Abstract

For coherent X-ray imaging of pure phase objects we study the reliability of linear relations in phase-retrieval algorithms based on a single intensity map after free-space propagation. For large phase changes and/or large propagation distances we propose two venues of working beyond linearity: Projection onto an effective, linear and local model in Fourier space and expansion of intensity contrast in powers of object-detector distance. We apply both algorithms successfully to simulated data.

© 2011 OSA

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  1. B. Henke, E. Gullikson, and J. Davis, “X-ray interactions: photoabsorption, scattering, transmission, and reflection at E=50-30000 eV, z=1–92,” At. Data. Nucl. Data Tables 54, 181–342 (1993).
    [CrossRef]
  2. A. Snigirev, I. Snigireva, V. Kohn, S. Kuznetsov, and I. Schelokov, “On the possibilities of X-ray phase contrast microimaging by coherent high-energy synchrotron radiation,” Rev. Sci. Instrum. 66, 5486–5492 (1995).
    [CrossRef]
  3. P. Cloetens, “Contribution to phase contrast imaging, reconstruction and tomography with hard synchrotron radiation,” PhD dissertation, Vrije Universiteit Brussel (1999).
  4. S. W. Wilkins, T. E. Gureyev, D. Gao, A. Pogany, and A. W. Stevenson, “Phase-contrast imaging using polychromatic hard X-rays,” Nature 384, 335–338 (1996).
    [CrossRef]
  5. K. A. Nugent, T. E. Gureyev, D. F. Cookson, D. M. Paganin, and Z. Barnea, “Quantitative phase imaging using hard X rays,” Phys. Rev. Lett. 77, 2961–2964 (1996).
    [CrossRef] [PubMed]
  6. A. Momose and J. Fukuda, “Phase-contrast radiographs of nonstained rat cerebellar specimen,” Med. Phys. 22, 375–379 (1995).
    [CrossRef] [PubMed]
  7. The cone-beam case relates to the parallel-beam case via a simple rescaling operation, see D. M. Paganin, Coherent X-Ray Optics (Oxford University Press, 2006).
    [CrossRef]
  8. M. R. Teague, “Deterministic phase retrieval: a Greens function solution,” J. Opt. Soc. Am. 73, 1434–1441 (1983).
    [CrossRef]
  9. J.-P. Guigay, “Fourier transform analysis of Fresnel diffraction patterns and in-line holograms,” Optik 49, 121–125 (1977).
  10. P. Cloetens, W. Ludwig, J. Baruchel, J.-P. Guigay, P. Rejmankova-Pernot, M. Salome, M. Schlenker, J. Y. Buffiere, E. Maire, and G. Peix, “Hard X-ray phase imaging using simple propagation of a coherent synchrotron radiation beam,” J. Phys. D: Appl. Phys. 32, A145–A151 (1999).
    [CrossRef]
  11. L. Turner, B. Dhal, J. Hayes, A. Mancuso, K. Nugent, D. Paterson, R. Scholten, C. Tran, and A. Peele, “X-ray phase imaging: Demonstration of extended conditions for homogeneous objects,” Opt. Express 12, 2960–2965 (2004).
    [CrossRef] [PubMed]
  12. T. E. Gureyev, D. M. Paganin, G. R. Myers, Y. I. Nesterets, and S. W. Wilkins, “Phase-and-amplitude computer tomography,” Appl. Phys. Lett. 89, 034102 (2006).
    [CrossRef]
  13. T. E. Gureyev, Y. Nesterets, D. Paganin, A. Pogany, and S. Wilkins, “Linear algorithms for phase retrieval in the Fresnel region. 2. partially coherent illumination,” Opt. Commun. 259, 569–580 (2006).
    [CrossRef]
  14. S. Zabler, P. Cloetens, J.-P. Guigay, J. Baruchel, and M. Schlenker, “Optimization of phase contrast imaging using hard X rays,” Rev. Sci. Instrum. 76, 073705 (2005).
    [CrossRef]
  15. L. Waller, L. Tian, and G. Barbastathis, “Transport of intensity phase-amplitude imaging with higher order intensity derivatives,” Opt. Express 18, 12552–12561 (2010).
    [CrossRef] [PubMed]
  16. J. Moosmann, R. Hofmann, A. V. Bronnikov, and T. Baumbach, “Nonlinear phase retrieval from single-distance radiograph,” Opt. Express 18, 25771–25785 (2010).
    [CrossRef] [PubMed]
  17. J. Moosmann, R. Hofmann, and T. Baumbach, “Nonlinear, single-distance phase retrieval and Schwinger regularization,” To appear in Phys. Status. Solidi A.
  18. L. D. Landau, “The theory of a Fermi liquid,” Sov. Phys. JETP 3, 920–925 (1957).
  19. M. Langer, P. Cloetens, and F. Peyrin, “Regularization of phase retrieval with phase-attenuation duality prior for 3-D holotomography,” IEEE Transactions on Image Processing 19, 2429–2436 (2010).
    [CrossRef]
  20. J. R. Fienup, “Phase retrieval algorithms: a comparison,” Appl. Opt. 21, 2758–2769 (1982).
    [CrossRef] [PubMed]
  21. V. Altapova, D. Hänschke, J. Moosmann, R. Hofmann, and T. Baumbach, “Imaging in evolutionary biology: Single-distance phase contrast, nonlinear-noniterative retrieval, and tomographic reconstruction,” In preparation (2011).

2010 (3)

2006 (2)

T. E. Gureyev, D. M. Paganin, G. R. Myers, Y. I. Nesterets, and S. W. Wilkins, “Phase-and-amplitude computer tomography,” Appl. Phys. Lett. 89, 034102 (2006).
[CrossRef]

T. E. Gureyev, Y. Nesterets, D. Paganin, A. Pogany, and S. Wilkins, “Linear algorithms for phase retrieval in the Fresnel region. 2. partially coherent illumination,” Opt. Commun. 259, 569–580 (2006).
[CrossRef]

2005 (1)

S. Zabler, P. Cloetens, J.-P. Guigay, J. Baruchel, and M. Schlenker, “Optimization of phase contrast imaging using hard X rays,” Rev. Sci. Instrum. 76, 073705 (2005).
[CrossRef]

2004 (1)

1999 (1)

P. Cloetens, W. Ludwig, J. Baruchel, J.-P. Guigay, P. Rejmankova-Pernot, M. Salome, M. Schlenker, J. Y. Buffiere, E. Maire, and G. Peix, “Hard X-ray phase imaging using simple propagation of a coherent synchrotron radiation beam,” J. Phys. D: Appl. Phys. 32, A145–A151 (1999).
[CrossRef]

1996 (2)

S. W. Wilkins, T. E. Gureyev, D. Gao, A. Pogany, and A. W. Stevenson, “Phase-contrast imaging using polychromatic hard X-rays,” Nature 384, 335–338 (1996).
[CrossRef]

K. A. Nugent, T. E. Gureyev, D. F. Cookson, D. M. Paganin, and Z. Barnea, “Quantitative phase imaging using hard X rays,” Phys. Rev. Lett. 77, 2961–2964 (1996).
[CrossRef] [PubMed]

1995 (2)

A. Momose and J. Fukuda, “Phase-contrast radiographs of nonstained rat cerebellar specimen,” Med. Phys. 22, 375–379 (1995).
[CrossRef] [PubMed]

A. Snigirev, I. Snigireva, V. Kohn, S. Kuznetsov, and I. Schelokov, “On the possibilities of X-ray phase contrast microimaging by coherent high-energy synchrotron radiation,” Rev. Sci. Instrum. 66, 5486–5492 (1995).
[CrossRef]

1993 (1)

B. Henke, E. Gullikson, and J. Davis, “X-ray interactions: photoabsorption, scattering, transmission, and reflection at E=50-30000 eV, z=1–92,” At. Data. Nucl. Data Tables 54, 181–342 (1993).
[CrossRef]

1983 (1)

1982 (1)

1977 (1)

J.-P. Guigay, “Fourier transform analysis of Fresnel diffraction patterns and in-line holograms,” Optik 49, 121–125 (1977).

1957 (1)

L. D. Landau, “The theory of a Fermi liquid,” Sov. Phys. JETP 3, 920–925 (1957).

Barbastathis, G.

Barnea, Z.

K. A. Nugent, T. E. Gureyev, D. F. Cookson, D. M. Paganin, and Z. Barnea, “Quantitative phase imaging using hard X rays,” Phys. Rev. Lett. 77, 2961–2964 (1996).
[CrossRef] [PubMed]

Baruchel, J.

S. Zabler, P. Cloetens, J.-P. Guigay, J. Baruchel, and M. Schlenker, “Optimization of phase contrast imaging using hard X rays,” Rev. Sci. Instrum. 76, 073705 (2005).
[CrossRef]

P. Cloetens, W. Ludwig, J. Baruchel, J.-P. Guigay, P. Rejmankova-Pernot, M. Salome, M. Schlenker, J. Y. Buffiere, E. Maire, and G. Peix, “Hard X-ray phase imaging using simple propagation of a coherent synchrotron radiation beam,” J. Phys. D: Appl. Phys. 32, A145–A151 (1999).
[CrossRef]

Baumbach, T.

Bronnikov, A. V.

Buffiere, J. Y.

P. Cloetens, W. Ludwig, J. Baruchel, J.-P. Guigay, P. Rejmankova-Pernot, M. Salome, M. Schlenker, J. Y. Buffiere, E. Maire, and G. Peix, “Hard X-ray phase imaging using simple propagation of a coherent synchrotron radiation beam,” J. Phys. D: Appl. Phys. 32, A145–A151 (1999).
[CrossRef]

Cloetens, P.

M. Langer, P. Cloetens, and F. Peyrin, “Regularization of phase retrieval with phase-attenuation duality prior for 3-D holotomography,” IEEE Transactions on Image Processing 19, 2429–2436 (2010).
[CrossRef]

S. Zabler, P. Cloetens, J.-P. Guigay, J. Baruchel, and M. Schlenker, “Optimization of phase contrast imaging using hard X rays,” Rev. Sci. Instrum. 76, 073705 (2005).
[CrossRef]

P. Cloetens, W. Ludwig, J. Baruchel, J.-P. Guigay, P. Rejmankova-Pernot, M. Salome, M. Schlenker, J. Y. Buffiere, E. Maire, and G. Peix, “Hard X-ray phase imaging using simple propagation of a coherent synchrotron radiation beam,” J. Phys. D: Appl. Phys. 32, A145–A151 (1999).
[CrossRef]

Cookson, D. F.

K. A. Nugent, T. E. Gureyev, D. F. Cookson, D. M. Paganin, and Z. Barnea, “Quantitative phase imaging using hard X rays,” Phys. Rev. Lett. 77, 2961–2964 (1996).
[CrossRef] [PubMed]

Davis, J.

B. Henke, E. Gullikson, and J. Davis, “X-ray interactions: photoabsorption, scattering, transmission, and reflection at E=50-30000 eV, z=1–92,” At. Data. Nucl. Data Tables 54, 181–342 (1993).
[CrossRef]

Dhal, B.

Fienup, J. R.

Fukuda, J.

A. Momose and J. Fukuda, “Phase-contrast radiographs of nonstained rat cerebellar specimen,” Med. Phys. 22, 375–379 (1995).
[CrossRef] [PubMed]

Gao, D.

S. W. Wilkins, T. E. Gureyev, D. Gao, A. Pogany, and A. W. Stevenson, “Phase-contrast imaging using polychromatic hard X-rays,” Nature 384, 335–338 (1996).
[CrossRef]

Guigay, J.-P.

S. Zabler, P. Cloetens, J.-P. Guigay, J. Baruchel, and M. Schlenker, “Optimization of phase contrast imaging using hard X rays,” Rev. Sci. Instrum. 76, 073705 (2005).
[CrossRef]

P. Cloetens, W. Ludwig, J. Baruchel, J.-P. Guigay, P. Rejmankova-Pernot, M. Salome, M. Schlenker, J. Y. Buffiere, E. Maire, and G. Peix, “Hard X-ray phase imaging using simple propagation of a coherent synchrotron radiation beam,” J. Phys. D: Appl. Phys. 32, A145–A151 (1999).
[CrossRef]

J.-P. Guigay, “Fourier transform analysis of Fresnel diffraction patterns and in-line holograms,” Optik 49, 121–125 (1977).

Gullikson, E.

B. Henke, E. Gullikson, and J. Davis, “X-ray interactions: photoabsorption, scattering, transmission, and reflection at E=50-30000 eV, z=1–92,” At. Data. Nucl. Data Tables 54, 181–342 (1993).
[CrossRef]

Gureyev, T. E.

T. E. Gureyev, D. M. Paganin, G. R. Myers, Y. I. Nesterets, and S. W. Wilkins, “Phase-and-amplitude computer tomography,” Appl. Phys. Lett. 89, 034102 (2006).
[CrossRef]

T. E. Gureyev, Y. Nesterets, D. Paganin, A. Pogany, and S. Wilkins, “Linear algorithms for phase retrieval in the Fresnel region. 2. partially coherent illumination,” Opt. Commun. 259, 569–580 (2006).
[CrossRef]

K. A. Nugent, T. E. Gureyev, D. F. Cookson, D. M. Paganin, and Z. Barnea, “Quantitative phase imaging using hard X rays,” Phys. Rev. Lett. 77, 2961–2964 (1996).
[CrossRef] [PubMed]

S. W. Wilkins, T. E. Gureyev, D. Gao, A. Pogany, and A. W. Stevenson, “Phase-contrast imaging using polychromatic hard X-rays,” Nature 384, 335–338 (1996).
[CrossRef]

Hayes, J.

Henke, B.

B. Henke, E. Gullikson, and J. Davis, “X-ray interactions: photoabsorption, scattering, transmission, and reflection at E=50-30000 eV, z=1–92,” At. Data. Nucl. Data Tables 54, 181–342 (1993).
[CrossRef]

Hofmann, R.

Kohn, V.

A. Snigirev, I. Snigireva, V. Kohn, S. Kuznetsov, and I. Schelokov, “On the possibilities of X-ray phase contrast microimaging by coherent high-energy synchrotron radiation,” Rev. Sci. Instrum. 66, 5486–5492 (1995).
[CrossRef]

Kuznetsov, S.

A. Snigirev, I. Snigireva, V. Kohn, S. Kuznetsov, and I. Schelokov, “On the possibilities of X-ray phase contrast microimaging by coherent high-energy synchrotron radiation,” Rev. Sci. Instrum. 66, 5486–5492 (1995).
[CrossRef]

Landau, L. D.

L. D. Landau, “The theory of a Fermi liquid,” Sov. Phys. JETP 3, 920–925 (1957).

Langer, M.

M. Langer, P. Cloetens, and F. Peyrin, “Regularization of phase retrieval with phase-attenuation duality prior for 3-D holotomography,” IEEE Transactions on Image Processing 19, 2429–2436 (2010).
[CrossRef]

Ludwig, W.

P. Cloetens, W. Ludwig, J. Baruchel, J.-P. Guigay, P. Rejmankova-Pernot, M. Salome, M. Schlenker, J. Y. Buffiere, E. Maire, and G. Peix, “Hard X-ray phase imaging using simple propagation of a coherent synchrotron radiation beam,” J. Phys. D: Appl. Phys. 32, A145–A151 (1999).
[CrossRef]

Maire, E.

P. Cloetens, W. Ludwig, J. Baruchel, J.-P. Guigay, P. Rejmankova-Pernot, M. Salome, M. Schlenker, J. Y. Buffiere, E. Maire, and G. Peix, “Hard X-ray phase imaging using simple propagation of a coherent synchrotron radiation beam,” J. Phys. D: Appl. Phys. 32, A145–A151 (1999).
[CrossRef]

Mancuso, A.

Momose, A.

A. Momose and J. Fukuda, “Phase-contrast radiographs of nonstained rat cerebellar specimen,” Med. Phys. 22, 375–379 (1995).
[CrossRef] [PubMed]

Moosmann, J.

Myers, G. R.

T. E. Gureyev, D. M. Paganin, G. R. Myers, Y. I. Nesterets, and S. W. Wilkins, “Phase-and-amplitude computer tomography,” Appl. Phys. Lett. 89, 034102 (2006).
[CrossRef]

Nesterets, Y.

T. E. Gureyev, Y. Nesterets, D. Paganin, A. Pogany, and S. Wilkins, “Linear algorithms for phase retrieval in the Fresnel region. 2. partially coherent illumination,” Opt. Commun. 259, 569–580 (2006).
[CrossRef]

Nesterets, Y. I.

T. E. Gureyev, D. M. Paganin, G. R. Myers, Y. I. Nesterets, and S. W. Wilkins, “Phase-and-amplitude computer tomography,” Appl. Phys. Lett. 89, 034102 (2006).
[CrossRef]

Nugent, K.

Nugent, K. A.

K. A. Nugent, T. E. Gureyev, D. F. Cookson, D. M. Paganin, and Z. Barnea, “Quantitative phase imaging using hard X rays,” Phys. Rev. Lett. 77, 2961–2964 (1996).
[CrossRef] [PubMed]

Paganin, D.

T. E. Gureyev, Y. Nesterets, D. Paganin, A. Pogany, and S. Wilkins, “Linear algorithms for phase retrieval in the Fresnel region. 2. partially coherent illumination,” Opt. Commun. 259, 569–580 (2006).
[CrossRef]

Paganin, D. M.

T. E. Gureyev, D. M. Paganin, G. R. Myers, Y. I. Nesterets, and S. W. Wilkins, “Phase-and-amplitude computer tomography,” Appl. Phys. Lett. 89, 034102 (2006).
[CrossRef]

K. A. Nugent, T. E. Gureyev, D. F. Cookson, D. M. Paganin, and Z. Barnea, “Quantitative phase imaging using hard X rays,” Phys. Rev. Lett. 77, 2961–2964 (1996).
[CrossRef] [PubMed]

The cone-beam case relates to the parallel-beam case via a simple rescaling operation, see D. M. Paganin, Coherent X-Ray Optics (Oxford University Press, 2006).
[CrossRef]

Paterson, D.

Peele, A.

Peix, G.

P. Cloetens, W. Ludwig, J. Baruchel, J.-P. Guigay, P. Rejmankova-Pernot, M. Salome, M. Schlenker, J. Y. Buffiere, E. Maire, and G. Peix, “Hard X-ray phase imaging using simple propagation of a coherent synchrotron radiation beam,” J. Phys. D: Appl. Phys. 32, A145–A151 (1999).
[CrossRef]

Peyrin, F.

M. Langer, P. Cloetens, and F. Peyrin, “Regularization of phase retrieval with phase-attenuation duality prior for 3-D holotomography,” IEEE Transactions on Image Processing 19, 2429–2436 (2010).
[CrossRef]

Pogany, A.

T. E. Gureyev, Y. Nesterets, D. Paganin, A. Pogany, and S. Wilkins, “Linear algorithms for phase retrieval in the Fresnel region. 2. partially coherent illumination,” Opt. Commun. 259, 569–580 (2006).
[CrossRef]

S. W. Wilkins, T. E. Gureyev, D. Gao, A. Pogany, and A. W. Stevenson, “Phase-contrast imaging using polychromatic hard X-rays,” Nature 384, 335–338 (1996).
[CrossRef]

Rejmankova-Pernot, P.

P. Cloetens, W. Ludwig, J. Baruchel, J.-P. Guigay, P. Rejmankova-Pernot, M. Salome, M. Schlenker, J. Y. Buffiere, E. Maire, and G. Peix, “Hard X-ray phase imaging using simple propagation of a coherent synchrotron radiation beam,” J. Phys. D: Appl. Phys. 32, A145–A151 (1999).
[CrossRef]

Salome, M.

P. Cloetens, W. Ludwig, J. Baruchel, J.-P. Guigay, P. Rejmankova-Pernot, M. Salome, M. Schlenker, J. Y. Buffiere, E. Maire, and G. Peix, “Hard X-ray phase imaging using simple propagation of a coherent synchrotron radiation beam,” J. Phys. D: Appl. Phys. 32, A145–A151 (1999).
[CrossRef]

Schelokov, I.

A. Snigirev, I. Snigireva, V. Kohn, S. Kuznetsov, and I. Schelokov, “On the possibilities of X-ray phase contrast microimaging by coherent high-energy synchrotron radiation,” Rev. Sci. Instrum. 66, 5486–5492 (1995).
[CrossRef]

Schlenker, M.

S. Zabler, P. Cloetens, J.-P. Guigay, J. Baruchel, and M. Schlenker, “Optimization of phase contrast imaging using hard X rays,” Rev. Sci. Instrum. 76, 073705 (2005).
[CrossRef]

P. Cloetens, W. Ludwig, J. Baruchel, J.-P. Guigay, P. Rejmankova-Pernot, M. Salome, M. Schlenker, J. Y. Buffiere, E. Maire, and G. Peix, “Hard X-ray phase imaging using simple propagation of a coherent synchrotron radiation beam,” J. Phys. D: Appl. Phys. 32, A145–A151 (1999).
[CrossRef]

Scholten, R.

Snigirev, A.

A. Snigirev, I. Snigireva, V. Kohn, S. Kuznetsov, and I. Schelokov, “On the possibilities of X-ray phase contrast microimaging by coherent high-energy synchrotron radiation,” Rev. Sci. Instrum. 66, 5486–5492 (1995).
[CrossRef]

Snigireva, I.

A. Snigirev, I. Snigireva, V. Kohn, S. Kuznetsov, and I. Schelokov, “On the possibilities of X-ray phase contrast microimaging by coherent high-energy synchrotron radiation,” Rev. Sci. Instrum. 66, 5486–5492 (1995).
[CrossRef]

Stevenson, A. W.

S. W. Wilkins, T. E. Gureyev, D. Gao, A. Pogany, and A. W. Stevenson, “Phase-contrast imaging using polychromatic hard X-rays,” Nature 384, 335–338 (1996).
[CrossRef]

Teague, M. R.

Tian, L.

Tran, C.

Turner, L.

Waller, L.

Wilkins, S.

T. E. Gureyev, Y. Nesterets, D. Paganin, A. Pogany, and S. Wilkins, “Linear algorithms for phase retrieval in the Fresnel region. 2. partially coherent illumination,” Opt. Commun. 259, 569–580 (2006).
[CrossRef]

Wilkins, S. W.

T. E. Gureyev, D. M. Paganin, G. R. Myers, Y. I. Nesterets, and S. W. Wilkins, “Phase-and-amplitude computer tomography,” Appl. Phys. Lett. 89, 034102 (2006).
[CrossRef]

S. W. Wilkins, T. E. Gureyev, D. Gao, A. Pogany, and A. W. Stevenson, “Phase-contrast imaging using polychromatic hard X-rays,” Nature 384, 335–338 (1996).
[CrossRef]

Zabler, S.

S. Zabler, P. Cloetens, J.-P. Guigay, J. Baruchel, and M. Schlenker, “Optimization of phase contrast imaging using hard X rays,” Rev. Sci. Instrum. 76, 073705 (2005).
[CrossRef]

Appl. Opt. (1)

Appl. Phys. Lett. (1)

T. E. Gureyev, D. M. Paganin, G. R. Myers, Y. I. Nesterets, and S. W. Wilkins, “Phase-and-amplitude computer tomography,” Appl. Phys. Lett. 89, 034102 (2006).
[CrossRef]

At. Data. Nucl. Data Tables (1)

B. Henke, E. Gullikson, and J. Davis, “X-ray interactions: photoabsorption, scattering, transmission, and reflection at E=50-30000 eV, z=1–92,” At. Data. Nucl. Data Tables 54, 181–342 (1993).
[CrossRef]

IEEE Transactions on Image Processing (1)

M. Langer, P. Cloetens, and F. Peyrin, “Regularization of phase retrieval with phase-attenuation duality prior for 3-D holotomography,” IEEE Transactions on Image Processing 19, 2429–2436 (2010).
[CrossRef]

J. Opt. Soc. Am. (1)

J. Phys. D: Appl. Phys. (1)

P. Cloetens, W. Ludwig, J. Baruchel, J.-P. Guigay, P. Rejmankova-Pernot, M. Salome, M. Schlenker, J. Y. Buffiere, E. Maire, and G. Peix, “Hard X-ray phase imaging using simple propagation of a coherent synchrotron radiation beam,” J. Phys. D: Appl. Phys. 32, A145–A151 (1999).
[CrossRef]

Med. Phys. (1)

A. Momose and J. Fukuda, “Phase-contrast radiographs of nonstained rat cerebellar specimen,” Med. Phys. 22, 375–379 (1995).
[CrossRef] [PubMed]

Nature (1)

S. W. Wilkins, T. E. Gureyev, D. Gao, A. Pogany, and A. W. Stevenson, “Phase-contrast imaging using polychromatic hard X-rays,” Nature 384, 335–338 (1996).
[CrossRef]

Opt. Commun. (1)

T. E. Gureyev, Y. Nesterets, D. Paganin, A. Pogany, and S. Wilkins, “Linear algorithms for phase retrieval in the Fresnel region. 2. partially coherent illumination,” Opt. Commun. 259, 569–580 (2006).
[CrossRef]

Opt. Express (3)

Optik (1)

J.-P. Guigay, “Fourier transform analysis of Fresnel diffraction patterns and in-line holograms,” Optik 49, 121–125 (1977).

Phys. Rev. Lett. (1)

K. A. Nugent, T. E. Gureyev, D. F. Cookson, D. M. Paganin, and Z. Barnea, “Quantitative phase imaging using hard X rays,” Phys. Rev. Lett. 77, 2961–2964 (1996).
[CrossRef] [PubMed]

Rev. Sci. Instrum. (2)

A. Snigirev, I. Snigireva, V. Kohn, S. Kuznetsov, and I. Schelokov, “On the possibilities of X-ray phase contrast microimaging by coherent high-energy synchrotron radiation,” Rev. Sci. Instrum. 66, 5486–5492 (1995).
[CrossRef]

S. Zabler, P. Cloetens, J.-P. Guigay, J. Baruchel, and M. Schlenker, “Optimization of phase contrast imaging using hard X rays,” Rev. Sci. Instrum. 76, 073705 (2005).
[CrossRef]

Sov. Phys. JETP (1)

L. D. Landau, “The theory of a Fermi liquid,” Sov. Phys. JETP 3, 920–925 (1957).

Other (4)

V. Altapova, D. Hänschke, J. Moosmann, R. Hofmann, and T. Baumbach, “Imaging in evolutionary biology: Single-distance phase contrast, nonlinear-noniterative retrieval, and tomographic reconstruction,” In preparation (2011).

J. Moosmann, R. Hofmann, and T. Baumbach, “Nonlinear, single-distance phase retrieval and Schwinger regularization,” To appear in Phys. Status. Solidi A.

P. Cloetens, “Contribution to phase contrast imaging, reconstruction and tomography with hard synchrotron radiation,” PhD dissertation, Vrije Universiteit Brussel (1999).

The cone-beam case relates to the parallel-beam case via a simple rescaling operation, see D. M. Paganin, Coherent X-Ray Optics (Oxford University Press, 2006).
[CrossRef]

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Figures (5)

Fig. 1
Fig. 1

(a) exact 2D zero-padded phase map (Lena test pattern) as input for intensity computation by free-space propagation; (b), (c), and (d): intensity contrast in Fourier space, log |ℱ gz |(ξ⃗), with maximal phase shift ϕ max for E = 10keV and z = 50cm. The colorbar relates to (b), (c), and (d) only. The effective linear pixel size Δξ in Fourier space is given as Δ ξ = 1 M Δ x where Δx denotes the effective linear pixel size of the detector, and M is the linear number of pixels. The yellow line in (a) depicts the line cut relating to the presentations in Figs. 2(a) and 2(b). The linear extents of the 2D position-space maps in Figs. 2,3,4, and 5 are all set by the yellow line in (a).

Fig. 2
Fig. 2

Linear phase retrieval from the computed intensity associated with Figs. 1(b), 1(c), see (a), (b), and 1(d), see (c), (d). Since the retrieved phase is undetermined up to an additive constant we subtract mean values to compare exact and retrieved phase. For ϕ max = 6, see Fig. 2(c), a line-cut presentation of CTF as in Figs. 2(a), 2(b) no longer is adequate since the texture fluctuations have a very large amplitude, compare with 2(d) where linearized TIE was applied yielding reasonable retrieval at reduced resolution.

Fig. 3
Fig. 3

2D phase maps from linear retrieval subject to intensity computed at ϕ max = 1. Intensity is subject to Poisson noise.

Fig. 4
Fig. 4

Intensity contrast in Fourier space after binary filtering with threshold ε and retrieved phase maps at ϕ max = 1, z = 50cm, E = 10keV. Intensity is subject to Poisson noise.

Fig. 5
Fig. 5

2D maps of the modulus of the difference between exact and retrieved phase. Parameter values are ϕ max = 6, z = 50cm, E = 10keV, and intensity is subject to Poisson noise. The colorbar applies to all images.

Equations (7)

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g z ( r ) z k 2 ϕ z = 0 ( r ) .
| ϕ z = 0 ( r π z k ξ ) ϕ z = 0 ( r + π z k ξ ) | 1 ,
( I z ) ( ξ ) = d 2 r exp ( 2 π i r ξ ) ψ z = 0 ( r π z k ξ ) ψ z = 0 * ( r + π z k ξ )
g z ( r ) 2 [ i = 1 1 ( 2 i 1 ) ! ( z 2 k ) 2 i 1 ( 2 ) 2 i 1 ] ϕ z = 0 ( r ) .
( g z ) ( ξ ) = 2 sin ( 2 π 2 z k ξ 2 ) ( ϕ z = 0 ) ( ξ ) ,
g z ( r ) z k 2 ϕ z = 0 ( r ) + z 2 2 k 2 [ ( 2 ϕ z = 0 ) 2 + 1 2 2 ( ϕ z = 0 ) 2 + ( 2 ϕ z = 0 ) ϕ z = 0 ] .
( I z ) ( ξ ) I z = 0 δ ( 2 ) ( ξ ) I z = 0 = d 2 r ϕ z = 0 ( r π z k ξ ) ϕ z = 0 ( r + π z k ξ ) .

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