Abstract

Coherent x-ray diffraction microscopy is a method of imaging nonperiodic isolated objects at resolutions limited, in principle, by only the wavelength and largest scattering angles recorded. We demonstrate x-ray diffraction imaging with high resolution in all three dimensions, as determined by a quantitative analysis of the reconstructed volume images. These images are retrieved from the three-dimensional diffraction data using no a priori knowledge about the shape or composition of the object, which has never before been demonstrated on a nonperiodic object. We also construct two-dimensional images of thick objects with greatly increased depth of focus (without loss of transverse spatial resolution). These methods can be used to image biological and materials science samples at high resolution with x-ray undulator radiation and establishes the techniques to be used in atomic-resolution ultrafast imaging at x-ray free-electron laser sources.

© 2006 Optical Society of America

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2005

M. R. Howells, T. Beetz, H. N. Chapman, C. Cui, J. M. Holton, C. J. Jacobsen, J. Kirz, E. Lima, S. Marchesini, H. Miao, D. Sayre, D. A. Shapiro, and J. C. H. Spence, "An assessment of the resolution limitation due to radiation-damage in x-ray diffraction microscopy," arxiv.org e-print archive, physics/0502059, February 11, 2005, http://arxiv.org/pdf/physics/0502059.

T. Beetz, M. Howells, C. Jacobsen, C. Kao, J. Kirz, E. Lima, T. Mentes, H. Miao, C. Sanchez-Hanke, D. Sayre, and D. Shapiro, "Apparatus for x-ray diffraction microscopy and tomography of cryo specimens," Nucl. Instrum. Methods Phys. Res. A 545, 459-468 (2005).
[CrossRef]

D. Shapiro, P. Thibault, T. Beetz, V. Elser, M. Howells, C. Jacobsen, J. Kirz, E. Lima, H. Miao, A. M. Neimann, and D. Sayre, "Biological imaging by soft x-ray diffraction microscopy," Proc. Natl. Acad. Sci. U.S.A. 102, 15343-15346 (2005).
[CrossRef] [PubMed]

J. R. Fienup, "Phase retrieval and support estimation in x-ray diffraction," in Coherence 2005: International Workshop on Phase Retrieval and Coherent Scattering (ESRF, 2005), http://www.esrf.fr/NewsAndEvents/Conferences/Coherence2005/Proceedings/files/Talks/Fienup.pdf.

D. R. Luke, "Relaxed averaged alternating reflections for diffraction imaging," Inverse Probl. 21, 37-50 (2005).
[CrossRef]

M. Frigo and S. G. Johnson, "The design and implementation of FFTW3," Proc. IEEE 93, 216-231 (2005). (Special issue on "Program Generation, Optimization, and Platform Adaptation").
[CrossRef]

G. Huldt, Biomedical Centre, Uppsala Universitet (personal communication, 2005).

2004

S. P. Hau-Riege, H. Szoke, H. N. Chapman, A. Szoke, S. Marchesini, A. Noy, H. He, M. Howells, U. Weierstall, and J. C. H. Spence, "SPEDEN: reconstructing single particles from their diffraction patterns," Acta Crystallogr., Sect. A: Found. Crystallogr. 60, 294-305 (2004).
[CrossRef]

R. Crandall, E. Jones, J. Klivington, and D. Kramer, "Gigaelement FFTs on Apple G5 clusters," Advanced Computation Group, Apple Computer (2004), http://images.apple.com/acg/pdf/20040827lowbarGigaFFT.pdf.

F. Natterer, "An error bound for the Born approximation," Inverse Probl. 20, 447-452 (2004).
[CrossRef]

J. C. H. Spence, U. Weierstall, and M. Howells, "Coherence and sampling requirements for diffractive imaging," Ultramicroscopy 101, 149-152 (2004).
[CrossRef] [PubMed]

J. C. H. Spence and R. B. Doak, "Single molecule diffraction," Phys. Rev. Lett. 92, 198102 (2004).
[CrossRef] [PubMed]

C. A. Larabell and M. A. Le Gros, "X-ray tomography generates 3-D reconstructions of the yeast, Saccharomyces cerevisiae, at 60-nm resolution," Mol. Biol. Cell 15, 957-962 (2004).
[CrossRef]

2003

G. J. Williams, M. A. Pfeifer, I. A. Vartanyants, and I. K. Robinson, "Three-dimensional imaging of microstructure in Au nanocrystals," Phys. Rev. Lett. 90, 175501 (2003).
[CrossRef] [PubMed]

S. Marchesini, H. He, H. N. Chapman, S. P. Hau-Riege, A. Noy, M. R. Howells, U. Weierstall, and J. C. H. Spence, "X-ray image reconstruction from a diffraction pattern alone," Phys. Rev. B 68, 140101 (2003).
[CrossRef]

S. Marchesini, H. N. Chapman, S. P. Hau-Riege, R. A. London, A. Szoke, H. He, M. R. Howells, H. Padmore, R. Rosen, J. C. H. Spence, and U. Weierstall, "Coherent x-ray diffractive imaging: applications and limitations," Opt. Express 11, 2344-2353 (2003).
[CrossRef] [PubMed]

H. He, S. Marchesini, M. Howells, U. Weierstall, H. Chapman, S. Hau-Riege, A. Noy, and J. C. H. Spence, "Inversion of x-ray diffuse scattering to images using prepared objects," Phys. Rev. B 67, 174114 (2003).
[CrossRef]

J. Miao, T. Ishikawa, E. H. Anderson, and K. O. Hodgson, "Phase retrieval of diffraction patterns from noncrystalline samples using the oversampling method," Phys. Rev. B 67, 174104 (2003).
[CrossRef]

Y. Nishino, J. Miao, and T. Ishikawa, "Image reconstruction of nanostructured nonperiodic objects only from oversampled hard x-ray diffraction intensities," Phys. Rev. B 68, 220101 (2003).
[CrossRef]

H. He, S. Marchesini, M. Howells, U. Weierstall, G. Hembree, and J. C. H. Spence, "Experimental lensless soft-x-ray imaging using iterative algorithms: phasing diffuse scattering," Acta Crystallogr., Sect. A: Found. Crystallogr. 59, 143-152 (2003).
[CrossRef]

V. Elser, "Phase retrieval by iterated projections," J. Opt. Soc. Am. A 20, 40-55 (2003).
[CrossRef]

2002

J. C. H. Spence, U. Weierstall, and M. Howells, "Phase recovery and lensless imaging by iterative methods in optical, x-ray and electron diffraction," Philos. Trans. R. Soc. London, Ser. A 360, 875-895 (2002).
[CrossRef]

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge U. Press, 2002).

M. R. Howells, P. Charalambous, H. He, S. Marchesini, and J. C. H. Spence, "An off-axis zone-plate monochromator for high-power undulator radiation," in Design and Microfabrication of Novel X-Ray Optics, D.C.Mancini, ed., Proc. SPIE 4783, 65-73 (2002).

2001

I. K. Robinson, I. A. Vartanyants, G. J. Williams, M. A. Pfeifer, and J. A. Pitney, "Reconstruction of the shapes of gold nanocrystals using coherent x-ray diffraction," Phys. Rev. Lett. 87, 195505 (2001).
[CrossRef] [PubMed]

J. Miao, K. O. Hodgson, and D. Sayre, "An approach to three-dimensional structures of biomolecules by using single-molecule diffraction images," Proc. Natl. Acad. Sci. U.S.A. 98, 6641-6645 (2001).
[CrossRef] [PubMed]

F. Natterer, The Mathematics of Computerized Tomography (SIAM, 2001).
[CrossRef]

D. Potts, G. Steidl, and M. Tasche, "Fast Fourier transforms for nonequispaced data: a tutorial," in Modern Sampling Theory: Mathematics and Applications, J.J.Benedetto and P.Ferreira, eds. (Springer, 2001), Chap. 12, pp. 249-274.

2000

D. Weiss, G. Schneider, B. Niemann, P. Guttmann, D. Rudolph, and G. Schmahl, "Computed tomography of cryogenic biological specimens based on x-ray microscopic images," Ultramicroscopy 84, 185-197 (2000).
[CrossRef] [PubMed]

R. Neutze, R. Wouts, D. van der Spoel, E. Weckert, and J. Hajdu, "Potential for biomolecular imaging with femtosecond x-ray pulses," Nature (London) 406, 753-757 (2000).
[CrossRef]

1999

J. Miao, P. Charalambous, J. Kirz, and D. Sayre, "Extending the methodology of x-ray crystallography to allow imaging of micrometre-sized non-crystalline specimens," Nature (London) 400, 342-344 (1999).
[CrossRef]

1998

D. Sayre, H. N. Chapman, and J. Miao, "On the extendibility of x-ray crystallography to noncrystals," Acta Crystallogr., Sect. A: Found. Crystallogr. 54, 232-239 (1998).
[CrossRef]

C. Giacovazzo, Direct Phasing in Crystallography (Oxford U. Press, 1998), p. 468.

E. Salerno, "Superresolution capabilities of the Gerchberg method in the band-pass case: an eigenvalue analysis," Int. J. Imaging Syst. Technol. 90, 181-188 (1998).
[CrossRef]

J. Miao, D. Sayre, and 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]

H. Choi and D. C. Munson, Jr., "Direct-Fourier reconstruction in tomography and synthetic aperture radar," Int. J. Imaging Syst. Technol. 9, 1-13 (1998).
[CrossRef]

A. J. Ladd, J. H. Kinney, D. L. Haupt, and S. A. Goldstein, "Finite-element modeling of trabecular bone: comparison with mechanical testing and determination of tissue modulus," J. Orthop. Res. 16, 622-628 (1998).
[CrossRef] [PubMed]

1997

A. Szoke, H. Szoke, and J. R. Somoza, "Holographic methods in x-ray crystallography. V. Multiple isomorphous replacement, multiple anomalous dispersion and noncrystallographic symmetry," Acta Crystallogr., Sect. A: Found. Crystallogr. 53, 291-313 (1997).
[CrossRef]

J. R. Fienup, "Invariant error metrics for image reconstruction," Appl. Opt. 36, 8352-8357 (1997).
[CrossRef]

1996

J. Frank, Three-Dimensional Electron Microscopy of Macromolecular Assemblies (Academic, 1996).

J. W. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, 1996).

1995

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

D. Sayre and H. N. Chapman, "X-ray microscopy," Acta Crystallogr., Sect. A: Found. Crystallogr. 51, 237-252 (1995).
[CrossRef]

1994

W. S. Haddad, I. McNulty, J. Trebes, E. Anderson, R. Levesque, and L. Yang, "Ultrahigh-resolution x-ray tomography," Science 266, 1213-1215 (1994).
[CrossRef] [PubMed]

1990

1987

1986

J. R. Fienup and C. Wackerman, "Phase-retrieval stagnation problems and solutions," J. Opt. Soc. Am. A 3, 1897-1907 (1986).
[CrossRef]

R. N. Bracewell, The Fourier Transform and Its Applications, 2nd ed. (McGraw-Hill, 1986).

1985

1983

S. Pan and A. Kak, "A computational study of reconstruction algorithms for diffraction tomography: interpolation versus filtered-backpropagation," IEEE Trans. Acoust., Speech, Signal Process. SP- 31, 1262-1275 (1983).
[CrossRef]

1982

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

A. J. Devaney, "A filtered backpropagation algorithm for diffraction tomography," Ultrason. Imaging 4, 336-350 (1982).
[CrossRef] [PubMed]

M. Bertero and E. R. Pike, "Resolution in diffraction-limited imaging, a singular value analysis. I. The case of coherent illumination," Opt. Acta 29, 727-746 (1982).
[CrossRef]

1981

1970

R. Crowther, D. DeRosier, and A. Klug, "The reconstruction of a three-dimensional structure from its projections and its applications to electron microscopy," Proc. R. Soc. London A317, 319-340 (1970).

1969

E. Wolf, "Three-dimensional structure determination of semi-transparent objects from holographic data," Opt. Commun. 1, 153-156 (1969).
[CrossRef]

1962

R. W. James, The Optical Principles of the Diffraction of X-Rays (Bell, 1962).

Anderson, E.

W. S. Haddad, I. McNulty, J. Trebes, E. Anderson, R. Levesque, and L. Yang, "Ultrahigh-resolution x-ray tomography," Science 266, 1213-1215 (1994).
[CrossRef] [PubMed]

Anderson, E. H.

J. Miao, T. Ishikawa, E. H. Anderson, and K. O. Hodgson, "Phase retrieval of diffraction patterns from noncrystalline samples using the oversampling method," Phys. Rev. B 67, 174104 (2003).
[CrossRef]

Beetz, T.

T. Beetz, M. Howells, C. Jacobsen, C. Kao, J. Kirz, E. Lima, T. Mentes, H. Miao, C. Sanchez-Hanke, D. Sayre, and D. Shapiro, "Apparatus for x-ray diffraction microscopy and tomography of cryo specimens," Nucl. Instrum. Methods Phys. Res. A 545, 459-468 (2005).
[CrossRef]

M. R. Howells, T. Beetz, H. N. Chapman, C. Cui, J. M. Holton, C. J. Jacobsen, J. Kirz, E. Lima, S. Marchesini, H. Miao, D. Sayre, D. A. Shapiro, and J. C. H. Spence, "An assessment of the resolution limitation due to radiation-damage in x-ray diffraction microscopy," arxiv.org e-print archive, physics/0502059, February 11, 2005, http://arxiv.org/pdf/physics/0502059.

D. Shapiro, P. Thibault, T. Beetz, V. Elser, M. Howells, C. Jacobsen, J. Kirz, E. Lima, H. Miao, A. M. Neimann, and D. Sayre, "Biological imaging by soft x-ray diffraction microscopy," Proc. Natl. Acad. Sci. U.S.A. 102, 15343-15346 (2005).
[CrossRef] [PubMed]

Bertero, M.

M. Bertero and E. R. Pike, "Resolution in diffraction-limited imaging, a singular value analysis. I. The case of coherent illumination," Opt. Acta 29, 727-746 (1982).
[CrossRef]

Born, M.

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge U. Press, 2002).

Bracewell, R. N.

R. N. Bracewell, The Fourier Transform and Its Applications, 2nd ed. (McGraw-Hill, 1986).

Chapman, H.

H. He, S. Marchesini, M. Howells, U. Weierstall, H. Chapman, S. Hau-Riege, A. Noy, and J. C. H. Spence, "Inversion of x-ray diffuse scattering to images using prepared objects," Phys. Rev. B 67, 174114 (2003).
[CrossRef]

Chapman, H. N.

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D. Shapiro, P. Thibault, T. Beetz, V. Elser, M. Howells, C. Jacobsen, J. Kirz, E. Lima, H. Miao, A. M. Neimann, and D. Sayre, "Biological imaging by soft x-ray diffraction microscopy," Proc. Natl. Acad. Sci. U.S.A. 102, 15343-15346 (2005).
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T. Beetz, M. Howells, C. Jacobsen, C. Kao, J. Kirz, E. Lima, T. Mentes, H. Miao, C. Sanchez-Hanke, D. Sayre, and D. Shapiro, "Apparatus for x-ray diffraction microscopy and tomography of cryo specimens," Nucl. Instrum. Methods Phys. Res. A 545, 459-468 (2005).
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M. R. Howells, T. Beetz, H. N. Chapman, C. Cui, J. M. Holton, C. J. Jacobsen, J. Kirz, E. Lima, S. Marchesini, H. Miao, D. Sayre, D. A. Shapiro, and J. C. H. Spence, "An assessment of the resolution limitation due to radiation-damage in x-ray diffraction microscopy," arxiv.org e-print archive, physics/0502059, February 11, 2005, http://arxiv.org/pdf/physics/0502059.

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R. Crandall, E. Jones, J. Klivington, and D. Kramer, "Gigaelement FFTs on Apple G5 clusters," Advanced Computation Group, Apple Computer (2004), http://images.apple.com/acg/pdf/20040827lowbarGigaFFT.pdf.

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T. Beetz, M. Howells, C. Jacobsen, C. Kao, J. Kirz, E. Lima, T. Mentes, H. Miao, C. Sanchez-Hanke, D. Sayre, and D. Shapiro, "Apparatus for x-ray diffraction microscopy and tomography of cryo specimens," Nucl. Instrum. Methods Phys. Res. A 545, 459-468 (2005).
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D. Shapiro, P. Thibault, T. Beetz, V. Elser, M. Howells, C. Jacobsen, J. Kirz, E. Lima, H. Miao, A. M. Neimann, and D. Sayre, "Biological imaging by soft x-ray diffraction microscopy," Proc. Natl. Acad. Sci. U.S.A. 102, 15343-15346 (2005).
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S. P. Hau-Riege, H. Szoke, H. N. Chapman, A. Szoke, S. Marchesini, A. Noy, H. He, M. Howells, U. Weierstall, and J. C. H. Spence, "SPEDEN: reconstructing single particles from their diffraction patterns," Acta Crystallogr., Sect. A: Found. Crystallogr. 60, 294-305 (2004).
[CrossRef]

S. Marchesini, H. He, H. N. Chapman, S. P. Hau-Riege, A. Noy, M. R. Howells, U. Weierstall, and J. C. H. Spence, "X-ray image reconstruction from a diffraction pattern alone," Phys. Rev. B 68, 140101 (2003).
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H. He, S. Marchesini, M. Howells, U. Weierstall, H. Chapman, S. Hau-Riege, A. Noy, and J. C. H. Spence, "Inversion of x-ray diffuse scattering to images using prepared objects," Phys. Rev. B 67, 174114 (2003).
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H. He, S. Marchesini, M. Howells, U. Weierstall, G. Hembree, and J. C. H. Spence, "Experimental lensless soft-x-ray imaging using iterative algorithms: phasing diffuse scattering," Acta Crystallogr., Sect. A: Found. Crystallogr. 59, 143-152 (2003).
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S. Marchesini, H. N. Chapman, S. P. Hau-Riege, R. A. London, A. Szoke, H. He, M. R. Howells, H. Padmore, R. Rosen, J. C. H. Spence, and U. Weierstall, "Coherent x-ray diffractive imaging: applications and limitations," Opt. Express 11, 2344-2353 (2003).
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T. Beetz, M. Howells, C. Jacobsen, C. Kao, J. Kirz, E. Lima, T. Mentes, H. Miao, C. Sanchez-Hanke, D. Sayre, and D. Shapiro, "Apparatus for x-ray diffraction microscopy and tomography of cryo specimens," Nucl. Instrum. Methods Phys. Res. A 545, 459-468 (2005).
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T. Beetz, M. Howells, C. Jacobsen, C. Kao, J. Kirz, E. Lima, T. Mentes, H. Miao, C. Sanchez-Hanke, D. Sayre, and D. Shapiro, "Apparatus for x-ray diffraction microscopy and tomography of cryo specimens," Nucl. Instrum. Methods Phys. Res. A 545, 459-468 (2005).
[CrossRef]

M. R. Howells, T. Beetz, H. N. Chapman, C. Cui, J. M. Holton, C. J. Jacobsen, J. Kirz, E. Lima, S. Marchesini, H. Miao, D. Sayre, D. A. Shapiro, and J. C. H. Spence, "An assessment of the resolution limitation due to radiation-damage in x-ray diffraction microscopy," arxiv.org e-print archive, physics/0502059, February 11, 2005, http://arxiv.org/pdf/physics/0502059.

D. Shapiro, P. Thibault, T. Beetz, V. Elser, M. Howells, C. Jacobsen, J. Kirz, E. Lima, H. Miao, A. M. Neimann, and D. Sayre, "Biological imaging by soft x-ray diffraction microscopy," Proc. Natl. Acad. Sci. U.S.A. 102, 15343-15346 (2005).
[CrossRef] [PubMed]

Miao, J.

Y. Nishino, J. Miao, and T. Ishikawa, "Image reconstruction of nanostructured nonperiodic objects only from oversampled hard x-ray diffraction intensities," Phys. Rev. B 68, 220101 (2003).
[CrossRef]

J. Miao, T. Ishikawa, E. H. Anderson, and K. O. Hodgson, "Phase retrieval of diffraction patterns from noncrystalline samples using the oversampling method," Phys. Rev. B 67, 174104 (2003).
[CrossRef]

J. Miao, K. O. Hodgson, and D. Sayre, "An approach to three-dimensional structures of biomolecules by using single-molecule diffraction images," Proc. Natl. Acad. Sci. U.S.A. 98, 6641-6645 (2001).
[CrossRef] [PubMed]

J. Miao, P. Charalambous, J. Kirz, and D. Sayre, "Extending the methodology of x-ray crystallography to allow imaging of micrometre-sized non-crystalline specimens," Nature (London) 400, 342-344 (1999).
[CrossRef]

J. Miao, D. Sayre, and 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]

D. Sayre, H. N. Chapman, and J. Miao, "On the extendibility of x-ray crystallography to noncrystals," Acta Crystallogr., Sect. A: Found. Crystallogr. 54, 232-239 (1998).
[CrossRef]

Munson, D. C.

H. Choi and D. C. Munson, Jr., "Direct-Fourier reconstruction in tomography and synthetic aperture radar," Int. J. Imaging Syst. Technol. 9, 1-13 (1998).
[CrossRef]

Natterer, F.

F. Natterer, "An error bound for the Born approximation," Inverse Probl. 20, 447-452 (2004).
[CrossRef]

F. Natterer, The Mathematics of Computerized Tomography (SIAM, 2001).
[CrossRef]

Neimann, A. M.

D. Shapiro, P. Thibault, T. Beetz, V. Elser, M. Howells, C. Jacobsen, J. Kirz, E. Lima, H. Miao, A. M. Neimann, and D. Sayre, "Biological imaging by soft x-ray diffraction microscopy," Proc. Natl. Acad. Sci. U.S.A. 102, 15343-15346 (2005).
[CrossRef] [PubMed]

Neutze, R.

R. Neutze, R. Wouts, D. van der Spoel, E. Weckert, and J. Hajdu, "Potential for biomolecular imaging with femtosecond x-ray pulses," Nature (London) 406, 753-757 (2000).
[CrossRef]

Niemann, B.

D. Weiss, G. Schneider, B. Niemann, P. Guttmann, D. Rudolph, and G. Schmahl, "Computed tomography of cryogenic biological specimens based on x-ray microscopic images," Ultramicroscopy 84, 185-197 (2000).
[CrossRef] [PubMed]

Nishino, Y.

Y. Nishino, J. Miao, and T. Ishikawa, "Image reconstruction of nanostructured nonperiodic objects only from oversampled hard x-ray diffraction intensities," Phys. Rev. B 68, 220101 (2003).
[CrossRef]

Norton, S. J.

Noy, A.

S. P. Hau-Riege, H. Szoke, H. N. Chapman, A. Szoke, S. Marchesini, A. Noy, H. He, M. Howells, U. Weierstall, and J. C. H. Spence, "SPEDEN: reconstructing single particles from their diffraction patterns," Acta Crystallogr., Sect. A: Found. Crystallogr. 60, 294-305 (2004).
[CrossRef]

S. Marchesini, H. He, H. N. Chapman, S. P. Hau-Riege, A. Noy, M. R. Howells, U. Weierstall, and J. C. H. Spence, "X-ray image reconstruction from a diffraction pattern alone," Phys. Rev. B 68, 140101 (2003).
[CrossRef]

H. He, S. Marchesini, M. Howells, U. Weierstall, H. Chapman, S. Hau-Riege, A. Noy, and J. C. H. Spence, "Inversion of x-ray diffuse scattering to images using prepared objects," Phys. Rev. B 67, 174114 (2003).
[CrossRef]

Padmore, H.

Pan, S.

S. Pan and A. Kak, "A computational study of reconstruction algorithms for diffraction tomography: interpolation versus filtered-backpropagation," IEEE Trans. Acoust., Speech, Signal Process. SP- 31, 1262-1275 (1983).
[CrossRef]

Pfeifer, M. A.

G. J. Williams, M. A. Pfeifer, I. A. Vartanyants, and I. K. Robinson, "Three-dimensional imaging of microstructure in Au nanocrystals," Phys. Rev. Lett. 90, 175501 (2003).
[CrossRef] [PubMed]

I. K. Robinson, I. A. Vartanyants, G. J. Williams, M. A. Pfeifer, and J. A. Pitney, "Reconstruction of the shapes of gold nanocrystals using coherent x-ray diffraction," Phys. Rev. Lett. 87, 195505 (2001).
[CrossRef] [PubMed]

Pike, E. R.

M. Bertero and E. R. Pike, "Resolution in diffraction-limited imaging, a singular value analysis. I. The case of coherent illumination," Opt. Acta 29, 727-746 (1982).
[CrossRef]

Pitney, J. A.

I. K. Robinson, I. A. Vartanyants, G. J. Williams, M. A. Pfeifer, and J. A. Pitney, "Reconstruction of the shapes of gold nanocrystals using coherent x-ray diffraction," Phys. Rev. Lett. 87, 195505 (2001).
[CrossRef] [PubMed]

Potts, D.

D. Potts, G. Steidl, and M. Tasche, "Fast Fourier transforms for nonequispaced data: a tutorial," in Modern Sampling Theory: Mathematics and Applications, J.J.Benedetto and P.Ferreira, eds. (Springer, 2001), Chap. 12, pp. 249-274.

Robinson, I. K.

G. J. Williams, M. A. Pfeifer, I. A. Vartanyants, and I. K. Robinson, "Three-dimensional imaging of microstructure in Au nanocrystals," Phys. Rev. Lett. 90, 175501 (2003).
[CrossRef] [PubMed]

I. K. Robinson, I. A. Vartanyants, G. J. Williams, M. A. Pfeifer, and J. A. Pitney, "Reconstruction of the shapes of gold nanocrystals using coherent x-ray diffraction," Phys. Rev. Lett. 87, 195505 (2001).
[CrossRef] [PubMed]

Rosen, R.

Rudolph, D.

D. Weiss, G. Schneider, B. Niemann, P. Guttmann, D. Rudolph, and G. Schmahl, "Computed tomography of cryogenic biological specimens based on x-ray microscopic images," Ultramicroscopy 84, 185-197 (2000).
[CrossRef] [PubMed]

Salerno, E.

E. Salerno, "Superresolution capabilities of the Gerchberg method in the band-pass case: an eigenvalue analysis," Int. J. Imaging Syst. Technol. 90, 181-188 (1998).
[CrossRef]

Sanchez-Hanke, C.

T. Beetz, M. Howells, C. Jacobsen, C. Kao, J. Kirz, E. Lima, T. Mentes, H. Miao, C. Sanchez-Hanke, D. Sayre, and D. Shapiro, "Apparatus for x-ray diffraction microscopy and tomography of cryo specimens," Nucl. Instrum. Methods Phys. Res. A 545, 459-468 (2005).
[CrossRef]

Sato, T.

Sayre, D.

D. Shapiro, P. Thibault, T. Beetz, V. Elser, M. Howells, C. Jacobsen, J. Kirz, E. Lima, H. Miao, A. M. Neimann, and D. Sayre, "Biological imaging by soft x-ray diffraction microscopy," Proc. Natl. Acad. Sci. U.S.A. 102, 15343-15346 (2005).
[CrossRef] [PubMed]

T. Beetz, M. Howells, C. Jacobsen, C. Kao, J. Kirz, E. Lima, T. Mentes, H. Miao, C. Sanchez-Hanke, D. Sayre, and D. Shapiro, "Apparatus for x-ray diffraction microscopy and tomography of cryo specimens," Nucl. Instrum. Methods Phys. Res. A 545, 459-468 (2005).
[CrossRef]

M. R. Howells, T. Beetz, H. N. Chapman, C. Cui, J. M. Holton, C. J. Jacobsen, J. Kirz, E. Lima, S. Marchesini, H. Miao, D. Sayre, D. A. Shapiro, and J. C. H. Spence, "An assessment of the resolution limitation due to radiation-damage in x-ray diffraction microscopy," arxiv.org e-print archive, physics/0502059, February 11, 2005, http://arxiv.org/pdf/physics/0502059.

J. Miao, K. O. Hodgson, and D. Sayre, "An approach to three-dimensional structures of biomolecules by using single-molecule diffraction images," Proc. Natl. Acad. Sci. U.S.A. 98, 6641-6645 (2001).
[CrossRef] [PubMed]

J. Miao, P. Charalambous, J. Kirz, and D. Sayre, "Extending the methodology of x-ray crystallography to allow imaging of micrometre-sized non-crystalline specimens," Nature (London) 400, 342-344 (1999).
[CrossRef]

J. Miao, D. Sayre, and 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]

D. Sayre, H. N. Chapman, and J. Miao, "On the extendibility of x-ray crystallography to noncrystals," Acta Crystallogr., Sect. A: Found. Crystallogr. 54, 232-239 (1998).
[CrossRef]

D. Sayre and H. N. Chapman, "X-ray microscopy," Acta Crystallogr., Sect. A: Found. Crystallogr. 51, 237-252 (1995).
[CrossRef]

Schmahl, G.

D. Weiss, G. Schneider, B. Niemann, P. Guttmann, D. Rudolph, and G. Schmahl, "Computed tomography of cryogenic biological specimens based on x-ray microscopic images," Ultramicroscopy 84, 185-197 (2000).
[CrossRef] [PubMed]

Schneider, G.

D. Weiss, G. Schneider, B. Niemann, P. Guttmann, D. Rudolph, and G. Schmahl, "Computed tomography of cryogenic biological specimens based on x-ray microscopic images," Ultramicroscopy 84, 185-197 (2000).
[CrossRef] [PubMed]

Shapiro, D.

T. Beetz, M. Howells, C. Jacobsen, C. Kao, J. Kirz, E. Lima, T. Mentes, H. Miao, C. Sanchez-Hanke, D. Sayre, and D. Shapiro, "Apparatus for x-ray diffraction microscopy and tomography of cryo specimens," Nucl. Instrum. Methods Phys. Res. A 545, 459-468 (2005).
[CrossRef]

D. Shapiro, P. Thibault, T. Beetz, V. Elser, M. Howells, C. Jacobsen, J. Kirz, E. Lima, H. Miao, A. M. Neimann, and D. Sayre, "Biological imaging by soft x-ray diffraction microscopy," Proc. Natl. Acad. Sci. U.S.A. 102, 15343-15346 (2005).
[CrossRef] [PubMed]

Shapiro, D. A.

M. R. Howells, T. Beetz, H. N. Chapman, C. Cui, J. M. Holton, C. J. Jacobsen, J. Kirz, E. Lima, S. Marchesini, H. Miao, D. Sayre, D. A. Shapiro, and J. C. H. Spence, "An assessment of the resolution limitation due to radiation-damage in x-ray diffraction microscopy," arxiv.org e-print archive, physics/0502059, February 11, 2005, http://arxiv.org/pdf/physics/0502059.

Somoza, J. R.

A. Szoke, H. Szoke, and J. R. Somoza, "Holographic methods in x-ray crystallography. V. Multiple isomorphous replacement, multiple anomalous dispersion and noncrystallographic symmetry," Acta Crystallogr., Sect. A: Found. Crystallogr. 53, 291-313 (1997).
[CrossRef]

Spence, J. C.

M. R. Howells, T. Beetz, H. N. Chapman, C. Cui, J. M. Holton, C. J. Jacobsen, J. Kirz, E. Lima, S. Marchesini, H. Miao, D. Sayre, D. A. Shapiro, and J. C. H. Spence, "An assessment of the resolution limitation due to radiation-damage in x-ray diffraction microscopy," arxiv.org e-print archive, physics/0502059, February 11, 2005, http://arxiv.org/pdf/physics/0502059.

S. P. Hau-Riege, H. Szoke, H. N. Chapman, A. Szoke, S. Marchesini, A. Noy, H. He, M. Howells, U. Weierstall, and J. C. H. Spence, "SPEDEN: reconstructing single particles from their diffraction patterns," Acta Crystallogr., Sect. A: Found. Crystallogr. 60, 294-305 (2004).
[CrossRef]

J. C. H. Spence and R. B. Doak, "Single molecule diffraction," Phys. Rev. Lett. 92, 198102 (2004).
[CrossRef] [PubMed]

J. C. H. Spence, U. Weierstall, and M. Howells, "Coherence and sampling requirements for diffractive imaging," Ultramicroscopy 101, 149-152 (2004).
[CrossRef] [PubMed]

H. He, S. Marchesini, M. Howells, U. Weierstall, G. Hembree, and J. C. H. Spence, "Experimental lensless soft-x-ray imaging using iterative algorithms: phasing diffuse scattering," Acta Crystallogr., Sect. A: Found. Crystallogr. 59, 143-152 (2003).
[CrossRef]

H. He, S. Marchesini, M. Howells, U. Weierstall, H. Chapman, S. Hau-Riege, A. Noy, and J. C. H. Spence, "Inversion of x-ray diffuse scattering to images using prepared objects," Phys. Rev. B 67, 174114 (2003).
[CrossRef]

S. Marchesini, H. He, H. N. Chapman, S. P. Hau-Riege, A. Noy, M. R. Howells, U. Weierstall, and J. C. H. Spence, "X-ray image reconstruction from a diffraction pattern alone," Phys. Rev. B 68, 140101 (2003).
[CrossRef]

S. Marchesini, H. N. Chapman, S. P. Hau-Riege, R. A. London, A. Szoke, H. He, M. R. Howells, H. Padmore, R. Rosen, J. C. H. Spence, and U. Weierstall, "Coherent x-ray diffractive imaging: applications and limitations," Opt. Express 11, 2344-2353 (2003).
[CrossRef] [PubMed]

M. R. Howells, P. Charalambous, H. He, S. Marchesini, and J. C. H. Spence, "An off-axis zone-plate monochromator for high-power undulator radiation," in Design and Microfabrication of Novel X-Ray Optics, D.C.Mancini, ed., Proc. SPIE 4783, 65-73 (2002).

J. C. H. Spence, U. Weierstall, and M. Howells, "Phase recovery and lensless imaging by iterative methods in optical, x-ray and electron diffraction," Philos. Trans. R. Soc. London, Ser. A 360, 875-895 (2002).
[CrossRef]

Steidl, G.

D. Potts, G. Steidl, and M. Tasche, "Fast Fourier transforms for nonequispaced data: a tutorial," in Modern Sampling Theory: Mathematics and Applications, J.J.Benedetto and P.Ferreira, eds. (Springer, 2001), Chap. 12, pp. 249-274.

Streibl, N.

Szoke, A.

S. P. Hau-Riege, H. Szoke, H. N. Chapman, A. Szoke, S. Marchesini, A. Noy, H. He, M. Howells, U. Weierstall, and J. C. H. Spence, "SPEDEN: reconstructing single particles from their diffraction patterns," Acta Crystallogr., Sect. A: Found. Crystallogr. 60, 294-305 (2004).
[CrossRef]

S. Marchesini, H. N. Chapman, S. P. Hau-Riege, R. A. London, A. Szoke, H. He, M. R. Howells, H. Padmore, R. Rosen, J. C. H. Spence, and U. Weierstall, "Coherent x-ray diffractive imaging: applications and limitations," Opt. Express 11, 2344-2353 (2003).
[CrossRef] [PubMed]

A. Szoke, H. Szoke, and J. R. Somoza, "Holographic methods in x-ray crystallography. V. Multiple isomorphous replacement, multiple anomalous dispersion and noncrystallographic symmetry," Acta Crystallogr., Sect. A: Found. Crystallogr. 53, 291-313 (1997).
[CrossRef]

Szoke, H.

S. P. Hau-Riege, H. Szoke, H. N. Chapman, A. Szoke, S. Marchesini, A. Noy, H. He, M. Howells, U. Weierstall, and J. C. H. Spence, "SPEDEN: reconstructing single particles from their diffraction patterns," Acta Crystallogr., Sect. A: Found. Crystallogr. 60, 294-305 (2004).
[CrossRef]

A. Szoke, H. Szoke, and J. R. Somoza, "Holographic methods in x-ray crystallography. V. Multiple isomorphous replacement, multiple anomalous dispersion and noncrystallographic symmetry," Acta Crystallogr., Sect. A: Found. Crystallogr. 53, 291-313 (1997).
[CrossRef]

Tasche, M.

D. Potts, G. Steidl, and M. Tasche, "Fast Fourier transforms for nonequispaced data: a tutorial," in Modern Sampling Theory: Mathematics and Applications, J.J.Benedetto and P.Ferreira, eds. (Springer, 2001), Chap. 12, pp. 249-274.

Thelen, B. J.

Thibault, P.

D. Shapiro, P. Thibault, T. Beetz, V. Elser, M. Howells, C. Jacobsen, J. Kirz, E. Lima, H. Miao, A. M. Neimann, and D. Sayre, "Biological imaging by soft x-ray diffraction microscopy," Proc. Natl. Acad. Sci. U.S.A. 102, 15343-15346 (2005).
[CrossRef] [PubMed]

Trebes, J.

W. S. Haddad, I. McNulty, J. Trebes, E. Anderson, R. Levesque, and L. Yang, "Ultrahigh-resolution x-ray tomography," Science 266, 1213-1215 (1994).
[CrossRef] [PubMed]

van der Spoel, D.

R. Neutze, R. Wouts, D. van der Spoel, E. Weckert, and J. Hajdu, "Potential for biomolecular imaging with femtosecond x-ray pulses," Nature (London) 406, 753-757 (2000).
[CrossRef]

Vartanyants, I. A.

G. J. Williams, M. A. Pfeifer, I. A. Vartanyants, and I. K. Robinson, "Three-dimensional imaging of microstructure in Au nanocrystals," Phys. Rev. Lett. 90, 175501 (2003).
[CrossRef] [PubMed]

I. K. Robinson, I. A. Vartanyants, G. J. Williams, M. A. Pfeifer, and J. A. Pitney, "Reconstruction of the shapes of gold nanocrystals using coherent x-ray diffraction," Phys. Rev. Lett. 87, 195505 (2001).
[CrossRef] [PubMed]

Wackerman, C.

Weckert, E.

R. Neutze, R. Wouts, D. van der Spoel, E. Weckert, and J. Hajdu, "Potential for biomolecular imaging with femtosecond x-ray pulses," Nature (London) 406, 753-757 (2000).
[CrossRef]

Weierstall, U.

S. P. Hau-Riege, H. Szoke, H. N. Chapman, A. Szoke, S. Marchesini, A. Noy, H. He, M. Howells, U. Weierstall, and J. C. H. Spence, "SPEDEN: reconstructing single particles from their diffraction patterns," Acta Crystallogr., Sect. A: Found. Crystallogr. 60, 294-305 (2004).
[CrossRef]

J. C. H. Spence, U. Weierstall, and M. Howells, "Coherence and sampling requirements for diffractive imaging," Ultramicroscopy 101, 149-152 (2004).
[CrossRef] [PubMed]

H. He, S. Marchesini, M. Howells, U. Weierstall, G. Hembree, and J. C. H. Spence, "Experimental lensless soft-x-ray imaging using iterative algorithms: phasing diffuse scattering," Acta Crystallogr., Sect. A: Found. Crystallogr. 59, 143-152 (2003).
[CrossRef]

H. He, S. Marchesini, M. Howells, U. Weierstall, H. Chapman, S. Hau-Riege, A. Noy, and J. C. H. Spence, "Inversion of x-ray diffuse scattering to images using prepared objects," Phys. Rev. B 67, 174114 (2003).
[CrossRef]

S. Marchesini, H. He, H. N. Chapman, S. P. Hau-Riege, A. Noy, M. R. Howells, U. Weierstall, and J. C. H. Spence, "X-ray image reconstruction from a diffraction pattern alone," Phys. Rev. B 68, 140101 (2003).
[CrossRef]

S. Marchesini, H. N. Chapman, S. P. Hau-Riege, R. A. London, A. Szoke, H. He, M. R. Howells, H. Padmore, R. Rosen, J. C. H. Spence, and U. Weierstall, "Coherent x-ray diffractive imaging: applications and limitations," Opt. Express 11, 2344-2353 (2003).
[CrossRef] [PubMed]

J. C. H. Spence, U. Weierstall, and M. Howells, "Phase recovery and lensless imaging by iterative methods in optical, x-ray and electron diffraction," Philos. Trans. R. Soc. London, Ser. A 360, 875-895 (2002).
[CrossRef]

Weiss, D.

D. Weiss, G. Schneider, B. Niemann, P. Guttmann, D. Rudolph, and G. Schmahl, "Computed tomography of cryogenic biological specimens based on x-ray microscopic images," Ultramicroscopy 84, 185-197 (2000).
[CrossRef] [PubMed]

Williams, G. J.

G. J. Williams, M. A. Pfeifer, I. A. Vartanyants, and I. K. Robinson, "Three-dimensional imaging of microstructure in Au nanocrystals," Phys. Rev. Lett. 90, 175501 (2003).
[CrossRef] [PubMed]

I. K. Robinson, I. A. Vartanyants, G. J. Williams, M. A. Pfeifer, and J. A. Pitney, "Reconstruction of the shapes of gold nanocrystals using coherent x-ray diffraction," Phys. Rev. Lett. 87, 195505 (2001).
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M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge U. Press, 2002).

E. Wolf, "Three-dimensional structure determination of semi-transparent objects from holographic data," Opt. Commun. 1, 153-156 (1969).
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R. Neutze, R. Wouts, D. van der Spoel, E. Weckert, and J. Hajdu, "Potential for biomolecular imaging with femtosecond x-ray pulses," Nature (London) 406, 753-757 (2000).
[CrossRef]

Yang, L.

W. S. Haddad, I. McNulty, J. Trebes, E. Anderson, R. Levesque, and L. Yang, "Ultrahigh-resolution x-ray tomography," Science 266, 1213-1215 (1994).
[CrossRef] [PubMed]

Acta Crystallogr., Sect. A: Found. Crystallogr.

D. Sayre and H. N. Chapman, "X-ray microscopy," Acta Crystallogr., Sect. A: Found. Crystallogr. 51, 237-252 (1995).
[CrossRef]

D. Sayre, H. N. Chapman, and J. Miao, "On the extendibility of x-ray crystallography to noncrystals," Acta Crystallogr., Sect. A: Found. Crystallogr. 54, 232-239 (1998).
[CrossRef]

A. Szoke, H. Szoke, and J. R. Somoza, "Holographic methods in x-ray crystallography. V. Multiple isomorphous replacement, multiple anomalous dispersion and noncrystallographic symmetry," Acta Crystallogr., Sect. A: Found. Crystallogr. 53, 291-313 (1997).
[CrossRef]

S. P. Hau-Riege, H. Szoke, H. N. Chapman, A. Szoke, S. Marchesini, A. Noy, H. He, M. Howells, U. Weierstall, and J. C. H. Spence, "SPEDEN: reconstructing single particles from their diffraction patterns," Acta Crystallogr., Sect. A: Found. Crystallogr. 60, 294-305 (2004).
[CrossRef]

H. He, S. Marchesini, M. Howells, U. Weierstall, G. Hembree, and J. C. H. Spence, "Experimental lensless soft-x-ray imaging using iterative algorithms: phasing diffuse scattering," Acta Crystallogr., Sect. A: Found. Crystallogr. 59, 143-152 (2003).
[CrossRef]

Appl. Opt.

IEEE Trans. Acoust., Speech, Signal Process.

S. Pan and A. Kak, "A computational study of reconstruction algorithms for diffraction tomography: interpolation versus filtered-backpropagation," IEEE Trans. Acoust., Speech, Signal Process. SP- 31, 1262-1275 (1983).
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Int. J. Imaging Syst. Technol.

H. Choi and D. C. Munson, Jr., "Direct-Fourier reconstruction in tomography and synthetic aperture radar," Int. J. Imaging Syst. Technol. 9, 1-13 (1998).
[CrossRef]

E. Salerno, "Superresolution capabilities of the Gerchberg method in the band-pass case: an eigenvalue analysis," Int. J. Imaging Syst. Technol. 90, 181-188 (1998).
[CrossRef]

Inverse Probl.

F. Natterer, "An error bound for the Born approximation," Inverse Probl. 20, 447-452 (2004).
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D. R. Luke, "Relaxed averaged alternating reflections for diffraction imaging," Inverse Probl. 21, 37-50 (2005).
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J. Opt. Soc. Am. A

J. Orthop. Res.

A. J. Ladd, J. H. Kinney, D. L. Haupt, and S. A. Goldstein, "Finite-element modeling of trabecular bone: comparison with mechanical testing and determination of tissue modulus," J. Orthop. Res. 16, 622-628 (1998).
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Mol. Biol. Cell

C. A. Larabell and M. A. Le Gros, "X-ray tomography generates 3-D reconstructions of the yeast, Saccharomyces cerevisiae, at 60-nm resolution," Mol. Biol. Cell 15, 957-962 (2004).
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Nature (London)

R. Neutze, R. Wouts, D. van der Spoel, E. Weckert, and J. Hajdu, "Potential for biomolecular imaging with femtosecond x-ray pulses," Nature (London) 406, 753-757 (2000).
[CrossRef]

J. Miao, P. Charalambous, J. Kirz, and D. Sayre, "Extending the methodology of x-ray crystallography to allow imaging of micrometre-sized non-crystalline specimens," Nature (London) 400, 342-344 (1999).
[CrossRef]

Nucl. Instrum. Methods Phys. Res. A

T. Beetz, M. Howells, C. Jacobsen, C. Kao, J. Kirz, E. Lima, T. Mentes, H. Miao, C. Sanchez-Hanke, D. Sayre, and D. Shapiro, "Apparatus for x-ray diffraction microscopy and tomography of cryo specimens," Nucl. Instrum. Methods Phys. Res. A 545, 459-468 (2005).
[CrossRef]

Opt. Acta

M. Bertero and E. R. Pike, "Resolution in diffraction-limited imaging, a singular value analysis. I. The case of coherent illumination," Opt. Acta 29, 727-746 (1982).
[CrossRef]

Opt. Commun.

E. Wolf, "Three-dimensional structure determination of semi-transparent objects from holographic data," Opt. Commun. 1, 153-156 (1969).
[CrossRef]

Opt. Express

Philos. Trans. R. Soc. London, Ser. A

J. C. H. Spence, U. Weierstall, and M. Howells, "Phase recovery and lensless imaging by iterative methods in optical, x-ray and electron diffraction," Philos. Trans. R. Soc. London, Ser. A 360, 875-895 (2002).
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Figures (11)

Fig. 1
Fig. 1

Scattering geometry for coherent x-ray diffraction imaging. The sample is rotated about the y axis by an angle ϕ.

Fig. 2
Fig. 2

(Color online) (a) SEM image of the pyramid test object, consisting of 50-nm-diameter gold spheres lining the inside of a pyramid-shaped indentation in a 100-nm-thick silicon nitride membrane. The membrane extends over a window of size 50 μ m × 1.7 mm , the pyramid base width is 2.5 μ m , and the height is 1.8 μ m . (b) Isosurface rendering of the reconstructed 3D image. (c) Extremely large depth-of-field x-ray projection image from a central section of the 3D diffraction data set, reconstructed with the Shrinkwrap algorithm. (d) Maximum value projection of the 3D reconstructed image (left) with a horizontal white line indicating the location of a tomographic slice (right). The scale-bar length is 1 μ m and applies to all images.

Fig. 3
Fig. 3

(a) Diffraction pattern for the ϕ = 0 ° orientation of the pyramid. (b) Autocorrelation image i ( x ) , formed by Fourier transforming the diffraction intensity pattern of (a) after filtering, displayed with a linear gray scale (white corresponds to highest intensity). The scale bar is 1 μ m . (c) Autocorrelation spectrogram of the same single-view diffraction pattern of the pyramid, displayed with a logarithmic gray scale. The central and rightmost images of the bottom row of (c) are redisplayed in (d) and (e), respectively.

Fig. 4
Fig. 4

(Color online) (a) Rendering of the entire 3D diffraction data set. (b) Central slice of the data set in the plane rotated by 24 ° about the y axis from the u x u y plane. (c) Central slice of the data set in the u x u z plane. All patterns are displayed on a logarithmic gray scale (white corresponds to highest intensity). The half-width of each pattern is u x , max = 0.048 nm 1 .

Fig. 5
Fig. 5

Maximum value projections along three orthogonal directions of the reconstructed 3D image. Projections were performed along (a) z, (b) x, and (c) y directions. (d) Enlarged region of (a) for comparison with Fig. 8 below. The 3D image was reconstructed with reality and positivity constraints. The scale bars are 500 nm.

Fig. 6
Fig. 6

Real part of the image reconstructed from (a) a single-view diffraction pattern, and real part of the image formed by numerically propagating (a) by (b) 0.5 μ m and (c) + 0.7 μ m . Lineouts from the image near (d) the pyramid center and (e) the arm extremity for a range of propagation from 2.5 to + 2.5 μ m . The locations of these lineouts are indicated by arrows in (a). The difference in the plane of best focus for these two image locations is apparent. The scale bars are 500 nm.

Fig. 7
Fig. 7

Distributions of the real-space complex amplitudes γ ¯ M , in the Argand plane, of simulated single-view coherent images for (a) a 2D and (b) a 3D object consisting of 50-nm-diameter gold balls for an x-ray wavelength of 1.6 nm. Distributions of complex amplitudes of images reconstructed from experimental data are given for (c) the 2D projection image shown in Fig. 8, (d) the single-view 2D image of Fig. 6, and (e) the full 3D image. Cases (c) and (d) were reconstructed by using P S , and (e) by using P S + .

Fig. 8
Fig. 8

Extremely large depth-of-focus projection images for the object orientation ϕ = 0 ° . (a) Reconstruction from a 2D central section interpolated from the 3D diffraction data set. The reconstruction was performed without a positivity constraint ( E S 2 = 0.167 ) . (b) Enlargement of the lower right arm of (a). (c) [and also Fig. 2c] Reconstruction from the 2D central section using a positivity constraint ( E S 2 = 0.072 ) . (d) Projected image formed by integrating the full 3D reconstructed image ( E S 2 = 0.113 ) . The scale bars are 500 nm.

Fig. 9
Fig. 9

3D diffraction intensities I ( u ) , averaged over shells of constant u, in units of average photon count per CCD pixel. The average over constant u of the 3D SNR of the measured intensities is shown as a dotted’s dashed curve.

Fig. 10
Fig. 10

(Color online) (a) Phase retrieval transfer function (PRTF), averaged over shells of constant u, for the real-positive 3D projection image (solid curve) and averaged over circles of constant u for the complex 2D image (dotted–dashed curves). The dotted–dashed curve with lower values is for the 2D projection image without correction of vortex phase modes. (b) Isosurface rendering of the 3D PRTF at a threshold level of 0.5. The tick marks on the u y and u z axes indicate 0.05 nm 1 .

Fig. 11
Fig. 11

Lineouts of the real part of the reconstructed complex amplitude 3D image, for three orthogonal directions (a) x, (b) y, and (c) z, through the isolated single ball at the pyramid apex. Coordinates are relative to the center of the 3D image array. Dotted curves show lineouts from a simulated 3D coherent image with a cube CTF with a 60° missing sector.

Tables (2)

Tables Icon

Table 1 Minimum Memory Footprint Required for Iterative 3D Phase Retrieval for Various Array Sizes a

Tables Icon

Table 2 Computing Times Using a Cluster-Based Fourier Transform and Reconstruction Code on 16 G5 Dual-Processor Xserve Compute Nodes a

Equations (25)

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O ( u ) = F { o ( x ) } o ( x ) exp ( 2 π i u x ) d x ,
I ( q ; Ω ) = I 0 Ω P O ( q ) 2 ,
o ( x ) = r e ρ ( x ) = π λ 2 [ 1 n 2 ( x ) ] .
o ( x ) 2 π λ 2 [ δ ( x ) + i β ( x ) ] = 2 π λ 2 Δ n ( x ) .
i ( x ) = F 1 { I ( q ) } o ( x ) o * ( x ) ,
Δ ϕ = Δ q q max = Δ x D ,
q i , j = k out k in = 1 λ ( p i , j + z D k ̂ p i , j 2 + z D 2 k ̂ ) ,
q i , j x = q i , j i ̂ = 1 λ p i p 2 ( i 2 + j 2 ) + z D 2 ,
q i , j z = q i , j k ̂ = 1 λ ( z D p 2 ( i 2 + j 2 ) + z D 2 1 ) .
u i , j , ϕ = R ϕ q i , j ,
g n + 1 = [ 1 2 β ( R S R M + I ) + ( 1 β ) P M ] g n = [ 2 β P S P M + ( 1 2 β ) P M + β ( P S I ) ] g n ,
P M g
= F 1 { G ( u ) G ( u ) + ϵ [ I ( u ) + σ I ( u ) ] if G ( u ) > I ( u ) + σ I ( u ) G ( u ) G ( u ) + ϵ [ I ( u ) σ I ( u ) ] if G ( u ) < I ( u ) σ I ( u ) G ( u ) otherwise , or u M } ,
P S g = { g ( x ) if x S 0 otherwise } .
P S + g = { R { g ( x ) } if x S and R { g ( x ) } > 0 0 otherwise } .
g n + 1 = { P M g n if x S ( I β P M ) g n otherwise } ,
E S 2 g n P S g n 2 P S g n 2 = x S g n ( x ) 2 x S g n ( x ) 2 .
E M 2 g n P M g n 2 P M g n 2 = G n I 2 I ,
γ M = P M g n
f ( q ) = { ( q 2 a ) 4 exp ( 2 q 2 2 a 2 ) if q < 2 a 1 otherwise } ,
q z = 1 λ 1 λ 2 q x 2 q y 2 λ 2 ( q x 2 + q y 2 ) ,
D < λ 2 NA 2 ,
γ ¯ M = γ M exp ( i ϕ 0 ) ,
PRTF ( u ) = F u { γ ¯ M } I ( u ) = Γ M ( u ) exp ( i ϕ 0 ) I ( u ) ,
α = k [ R { γ M ( 0 ) ( k ) exp ( i ϕ 0 ) } ] 2 = k { 2 γ M ( 0 ) ( k ) 2 + [ γ M ( 0 ) ( k ) ] 2 exp ( 2 i ϕ 0 ) + [ γ M ( 0 ) ( k ) ] * 2 exp ( 2 i ϕ 0 ) } 4

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