Abstract

We investigate why in free-space propagation single-distance phase retrieval based on a modified contrast-transfer function of linearized Fresnel theory yields good results for moderately strong pure-phase objects. Upscaling phase-variations in the exit plane, the growth of maxima of the modulus of the Fourier transformed intensity contrast dominates the minima. Cutting out small regions around the latter thus keeps information loss due to nonlocal, nonlinear effects negligible. This quasiparticle approach breaks down at a critical upscaling where the positions of the minima start to move rapidly. We apply our results to X-ray data of an early-stage Xenopus (frog) embryo.

© 2011 OSA

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  1. A. Snigirev, I. Snigireva, V. Kohn, S. Kuznetsov, and I. Schelokov, “On the possibilities of Xray phase contrast microimaging by coherent high-energy synchrotron radiation,” Rev. Sci. Instrum. 66, 5486–5492 (1995).
    [CrossRef]
  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]
  3. 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]
  4. P. Cloetens, “Contribution to phase contrast imaging, reconstruction and tomography with hard synchrotron radiation,” PhD dissertation, Vrije Universiteit Brussel (1999).
  5. 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]
  6. 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]
  7. M. R. Teague, “Deterministic phase retrieval: a Greens function solution,” J. Opt. Soc. Am. 73, 1434–1441 (1983).
    [CrossRef]
  8. J.-P. Guigay, “Fourier transform analysis of Fresnel diffraction patterns and in-line holograms,” Optik 49, 121–125 (1977).
  9. 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]
  10. J. Moosmann, R. Hofmann, and T. Baumbach, “Single-distance phase retrieval at large phase shifts,” Opt. Express 19, 12066–12073 (2011).
    [CrossRef] [PubMed]
  11. L. D. Landau, “The theory of a Fermi liquid,” Sov. Phys. JETP 3, 920–925 (1957).
  12. J. Goldstone, “Field theories with superconductor solutions,” Nuovo Cimento 19, 154–164 (1961).
    [CrossRef]
  13. J. Goldstone, A. Salam, and S. Weinberg, “Broken symmetries,” Phys. Rev. 127, 965–970 (1962).
    [CrossRef]
  14. Y. Nambu, “Quasiparticles and gauge invariance in the theory of superconductivity,” Phys. Rev. 117, 648–663 (1960).
    [CrossRef]
  15. F. Zernike, “Phase-contrast, a new method for microscopic observation of transparent objects. Part I.,” Physica 9, 686–698 (1942).
    [CrossRef]
  16. F. Zernike, “Phase-contrast, a new method for microscopic observation of transparent objects. Part II.,” Physica 9, 974–986 (1942).
    [CrossRef]
  17. J. Moosmann, V. Altapova, D. Hänschke, R. Hofmann, and T. Baumbach, “Nonlinear, noniterative, single-distance phase retrieval and developmental biology,” submitted to AIP Proceedings, ICXOM 21.

2011 (1)

2006 (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]

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]

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 (1)

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

1983 (1)

1977 (1)

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

1962 (1)

J. Goldstone, A. Salam, and S. Weinberg, “Broken symmetries,” Phys. Rev. 127, 965–970 (1962).
[CrossRef]

1961 (1)

J. Goldstone, “Field theories with superconductor solutions,” Nuovo Cimento 19, 154–164 (1961).
[CrossRef]

1960 (1)

Y. Nambu, “Quasiparticles and gauge invariance in the theory of superconductivity,” Phys. Rev. 117, 648–663 (1960).
[CrossRef]

1957 (1)

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

1942 (2)

F. Zernike, “Phase-contrast, a new method for microscopic observation of transparent objects. Part I.,” Physica 9, 686–698 (1942).
[CrossRef]

F. Zernike, “Phase-contrast, a new method for microscopic observation of transparent objects. Part II.,” Physica 9, 974–986 (1942).
[CrossRef]

Altapova, V.

J. Moosmann, V. Altapova, D. Hänschke, R. Hofmann, and T. Baumbach, “Nonlinear, noniterative, single-distance phase retrieval and developmental biology,” submitted to AIP Proceedings, ICXOM 21.

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.

J. Moosmann, R. Hofmann, and T. Baumbach, “Single-distance phase retrieval at large phase shifts,” Opt. Express 19, 12066–12073 (2011).
[CrossRef] [PubMed]

J. Moosmann, V. Altapova, D. Hänschke, R. Hofmann, and T. Baumbach, “Nonlinear, noniterative, single-distance phase retrieval and developmental biology,” submitted to AIP Proceedings, ICXOM 21.

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.

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]

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

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]

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]

Goldstone, J.

J. Goldstone, A. Salam, and S. Weinberg, “Broken symmetries,” Phys. Rev. 127, 965–970 (1962).
[CrossRef]

J. Goldstone, “Field theories with superconductor solutions,” Nuovo Cimento 19, 154–164 (1961).
[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).

Gureyev, T. E.

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]

Hänschke, D.

J. Moosmann, V. Altapova, D. Hänschke, R. Hofmann, and T. Baumbach, “Nonlinear, noniterative, single-distance phase retrieval and developmental biology,” submitted to AIP Proceedings, ICXOM 21.

Hofmann, R.

J. Moosmann, R. Hofmann, and T. Baumbach, “Single-distance phase retrieval at large phase shifts,” Opt. Express 19, 12066–12073 (2011).
[CrossRef] [PubMed]

J. Moosmann, V. Altapova, D. Hänschke, R. Hofmann, and T. Baumbach, “Nonlinear, noniterative, single-distance phase retrieval and developmental biology,” submitted to AIP Proceedings, ICXOM 21.

Kohn, V.

A. Snigirev, I. Snigireva, V. Kohn, S. Kuznetsov, and I. Schelokov, “On the possibilities of Xray 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 Xray 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).

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]

Moosmann, J.

J. Moosmann, R. Hofmann, and T. Baumbach, “Single-distance phase retrieval at large phase shifts,” Opt. Express 19, 12066–12073 (2011).
[CrossRef] [PubMed]

J. Moosmann, V. Altapova, D. Hänschke, R. Hofmann, and T. Baumbach, “Nonlinear, noniterative, single-distance phase retrieval and developmental biology,” submitted to AIP Proceedings, ICXOM 21.

Nambu, Y.

Y. Nambu, “Quasiparticles and gauge invariance in the theory of superconductivity,” Phys. Rev. 117, 648–663 (1960).
[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]

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.

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]

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]

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]

Salam, A.

J. Goldstone, A. Salam, and S. Weinberg, “Broken symmetries,” Phys. Rev. 127, 965–970 (1962).
[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 Xray 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]

Snigirev, A.

A. Snigirev, I. Snigireva, V. Kohn, S. Kuznetsov, and I. Schelokov, “On the possibilities of Xray 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 Xray 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.

Weinberg, S.

J. Goldstone, A. Salam, and S. Weinberg, “Broken symmetries,” Phys. Rev. 127, 965–970 (1962).
[CrossRef]

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.

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]

Zernike, F.

F. Zernike, “Phase-contrast, a new method for microscopic observation of transparent objects. Part I.,” Physica 9, 686–698 (1942).
[CrossRef]

F. Zernike, “Phase-contrast, a new method for microscopic observation of transparent objects. Part II.,” Physica 9, 974–986 (1942).
[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]

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]

Nuovo Cimento (1)

J. Goldstone, “Field theories with superconductor solutions,” Nuovo Cimento 19, 154–164 (1961).
[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 (1)

Optik (1)

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

Phys. Rev. (2)

J. Goldstone, A. Salam, and S. Weinberg, “Broken symmetries,” Phys. Rev. 127, 965–970 (1962).
[CrossRef]

Y. Nambu, “Quasiparticles and gauge invariance in the theory of superconductivity,” Phys. Rev. 117, 648–663 (1960).
[CrossRef]

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]

Physica (2)

F. Zernike, “Phase-contrast, a new method for microscopic observation of transparent objects. Part I.,” Physica 9, 686–698 (1942).
[CrossRef]

F. Zernike, “Phase-contrast, a new method for microscopic observation of transparent objects. Part II.,” Physica 9, 974–986 (1942).
[CrossRef]

Rev. Sci. Instrum. (2)

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]

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

Sov. Phys. JETP (1)

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

Other (2)

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

J. Moosmann, V. Altapova, D. Hänschke, R. Hofmann, and T. Baumbach, “Nonlinear, noniterative, single-distance phase retrieval and developmental biology,” submitted to AIP Proceedings, ICXOM 21.

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Figures (7)

Fig. 1
Fig. 1

(a): transverse position-space phase map at z = 0 (zero padded test pattern Lena), (b): g z ¯ ( | ξ | ) ( Δ x ) 2 at E = 10keV, z = 0.5m and for S = 200 and S = 450, see text. The solid line is a plot of 100|sin(2π2z|ξ⃗|2/k)|. Dots and crosses indicate the respective “data”, dashed-dotted and dashed lines are respective fits to F(t) ≡ c1ec2t2c3t |sin(t2)| + c4 + c5t + c6t2 + c7t3 + c8t4 + c9t5, where t π 2 z ξ 2 k and c1, …, c9 are real constants. The actual argument in the plots is the pixel number pξ = L|ξ⃗| in Fourier space where L = px,maxΔx, L2 is area of the field of view, and px,max, Δx denote the maximal pixel number, resolution in transverse position space, respectively.

Fig. 2
Fig. 2

Plot of the angular average of the modulus of the transfer function of the linear CTF approximation | g z ϕ z = 0 | for various values of upscaling, S = 1 (solid black dots), S = 100 (blue crosses), and S = 200 (red circles), where S = 1 is associated with the phase map ϕz=0,CTF (compare with Fig. 1(a)) at ϕmax = 0.01.

Fig. 3
Fig. 3

Plot of pixel number pξ, which belongs to |ξ⃗|min,1, as a function of S. Notice the onset of critical behavior (second-order like phase transition) to the right of Sc = 356. Notice also that the variance of |ξ⃗|min,1 for S < Sc practically is zero.

Fig. 4
Fig. 4

Plot of function R ( S ) v max , 1 v min , 1. (The “data” g z ¯ ( | ξ | max , 1 ) ( S ) and g z ¯ ( | ξ | min , 1 ) ( S ) was fitted to 9th-degree polynomials, and the derivatives defining vmax,1 and vmin,1 were taken of these polynomials.)

Fig. 5
Fig. 5

The CTF situation: (a) phase retrieval of a projection through the four-cell stage of a Xenopus embryo. The size of the projection is 1725 × 1338 pixel2, or 1.3 × 1.0 mm2. (b) 2-D slice of the tomographic reconstruction of the electron density. The data, compare with Fig. 6(a),(d), was taken at the ID19 beamline at ESRF with E = 20keV (monochromatized to Δ E E = 10 4 using double Si 111 crystals), 1599 projections per tomogram, an exposure time per frame of 2 s, an effective pixel size of 0.745μm, and an object-detector distance of z = 0.945m. The size of the slice is 1532 × 1691 pixel2, or 1.14 × 1.25 mm2. In both images large-scale variations were subtracted for better visibility.

Fig. 6
Fig. 6

Same object and experimental conditions as in Fig. 5. Shown is a 2-D slice of a tomographic reconstruction based on (a) intensity contrast, and on the phase retrieved according to (b) linearized TIE and (c) projected CTF with ɛ = 0.01. The images in the second row depict the quadratic regions of interest (ROIs) selected from the images in the first row. The width of the ROIs is 150 pixel, or 112 μm. In all images large-scale variations (mainly arising from small absorptive effects) were subtracted for better visibility.

Fig. 7
Fig. 7

Plots of function g z S ¯ ( | ξ | ) in dependence of pξ. The black solid line is a fit (same fit function as in Fig. 1) to the data associated with Fig. 6(a),(d) (S = 1, data are black dots), and the dashed blue line is a fit to g z S ¯ ( | ξ | ), obtained by an S = 2 (data are blue crosses) upscaling of retrieved phase using projected CTF, and a subsequent Fresnel forward propagation.

Equations (6)

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| 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 ) ( ξ ) = 2 sin ( s ) ( ϕ z = 0 ) ( ξ ) cos ( s ) d 2 ξ ( ϕ z = 0 ) ( ξ ) ( ϕ z = 0 ) ( ξ ξ ) + e i s d 2 ξ e 4 π 2 i z ξ ξ k ( ϕ z = 0 ) ( ξ ) ( ϕ z = 0 ) ( ξ ξ ) + O ( ( ϕ z = 0 ) 3 ) ,
( g z GM ) ( ξ ) = σ 2 π [ 4 sin ( 2 π 2 z k ξ 2 ) e 2 π 2 σ 2 ξ 2 S e π 2 σ 2 ξ 2 ( cos ( 2 π 2 z k ξ 2 ) e π 2 z 2 k 2 σ 2 ξ 2 ) S 2 ] .
φ 1 4 S 1 e 2 π 2 σ 2 ξ 2 + S 2 16 | ξ 2 = ξ 1 2 = k 2 π z 1 4 S .
( g z ) ( ξ ) Θ ( | sin ( 2 π 2 z k ξ 2 ) | ɛ ) × ( g z ) ( ξ ) ,

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