Abstract

It is suggested that, given the magnitude of Fourier transforms sampled at the Bragg density, the phase problem is underdetermined by a factor of 2 for 1D, 2D, and 3D objects. It is therefore unnecessary to oversample the magnitude of Fourier transforms by 2× in each dimension (i.e., oversampling by 4× for 2D and 8× for 3D) in retrieving the phase of 2D and 3D objects. Our computer phasing experiments accurately retrieved the phase from the magnitude of the Fourier transforms of 2D and 3D complex-valued objects by using positivity constraints on the imaginary part of the objects and loose supports, with the oversampling factor much less than 4 for 2D and 8 for 3D objects. Under the same conditions we also obtained reasonably good reconstructions of 2D and 3D complex-valued objects from the magnitude of their Fourier transforms with added noise and a central stop.

© 1998 Optical Society of America

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References

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  1. R. W. Gerchberg, W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik (Stuttgart) 35, 237–246 (1972).
  2. J. R. Fienup, “Reconstruction of an object from the modulus of its Fourier transform,” Opt. Lett. 3, 27–29 (1978).
    [CrossRef] [PubMed]
  3. J. R. Fienup, “Phase retrieval algorithm: a comparison,” Appl. Opt. 21, 2758–2769 (1982).
    [CrossRef] [PubMed]
  4. R. H. T. Bates, W. R. Fright, “Composite two-dimensional phase reconstruction procedure,” J. Opt. Soc. Am. 73, 358–365 (1983).
    [CrossRef]
  5. J. R. Fienup, C. C. Wackerman, “Phase-retrieval stagnation problems and solutions,” J. Opt. Soc. Am. A 3, 1897–1907 (1986).
    [CrossRef]
  6. R. H. T. Bates, M. J. McDonnell, Image restoration and reconstruction (Oxford U. Press, Oxford, UK, 1986).
  7. D. Sayre, H. N. Chapman, J. Miao, “On the possible extension of x-ray crystallography to non-crystals,” Acta Crystallogr., Sect. A: Cryst. Phys., Diffr., Theor. Gen. Crystallogr. (to be published).
  8. J. Miao, H. N. Chapman, D. Sayre, “Image reconstruction from the oversampled diffraction pattern,” Microscopy Microanalysis 3 (Suppl. 2), 1155–1156 (1997).
  9. S. Lindaas, M. Howells, C. Jacobsen, A. Kalinovsky, “X-ray holographic microscopy by means of photoresist recording and atomic-force microscope readout,” J. Opt. Soc. Am. A 13, 1788–1800 (1996).
    [CrossRef]
  10. Yu. M. Bruck, L. G. Sodin, “On the ambiguity of the image reconstruction problem,” Opt. Commun. 30, 304–308 (1979).
    [CrossRef]
  11. M. H. Hayes, “The reconstruction of a multidimensional sequence from the phase or magnitude of its Fourier transform,” IEEE Trans. Acoust., Speech, Signal Process. ASSP-30, 140–154 (1982).
    [CrossRef]
  12. J. Boyes-Watson, K. Davidson, M. F. Perutz, “An x-ray study of horse methaemoglobin. I,” Proc. R. Soc. London, Ser. A 191, 83–137 (1947).
    [CrossRef]
  13. D. Sayre, “Some implications of a theorem due to Shannon,” Acta Crystallogr. 5, 843 (1952).
    [CrossRef]
  14. R. H. T. Bates, “Fourier phase problems are uniquely soluble in more than one dimension. I: underlying theory,” Optik (Stuttgart) 61, 247–262 (1982).
  15. R. H. T. Bates, “Uniqueness of solution to two-dimensional Fourier phase problems of localized and positive images,” Comput. Vis. Graph. Image Process. 25, 205–217 (1984).
    [CrossRef]
  16. R. P. Millane, W. J. Stroud, “Reconstructing symmetric images from their undersampled Fourier intensities,” J. Opt. Soc. Am. A 14, 568–579 (1997).
    [CrossRef]
  17. A. Szöke, University of California, Lawrence Livermore National Laboratory, P.O. Box 808, L-41, Livermore, Calif. 94551 (personal communication, December1995). Szöke said that he thought 2× oversampling should in theory suffice for any dimensionality ⩾2.
  18. R. Barakat, G. Newsam, “Necessary conditions for a unique solution to two-dimensional phase recovery,” J. Math. Phys. 25, 3190–3193 (1984).
    [CrossRef]
  19. R. H. Bates, D. G. H. Tan, “Fourier phase retrieval when the image is complex,” in Inverse Optics II, A. J. Devaney, R. H. Bates, eds., Proc. SPIE558, 54–59 (1985).
    [CrossRef]
  20. R. G. Lane, “Recovery of complex images from Fourier magnitude,” Opt. Commun. 63, 6–10 (1987).
    [CrossRef]
  21. 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]
  22. D. Sayre, H. N. Chapman, “X-ray microscopy,” Acta Crystallogr. Sect. A 51, 237–252 (1995).
    [CrossRef]
  23. G. H. Stout, L. H. Jensen, X-Ray Structure Determination (Wiley, New York, 1989).
  24. B. L. Henke, E. M. Gullikson, J. C. Davis, “X-ray interactions: photoabsorption, scattering, transmission, and reflection at E=50,-30,000 eV,Z=1-92,” At. Data Nucl. Data Tables 54, 181–342 (1993).
    [CrossRef]
  25. J. Kirz, C. Jacobsen, M. Howells, “Soft x-ray microscopes and their biological applications,” Q. Rev. Biophys. 28, 33–130 (1995).
    [CrossRef] [PubMed]
  26. The exception is where the material amplifies the incident x-ray beam, as with the x-ray laser amplifier.

1997

J. Miao, H. N. Chapman, D. Sayre, “Image reconstruction from the oversampled diffraction pattern,” Microscopy Microanalysis 3 (Suppl. 2), 1155–1156 (1997).

R. P. Millane, W. J. Stroud, “Reconstructing symmetric images from their undersampled Fourier intensities,” J. Opt. Soc. Am. A 14, 568–579 (1997).
[CrossRef]

1996

1995

J. Kirz, C. Jacobsen, M. Howells, “Soft x-ray microscopes and their biological applications,” Q. Rev. Biophys. 28, 33–130 (1995).
[CrossRef] [PubMed]

D. Sayre, H. N. Chapman, “X-ray microscopy,” Acta Crystallogr. Sect. A 51, 237–252 (1995).
[CrossRef]

1993

B. L. Henke, E. M. Gullikson, J. C. Davis, “X-ray interactions: photoabsorption, scattering, transmission, and reflection at E=50,-30,000 eV,Z=1-92,” At. Data Nucl. Data Tables 54, 181–342 (1993).
[CrossRef]

1987

1986

1984

R. H. T. Bates, “Uniqueness of solution to two-dimensional Fourier phase problems of localized and positive images,” Comput. Vis. Graph. Image Process. 25, 205–217 (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

1982

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

R. H. T. Bates, “Fourier phase problems are uniquely soluble in more than one dimension. I: underlying theory,” Optik (Stuttgart) 61, 247–262 (1982).

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

1979

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

1978

1972

R. W. Gerchberg, W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik (Stuttgart) 35, 237–246 (1972).

1952

D. Sayre, “Some implications of a theorem due to Shannon,” Acta Crystallogr. 5, 843 (1952).
[CrossRef]

1947

J. Boyes-Watson, K. Davidson, M. F. Perutz, “An x-ray study of horse methaemoglobin. I,” Proc. R. Soc. London, Ser. A 191, 83–137 (1947).
[CrossRef]

Barakat, R.

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

Bates, R. H.

R. H. Bates, D. G. H. Tan, “Fourier phase retrieval when the image is complex,” in Inverse Optics II, A. J. Devaney, R. H. Bates, eds., Proc. SPIE558, 54–59 (1985).
[CrossRef]

Bates, R. H. T.

R. H. T. Bates, “Uniqueness of solution to two-dimensional Fourier phase problems of localized and positive images,” Comput. Vis. Graph. Image Process. 25, 205–217 (1984).
[CrossRef]

R. H. T. Bates, W. R. Fright, “Composite two-dimensional phase reconstruction procedure,” J. Opt. Soc. Am. 73, 358–365 (1983).
[CrossRef]

R. H. T. Bates, “Fourier phase problems are uniquely soluble in more than one dimension. I: underlying theory,” Optik (Stuttgart) 61, 247–262 (1982).

R. H. T. Bates, M. J. McDonnell, Image restoration and reconstruction (Oxford U. Press, Oxford, UK, 1986).

Boyes-Watson, J.

J. Boyes-Watson, K. Davidson, M. F. Perutz, “An x-ray study of horse methaemoglobin. I,” Proc. R. Soc. London, Ser. A 191, 83–137 (1947).
[CrossRef]

Bruck, Yu. M.

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

Chapman, H. N.

J. Miao, H. N. Chapman, D. Sayre, “Image reconstruction from the oversampled diffraction pattern,” Microscopy Microanalysis 3 (Suppl. 2), 1155–1156 (1997).

D. Sayre, H. N. Chapman, “X-ray microscopy,” Acta Crystallogr. Sect. A 51, 237–252 (1995).
[CrossRef]

D. Sayre, H. N. Chapman, J. Miao, “On the possible extension of x-ray crystallography to non-crystals,” Acta Crystallogr., Sect. A: Cryst. Phys., Diffr., Theor. Gen. Crystallogr. (to be published).

Davidson, K.

J. Boyes-Watson, K. Davidson, M. F. Perutz, “An x-ray study of horse methaemoglobin. I,” Proc. R. Soc. London, Ser. A 191, 83–137 (1947).
[CrossRef]

Davis, J. C.

B. L. Henke, E. M. Gullikson, J. C. Davis, “X-ray interactions: photoabsorption, scattering, transmission, and reflection at E=50,-30,000 eV,Z=1-92,” At. Data Nucl. Data Tables 54, 181–342 (1993).
[CrossRef]

Fienup, J. R.

Fright, W. R.

Gerchberg, R. W.

R. W. Gerchberg, W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik (Stuttgart) 35, 237–246 (1972).

Gullikson, E. M.

B. L. Henke, E. M. Gullikson, J. C. Davis, “X-ray interactions: photoabsorption, scattering, transmission, and reflection at E=50,-30,000 eV,Z=1-92,” At. Data Nucl. Data Tables 54, 181–342 (1993).
[CrossRef]

Hayes, M. H.

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

Henke, B. L.

B. L. Henke, E. M. Gullikson, J. C. Davis, “X-ray interactions: photoabsorption, scattering, transmission, and reflection at E=50,-30,000 eV,Z=1-92,” At. Data Nucl. Data Tables 54, 181–342 (1993).
[CrossRef]

Howells, M.

Jacobsen, C.

Jensen, L. H.

G. H. Stout, L. H. Jensen, X-Ray Structure Determination (Wiley, New York, 1989).

Kalinovsky, A.

Kirz, J.

J. Kirz, C. Jacobsen, M. Howells, “Soft x-ray microscopes and their biological applications,” Q. Rev. Biophys. 28, 33–130 (1995).
[CrossRef] [PubMed]

Lane, R. G.

R. G. Lane, “Recovery of complex images from Fourier magnitude,” Opt. Commun. 63, 6–10 (1987).
[CrossRef]

Lindaas, S.

McDonnell, M. J.

R. H. T. Bates, M. J. McDonnell, Image restoration and reconstruction (Oxford U. Press, Oxford, UK, 1986).

Miao, J.

J. Miao, H. N. Chapman, D. Sayre, “Image reconstruction from the oversampled diffraction pattern,” Microscopy Microanalysis 3 (Suppl. 2), 1155–1156 (1997).

D. Sayre, H. N. Chapman, J. Miao, “On the possible extension of x-ray crystallography to non-crystals,” Acta Crystallogr., Sect. A: Cryst. Phys., Diffr., Theor. Gen. Crystallogr. (to be published).

Millane, R. P.

Newsam, G.

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

Perutz, M. F.

J. Boyes-Watson, K. Davidson, M. F. Perutz, “An x-ray study of horse methaemoglobin. I,” Proc. R. Soc. London, Ser. A 191, 83–137 (1947).
[CrossRef]

Saxton, W. O.

R. W. Gerchberg, W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik (Stuttgart) 35, 237–246 (1972).

Sayre, D.

J. Miao, H. N. Chapman, D. Sayre, “Image reconstruction from the oversampled diffraction pattern,” Microscopy Microanalysis 3 (Suppl. 2), 1155–1156 (1997).

D. Sayre, H. N. Chapman, “X-ray microscopy,” Acta Crystallogr. Sect. A 51, 237–252 (1995).
[CrossRef]

D. Sayre, “Some implications of a theorem due to Shannon,” Acta Crystallogr. 5, 843 (1952).
[CrossRef]

D. Sayre, H. N. Chapman, J. Miao, “On the possible extension of x-ray crystallography to non-crystals,” Acta Crystallogr., Sect. A: Cryst. Phys., Diffr., Theor. Gen. Crystallogr. (to be published).

Sodin, L. G.

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

Stout, G. H.

G. H. Stout, L. H. Jensen, X-Ray Structure Determination (Wiley, New York, 1989).

Stroud, W. J.

Szöke, A.

A. Szöke, University of California, Lawrence Livermore National Laboratory, P.O. Box 808, L-41, Livermore, Calif. 94551 (personal communication, December1995). Szöke said that he thought 2× oversampling should in theory suffice for any dimensionality ⩾2.

Tan, D. G. H.

R. H. Bates, D. G. H. Tan, “Fourier phase retrieval when the image is complex,” in Inverse Optics II, A. J. Devaney, R. H. Bates, eds., Proc. SPIE558, 54–59 (1985).
[CrossRef]

Wackerman, C. C.

Acta Crystallogr.

D. Sayre, “Some implications of a theorem due to Shannon,” Acta Crystallogr. 5, 843 (1952).
[CrossRef]

Acta Crystallogr. Sect. A

D. Sayre, H. N. Chapman, “X-ray microscopy,” Acta Crystallogr. Sect. A 51, 237–252 (1995).
[CrossRef]

Appl. Opt.

At. Data Nucl. Data Tables

B. L. Henke, E. M. Gullikson, J. C. Davis, “X-ray interactions: photoabsorption, scattering, transmission, and reflection at E=50,-30,000 eV,Z=1-92,” At. Data Nucl. Data Tables 54, 181–342 (1993).
[CrossRef]

Comput. Vis. Graph. Image Process.

R. H. T. Bates, “Uniqueness of solution to two-dimensional Fourier phase problems of localized and positive images,” Comput. Vis. Graph. Image Process. 25, 205–217 (1984).
[CrossRef]

IEEE Trans. Acoust., Speech, Signal Process.

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

J. Math. Phys.

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.

J. Opt. Soc. Am. A

Microscopy Microanalysis

J. Miao, H. N. Chapman, D. Sayre, “Image reconstruction from the oversampled diffraction pattern,” Microscopy Microanalysis 3 (Suppl. 2), 1155–1156 (1997).

Opt. Commun.

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

R. G. Lane, “Recovery of complex images from Fourier magnitude,” Opt. Commun. 63, 6–10 (1987).
[CrossRef]

Opt. Lett.

Optik (Stuttgart)

R. W. Gerchberg, W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik (Stuttgart) 35, 237–246 (1972).

R. H. T. Bates, “Fourier phase problems are uniquely soluble in more than one dimension. I: underlying theory,” Optik (Stuttgart) 61, 247–262 (1982).

Proc. R. Soc. London, Ser. A

J. Boyes-Watson, K. Davidson, M. F. Perutz, “An x-ray study of horse methaemoglobin. I,” Proc. R. Soc. London, Ser. A 191, 83–137 (1947).
[CrossRef]

Q. Rev. Biophys.

J. Kirz, C. Jacobsen, M. Howells, “Soft x-ray microscopes and their biological applications,” Q. Rev. Biophys. 28, 33–130 (1995).
[CrossRef] [PubMed]

Other

The exception is where the material amplifies the incident x-ray beam, as with the x-ray laser amplifier.

G. H. Stout, L. H. Jensen, X-Ray Structure Determination (Wiley, New York, 1989).

R. H. Bates, D. G. H. Tan, “Fourier phase retrieval when the image is complex,” in Inverse Optics II, A. J. Devaney, R. H. Bates, eds., Proc. SPIE558, 54–59 (1985).
[CrossRef]

A. Szöke, University of California, Lawrence Livermore National Laboratory, P.O. Box 808, L-41, Livermore, Calif. 94551 (personal communication, December1995). Szöke said that he thought 2× oversampling should in theory suffice for any dimensionality ⩾2.

R. H. T. Bates, M. J. McDonnell, Image restoration and reconstruction (Oxford U. Press, Oxford, UK, 1986).

D. Sayre, H. N. Chapman, J. Miao, “On the possible extension of x-ray crystallography to non-crystals,” Acta Crystallogr., Sect. A: Cryst. Phys., Diffr., Theor. Gen. Crystallogr. (to be published).

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Figures (5)

Fig. 1
Fig. 1

Examples of image reconstruction from the magnitude of the Fourier transforms of complex-valued objects by use of the positivity constraints on the imaginary part and zero-valued pixel constraints. (a) Magnitude of an original 2D complex-valued object with σ=4. (b), (c), (d), and (e) Magnitude of the reconstructed objects with σ=4, 3, 2.6, and 2.5, respectively. Note that σ is the ratio of the total array area to the area other than the square hole and that for (b), (c), (d), and (e) the square hole in the object grows smaller.

Fig. 2
Fig. 2

Reconstruction error versus iteration number for the reconstruction of Fig. 1. Curves 1B, 1C, 1D, and 1E correspond to the reconstruction of Figs. 1(b), 1(c), 1(d), and 1(e), respectively.

Fig. 3
Fig. 3

Examples of image reconstruction from the magnitude of the Fourier transforms of complex-valued objects by oversampling; positivity constraints on the imaginary part were used together with loose support constraints. (a) Magnitude of an original 2D complex-valued object. (b), (c), (d), and (e) Magnitude of the reconstructed objects with σ=4, 3, 2.6, and 2.5, respectively. The “corner” symbols in (b)–(e) outline the region of added zero pixels. (f) Isodensity of the magnitude of an original 3D complex-valued object. (g), (h), (i), and (j) Isodensity of the magnitude of the reconstructed objects with σ=7.8, 4, 2.57, and 2.3, respectively.

Fig. 4
Fig. 4

Reconstruction error versus iteration number for the reconstruction of Fig. 3. Curves 3B, 3C, 3D, 3E, 3G, 3H, 3I, and 3J correspond to the reconstruction of Figs. 3(b), 3(c), 3(d), 3(e), 3(g), 3(h), 3(i), and 3(j), respectively.

Fig. 5
Fig. 5

Examples of image reconstruction from the magnitude of the Fourier transforms of complex-valued objects with noise and a central stop, with oversampling and the use of positivity constraints on the imaginary part and loose supports. (a), (b), and (c) are reconstructed from the magnitude of the Fourier transform of the original 2D complex-valued object shown in Fig. 3(a). (a) Reconstructed with SNR=20 and σ=2.6, (b) reconstructed with SNR=10 and σ=2.6, (c) reconstructed with SNR=20, σ=2.6, and 11×11 pixel central stop. (d) and (e) illustrate the isodensity of the magnitude of a 3D complex-valued object reconstructed from the magnitude of the Fourier transform of the original 3D complex-valued object shown in Fig. 3(f). (d) Reconstructed with SNR=40 and σ=2.57, (e) reconstructed with SNR=20 and σ=2.57.

Equations (7)

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

F(k)=-f(x)exp(2πik·x)dx,
F(k)=x=0N-1f(x)exp(2πik·x/N),
|F(k)|=x=0N-1f(x)exp(2πik·x/N).
σ=totalpixelnumberunknown-valuedpixelnumber,
fj+1(x)=fj(x)x, fj(x)Sfj(x)-βfj(x)x, fj(x)S,
Ej=xs|fj(x)|2xs|fj(x)|21/2.
noise=signalSNR×random

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