Abstract

A recently introduced approach to phase-retrieval problems is applied to present a unified discussion of support information and zero locations in the reconstruction of a discrete complex image from Fourier-transform phaseless data. The choice of the square-modulus function of the Fourier transform of the unknown as the problem datum results in a quadratic operator that has to be inverted, i.e., a simple nonlinearity. This circumstance makes it possible to consider and to point out some relevant factors that affect the local minima problem that arises in the solution procedure (which amounts to minimizing a quartic functional). Simple modifications of the basic procedure help to explain the role of support information and zeros in the data and to develop suitable strategies for avoiding the local minima problem. All results can be summarized by reference to the ratio between the effective dimensions of the data space and the space of unknowns. Numerical results identify the approach’s considerable robustness against false solutions, starting from completely random first guesses, if the above ratio is larger than 3. The algorithm also ensures robust performance in the presence of noise in the data.

© 1999 Optical Society of America

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    [CrossRef]
  2. R. P. Millane, “Image reconstruction from cylindrically averaged diffraction intensities,” in Digital Image Recovery and Synthesis II, P. S. Idell, ed., Proc. SPIE2029, 137–143 (1993).
    [CrossRef]
  3. T. Isernia, G. Leone, R. Pierri, “Phase retrieval of radiated fields,” Inverse Probl. 11, 183–203 (1995).
    [CrossRef]
  4. T. Isernia, G. Leone, R. Pierri, “Radiation pattern evaluation from near field intensities over two planes,” IEEE Trans. Antennas Propag. 44, 701–710 (1996).
    [CrossRef]
  5. G. Leone, R. Pierri, F. Soldovieri, “Reconstruction of complex signals from intensities of Fourier transform pairs,” J. Opt. Soc. Am. A 13, 1546–1556 (1996).
    [CrossRef]
  6. I. Sabba Stefanescu, “On the phase retrieval problem in two dimensions,” J. Math. Phys. 26, 2141–2160 (1985).
    [CrossRef]
  7. Y. M. Bruck, L. G. Sodin, “On the ambiguity of the image reconstruction problem,” Opt. Commun. 30, 304–308 (1979).
    [CrossRef]
  8. T. R. Crimmins, J. R. Fienup, “Uniqueness of phase retrieval for function with sufficient disconnected support,” J. Opt. Soc. Am. A 73, 218–221 (1983).
    [CrossRef]
  9. M. A. Fiddy, B. J. Brames, J. C. Dainty, “Enforcing irreducibility for phase retrieval in two dimensions,” Opt. Lett. 8, 96–98 (1983).
    [CrossRef] [PubMed]
  10. J. R. Fienup, “Reconstruction of objects having latent reference points,” J. Opt. Soc. Am. A 73, 1421–1426 (1983).
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  11. B. J. Brames, “Unique phase retrieval with explicit support information,” Opt. Lett. 11, 61–63 (1986).
    [CrossRef] [PubMed]
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    [CrossRef]
  13. B. Blaschke-Kaltenbaker, H. W. Engl, “Regularization methods for nonlinear ill-posed problems with application to phase reconstruction,” in Inverse Problems in Medical Imaging and Nondestructive Testing, H. W. Engl, ed. (Springer, New York, 1997), pp. 17–35.
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    [CrossRef]
  16. R. W. Gerchberg, W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik 35, 227–246 (1972).
  17. J. R. Fienup, “Phase retrieval algorithms: a comparison,” Appl. Opt. 21, 2758–2769 (1982).
    [CrossRef] [PubMed]
  18. J. R. Fienup, “Phase retrieval algorithms for a complicated optical system,” Appl. Opt. 32, 1737–1746 (1993).
    [CrossRef] [PubMed]
  19. R. Barakat, G. Newsam, “Algorithms for reconstruction of partially known, bandlimited Fourier transform pairs from noisy data. II. The nonlinear problem of phase retrieval,” J. Integr. Eq. 9, 77–125 (1985).
  20. 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]
  21. D. C. Dobson, “Phase reconstruction via nonlinear least squares,” Inverse Probl. 8, 541–557 (1992).
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  22. T. Isernia, V. Pascazio, R. Pierri, G. Schirinzi, “Image reconstruction from Fourier transform magnitude with applications to SAR imaging,” J. Opt. Soc. Am. A 13, 922–934 (1996).
    [CrossRef]
  23. P. T. Chen, M. A. Fiddy, C. W. Liao, D. A. Pommel, “Blind deconvolution and phase retrieval from point zeros,” J. Opt. Soc. Am. A 13, 1524–1531 (1996).
    [CrossRef]
  24. C. C. Wakerman, A. E. Yagle, “Phase retrieval and estimation with use of real plane zeros,” J. Opt. Soc. Am. A 11, 2016–2026 (1994).
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  25. Z. Mou-yan, R. Unbehauen, “Methods for reconstruction of 2-D sequences from Fourier transform magnitude,” IEEE Trans. Image Process. 6, 222–233 (1997).
    [CrossRef]
  26. T. Isernia, G. Leone, R. Pierri, F. Soldovieri, “On the local minima in phase reconstruction algorithms,” Radio Sci. 31, 1887–1899 (1996).
    [CrossRef]
  27. G. Leone, R. Pierri, F. Soldovieri, “On the performances of two algorithms in phaseless antenna measurements,” in Proceedings of the 10th International Conference on Antennas and Propagation, Conf. Publ. 436 IEE (Institute of Electrical Engineers, London, 1997), pp. 1.136–1.141.
  28. R. Pierri, A. Tamburrino, “On the local minima problem in conductivity imaging via a quadratic approach,” Inverse Probl. 13, 1–22 (1998).
  29. A. E. Taylor, D. L. Kay, Introduction to Functional Analysis, 2nd ed. (Krieger, Malabar, Fla., 1980).
  30. R. Pierri, G. Leone, “The phase retrieval as a quadratic inversion in microwave applications,” presented at the 25th European Microwave Conference, Bologna, Italy, 1995.
  31. R. Barakat, G. Newsam, “Necessary conditions for an unique solution to two-dimensional phase recovery,” J. Math. Phys. 25, 3190–3193 (1984).
    [CrossRef]
  32. M. H. Hayes, J. H. McClellan, “Reducible polynomials in two or more variables,” Proc. IEEE 70, 197–198 (1982).
    [CrossRef]
  33. 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]
  34. J. H. Seldin, J. R. Fienup, “Numerical investigation of the uniqueness of phase retrieval,” J. Opt. Soc. Am. A 7, 412–427 (1990).
    [CrossRef]
  35. J. M. Ortega, W. C. Rheinboldt, Iterative Solution of Nonlinear Equations in Several Variables (Academic, San Diego, Calif., 1970).
  36. Note that, if we fix, without any loss of generality, the modulus and the phase of a sample of F, the sequence f cannot assume a null value.
  37. D. Luenberger, Linear and Nonlinear Programming (Addison-Wesley, Reading, Mass., 1987).
  38. W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery, Numerical Recipes, 2nd ed. (Cambridge U. Press, Cambridge, UK, 1992).
  39. A different kind of support would imply modification also of the essential dimension of data space (see Subsection 4.B).
  40. J. R. Fienup, “Gradient-search phase retrieval algorithm for inverse synthetic-aperture radar,” Opt. Eng. 33, 3237–3242 (1994).
    [CrossRef]
  41. T. Isernia, V. Pascazio, R. Pierri, G. Schirinzi, “Synthetic aperture radar imaging from phase-corrupted data,” IEE Proc. Radar Sonar Navigation 143, 268–274 (1996).
    [CrossRef]
  42. B. R. Hunt, T. L. Overman, P. Gough, “Image reconstruction of Fourier-transform magnitude,” Opt. Lett. 23, 1123–1125 (1998).
    [CrossRef]
  43. J. Miao, D. Sayre, H. N. Chapman, “Phase retrieval from the magnitude of the Fourier transforms of nonperiodic objects,” J. Opt. Soc. Am. A 15, 1662–1669 (1998).
    [CrossRef]

1998 (3)

1997 (2)

Z. Mou-yan, R. Unbehauen, “Methods for reconstruction of 2-D sequences from Fourier transform magnitude,” IEEE Trans. Image Process. 6, 222–233 (1997).
[CrossRef]

D. S. Weile, E. Michielsen, “Genetic algorithm optimization applied to electromagnetics,” IEEE Trans. Antennas Propag. 45, 343–353 (1997).
[CrossRef]

1996 (6)

T. Isernia, G. Leone, R. Pierri, “Radiation pattern evaluation from near field intensities over two planes,” IEEE Trans. Antennas Propag. 44, 701–710 (1996).
[CrossRef]

T. Isernia, G. Leone, R. Pierri, F. Soldovieri, “On the local minima in phase reconstruction algorithms,” Radio Sci. 31, 1887–1899 (1996).
[CrossRef]

T. Isernia, V. Pascazio, R. Pierri, G. Schirinzi, “Synthetic aperture radar imaging from phase-corrupted data,” IEE Proc. Radar Sonar Navigation 143, 268–274 (1996).
[CrossRef]

T. Isernia, V. Pascazio, R. Pierri, G. Schirinzi, “Image reconstruction from Fourier transform magnitude with applications to SAR imaging,” J. Opt. Soc. Am. A 13, 922–934 (1996).
[CrossRef]

P. T. Chen, M. A. Fiddy, C. W. Liao, D. A. Pommel, “Blind deconvolution and phase retrieval from point zeros,” J. Opt. Soc. Am. A 13, 1524–1531 (1996).
[CrossRef]

G. Leone, R. Pierri, F. Soldovieri, “Reconstruction of complex signals from intensities of Fourier transform pairs,” J. Opt. Soc. Am. A 13, 1546–1556 (1996).
[CrossRef]

1995 (1)

T. Isernia, G. Leone, R. Pierri, “Phase retrieval of radiated fields,” Inverse Probl. 11, 183–203 (1995).
[CrossRef]

1994 (2)

C. C. Wakerman, A. E. Yagle, “Phase retrieval and estimation with use of real plane zeros,” J. Opt. Soc. Am. A 11, 2016–2026 (1994).
[CrossRef]

J. R. Fienup, “Gradient-search phase retrieval algorithm for inverse synthetic-aperture radar,” Opt. Eng. 33, 3237–3242 (1994).
[CrossRef]

1993 (1)

1992 (1)

D. C. Dobson, “Phase reconstruction via nonlinear least squares,” Inverse Probl. 8, 541–557 (1992).
[CrossRef]

1990 (2)

1988 (1)

1987 (2)

1986 (1)

1985 (2)

R. Barakat, G. Newsam, “Algorithms for reconstruction of partially known, bandlimited Fourier transform pairs from noisy data. II. The nonlinear problem of phase retrieval,” J. Integr. Eq. 9, 77–125 (1985).

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

1984 (1)

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

1983 (4)

T. R. Crimmins, J. R. Fienup, “Uniqueness of phase retrieval for function with sufficient disconnected support,” J. Opt. Soc. Am. A 73, 218–221 (1983).
[CrossRef]

J. R. Fienup, “Reconstruction of objects having latent reference points,” J. Opt. Soc. Am. A 73, 1421–1426 (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]

M. A. Fiddy, B. J. Brames, J. C. Dainty, “Enforcing irreducibility for phase retrieval in two dimensions,” Opt. Lett. 8, 96–98 (1983).
[CrossRef] [PubMed]

1982 (2)

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 two or more variables,” Proc. IEEE 70, 197–198 (1982).
[CrossRef]

1979 (1)

Y. M. Bruck, L. G. Sodin, “On the ambiguity of the image reconstruction problem,” Opt. Commun. 30, 304–308 (1979).
[CrossRef]

1972 (1)

R. W. Gerchberg, W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik 35, 227–246 (1972).

Barakat, R.

R. Barakat, G. Newsam, “Algorithms for reconstruction of partially known, bandlimited Fourier transform pairs from noisy data. II. The nonlinear problem of phase retrieval,” J. Integr. Eq. 9, 77–125 (1985).

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

Blaschke-Kaltenbaker, B.

B. Blaschke-Kaltenbaker, H. W. Engl, “Regularization methods for nonlinear ill-posed problems with application to phase reconstruction,” in Inverse Problems in Medical Imaging and Nondestructive Testing, H. W. Engl, ed. (Springer, New York, 1997), pp. 17–35.

Brames, B. J.

Bruck, Y. M.

Y. M. Bruck, L. G. Sodin, “On the ambiguity of the image reconstruction problem,” Opt. Commun. 30, 304–308 (1979).
[CrossRef]

Chapman, H. N.

Chen, P. T.

Crimmins, T. R.

T. R. Crimmins, “Phase retrieval for discrete functions with support constraints,” J. Opt. Soc. Am. A 4, 124–134 (1987).
[CrossRef]

T. R. Crimmins, J. R. Fienup, “Uniqueness of phase retrieval for function with sufficient disconnected support,” J. Opt. Soc. Am. A 73, 218–221 (1983).
[CrossRef]

Dainty, J. C.

Dobson, D. C.

D. C. Dobson, “Phase reconstruction via nonlinear least squares,” Inverse Probl. 8, 541–557 (1992).
[CrossRef]

Engl, H. W.

B. Blaschke-Kaltenbaker, H. W. Engl, “Regularization methods for nonlinear ill-posed problems with application to phase reconstruction,” in Inverse Problems in Medical Imaging and Nondestructive Testing, H. W. Engl, ed. (Springer, New York, 1997), pp. 17–35.

Fiddy, M. A.

Fienup, J. R.

J. R. Fienup, “Gradient-search phase retrieval algorithm for inverse synthetic-aperture radar,” Opt. Eng. 33, 3237–3242 (1994).
[CrossRef]

J. R. Fienup, “Phase retrieval algorithms for a complicated optical system,” Appl. Opt. 32, 1737–1746 (1993).
[CrossRef] [PubMed]

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. 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]

J. R. Fienup, “Reconstruction of objects having latent reference points,” J. Opt. Soc. Am. A 73, 1421–1426 (1983).
[CrossRef]

T. R. Crimmins, J. R. Fienup, “Uniqueness of phase retrieval for function with sufficient disconnected support,” J. Opt. Soc. Am. A 73, 218–221 (1983).
[CrossRef]

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

Flannery, B. P.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery, Numerical Recipes, 2nd ed. (Cambridge U. Press, Cambridge, UK, 1992).

Fuentes, F. J.

Gerchberg, R. W.

R. W. Gerchberg, W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik 35, 227–246 (1972).

Gough, P.

Hayes, M. H.

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

Huang, T. S.

Hunt, B. R.

Isernia, T.

T. Isernia, G. Leone, R. Pierri, “Radiation pattern evaluation from near field intensities over two planes,” IEEE Trans. Antennas Propag. 44, 701–710 (1996).
[CrossRef]

T. Isernia, V. Pascazio, R. Pierri, G. Schirinzi, “Synthetic aperture radar imaging from phase-corrupted data,” IEE Proc. Radar Sonar Navigation 143, 268–274 (1996).
[CrossRef]

T. Isernia, G. Leone, R. Pierri, F. Soldovieri, “On the local minima in phase reconstruction algorithms,” Radio Sci. 31, 1887–1899 (1996).
[CrossRef]

T. Isernia, V. Pascazio, R. Pierri, G. Schirinzi, “Image reconstruction from Fourier transform magnitude with applications to SAR imaging,” J. Opt. Soc. Am. A 13, 922–934 (1996).
[CrossRef]

T. Isernia, G. Leone, R. Pierri, “Phase retrieval of radiated fields,” Inverse Probl. 11, 183–203 (1995).
[CrossRef]

Kay, D. L.

A. E. Taylor, D. L. Kay, Introduction to Functional Analysis, 2nd ed. (Krieger, Malabar, Fla., 1980).

Leone, G.

T. Isernia, G. Leone, R. Pierri, F. Soldovieri, “On the local minima in phase reconstruction algorithms,” Radio Sci. 31, 1887–1899 (1996).
[CrossRef]

G. Leone, R. Pierri, F. Soldovieri, “Reconstruction of complex signals from intensities of Fourier transform pairs,” J. Opt. Soc. Am. A 13, 1546–1556 (1996).
[CrossRef]

T. Isernia, G. Leone, R. Pierri, “Radiation pattern evaluation from near field intensities over two planes,” IEEE Trans. Antennas Propag. 44, 701–710 (1996).
[CrossRef]

T. Isernia, G. Leone, R. Pierri, “Phase retrieval of radiated fields,” Inverse Probl. 11, 183–203 (1995).
[CrossRef]

G. Leone, R. Pierri, F. Soldovieri, “On the performances of two algorithms in phaseless antenna measurements,” in Proceedings of the 10th International Conference on Antennas and Propagation, Conf. Publ. 436 IEE (Institute of Electrical Engineers, London, 1997), pp. 1.136–1.141.

R. Pierri, G. Leone, “The phase retrieval as a quadratic inversion in microwave applications,” presented at the 25th European Microwave Conference, Bologna, Italy, 1995.

Liao, C. W.

Luenberger, D.

D. Luenberger, Linear and Nonlinear Programming (Addison-Wesley, Reading, Mass., 1987).

McClellan, J. H.

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

Miao, J.

Michielsen, E.

D. S. Weile, E. Michielsen, “Genetic algorithm optimization applied to electromagnetics,” IEEE Trans. Antennas Propag. 45, 343–353 (1997).
[CrossRef]

Millane, R. P.

R. P. Millane, “Phase retrieval in crystallography and optics,” J. Opt. Soc. Am. A 7, 394–411 (1990).
[CrossRef]

R. P. Millane, “Image reconstruction from cylindrically averaged diffraction intensities,” in Digital Image Recovery and Synthesis II, P. S. Idell, ed., Proc. SPIE2029, 137–143 (1993).
[CrossRef]

Mou-yan, Z.

Z. Mou-yan, R. Unbehauen, “Methods for reconstruction of 2-D sequences from Fourier transform magnitude,” IEEE Trans. Image Process. 6, 222–233 (1997).
[CrossRef]

Navarro, R.

Newsam, G.

R. Barakat, G. Newsam, “Algorithms for reconstruction of partially known, bandlimited Fourier transform pairs from noisy data. II. The nonlinear problem of phase retrieval,” J. Integr. Eq. 9, 77–125 (1985).

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

Nieto-Vesperinas, M.

Ortega, J. M.

J. M. Ortega, W. C. Rheinboldt, Iterative Solution of Nonlinear Equations in Several Variables (Academic, San Diego, Calif., 1970).

Overman, T. L.

Pascazio, V.

T. Isernia, V. Pascazio, R. Pierri, G. Schirinzi, “Image reconstruction from Fourier transform magnitude with applications to SAR imaging,” J. Opt. Soc. Am. A 13, 922–934 (1996).
[CrossRef]

T. Isernia, V. Pascazio, R. Pierri, G. Schirinzi, “Synthetic aperture radar imaging from phase-corrupted data,” IEE Proc. Radar Sonar Navigation 143, 268–274 (1996).
[CrossRef]

Pierri, R.

R. Pierri, A. Tamburrino, “On the local minima problem in conductivity imaging via a quadratic approach,” Inverse Probl. 13, 1–22 (1998).

T. Isernia, V. Pascazio, R. Pierri, G. Schirinzi, “Synthetic aperture radar imaging from phase-corrupted data,” IEE Proc. Radar Sonar Navigation 143, 268–274 (1996).
[CrossRef]

T. Isernia, G. Leone, R. Pierri, F. Soldovieri, “On the local minima in phase reconstruction algorithms,” Radio Sci. 31, 1887–1899 (1996).
[CrossRef]

T. Isernia, V. Pascazio, R. Pierri, G. Schirinzi, “Image reconstruction from Fourier transform magnitude with applications to SAR imaging,” J. Opt. Soc. Am. A 13, 922–934 (1996).
[CrossRef]

G. Leone, R. Pierri, F. Soldovieri, “Reconstruction of complex signals from intensities of Fourier transform pairs,” J. Opt. Soc. Am. A 13, 1546–1556 (1996).
[CrossRef]

T. Isernia, G. Leone, R. Pierri, “Radiation pattern evaluation from near field intensities over two planes,” IEEE Trans. Antennas Propag. 44, 701–710 (1996).
[CrossRef]

T. Isernia, G. Leone, R. Pierri, “Phase retrieval of radiated fields,” Inverse Probl. 11, 183–203 (1995).
[CrossRef]

G. Leone, R. Pierri, F. Soldovieri, “On the performances of two algorithms in phaseless antenna measurements,” in Proceedings of the 10th International Conference on Antennas and Propagation, Conf. Publ. 436 IEE (Institute of Electrical Engineers, London, 1997), pp. 1.136–1.141.

R. Pierri, G. Leone, “The phase retrieval as a quadratic inversion in microwave applications,” presented at the 25th European Microwave Conference, Bologna, Italy, 1995.

Pommel, D. A.

Press, W. H.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery, Numerical Recipes, 2nd ed. (Cambridge U. Press, Cambridge, UK, 1992).

Rheinboldt, W. C.

J. M. Ortega, W. C. Rheinboldt, Iterative Solution of Nonlinear Equations in Several Variables (Academic, San Diego, Calif., 1970).

Sabba Stefanescu, I.

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

Sanz, J. L. C.

Saxton, W. O.

R. W. Gerchberg, W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik 35, 227–246 (1972).

Sayre, D.

Schirinzi, G.

T. Isernia, V. Pascazio, R. Pierri, G. Schirinzi, “Image reconstruction from Fourier transform magnitude with applications to SAR imaging,” J. Opt. Soc. Am. A 13, 922–934 (1996).
[CrossRef]

T. Isernia, V. Pascazio, R. Pierri, G. Schirinzi, “Synthetic aperture radar imaging from phase-corrupted data,” IEE Proc. Radar Sonar Navigation 143, 268–274 (1996).
[CrossRef]

Seldin, J. H.

Sodin, L. G.

Y. M. Bruck, L. G. Sodin, “On the ambiguity of the image reconstruction problem,” Opt. Commun. 30, 304–308 (1979).
[CrossRef]

Soldovieri, F.

T. Isernia, G. Leone, R. Pierri, F. Soldovieri, “On the local minima in phase reconstruction algorithms,” Radio Sci. 31, 1887–1899 (1996).
[CrossRef]

G. Leone, R. Pierri, F. Soldovieri, “Reconstruction of complex signals from intensities of Fourier transform pairs,” J. Opt. Soc. Am. A 13, 1546–1556 (1996).
[CrossRef]

G. Leone, R. Pierri, F. Soldovieri, “On the performances of two algorithms in phaseless antenna measurements,” in Proceedings of the 10th International Conference on Antennas and Propagation, Conf. Publ. 436 IEE (Institute of Electrical Engineers, London, 1997), pp. 1.136–1.141.

Tamburrino, A.

R. Pierri, A. Tamburrino, “On the local minima problem in conductivity imaging via a quadratic approach,” Inverse Probl. 13, 1–22 (1998).

Taylor, A. E.

A. E. Taylor, D. L. Kay, Introduction to Functional Analysis, 2nd ed. (Krieger, Malabar, Fla., 1980).

Teukolsky, S. A.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery, Numerical Recipes, 2nd ed. (Cambridge U. Press, Cambridge, UK, 1992).

Unbehauen, R.

Z. Mou-yan, R. Unbehauen, “Methods for reconstruction of 2-D sequences from Fourier transform magnitude,” IEEE Trans. Image Process. 6, 222–233 (1997).
[CrossRef]

Vetterling, W. T.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery, Numerical Recipes, 2nd ed. (Cambridge U. Press, Cambridge, UK, 1992).

Wakerman, C. C.

Weile, D. S.

D. S. Weile, E. Michielsen, “Genetic algorithm optimization applied to electromagnetics,” IEEE Trans. Antennas Propag. 45, 343–353 (1997).
[CrossRef]

Yagle, A. E.

Appl. Opt. (2)

IEE Proc. Radar Sonar Navigation (1)

T. Isernia, V. Pascazio, R. Pierri, G. Schirinzi, “Synthetic aperture radar imaging from phase-corrupted data,” IEE Proc. Radar Sonar Navigation 143, 268–274 (1996).
[CrossRef]

IEEE Trans. Antennas Propag. (2)

T. Isernia, G. Leone, R. Pierri, “Radiation pattern evaluation from near field intensities over two planes,” IEEE Trans. Antennas Propag. 44, 701–710 (1996).
[CrossRef]

D. S. Weile, E. Michielsen, “Genetic algorithm optimization applied to electromagnetics,” IEEE Trans. Antennas Propag. 45, 343–353 (1997).
[CrossRef]

IEEE Trans. Image Process. (1)

Z. Mou-yan, R. Unbehauen, “Methods for reconstruction of 2-D sequences from Fourier transform magnitude,” IEEE Trans. Image Process. 6, 222–233 (1997).
[CrossRef]

Inverse Probl. (3)

R. Pierri, A. Tamburrino, “On the local minima problem in conductivity imaging via a quadratic approach,” Inverse Probl. 13, 1–22 (1998).

D. C. Dobson, “Phase reconstruction via nonlinear least squares,” Inverse Probl. 8, 541–557 (1992).
[CrossRef]

T. Isernia, G. Leone, R. Pierri, “Phase retrieval of radiated fields,” Inverse Probl. 11, 183–203 (1995).
[CrossRef]

J. Integr. Eq. (1)

R. Barakat, G. Newsam, “Algorithms for reconstruction of partially known, bandlimited Fourier transform pairs from noisy data. II. The nonlinear problem of phase retrieval,” J. Integr. Eq. 9, 77–125 (1985).

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Note that, if we fix, without any loss of generality, the modulus and the phase of a sample of F, the sequence f cannot assume a null value.

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Figures (5)

Fig. 1
Fig. 1

(a) Modulus and (b) phase distribution of a test object that has a framelike support. The modulus distribution is given in decibels.

Fig. 2
Fig. 2

(a) Modulus and (b) phase reconstruction of the reference object of Fig. 1.

Fig. 3
Fig. 3

Behavior of the eigenvalues of the Hessian of Φ in fˆ: Because all the eigenvalues are positive, a local minimum has been reached.

Fig. 4
Fig. 4

Comparison of the behavior of the eigenvalues of the Hessians of Φ (solid curve) and Ψ (dotted curve) in the solution f0.

Fig. 5
Fig. 5

Reconstruction of the modulus distribution of an object that has an L-shaped support (Q=9) by (a) not using zeros information and (b) using zeros information.

Tables (2)

Tables Icon

Table 1 Reliability Properties of the Solution Algorithm for the Framelike Support Objects

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Table 2 Reliability Properties of the Solution Algorithm for the L-Shaped Support Objects

Equations (50)

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F(u, v)=n=-NNm=-NNfnm exp[-jΔ(nu+mv)]=n=-NNm=-NN(ξnm+jγnm)exp[-jΔ(nu+mv)],
B(f)=M2(u, v),
F(z, w)=n=-NNm=-NNfnmznwm.
F(z, w)=z-nzw-nwk=1PAk(z, w),
f(x, y)=g(x, y)*h(x, y),
f1(x, y)=h(x, y)*g(-x, -y)*,
f2(x, y)=h(-x, -y)**g(-x, -y)*,
f3(x, y)=h(-x, -y)**g(x, y).
M2(u, v)=ν=-2N2Nμ=-2N2Ncνμ exp[-jΔ(νu+μv)].
DN(t)=sin[(N+1/2)tΔ](2N+1)sin(tΔ/2)
M2(u, v)=k=-2N2Nl=-2N2NM2(uk, vl)D2N(u-uk)×D2N(v-vl),
F(uk, vl)=Fkl=n=-NNm=-NNfnm×exp[-j2π(kn+lm)/(4N+1)],
Φ(f)=B(f)-M2(u, v)2=k=-2N2Nl=-2N2N(|Fkl|2-Mkl2)2,
Φ(f+λn)=aλ4+bλ3+cλ2+dλ+e
Φ(λ)=Φ(f0+λn)=λ2(a1λ2+b1λ+c1),
a1=k=-2N2N l=-2N2N|Nkl|4,
b1=4k=-2N2N l=-2N2N|Nkl|2 Re(F0klNkl*),
c1=4k=-2N2N l=-2N2N|Re(F0klNkl*)|2,
b12a1c1<329.
Φˆ=Φ+Φs=Φ+α2n=-NNm=-NNanm(|ξnm|2+|γnm|2),
cˆ=c1+cs=c1+α2n=-NNm=-NNanm|nnm|2,
b12a1(c1+cs)<329.
fnm=0,(n, m)S.
Φ˜=Φ+Φz=Φ+β2(ui, vj)Z|F(ui, vj)|2,
F(ui, vj)=0,(ui, vj)Z.
Φ˜ˆ=Φ+Φz+Φs.
Ψ(f)=k=-2N2Nl=-2N2N (|Fkl|2-Mkl2)2Mkl2.
Ψ(f0+λn)=λ2(a˜λ2+b˜λ+c˜),
a˜=k=-2N2Nl=-2N2N |Nkl|4Mkl2,
b˜=4k=-2N2Nl=-2N2N |Nkl|2 Re(F0klNkl*)Mkl2,
c˜=4k=-2N2Nl=-2N2N |Re(F0klNkl*)|2Mkl2
Φnm=Φξnm+j Φγnm
Φξnm+j Φγnm=4kl(|Fkl|2-Mkl2)Fkl×exp[j2π(kn+lm)/M].
1=10 logkl(|Fkl|2-Mkl2)2klMkl4,
δ=10 logn=-NNm=-NN|fnm-f0nm|2/
m=-NNm=-NN|f0nm|2,
r=(4N+1)22×[(2N+1)2-(100-Q2)].
r=(4N+1)2-2Q22[(2N+1)2-Q2],
Nkl=n=-NNm=-NNnnm exp[-j2π(kn+lm)/(4N+1)],
a=kl|Nkl|4,
b=4kl|Nkl|2 Re(FklNkl*),
c=kl4|Re(FklNkl*)|2+2|Nkl|2(|Fkl|2-Mkl2),
d=4kl(|Fkl|2-Mkl2)Re(FklNkl*),
e=kl(|Fkl|2-Mkl2)2=Φ(f).
αnmst=kl|Fkl|2 exp[j2π[k(n-s)+l(m-t)]/M],
βnmst=klFkl2 exp{j2π[k(n+s)+l(m+t)]/M},
ωnmst=kl(|Fkl|2-Mkl2)exp{j2π[k(n-s)+l(m-t)]/M},
2Φξnmξst=4 Re(αnmst+βnmst)+4 Re(ωnmst),
2Φγnmγst=-4 Re(αnmst+βnmst)+4 Re(ωnmst),
2Φξnmγst=4 Im(αnmst+βnmst)-4 Im(ωnmst).

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