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]

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]

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]

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]

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]

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]

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

[CrossRef]

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

[CrossRef]
[PubMed]

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

[CrossRef]

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

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

[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, 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).

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]

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]

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]

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]

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.

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

[CrossRef]

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

[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]

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]

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

[CrossRef]

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

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

[CrossRef]

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

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

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

[CrossRef]

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

[CrossRef]

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

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, 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]

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

[CrossRef]

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

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]

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]

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]

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

[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]

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]

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

[CrossRef]

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

[CrossRef]

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]

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]

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]

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

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.