Abstract

We consider the problem of restoring an image from the phase-sign function (the so-called “one-bit of phase”) and the Fourier magnitude by using the method of generalized projections. The method of generalized projections is an extension of the method of convex projections and can be used when constraint sets are nonconvex, as is the case here. While a structurally similar algorithm exists for retrieval from a signed magnitude, the advantage of generalized projections is that a certain type of error-reduction property is ensured. To the best of our knowledge this claim is not made for the other algorithm. Computer simulations of restorations in one and two dimensions are furnished. The recovered images are of good quality and do not exhibit the severe stagnation that is sometimes seen in magnitude-only restorations. In addition to nonconvex sets, signed-magnitude recovery involves nonclosed sets as well. We discuss this theoretical problem in some detail.

© 1987 Optical Society of America

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References

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  1. See, for example, Refs. 2–4 below, where further references may be found.
  2. A. Levi, H. Stark, “Signal restoration from phase by projections onto convex sets,”J. Opt. Soc. Am. 73, 810–822 (1983).
    [CrossRef]
  3. A. Levi, H. Stark, “Image restoration by the method of generalized projections with applications to restoration from magnitude,” J. Opt. Soc. Am. A 1, 932–943 (1984).
    [CrossRef]
  4. We work in a Hilbert space ℋ with the norm of g ∈ ℋ denoted by ||g|| and the inner product of g ∈ ℋ denoted by (g, f). For more details refer to Refs. 2, 3, 6, and 7.
  5. M. H. Hayes, “The unique reconstruction of multidimensional sequences from Fourier transform magnitude or phase,” in Image Recovery: Theory and Applications, H. Stark, ed. (Academic, New York, to be published), Chap. 6.
  6. D. C. Youla, H. Webb, “Image restoration by the method of projections onto convex sets. Part I,”IEEE Trans. Med. Imaging TMI-1, 81–94 (1982).
    [CrossRef]
  7. M. I. Sezan, H. Stark, “Image restoration by the method of projections onto convex sets. Part II,”IEEE Trans. Med. Imaging TMI-1, 95–101 (1982).
    [CrossRef]
  8. M. H. Hayes, “The reconstruction of a multidimensional sequence from the phase or magnitude of its Fourier transform,”IEEE Trans. Acoustics Speech Signal Process. ASSP-30, 140–154 (1982).
    [CrossRef]
  9. S. R. Curtis, J. S. Lim, A. V. Oppenheim, “Signal reconstruction from Fourier transform sign information,”IEEE Trans. Acoustics Speech Signal Process. ASSP-33, 643–657 (1985).
    [CrossRef]
  10. P. L. Van Hove, M. H. Hayes, J. L. Lim, A. V. Oppenheim, “Signal reconstruction. from signed Fourier transform magnitude,”IEEE Trans. Acoustics ASSP-31, 1286–1293 (1983).
    [CrossRef]
  11. A set C⊆ ℋ is said to be closed if given a sequence {fn}, fn∈ C∀ n, limn→∞||fn− f|| = 0 implies that f ∈ C, i.e., C is closed if it contains all its limit functions.
  12. Let C′ be the set of all limit points of C in ℋ; then the closure of C(denoted by C¯) is C¯ = C∪ C′. A set is closed if and only if C=C¯.
  13. m(A) denotes the (Lebesgue) measure of the set A, where A is any set of real numbers. (The Lebesgue measure of an interval is its length. The measure of an isolated point is zero. The measure of the real line is infinite.)
  14. L2is the space of all square-integrable functions over −∞ < t< ∞.
  15. The theorem that states this result uses a more general definition of the signum function that permits different partitions of the complex plane. For the case under consideration in this paper (the standard signum function) the theorem requires, in addition, that the sequence vanish at the origin.
  16. A (complex) function is entire if it is analytic everywhere. The FT of an integrable function with finite support is entire.
  17. See, for example, R. V. Churchill, Complex Variables and Applications (McGraw-Hill, New York, 1948), p. 194.
  18. An alternative and more concise proof is that every causal function may be written as a sum of an even function fe and an odd function fo. Now, we prove the statement by contradiction. Assume that Im F(ω) = 0. Since Im Fe(ω) = 0 (an even function) we conclude that Im Fo(ω) = 0. But Re Fo(ω) = 0 (an odd function). Thus fo= 0 and f= fe, which contradicts our assumptions on f(f causal and f≠ 0).

1985

S. R. Curtis, J. S. Lim, A. V. Oppenheim, “Signal reconstruction from Fourier transform sign information,”IEEE Trans. Acoustics Speech Signal Process. ASSP-33, 643–657 (1985).
[CrossRef]

1984

1983

A. Levi, H. Stark, “Signal restoration from phase by projections onto convex sets,”J. Opt. Soc. Am. 73, 810–822 (1983).
[CrossRef]

P. L. Van Hove, M. H. Hayes, J. L. Lim, A. V. Oppenheim, “Signal reconstruction. from signed Fourier transform magnitude,”IEEE Trans. Acoustics ASSP-31, 1286–1293 (1983).
[CrossRef]

1982

D. C. Youla, H. Webb, “Image restoration by the method of projections onto convex sets. Part I,”IEEE Trans. Med. Imaging TMI-1, 81–94 (1982).
[CrossRef]

M. I. Sezan, H. Stark, “Image restoration by the method of projections onto convex sets. Part II,”IEEE Trans. Med. Imaging TMI-1, 95–101 (1982).
[CrossRef]

M. H. Hayes, “The reconstruction of a multidimensional sequence from the phase or magnitude of its Fourier transform,”IEEE Trans. Acoustics Speech Signal Process. ASSP-30, 140–154 (1982).
[CrossRef]

Churchill, R. V.

See, for example, R. V. Churchill, Complex Variables and Applications (McGraw-Hill, New York, 1948), p. 194.

Curtis, S. R.

S. R. Curtis, J. S. Lim, A. V. Oppenheim, “Signal reconstruction from Fourier transform sign information,”IEEE Trans. Acoustics Speech Signal Process. ASSP-33, 643–657 (1985).
[CrossRef]

Hayes, M. H.

P. L. Van Hove, M. H. Hayes, J. L. Lim, A. V. Oppenheim, “Signal reconstruction. from signed Fourier transform magnitude,”IEEE Trans. Acoustics ASSP-31, 1286–1293 (1983).
[CrossRef]

M. H. Hayes, “The reconstruction of a multidimensional sequence from the phase or magnitude of its Fourier transform,”IEEE Trans. Acoustics Speech Signal Process. ASSP-30, 140–154 (1982).
[CrossRef]

M. H. Hayes, “The unique reconstruction of multidimensional sequences from Fourier transform magnitude or phase,” in Image Recovery: Theory and Applications, H. Stark, ed. (Academic, New York, to be published), Chap. 6.

Levi, A.

Lim, J. L.

P. L. Van Hove, M. H. Hayes, J. L. Lim, A. V. Oppenheim, “Signal reconstruction. from signed Fourier transform magnitude,”IEEE Trans. Acoustics ASSP-31, 1286–1293 (1983).
[CrossRef]

Lim, J. S.

S. R. Curtis, J. S. Lim, A. V. Oppenheim, “Signal reconstruction from Fourier transform sign information,”IEEE Trans. Acoustics Speech Signal Process. ASSP-33, 643–657 (1985).
[CrossRef]

Oppenheim, A. V.

S. R. Curtis, J. S. Lim, A. V. Oppenheim, “Signal reconstruction from Fourier transform sign information,”IEEE Trans. Acoustics Speech Signal Process. ASSP-33, 643–657 (1985).
[CrossRef]

P. L. Van Hove, M. H. Hayes, J. L. Lim, A. V. Oppenheim, “Signal reconstruction. from signed Fourier transform magnitude,”IEEE Trans. Acoustics ASSP-31, 1286–1293 (1983).
[CrossRef]

Sezan, M. I.

M. I. Sezan, H. Stark, “Image restoration by the method of projections onto convex sets. Part II,”IEEE Trans. Med. Imaging TMI-1, 95–101 (1982).
[CrossRef]

Stark, H.

Van Hove, P. L.

P. L. Van Hove, M. H. Hayes, J. L. Lim, A. V. Oppenheim, “Signal reconstruction. from signed Fourier transform magnitude,”IEEE Trans. Acoustics ASSP-31, 1286–1293 (1983).
[CrossRef]

Webb, H.

D. C. Youla, H. Webb, “Image restoration by the method of projections onto convex sets. Part I,”IEEE Trans. Med. Imaging TMI-1, 81–94 (1982).
[CrossRef]

Youla, D. C.

D. C. Youla, H. Webb, “Image restoration by the method of projections onto convex sets. Part I,”IEEE Trans. Med. Imaging TMI-1, 81–94 (1982).
[CrossRef]

IEEE Trans. Acoustics

P. L. Van Hove, M. H. Hayes, J. L. Lim, A. V. Oppenheim, “Signal reconstruction. from signed Fourier transform magnitude,”IEEE Trans. Acoustics ASSP-31, 1286–1293 (1983).
[CrossRef]

IEEE Trans. Acoustics Speech Signal Process.

M. H. Hayes, “The reconstruction of a multidimensional sequence from the phase or magnitude of its Fourier transform,”IEEE Trans. Acoustics Speech Signal Process. ASSP-30, 140–154 (1982).
[CrossRef]

S. R. Curtis, J. S. Lim, A. V. Oppenheim, “Signal reconstruction from Fourier transform sign information,”IEEE Trans. Acoustics Speech Signal Process. ASSP-33, 643–657 (1985).
[CrossRef]

IEEE Trans. Med. Imaging

D. C. Youla, H. Webb, “Image restoration by the method of projections onto convex sets. Part I,”IEEE Trans. Med. Imaging TMI-1, 81–94 (1982).
[CrossRef]

M. I. Sezan, H. Stark, “Image restoration by the method of projections onto convex sets. Part II,”IEEE Trans. Med. Imaging TMI-1, 95–101 (1982).
[CrossRef]

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

Other

We work in a Hilbert space ℋ with the norm of g ∈ ℋ denoted by ||g|| and the inner product of g ∈ ℋ denoted by (g, f). For more details refer to Refs. 2, 3, 6, and 7.

M. H. Hayes, “The unique reconstruction of multidimensional sequences from Fourier transform magnitude or phase,” in Image Recovery: Theory and Applications, H. Stark, ed. (Academic, New York, to be published), Chap. 6.

A set C⊆ ℋ is said to be closed if given a sequence {fn}, fn∈ C∀ n, limn→∞||fn− f|| = 0 implies that f ∈ C, i.e., C is closed if it contains all its limit functions.

Let C′ be the set of all limit points of C in ℋ; then the closure of C(denoted by C¯) is C¯ = C∪ C′. A set is closed if and only if C=C¯.

m(A) denotes the (Lebesgue) measure of the set A, where A is any set of real numbers. (The Lebesgue measure of an interval is its length. The measure of an isolated point is zero. The measure of the real line is infinite.)

L2is the space of all square-integrable functions over −∞ < t< ∞.

The theorem that states this result uses a more general definition of the signum function that permits different partitions of the complex plane. For the case under consideration in this paper (the standard signum function) the theorem requires, in addition, that the sequence vanish at the origin.

A (complex) function is entire if it is analytic everywhere. The FT of an integrable function with finite support is entire.

See, for example, R. V. Churchill, Complex Variables and Applications (McGraw-Hill, New York, 1948), p. 194.

An alternative and more concise proof is that every causal function may be written as a sum of an even function fe and an odd function fo. Now, we prove the statement by contradiction. Assume that Im F(ω) = 0. Since Im Fe(ω) = 0 (an even function) we conclude that Im Fo(ω) = 0. But Re Fo(ω) = 0 (an odd function). Thus fo= 0 and f= fe, which contradicts our assumptions on f(f causal and f≠ 0).

See, for example, Refs. 2–4 below, where further references may be found.

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Figures (7)

Fig. 1
Fig. 1

Original one-dimensional signal.

Fig. 2
Fig. 2

Original image: Test Pattern (denoted as I1).

Fig. 3
Fig. 3

Original image: Building (denoted as I2).

Fig. 4
Fig. 4

Restoration of one-dimensional signal with fo = 1.0, λ = 1.0. (a) After 5 iterations. (b) After 10 iterations. (c) After 20 iterations. (d) MSE as a function of the iteration number.

Fig. 5
Fig. 5

Restoration of Test Pattern (I1) with fo = 1.0, λ = 1.0. (a) After 4 iterations. (b) After 10 iterations. (c) After 30 iterations.

Fig. 6
Fig. 6

Restoration of Building (I2) with fo = 1.0, λ = 1.0. (a) After 4 iterations. (b) Absolute value of difference image for (a). (c) After 30 iterations. (d) Absolute value of difference image for (c). (e) After 90 iterations. (f) Absolute value of difference image for (e). (g) MSE as a function of the iteration number.

Fig. 7
Fig. 7

Restoration of Building (I2) with fo = 1.0, λ = 1.9. (a) After 30 iterations. (b) Absolute value of difference image for (a). (c) After 50 iterations. (d) Absolute value of difference image for (c). (e) MSE as a function of the iteration number.

Tables (2)

Tables Icon

Table 1 Signals for Experiments

Tables Icon

Table 2 Summary of Experimental Results

Equations (49)

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g - h = min y - h             over all y C i ,             i = 1 , 2 , , m .
f n + 1 = P 1 P 2 P m f n ,             f o arbitrary
f n + 1 = T 1 T 2 T m f n ,             f o arbitrary ,
T i = 1 + λ i ( P i - 1 )             i = 1 , 2 , , m ,             0 < λ i < 2.
C M = { g ( x ) G ( ω ) : G ( ω ) = M ( ω ) for all ω }
C L = { g ( x ) : g ( x ) = 0 for x < 0 , x > a } ,
f n + 1 = T L T M f n ,             f o arbitrary ,
J ( f n ) = P L f n - f n + P M f n - f n ,
J ( f n + 1 ) J ( f n ) .
C L = { g ( x ) : g ( x ) = 0 for x < 0 , x > a }
C S P = { g ( x ) : α g ( ω ) = θ ( ω ) } ,
α g ( ω ) sgn [ ϕ g ( ω ) ]
sgn ( x ) = { 1 x 0 - 1 x < 0 .
sgn [ ϕ f ( ω ) ] = sgn { sin [ ϕ f ( ω ) ] } = sgn [ Im F ( ω ) ]
K S P = { g ( x ) : m [ { ω : Im G ( ω ) = 0 } { ω : θ ( ω ) = - 1 } ] > 0 } .
P L g = { g ( x ) 0 x a 0 otherwise .
P S P g FT { G ( ω ) if α g ( ω ) = θ ( ω ) Re G ( ω ) if α g ( ω ) = - θ ( ω ) ,
f n + 1 = T S P T L f n ,             0 f o , arbitrary ,
T i 1 + λ i ( P i - 1 ) ,             i { S P , L } .
C S M = { g ( x ) : A g ( ω ) = P ( ω ) } ,
A g ( ω ) G ( ω ) α g ( ω ) ,
P S M g FT { P ( ω ) exp [ j ϕ g ( ω ) ] if α g ( ω ) = θ ( ω ) P ( ω ) sgn { Re [ G ( ω ) ] } if α g ( ω ) = - θ ( ω ) ,
C L L = { g ( x ) : g ( x ) = 0 for x < 0 , x > a and 0 g ( x ) g max for 0 x a } ,
P L L g = { max ( min [ g ( x ) , g max ] , 0 ) 0 x a 0 otherwise
f n + 1 = P L L T S M f n ,             f o arbitrary ,
T S M = 1 + λ ( P S M - 1 ) .
MSE = 100 f n - f f ,
S ( x ) = { + 1 - π / 2 x π / 2 - 1 otherwise .
C S P = { g ( x ) : α g ( ω ) = - sgn ω } .
F n ( ω ) = 1 n + j ω = n n 2 + ω 2 - j ω n 2 + ω 2 .
sgn [ Im F n ( ω ) ] = - sgn ω ,
- [ B n ( ω ) - B ( ω ) ] 2 d ω 0 ,             n .
Γ A = { ω : sgn [ B ( ω ) ] sgn [ B n ( ω ) ] } .
Γ A [ B n ( ω ) - B ( ω ) ] 2 d ω 0 ,             n .
sgn [ B ( ω ) ] = sgn ( 0 ) = 1 a . e . for ω Γ A
sgn [ B n ( ω ) ] = sgn [ B ( ω ) ] = - 1 a . e . for ω Γ A .
g - f 2 = 1 2 π G ( ω ) - F ( ω ) 2 d ω .
I = [ Re G ( ω ) - Re F ( ω ) ] 2 + [ Im G ( ω ) - Im F ( ω ) ] 2 .
α f ( ω ) sgn [ Im F ( ω ) ] = θ ( ω ) .
Re F ( ω ) = Re G ( ω ) ,
Im F ( ω ) = { Im G ( ω ) if α g ( ω ) = α f ( ω ) = θ ( ω ) 0 if α g ( ω ) = - α f ( ω ) = - θ ( ω ) .
P S P g FT { G ( ω ) if α g ( ω ) = θ ( ω ) Re G ( ω ) if α g ( ω ) = - θ ( ω ) .
g - h 2 = 1 2 π G ( ω ) exp [ j ϕ g ( ω ) ] - P ( ω ) exp [ j ϕ h ( ω ) ] 2 d ω = 1 2 π { G ( ω ) 2 + P ( ω ) 2 - 2 G ( ω ) P ( ω ) × cos [ ϕ g ( ω ) - ϕ h ( ω ) ] } d ω ,
I 1 = cos [ ϕ g ( ω ) - ϕ h ( ω ) ] .
I 1 = cos ( ϕ g ( ω ) sgn [ ϕ g ( ω ) ] - ϕ h ( ω ) sgn [ ϕ h ( ω ) ] ) .
H ( ω ) = P ( ω ) exp [ j ϕ g ( ω ) ]             if α g ( ω ) = θ ( ω ) .
ϕ h ( ω ) = { 0 for ϕ g ( ω ) π / 2 π for ϕ g ( ω ) > π / 2 ,
H ( ω ) = { P ( ω ) for ϕ g ( ω ) π / 2 - P ( ω ) for ϕ g ( ω ) > π / 2 ,
H ( ω ) = P ( ω ) sgn [ Re G ( ω ) ]             if α g ( ω ) = - θ ( ω ) .

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