Abstract

From finite complex spectral data one can construct a continuous object with a given support that is consistent with the data. Given Fourier magnitude data only, one can choose the phases arbitrarily in the above construction. The energy in the extrapolated spectrum is phase dependent and provides a cost function to be used in phase retrieval. The minimization process is performed iteratively, using an algorithm that can be viewed as a combination of Gerchberg–Papoulis and Fienup error reduction.

© 1987 Optical Society of America

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References

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  1. H. A. Ferwerda, “The phase reconstruction problem for wave amplitudes and coherence functions,” in Inverse Source Problems in Optics, H. P. Baltes, ed. (Springer-Verlag, Berlin, 1978), Vol. 9, p. 13.
    [CrossRef]
  2. L. S. Taylor, “The phase retrieval problem,”IEEE Trans Antennas Propag. AP-29, 317–343 (1981).
  3. M. A. Fiddy, “The phase retrieval problem,” in Inverse Optics, A. J. Devaney, ed., Proc. Soc. Photo-Opt. Instrum. Eng.413, 176–181 (1983).
    [CrossRef]
  4. J. Rosenblatt, “Phase retrieval,” Commun. Math. Phys. 95, 317–343 (1984).
    [CrossRef]
  5. J. L. C. Sanz, “Mathematical considerations for the problem of Fourier transform phase retrieval from magnitude,” SIAM J. Appl. Math. 45, 651–664 (1985).
    [CrossRef]
  6. I. S. Stefanescu, “On the phase retrieval problem in two dimensions,”J. Math. Phys. 26, 2141–2160 (1985).
    [CrossRef]
  7. 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]
  8. M. H. Hayes, J. H. McClellan, “Reducible polynomials in more than one variable,” Proc. IEEE 70, 197–198 (1982).
    [CrossRef]
  9. R. Barakat, G. Newsam, “Necessary conditions for a unique solution to two dimensional phase recovery,”J. Math. Phys. 25, 3190–3193 (1984).
    [CrossRef]
  10. W. Lawton “Uniqueness results for the phase retrieval problem for radial functions,”J. Opt. Soc. Am. 71, 1519–1522 (1981).
    [CrossRef]
  11. J. L. C. Sanz, T. S. Huang, “Unique reconstruction of a band-limited multidimensional signal from its phase or magnitude,”J. Opt. Soc. Am. 73, 1446–1450 (1983).
    [CrossRef]
  12. T. S. Huang, ed., Advances in Computer Vision and Image Processing (JAI, New York, 1984), Vol. 1.
  13. C. R. Smith, W. T. Grandy, Maximum Entropy and Bayesian Methods in Inverse Problems (Reidel, Dordrecht, The Netherlands, 1985).
    [CrossRef]
  14. A. M. Darling, H. V. Deighton, M. A. Fiddy, “Phase ambiguities in more than one dimension,” in Inverse Optics, A. J. Devaney, ed., Proc. Soc. Photo-Opt. Instrum. Eng.413, 197–201 (1983).
    [CrossRef]
  15. C. L. Byrne, R. M. Fitzgerald, “Spectral estimates that extend the maximum entropy and maximum likelihood methods,” SIAM J. Appl. Math 44, 425–442 (1984).
    [CrossRef]
  16. C. L. Byrne, R. M. Fitzgerald, M. A. Fiddy, T. J. Hall, A. M. Darling, “Image restoration and resolution enhancement,”J. Opt. Soc. Am. 73, 1481–1487 (1983).
    [CrossRef]
  17. A. M. Darling, T. J. Hall, M. A. Fiddy, “Stable, noniterative object reconstruction from incomplete data using prior knowledge,”J. Opt. Soc. Am. 73, 1466–1469 (1983).
    [CrossRef]
  18. R. W. Gerchberg, “Super-resolution through error energy reduction,” Opt. Acta 21, 709–720 (1974).
    [CrossRef]
  19. A. Papoulis, “A new algorithm in spectral analysis and band-limited extrapolation,”IEEE Trans. Circuits Syst. CAS-22, 733–742 (1975).
  20. C. L. Byrne, L. K. Jones, SIAM J. Appl. Math. (to be published).
  21. D. C. Youla, “Generalized image restoration by the method of alternating orthogonal projections,”IEEE Trans. Circuits Syst. CAS-25, 694–702 (1978).
    [CrossRef]
  22. D. C. Youla, H. Webb, “Image restoration by the method of projections onto convex sets,”IEEE Trans. Med. Imaging TMI-1, 81–94 (1982).
    [CrossRef]
  23. A. Levi, H. Stark, “Image restoration by the method of generalized projections with application to restoration from magnitude,” J. Opt. Soc. Am. A 1, 932–943 (1984).
    [CrossRef]
  24. J. R. Fienup, “Phase retrieval algorithms: a comparison,” Appl. Opt. 21, 2758–2769 (1982).
    [CrossRef] [PubMed]
  25. R. L. Frost, C. K. Rushforth, B. S. Baxter, “High resolution astronomical imaging through the turbulent atmosphere,” (University of Utah, Salt Lake City, 1979). Frost et al. developed an algorithm with some similarities to that presented here. Their procedure did not correct the magnitude spectrum to the measured magnitude each iteration, as in Fienup’s error-reduction algorithm. Instead, the magnitude and phase spectra were alternately extended through the frequency space. The phase-spectrum extension was allowed to converge, and then the new magnitude spectrum was scaled to agree with the measured magnitude below the old cutoff frequency. The reason for doing this was to limit instabilities in the extrapolation due to reintroducing the noise present in the known portion of the spectrum. They studied the consequences of varying the rate of extrapolation on the relative contribution of the noise on the final extrapolated estimate. Results presented by them looked promising, but further development and optimization of the procedure were not pursued.

1985 (3)

J. L. C. Sanz, “Mathematical considerations for the problem of Fourier transform phase retrieval from magnitude,” SIAM J. Appl. Math. 45, 651–664 (1985).
[CrossRef]

I. S. Stefanescu, “On the phase retrieval problem in two dimensions,”J. Math. Phys. 26, 2141–2160 (1985).
[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]

1984 (4)

C. L. Byrne, R. M. Fitzgerald, “Spectral estimates that extend the maximum entropy and maximum likelihood methods,” SIAM J. Appl. Math 44, 425–442 (1984).
[CrossRef]

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

J. Rosenblatt, “Phase retrieval,” Commun. Math. Phys. 95, 317–343 (1984).
[CrossRef]

R. Barakat, G. Newsam, “Necessary conditions for a unique solution to two dimensional phase recovery,”J. Math. Phys. 25, 3190–3193 (1984).
[CrossRef]

1983 (3)

1982 (3)

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

J. R. Fienup, “Phase retrieval algorithms: a comparison,” Appl. Opt. 21, 2758–2769 (1982).
[CrossRef] [PubMed]

M. H. Hayes, J. H. McClellan, “Reducible polynomials in more than one variable,” Proc. IEEE 70, 197–198 (1982).
[CrossRef]

1981 (2)

L. S. Taylor, “The phase retrieval problem,”IEEE Trans Antennas Propag. AP-29, 317–343 (1981).

W. Lawton “Uniqueness results for the phase retrieval problem for radial functions,”J. Opt. Soc. Am. 71, 1519–1522 (1981).
[CrossRef]

1978 (1)

D. C. Youla, “Generalized image restoration by the method of alternating orthogonal projections,”IEEE Trans. Circuits Syst. CAS-25, 694–702 (1978).
[CrossRef]

1975 (1)

A. Papoulis, “A new algorithm in spectral analysis and band-limited extrapolation,”IEEE Trans. Circuits Syst. CAS-22, 733–742 (1975).

1974 (1)

R. W. Gerchberg, “Super-resolution through error energy reduction,” Opt. Acta 21, 709–720 (1974).
[CrossRef]

Barakat, R.

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]

R. Barakat, G. Newsam, “Necessary conditions for a unique solution to two dimensional phase recovery,”J. Math. Phys. 25, 3190–3193 (1984).
[CrossRef]

Baxter, B. S.

R. L. Frost, C. K. Rushforth, B. S. Baxter, “High resolution astronomical imaging through the turbulent atmosphere,” (University of Utah, Salt Lake City, 1979). Frost et al. developed an algorithm with some similarities to that presented here. Their procedure did not correct the magnitude spectrum to the measured magnitude each iteration, as in Fienup’s error-reduction algorithm. Instead, the magnitude and phase spectra were alternately extended through the frequency space. The phase-spectrum extension was allowed to converge, and then the new magnitude spectrum was scaled to agree with the measured magnitude below the old cutoff frequency. The reason for doing this was to limit instabilities in the extrapolation due to reintroducing the noise present in the known portion of the spectrum. They studied the consequences of varying the rate of extrapolation on the relative contribution of the noise on the final extrapolated estimate. Results presented by them looked promising, but further development and optimization of the procedure were not pursued.

Byrne, C. L.

C. L. Byrne, R. M. Fitzgerald, “Spectral estimates that extend the maximum entropy and maximum likelihood methods,” SIAM J. Appl. Math 44, 425–442 (1984).
[CrossRef]

C. L. Byrne, R. M. Fitzgerald, M. A. Fiddy, T. J. Hall, A. M. Darling, “Image restoration and resolution enhancement,”J. Opt. Soc. Am. 73, 1481–1487 (1983).
[CrossRef]

C. L. Byrne, L. K. Jones, SIAM J. Appl. Math. (to be published).

Darling, A. M.

Deighton, H. V.

A. M. Darling, H. V. Deighton, M. A. Fiddy, “Phase ambiguities in more than one dimension,” in Inverse Optics, A. J. Devaney, ed., Proc. Soc. Photo-Opt. Instrum. Eng.413, 197–201 (1983).
[CrossRef]

Ferwerda, H. A.

H. A. Ferwerda, “The phase reconstruction problem for wave amplitudes and coherence functions,” in Inverse Source Problems in Optics, H. P. Baltes, ed. (Springer-Verlag, Berlin, 1978), Vol. 9, p. 13.
[CrossRef]

Fiddy, M. A.

C. L. Byrne, R. M. Fitzgerald, M. A. Fiddy, T. J. Hall, A. M. Darling, “Image restoration and resolution enhancement,”J. Opt. Soc. Am. 73, 1481–1487 (1983).
[CrossRef]

A. M. Darling, T. J. Hall, M. A. Fiddy, “Stable, noniterative object reconstruction from incomplete data using prior knowledge,”J. Opt. Soc. Am. 73, 1466–1469 (1983).
[CrossRef]

A. M. Darling, H. V. Deighton, M. A. Fiddy, “Phase ambiguities in more than one dimension,” in Inverse Optics, A. J. Devaney, ed., Proc. Soc. Photo-Opt. Instrum. Eng.413, 197–201 (1983).
[CrossRef]

M. A. Fiddy, “The phase retrieval problem,” in Inverse Optics, A. J. Devaney, ed., Proc. Soc. Photo-Opt. Instrum. Eng.413, 176–181 (1983).
[CrossRef]

Fienup, J. R.

Fitzgerald, R. M.

C. L. Byrne, R. M. Fitzgerald, “Spectral estimates that extend the maximum entropy and maximum likelihood methods,” SIAM J. Appl. Math 44, 425–442 (1984).
[CrossRef]

C. L. Byrne, R. M. Fitzgerald, M. A. Fiddy, T. J. Hall, A. M. Darling, “Image restoration and resolution enhancement,”J. Opt. Soc. Am. 73, 1481–1487 (1983).
[CrossRef]

Frost, R. L.

R. L. Frost, C. K. Rushforth, B. S. Baxter, “High resolution astronomical imaging through the turbulent atmosphere,” (University of Utah, Salt Lake City, 1979). Frost et al. developed an algorithm with some similarities to that presented here. Their procedure did not correct the magnitude spectrum to the measured magnitude each iteration, as in Fienup’s error-reduction algorithm. Instead, the magnitude and phase spectra were alternately extended through the frequency space. The phase-spectrum extension was allowed to converge, and then the new magnitude spectrum was scaled to agree with the measured magnitude below the old cutoff frequency. The reason for doing this was to limit instabilities in the extrapolation due to reintroducing the noise present in the known portion of the spectrum. They studied the consequences of varying the rate of extrapolation on the relative contribution of the noise on the final extrapolated estimate. Results presented by them looked promising, but further development and optimization of the procedure were not pursued.

Gerchberg, R. W.

R. W. Gerchberg, “Super-resolution through error energy reduction,” Opt. Acta 21, 709–720 (1974).
[CrossRef]

Grandy, W. T.

C. R. Smith, W. T. Grandy, Maximum Entropy and Bayesian Methods in Inverse Problems (Reidel, Dordrecht, The Netherlands, 1985).
[CrossRef]

Hall, T. J.

Hayes, M. H.

M. H. Hayes, J. H. McClellan, “Reducible polynomials in more than one variable,” Proc. IEEE 70, 197–198 (1982).
[CrossRef]

Huang, T. S.

Jones, L. K.

C. L. Byrne, L. K. Jones, SIAM J. Appl. Math. (to be published).

Lawton, W.

Levi, A.

McClellan, J. H.

M. H. Hayes, J. H. McClellan, “Reducible polynomials in more than one variable,” Proc. IEEE 70, 197–198 (1982).
[CrossRef]

Newsam, G.

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]

R. Barakat, G. Newsam, “Necessary conditions for a unique solution to two dimensional phase recovery,”J. Math. Phys. 25, 3190–3193 (1984).
[CrossRef]

Papoulis, A.

A. Papoulis, “A new algorithm in spectral analysis and band-limited extrapolation,”IEEE Trans. Circuits Syst. CAS-22, 733–742 (1975).

Rosenblatt, J.

J. Rosenblatt, “Phase retrieval,” Commun. Math. Phys. 95, 317–343 (1984).
[CrossRef]

Rushforth, C. K.

R. L. Frost, C. K. Rushforth, B. S. Baxter, “High resolution astronomical imaging through the turbulent atmosphere,” (University of Utah, Salt Lake City, 1979). Frost et al. developed an algorithm with some similarities to that presented here. Their procedure did not correct the magnitude spectrum to the measured magnitude each iteration, as in Fienup’s error-reduction algorithm. Instead, the magnitude and phase spectra were alternately extended through the frequency space. The phase-spectrum extension was allowed to converge, and then the new magnitude spectrum was scaled to agree with the measured magnitude below the old cutoff frequency. The reason for doing this was to limit instabilities in the extrapolation due to reintroducing the noise present in the known portion of the spectrum. They studied the consequences of varying the rate of extrapolation on the relative contribution of the noise on the final extrapolated estimate. Results presented by them looked promising, but further development and optimization of the procedure were not pursued.

Sanz, J. L. C.

J. L. C. Sanz, “Mathematical considerations for the problem of Fourier transform phase retrieval from magnitude,” SIAM J. Appl. Math. 45, 651–664 (1985).
[CrossRef]

J. L. C. Sanz, T. S. Huang, “Unique reconstruction of a band-limited multidimensional signal from its phase or magnitude,”J. Opt. Soc. Am. 73, 1446–1450 (1983).
[CrossRef]

Smith, C. R.

C. R. Smith, W. T. Grandy, Maximum Entropy and Bayesian Methods in Inverse Problems (Reidel, Dordrecht, The Netherlands, 1985).
[CrossRef]

Stark, H.

Stefanescu, I. S.

I. S. Stefanescu, “On the phase retrieval problem in two dimensions,”J. Math. Phys. 26, 2141–2160 (1985).
[CrossRef]

Taylor, L. S.

L. S. Taylor, “The phase retrieval problem,”IEEE Trans Antennas Propag. AP-29, 317–343 (1981).

Webb, H.

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

D. C. Youla, “Generalized image restoration by the method of alternating orthogonal projections,”IEEE Trans. Circuits Syst. CAS-25, 694–702 (1978).
[CrossRef]

Appl. Opt. (1)

Commun. Math. Phys. (1)

J. Rosenblatt, “Phase retrieval,” Commun. Math. Phys. 95, 317–343 (1984).
[CrossRef]

IEEE Trans Antennas Propag. (1)

L. S. Taylor, “The phase retrieval problem,”IEEE Trans Antennas Propag. AP-29, 317–343 (1981).

IEEE Trans. Circuits Syst. (2)

A. Papoulis, “A new algorithm in spectral analysis and band-limited extrapolation,”IEEE Trans. Circuits Syst. CAS-22, 733–742 (1975).

D. C. Youla, “Generalized image restoration by the method of alternating orthogonal projections,”IEEE Trans. Circuits Syst. CAS-25, 694–702 (1978).
[CrossRef]

IEEE Trans. Med. Imaging (1)

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

J. Math. Phys. (2)

I. S. Stefanescu, “On the phase retrieval problem in two dimensions,”J. Math. Phys. 26, 2141–2160 (1985).
[CrossRef]

R. Barakat, G. Newsam, “Necessary conditions for a unique solution to two dimensional phase recovery,”J. Math. Phys. 25, 3190–3193 (1984).
[CrossRef]

J. Opt. Soc. Am. (4)

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

Opt. Acta (1)

R. W. Gerchberg, “Super-resolution through error energy reduction,” Opt. Acta 21, 709–720 (1974).
[CrossRef]

Proc. IEEE (1)

M. H. Hayes, J. H. McClellan, “Reducible polynomials in more than one variable,” Proc. IEEE 70, 197–198 (1982).
[CrossRef]

SIAM J. Appl. Math (1)

C. L. Byrne, R. M. Fitzgerald, “Spectral estimates that extend the maximum entropy and maximum likelihood methods,” SIAM J. Appl. Math 44, 425–442 (1984).
[CrossRef]

SIAM J. Appl. Math. (1)

J. L. C. Sanz, “Mathematical considerations for the problem of Fourier transform phase retrieval from magnitude,” SIAM J. Appl. Math. 45, 651–664 (1985).
[CrossRef]

Other (7)

M. A. Fiddy, “The phase retrieval problem,” in Inverse Optics, A. J. Devaney, ed., Proc. Soc. Photo-Opt. Instrum. Eng.413, 176–181 (1983).
[CrossRef]

T. S. Huang, ed., Advances in Computer Vision and Image Processing (JAI, New York, 1984), Vol. 1.

C. R. Smith, W. T. Grandy, Maximum Entropy and Bayesian Methods in Inverse Problems (Reidel, Dordrecht, The Netherlands, 1985).
[CrossRef]

A. M. Darling, H. V. Deighton, M. A. Fiddy, “Phase ambiguities in more than one dimension,” in Inverse Optics, A. J. Devaney, ed., Proc. Soc. Photo-Opt. Instrum. Eng.413, 197–201 (1983).
[CrossRef]

H. A. Ferwerda, “The phase reconstruction problem for wave amplitudes and coherence functions,” in Inverse Source Problems in Optics, H. P. Baltes, ed. (Springer-Verlag, Berlin, 1978), Vol. 9, p. 13.
[CrossRef]

R. L. Frost, C. K. Rushforth, B. S. Baxter, “High resolution astronomical imaging through the turbulent atmosphere,” (University of Utah, Salt Lake City, 1979). Frost et al. developed an algorithm with some similarities to that presented here. Their procedure did not correct the magnitude spectrum to the measured magnitude each iteration, as in Fienup’s error-reduction algorithm. Instead, the magnitude and phase spectra were alternately extended through the frequency space. The phase-spectrum extension was allowed to converge, and then the new magnitude spectrum was scaled to agree with the measured magnitude below the old cutoff frequency. The reason for doing this was to limit instabilities in the extrapolation due to reintroducing the noise present in the known portion of the spectrum. They studied the consequences of varying the rate of extrapolation on the relative contribution of the noise on the final extrapolated estimate. Results presented by them looked promising, but further development and optimization of the procedure were not pursued.

C. L. Byrne, L. K. Jones, SIAM J. Appl. Math. (to be published).

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Figures (2)

Fig. 1
Fig. 1

The dependence of the norm [Eq. (8)] on phase errors ranging from a, 0 to 50% and b, 0 to 0.01%, for a two-dimensional object function.

Fig. 2
Fig. 2

a, Original 17 × 17 (nonzero) object embedded in 256 × 256 zero array; b, DFT of 32 × 32 complex Fourier samples; c, PDFT calculated from Eqs. (6) and (7); d, an approximation to c after only 100 iterations and that converges to the latter; e, the reconstruction from Fourier magnitude alone after 100 iterations.

Equations (20)

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F ( x ) = - π π f ( t ) exp ( i x t ) d t / 2 π .
f ( t ) = n = - F ( n ) exp ( - i n t ) ,             t π .
DFT ( t ) = n = - N N F ( n ) exp ( - i n t ) ,             t π .
- v v DFT ( t ) 2 d t = F SF .
1 - F SF / F F 1 - λ max > 0 ,
PDFT ( t ) = p ( t ) n = - N N a n exp ( - i n t ) ,             t π ,
F ^ ( m ) = n = - N N a n P ( m - n ) ,             m > N ,
- v v PDFT ( t ) 2 d t = F S - 1 F = a Sa .
E ( θ ) = G ( θ ) S - 1 G ( θ ) .
{ average ( true phase - noisy phase ) / average ( true phase ) } · ( 100 ) % ,
MCE ( t ) = p ( t ) exp [ n = - N N b n exp ( - i n t ) ] ,
H = - v v MCE ( t ) [ n = - N N b n exp ( - i n t ) ] - v v | [ n = - N N b n exp ( - i n t ) ] | 2 ,
g n + 1 = V - 1 ( I - A ) g n + V - 1 DA g n .
P 1 g = V g ,             P 2 g = - 1 ( I - A ) g + - 1 DA g ,
g n + 1 = P 1 P 2 g n .
J ( g ) = P 1 g - g + P 2 g - g ,             the L 2 ( - π , π ) norm
g n + 1 = g n - V - 1 A g n + V - 1 DA g n
g n + 1 = V - 1 D g n ,
a n + 1 = a n - S a n + DS a n ,
C n + 1 = C n - S C n S + DS C n S ,

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