Abstract

We examine the statistical basis of the shift-and-add technique, by which true images of objects (viewed through fluctuating, distorting media) are formed by a particular kind of averaging of speckle images Our theory agrees with experimental results for objects consisting of isolated unresolvable points We develop a preliminary analysis for extended objects (i e, objects having continuous, resolvable brightness distributions)

© 1983 Optical Society of America

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References

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  1. A Labeyrie, “Attainment of diffraction-limited resolution in large telescopes by Fourier analyzing speckle patterns in star images,” Astron Astrophys 6, 85–87 (1970)
  2. R H T Bates and F M Cady, “Towards true imaging by wideband speckle interferometry,” Opt Commun 32, 365–369 (1980),F M Cady and R H T Bates, “Speckle processing gives diffraction-limited true images from severely aberrated instruments,” Opt Lett 5, 438–440 (1980)
    [CrossRef] [PubMed]
  3. J C Dainty, Institute of Optics, University of Rochester, Rochester, New York 14627
  4. C R Lynds, S P Worden, and J W Harvey, Astrophys J 207, 174–179 (1976)
    [CrossRef]
  5. K T Knox and B J Thompson, “Recovery of images from atmospherically-degraded short-exposure photographs,” Astrophys J 193, L45–L48 (1974)
    [CrossRef]
  6. W T Rhodes and J W Goodman, “Interferometric technique for recording and restoring images degraded by unknown aberrations,” J Opt Soc Am 63, 647–657 (1973)
    [CrossRef]
  7. 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)
  8. J W Goodman, “Statistical properties of laser speckle patterns,” in Laser Speckle and Related Phenomena, Vol 9 of Topics in Applied Physics (Springer-Verlag, Berlin, 1975), Chap 2
    [CrossRef]
  9. A Papoullis, Probability, Random Variables, and Stochastic Processes (McGraw-Hill, New York, 1965)
  10. J C Dainty, “Stellar speckle interferometry,” in Laser Speckle and Related Phenomena, Vol 9 of Topics in Applied Physics (Springer-Verlag, Berlin, 1975), Chap 7
    [CrossRef]
  11. E J Gumbel, Statistics of Extreme Values (Columbia U Press, New York, 1958)
  12. F Mosteller and J W Tukey, Data Analysis and Regression (Addison-Wesley, New York, 1976)
  13. H Andrews and B R Hunt, Digital Image Restoration (Prentice-Hall, Englewood Cliffs, N J, 1977)
  14. I Gradshteyn and S Rizhik, Table of Integrals, Series, and Products (Academic, New York, 1965), p 721

1982 (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)

1980 (1)

R H T Bates and F M Cady, “Towards true imaging by wideband speckle interferometry,” Opt Commun 32, 365–369 (1980),F M Cady and R H T Bates, “Speckle processing gives diffraction-limited true images from severely aberrated instruments,” Opt Lett 5, 438–440 (1980)
[CrossRef] [PubMed]

1976 (1)

C R Lynds, S P Worden, and J W Harvey, Astrophys J 207, 174–179 (1976)
[CrossRef]

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]

1973 (1)

W T Rhodes and J W Goodman, “Interferometric technique for recording and restoring images degraded by unknown aberrations,” J Opt Soc Am 63, 647–657 (1973)
[CrossRef]

1970 (1)

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

Andrews, H

H Andrews and B R Hunt, Digital Image Restoration (Prentice-Hall, Englewood Cliffs, N J, 1977)

Bates, R H T

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)

R H T Bates and F M Cady, “Towards true imaging by wideband speckle interferometry,” Opt Commun 32, 365–369 (1980),F M Cady and R H T Bates, “Speckle processing gives diffraction-limited true images from severely aberrated instruments,” Opt Lett 5, 438–440 (1980)
[CrossRef] [PubMed]

Cady, F M

R H T Bates and F M Cady, “Towards true imaging by wideband speckle interferometry,” Opt Commun 32, 365–369 (1980),F M Cady and R H T Bates, “Speckle processing gives diffraction-limited true images from severely aberrated instruments,” Opt Lett 5, 438–440 (1980)
[CrossRef] [PubMed]

Dainty, J C

J C Dainty, Institute of Optics, University of Rochester, Rochester, New York 14627

J C Dainty, “Stellar speckle interferometry,” in Laser Speckle and Related Phenomena, Vol 9 of Topics in Applied Physics (Springer-Verlag, Berlin, 1975), Chap 7
[CrossRef]

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)

Goodman, J W

W T Rhodes and J W Goodman, “Interferometric technique for recording and restoring images degraded by unknown aberrations,” J Opt Soc Am 63, 647–657 (1973)
[CrossRef]

J W Goodman, “Statistical properties of laser speckle patterns,” in Laser Speckle and Related Phenomena, Vol 9 of Topics in Applied Physics (Springer-Verlag, Berlin, 1975), Chap 2
[CrossRef]

Gradshteyn, I

I Gradshteyn and S Rizhik, Table of Integrals, Series, and Products (Academic, New York, 1965), p 721

Gumbel, E J

E J Gumbel, Statistics of Extreme Values (Columbia U Press, New York, 1958)

Harvey, J W

C R Lynds, S P Worden, and J W Harvey, Astrophys J 207, 174–179 (1976)
[CrossRef]

Hunt, B R

H Andrews and B R Hunt, Digital Image Restoration (Prentice-Hall, Englewood Cliffs, N J, 1977)

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]

Labeyrie, A

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

Lynds, C R

C R Lynds, S P Worden, and J W Harvey, Astrophys J 207, 174–179 (1976)
[CrossRef]

Mosteller, F

F Mosteller and J W Tukey, Data Analysis and Regression (Addison-Wesley, New York, 1976)

Papoullis, A

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

Rhodes, W T

W T Rhodes and J W Goodman, “Interferometric technique for recording and restoring images degraded by unknown aberrations,” J Opt Soc Am 63, 647–657 (1973)
[CrossRef]

Rizhik, S

I Gradshteyn and S Rizhik, Table of Integrals, Series, and Products (Academic, New York, 1965), p 721

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]

Tukey, J W

F Mosteller and J W Tukey, Data Analysis and Regression (Addison-Wesley, New York, 1976)

Worden, S P

C R Lynds, S P Worden, and J W Harvey, Astrophys J 207, 174–179 (1976)
[CrossRef]

Astron Astrophys (1)

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

Astrophys J (2)

C R Lynds, S P Worden, and J W Harvey, Astrophys J 207, 174–179 (1976)
[CrossRef]

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

J Opt Soc Am (1)

W T Rhodes and J W Goodman, “Interferometric technique for recording and restoring images degraded by unknown aberrations,” J Opt Soc Am 63, 647–657 (1973)
[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)

R H T Bates and F M Cady, “Towards true imaging by wideband speckle interferometry,” Opt Commun 32, 365–369 (1980),F M Cady and R H T Bates, “Speckle processing gives diffraction-limited true images from severely aberrated instruments,” Opt Lett 5, 438–440 (1980)
[CrossRef] [PubMed]

Other (8)

J C Dainty, Institute of Optics, University of Rochester, Rochester, New York 14627

J W Goodman, “Statistical properties of laser speckle patterns,” in Laser Speckle and Related Phenomena, Vol 9 of Topics in Applied Physics (Springer-Verlag, Berlin, 1975), Chap 2
[CrossRef]

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

J C Dainty, “Stellar speckle interferometry,” in Laser Speckle and Related Phenomena, Vol 9 of Topics in Applied Physics (Springer-Verlag, Berlin, 1975), Chap 7
[CrossRef]

E J Gumbel, Statistics of Extreme Values (Columbia U Press, New York, 1958)

F Mosteller and J W Tukey, Data Analysis and Regression (Addison-Wesley, New York, 1976)

H Andrews and B R Hunt, Digital Image Restoration (Prentice-Hall, Englewood Cliffs, N J, 1977)

I Gradshteyn and S Rizhik, Table of Integrals, Series, and Products (Academic, New York, 1965), p 721

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Figures (5)

Fig 1
Fig 1

One-dimensional image of object consisting of three unresolved points

Fig 2
Fig 2

Shift-and-add image of object shown in Fig 1 formed from 512 independent, measured speckle images

Fig 3
Fig 3

Typical measured speckle image of object shown in Fig 1

Fig 4
Fig 4

Average of the autocorrelations of 512 independent, measured speckle images of an object consisting of a single unresolvable point

Fig 5
Fig 5

Speckle processing of an extended object (a) the object, (b) the curve SA(x) = Rl(x) + Rd(x), which is the spatial autocorrelation of the object, (c) average of the autocorrelations of speckle images of an object consisting of a single unresolvable point [note that the central spike is a(x), whereas the broad hump is l(x)], (d) convolution of (x) with SA(x), (e) convolution of a(x) with SA(x)

Equations (81)

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s m ( x ) = h m ( x ) * f ( x ) + c m ( x ) ,
s ( x ) = 1 M m = 1 M s m ( x + ξ m )
f ( x ) = J = 1 T a J δ ( x n J )
s m ( x ) = J = 1 T a J h m ( x n J )
σ m ( x ) = s m ( x + ξ m )
s ( x ) = 1 M m = 1 M σ m ( x )
lim M s ( x ) = 1 M σ m ( x ) E [ σ m ( x ) | σ m ( 0 ) ] ,
s ( x ) = E [ σ m ( x ) | σ m ( 0 ) ]
p ( I ) = 1 I exp ( I I ) ,
E [ σ m ( 0 ) ] = s max ,
1 M m = 1 M σ m ( 0 ) = s max
I 1 = σ m ( 0 ) , I 2 = σ m ( x )
E [ σ m ( x ) | σ m ( 0 ) ] = E [ I 2 | I 1 ] = 0 I 2 P ( I 2 | I 1 ) d I 2
p ( I 2 | I 1 ) = exp [ | μ A | 2 I 1 + I 2 I ( 1 | μ A | 2 ) ] I ( 1 | μ A | 2 ) I 0 [ 2 I 1 I 2 | μ A | I ( 1 | μ A | 2 ) ] ,
R s ( x ) = I 2 [ 1 + | μ A ( x ) | 2 ] ,
R s ( x ) = E [ s m ( x 1 ) s m ( x 1 + x ) ]
E [ I 2 | I 1 ] = 2 I + R s ( x ) I 2 ( I 1 I ) I 1
E [ σ m ( x ) | σ m ( 0 ) ] = 2 I + R s ( x ) I 2 [ σ m ( 0 ) I ] σ m ( 0 ) ,
lim M s ( 0 ) E [ σ m ( 0 ) ] = s max ,
lim M s ( x ) = 2 I + R s ( x ) I 2 ( s max I ) s max ,
W ( u ) = F [ R s ( x ) ] | T ( u ) | 2 + k T D ( u ) ,
R s ( x ) = F 1 [ W ( u ) ] l ( x ) + k K a ( x ) ,
R s ( x ) = I 2 + I 2 a ( x )
R s ( 0 ) = V s + μ s 2 ,
μ s 2 = I 2
V s = I 2 I 2 = 2 I 2 I 2 = I 2 ,
R s ( 0 ) = 2 I 2 ,
a ( 0 ) = 1 ,
a ( x ) = a ( x ) a ( 0 )
a ( x ) = | μ A ( x ) | 2 ,
E ( largest value in N independent observations ) ln N + γ ,
γ = 05772156649
E [ σ m ( 0 ) ] = s max I ln N ,
s k ( x ) = 2 I k + R s ( x η k ) I k 2 [ s max ( k ) I k ] s max ( k ) ,
s max ( k ) = I k l N
I = T = 1 T I k
s ( x ) = k = 1 T s k ( x ) = k = 1 T { 2 I k + R s ( x η k ) I k 2 [ s max ( k ) I k ] s max ( k ) } = 2 I + k = 1 T { R s ( x η k ) I k 2 [ s max ( k ) I k ] s max ( k ) }
s m ( x ) = a 1 h m ( x η 1 ) + a 2 h m ( x η 2 ) ,
p ( I 1 ) = exp ( I 1 / I 1 ) I 1 ,
ω = I 2 I 1 = a 2 a 1
u = z + rv ( 2 ) , υ = ω z + rv ( 1 )
Q 1 = Prob ( choose η 1 peak location instead of η 2 ) = Prob ( u > υ ) ,
Q 2 = Prob ( choose η 2 peak location instead of η 1 ) = 1 Prob ( u > υ ) = Prob ( u < υ )
Q 3 = Prob ( choose neither η 1 nor η 2 peak location ) ,
p ( u , υ | z ) = p ( u | z ) p ( υ | z ) ,
p ( u , υ | z ) = 1 I 1 I 2 exp ( u z I 2 ) exp ( υ ω z I 1 ) for u z , υ ω z ,
Prob ( u > υ ) = z [ ω z u p ( u , υ | z ) d υ ] d u = z { ω z u exp [ ( υ ω z I 1 ) ] I 1 d υ } × exp [ ( u z I 2 ) ] I 2 d u
P ( u < υ ) = 1 P ( u > υ ) ,
Prob ( u > υ | z ) = Prob ( choose η 2 peak instead of η 1 | z ) = I 1 I 1 + I 2 exp ( z ( 1 ω ) I 1 ) .
I 2 = ω I 1
Prob ( Wrong peak selected | E [ x ] I 1 l η N ) = Q 2 / ( Q 1 + Q 2 ) = 1 1 + ω exp [ ( l η N ) ( 1 ω ) ] ,
s m ( x ) = f ( x x 1 ) h m ( x 1 ) d x 1 + c m ( x )
p ( I 2 | I 1 ) = 1 α e [ 2 π ( 1 r 2 ) ] 1 / 2 exp { 1 2 α e 2 ( 1 r 2 ) × [ ( I 2 I e ) r ( I 1 I e ) ] 2 } ,
r = r ( x ) = R e ( x ) α e 2
r ( x ) = E [ ( I 2 I e ) ( I 1 I e ) ] α e 2 ,
lim M s e ( x ) = I e + R e ( x ) α e 2 ( s max I e ) ,
S m ( u ) = F ( u ) H m ( u ) + C m ( u )
Φ s ( u ) = | F ( u ) | 2 Φ H ( u ) + Φ c ( u ) ,
R e ( x ) = R h ( x ) * F 1 [ | F ( u ) | 2 ] + R x ( x ) ,
R e ( x ) = R e ( x ) + I e 2
| F ( u ) | 2 = | F l ( u ) | 2 + | F d ( u ) | 2
R e ( x ) = R h ( x ) * [ R l ( x ) + R d ( x ) ] + R c ( x ) ,
R h ( x ) = R S ( x ) l ( x ) + k K a ( x ) ,
R e ( x ) = l ( x ) * [ R l ( x ) + R d ( x ) ] + k K a ( x ) * [ R l ( x ) + R d ( x ) ] + R c ( x )
R e ( x ) = l ( x ) * [ R l ( x ) + R d ( x ) ] + k K a ( x ) * [ R l ( x ) + R d ( x ) ] + R c ( x ) I e 2
f ( x ) = J = 1 P f J δ ( x η J ) ,
s f ( x ) = J = 1 p f J s e ( x η J )
s f ( x ) = f ( x 1 ) s e ( x x 1 ) d x 1 ,
0 I 2 p ( I 2 / I 1 ) d I 2 = 0 I 2 { exp [ | μ A | 2 I 1 + I 2 I ( 1 | μ A | 2 ) ] I ( 1 | μ A | 2 ) × I 0 [ 2 I 1 I 2 | μ A | I ( 1 | μ A | 2 ] } d I 2
exp [ | μ A | 2 I 1 I ( 1 | μ A | 2 ) ] I ( 1 | μ A | 2 ) 0 I 2 exp [ I 2 I ( 1 | μ A | 2 ) ] × I 0 [ 2 I 1 I 2 | μ A | I ( 1 | μ A | 2 ) ] d I 2
I ν ( x ) = ι ν J ν ( ι x ) ,
I 0 [ 2 I 1 I 2 | μ A | I ( 1 | μ A | 2 ) ] = J 0 [ ι 2 I 1 I 2 | μ A | I ( 1 | μ A | 2 ]
0 x η + ν / 2 exp ( α x ) J ν ( 2 β x ) d x = η ! β ν exp ( β 2 α ) α η ν 1 L η ν ( β 2 α ) ,
x = I 2 , ν = 0 , η = 1 , α = 1 I ( 1 | η A | 2 ) , β = ι I 1 | μ A | I ( 1 | μ A | 2 ) ,
β 2 A = | μ A | 2 I 1 I ( 1 | μ A | 2 )
0 I 2 p ( I 2 | I 1 ) d I 2 = × exp [ | μ A | 2 I 1 I ( 1 | μ A | 2 ) ] I ( 1 | μ A | 2 ) { exp [ | μ A | 2 I 1 I ( 1 | μ A | 2 ) ] × [ I ( 1 | μ A | 2 ) ] 2 L 1 0 [ | μ A | 2 I 1 I ( 1 | μ A | 2 ) ] } .
L η γ ( x ) = m = 0 n ( 1 ) m ( n + γ n m ) x m m ! ,
L 1 0 ( β 2 α ) = 1 + ( 1 ) [ | μ A | 2 I 1 I ( 1 | μ A | 2 ) ] = I ( 1 | μ A | 2 ) + | μ A | 2 I 1 I ( 1 | μ A | 2 ) .
0 I 2 p ( I 2 | I 1 ) d I 2 = I ( 1 | μ A | 2 ) + | μ A | 2 I 1
| μ A ( Δ x ) | 2 = R s ( Δ x ) I 2 1 ,
0 I 2 p ( I 2 | I 1 ) d I 2 = 2 I + R s ( Δ x ) I 2 ( I 1 I ) I 1