Abstract

New methods are described for determining tighter upper bounds on the support of an object, given the support of its autocorrelation. These upper bounds are shown, in a digital experiment, to be useful as object-support constraints used with the iterative transform algorithm for solving the phase-retrieval problem.

© 1990 Optical Society of America

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References

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  1. J. R. Fienup, “Reconstruction of an object from the modulus of its Fourier transform,” Opt. Lett. 3, 27–29 (1978).
    [CrossRef] [PubMed]
  2. J. R. Fienup, “Phase retrieval algorithms: a comparison,” Appl. Opt. 21, 2758–2769 (1982).
    [CrossRef] [PubMed]
  3. J. R. Fienup, C. C. Wackerman, “Phase-retrieval stagnation problems and solutions,”J. Opt. Soc. Am. A 3, 1897–1907 (1986).
    [CrossRef]
  4. J. R. Fienup, “Reconstruction of a complex-valued object from the modulus of its Fourier transform using a support constraint,” J. Opt. Soc. Am. A 4, 118–123 (1987).
    [CrossRef]
  5. R. H. T. Bates, D. G. H. Tan, “Fourier phase retrieval when the image is complex,” in Inverse Optics II, A. J. Devaney, R. H. T. Bates, eds., Proc. Soc. Photo-Opt. Instrum. Eng.558, 54–59 (1985).
    [CrossRef]
  6. B. J. Brames, “Efficient method of support reduction,” Opt. Commun. 64, 333–337 (1987).
    [CrossRef]
  7. J. R. Fienup, T. R. Crimmins, W. Holsztynski, “Reconstruction of the support of an object from the support of its autocorrelation,” J. Opt. Soc. Am. 72, 610–624 (1982).
    [CrossRef]
  8. A more general class of object representations would be the class of all compactly supported complex measures on R2.See W. Rudin, Real and Complex Analysis (McGraw-Hill, New York, 1974) for more details regarding complex measures. Essentially the whole discussion in this section also applies to this more general class of object representations, but it requires substantially more technical detail and really is not particularly helpful for understanding the basic ideas presented in this discussion.
  9. J. Donoghue, Distributions and Fourier TransformsAcademic, New York, 1969).

1987 (2)

1986 (1)

1982 (2)

1978 (1)

Bates, R. H. T.

R. H. T. Bates, D. G. H. Tan, “Fourier phase retrieval when the image is complex,” in Inverse Optics II, A. J. Devaney, R. H. T. Bates, eds., Proc. Soc. Photo-Opt. Instrum. Eng.558, 54–59 (1985).
[CrossRef]

Brames, B. J.

B. J. Brames, “Efficient method of support reduction,” Opt. Commun. 64, 333–337 (1987).
[CrossRef]

Crimmins, T. R.

Donoghue, J.

J. Donoghue, Distributions and Fourier TransformsAcademic, New York, 1969).

Fienup, J. R.

Holsztynski, W.

Rudin, W.

A more general class of object representations would be the class of all compactly supported complex measures on R2.See W. Rudin, Real and Complex Analysis (McGraw-Hill, New York, 1974) for more details regarding complex measures. Essentially the whole discussion in this section also applies to this more general class of object representations, but it requires substantially more technical detail and really is not particularly helpful for understanding the basic ideas presented in this discussion.

Tan, D. G. H.

R. H. T. Bates, D. G. H. Tan, “Fourier phase retrieval when the image is complex,” in Inverse Optics II, A. J. Devaney, R. H. T. Bates, eds., Proc. Soc. Photo-Opt. Instrum. Eng.558, 54–59 (1985).
[CrossRef]

Wackerman, C. C.

Appl. Opt. (1)

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (2)

Opt. Commun. (1)

B. J. Brames, “Efficient method of support reduction,” Opt. Commun. 64, 333–337 (1987).
[CrossRef]

Opt. Lett. (1)

Other (3)

A more general class of object representations would be the class of all compactly supported complex measures on R2.See W. Rudin, Real and Complex Analysis (McGraw-Hill, New York, 1974) for more details regarding complex measures. Essentially the whole discussion in this section also applies to this more general class of object representations, but it requires substantially more technical detail and really is not particularly helpful for understanding the basic ideas presented in this discussion.

J. Donoghue, Distributions and Fourier TransformsAcademic, New York, 1969).

R. H. T. Bates, D. G. H. Tan, “Fourier phase retrieval when the image is complex,” in Inverse Optics II, A. J. Devaney, R. H. T. Bates, eds., Proc. Soc. Photo-Opt. Instrum. Eng.558, 54–59 (1985).
[CrossRef]

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Figures (15)

Fig. 1
Fig. 1

Example of maximal and locally maximal points. E(B, u) = {x1, x2}, El(B, u) = {x1, x2}, E(B, −u) = {x3}, and El(b, −u) = {x3, x4}.

Fig. 2
Fig. 2

Example of a set S such that El(S, u) − El(S, −u) ≠ El(A, u). (a) Set S, where E(S, u) = {x1}, E(S, −u) = {x4}, El(S, u) = {x1, x3}, and El(S, −u) = {x2, x4} (the shaded area is not considered to be part of S); (b) set A = SS, which is a disk, where E(A, u) = {x1x4} = El(A, u) and El(S, u) − El(S, −u) = {x1x2, x1x4, x3x2, x3x4}, which is not equal to El(A, u).

Fig. 3
Fig. 3

Example of the two-point rule, (a) Set S, (b) set A = SS with E(A, u) = {a1, a2} and E(A, υ) = {a3, a4}, (c) single-sided locator set L1 = A ∩ (A + a1) ∩ (A + a2), (d) single-sided locator set L2 = A ∩ (A + a3) ∩ (A + a4) (indicated by the shaded areas).

Fig. 4
Fig. 4

Example of the two-point rule, (a) Set S, (b) set A = SS and E(A, u) = {a1, a2}, (c) shaded area is the single-sided locator set L = A ∩ (A + a1) ∩ (A + a2), which is identical to a translation of −S.

Fig. 5
Fig. 5

Example of the two-point rule, (a) Set S consisting of discrete points; (b) set A = SS with E(A, u) = {a1, a2}, E(A, υ) = {a3, a4}, and E(A, w) = {a5, a6}; (c) single-sided locator set L1 = A ∩ (A + a1) ∩ (A + a2); (d) single-sided locator set L2 = A ∩ (A + a3) ∩ (A + a4); (e) single-sided locator set L3 = A ∩ (A + a3) ∩ (A + a6).

Fig. 6
Fig. 6

Example of the asymmetric three-point rule, (a) Set S, (b) set A = SS and E(A, u) = {a1, a2, a3,}. In this case the single-sided locator set L = A ∩(A + a1) ∩(A + a2) ∩(A + a3), based on the three-point rule where the three points are asymmetric, is identical to a translation of the original support S.

Fig. 7
Fig. 7

Example of the totally asymmetric rule, (a) Set S consisting of discrete points, (b) set A = SS and E(A, u) = {a1, a2, a3, a4, a5}, (c) a single-sided locator set L = A [ 1 5 ( A + a i ) ], as determined by the asymmetric rule. Note that L is the same as a translation of the original support S except for one extra point.

Fig. 8
Fig. 8

Examples of u-convex and non-u-convex sets, (a) Set B1 is not u convex, (b) set B2 is u convex.

Fig. 9
Fig. 9

Example of the u-convex rule, (a) Set S; (b) set A = SS, where a1 and m are an endpoint and the midpoint, respectively, of E(A, u); (c) a single-sided locator set L = A ∩ (A + a1) ∩ (A + m) determined by the u-convex rule (indicated by the shaded area).

Fig. 10
Fig. 10

Example showing the insufficiency of E(A, u) = [a1,a2]. (a) Set S; (b) set A = SS, where a1 and m are an endpoint and the midpoint, respectively, of E(A, u) = [a1,a2] (the shaded regions, which are open sets, are not part of A); (c) set L = A ∩ (A + m) ∩ (A + a1). Note that L is not a single-sided locator set since no translation of S or −S is contained in L.

Fig. 11
Fig. 11

Example of the one-point rule. (a)Set S; (b) set A = SS and E(A, u) = {a}; (c) single-sided locator set L = A ∩ (A + a) ∩ H, where H is determined by the displayed u, υ, and the one-point rule as in Theorem 6.

Fig. 12
Fig. 12

Example of the rule in Theorem 7. (a) Set S consisting of discrete points [same as Fig. 7(a)]; (b) set A = SS and E(A, u) = {a1, a2, a3, a4, a5}, where a′ = a6, ∈ El(A, u) [actually all points in A are in El(A, u)]. The single-sided locator set L = A [ 1 6 ( A + a i ) ] as determined by the rule in Theorem 7, is identical to a translation of the original object support shown in (a).

Fig. 13
Fig. 13

Example of combining single-side locator sets, (a), (b) Single-sided locator sets from Fig. 3; (c) a single-sided locator set L obtained by combining the single-sided locator sets in (a) and (b), using Theorem 8. Note that the single-sided locator set is identical to a translation of the original support S, shown in Fig. 3(a).

Fig. 14
Fig. 14

Example of combining single-sided locator sets, (a) Set S, consisting of discrete points (same as in Fig. 5); (b), (c) single-sided locator sets L2 and L1 from Fig. 5; (d) a single-sided locator set L obtained by combining L1 and L2, using Theorem 8. Note that L contains only three more points, x1, x2, and x3, than the original support S. Since L\{x1} − L\{x1} = A, i.e., L\{x1} generates A, the points x2 and x3 must be contained in any single-sided locator set.

Fig. 15
Fig. 15

Computer reconstruction example using single-sided locator set. (A) Object, (B) Fourier modulus, (C) thresholded autocorrelation (estimated autocorrelation support), (D) single-sided locator set computed from (C), (E) image reconstructed from Fourier modulus with ten iterations of the iterative Fourier-transform algorithm using the single-sided locator set in (D) as a support constraint, (F) enlarged support constraint, (G) image reconstructed from ten more iterations using (F) as a support constraint.

Equations (37)

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F ( u ) = | F ( u ) | exp [ i ψ ( u ) ] = | f | = f ( x ) exp ( i 2 π u x ) d x ,
r ( x ) = f ( x ) f * ( x x ) d x = 1 [ | F | 2 ] ,
f ( x ) = 1 N α i δ ( x x i ) + f c ( x ) ,
S f = 1 N { x i } supp ( f c ) ,
S f f S f S f = { x y : x , y S f } .
B { S 0 : S 0 S 0 } .
B S x or B x S for some x R 2 .
E ( A , u ) = E ( S , u ) E ( S , u )
E l ( A , u ) E l ( S , u ) E l ( S , u ) .
B x E ( S , u ) for some x E ( S , u ) .
B E ( S , u ) x for some x E ( S , u ) .
( x , y ) = { { z : z = tx + ( 1 t ) y , t ( 0 , 1 ) } if x y if x = y }
[ x , y ] = { z : z = tx + ( 1 t ) y , t [ 0 , 1 ] } ,
a 1 + a 2 [ E ( A , u ) \ { a 1 , a 2 } ] + [ E ( A , u ) \ { a 1 + a 2 } ] ,
d ( B , u ) = sup { x y , u : x , y B } ,
P ( B ) = { ( b 1 , b 2 , b 3 , b 4 : b i B for i = 1 , 2 , 3 , 4 ; b 1 b j for j = 2 , 3 , 4 ; and b 1 + b 3 = b 2 + b 4 } .
P = { ( a 1 , a 2 , a 3 , a 4 ) P ( I ) : a 1 E ( A , u ) , a 3 = a } = ,
sup { x , u : x B } = inf { x , u : x B } = d ( B , u ) 2
sup { x , υ : x B } = inf { x , υ : x B } = d ( B , υ ) 2 ,
S + x S 0 = S 0 y + y S 0 S 0 + y = A + y .
S + x { A + y : y S 0 } { A + y : y B } = L .
x y , u = x , u + y , y x , u + y , y = x y , u .
( x 1 y 2 ) + ( x 2 y 1 ) = a 1 + a 2 .
C u ( B ) = { C : B C ; C is u convex } .
C u ( B ) = { [ x , y ] : x , y B , ( x y ) u } .
x 1 y 2 = ( x 1 b 1 ) + ( b 1 b 2 ) + ( b 2 y 2 ) u .
[ x 1 , y 2 ] , [ x 1 , x 2 ] , [ y 1 , y 2 ] , [ x 2 , y 1 ] C 0 .
m = a 1 + a 2 2 = x 1 y 1 + x 2 y 2 2 = x 1 y 1 + x 1 + β υ y 1 + γ υ 2 = x 1 + β + γ 2 υ y 1 .
d ( A , u ) = 2 d ( S , u ) .
E [ A ( A + a ) , u ] = { a } .
E [ A ( A + a ) , u ] = { 0 } ,
d [ A ( A + a ) , u ] = d ( A , u ) 2 = d ( S , u ) .
x y x y 2 , υ < d 4
y + x x y 2 , υ < d 4 .
y x , υ > d 2 ,
sup { x + x 0 , u : x B 1 } = x 0 , u + d ( B 1 , u ) 2 > d ( B 2 , u ) 2
inf { x + x 0 , u : x B 1 } = d ( B 1 , u ) 2 x 0 , u < d ( B 2 , u ) 2

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