, “Phase-gradient stellar image reconstruction,” Opt. Commun. 56, 379–383 (1986).
[CrossRef]

Ayers, G. R.

Belsher, J. F.

J. W. Goodman, J. F. Belsher, “Fundamental limitations in linear invariant restoration of atmospherically degraded images,” in Imaging through the Atmosphere, J. C. Wyant, ed., Proc. Soc. Photo-Opt. Instrum. Eng.75, 141–154 (1976).
[CrossRef]

Dainty, J. C.

Ebstein, S. E.

S. E. Ebstein, “Speckle imaging of active galactic nuclei: NGC 1068 and NGC 4151,” doctoral dissertation (Harvard University, Cambridge, Mass., 1987).

Goodman, J. W.

J. W. Goodman, J. F. Belsher, “Fundamental limitations in linear invariant restoration of atmospherically degraded images,” in Imaging through the Atmosphere, J. C. Wyant, ed., Proc. Soc. Photo-Opt. Instrum. Eng.75, 141–154 (1976).
[CrossRef]

Houtman, R.

G. J. M. Aitken, R. Johnson, R. Houtman, “Phase-gradient stellar image reconstruction,” Opt. Commun. 56, 379–383 (1986).
[CrossRef]

Johnson, R.

G. J. M. Aitken, R. Johnson, “Phase-gradient reconstruction from photon-limited stellar speckle images,” Appl. Opt. 26, 4246–4249 (1987).
[CrossRef] [PubMed]

G. J. M. Aitken, R. Johnson, R. Houtman, “Phase-gradient stellar image reconstruction,” Opt. Commun. 56, 379–383 (1986).
[CrossRef]

Knox, K. T.

Labeyrie, A.

A. Labeyrie, “Attainment of diffraction-limited resolution in large telescopes by Fourier analysing the speckle patterns in star images,” Astron. Astrophys. 6, 85–87 (1970).

Northcott, M. J.

Papoulis, A.

A. Papoulis, Probability, Random Variables and Stochastic Processes (McGraw-Hill, New York, 1965).

Roddier, F.

F. Roddier, “The effects of atmospheric turbulence in optical astronomy,” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1981), Vol. 19, pp. 281–376.
[CrossRef]

Appl. Opt. (1)

Astron. Astrophys. (1)

A. Labeyrie, “Attainment of diffraction-limited resolution in large telescopes by Fourier analysing the speckle patterns in star images,” Astron. Astrophys. 6, 85–87 (1970).

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (1)

Opt. Commun. (1)

G. J. M. Aitken, R. Johnson, R. Houtman, “Phase-gradient stellar image reconstruction,” Opt. Commun. 56, 379–383 (1986).
[CrossRef]

Other (4)

F. Roddier, “The effects of atmospheric turbulence in optical astronomy,” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1981), Vol. 19, pp. 281–376.
[CrossRef]

S. E. Ebstein, “Speckle imaging of active galactic nuclei: NGC 1068 and NGC 4151,” doctoral dissertation (Harvard University, Cambridge, Mass., 1987).

J. W. Goodman, J. F. Belsher, “Fundamental limitations in linear invariant restoration of atmospherically degraded images,” in Imaging through the Atmosphere, J. C. Wyant, ed., Proc. Soc. Photo-Opt. Instrum. Eng.75, 141–154 (1976).
[CrossRef]

A. Papoulis, Probability, Random Variables and Stochastic Processes (McGraw-Hill, New York, 1965).

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Equations (57)

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I n ξ = d I n / d f ξ = j 2 π FT ( ξ i n ) ,
D n ξ = Re { - j I n ξ I n * } ,
D n ξ = - 2 π Re { FT ( i n ξ ) i n } .
α ξ = D n ξ / I n 2 .
i n ( r ) = k = 1 N n δ ( r - r k ) ,
I n ( f ) = k = 1 N n exp ( - j 2 π f · r k ) ,
D n ξ = - 2 π Re { FT l = 1 k l N n k = 1 N n ξ l δ ( r l - r k - r ) } .
I n 2 = FT l = 1 l k N n k = 1 N n δ ( r l - r k - r )
P n ξ = I n ( f + Δ f ξ ) I n * ( f ) ,
P n ξ = FT [ i n exp ( - j 2 π Δ f ξ ξ ) ] i n ,
P n ξ = FT l = 1 l k N n k = 1 N n exp ( - j 2 π Δ f ξ ξ l ) δ ( r l - r k - r )
I n w ( f ) = I n ( f ) exp ( - j 2 π f · w n ) .
P n c ξ = N n ( N n - 1 ) I ˜ ( f + Δ f ξ - Δ f ξ N n ) × I ˜ * ( f + Δ f ξ N n ) I ˜ * ( Δ f ξ N n ) N n - 2 ,
P n ξ = N n ( N n - 1 ) I ˜ ( f + Δ f ξ ) I ˜ * ( f ) ,
I n w ξ = ( I n ξ - j 2 π w n ξ I n ) exp ( - j 2 π f · w n ) .
D n w ξ = Re { - j I n w ξ I n w * } = D n ξ - 2 π w n ξ I n 2 ,
c n = { 1 n n p = 1 N n r p , N n 0 0 , N n = 0 .
Re { - j I n ξ I n * } = Re ( k = 1 N n l = 1 N n ( - 2 π ) [ ξ k - 1 N n ( ξ k + ξ l + p = 1 p k l N n ξ p ) ] × exp { - j 2 π f · [ r k - 1 N n ( r k + p = 1 p k N n r p ) ] } × exp { j 2 π f · [ r l - 1 N n ( r l + p = 1 p l N n r p ) ] } ) .
p ( r k ) = i n ( r k ) / I n ( 0 ) .
D n c ξ = ( N n - 1 ) ( N n - 2 ) Re { - j [ I ˜ n ξ I ˜ n ξ * - I ˜ n ξ ( 0 ) I ˜ n 2 ] } m ,
D n c ξ = { N ¯ 2 - 2 N ¯ + 2 [ 1 - exp ( - N ¯ ) ] } × ( D ˜ n ξ + 2 π c n ξ I n 2 ) .
D n ξ = N n ( N n - 1 ) Re { - j I ˜ n ξ I ˜ n * } = N ¯ 2 D ˜ n ξ ,
σ D ξ 2 = D n ξ 2 - D n ξ 2 .
σ D ξ 2 = [ π 2 i j k l ( x k x i ) × ( { exp [ j 2 π f · ( r l - r k + r j - r i ) ] + c . c } + { exp [ j 2 π f · ( r l - r k - r j + r i ) ] + c . c . } ) ] - ( { π k i j , l k l ( - x k ) exp [ j 2 π f · ( r l - r k ) ] + c . c . } 2 ) ,
σ D ξ 2 = ( 1 / 2 ) N ¯ 2 [ - I ˜ n ξ ( 0 ) + I ˜ n ξ ( 0 ) 2 ] .
σ α ξ 2 = I n 2 - 2 [ σ D ξ 2 + ( D n ξ / I n 2 ) 2 σ I 2 ] N - 1 = I n 2 - 2 [ σ D ξ 2 + ( α ξ ) 2 σ I 2 ] N - 1 ,
I ˜ ( f ) = exp [ - 4.4 ( f / f l ) 2 ] ,
I n 2 = N ¯ 2 ( r 0 / d ) 2 T ( f ) ,
σ α ξ = [ ( 2.1 / f l ) + α ξ ] / [ N ¯ ( r 0 / d ) 2 T ( f ) N 1 / 2 ] .
σ Δ α PG = 2.1 ( Δ f / f l ) / [ N ps T ( f ) N 1 / 2 ] .
σ P ξ 2 = P n ξ 2 - P n ξ 2 ,
σ Δ α KT 2 = σ P ξ 2 / P n ξ 2 N .
σ Δ α KT = 1 / [ N p s T ( f ) N 1 / 2 ] .
p s ( r ) = a ( r ) / a ( r ) d r = a ˜ ( r ) ,
i s n = k = 1 N n δ ( r - r k ) + q = 1 Q n δ ( r - r q ) ,
I s n 2 = I n 2 + k = 1 N n p = 1 Q n exp [ - j 2 π f · ( r k - r p ) ] + l = 1 N n q = 1 Q n exp [ - j 2 π f · ( r q - r l ) ] + q = 1 Q n p = 1 Q n exp [ - j 2 π f · ( r q - r p ) ] ,             k 1 ,             q p .
I s n 2 = N ¯ 2 I ˜ n 2 + 2 N Q ¯ Re { I ˜ n ( f ) } A ˜ ( f ) } + Q ¯ 2 A ˜ ( f ) 2 .
D n s ξ = N ¯ 2 D ˜ n ξ + N ¯ Q ¯ Im { I ˜ n ξ ( f ) A ˜ * ( f ) + I ˜ n * ( f ) A ˜ ξ ( f ) } + Q ¯ 2 Im { A ˜ ξ ( f ) A ˜ * ( f ) } .
P n s ξ = N ¯ 2 P ˜ n ξ + N ¯ Q ¯ [ I ˜ n ( f + Δ f ξ ) A ˜ * ( f ) + I ˜ n * ( f ) A ˜ ( f + Δ f ) ] + Q ¯ 2 A ˜ ( f + Δ f ξ ) A ˜ * ( f ) .
A ˜ ( i , j ) = { 1 , i = j = 0 0 , i 0 , j 0
A ˜ ξ ( 0 , 0 ) = 0.
B PG ( i , j ) = { N ¯ Q ¯ Im { I ˜ n ξ ( 0 , 0 ) } , i = j = 0 N ¯ Q ¯ Im { I ˜ n ( p , q ) A ˜ ξ ( p , q ) } , otherwise ,
B KT ( i , j ) = { N ¯ Q ¯ I ˜ n ( 0 , Δ f q ) , i = j = 0 N ¯ Q ¯ I ˜ n * ( 0 , - Δ f 1 ) , i = 0 ,             j = - Δ f q 0 , otherwise ,
B I ( i , j ) = { N ¯ Q ¯ + Q ¯ 2 , i = j = 0 0 , otherwise .
exp ( - j 2 π f · r k ) = [ 1 / I ( 0 ) ] - i ( r k ) exp ( - j 2 π f · r k ) d r k = I ( f ) / I ( 0 ) = I ˜ ( f ) ,
( j 2 π ) ξ k exp ( - j 2 π f · r k ) = - I ˜ ξ ( f ) ,
( 4 π 2 ) ξ k 2 exp ( - j 2 π f · r k ) = - I ˜ ξ ( f ) ,
( 4 π 2 ) ξ k ξ l exp [ - j 2 π f · ( r k + r l ) ] = - [ I ˜ ξ ( f ) ] 2 ,
( 4 π 2 ) ξ k ξ l exp [ - j 2 π f · ( r k - r l ) ] = I ˜ ξ ( f ) 2 .
N n ( N n - 1 ) ( N n - k + 1 ) = N ¯ k ,
( a )             ( 1 / 2 ) N ¯ 2 [ - Re { I ˜ ξ ( 2 f ) I ˜ * ( 2 f ) } - I ˜ ξ ( 0 ) ] ,
( b )             ( 1 / 2 ) N ¯ 2 [ I ˜ ξ ( 0 ) 2 + I ˜ ξ ( 2 f ) 2 ] ,
( c )             ( 1 / 2 ) N ¯ 3 [ - Re { I ˜ ξ ( 2 f ) [ I ˜ * ( 2 f ) ] 2 } - I ˜ ξ ( 0 ) I ˜ ( f ) 2 ] ,
( d ) , ( e )             N ¯ 3 [ - Re { I ˜ ξ ( 0 ) I ˜ ξ ( f ) I ˜ * ( f ) + I ˜ ξ ( 2 f ) I ˜ ξ * ( f ) I ˜ * ( f ) } ] ,
( f )             ( 1 / 2 ) N ¯ 3 [ - Re { I ˜ * ( 2 f ) [ I ˜ ξ ( f ) ] 2 } + I ˜ ξ ( f ) 2 ] ,
( g )             ( 1 / 2 ) N ¯ 4 [ - Re { I ˜ * ( f ) I ˜ ξ ( f ) ] 2 } + I ˜ ( f ) 2 I ˜ ξ ( f ) 2 ] .
[ N ¯ 2 Re { - j I ˜ ξ I ˜ * } ] 2 .

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