Abstract

It is shown that phase retrieval for two-dimensional discrete object functions satisfying certain support constraints is unique among all object functions satisfying the same support constraint. Closed-form reconstruction algorithms for reconstructing such object functions from their autocorrelation functions are defined. Also, an algorithm for generating these reconstruction algorithms is described.

© 1987 Optical Society of America

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References

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  1. J. R. Fienup, T. R. Crimmins, W. Holsztynski, “Reconstruction of the support of an object from the support of its autocorrelation function,”J. Opt. Soc. Am. 72, 610–624 (1982).
    [CrossRef]
  2. M. H. Hayes, T. F. Quatieri, “Recursive phase retrieval using boundary conditions,”J. Opt. Soc. Am. 73, 1427–1433 (1983).
    [CrossRef]
  3. Y. M. Bruck, L. G. Sodin, “On the ambiguity of the image reconstruction problem,” Opt. Commun. 30, 304–308 (1979).
    [CrossRef]
  4. M. A. Fiddy, B. J. Brames, J. C. Dainty, “Enforcing irreducibility for phase retrieval in two dimensions,” Opt. Lett. 8, 96–98 (1983).
    [CrossRef] [PubMed]
  5. J. R. Fienup, “Phase retrieval algorithms: a comparison,” Appl. Opt. 21, 2758–2769 (1982).
    [CrossRef] [PubMed]
  6. J. R. Fienup, “Reconstruction of an object from the modulus of its Fourier transform,” Opt. Lett. 3, 27–29 (1978).
    [CrossRef] [PubMed]
  7. J. R. Fienup, “Space object imaging through the turbulent atmosphere,” Opt. Eng. 18, 529–534 (1979).
    [CrossRef]
  8. J. R. Fienup, “Reconstruction of objects having latent reference points,”J. Opt. Soc. Am. 73, 1421–1426 (1983).
    [CrossRef]
  9. B. J. Brames, “Unique phase retrieval with explicit support information,” Opt. Lett. 11, 61–63 (1986).
    [CrossRef] [PubMed]
  10. J. W. Goodman, “Analogy between holography and interferometric image formation,”J. Opt. Soc. Am. 60, 506–509 (1970).
    [CrossRef]
  11. C. Y. C. Liu, A. W. Lohman, “High resolution image formation through the turbulent atmosphere,” Opt. Commun. 8, 372–377 (1973).
    [CrossRef]

1986 (1)

1983 (3)

1982 (2)

1979 (2)

Y. M. Bruck, L. G. Sodin, “On the ambiguity of the image reconstruction problem,” Opt. Commun. 30, 304–308 (1979).
[CrossRef]

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

1978 (1)

1973 (1)

C. Y. C. Liu, A. W. Lohman, “High resolution image formation through the turbulent atmosphere,” Opt. Commun. 8, 372–377 (1973).
[CrossRef]

1970 (1)

Brames, B. J.

Bruck, Y. M.

Y. M. Bruck, L. G. Sodin, “On the ambiguity of the image reconstruction problem,” Opt. Commun. 30, 304–308 (1979).
[CrossRef]

Crimmins, T. R.

Dainty, J. C.

Fiddy, M. A.

Fienup, J. R.

Goodman, J. W.

Hayes, M. H.

Holsztynski, W.

Liu, C. Y. C.

C. Y. C. Liu, A. W. Lohman, “High resolution image formation through the turbulent atmosphere,” Opt. Commun. 8, 372–377 (1973).
[CrossRef]

Lohman, A. W.

C. Y. C. Liu, A. W. Lohman, “High resolution image formation through the turbulent atmosphere,” Opt. Commun. 8, 372–377 (1973).
[CrossRef]

Quatieri, T. F.

Sodin, L. G.

Y. M. Bruck, L. G. Sodin, “On the ambiguity of the image reconstruction problem,” Opt. Commun. 30, 304–308 (1979).
[CrossRef]

Appl. Opt. (1)

J. Opt. Soc. Am. (4)

Opt. Commun. (2)

Y. M. Bruck, L. G. Sodin, “On the ambiguity of the image reconstruction problem,” Opt. Commun. 30, 304–308 (1979).
[CrossRef]

C. Y. C. Liu, A. W. Lohman, “High resolution image formation through the turbulent atmosphere,” Opt. Commun. 8, 372–377 (1973).
[CrossRef]

Opt. Eng. (1)

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

Opt. Lett. (3)

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Figures (10)

Fig. 1
Fig. 1

M1 is a mask. M2 is not a mask, since [M2] has two parallel sides.

Fig. 2
Fig. 2

The vertex v is opposite the side s.

Fig. 3
Fig. 3

The circled vertices of [M] are the reference points of the mask M.

Fig. 4
Fig. 4

The numberings of the vertices and reference points of a mask. Here S = 8, T = 5, and K = 2.

Fig. 5
Fig. 5

An illustration of the vectors wn and Uwn. Here n = 0 and the origin in 2 is denoted by 0.

Fig. 6
Fig. 6

An illustration of the vectors yn. Here S = 8, T = 5, n = 4, k4 = 1, (k4 = 1)mod S = 0, (n + 1)mod T = 0, and k0 = 4.

Fig. 7
Fig. 7

The distance from an arbitrary point x in D to the line through β and perpendicular to yn is d = |hn(x)|/||yn||. Here S = 8, T = 5, and n = 4.

Fig. 8
Fig. 8

Example of a mask, a numbering of its vertices, and the resulting numberings of its reference points.

Fig. 9
Fig. 9

A reconstruction algorithm (q, m) generated for the mask and vertex numbering shown in Fig. 8. The numbers under the points of the mask indicate the ordering q = (q0, …, q15) and m = (0, 1, 4, 0, 1, 2, 3, 4, 0, 1, 2).

Fig. 10
Fig. 10

The set [R(M)] is the convex polygon with sides ti, i = 0, …, 4.

Equations (90)

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r ( x ) = y Z 2 f ( y ) f ( y - x ) * ,
r ( q ( n + 1 ) mod T - q n ) = f ( q ( n + 1 ) mod T ) f ( q n ) * ,
f ( q 0 ) 2 = n = 0 K r ( q 2 n - q ( 2 n + 1 ) mod T ) n = 0 K - 1 r ( q 2 n + 2 - q 2 n + 1 ) .
y n = U w ( k n - 1 ) mod S - U w k ( n + 1 ) mod T ;
p j , U u n < p m , U u n .
U w ( k - 1 ) mod S , v i - v a < 0 ,
U w k , v i - v b < 0.
v ( k - 1 ) mod S - p m , U u n = v ( k - 1 ) mod S - v k , U ( p ( n + 1 ) mod T - p n ) = - w ( k - 1 ) mod S , U ( v b - v a ) = U w ( k - 1 ) mod S , v b - v a < 0.
v ( k + 1 ) mod S - p m , U u n = v ( k + 1 ) mod S - V k , U ( p ( n + 1 ) mod T - p n ) = w k , U ( v b - v a ) = U w k , v a - v b < 0.
p - p m = α ( v ( k - 1 ) mod S - p m ) + β ( v ( k + 1 ) mod S - p m ) .
0 < p - v k , U w k = p - p m , U ( v ( k + 1 ) mod S - p m ) = α v ( k - 1 ) mod S - p m , U ( v ( k + 1 ) mod S - p m ) + β v ( k + 1 ) mod S - p m , U ( v ( k + 1 ) mod S - p m ) = α v ( k - 1 ) mod S - p m , U ( v ( k + 1 ) mod S - p m ) = - α w ( k - 1 ) mod S , U ( v ( k + 1 ) mod S - v k ) = α U w ( k - 1 ) mod S , v ( k + 1 ) mod S - v k .
0 < p - v k , U w ( k - 1 ) mod S = p - p m , U ( p m - v ( k - 1 ) mod S ) = α v ( k - 1 ) mod S - p m , U ( p m - v ( k - 1 ) mod S ) + β v ( k + 1 ) mod S - p m , U ( p m - v ( k - 1 ) mod S ) = - α p m - v ( k - 1 ) mod S , U ( p m - v ( k - 1 ) mod S ) + β v ( k + 1 ) mod S - p m , U ( p m - v ( k - 1 ) mod S ) = β v ( k + 1 ) mod S - p m , U ( p m - v ( k - 1 ) mod S ) = β v ( k + 1 ) mod S - v k , U w ( k - 1 ) mod S .
p - p m , U u n = α v ( k - 1 ) mod S - p m , U u n + β v ( k + 1 ) mod S - p m , U u n < 0.
q ( j + 1 ) mod T , y j - v , y j = q ( j + 1 ) mod T - v , y j = q ( j + 1 ) mod T - v , U w m - U w n = q ( j + 1 ) mod T - v , U w m - q ( j + 1 ) mod T - v , U w n = q ( j + 1 ) mod T - v , U w m - v n - v , U w n = q ( j + 1 ) mod T - v , U w m + v - v n , U w n > 0.
v , y j - q j , y j = v - q j , y j = v - q j , U w m - U w n = v - q j , U w m + q j - v , U w n = v - v k j , U w m + q j - v , U w n = v - v ( m + 1 ) mod S , U w m + q j - v , U w n > 0.
x 1 - x 2 , y j = x 1 , y j - x 2 , y j < q j 1 , y j - q j , y j = q j 1 - q j , y j ,
f ( x ) = u Z 2 g ( u ) h ( x - u ) .
x , y < x , y .
x 1 , y + x 2 , y = x 1 + x 2 , y = x , y = x 1 + x 2 , y = x 1 , y + x 2 , y .
x , y = x 1 , y + x 2 , y x 1 , y + x 2 , y = x , y .
q j = a j + b j ,             j = 0 , , T - 1.
x , y j = x , y j + b j , y j a j , y j + b j , y j = q j , y j .
a j , y j < x , y j < a j 1 , y j .
b j , y j < x , y j < b j 1 , y j .
[ S ( f 1 ) ] = [ S ( g ) - S ( h ) ] + d .
a j - b j 1 + d M .
x , y j = x 1 , y j - x 2 , y j > a j , y j - b j 1 , y j = a j - b j + 1 , y j .
a j 1 - b j + d M .
( a j - b j 1 + d ) - ( a j 1 - b j + d ) = ( a j + b j ) - ( a j 1 + b j 1 ) = q j - q j 1 .
b j + b j 1 = d .
b 0 = b 1 = = b T - 1 = d / 2.
f ( x ) = g * h ( x ) = h ( d / 2 ) g ( x - d / 2 ) .
f 1 ( x ) = g * h 1 ( x - d ) = h ( d / 2 ) * g ( x - d + d / 2 ) = h ( d / 2 ) * g ( x - d / 2 ) = α f ( x ) ,
α = h ( d / 2 ) * h ( d / 2 ) .
r ( q n - q m n ) = y Z 2 f ( y ) f ( y - q n + q m n ) * .
y M ( M + q n - q m n ) { q 0 , , q n } .
r ( q n - q m n ) = k = 0 n f ( q k ) f ( q k - q n + q m n ) * = f ( q n ) f ( q m n ) * + k = 0 n - 1 f ( q k ) f ( q k - q n + q m n ) * .
f ( q n ) = 1 f ( q m n ) * [ r ( q n - q m n ) - k = 0 n - 1 f ( q k ) f ( q k - q n + q m n ) * ] .
H ( n ) :             f ( q j ) is computed correctly for 0 j n .
int [ x , y , z ] = { a x + b y + c z : a , b , c > 0 and a + b + c = 1 } ;
σ 1 μ 1 + σ 2 μ 2 + σ 3 μ 3 = 0.
μ 2 = - ( σ 3 / σ 2 ) μ 3 ;
0 < μ 2 , ν 2 = - ( σ 3 / σ 2 ) μ 3 , ν 2 .
- σ 3 / σ 2 < 0.
0 > μ 2 , ν 1 = - ( σ 3 / σ 2 ) μ 3 , ν 1 ,
σ 1 μ 1 , ν 1 + σ 2 μ 2 , ν 1 + σ 3 μ 3 , ν 1 = σ 1 μ 1 + σ 2 μ 2 + σ 3 μ 3 , ν 1 = 0.
σ 2 μ 2 , ν 1 + σ 3 μ 3 , ν 1 = - σ 1 μ 1 , ν 1 < 0.
σ 1 μ 1 , ν 3 + σ 2 μ 2 , ν 3 + σ 3 μ 3 , ν 3 = σ 1 μ 1 + σ 2 μ 2 + σ 3 μ 3 , ν 3 = 0.
σ 3 μ 3 , ν 3 = - σ 1 μ 1 , ν 3 - σ 2 μ 2 , ν 3 > 0.
μ 1 = q j 1 - q j 3 ,             μ 2 = q j 2 - q j 1 ,             μ 3 = q j - q j 2 , ν 1 = y j ,             ν 2 = y j 1 ,             ν 3 = y j 2 .
σ 1 y j + σ 2 y j 1 + σ 3 y j 2 = 0.
τ 1 y j 1 + τ 2 y j 2 + τ 3 ( - 1 ) k - 1 y j k = 0.
λ 1 y j + λ 2 y j 1 + λ 3 ( - 1 ) k y j k = 0 ,
λ 2 y j 1 = - λ 1 y j + λ 3 ( - ) k - 1 y j k .
λ 2 y j 1 = - λ 1 y j + λ 3 y j k ;
λ 2 q j k 1 - q j 1 , y j 1 = - λ 1 q j k 1 - q j 1 , y j + λ 3 q j k 1 - q j 1 , y j k = λ 1 q j 1 - q j k 1 , y j + λ 3 q j k 1 - q j 1 , y j k > 0.
λ 2 y j 1 = - λ 1 y j - λ 3 y j k ;
λ 2 q j k - q j 1 , y j 1 = λ 1 q j 1 - q j k , y j + λ 3 q j 1 - q j k , y j k > 0.
b a = min { j : 0 j N - T - 1 and ϕ ( d k a , j ) = 1 }
ϕ ( d k a , b a ) = 1
ϕ ( α k a - d k a , b a ) = 1
x a = { d k a , b a if h k a ( d k a , b a ) 0 α k a - d k a , b a otherwise
h k a ( x a ) = h k a ( d k a , b a ) .
ϕ ( x ) = 1 h k a ( x ) h k a ( x a ) .
( - 1 ) a x a - q k a 1 , y k ( T - 1 ) > 0.
α k - x 0 , y k ( T - 1 ) < q k , y k ( T - 1 ) ,
x 0 - q k 1 , y k ( T - 1 ) > 0.
h k ( a - 1 ) ( α k a - x a ) h k ( a - 1 ) ( α k a - x a ) h ( a - 1 ) ( x a - 1 ) ,
x a - 1 - α k a + x a , y k ( a - 1 ) 0.
- h k a ( x a - 1 ) h k a ( x a - 1 ) h k a ( x a ) = - h k a ( α k a - x a ) ,
h k a ( a k a - x a ) h k a ( x a - 1 ) ,
x a - 1 - α k a + x a , y k a 0.
σ 1 y k ( a - 1 ) + σ 2 y k a + σ 3 ( - 1 ) T - a y k ( T - 1 ) = 0.
σ 3 ( - 1 ) a y k ( T - 1 ) = σ 1 y k ( a - 1 ) + σ 2 y k a .
σ 3 ( - 1 ) a x a - 1 - α k a + x a , y k ( T - 1 ) = σ 1 x a - 1 - α k a + x a , y k ( a - 1 ) + σ 2 x a - 1 - α k a + x a , y k a 0 ,
( - 1 ) a x a - 1 - α k a + x a , y k ( T - 1 ) 0.
( - 1 ) a x a - q k a 1 , y k ( T - 1 ) - ( - 1 ) a x a - 1 - q k a , y k ( T - 1 ) = ( - 1 ) a - 1 x a - 1 - q k a , y k ( T - 1 ) > 0.
x T - 2 , y k ( T - 1 ) < q k ( T - 1 ) , y k ( T - 1 ) .
h k ( d k , b ) < h k ( x ) ϕ ( x ) = 0.
ϕ ( x ) = 0 x { q 0 , , q n - 1 } .
M ( M + q n - q m n ) { q 0 , , q n } .
h k ( d k , b ) < h k ( x ) .
h k ( d k , b ) 0.
q k , y k < z , y k = x , y k - d k , b , y k + q k , y k ,
h k ( d k , b ) < 0.
x , y k - d k , b , y k + q k 1 , y k = z , y k < q k 1 , y k ,
M ( M - d k , b + q m n ) { q 0 , , q n - 1 } .
x , y k + d k , b , y k - q k , y k = z , y k < q k 1 , y k .
d k , b - β k , y k < - x + β k , y k ,
q k , y k < z , y k = x , y k + d k , b , y k - q k 1 , y k ,

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