Abstract

Signal reconstruction based on knowing either the magnitude or the phase of the Fourier transform of the signal is important in numerous applications, and the problem of signal reconstruction from noisy-phase and noisy-magnitude data is addressed. The proposed procedure relates to the deviations of the available magnitude and phase estimates from their exact values in the reconstruction algorithm by use of spectral prototype constraint sets. The properties of these new constraint sets for the magnitude and the phase of the Fourier transform are analyzed, and the corresponding projection operators are constructed. Simulation results indicate improvement of the performance of reconstructions from noisy-phase and -magnitude values based on these sets.

© 1994 Optical Society of America

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  1. N. E. Hurt, Phase Retrieval and Zero Crossings: Mathematical Methods in Image Reconstruction (Kluwer Academic, Norwell, Mass, 1989), pp. 13–114.
  2. A. Tikhonov, V. Arsenin, Solutions of Ill-Posed Problems (Holt, Rinehart & Winston, New York, 1977), pp. 1–258.
  3. J. R. Fienup, “Reconstruction of a complex-values object from the modulus of its Fourier transform using support constraint,” J. Opt. Soc. Am. A 4, 118–123 (1987).
    [CrossRef]
  4. A Levi, H. Stark, “Image restoration by the method of convex projections with application to restoration from amplitude,” J. Opt. Soc. Am. A 1, 932–943 (1984).
    [CrossRef]
  5. M. H. Hayes, “The unique reconstruction of multidimensional sequences from Fourier-transform magnitude or phase,” in Image Recovery, H. Stark, ed. (Academic, New York, 1987), pp. 195–230.
  6. A. Levi, H. Stark, “Restoration from phase and magnitude by generalized projections,” in Image Recovery, H. Stark, ed. Academic, New York, 1987), pp. 277–320.
  7. J. C. Dainty, J. R. Fienup, “Phase retrieval and image reconstruction for astronomy,” in Image Recovery, H. Stark, ed. (Academic, New York, 1987), pp. 231–275.
  8. J. H. Seldin, J. R. Fienup, “Numerical investigation of the uniqueness of phase retrieval,” J. Opt. Soc. Am. A 7, 412–427 (1990).
    [CrossRef]
  9. C. A. Haniff, “Least-squares Fourier phase estimation from the modulo 2 bispectrum phase,” J. Opt. Soc. Am. A 8, 134–141 (1991).
    [CrossRef]
  10. J. R. Fienup, A. M. Kowalczyk, “Phase retrieval for a complex-valued object by using a low-resolution image,” J. Opt. Soc. Am. A 7, 450–459 (1990).
    [CrossRef]
  11. J. H. Seldin, J. R. Fienup, “Iterative blind deconvolution algorithm applied to phase retrieval,” J. Opt. Soc. Am. A 7, 428–433 (1990).
    [CrossRef]
  12. D. C. Youla, “Mathematical theory of image restoration by the method of convex projections,” in Image Recovery, H. Stark, ed. (Academic, New York, 1987), pp. 29–77.
  13. H. J. Trussel, M. R. Civanlar, “The feasible solution on signal restoration,” IEEE Trans. Acoust. Speech Signal Process. ASSP-32, 201–212 (1984).
    [CrossRef]
  14. R. Barakat, G. Newsam, “Algorithms for reconstruction of partially known, band limited Fourier-transform pairs from noisy data,” J. Opt. Soc. Am. A 2, 2027–2039 (1985).
    [CrossRef]
  15. D. C. Youla, H. Webb, “Image restoration by the method of convex projections. Part 1. Theory,” IEEE Trans. Med. Imag. MI-2, 81–94 (1982).
    [CrossRef]
  16. N. E. Hurt, “Signal enhancement and the method of successive projections,” Acta Appl. Math. 23, 145–162 (1991).
    [CrossRef]
  17. P. Combettes, H. J. Trussel, “A method of successive projections for finding a common point of sets in metric spaces,” J. Opt. Theory Appl. 67, 487–507 (1990).
    [CrossRef]
  18. J. C. Marron, P. P. Sanchez, R. C. Sullivan, “Unwrapping algorithm for least-squares phase recovery from the modulo 2π bispectrum phase,” J. Opt. Soc. Am. A 7, 14–20 (1990).
    [CrossRef]
  19. X. Shi, R. K. Ward, “Restoration of images degraded by atmospheric turbulence and detection noise,” J. Opt. Soc. Am. A 9, 364–370 (1992).
    [CrossRef]
  20. G. R. Ayers, M. J. Northcott, J. C. Dainty, “Knox–Thompson and triple-correlation imaging through atmospheric turbulence,” J. Opt. Soc. Am. A 7, 963–985 (1988).
    [CrossRef]
  21. S. Jin, S. Wear, M. R. Raghuveer, “Reconstruction of speckled images using bispectra,” J. Opt. Soc. Am. A 9, 371–376 (1992).
    [CrossRef]
  22. M. I. Sezan, H. J. Trussel, “Prototype constraints for set-theoretic image restoration,” IEEE Trans. Acoust. Speech Signal Process. 39, 2275–2285 (1991).
  23. A. Glindemann, R. G. Lane, J. C. Dainty, “Estimation of binary star parameters by model fitting the bispectrum phase,” J. Opt. Soc. Am A 9, 543–548 (1992).
    [CrossRef]
  24. J. R. Fienup, “Image reconstruction using the phase variance algorithm,” in Digital Image Synthesis and Inverse Optics, A. F. Gmitro, P. S. Idell, I. J. Lattaie, eds., Proc. Soc. Photo-Opt. Instrum. Eng. 1351, 652–660 (1990).
  25. B. M. Sadler, G. B. Giannakis, “Shift- and rotation-invariant object reconstruction using the bispectrum,” J. Opt. Soc. Am. A 9, 57–69 (1992).
    [CrossRef]

1992 (4)

1991 (3)

C. A. Haniff, “Least-squares Fourier phase estimation from the modulo 2 bispectrum phase,” J. Opt. Soc. Am. A 8, 134–141 (1991).
[CrossRef]

M. I. Sezan, H. J. Trussel, “Prototype constraints for set-theoretic image restoration,” IEEE Trans. Acoust. Speech Signal Process. 39, 2275–2285 (1991).

N. E. Hurt, “Signal enhancement and the method of successive projections,” Acta Appl. Math. 23, 145–162 (1991).
[CrossRef]

1990 (5)

1988 (1)

G. R. Ayers, M. J. Northcott, J. C. Dainty, “Knox–Thompson and triple-correlation imaging through atmospheric turbulence,” J. Opt. Soc. Am. A 7, 963–985 (1988).
[CrossRef]

1987 (1)

1985 (1)

1984 (2)

A Levi, H. Stark, “Image restoration by the method of convex projections with application to restoration from amplitude,” J. Opt. Soc. Am. A 1, 932–943 (1984).
[CrossRef]

H. J. Trussel, M. R. Civanlar, “The feasible solution on signal restoration,” IEEE Trans. Acoust. Speech Signal Process. ASSP-32, 201–212 (1984).
[CrossRef]

1982 (1)

D. C. Youla, H. Webb, “Image restoration by the method of convex projections. Part 1. Theory,” IEEE Trans. Med. Imag. MI-2, 81–94 (1982).
[CrossRef]

Arsenin, V.

A. Tikhonov, V. Arsenin, Solutions of Ill-Posed Problems (Holt, Rinehart & Winston, New York, 1977), pp. 1–258.

Ayers, G. R.

G. R. Ayers, M. J. Northcott, J. C. Dainty, “Knox–Thompson and triple-correlation imaging through atmospheric turbulence,” J. Opt. Soc. Am. A 7, 963–985 (1988).
[CrossRef]

Barakat, R.

Civanlar, M. R.

H. J. Trussel, M. R. Civanlar, “The feasible solution on signal restoration,” IEEE Trans. Acoust. Speech Signal Process. ASSP-32, 201–212 (1984).
[CrossRef]

Combettes, P.

P. Combettes, H. J. Trussel, “A method of successive projections for finding a common point of sets in metric spaces,” J. Opt. Theory Appl. 67, 487–507 (1990).
[CrossRef]

Dainty, J. C.

A. Glindemann, R. G. Lane, J. C. Dainty, “Estimation of binary star parameters by model fitting the bispectrum phase,” J. Opt. Soc. Am A 9, 543–548 (1992).
[CrossRef]

G. R. Ayers, M. J. Northcott, J. C. Dainty, “Knox–Thompson and triple-correlation imaging through atmospheric turbulence,” J. Opt. Soc. Am. A 7, 963–985 (1988).
[CrossRef]

J. C. Dainty, J. R. Fienup, “Phase retrieval and image reconstruction for astronomy,” in Image Recovery, H. Stark, ed. (Academic, New York, 1987), pp. 231–275.

Fienup, J. R.

Giannakis, G. B.

Glindemann, A.

A. Glindemann, R. G. Lane, J. C. Dainty, “Estimation of binary star parameters by model fitting the bispectrum phase,” J. Opt. Soc. Am A 9, 543–548 (1992).
[CrossRef]

Haniff, C. A.

Hayes, M. H.

M. H. Hayes, “The unique reconstruction of multidimensional sequences from Fourier-transform magnitude or phase,” in Image Recovery, H. Stark, ed. (Academic, New York, 1987), pp. 195–230.

Hurt, N. E.

N. E. Hurt, “Signal enhancement and the method of successive projections,” Acta Appl. Math. 23, 145–162 (1991).
[CrossRef]

N. E. Hurt, Phase Retrieval and Zero Crossings: Mathematical Methods in Image Reconstruction (Kluwer Academic, Norwell, Mass, 1989), pp. 13–114.

Jin, S.

Kowalczyk, A. M.

Lane, R. G.

A. Glindemann, R. G. Lane, J. C. Dainty, “Estimation of binary star parameters by model fitting the bispectrum phase,” J. Opt. Soc. Am A 9, 543–548 (1992).
[CrossRef]

Levi, A

Levi, A.

A. Levi, H. Stark, “Restoration from phase and magnitude by generalized projections,” in Image Recovery, H. Stark, ed. Academic, New York, 1987), pp. 277–320.

Marron, J. C.

Newsam, G.

Northcott, M. J.

G. R. Ayers, M. J. Northcott, J. C. Dainty, “Knox–Thompson and triple-correlation imaging through atmospheric turbulence,” J. Opt. Soc. Am. A 7, 963–985 (1988).
[CrossRef]

Raghuveer, M. R.

Sadler, B. M.

Sanchez, P. P.

Seldin, J. H.

Sezan, M. I.

M. I. Sezan, H. J. Trussel, “Prototype constraints for set-theoretic image restoration,” IEEE Trans. Acoust. Speech Signal Process. 39, 2275–2285 (1991).

Shi, X.

Stark, H.

A Levi, H. Stark, “Image restoration by the method of convex projections with application to restoration from amplitude,” J. Opt. Soc. Am. A 1, 932–943 (1984).
[CrossRef]

A. Levi, H. Stark, “Restoration from phase and magnitude by generalized projections,” in Image Recovery, H. Stark, ed. Academic, New York, 1987), pp. 277–320.

Sullivan, R. C.

Tikhonov, A.

A. Tikhonov, V. Arsenin, Solutions of Ill-Posed Problems (Holt, Rinehart & Winston, New York, 1977), pp. 1–258.

Trussel, H. J.

M. I. Sezan, H. J. Trussel, “Prototype constraints for set-theoretic image restoration,” IEEE Trans. Acoust. Speech Signal Process. 39, 2275–2285 (1991).

P. Combettes, H. J. Trussel, “A method of successive projections for finding a common point of sets in metric spaces,” J. Opt. Theory Appl. 67, 487–507 (1990).
[CrossRef]

H. J. Trussel, M. R. Civanlar, “The feasible solution on signal restoration,” IEEE Trans. Acoust. Speech Signal Process. ASSP-32, 201–212 (1984).
[CrossRef]

Ward, R. K.

Wear, S.

Webb, H.

D. C. Youla, H. Webb, “Image restoration by the method of convex projections. Part 1. Theory,” IEEE Trans. Med. Imag. MI-2, 81–94 (1982).
[CrossRef]

Youla, D. C.

D. C. Youla, H. Webb, “Image restoration by the method of convex projections. Part 1. Theory,” IEEE Trans. Med. Imag. MI-2, 81–94 (1982).
[CrossRef]

D. C. Youla, “Mathematical theory of image restoration by the method of convex projections,” in Image Recovery, H. Stark, ed. (Academic, New York, 1987), pp. 29–77.

Acta Appl. Math. (1)

N. E. Hurt, “Signal enhancement and the method of successive projections,” Acta Appl. Math. 23, 145–162 (1991).
[CrossRef]

IEEE Trans. Acoust. Speech Signal Process. (2)

H. J. Trussel, M. R. Civanlar, “The feasible solution on signal restoration,” IEEE Trans. Acoust. Speech Signal Process. ASSP-32, 201–212 (1984).
[CrossRef]

M. I. Sezan, H. J. Trussel, “Prototype constraints for set-theoretic image restoration,” IEEE Trans. Acoust. Speech Signal Process. 39, 2275–2285 (1991).

IEEE Trans. Med. Imag. (1)

D. C. Youla, H. Webb, “Image restoration by the method of convex projections. Part 1. Theory,” IEEE Trans. Med. Imag. MI-2, 81–94 (1982).
[CrossRef]

J. Opt. Soc. Am A (1)

A. Glindemann, R. G. Lane, J. C. Dainty, “Estimation of binary star parameters by model fitting the bispectrum phase,” J. Opt. Soc. Am A 9, 543–548 (1992).
[CrossRef]

J. Opt. Soc. Am. A (12)

G. R. Ayers, M. J. Northcott, J. C. Dainty, “Knox–Thompson and triple-correlation imaging through atmospheric turbulence,” J. Opt. Soc. Am. A 7, 963–985 (1988).
[CrossRef]

A Levi, H. Stark, “Image restoration by the method of convex projections with application to restoration from amplitude,” J. Opt. Soc. Am. A 1, 932–943 (1984).
[CrossRef]

R. Barakat, G. Newsam, “Algorithms for reconstruction of partially known, band limited Fourier-transform pairs from noisy data,” J. Opt. Soc. Am. A 2, 2027–2039 (1985).
[CrossRef]

J. R. Fienup, “Reconstruction of a complex-values object from the modulus of its Fourier transform using support constraint,” J. Opt. Soc. Am. A 4, 118–123 (1987).
[CrossRef]

J. C. Marron, P. P. Sanchez, R. C. Sullivan, “Unwrapping algorithm for least-squares phase recovery from the modulo 2π bispectrum phase,” J. Opt. Soc. Am. A 7, 14–20 (1990).
[CrossRef]

J. H. Seldin, J. R. Fienup, “Numerical investigation of the uniqueness of phase retrieval,” J. Opt. Soc. Am. A 7, 412–427 (1990).
[CrossRef]

J. H. Seldin, J. R. Fienup, “Iterative blind deconvolution algorithm applied to phase retrieval,” J. Opt. Soc. Am. A 7, 428–433 (1990).
[CrossRef]

J. R. Fienup, A. M. Kowalczyk, “Phase retrieval for a complex-valued object by using a low-resolution image,” J. Opt. Soc. Am. A 7, 450–459 (1990).
[CrossRef]

C. A. Haniff, “Least-squares Fourier phase estimation from the modulo 2 bispectrum phase,” J. Opt. Soc. Am. A 8, 134–141 (1991).
[CrossRef]

B. M. Sadler, G. B. Giannakis, “Shift- and rotation-invariant object reconstruction using the bispectrum,” J. Opt. Soc. Am. A 9, 57–69 (1992).
[CrossRef]

X. Shi, R. K. Ward, “Restoration of images degraded by atmospheric turbulence and detection noise,” J. Opt. Soc. Am. A 9, 364–370 (1992).
[CrossRef]

S. Jin, S. Wear, M. R. Raghuveer, “Reconstruction of speckled images using bispectra,” J. Opt. Soc. Am. A 9, 371–376 (1992).
[CrossRef]

J. Opt. Theory Appl. (1)

P. Combettes, H. J. Trussel, “A method of successive projections for finding a common point of sets in metric spaces,” J. Opt. Theory Appl. 67, 487–507 (1990).
[CrossRef]

Other (7)

D. C. Youla, “Mathematical theory of image restoration by the method of convex projections,” in Image Recovery, H. Stark, ed. (Academic, New York, 1987), pp. 29–77.

M. H. Hayes, “The unique reconstruction of multidimensional sequences from Fourier-transform magnitude or phase,” in Image Recovery, H. Stark, ed. (Academic, New York, 1987), pp. 195–230.

A. Levi, H. Stark, “Restoration from phase and magnitude by generalized projections,” in Image Recovery, H. Stark, ed. Academic, New York, 1987), pp. 277–320.

J. C. Dainty, J. R. Fienup, “Phase retrieval and image reconstruction for astronomy,” in Image Recovery, H. Stark, ed. (Academic, New York, 1987), pp. 231–275.

N. E. Hurt, Phase Retrieval and Zero Crossings: Mathematical Methods in Image Reconstruction (Kluwer Academic, Norwell, Mass, 1989), pp. 13–114.

A. Tikhonov, V. Arsenin, Solutions of Ill-Posed Problems (Holt, Rinehart & Winston, New York, 1977), pp. 1–258.

J. R. Fienup, “Image reconstruction using the phase variance algorithm,” in Digital Image Synthesis and Inverse Optics, A. F. Gmitro, P. S. Idell, I. J. Lattaie, eds., Proc. Soc. Photo-Opt. Instrum. Eng. 1351, 652–660 (1990).

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Figures (4)

Fig. 1
Fig. 1

Signal restoration from noisy spectral modulus data: (a) original object, (b) noisy estimate based on the modulus data with ∊ = 0.2, (c) reconstruction with noisy magnitude, (d) signal reconstruction obtained by projection on the M2 set of constraints.

Fig. 2
Fig. 2

Signal reconstruction from a noisy spectrum phase estimate: (a) noisy estimated based on the phase data with additive noise, σ = 0.06; (b) restoration from the noisy phase; (c) reconstruction obtained by projections on the P2 set of constraints, σ = 0.06; (d) reconstruction obtained by projection on the P1 set, σ = π/3. The original signal is given in Fig. 1(a).

Fig. 3
Fig. 3

Signal reconstruction from noisy-modulus and noisy-phase data: (a) noisy estimate based on noisy-modulus data and noisy-phase data; (b) restoration obtained by projection on the M2 set, ∊ = 0.35; (c) restoration obtained by projection on the P2 set of constraints, σ = 0.12; (d) combined spectral synthesis obtained by the M2 and P2 sets. The original signal is given in Fig. 1(a)

Fig. 4
Fig. 4

Two-dimensional simulation of signal reconstruction from noisy-modulus data: (a) original object consisting of 128 × 128 pixels, (b) noisy estimate based on the modulus data with ∊ = 0.2, (c) reconstruction with noisy magnitude, (d) image reconstruction obtained by projection on the M2 set of constraints.

Tables (4)

Tables Icon

Table 1 Performance of the Iterative Algorithm [Eq. (18)] Utilizing the M2 Constraint Set for Various Values of ∊

Tables Icon

Table 2 Performance of the Iterative Algorithm [Eq. (18)] Utilizing the M1 Constraint Set for Various Values of ∊

Tables Icon

Table 3 Performance of the Iterative Algorithm [Eq. (17)] Utilizing the P1 Constraint Set for Various Values of σ

Tables Icon

Table 4 Performance of the Iterative Algorithm [Eq. (17)] Utilizing the P2 Constraint Set for Various Values of σ

Equations (28)

Equations on this page are rendered with MathJax. Learn more.

F { f ( x ) } = F ( w ) exp [ i φ ( w ) ] ,
f k + 1 = P 1 P 2 P N f k .
G i ( w ) = H i ( w ) F ( w ) + N i ( w ) ,
G ( w ) 2 = H ( w ) 2 F ( w ) 2 + E ( w ) ,
F ( w ) 2 = G ( w ) 2 - E ( w ) H ( w ) 2 .
F e ( w ) = [ F ( w ) ] + δ E ( w ) H ( w ) 2 .
F ( w ) - F e ( w ) .
F ( w ) - F e ( w ) .
P M f = F - 1 { M F ( w ) exp [ i φ ( w ) ] } ,
M 1 F ( w ) = { F e ( w ) + F ( w ) > F e ( w ) + F e ( w ) - F ( w ) < F e ( w ) - F e ( w ) F ( w ) - F e ( w ) ,
M 2 F ( w ) = { F ( w ) F ( w ) - F e ( w ) F e ( w ) + F ( w ) - F e ( w ) [ F ( w ) - F e ( w ) ] F ( w ) - F e ( w ) > .
φ ( w ) - φ e ( w ) σ .
φ ( w ) - φ e ( w ) σ .
P P f = F - 1 { F ( w ) cos [ P φ ( w ) - φ ( w ) ] exp [ i P φ ( w ) ] } if cos [ P φ ( w ) - φ ( w ) ] 0 , P P f = 0 if cos [ P φ ( w ) - φ ( w ) ] < 0
P 1 φ ( w ) = { φ ( w ) φ ( w ) - φ e ( w ) σ φ e ( w ) + σ φ ( w ) > φ e ( w ) + σ φ e ( w ) - σ φ ( w ) < φ e ( w ) - σ ,
P 2 φ ( w ) = { φ ( w ) φ ( w ) - φ e ( w ) σ φ e ( w ) + σ φ ( w ) - φ e ( w ) [ φ ( w ) - φ e ( w ) ] φ ( w ) - φ e ( w ) > σ .
f k + 1 = P P 1 , 2 S f k .
f k + 1 = P M 1 , 2 S f k .
f k + 1 = P S [ f k + λ ( 1 N i = 1 N P i f k - f k ) ] ,
p - f 2 = - { [ P ( w ) 2 + F ( w ) 2 - 2 P ( w ) F ( w ) × cos [ p ( w ) - φ ( w ) ] } d w .
F n ( w ) exp [ i φ n ( w ) ] F ( w ) exp [ i φ ( w ) ] .
φ ( w ) = φ 1 ( w ) + arctan { ( 1 - u ) F 2 ( w ) sin [ φ 2 ( w ) - φ 1 ( w ) ] μ F 1 ( w ) + ( 1 - μ ) F 2 ( w ) cos [ φ 2 ( w ) - φ 1 ( w ) ] } .
φ ( w ) = φ 1 ( w ) + k [ φ 2 ( w ) - φ 1 ( w ) ] ,             0 k 1.
φ ( w ) - φ e ( w ) ( 1 - k ) φ 1 ( w ) - φ e ( w ) + k φ 2 ( w ) - φ e ( w ) σ .
φ ( w ) - φ e ( w ) ( 1 - k ) φ 1 ( w ) - φ e ( w ) + k φ 2 ( w ) - φ e ( w ) σ .
P ( w ) = { F ( w ) cos [ a ( w ) - φ ( w ) ] cos [ a ( w ) - φ ( w ) 0 0 cos [ a ( w ) - φ ( w ) ] < 0 .
min a ( w ) { - F ( w ) 2 sin 2 [ a ( w ) - φ ( w ) ] d w + λ [ a ( w ) - φ e ( w ) 2 - σ 2 ] } ,
min a ( w ) { a ( w ) - ( w ) 2 + λ [ a ( w ) - φ e ( w ) 2 - σ 2 ] } .

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