Abstract

In x-ray diffraction microscopy, iterative algorithms retrieve reciprocal space phase information, and a real space image, from an object’s coherent diffraction intensities through the use of a priori information such as a finite support constraint. In many experiments, the object’s shape or support is not well known, and the diffraction pattern is incompletely measured. We describe here computer simulations to look at the effects of both of these possible errors when using several common reconstruction algorithms. Overly tight object supports prevent successful convergence; however, we show that this can often be recognized through pathological behavior of the phase retrieval transfer function. Dynamic range limitations often make it difficult to record the central speckles of the diffraction pattern. We show that this leads to increasing artifacts in the image when the number of missing central speckles exceeds about 10, and that the removal of unconstrained modes from the reconstructed image is helpful only when the number of missing central speckles is less than about 50. This simulation study helps in judging the reconstructability of experimentally recorded coherent diffraction patterns.

© 2010 Optical Society of America

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References

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  1. D. Sayre, “Prospects for long-wavelength X-ray microscopy and diffraction,” in Imaging Processes and Coherence in Physics, M. Schlenker, M. Fink, J. Goedgebuer, C. Malgrange, J. Viénot, and R. Wade, eds., vol. 112 of Lecture Notes in Physics, pp. 229–235 (Springer-Verlag, 1980).
    [Crossref]
  2. J. Miao, P. Charalambous, J. Kirz, and D. Sayre, “An extension of the methods of x-ray crystallography to allow imaging of micron-size non-crystalline specimens,” Nature 400, 342–344 (1999).
    [Crossref]
  3. J. Fienup, “Phase retrieval algorithms: a comparison,” Appl. Opt. 21, 2758–2769 (1982).
    [Crossref] [PubMed]
  4. V. Elser, “Phase retrieval by iterated projections,” J. Opt. Soc. Am. A 20, 40–55 (2003).
    [Crossref]
  5. J. P. Abrahams and A. W. G. Leslie, “Methods used in the structure determination of bovine mitochondrial F1 ATPase,” Acta Crystallogr. D Biol. Crystallogr. 52, 30–42 (1996).
    [Crossref] [PubMed]
  6. H. H. Bauschke, P. L. Combettes, and D. R. Luke, “Phase retrieval, error reduction algorithm, and Fienup variants: a view from convex optimization,” J. Opt. Soc. Am. A 19, 1334–1345 (2002).
    [Crossref]
  7. H. H. Bauschke, P. L. Combettes, and D. R. Luke, “Hybrid projection-reflection method for phase retrieval,” J. Opt. Soc. Am. A 20, 1025–1034 (2003).
    [Crossref]
  8. D. R. Luke, “Relaxed averaged alternating reflections for diffraction imaging,” Inverse Probl. 21, 37–50 (2005).
    [Crossref]
  9. R. H. T. Bates, “Fourier phase problems are uniquely solvable in more than one dimension. I. Underlying theory,” Optik 61, 247–262 (1982).
  10. S. Marchesini, H. He, H. Chapman, S. Hau-Riege, A. Noy, M. Howells, U. Weierstall, and J. Spence, “X-ray image reconstruction from a diffraction pattern alone,” Phys. Rev. B 68, 140,101 (2003).
    [Crossref]
  11. H. Jiang, C. Song, C. Chen, R. Xu, K. Raines, B. Fahimian, C. Lu, T. Lee, A. Nakashima, J. Urano, T. Ishikawa, F. Tamanoi, and J. Miao, “Quantitative 3D imaging of whole, unstained cells by using X-ray diffraction microscopy,” Proc. Natl. Acad. Sci. U.S.A. 107, 11,234–11,239 (2010).
    [Crossref]
  12. G. Williams, M. Pfeifer, I. Vartanyants, and I. Robinson, “Effectiveness of iterative algorithms in recovering phase in the presence of noise,” Acta Crystallogr. A 63, 36–42 (2007).
    [Crossref]
  13. S. Marchesini, “A unified evaluation of iterative projection algorithms for phase retrieval,” Rev. Sci. Instrum. 78, 011,301 (2007).
  14. R. W. Gerchberg and W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik 35, 237–246 (1972).
  15. X. Huang, H. Miao, J. Steinbrener, J. Nelson, A. Stewart, D. Shapiro, and C. Jacobsen, “Signal to noise considerations in diffraction and conventional microscopy,” Opt. Express 17, 13,541–13,553 (2009).
    [Crossref]
  16. B. L. Henke, E. M. Gullikson, and 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]
  17. R. A. London, M. D. Rosen, and J. E. Trebes, “Wavelength choice for soft x-ray laser holography of biological samples,” Appl. Opt. 28, 3397–3404 (1989).
    [Crossref] [PubMed]
  18. J. M. Cowley and A. F. Moodie, “Fourier Images. I. The Point Source,” Proc. Phys. Soc. B 70, 486–496 (1957).
    [Crossref]
  19. P. Thibault, V. Elser, C. Jacobsen, D. Shapiro, and D. Sayre, “Reconstruction of a yeast cell from x-ray diffraction data,” Acta Crystallogr. A 62, 248–261 (2006).
    [Crossref] [PubMed]
  20. C.-C. Chen, J. Miao, C. W. Wang, and T. K. Lee, “Application of optimization technique to noncrystalline x-ray diffraction microscopy: Guided hybrid input-output method,” Phys. Rev. B 76, 064113 (2007).
    [Crossref]
  21. J. Miao, Y. Nishino, Y. Kohmura, B. Johnson, C. Song, S. Risbud, and T. Ishikawa, “Quantitative Image Reconstruction of GaN Quantum Dots from Oversampled Diffraction Intensities Alone,” Phys. Rev. Lett. 95, 085,503 (2005).
    [Crossref]
  22. J. Steinbrener, J. Nelson, X. Huang, S. Marchesini, D. Shapiro, J. Turner, and C. Jacobsen, “Data preparation and evaluation techniques for x-ray diffraction microscopy,” Opt. Express 18, 18,598–18,614 (2010).
    [Crossref]

2010 (2)

H. Jiang, C. Song, C. Chen, R. Xu, K. Raines, B. Fahimian, C. Lu, T. Lee, A. Nakashima, J. Urano, T. Ishikawa, F. Tamanoi, and J. Miao, “Quantitative 3D imaging of whole, unstained cells by using X-ray diffraction microscopy,” Proc. Natl. Acad. Sci. U.S.A. 107, 11,234–11,239 (2010).
[Crossref]

J. Steinbrener, J. Nelson, X. Huang, S. Marchesini, D. Shapiro, J. Turner, and C. Jacobsen, “Data preparation and evaluation techniques for x-ray diffraction microscopy,” Opt. Express 18, 18,598–18,614 (2010).
[Crossref]

2009 (1)

X. Huang, H. Miao, J. Steinbrener, J. Nelson, A. Stewart, D. Shapiro, and C. Jacobsen, “Signal to noise considerations in diffraction and conventional microscopy,” Opt. Express 17, 13,541–13,553 (2009).
[Crossref]

2007 (3)

G. Williams, M. Pfeifer, I. Vartanyants, and I. Robinson, “Effectiveness of iterative algorithms in recovering phase in the presence of noise,” Acta Crystallogr. A 63, 36–42 (2007).
[Crossref]

S. Marchesini, “A unified evaluation of iterative projection algorithms for phase retrieval,” Rev. Sci. Instrum. 78, 011,301 (2007).

C.-C. Chen, J. Miao, C. W. Wang, and T. K. Lee, “Application of optimization technique to noncrystalline x-ray diffraction microscopy: Guided hybrid input-output method,” Phys. Rev. B 76, 064113 (2007).
[Crossref]

2006 (1)

P. Thibault, V. Elser, C. Jacobsen, D. Shapiro, and D. Sayre, “Reconstruction of a yeast cell from x-ray diffraction data,” Acta Crystallogr. A 62, 248–261 (2006).
[Crossref] [PubMed]

2005 (2)

J. Miao, Y. Nishino, Y. Kohmura, B. Johnson, C. Song, S. Risbud, and T. Ishikawa, “Quantitative Image Reconstruction of GaN Quantum Dots from Oversampled Diffraction Intensities Alone,” Phys. Rev. Lett. 95, 085,503 (2005).
[Crossref]

D. R. Luke, “Relaxed averaged alternating reflections for diffraction imaging,” Inverse Probl. 21, 37–50 (2005).
[Crossref]

2003 (3)

S. Marchesini, H. He, H. Chapman, S. Hau-Riege, A. Noy, M. Howells, U. Weierstall, and J. Spence, “X-ray image reconstruction from a diffraction pattern alone,” Phys. Rev. B 68, 140,101 (2003).
[Crossref]

V. Elser, “Phase retrieval by iterated projections,” J. Opt. Soc. Am. A 20, 40–55 (2003).
[Crossref]

H. H. Bauschke, P. L. Combettes, and D. R. Luke, “Hybrid projection-reflection method for phase retrieval,” J. Opt. Soc. Am. A 20, 1025–1034 (2003).
[Crossref]

2002 (1)

1999 (1)

J. Miao, P. Charalambous, J. Kirz, and D. Sayre, “An extension of the methods of x-ray crystallography to allow imaging of micron-size non-crystalline specimens,” Nature 400, 342–344 (1999).
[Crossref]

1996 (1)

J. P. Abrahams and A. W. G. Leslie, “Methods used in the structure determination of bovine mitochondrial F1 ATPase,” Acta Crystallogr. D Biol. Crystallogr. 52, 30–42 (1996).
[Crossref] [PubMed]

1993 (1)

B. L. Henke, E. M. Gullikson, and 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]

1989 (1)

1982 (2)

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

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

1972 (1)

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

1957 (1)

J. M. Cowley and A. F. Moodie, “Fourier Images. I. The Point Source,” Proc. Phys. Soc. B 70, 486–496 (1957).
[Crossref]

Abrahams, J. P.

J. P. Abrahams and A. W. G. Leslie, “Methods used in the structure determination of bovine mitochondrial F1 ATPase,” Acta Crystallogr. D Biol. Crystallogr. 52, 30–42 (1996).
[Crossref] [PubMed]

Bates, R. H. T.

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

Bauschke, H. H.

Chapman, H.

S. Marchesini, H. He, H. Chapman, S. Hau-Riege, A. Noy, M. Howells, U. Weierstall, and J. Spence, “X-ray image reconstruction from a diffraction pattern alone,” Phys. Rev. B 68, 140,101 (2003).
[Crossref]

Charalambous, P.

J. Miao, P. Charalambous, J. Kirz, and D. Sayre, “An extension of the methods of x-ray crystallography to allow imaging of micron-size non-crystalline specimens,” Nature 400, 342–344 (1999).
[Crossref]

Chen, C.

H. Jiang, C. Song, C. Chen, R. Xu, K. Raines, B. Fahimian, C. Lu, T. Lee, A. Nakashima, J. Urano, T. Ishikawa, F. Tamanoi, and J. Miao, “Quantitative 3D imaging of whole, unstained cells by using X-ray diffraction microscopy,” Proc. Natl. Acad. Sci. U.S.A. 107, 11,234–11,239 (2010).
[Crossref]

Chen, C.-C.

C.-C. Chen, J. Miao, C. W. Wang, and T. K. Lee, “Application of optimization technique to noncrystalline x-ray diffraction microscopy: Guided hybrid input-output method,” Phys. Rev. B 76, 064113 (2007).
[Crossref]

Combettes, P. L.

Cowley, J. M.

J. M. Cowley and A. F. Moodie, “Fourier Images. I. The Point Source,” Proc. Phys. Soc. B 70, 486–496 (1957).
[Crossref]

Davis, J. C.

B. L. Henke, E. M. Gullikson, and 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]

Elser, V.

P. Thibault, V. Elser, C. Jacobsen, D. Shapiro, and D. Sayre, “Reconstruction of a yeast cell from x-ray diffraction data,” Acta Crystallogr. A 62, 248–261 (2006).
[Crossref] [PubMed]

V. Elser, “Phase retrieval by iterated projections,” J. Opt. Soc. Am. A 20, 40–55 (2003).
[Crossref]

Fahimian, B.

H. Jiang, C. Song, C. Chen, R. Xu, K. Raines, B. Fahimian, C. Lu, T. Lee, A. Nakashima, J. Urano, T. Ishikawa, F. Tamanoi, and J. Miao, “Quantitative 3D imaging of whole, unstained cells by using X-ray diffraction microscopy,” Proc. Natl. Acad. Sci. U.S.A. 107, 11,234–11,239 (2010).
[Crossref]

Fienup, J.

Gerchberg, R. W.

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

Gullikson, E. M.

B. L. Henke, E. M. Gullikson, and 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]

Hau-Riege, S.

S. Marchesini, H. He, H. Chapman, S. Hau-Riege, A. Noy, M. Howells, U. Weierstall, and J. Spence, “X-ray image reconstruction from a diffraction pattern alone,” Phys. Rev. B 68, 140,101 (2003).
[Crossref]

He, H.

S. Marchesini, H. He, H. Chapman, S. Hau-Riege, A. Noy, M. Howells, U. Weierstall, and J. Spence, “X-ray image reconstruction from a diffraction pattern alone,” Phys. Rev. B 68, 140,101 (2003).
[Crossref]

Henke, B. L.

B. L. Henke, E. M. Gullikson, and 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.

S. Marchesini, H. He, H. Chapman, S. Hau-Riege, A. Noy, M. Howells, U. Weierstall, and J. Spence, “X-ray image reconstruction from a diffraction pattern alone,” Phys. Rev. B 68, 140,101 (2003).
[Crossref]

Huang, X.

J. Steinbrener, J. Nelson, X. Huang, S. Marchesini, D. Shapiro, J. Turner, and C. Jacobsen, “Data preparation and evaluation techniques for x-ray diffraction microscopy,” Opt. Express 18, 18,598–18,614 (2010).
[Crossref]

X. Huang, H. Miao, J. Steinbrener, J. Nelson, A. Stewart, D. Shapiro, and C. Jacobsen, “Signal to noise considerations in diffraction and conventional microscopy,” Opt. Express 17, 13,541–13,553 (2009).
[Crossref]

Ishikawa, T.

H. Jiang, C. Song, C. Chen, R. Xu, K. Raines, B. Fahimian, C. Lu, T. Lee, A. Nakashima, J. Urano, T. Ishikawa, F. Tamanoi, and J. Miao, “Quantitative 3D imaging of whole, unstained cells by using X-ray diffraction microscopy,” Proc. Natl. Acad. Sci. U.S.A. 107, 11,234–11,239 (2010).
[Crossref]

J. Miao, Y. Nishino, Y. Kohmura, B. Johnson, C. Song, S. Risbud, and T. Ishikawa, “Quantitative Image Reconstruction of GaN Quantum Dots from Oversampled Diffraction Intensities Alone,” Phys. Rev. Lett. 95, 085,503 (2005).
[Crossref]

Jacobsen, C.

J. Steinbrener, J. Nelson, X. Huang, S. Marchesini, D. Shapiro, J. Turner, and C. Jacobsen, “Data preparation and evaluation techniques for x-ray diffraction microscopy,” Opt. Express 18, 18,598–18,614 (2010).
[Crossref]

X. Huang, H. Miao, J. Steinbrener, J. Nelson, A. Stewart, D. Shapiro, and C. Jacobsen, “Signal to noise considerations in diffraction and conventional microscopy,” Opt. Express 17, 13,541–13,553 (2009).
[Crossref]

P. Thibault, V. Elser, C. Jacobsen, D. Shapiro, and D. Sayre, “Reconstruction of a yeast cell from x-ray diffraction data,” Acta Crystallogr. A 62, 248–261 (2006).
[Crossref] [PubMed]

Jiang, H.

H. Jiang, C. Song, C. Chen, R. Xu, K. Raines, B. Fahimian, C. Lu, T. Lee, A. Nakashima, J. Urano, T. Ishikawa, F. Tamanoi, and J. Miao, “Quantitative 3D imaging of whole, unstained cells by using X-ray diffraction microscopy,” Proc. Natl. Acad. Sci. U.S.A. 107, 11,234–11,239 (2010).
[Crossref]

Johnson, B.

J. Miao, Y. Nishino, Y. Kohmura, B. Johnson, C. Song, S. Risbud, and T. Ishikawa, “Quantitative Image Reconstruction of GaN Quantum Dots from Oversampled Diffraction Intensities Alone,” Phys. Rev. Lett. 95, 085,503 (2005).
[Crossref]

Kirz, J.

J. Miao, P. Charalambous, J. Kirz, and D. Sayre, “An extension of the methods of x-ray crystallography to allow imaging of micron-size non-crystalline specimens,” Nature 400, 342–344 (1999).
[Crossref]

Kohmura, Y.

J. Miao, Y. Nishino, Y. Kohmura, B. Johnson, C. Song, S. Risbud, and T. Ishikawa, “Quantitative Image Reconstruction of GaN Quantum Dots from Oversampled Diffraction Intensities Alone,” Phys. Rev. Lett. 95, 085,503 (2005).
[Crossref]

Lee, T.

H. Jiang, C. Song, C. Chen, R. Xu, K. Raines, B. Fahimian, C. Lu, T. Lee, A. Nakashima, J. Urano, T. Ishikawa, F. Tamanoi, and J. Miao, “Quantitative 3D imaging of whole, unstained cells by using X-ray diffraction microscopy,” Proc. Natl. Acad. Sci. U.S.A. 107, 11,234–11,239 (2010).
[Crossref]

Lee, T. K.

C.-C. Chen, J. Miao, C. W. Wang, and T. K. Lee, “Application of optimization technique to noncrystalline x-ray diffraction microscopy: Guided hybrid input-output method,” Phys. Rev. B 76, 064113 (2007).
[Crossref]

Leslie, A. W. G.

J. P. Abrahams and A. W. G. Leslie, “Methods used in the structure determination of bovine mitochondrial F1 ATPase,” Acta Crystallogr. D Biol. Crystallogr. 52, 30–42 (1996).
[Crossref] [PubMed]

London, R. A.

Lu, C.

H. Jiang, C. Song, C. Chen, R. Xu, K. Raines, B. Fahimian, C. Lu, T. Lee, A. Nakashima, J. Urano, T. Ishikawa, F. Tamanoi, and J. Miao, “Quantitative 3D imaging of whole, unstained cells by using X-ray diffraction microscopy,” Proc. Natl. Acad. Sci. U.S.A. 107, 11,234–11,239 (2010).
[Crossref]

Luke, D. R.

Marchesini, S.

J. Steinbrener, J. Nelson, X. Huang, S. Marchesini, D. Shapiro, J. Turner, and C. Jacobsen, “Data preparation and evaluation techniques for x-ray diffraction microscopy,” Opt. Express 18, 18,598–18,614 (2010).
[Crossref]

S. Marchesini, “A unified evaluation of iterative projection algorithms for phase retrieval,” Rev. Sci. Instrum. 78, 011,301 (2007).

S. Marchesini, H. He, H. Chapman, S. Hau-Riege, A. Noy, M. Howells, U. Weierstall, and J. Spence, “X-ray image reconstruction from a diffraction pattern alone,” Phys. Rev. B 68, 140,101 (2003).
[Crossref]

Miao, H.

X. Huang, H. Miao, J. Steinbrener, J. Nelson, A. Stewart, D. Shapiro, and C. Jacobsen, “Signal to noise considerations in diffraction and conventional microscopy,” Opt. Express 17, 13,541–13,553 (2009).
[Crossref]

Miao, J.

H. Jiang, C. Song, C. Chen, R. Xu, K. Raines, B. Fahimian, C. Lu, T. Lee, A. Nakashima, J. Urano, T. Ishikawa, F. Tamanoi, and J. Miao, “Quantitative 3D imaging of whole, unstained cells by using X-ray diffraction microscopy,” Proc. Natl. Acad. Sci. U.S.A. 107, 11,234–11,239 (2010).
[Crossref]

C.-C. Chen, J. Miao, C. W. Wang, and T. K. Lee, “Application of optimization technique to noncrystalline x-ray diffraction microscopy: Guided hybrid input-output method,” Phys. Rev. B 76, 064113 (2007).
[Crossref]

J. Miao, Y. Nishino, Y. Kohmura, B. Johnson, C. Song, S. Risbud, and T. Ishikawa, “Quantitative Image Reconstruction of GaN Quantum Dots from Oversampled Diffraction Intensities Alone,” Phys. Rev. Lett. 95, 085,503 (2005).
[Crossref]

J. Miao, P. Charalambous, J. Kirz, and D. Sayre, “An extension of the methods of x-ray crystallography to allow imaging of micron-size non-crystalline specimens,” Nature 400, 342–344 (1999).
[Crossref]

Moodie, A. F.

J. M. Cowley and A. F. Moodie, “Fourier Images. I. The Point Source,” Proc. Phys. Soc. B 70, 486–496 (1957).
[Crossref]

Nakashima, A.

H. Jiang, C. Song, C. Chen, R. Xu, K. Raines, B. Fahimian, C. Lu, T. Lee, A. Nakashima, J. Urano, T. Ishikawa, F. Tamanoi, and J. Miao, “Quantitative 3D imaging of whole, unstained cells by using X-ray diffraction microscopy,” Proc. Natl. Acad. Sci. U.S.A. 107, 11,234–11,239 (2010).
[Crossref]

Nelson, J.

J. Steinbrener, J. Nelson, X. Huang, S. Marchesini, D. Shapiro, J. Turner, and C. Jacobsen, “Data preparation and evaluation techniques for x-ray diffraction microscopy,” Opt. Express 18, 18,598–18,614 (2010).
[Crossref]

X. Huang, H. Miao, J. Steinbrener, J. Nelson, A. Stewart, D. Shapiro, and C. Jacobsen, “Signal to noise considerations in diffraction and conventional microscopy,” Opt. Express 17, 13,541–13,553 (2009).
[Crossref]

Nishino, Y.

J. Miao, Y. Nishino, Y. Kohmura, B. Johnson, C. Song, S. Risbud, and T. Ishikawa, “Quantitative Image Reconstruction of GaN Quantum Dots from Oversampled Diffraction Intensities Alone,” Phys. Rev. Lett. 95, 085,503 (2005).
[Crossref]

Noy, A.

S. Marchesini, H. He, H. Chapman, S. Hau-Riege, A. Noy, M. Howells, U. Weierstall, and J. Spence, “X-ray image reconstruction from a diffraction pattern alone,” Phys. Rev. B 68, 140,101 (2003).
[Crossref]

Pfeifer, M.

G. Williams, M. Pfeifer, I. Vartanyants, and I. Robinson, “Effectiveness of iterative algorithms in recovering phase in the presence of noise,” Acta Crystallogr. A 63, 36–42 (2007).
[Crossref]

Raines, K.

H. Jiang, C. Song, C. Chen, R. Xu, K. Raines, B. Fahimian, C. Lu, T. Lee, A. Nakashima, J. Urano, T. Ishikawa, F. Tamanoi, and J. Miao, “Quantitative 3D imaging of whole, unstained cells by using X-ray diffraction microscopy,” Proc. Natl. Acad. Sci. U.S.A. 107, 11,234–11,239 (2010).
[Crossref]

Risbud, S.

J. Miao, Y. Nishino, Y. Kohmura, B. Johnson, C. Song, S. Risbud, and T. Ishikawa, “Quantitative Image Reconstruction of GaN Quantum Dots from Oversampled Diffraction Intensities Alone,” Phys. Rev. Lett. 95, 085,503 (2005).
[Crossref]

Robinson, I.

G. Williams, M. Pfeifer, I. Vartanyants, and I. Robinson, “Effectiveness of iterative algorithms in recovering phase in the presence of noise,” Acta Crystallogr. A 63, 36–42 (2007).
[Crossref]

Rosen, M. D.

Saxton, W. O.

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

Sayre, D.

P. Thibault, V. Elser, C. Jacobsen, D. Shapiro, and D. Sayre, “Reconstruction of a yeast cell from x-ray diffraction data,” Acta Crystallogr. A 62, 248–261 (2006).
[Crossref] [PubMed]

J. Miao, P. Charalambous, J. Kirz, and D. Sayre, “An extension of the methods of x-ray crystallography to allow imaging of micron-size non-crystalline specimens,” Nature 400, 342–344 (1999).
[Crossref]

D. Sayre, “Prospects for long-wavelength X-ray microscopy and diffraction,” in Imaging Processes and Coherence in Physics, M. Schlenker, M. Fink, J. Goedgebuer, C. Malgrange, J. Viénot, and R. Wade, eds., vol. 112 of Lecture Notes in Physics, pp. 229–235 (Springer-Verlag, 1980).
[Crossref]

Shapiro, D.

J. Steinbrener, J. Nelson, X. Huang, S. Marchesini, D. Shapiro, J. Turner, and C. Jacobsen, “Data preparation and evaluation techniques for x-ray diffraction microscopy,” Opt. Express 18, 18,598–18,614 (2010).
[Crossref]

X. Huang, H. Miao, J. Steinbrener, J. Nelson, A. Stewart, D. Shapiro, and C. Jacobsen, “Signal to noise considerations in diffraction and conventional microscopy,” Opt. Express 17, 13,541–13,553 (2009).
[Crossref]

P. Thibault, V. Elser, C. Jacobsen, D. Shapiro, and D. Sayre, “Reconstruction of a yeast cell from x-ray diffraction data,” Acta Crystallogr. A 62, 248–261 (2006).
[Crossref] [PubMed]

Song, C.

H. Jiang, C. Song, C. Chen, R. Xu, K. Raines, B. Fahimian, C. Lu, T. Lee, A. Nakashima, J. Urano, T. Ishikawa, F. Tamanoi, and J. Miao, “Quantitative 3D imaging of whole, unstained cells by using X-ray diffraction microscopy,” Proc. Natl. Acad. Sci. U.S.A. 107, 11,234–11,239 (2010).
[Crossref]

J. Miao, Y. Nishino, Y. Kohmura, B. Johnson, C. Song, S. Risbud, and T. Ishikawa, “Quantitative Image Reconstruction of GaN Quantum Dots from Oversampled Diffraction Intensities Alone,” Phys. Rev. Lett. 95, 085,503 (2005).
[Crossref]

Spence, J.

S. Marchesini, H. He, H. Chapman, S. Hau-Riege, A. Noy, M. Howells, U. Weierstall, and J. Spence, “X-ray image reconstruction from a diffraction pattern alone,” Phys. Rev. B 68, 140,101 (2003).
[Crossref]

Steinbrener, J.

J. Steinbrener, J. Nelson, X. Huang, S. Marchesini, D. Shapiro, J. Turner, and C. Jacobsen, “Data preparation and evaluation techniques for x-ray diffraction microscopy,” Opt. Express 18, 18,598–18,614 (2010).
[Crossref]

X. Huang, H. Miao, J. Steinbrener, J. Nelson, A. Stewart, D. Shapiro, and C. Jacobsen, “Signal to noise considerations in diffraction and conventional microscopy,” Opt. Express 17, 13,541–13,553 (2009).
[Crossref]

Stewart, A.

X. Huang, H. Miao, J. Steinbrener, J. Nelson, A. Stewart, D. Shapiro, and C. Jacobsen, “Signal to noise considerations in diffraction and conventional microscopy,” Opt. Express 17, 13,541–13,553 (2009).
[Crossref]

Tamanoi, F.

H. Jiang, C. Song, C. Chen, R. Xu, K. Raines, B. Fahimian, C. Lu, T. Lee, A. Nakashima, J. Urano, T. Ishikawa, F. Tamanoi, and J. Miao, “Quantitative 3D imaging of whole, unstained cells by using X-ray diffraction microscopy,” Proc. Natl. Acad. Sci. U.S.A. 107, 11,234–11,239 (2010).
[Crossref]

Thibault, P.

P. Thibault, V. Elser, C. Jacobsen, D. Shapiro, and D. Sayre, “Reconstruction of a yeast cell from x-ray diffraction data,” Acta Crystallogr. A 62, 248–261 (2006).
[Crossref] [PubMed]

Trebes, J. E.

Turner, J.

J. Steinbrener, J. Nelson, X. Huang, S. Marchesini, D. Shapiro, J. Turner, and C. Jacobsen, “Data preparation and evaluation techniques for x-ray diffraction microscopy,” Opt. Express 18, 18,598–18,614 (2010).
[Crossref]

Urano, J.

H. Jiang, C. Song, C. Chen, R. Xu, K. Raines, B. Fahimian, C. Lu, T. Lee, A. Nakashima, J. Urano, T. Ishikawa, F. Tamanoi, and J. Miao, “Quantitative 3D imaging of whole, unstained cells by using X-ray diffraction microscopy,” Proc. Natl. Acad. Sci. U.S.A. 107, 11,234–11,239 (2010).
[Crossref]

Vartanyants, I.

G. Williams, M. Pfeifer, I. Vartanyants, and I. Robinson, “Effectiveness of iterative algorithms in recovering phase in the presence of noise,” Acta Crystallogr. A 63, 36–42 (2007).
[Crossref]

Wang, C. W.

C.-C. Chen, J. Miao, C. W. Wang, and T. K. Lee, “Application of optimization technique to noncrystalline x-ray diffraction microscopy: Guided hybrid input-output method,” Phys. Rev. B 76, 064113 (2007).
[Crossref]

Weierstall, U.

S. Marchesini, H. He, H. Chapman, S. Hau-Riege, A. Noy, M. Howells, U. Weierstall, and J. Spence, “X-ray image reconstruction from a diffraction pattern alone,” Phys. Rev. B 68, 140,101 (2003).
[Crossref]

Williams, G.

G. Williams, M. Pfeifer, I. Vartanyants, and I. Robinson, “Effectiveness of iterative algorithms in recovering phase in the presence of noise,” Acta Crystallogr. A 63, 36–42 (2007).
[Crossref]

Xu, R.

H. Jiang, C. Song, C. Chen, R. Xu, K. Raines, B. Fahimian, C. Lu, T. Lee, A. Nakashima, J. Urano, T. Ishikawa, F. Tamanoi, and J. Miao, “Quantitative 3D imaging of whole, unstained cells by using X-ray diffraction microscopy,” Proc. Natl. Acad. Sci. U.S.A. 107, 11,234–11,239 (2010).
[Crossref]

Acta Crystallogr. A (2)

G. Williams, M. Pfeifer, I. Vartanyants, and I. Robinson, “Effectiveness of iterative algorithms in recovering phase in the presence of noise,” Acta Crystallogr. A 63, 36–42 (2007).
[Crossref]

P. Thibault, V. Elser, C. Jacobsen, D. Shapiro, and D. Sayre, “Reconstruction of a yeast cell from x-ray diffraction data,” Acta Crystallogr. A 62, 248–261 (2006).
[Crossref] [PubMed]

Acta Crystallogr. D Biol. Crystallogr. (1)

J. P. Abrahams and A. W. G. Leslie, “Methods used in the structure determination of bovine mitochondrial F1 ATPase,” Acta Crystallogr. D Biol. Crystallogr. 52, 30–42 (1996).
[Crossref] [PubMed]

Appl. Opt. (2)

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B. L. Henke, E. M. Gullikson, and 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]

Inverse Probl. (1)

D. R. Luke, “Relaxed averaged alternating reflections for diffraction imaging,” Inverse Probl. 21, 37–50 (2005).
[Crossref]

J. Opt. Soc. Am. A (3)

Nature (1)

J. Miao, P. Charalambous, J. Kirz, and D. Sayre, “An extension of the methods of x-ray crystallography to allow imaging of micron-size non-crystalline specimens,” Nature 400, 342–344 (1999).
[Crossref]

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X. Huang, H. Miao, J. Steinbrener, J. Nelson, A. Stewart, D. Shapiro, and C. Jacobsen, “Signal to noise considerations in diffraction and conventional microscopy,” Opt. Express 17, 13,541–13,553 (2009).
[Crossref]

J. Steinbrener, J. Nelson, X. Huang, S. Marchesini, D. Shapiro, J. Turner, and C. Jacobsen, “Data preparation and evaluation techniques for x-ray diffraction microscopy,” Opt. Express 18, 18,598–18,614 (2010).
[Crossref]

Optik (2)

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

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Phys. Rev. B (2)

S. Marchesini, H. He, H. Chapman, S. Hau-Riege, A. Noy, M. Howells, U. Weierstall, and J. Spence, “X-ray image reconstruction from a diffraction pattern alone,” Phys. Rev. B 68, 140,101 (2003).
[Crossref]

C.-C. Chen, J. Miao, C. W. Wang, and T. K. Lee, “Application of optimization technique to noncrystalline x-ray diffraction microscopy: Guided hybrid input-output method,” Phys. Rev. B 76, 064113 (2007).
[Crossref]

Phys. Rev. Lett. (1)

J. Miao, Y. Nishino, Y. Kohmura, B. Johnson, C. Song, S. Risbud, and T. Ishikawa, “Quantitative Image Reconstruction of GaN Quantum Dots from Oversampled Diffraction Intensities Alone,” Phys. Rev. Lett. 95, 085,503 (2005).
[Crossref]

Proc. Natl. Acad. Sci. U.S.A. (1)

H. Jiang, C. Song, C. Chen, R. Xu, K. Raines, B. Fahimian, C. Lu, T. Lee, A. Nakashima, J. Urano, T. Ishikawa, F. Tamanoi, and J. Miao, “Quantitative 3D imaging of whole, unstained cells by using X-ray diffraction microscopy,” Proc. Natl. Acad. Sci. U.S.A. 107, 11,234–11,239 (2010).
[Crossref]

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J. M. Cowley and A. F. Moodie, “Fourier Images. I. The Point Source,” Proc. Phys. Soc. B 70, 486–496 (1957).
[Crossref]

Rev. Sci. Instrum. (1)

S. Marchesini, “A unified evaluation of iterative projection algorithms for phase retrieval,” Rev. Sci. Instrum. 78, 011,301 (2007).

Other (1)

D. Sayre, “Prospects for long-wavelength X-ray microscopy and diffraction,” in Imaging Processes and Coherence in Physics, M. Schlenker, M. Fink, J. Goedgebuer, C. Malgrange, J. Viénot, and R. Wade, eds., vol. 112 of Lecture Notes in Physics, pp. 229–235 (Springer-Verlag, 1980).
[Crossref]

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Figures (7)

Fig. 1
Fig. 1 (a) The exit wave magnitude of the simulated fake cell. (b) Simulated diffraction pattern with photon noise and 2×2 pixel missing center.

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Fig. 2
Fig. 2 Illustration of the various support errors studied. A blue area denotes a region added to the correct support. A red area denotes a region removed from the correct support. (a) Magnitude of a reconstructed cell image from diffraction pattern with 2×2 pixel missing center. (b) The correct support. (c) The “bump-out” support generated by including 1439 pixels (out of 400×400) outside the correct support at a local area. (d) The “bite-in” support generated by excluding 1382 pixels inside the correct support at a local area. (e) The “loose” support generated by increasing the size of the correct support uniformly by 2 pixels which adds 1477 pixels. (f) The “tight” support generated by reducing the correct support size uniformly by 2 pixels which removes 1439 pixels.

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Fig. 3
Fig. 3 Reconstructed image magnitudes with HIO, DM, RAAR and ER algorithms using correct, “bump-out”, “bite-in”, “loose” and “tight” supports. Only the central part of images are displayed.

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Fig. 4
Fig. 4 Wiener-filtered phase retrieval transfer functions: (a) from DM reconstructions with different supports, (b) from different algorithms with the correct support.

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Fig. 5
Fig. 5 Magnitudes of reconstructed images of the σ = 4.6 simulated cell from different algorithms with varying sizes of missing centers. Only the central part of diffraction patterns and reconstructed images are displayed. The green line indicates where artifacts becomes noticeable in reconstructed images.

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Fig. 6
Fig. 6 Image magnitudes before and after unconstrained modes were removed from HIO reconstructions with various missing center sizes for σ = 4.6 cell (top row) and σ = 7.5 cell (bottom row). Up to 13 missing speckles, the artifacts in reconstructed images are negligible in both cases. Unconstrained mode removal works well up to 21 missing speckles for the σ = 4.6 cell, and 30 for the σ = 7.5 cell. Above that level, artifacts cannot be completely removed. When the missing speckle number is greater than 48, permanent artifacts are present in images in both cases.

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Fig. 7
Fig. 7 Measurements of the effects of missing central speckles in the measured diffraction intensities. Shown here are both the signal-to-noise ratio with unconstrained modes removed (red curve), and Rreal calculated before (grey curve) and after (black curve) removal of unconstrained missing modes. The results are shown for the same size simulated cell in a smaller array (σ = 4.6) at left, and a larger array (σ = 7.5) at right.

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Tables (1)

Tables Icon

Table 1 Real space r factor Rreal and signal-to-noise ratio SNR of reconstructed images with different supports. Smaller Rreal and larger SNR indicate more reliable reconstructions. The best image quality is highlighted for each case

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Equations (2)

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R real = 〈 ( M 1 − M 2 ) 2 〉 〈 ( M 1 + M 2 ) 2 〉 ,
SNR = r 1 − r , with   r   = 〈 ( I 1 − 〈 I 1 〉 ) ( I 2 − 〈 I 2 〉 ) * 〉 〈 ( I 1 − 〈 I 1 〉 ) 2 〈 ( I 2 − 〈 I 2 〉 ) 2 〉 ,

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