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

[CrossRef]

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

[CrossRef]
[PubMed]

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]

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]

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

[CrossRef]
[PubMed]

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]

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

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

[CrossRef]

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).

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

[CrossRef]

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]

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

[CrossRef]

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]

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).

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]

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

[CrossRef]

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).

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]

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]

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, C. C. Wackerman, “Phase-retrieval stagnation problems and solutions,” J. Opt. Soc. Am. A 3, 1897–1907 (1986).

[CrossRef]

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

[CrossRef]
[PubMed]

J. R. Fienup, “Reconstruction of an object from the modulus of its Fourier transform,” Opt. Lett. 3, 27–29 (1978).

[CrossRef]
[PubMed]

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).

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]

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]

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]

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]

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

[CrossRef]
[PubMed]

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]

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

[CrossRef]
[PubMed]

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

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

[CrossRef]
[PubMed]

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

[CrossRef]

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

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).

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

[CrossRef]

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]

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).

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).

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

[CrossRef]

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

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. 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]

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

[CrossRef]

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

[CrossRef]

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]

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]

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]

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

[CrossRef]

J. R. Fienup, C. C. Wackerman, “Phase-retrieval stagnation problems and solutions,” J. Opt. Soc. Am. A 3, 1897–1907 (1986).

[CrossRef]

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]

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

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

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

[CrossRef]

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

[CrossRef]

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).

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]

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

[CrossRef]
[PubMed]

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).

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. 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]

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).