Abstract

Single-particle imaging experiments of biomolecules at x-ray free-electron lasers (XFELs) require processing hundreds of thousands of images that contain very few x-rays. Each low-fluence image of the diffraction pattern is produced by a single, randomly oriented particle, such as a protein. We demonstrate the feasibility of recovering structural information at these extremes using low-fluence images of a randomly oriented 2D x-ray mask. Successful reconstruction is obtained with images averaging only 2.5 photons per frame, where it seems doubtful there could be information about the state of rotation, let alone the image contrast. This is accomplished with an expectation maximization algorithm that processes the low-fluence data in aggregate, and without any prior knowledge of the object or its orientation. The versatility of the method promises, more generally, to redefine what measurement scenarios can provide useful signal.

© 2012 OSA

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  1. R. Neutze, R. Wouts, D. van der Spoel, E. Weckert, and J. Hajdu, “Potential for biomolecular imaging with femtosecond x-ray pulses,” Nature 406, 752–757 (2000).
    [CrossRef] [PubMed]
  2. V. Elser, “Noise limits on reconstructing diffraction signals from random tomographs,” IEEE Trans. Inf. Theory 55, 4715–4722 (2009).
    [CrossRef]
  3. N.-T. D. Loh and V. Elser, “Reconstruction algorithm for single-particle diffraction imaging experiments,” Phys. Rev. E 80, 026705 (2009).
    [CrossRef]
  4. N. D. Loh, M. J. Bogan, V. Elser, A. Barty, S. Boutet, S. Bajt, J. Hajdu, T. Ekeberg, F. R. N. C. Maia, J. Schulz, M. M. Seibert, B. Iwan, N. Timneanu, S. Marchesini, I. Schlichting, R. L. Shoeman, L. Lomb, M. Frank, M. Liang, and H. N. Chapman, “Cryptotomography: reconstructing 3D Fourier intensities from randomly oriented single-shot diffraction patterns,” Phys. Rev. Lett. 104, 225501 (2010).
    [CrossRef] [PubMed]
  5. H. T. Philipp, L. J. Koerner, M. S. Hromalik, M. W. Tate, and S. M. Gruner, “Femtosecond radiation experiment detector for x-ray free-electron laser (XFEL) coherent x-ray imaging,” IEEE Trans. Nucl. Sci. 57, 3795–3799 (2010).
  6. H. T. Philipp, M. W. Tate, and S. M. Gruner, “Low-flux measurements with Cornell’s LCLS integrating pixel array detector.” J. Inst. 6, C11006 (2011).
    [CrossRef]
  7. L. E. Baum, T. Petrie, G. Soules, and N. Weiss, “A maximization technique occurring in the statistical analysis of probabilistic functions of Markov chains,” Ann. Math. Statist. 41, 164–171 (1970).
    [CrossRef]
  8. G. Huldt, A. Szoke, and J. Hajdu, “Diffraction imaging of single particles and biomolecules,” J. Struct. Biol. 144, 219–227 (2003).
    [CrossRef] [PubMed]
  9. R. R. Coifman, Y. Shkolnisky, F. J. Sigworth, and A. Singer, “Graph Laplacian tomography from unknown random projections,” IEEE Trans. Image Proc. 17, 1891–1899 (2008).
    [CrossRef]
  10. D. Giannakis, P. Schwander, and A. Ourmazd, “The symmetries of image formation by scattering. I. Theoretical framework,” arXiv:1009.5035 (2010).
  11. P. Thibault, M. Dierolf, A. Menzel, O. Bunk, C. David, and F. Pfeiffer, “High-resolution scanning x-ray diffraction microscopy,” Science 321, 379–382 (2008).
    [CrossRef] [PubMed]
  12. D. H. Bilderback, J. D. Brock, D. S. Dale, K. D. Finkelstein, M. A. Pfeifer, and S. M. Gruner, “Energy recovery linac (ERL) coherent hard x-ray sources,” New J. Phys. 12, 035011 (2010).
    [CrossRef]
  13. C. E. Shannon, “A Mathematical Theory of Communication,” Bell Syst. Tech. J., 27, 379–423, 623–656 (1948).

2011

H. T. Philipp, M. W. Tate, and S. M. Gruner, “Low-flux measurements with Cornell’s LCLS integrating pixel array detector.” J. Inst. 6, C11006 (2011).
[CrossRef]

2010

N. D. Loh, M. J. Bogan, V. Elser, A. Barty, S. Boutet, S. Bajt, J. Hajdu, T. Ekeberg, F. R. N. C. Maia, J. Schulz, M. M. Seibert, B. Iwan, N. Timneanu, S. Marchesini, I. Schlichting, R. L. Shoeman, L. Lomb, M. Frank, M. Liang, and H. N. Chapman, “Cryptotomography: reconstructing 3D Fourier intensities from randomly oriented single-shot diffraction patterns,” Phys. Rev. Lett. 104, 225501 (2010).
[CrossRef] [PubMed]

H. T. Philipp, L. J. Koerner, M. S. Hromalik, M. W. Tate, and S. M. Gruner, “Femtosecond radiation experiment detector for x-ray free-electron laser (XFEL) coherent x-ray imaging,” IEEE Trans. Nucl. Sci. 57, 3795–3799 (2010).

D. H. Bilderback, J. D. Brock, D. S. Dale, K. D. Finkelstein, M. A. Pfeifer, and S. M. Gruner, “Energy recovery linac (ERL) coherent hard x-ray sources,” New J. Phys. 12, 035011 (2010).
[CrossRef]

2009

V. Elser, “Noise limits on reconstructing diffraction signals from random tomographs,” IEEE Trans. Inf. Theory 55, 4715–4722 (2009).
[CrossRef]

N.-T. D. Loh and V. Elser, “Reconstruction algorithm for single-particle diffraction imaging experiments,” Phys. Rev. E 80, 026705 (2009).
[CrossRef]

2008

R. R. Coifman, Y. Shkolnisky, F. J. Sigworth, and A. Singer, “Graph Laplacian tomography from unknown random projections,” IEEE Trans. Image Proc. 17, 1891–1899 (2008).
[CrossRef]

P. Thibault, M. Dierolf, A. Menzel, O. Bunk, C. David, and F. Pfeiffer, “High-resolution scanning x-ray diffraction microscopy,” Science 321, 379–382 (2008).
[CrossRef] [PubMed]

2003

G. Huldt, A. Szoke, and J. Hajdu, “Diffraction imaging of single particles and biomolecules,” J. Struct. Biol. 144, 219–227 (2003).
[CrossRef] [PubMed]

2000

R. Neutze, R. Wouts, D. van der Spoel, E. Weckert, and J. Hajdu, “Potential for biomolecular imaging with femtosecond x-ray pulses,” Nature 406, 752–757 (2000).
[CrossRef] [PubMed]

1970

L. E. Baum, T. Petrie, G. Soules, and N. Weiss, “A maximization technique occurring in the statistical analysis of probabilistic functions of Markov chains,” Ann. Math. Statist. 41, 164–171 (1970).
[CrossRef]

1948

C. E. Shannon, “A Mathematical Theory of Communication,” Bell Syst. Tech. J., 27, 379–423, 623–656 (1948).

Bajt, S.

N. D. Loh, M. J. Bogan, V. Elser, A. Barty, S. Boutet, S. Bajt, J. Hajdu, T. Ekeberg, F. R. N. C. Maia, J. Schulz, M. M. Seibert, B. Iwan, N. Timneanu, S. Marchesini, I. Schlichting, R. L. Shoeman, L. Lomb, M. Frank, M. Liang, and H. N. Chapman, “Cryptotomography: reconstructing 3D Fourier intensities from randomly oriented single-shot diffraction patterns,” Phys. Rev. Lett. 104, 225501 (2010).
[CrossRef] [PubMed]

Barty, A.

N. D. Loh, M. J. Bogan, V. Elser, A. Barty, S. Boutet, S. Bajt, J. Hajdu, T. Ekeberg, F. R. N. C. Maia, J. Schulz, M. M. Seibert, B. Iwan, N. Timneanu, S. Marchesini, I. Schlichting, R. L. Shoeman, L. Lomb, M. Frank, M. Liang, and H. N. Chapman, “Cryptotomography: reconstructing 3D Fourier intensities from randomly oriented single-shot diffraction patterns,” Phys. Rev. Lett. 104, 225501 (2010).
[CrossRef] [PubMed]

Baum, L. E.

L. E. Baum, T. Petrie, G. Soules, and N. Weiss, “A maximization technique occurring in the statistical analysis of probabilistic functions of Markov chains,” Ann. Math. Statist. 41, 164–171 (1970).
[CrossRef]

Bilderback, D. H.

D. H. Bilderback, J. D. Brock, D. S. Dale, K. D. Finkelstein, M. A. Pfeifer, and S. M. Gruner, “Energy recovery linac (ERL) coherent hard x-ray sources,” New J. Phys. 12, 035011 (2010).
[CrossRef]

Bogan, M. J.

N. D. Loh, M. J. Bogan, V. Elser, A. Barty, S. Boutet, S. Bajt, J. Hajdu, T. Ekeberg, F. R. N. C. Maia, J. Schulz, M. M. Seibert, B. Iwan, N. Timneanu, S. Marchesini, I. Schlichting, R. L. Shoeman, L. Lomb, M. Frank, M. Liang, and H. N. Chapman, “Cryptotomography: reconstructing 3D Fourier intensities from randomly oriented single-shot diffraction patterns,” Phys. Rev. Lett. 104, 225501 (2010).
[CrossRef] [PubMed]

Boutet, S.

N. D. Loh, M. J. Bogan, V. Elser, A. Barty, S. Boutet, S. Bajt, J. Hajdu, T. Ekeberg, F. R. N. C. Maia, J. Schulz, M. M. Seibert, B. Iwan, N. Timneanu, S. Marchesini, I. Schlichting, R. L. Shoeman, L. Lomb, M. Frank, M. Liang, and H. N. Chapman, “Cryptotomography: reconstructing 3D Fourier intensities from randomly oriented single-shot diffraction patterns,” Phys. Rev. Lett. 104, 225501 (2010).
[CrossRef] [PubMed]

Brock, J. D.

D. H. Bilderback, J. D. Brock, D. S. Dale, K. D. Finkelstein, M. A. Pfeifer, and S. M. Gruner, “Energy recovery linac (ERL) coherent hard x-ray sources,” New J. Phys. 12, 035011 (2010).
[CrossRef]

Bunk, O.

P. Thibault, M. Dierolf, A. Menzel, O. Bunk, C. David, and F. Pfeiffer, “High-resolution scanning x-ray diffraction microscopy,” Science 321, 379–382 (2008).
[CrossRef] [PubMed]

Chapman, H. N.

N. D. Loh, M. J. Bogan, V. Elser, A. Barty, S. Boutet, S. Bajt, J. Hajdu, T. Ekeberg, F. R. N. C. Maia, J. Schulz, M. M. Seibert, B. Iwan, N. Timneanu, S. Marchesini, I. Schlichting, R. L. Shoeman, L. Lomb, M. Frank, M. Liang, and H. N. Chapman, “Cryptotomography: reconstructing 3D Fourier intensities from randomly oriented single-shot diffraction patterns,” Phys. Rev. Lett. 104, 225501 (2010).
[CrossRef] [PubMed]

Coifman, R. R.

R. R. Coifman, Y. Shkolnisky, F. J. Sigworth, and A. Singer, “Graph Laplacian tomography from unknown random projections,” IEEE Trans. Image Proc. 17, 1891–1899 (2008).
[CrossRef]

Dale, D. S.

D. H. Bilderback, J. D. Brock, D. S. Dale, K. D. Finkelstein, M. A. Pfeifer, and S. M. Gruner, “Energy recovery linac (ERL) coherent hard x-ray sources,” New J. Phys. 12, 035011 (2010).
[CrossRef]

David, C.

P. Thibault, M. Dierolf, A. Menzel, O. Bunk, C. David, and F. Pfeiffer, “High-resolution scanning x-ray diffraction microscopy,” Science 321, 379–382 (2008).
[CrossRef] [PubMed]

Dierolf, M.

P. Thibault, M. Dierolf, A. Menzel, O. Bunk, C. David, and F. Pfeiffer, “High-resolution scanning x-ray diffraction microscopy,” Science 321, 379–382 (2008).
[CrossRef] [PubMed]

Ekeberg, T.

N. D. Loh, M. J. Bogan, V. Elser, A. Barty, S. Boutet, S. Bajt, J. Hajdu, T. Ekeberg, F. R. N. C. Maia, J. Schulz, M. M. Seibert, B. Iwan, N. Timneanu, S. Marchesini, I. Schlichting, R. L. Shoeman, L. Lomb, M. Frank, M. Liang, and H. N. Chapman, “Cryptotomography: reconstructing 3D Fourier intensities from randomly oriented single-shot diffraction patterns,” Phys. Rev. Lett. 104, 225501 (2010).
[CrossRef] [PubMed]

Elser, V.

N. D. Loh, M. J. Bogan, V. Elser, A. Barty, S. Boutet, S. Bajt, J. Hajdu, T. Ekeberg, F. R. N. C. Maia, J. Schulz, M. M. Seibert, B. Iwan, N. Timneanu, S. Marchesini, I. Schlichting, R. L. Shoeman, L. Lomb, M. Frank, M. Liang, and H. N. Chapman, “Cryptotomography: reconstructing 3D Fourier intensities from randomly oriented single-shot diffraction patterns,” Phys. Rev. Lett. 104, 225501 (2010).
[CrossRef] [PubMed]

V. Elser, “Noise limits on reconstructing diffraction signals from random tomographs,” IEEE Trans. Inf. Theory 55, 4715–4722 (2009).
[CrossRef]

N.-T. D. Loh and V. Elser, “Reconstruction algorithm for single-particle diffraction imaging experiments,” Phys. Rev. E 80, 026705 (2009).
[CrossRef]

Finkelstein, K. D.

D. H. Bilderback, J. D. Brock, D. S. Dale, K. D. Finkelstein, M. A. Pfeifer, and S. M. Gruner, “Energy recovery linac (ERL) coherent hard x-ray sources,” New J. Phys. 12, 035011 (2010).
[CrossRef]

Frank, M.

N. D. Loh, M. J. Bogan, V. Elser, A. Barty, S. Boutet, S. Bajt, J. Hajdu, T. Ekeberg, F. R. N. C. Maia, J. Schulz, M. M. Seibert, B. Iwan, N. Timneanu, S. Marchesini, I. Schlichting, R. L. Shoeman, L. Lomb, M. Frank, M. Liang, and H. N. Chapman, “Cryptotomography: reconstructing 3D Fourier intensities from randomly oriented single-shot diffraction patterns,” Phys. Rev. Lett. 104, 225501 (2010).
[CrossRef] [PubMed]

Giannakis, D.

D. Giannakis, P. Schwander, and A. Ourmazd, “The symmetries of image formation by scattering. I. Theoretical framework,” arXiv:1009.5035 (2010).

Gruner, S. M.

H. T. Philipp, M. W. Tate, and S. M. Gruner, “Low-flux measurements with Cornell’s LCLS integrating pixel array detector.” J. Inst. 6, C11006 (2011).
[CrossRef]

D. H. Bilderback, J. D. Brock, D. S. Dale, K. D. Finkelstein, M. A. Pfeifer, and S. M. Gruner, “Energy recovery linac (ERL) coherent hard x-ray sources,” New J. Phys. 12, 035011 (2010).
[CrossRef]

H. T. Philipp, L. J. Koerner, M. S. Hromalik, M. W. Tate, and S. M. Gruner, “Femtosecond radiation experiment detector for x-ray free-electron laser (XFEL) coherent x-ray imaging,” IEEE Trans. Nucl. Sci. 57, 3795–3799 (2010).

Hajdu, J.

N. D. Loh, M. J. Bogan, V. Elser, A. Barty, S. Boutet, S. Bajt, J. Hajdu, T. Ekeberg, F. R. N. C. Maia, J. Schulz, M. M. Seibert, B. Iwan, N. Timneanu, S. Marchesini, I. Schlichting, R. L. Shoeman, L. Lomb, M. Frank, M. Liang, and H. N. Chapman, “Cryptotomography: reconstructing 3D Fourier intensities from randomly oriented single-shot diffraction patterns,” Phys. Rev. Lett. 104, 225501 (2010).
[CrossRef] [PubMed]

G. Huldt, A. Szoke, and J. Hajdu, “Diffraction imaging of single particles and biomolecules,” J. Struct. Biol. 144, 219–227 (2003).
[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 406, 752–757 (2000).
[CrossRef] [PubMed]

Hromalik, M. S.

H. T. Philipp, L. J. Koerner, M. S. Hromalik, M. W. Tate, and S. M. Gruner, “Femtosecond radiation experiment detector for x-ray free-electron laser (XFEL) coherent x-ray imaging,” IEEE Trans. Nucl. Sci. 57, 3795–3799 (2010).

Huldt, G.

G. Huldt, A. Szoke, and J. Hajdu, “Diffraction imaging of single particles and biomolecules,” J. Struct. Biol. 144, 219–227 (2003).
[CrossRef] [PubMed]

Iwan, B.

N. D. Loh, M. J. Bogan, V. Elser, A. Barty, S. Boutet, S. Bajt, J. Hajdu, T. Ekeberg, F. R. N. C. Maia, J. Schulz, M. M. Seibert, B. Iwan, N. Timneanu, S. Marchesini, I. Schlichting, R. L. Shoeman, L. Lomb, M. Frank, M. Liang, and H. N. Chapman, “Cryptotomography: reconstructing 3D Fourier intensities from randomly oriented single-shot diffraction patterns,” Phys. Rev. Lett. 104, 225501 (2010).
[CrossRef] [PubMed]

Koerner, L. J.

H. T. Philipp, L. J. Koerner, M. S. Hromalik, M. W. Tate, and S. M. Gruner, “Femtosecond radiation experiment detector for x-ray free-electron laser (XFEL) coherent x-ray imaging,” IEEE Trans. Nucl. Sci. 57, 3795–3799 (2010).

Liang, M.

N. D. Loh, M. J. Bogan, V. Elser, A. Barty, S. Boutet, S. Bajt, J. Hajdu, T. Ekeberg, F. R. N. C. Maia, J. Schulz, M. M. Seibert, B. Iwan, N. Timneanu, S. Marchesini, I. Schlichting, R. L. Shoeman, L. Lomb, M. Frank, M. Liang, and H. N. Chapman, “Cryptotomography: reconstructing 3D Fourier intensities from randomly oriented single-shot diffraction patterns,” Phys. Rev. Lett. 104, 225501 (2010).
[CrossRef] [PubMed]

Loh, N. D.

N. D. Loh, M. J. Bogan, V. Elser, A. Barty, S. Boutet, S. Bajt, J. Hajdu, T. Ekeberg, F. R. N. C. Maia, J. Schulz, M. M. Seibert, B. Iwan, N. Timneanu, S. Marchesini, I. Schlichting, R. L. Shoeman, L. Lomb, M. Frank, M. Liang, and H. N. Chapman, “Cryptotomography: reconstructing 3D Fourier intensities from randomly oriented single-shot diffraction patterns,” Phys. Rev. Lett. 104, 225501 (2010).
[CrossRef] [PubMed]

Loh, N.-T. D.

N.-T. D. Loh and V. Elser, “Reconstruction algorithm for single-particle diffraction imaging experiments,” Phys. Rev. E 80, 026705 (2009).
[CrossRef]

Lomb, L.

N. D. Loh, M. J. Bogan, V. Elser, A. Barty, S. Boutet, S. Bajt, J. Hajdu, T. Ekeberg, F. R. N. C. Maia, J. Schulz, M. M. Seibert, B. Iwan, N. Timneanu, S. Marchesini, I. Schlichting, R. L. Shoeman, L. Lomb, M. Frank, M. Liang, and H. N. Chapman, “Cryptotomography: reconstructing 3D Fourier intensities from randomly oriented single-shot diffraction patterns,” Phys. Rev. Lett. 104, 225501 (2010).
[CrossRef] [PubMed]

Maia, F. R. N. C.

N. D. Loh, M. J. Bogan, V. Elser, A. Barty, S. Boutet, S. Bajt, J. Hajdu, T. Ekeberg, F. R. N. C. Maia, J. Schulz, M. M. Seibert, B. Iwan, N. Timneanu, S. Marchesini, I. Schlichting, R. L. Shoeman, L. Lomb, M. Frank, M. Liang, and H. N. Chapman, “Cryptotomography: reconstructing 3D Fourier intensities from randomly oriented single-shot diffraction patterns,” Phys. Rev. Lett. 104, 225501 (2010).
[CrossRef] [PubMed]

Marchesini, S.

N. D. Loh, M. J. Bogan, V. Elser, A. Barty, S. Boutet, S. Bajt, J. Hajdu, T. Ekeberg, F. R. N. C. Maia, J. Schulz, M. M. Seibert, B. Iwan, N. Timneanu, S. Marchesini, I. Schlichting, R. L. Shoeman, L. Lomb, M. Frank, M. Liang, and H. N. Chapman, “Cryptotomography: reconstructing 3D Fourier intensities from randomly oriented single-shot diffraction patterns,” Phys. Rev. Lett. 104, 225501 (2010).
[CrossRef] [PubMed]

Menzel, A.

P. Thibault, M. Dierolf, A. Menzel, O. Bunk, C. David, and F. Pfeiffer, “High-resolution scanning x-ray diffraction microscopy,” Science 321, 379–382 (2008).
[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 406, 752–757 (2000).
[CrossRef] [PubMed]

Ourmazd, A.

D. Giannakis, P. Schwander, and A. Ourmazd, “The symmetries of image formation by scattering. I. Theoretical framework,” arXiv:1009.5035 (2010).

Petrie, T.

L. E. Baum, T. Petrie, G. Soules, and N. Weiss, “A maximization technique occurring in the statistical analysis of probabilistic functions of Markov chains,” Ann. Math. Statist. 41, 164–171 (1970).
[CrossRef]

Pfeifer, M. A.

D. H. Bilderback, J. D. Brock, D. S. Dale, K. D. Finkelstein, M. A. Pfeifer, and S. M. Gruner, “Energy recovery linac (ERL) coherent hard x-ray sources,” New J. Phys. 12, 035011 (2010).
[CrossRef]

Pfeiffer, F.

P. Thibault, M. Dierolf, A. Menzel, O. Bunk, C. David, and F. Pfeiffer, “High-resolution scanning x-ray diffraction microscopy,” Science 321, 379–382 (2008).
[CrossRef] [PubMed]

Philipp, H. T.

H. T. Philipp, M. W. Tate, and S. M. Gruner, “Low-flux measurements with Cornell’s LCLS integrating pixel array detector.” J. Inst. 6, C11006 (2011).
[CrossRef]

H. T. Philipp, L. J. Koerner, M. S. Hromalik, M. W. Tate, and S. M. Gruner, “Femtosecond radiation experiment detector for x-ray free-electron laser (XFEL) coherent x-ray imaging,” IEEE Trans. Nucl. Sci. 57, 3795–3799 (2010).

Schlichting, I.

N. D. Loh, M. J. Bogan, V. Elser, A. Barty, S. Boutet, S. Bajt, J. Hajdu, T. Ekeberg, F. R. N. C. Maia, J. Schulz, M. M. Seibert, B. Iwan, N. Timneanu, S. Marchesini, I. Schlichting, R. L. Shoeman, L. Lomb, M. Frank, M. Liang, and H. N. Chapman, “Cryptotomography: reconstructing 3D Fourier intensities from randomly oriented single-shot diffraction patterns,” Phys. Rev. Lett. 104, 225501 (2010).
[CrossRef] [PubMed]

Schulz, J.

N. D. Loh, M. J. Bogan, V. Elser, A. Barty, S. Boutet, S. Bajt, J. Hajdu, T. Ekeberg, F. R. N. C. Maia, J. Schulz, M. M. Seibert, B. Iwan, N. Timneanu, S. Marchesini, I. Schlichting, R. L. Shoeman, L. Lomb, M. Frank, M. Liang, and H. N. Chapman, “Cryptotomography: reconstructing 3D Fourier intensities from randomly oriented single-shot diffraction patterns,” Phys. Rev. Lett. 104, 225501 (2010).
[CrossRef] [PubMed]

Schwander, P.

D. Giannakis, P. Schwander, and A. Ourmazd, “The symmetries of image formation by scattering. I. Theoretical framework,” arXiv:1009.5035 (2010).

Seibert, M. M.

N. D. Loh, M. J. Bogan, V. Elser, A. Barty, S. Boutet, S. Bajt, J. Hajdu, T. Ekeberg, F. R. N. C. Maia, J. Schulz, M. M. Seibert, B. Iwan, N. Timneanu, S. Marchesini, I. Schlichting, R. L. Shoeman, L. Lomb, M. Frank, M. Liang, and H. N. Chapman, “Cryptotomography: reconstructing 3D Fourier intensities from randomly oriented single-shot diffraction patterns,” Phys. Rev. Lett. 104, 225501 (2010).
[CrossRef] [PubMed]

Shannon, C. E.

C. E. Shannon, “A Mathematical Theory of Communication,” Bell Syst. Tech. J., 27, 379–423, 623–656 (1948).

Shkolnisky, Y.

R. R. Coifman, Y. Shkolnisky, F. J. Sigworth, and A. Singer, “Graph Laplacian tomography from unknown random projections,” IEEE Trans. Image Proc. 17, 1891–1899 (2008).
[CrossRef]

Shoeman, R. L.

N. D. Loh, M. J. Bogan, V. Elser, A. Barty, S. Boutet, S. Bajt, J. Hajdu, T. Ekeberg, F. R. N. C. Maia, J. Schulz, M. M. Seibert, B. Iwan, N. Timneanu, S. Marchesini, I. Schlichting, R. L. Shoeman, L. Lomb, M. Frank, M. Liang, and H. N. Chapman, “Cryptotomography: reconstructing 3D Fourier intensities from randomly oriented single-shot diffraction patterns,” Phys. Rev. Lett. 104, 225501 (2010).
[CrossRef] [PubMed]

Sigworth, F. J.

R. R. Coifman, Y. Shkolnisky, F. J. Sigworth, and A. Singer, “Graph Laplacian tomography from unknown random projections,” IEEE Trans. Image Proc. 17, 1891–1899 (2008).
[CrossRef]

Singer, A.

R. R. Coifman, Y. Shkolnisky, F. J. Sigworth, and A. Singer, “Graph Laplacian tomography from unknown random projections,” IEEE Trans. Image Proc. 17, 1891–1899 (2008).
[CrossRef]

Soules, G.

L. E. Baum, T. Petrie, G. Soules, and N. Weiss, “A maximization technique occurring in the statistical analysis of probabilistic functions of Markov chains,” Ann. Math. Statist. 41, 164–171 (1970).
[CrossRef]

Szoke, A.

G. Huldt, A. Szoke, and J. Hajdu, “Diffraction imaging of single particles and biomolecules,” J. Struct. Biol. 144, 219–227 (2003).
[CrossRef] [PubMed]

Tate, M. W.

H. T. Philipp, M. W. Tate, and S. M. Gruner, “Low-flux measurements with Cornell’s LCLS integrating pixel array detector.” J. Inst. 6, C11006 (2011).
[CrossRef]

H. T. Philipp, L. J. Koerner, M. S. Hromalik, M. W. Tate, and S. M. Gruner, “Femtosecond radiation experiment detector for x-ray free-electron laser (XFEL) coherent x-ray imaging,” IEEE Trans. Nucl. Sci. 57, 3795–3799 (2010).

Thibault, P.

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Supplementary Material (1)

» Media 1: MPG (3102 KB)     

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Figures (4)

Fig. 1
Fig. 1

(a) The lead x-ray mask mounted within an aperture in an aluminum disk. (b) A static x-ray image of the pattern collected as 432 individual frames with approximately 1/5 photon per pixel per frame. The frames were thresholded and averaged. (c) A reconstruction using randomly-oriented data having an average 11.5 photons/frame and 1.2 million recorded photons. (d) A reconstruction using randomly-oriented data having an average 2.5 photons/frame and 1.2 million recorded photons ( Media 1).

Fig. 2
Fig. 2

(a–c) Three sample frames from the 2.5 photon/frame data set with detected x-ray photons circled. (d) Occupancy histogram compared with the Poisson distribution. (e) The sum of all thresholded frames from the 2.5 photon/frame data set showing a uniform angular distribution of data.

Fig. 3
Fig. 3

Effect of background on reconstruction quality. (a) Reconstruction from 2.5 photons/frame data set and no added background. This is the same as Fig. 1(d). (b) Reconstruction from the 11.5 photons/frame data set with an average of 11.5 photons of background added per frame ‘by hand’ with a Poisson distribution. The background level was subtracted before rendering to facilitate comparison to (a). As can be seen, the quality of the reconstructions is about the same, and much reduced from the original 11.5 photons/frame data (Fig. 1(c)).

Fig. 4
Fig. 4

Reduction in the rate at which pixels measuring photons acquire information as a function of signal-to-noise. Plotted is the ratio R of the information rate, acquired with and without background. SN is the ratio of signal to background photon counts. This plot applies to the model where half the pixels receive only background (covered by mask) and the other half receive signal and background (not covered by mask).

Equations (13)

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p f ( r ) i w ( i + r ) k f ( i ) k f ( i ) ! e w ( i + r ) i I f w ( i + r ) ,
w ( i ) = r p f ( r ) k f ( i r ) f .
c f f = i k f ( i ) k f ( i ) .
R = σ ( 1 + SN 1 ) log ( 1 + SN ) ( σ + SN 1 ) log ( 1 + σ SN ) σ log σ .
p ( w ) = ( 1 σ ) δ ( w ν ) + σ δ ( w ν μ ) ,
p ( k | w ) = w k k ! e w .
p ( w , k ) = p ( w ) p ( k | w ) .
I = H k H k | w ,
H k = k p ( k ) log p ( k )
H k | w = d w p ( w ) k p ( k | w ) log p ( k | w )
I ( μ , ν ) = μ ( σ ( 1 + SN 1 ) log ( 1 + SN ) ( σ + SN 1 ) log ( 1 + σ SN ) + O ( μ 2 ) .
I ( μ , 0 ) = μ σ log σ + O ( μ 2 ) .
I ( μ , ν ) I ( μ , 0 ) = R ( SN ) = σ ( 1 + SN 1 ) log ( 1 + SN ) ( σ + SN 1 ) log ( 1 + σ SN ) σ log σ .

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