Abstract

The interactions of a beam of hard and spatio-temporally coherent X-rays with a soft-matter sample primarily induce a transverse distribution of exit phase variations δϕ (retardations or advancements in pieces of the wave front exiting the object compared to the incoming wave front) whose free-space propagation over a distance z gives rise to intensity contrast gz. For single-distance image detection and |δϕ| ≪ 1 all-order-in-z phase-intensity contrast transfer is linear in δϕ. Here we show that ideal coherence implies a decay of the (shot-)noise-to-signal ratio in gz and of the associated phase noise as z−1/2 and z−1, respectively. Limits on X-ray dose thus favor large values of z. We discuss how a phase-scaling symmetry, exact in the limit δϕ → 0 and dynamically unbroken up to |δϕ| ∼ 1, suggests a filtering of gz in Fourier space, preserving non-iterative quasi-linear phase retrieval for phase variations up to order unity if induced by multi-scale objects inducing phase variations δϕ of a broad spatial frequency spectrum. Such an approach continues to be applicable under an assumed phase-attenuation duality. Using synchrotron radiation, ex and in vivo microtomography on frog embryos exemplifies improved resolution compared to a conventional single-distance phase-retrieval algorithm.

© 2016 Optical Society of America

Full Article  |  PDF Article
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References

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    [Crossref] [PubMed]
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    [Crossref] [PubMed]
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2015 (1)

B. Sixou, “Regularization Methods for Phase Retrieval and Phase Contrast Tomography,” Abstr. Appl. Anal. 2015, 943501 (2015).

2014 (4)

A. Ruhlandt, M. Krenkel, M. Bartels, and T. Salditt, “Three-dimensional phase retrieval in propagation-based phase-contrast imaging,” Phys. Rev. A 89, 033847 (2014).
[Crossref]

M. Langer, P. Cloetens, B. Hesse, H. Suhonen, A. Pacureanu, and F. Peyrin, “Priors for X-ray in-line phase contrast tomography of heterogeneous objects,” Phil. Trans. R. Soc. A 372, 20130129 (2014).
[Crossref]

J. Moosmann, A. Ershov, V. Weinhardt, T. Baumbach, M. S. Prasad, C. LaBonne, X. Xiao, J. Kashef, and R. Hofmann, “Time-lapse X-ray phase-contrast microtomography for in vivo imaging and analysis of morphogenesis,” Nat. Protoc. 9, 294–304 (2014).
[Crossref] [PubMed]

M. Mickoleit, B. Schmid, M. Weber, F. O. Fahrbach, S. Hombach, S. Reischauer, and J. Huisken, “High-resolution reconstruction of the beating zebrafish heart, ” Nat. Methods 11, 919–922 (2014).
[Crossref] [PubMed]

2013 (2)

J. Moosmann, A. Ershov, V. Altapova, T. Baumbach, M. S. Prasad, C. LaBonne, X. Xiao, J. Kashef, and R. Hofmann, “X-ray phase-contrast in vivo microtomography probes novel aspects of Xenopus gastrulation,” Nature 497, 374–377 (2013).
[Crossref] [PubMed]

S. F. S. Becker, R. Mayor, and J. Kashef, “Cadherin-11 Mediates Contact Inhibition of Locomotion during Xenopus Neural Crest Cell Migration,” PLoS ONE 8(12), e85717 (2013).
[PubMed]

2012 (1)

2011 (4)

2010 (2)

2006 (1)

F. Pfeiffer, T. Weitkamp, O. Bunk, and C. David, “Phase retrieval and differential phase-contrast imaging with low-brilliance X-ray sources,” Nat. Phys. 2, 258–261 (2006).
[Crossref]

2005 (2)

2002 (2)

D. M. Paganin, S. C. Mayo, T. E. Gureyev, P. R. Miller, and S. W. Wilkins, “Simultaneous phase and amplitude extraction from a single defocused image of a homogeneous object,” J. Microsc. 206, 33–40 (2002).
[Crossref] [PubMed]

C. David, B. Nöhammer, H. H. Solak, and E. Ziegler, “Differential phase-contrast imaging using a shearing interferometer,” Appl. Phys. Lett. 81, 3287–3289 (2002).
[Crossref]

1996 (3)

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]

A. Momose, T. Takeda, Y. Itai, and K. Hirano, “Phase-contrast X-ray computed tomography for observing biological soft tissues,” Nat. Medicine 2, 473–475 (1996).
[Crossref]

1995 (1)

A. Snigirev, I. Snigireva I., 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, 5486–5493 (1995).
[Crossref]

1993 (1)

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

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 the Fermi liquid,” Sov. J. Phys. JETP 3, 920 (1957).

1948 (1)

P. Kirkpatrick and A. V. Baez, “Formation of Optical Images by X-Rays,” J. Opt. Soc. Amer. 38, 766–774 (1948).
[Crossref]

Altapova, V.

J. Moosmann, A. Ershov, V. Altapova, T. Baumbach, M. S. Prasad, C. LaBonne, X. Xiao, J. Kashef, and R. Hofmann, “X-ray phase-contrast in vivo microtomography probes novel aspects of Xenopus gastrulation,” Nature 497, 374–377 (2013).
[Crossref] [PubMed]

Baez, A. V.

P. Kirkpatrick and A. V. Baez, “Formation of Optical Images by X-Rays,” J. Opt. Soc. Amer. 38, 766–774 (1948).
[Crossref]

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]

Bartels, M.

A. Ruhlandt, M. Krenkel, M. Bartels, and T. Salditt, “Three-dimensional phase retrieval in propagation-based phase-contrast imaging,” Phys. Rev. A 89, 033847 (2014).
[Crossref]

Baumbach, T.

J. Moosmann, A. Ershov, V. Weinhardt, T. Baumbach, M. S. Prasad, C. LaBonne, X. Xiao, J. Kashef, and R. Hofmann, “Time-lapse X-ray phase-contrast microtomography for in vivo imaging and analysis of morphogenesis,” Nat. Protoc. 9, 294–304 (2014).
[Crossref] [PubMed]

J. Moosmann, A. Ershov, V. Altapova, T. Baumbach, M. S. Prasad, C. LaBonne, X. Xiao, J. Kashef, and R. Hofmann, “X-ray phase-contrast in vivo microtomography probes novel aspects of Xenopus gastrulation,” Nature 497, 374–377 (2013).
[Crossref] [PubMed]

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

R. Hofmann, J. Moosmann, and T. Baumbach, “Criticality in single-distance phase retrieval,” Opt. Express 19, 25881–25890 (2011).
[Crossref]

J. Moosmann, R. Hofmann, and T. Baumbach, “Nonlinear phase retrieval from single-distance radiograph,” Opt. Express 18, 25771–25785 (2010).
[Crossref] [PubMed]

S. Hahn, R. Hofmann, J. Moosmann, O. Öktem, L. Helfen, J.-P. Guigay, Th. van de Kamp, and T. Baumbach, “Contrast transfer in propagation based X-ray phase-contrast imaging,” (to be published in Phys. Rev. A).

Becker, S. F. S.

S. F. S. Becker, R. Mayor, and J. Kashef, “Cadherin-11 Mediates Contact Inhibition of Locomotion during Xenopus Neural Crest Cell Migration,” PLoS ONE 8(12), e85717 (2013).
[PubMed]

Born, M.

M. Born and E. Wolf, Principles of Optics, 7th edition (Cambridge University, 1999).
[Crossref]

Bunk, O.

F. Pfeiffer, T. Weitkamp, O. Bunk, and C. David, “Phase retrieval and differential phase-contrast imaging with low-brilliance X-ray sources,” Nat. Phys. 2, 258–261 (2006).
[Crossref]

Canty, L.

N. Rohani, L. Canty, O. Luu, F. Fagotto, and R. Winklbauer, “EphrinB/EphB Signalling Controls Embryonic Germ layer Separation by Contact-Induced Cell Detachment,” PLoS Biol. 9, e1000597 (2011).
[Crossref]

Castelli, E.

Chen, R. C.

Cloetens, P.

M. Langer, P. Cloetens, B. Hesse, H. Suhonen, A. Pacureanu, and F. Peyrin, “Priors for X-ray in-line phase contrast tomography of heterogeneous objects,” Phil. Trans. R. Soc. A 372, 20130129 (2014).
[Crossref]

M. Langer, P. Cloetens, A. Pacureanu, and F. Peyrin, “X-ray in-line phase tomography of multimaterial objects,” Opt. Lett. 37, 2151–2153 (2012).
[Crossref] [PubMed]

T. Weitkamp, A. Diaz, C. David, F. Pfeiffer, M. Stampanoni, P. Cloetens, and E. Ziegler, “X-ray phase imaging with a grating interferometer,” Opt. Express 13, 6296–6304 (2005).
[Crossref] [PubMed]

P. Cloetens, Contribution to Phase Contrast Imaging, Reconstruction and Tomography with Hard Synchrotron Radiation: Principles, Implementation and Applications (PHD thesis 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]

David, C.

F. Pfeiffer, T. Weitkamp, O. Bunk, and C. David, “Phase retrieval and differential phase-contrast imaging with low-brilliance X-ray sources,” Nat. Phys. 2, 258–261 (2006).
[Crossref]

T. Weitkamp, A. Diaz, C. David, F. Pfeiffer, M. Stampanoni, P. Cloetens, and E. Ziegler, “X-ray phase imaging with a grating interferometer,” Opt. Express 13, 6296–6304 (2005).
[Crossref] [PubMed]

C. David, B. Nöhammer, H. H. Solak, and E. Ziegler, “Differential phase-contrast imaging using a shearing interferometer,” Appl. Phys. Lett. 81, 3287–3289 (2002).
[Crossref]

Davis, J. C.

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

Diaz, A.

Ershov, A.

J. Moosmann, A. Ershov, V. Weinhardt, T. Baumbach, M. S. Prasad, C. LaBonne, X. Xiao, J. Kashef, and R. Hofmann, “Time-lapse X-ray phase-contrast microtomography for in vivo imaging and analysis of morphogenesis,” Nat. Protoc. 9, 294–304 (2014).
[Crossref] [PubMed]

J. Moosmann, A. Ershov, V. Altapova, T. Baumbach, M. S. Prasad, C. LaBonne, X. Xiao, J. Kashef, and R. Hofmann, “X-ray phase-contrast in vivo microtomography probes novel aspects of Xenopus gastrulation,” Nature 497, 374–377 (2013).
[Crossref] [PubMed]

Fagotto, F.

N. Rohani, L. Canty, O. Luu, F. Fagotto, and R. Winklbauer, “EphrinB/EphB Signalling Controls Embryonic Germ layer Separation by Contact-Induced Cell Detachment,” PLoS Biol. 9, e1000597 (2011).
[Crossref]

Fahrbach, F. O.

M. Mickoleit, B. Schmid, M. Weber, F. O. Fahrbach, S. Hombach, S. Reischauer, and J. Huisken, “High-resolution reconstruction of the beating zebrafish heart, ” Nat. Methods 11, 919–922 (2014).
[Crossref] [PubMed]

Fresnel, A.-J.

C. Huygens, T. Young, and A.-J. Fresnel, The Wave Theory of Light: Memoirs of Huygens, Young, and Fresnel (American Book Company, 1900).

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.

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

S. Hahn, R. Hofmann, J. Moosmann, O. Öktem, L. Helfen, J.-P. Guigay, Th. van de Kamp, and T. Baumbach, “Contrast transfer in propagation based X-ray phase-contrast imaging,” (to be published in Phys. Rev. A).

Gullikson, E. M.

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

D. M. Paganin, S. C. Mayo, T. E. Gureyev, P. R. Miller, and S. W. Wilkins, “Simultaneous phase and amplitude extraction from a single defocused image of a homogeneous object,” J. Microsc. 206, 33–40 (2002).
[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]

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]

Hahn, S.

S. Hahn, R. Hofmann, J. Moosmann, O. Öktem, L. Helfen, J.-P. Guigay, Th. van de Kamp, and T. Baumbach, “Contrast transfer in propagation based X-ray phase-contrast imaging,” (to be published in Phys. Rev. A).

Helfen, L.

S. Hahn, R. Hofmann, J. Moosmann, O. Öktem, L. Helfen, J.-P. Guigay, Th. van de Kamp, and T. Baumbach, “Contrast transfer in propagation based X-ray phase-contrast imaging,” (to be published in Phys. Rev. A).

Henke, B. L.

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

Hesse, B.

M. Langer, P. Cloetens, B. Hesse, H. Suhonen, A. Pacureanu, and F. Peyrin, “Priors for X-ray in-line phase contrast tomography of heterogeneous objects,” Phil. Trans. R. Soc. A 372, 20130129 (2014).
[Crossref]

Hirano, K.

A. Momose, T. Takeda, Y. Itai, and K. Hirano, “Phase-contrast X-ray computed tomography for observing biological soft tissues,” Nat. Medicine 2, 473–475 (1996).
[Crossref]

Hofmann, R.

J. Moosmann, A. Ershov, V. Weinhardt, T. Baumbach, M. S. Prasad, C. LaBonne, X. Xiao, J. Kashef, and R. Hofmann, “Time-lapse X-ray phase-contrast microtomography for in vivo imaging and analysis of morphogenesis,” Nat. Protoc. 9, 294–304 (2014).
[Crossref] [PubMed]

J. Moosmann, A. Ershov, V. Altapova, T. Baumbach, M. S. Prasad, C. LaBonne, X. Xiao, J. Kashef, and R. Hofmann, “X-ray phase-contrast in vivo microtomography probes novel aspects of Xenopus gastrulation,” Nature 497, 374–377 (2013).
[Crossref] [PubMed]

R. Hofmann, J. Moosmann, and T. Baumbach, “Criticality in single-distance phase retrieval,” Opt. Express 19, 25881–25890 (2011).
[Crossref]

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, R. Hofmann, and T. Baumbach, “Nonlinear phase retrieval from single-distance radiograph,” Opt. Express 18, 25771–25785 (2010).
[Crossref] [PubMed]

S. Hahn, R. Hofmann, J. Moosmann, O. Öktem, L. Helfen, J.-P. Guigay, Th. van de Kamp, and T. Baumbach, “Contrast transfer in propagation based X-ray phase-contrast imaging,” (to be published in Phys. Rev. A).

R. Hofmann, The Thermodynamics of Quantum Yang-Mills Theory: Theory and Applications (World Scientific Publishing Co., 2012).

Hombach, S.

M. Mickoleit, B. Schmid, M. Weber, F. O. Fahrbach, S. Hombach, S. Reischauer, and J. Huisken, “High-resolution reconstruction of the beating zebrafish heart, ” Nat. Methods 11, 919–922 (2014).
[Crossref] [PubMed]

Huisken, J.

M. Mickoleit, B. Schmid, M. Weber, F. O. Fahrbach, S. Hombach, S. Reischauer, and J. Huisken, “High-resolution reconstruction of the beating zebrafish heart, ” Nat. Methods 11, 919–922 (2014).
[Crossref] [PubMed]

Huygens, C.

C. Huygens, T. Young, and A.-J. Fresnel, The Wave Theory of Light: Memoirs of Huygens, Young, and Fresnel (American Book Company, 1900).

Itai, Y.

A. Momose, T. Takeda, Y. Itai, and K. Hirano, “Phase-contrast X-ray computed tomography for observing biological soft tissues,” Nat. Medicine 2, 473–475 (1996).
[Crossref]

Kashef, J.

J. Moosmann, A. Ershov, V. Weinhardt, T. Baumbach, M. S. Prasad, C. LaBonne, X. Xiao, J. Kashef, and R. Hofmann, “Time-lapse X-ray phase-contrast microtomography for in vivo imaging and analysis of morphogenesis,” Nat. Protoc. 9, 294–304 (2014).
[Crossref] [PubMed]

J. Moosmann, A. Ershov, V. Altapova, T. Baumbach, M. S. Prasad, C. LaBonne, X. Xiao, J. Kashef, and R. Hofmann, “X-ray phase-contrast in vivo microtomography probes novel aspects of Xenopus gastrulation,” Nature 497, 374–377 (2013).
[Crossref] [PubMed]

S. F. S. Becker, R. Mayor, and J. Kashef, “Cadherin-11 Mediates Contact Inhibition of Locomotion during Xenopus Neural Crest Cell Migration,” PLoS ONE 8(12), e85717 (2013).
[PubMed]

Kirkpatrick, P.

P. Kirkpatrick and A. V. Baez, “Formation of Optical Images by X-Rays,” J. Opt. Soc. Amer. 38, 766–774 (1948).
[Crossref]

Kohn, V.

A. Snigirev, I. Snigireva I., 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, 5486–5493 (1995).
[Crossref]

Krenkel, M.

A. Ruhlandt, M. Krenkel, M. Bartels, and T. Salditt, “Three-dimensional phase retrieval in propagation-based phase-contrast imaging,” Phys. Rev. A 89, 033847 (2014).
[Crossref]

Kuznetsov, S.

A. Snigirev, I. Snigireva I., 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, 5486–5493 (1995).
[Crossref]

LaBonne, C.

J. Moosmann, A. Ershov, V. Weinhardt, T. Baumbach, M. S. Prasad, C. LaBonne, X. Xiao, J. Kashef, and R. Hofmann, “Time-lapse X-ray phase-contrast microtomography for in vivo imaging and analysis of morphogenesis,” Nat. Protoc. 9, 294–304 (2014).
[Crossref] [PubMed]

J. Moosmann, A. Ershov, V. Altapova, T. Baumbach, M. S. Prasad, C. LaBonne, X. Xiao, J. Kashef, and R. Hofmann, “X-ray phase-contrast in vivo microtomography probes novel aspects of Xenopus gastrulation,” Nature 497, 374–377 (2013).
[Crossref] [PubMed]

Landau, L. D.

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

Langer, M.

M. Langer, P. Cloetens, B. Hesse, H. Suhonen, A. Pacureanu, and F. Peyrin, “Priors for X-ray in-line phase contrast tomography of heterogeneous objects,” Phil. Trans. R. Soc. A 372, 20130129 (2014).
[Crossref]

M. Langer, P. Cloetens, A. Pacureanu, and F. Peyrin, “X-ray in-line phase tomography of multimaterial objects,” Opt. Lett. 37, 2151–2153 (2012).
[Crossref] [PubMed]

Liu, H.

Longo, R.

Luu, O.

N. Rohani, L. Canty, O. Luu, F. Fagotto, and R. Winklbauer, “EphrinB/EphB Signalling Controls Embryonic Germ layer Separation by Contact-Induced Cell Detachment,” PLoS Biol. 9, e1000597 (2011).
[Crossref]

Maxwell, J. C.

J. C. Maxwell, A Treatise on Electricity and Magnetism (Oxford Clarendon Press, 1873).

Mayo, S. C.

D. M. Paganin, S. C. Mayo, T. E. Gureyev, P. R. Miller, and S. W. Wilkins, “Simultaneous phase and amplitude extraction from a single defocused image of a homogeneous object,” J. Microsc. 206, 33–40 (2002).
[Crossref] [PubMed]

Mayor, R.

S. F. S. Becker, R. Mayor, and J. Kashef, “Cadherin-11 Mediates Contact Inhibition of Locomotion during Xenopus Neural Crest Cell Migration,” PLoS ONE 8(12), e85717 (2013).
[PubMed]

Mickoleit, M.

M. Mickoleit, B. Schmid, M. Weber, F. O. Fahrbach, S. Hombach, S. Reischauer, and J. Huisken, “High-resolution reconstruction of the beating zebrafish heart, ” Nat. Methods 11, 919–922 (2014).
[Crossref] [PubMed]

Miller, P. R.

D. M. Paganin, S. C. Mayo, T. E. Gureyev, P. R. Miller, and S. W. Wilkins, “Simultaneous phase and amplitude extraction from a single defocused image of a homogeneous object,” J. Microsc. 206, 33–40 (2002).
[Crossref] [PubMed]

Momose, A.

A. Momose, T. Takeda, Y. Itai, and K. Hirano, “Phase-contrast X-ray computed tomography for observing biological soft tissues,” Nat. Medicine 2, 473–475 (1996).
[Crossref]

Moosmann, J.

J. Moosmann, A. Ershov, V. Weinhardt, T. Baumbach, M. S. Prasad, C. LaBonne, X. Xiao, J. Kashef, and R. Hofmann, “Time-lapse X-ray phase-contrast microtomography for in vivo imaging and analysis of morphogenesis,” Nat. Protoc. 9, 294–304 (2014).
[Crossref] [PubMed]

J. Moosmann, A. Ershov, V. Altapova, T. Baumbach, M. S. Prasad, C. LaBonne, X. Xiao, J. Kashef, and R. Hofmann, “X-ray phase-contrast in vivo microtomography probes novel aspects of Xenopus gastrulation,” Nature 497, 374–377 (2013).
[Crossref] [PubMed]

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

R. Hofmann, J. Moosmann, and T. Baumbach, “Criticality in single-distance phase retrieval,” Opt. Express 19, 25881–25890 (2011).
[Crossref]

J. Moosmann, R. Hofmann, and T. Baumbach, “Nonlinear phase retrieval from single-distance radiograph,” Opt. Express 18, 25771–25785 (2010).
[Crossref] [PubMed]

S. Hahn, R. Hofmann, J. Moosmann, O. Öktem, L. Helfen, J.-P. Guigay, Th. van de Kamp, and T. Baumbach, “Contrast transfer in propagation based X-ray phase-contrast imaging,” (to be published in Phys. Rev. A).

Nöhammer, B.

C. David, B. Nöhammer, H. H. Solak, and E. Ziegler, “Differential phase-contrast imaging using a shearing interferometer,” Appl. Phys. Lett. 81, 3287–3289 (2002).
[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]

Öktem, O.

S. Hahn, R. Hofmann, J. Moosmann, O. Öktem, L. Helfen, J.-P. Guigay, Th. van de Kamp, and T. Baumbach, “Contrast transfer in propagation based X-ray phase-contrast imaging,” (to be published in Phys. Rev. A).

Pacureanu, A.

M. Langer, P. Cloetens, B. Hesse, H. Suhonen, A. Pacureanu, and F. Peyrin, “Priors for X-ray in-line phase contrast tomography of heterogeneous objects,” Phil. Trans. R. Soc. A 372, 20130129 (2014).
[Crossref]

M. Langer, P. Cloetens, A. Pacureanu, and F. Peyrin, “X-ray in-line phase tomography of multimaterial objects,” Opt. Lett. 37, 2151–2153 (2012).
[Crossref] [PubMed]

Paganin, D. M.

D. M. Paganin, S. C. Mayo, T. E. Gureyev, P. R. Miller, and S. W. Wilkins, “Simultaneous phase and amplitude extraction from a single defocused image of a homogeneous object,” J. Microsc. 206, 33–40 (2002).
[Crossref] [PubMed]

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]

D. M. Paganin, Coherent X-Ray Optics (Oxford University, 2006).
[Crossref]

Peyrin, F.

M. Langer, P. Cloetens, B. Hesse, H. Suhonen, A. Pacureanu, and F. Peyrin, “Priors for X-ray in-line phase contrast tomography of heterogeneous objects,” Phil. Trans. R. Soc. A 372, 20130129 (2014).
[Crossref]

M. Langer, P. Cloetens, A. Pacureanu, and F. Peyrin, “X-ray in-line phase tomography of multimaterial objects,” Opt. Lett. 37, 2151–2153 (2012).
[Crossref] [PubMed]

Pfeiffer, F.

F. Pfeiffer, T. Weitkamp, O. Bunk, and C. David, “Phase retrieval and differential phase-contrast imaging with low-brilliance X-ray sources,” Nat. Phys. 2, 258–261 (2006).
[Crossref]

T. Weitkamp, A. Diaz, C. David, F. Pfeiffer, M. Stampanoni, P. Cloetens, and E. Ziegler, “X-ray phase imaging with a grating interferometer,” Opt. Express 13, 6296–6304 (2005).
[Crossref] [PubMed]

Pogany, A.

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]

Prasad, M. S.

J. Moosmann, A. Ershov, V. Weinhardt, T. Baumbach, M. S. Prasad, C. LaBonne, X. Xiao, J. Kashef, and R. Hofmann, “Time-lapse X-ray phase-contrast microtomography for in vivo imaging and analysis of morphogenesis,” Nat. Protoc. 9, 294–304 (2014).
[Crossref] [PubMed]

J. Moosmann, A. Ershov, V. Altapova, T. Baumbach, M. S. Prasad, C. LaBonne, X. Xiao, J. Kashef, and R. Hofmann, “X-ray phase-contrast in vivo microtomography probes novel aspects of Xenopus gastrulation,” Nature 497, 374–377 (2013).
[Crossref] [PubMed]

Reischauer, S.

M. Mickoleit, B. Schmid, M. Weber, F. O. Fahrbach, S. Hombach, S. Reischauer, and J. Huisken, “High-resolution reconstruction of the beating zebrafish heart, ” Nat. Methods 11, 919–922 (2014).
[Crossref] [PubMed]

Rogon, L.

Rohani, N.

N. Rohani, L. Canty, O. Luu, F. Fagotto, and R. Winklbauer, “EphrinB/EphB Signalling Controls Embryonic Germ layer Separation by Contact-Induced Cell Detachment,” PLoS Biol. 9, e1000597 (2011).
[Crossref]

Ruhlandt, A.

A. Ruhlandt, M. Krenkel, M. Bartels, and T. Salditt, “Three-dimensional phase retrieval in propagation-based phase-contrast imaging,” Phys. Rev. A 89, 033847 (2014).
[Crossref]

Salditt, T.

A. Ruhlandt, M. Krenkel, M. Bartels, and T. Salditt, “Three-dimensional phase retrieval in propagation-based phase-contrast imaging,” Phys. Rev. A 89, 033847 (2014).
[Crossref]

Schelokov, I.

A. Snigirev, I. Snigireva I., 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, 5486–5493 (1995).
[Crossref]

Schmid, B.

M. Mickoleit, B. Schmid, M. Weber, F. O. Fahrbach, S. Hombach, S. Reischauer, and J. Huisken, “High-resolution reconstruction of the beating zebrafish heart, ” Nat. Methods 11, 919–922 (2014).
[Crossref] [PubMed]

Sixou, B.

B. Sixou, “Regularization Methods for Phase Retrieval and Phase Contrast Tomography,” Abstr. Appl. Anal. 2015, 943501 (2015).

Snigirev, A.

A. Snigirev, I. Snigireva I., 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, 5486–5493 (1995).
[Crossref]

Snigireva I., I.

A. Snigirev, I. Snigireva I., 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, 5486–5493 (1995).
[Crossref]

Solak, H. H.

C. David, B. Nöhammer, H. H. Solak, and E. Ziegler, “Differential phase-contrast imaging using a shearing interferometer,” Appl. Phys. Lett. 81, 3287–3289 (2002).
[Crossref]

Stampanoni, M.

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]

Suhonen, H.

M. Langer, P. Cloetens, B. Hesse, H. Suhonen, A. Pacureanu, and F. Peyrin, “Priors for X-ray in-line phase contrast tomography of heterogeneous objects,” Phil. Trans. R. Soc. A 372, 20130129 (2014).
[Crossref]

Takeda, T.

A. Momose, T. Takeda, Y. Itai, and K. Hirano, “Phase-contrast X-ray computed tomography for observing biological soft tissues,” Nat. Medicine 2, 473–475 (1996).
[Crossref]

Tian, L.

van de Kamp, Th.

S. Hahn, R. Hofmann, J. Moosmann, O. Öktem, L. Helfen, J.-P. Guigay, Th. van de Kamp, and T. Baumbach, “Contrast transfer in propagation based X-ray phase-contrast imaging,” (to be published in Phys. Rev. A).

Waller, L.

Weber, M.

M. Mickoleit, B. Schmid, M. Weber, F. O. Fahrbach, S. Hombach, S. Reischauer, and J. Huisken, “High-resolution reconstruction of the beating zebrafish heart, ” Nat. Methods 11, 919–922 (2014).
[Crossref] [PubMed]

Weinhardt, V.

J. Moosmann, A. Ershov, V. Weinhardt, T. Baumbach, M. S. Prasad, C. LaBonne, X. Xiao, J. Kashef, and R. Hofmann, “Time-lapse X-ray phase-contrast microtomography for in vivo imaging and analysis of morphogenesis,” Nat. Protoc. 9, 294–304 (2014).
[Crossref] [PubMed]

Weitkamp, T.

F. Pfeiffer, T. Weitkamp, O. Bunk, and C. David, “Phase retrieval and differential phase-contrast imaging with low-brilliance X-ray sources,” Nat. Phys. 2, 258–261 (2006).
[Crossref]

T. Weitkamp, A. Diaz, C. David, F. Pfeiffer, M. Stampanoni, P. Cloetens, and E. Ziegler, “X-ray phase imaging with a grating interferometer,” Opt. Express 13, 6296–6304 (2005).
[Crossref] [PubMed]

Wilkins, S. W.

D. M. Paganin, S. C. Mayo, T. E. Gureyev, P. R. Miller, and S. W. Wilkins, “Simultaneous phase and amplitude extraction from a single defocused image of a homogeneous object,” J. Microsc. 206, 33–40 (2002).
[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]

Winklbauer, R.

N. Rohani, L. Canty, O. Luu, F. Fagotto, and R. Winklbauer, “EphrinB/EphB Signalling Controls Embryonic Germ layer Separation by Contact-Induced Cell Detachment,” PLoS Biol. 9, e1000597 (2011).
[Crossref]

Wolf, E.

M. Born and E. Wolf, Principles of Optics, 7th edition (Cambridge University, 1999).
[Crossref]

E. Wolf, Introduction to the Theory of Coherence and Polarization of Light (Cambridge University, 2007).

Wu, X.

Xiao, T. Q.

Xiao, X.

J. Moosmann, A. Ershov, V. Weinhardt, T. Baumbach, M. S. Prasad, C. LaBonne, X. Xiao, J. Kashef, and R. Hofmann, “Time-lapse X-ray phase-contrast microtomography for in vivo imaging and analysis of morphogenesis,” Nat. Protoc. 9, 294–304 (2014).
[Crossref] [PubMed]

J. Moosmann, A. Ershov, V. Altapova, T. Baumbach, M. S. Prasad, C. LaBonne, X. Xiao, J. Kashef, and R. Hofmann, “X-ray phase-contrast in vivo microtomography probes novel aspects of Xenopus gastrulation,” Nature 497, 374–377 (2013).
[Crossref] [PubMed]

Xie, H. L.

Yan, A.

Young, T.

C. Huygens, T. Young, and A.-J. Fresnel, The Wave Theory of Light: Memoirs of Huygens, Young, and Fresnel (American Book Company, 1900).

Ziegler, E.

T. Weitkamp, A. Diaz, C. David, F. Pfeiffer, M. Stampanoni, P. Cloetens, and E. Ziegler, “X-ray phase imaging with a grating interferometer,” Opt. Express 13, 6296–6304 (2005).
[Crossref] [PubMed]

C. David, B. Nöhammer, H. H. Solak, and E. Ziegler, “Differential phase-contrast imaging using a shearing interferometer,” Appl. Phys. Lett. 81, 3287–3289 (2002).
[Crossref]

Abstr. Appl. Anal. (1)

B. Sixou, “Regularization Methods for Phase Retrieval and Phase Contrast Tomography,” Abstr. Appl. Anal. 2015, 943501 (2015).

Appl. Phys. Lett. (1)

C. David, B. Nöhammer, H. H. Solak, and E. Ziegler, “Differential phase-contrast imaging using a shearing interferometer,” Appl. Phys. Lett. 81, 3287–3289 (2002).
[Crossref]

At. Data. Nucl. Data Tables (1)

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

J. Microsc. (1)

D. M. Paganin, S. C. Mayo, T. E. Gureyev, P. R. Miller, and S. W. Wilkins, “Simultaneous phase and amplitude extraction from a single defocused image of a homogeneous object,” J. Microsc. 206, 33–40 (2002).
[Crossref] [PubMed]

J. Opt. Soc. Amer. (1)

P. Kirkpatrick and A. V. Baez, “Formation of Optical Images by X-Rays,” J. Opt. Soc. Amer. 38, 766–774 (1948).
[Crossref]

Nat. Medicine (1)

A. Momose, T. Takeda, Y. Itai, and K. Hirano, “Phase-contrast X-ray computed tomography for observing biological soft tissues,” Nat. Medicine 2, 473–475 (1996).
[Crossref]

Nat. Methods (1)

M. Mickoleit, B. Schmid, M. Weber, F. O. Fahrbach, S. Hombach, S. Reischauer, and J. Huisken, “High-resolution reconstruction of the beating zebrafish heart, ” Nat. Methods 11, 919–922 (2014).
[Crossref] [PubMed]

Nat. Phys. (1)

F. Pfeiffer, T. Weitkamp, O. Bunk, and C. David, “Phase retrieval and differential phase-contrast imaging with low-brilliance X-ray sources,” Nat. Phys. 2, 258–261 (2006).
[Crossref]

Nat. Protoc. (1)

J. Moosmann, A. Ershov, V. Weinhardt, T. Baumbach, M. S. Prasad, C. LaBonne, X. Xiao, J. Kashef, and R. Hofmann, “Time-lapse X-ray phase-contrast microtomography for in vivo imaging and analysis of morphogenesis,” Nat. Protoc. 9, 294–304 (2014).
[Crossref] [PubMed]

Nature (2)

J. Moosmann, A. Ershov, V. Altapova, T. Baumbach, M. S. Prasad, C. LaBonne, X. Xiao, J. Kashef, and R. Hofmann, “X-ray phase-contrast in vivo microtomography probes novel aspects of Xenopus gastrulation,” Nature 497, 374–377 (2013).
[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]

Opt. Express (5)

Opt. Lett. (3)

Optik (1)

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

Phil. Trans. R. Soc. A (1)

M. Langer, P. Cloetens, B. Hesse, H. Suhonen, A. Pacureanu, and F. Peyrin, “Priors for X-ray in-line phase contrast tomography of heterogeneous objects,” Phil. Trans. R. Soc. A 372, 20130129 (2014).
[Crossref]

Phys. Rev. A (1)

A. Ruhlandt, M. Krenkel, M. Bartels, and T. Salditt, “Three-dimensional phase retrieval in propagation-based phase-contrast imaging,” Phys. Rev. A 89, 033847 (2014).
[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]

PLoS Biol. (1)

N. Rohani, L. Canty, O. Luu, F. Fagotto, and R. Winklbauer, “EphrinB/EphB Signalling Controls Embryonic Germ layer Separation by Contact-Induced Cell Detachment,” PLoS Biol. 9, e1000597 (2011).
[Crossref]

PLoS ONE (1)

S. F. S. Becker, R. Mayor, and J. Kashef, “Cadherin-11 Mediates Contact Inhibition of Locomotion during Xenopus Neural Crest Cell Migration,” PLoS ONE 8(12), e85717 (2013).
[PubMed]

Rev. Sci. Instrum. (1)

A. Snigirev, I. Snigireva I., 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, 5486–5493 (1995).
[Crossref]

Sov. J. Phys. JETP (1)

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

Other (9)

R. Hofmann, The Thermodynamics of Quantum Yang-Mills Theory: Theory and Applications (World Scientific Publishing Co., 2012).

S. Hahn, R. Hofmann, and T. Baumbach are preparing a manuscript to be called “Analysis of Fresnel diffraction: the Gaussian phase.”

D. M. Paganin, Coherent X-Ray Optics (Oxford University, 2006).
[Crossref]

S. Hahn, R. Hofmann, J. Moosmann, O. Öktem, L. Helfen, J.-P. Guigay, Th. van de Kamp, and T. Baumbach, “Contrast transfer in propagation based X-ray phase-contrast imaging,” (to be published in Phys. Rev. A).

M. Born and E. Wolf, Principles of Optics, 7th edition (Cambridge University, 1999).
[Crossref]

E. Wolf, Introduction to the Theory of Coherence and Polarization of Light (Cambridge University, 2007).

P. Cloetens, Contribution to Phase Contrast Imaging, Reconstruction and Tomography with Hard Synchrotron Radiation: Principles, Implementation and Applications (PHD thesis Vrije Universiteit Brussel, 1999).

J. C. Maxwell, A Treatise on Electricity and Magnetism (Oxford Clarendon Press, 1873).

C. Huygens, T. Young, and A.-J. Fresnel, The Wave Theory of Light: Memoirs of Huygens, Young, and Fresnel (American Book Company, 1900).

Supplementary Material (8)

NameDescription
» Visualization 1: MP4 (16150 KB)      Slicing through volume of fixed 4-cell stage
» Visualization 2: MP4 (13588 KB)      Slicing through volume of fixed 4-cell stage
» Visualization 3: MP4 (13583 KB)      Slicing through volume of stage-19 in vivo, APS, early
» Visualization 4: MP4 (71 KB)      Comparison ROI between Paganin and QP phase retrieval, fixed 4-cell stage
» Visualization 5: MP4 (362 KB)      Comparison ROI between Paganin and QP phase retrieval, fixed 4-cell stage
» Visualization 6: MP4 (35 KB)      Comparison ROI between Paganin and modQP phase retrieval, stage-19 in vivo, APS
» Visualization 7: MP4 (11226 KB)      Comparison of slicings between Paganin and modQP phase retrieval, stage-19 in vivo, early, ESRF
» Visualization 8: MP4 (11230 KB)      Comparison of slicings between Paganin and modQP phase retrieval, stage-19 in vivo, early, ESRF

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Figures (6)

Fig. 1
Fig. 1

Principle of image formation in propagation based phase-contrast X-ray radiography. (a) snapshot of a 2D slice through a set-up where a perfect plane wave of constant intensity Iinc impinges on an object. Locally curved and attenuated wave fronts exit the object at z = 0 with intensity Iz=0. Even without object attenuation self-interference induces inhomogeneous intensity Iz in the detector plane at z > 0 (propagated projections at varying z of a fixed stage-12 Xenopus laevis embryo, see Appendix B). (b) representation of wave amplitude along dashed line Δ in (a). Object induced attenuation is indicated by a drop of amplitude at z = 0.

Fig. 2
Fig. 2

Image formation in full Fresnel theory and linear phase retrieval for the pure-phase case subject to statistical noise in dependence of propagation distance z at E = 20keV and Δx = 1μm using Lena as an exit phase map which is representative of broad class of pure-phase samples with a sufficiently broad frequency spectrum. A maximum exit phase variation of 1 constitutes the input for the simulated forward propagation used in (a) through (c) and of 0.01 (linear case) in (d) through (f). (a) z and first, second radial moments, M1, M2 (for definition see Eq. (7)), in dependence of z. The peak position of M1 and M2 indicates maximum edge enhancement. Here the right-hand side ordinate depicts the values of z while the left-hand side ordinate associates with the values of M1 and M2. (b) radial spectra �� g z ˜ (for definition see Eq. (4)) for three distinct values of z and a Poisson noise level of 1.5 % on Iz. Note that there is no relevant z dependence of these spectra. (c) logarithm of R g z (for definition of R g z , see Eq. (6)), (d) radial spectra of retrieved phase, ϕ̃, from noiseless gz for three distinct values of z. There is no dependence on z. (e) logarithm of the radial spectra of noise in retrieved phase, �� ϕ ˜ , induced by 0.1 % Poisson noise on Iz, for three distinct values of z. (f) transverse average of modulus of retrieved phase’s noise (0.1 % Poisson noise on Iz), �� ϕ ¯ , as a function of z, see Eq. (5).

Fig. 3
Fig. 3

Quasi-particle behavior of intensity contrast versus phase up to criticality under simulated Fresnel forward propagation at E = 20keV. A maximum exit phase variation of 0.01 (linear case: S = 1), 2 (non-linear case: S = 200) is the input for (a) and (b), (e) through (h), respectively. In (a) through (e) z = 1m was used. (a) modulus of Fourier transform of intensity contrast. (b) radial spectrum z of gz. First three minima |ξ|1, |ξ|2, and |ξ|3 are clearly discernible. (c) positions of |ξ|1, |ξ|2, and |ξ|3 in dependence of S, upscaling the linear case. For the hypothetical (infinite-resolution) limit m → ∞, S c m appears to converge to a finite value ∼ 200. (d) S dependences of first three maxima and minima normalized to first minimum at S = 1 (growth ratios). Note degeneracy of curves for growth of minima. (e) modulus of Fourier transformed modified intensity contrast, defined by Eq. (9), for δ = 0.01. (f) radial spectra of phase retrieved from noiseless gz. Note an approximate z independence. (g), logarithm of the radial spectra of noise in retrieved phase, subject to 1 % Poisson noise on Iz, for three distinct values of z. (h), transverse average of modulus of retrieved phase’s noise (1 % Poisson noise on Iz) as a function of z.

Fig. 4
Fig. 4

Global phase-attenuation duality vs. upscaling and confrontation with experimental data on fixed frog embryo. (b) through (f) are based on tomographic data, representing a fixed, 4-cell stage Xenopus laevis embryo, see Appendices A, C and Visualization 1. (a) dependence of 1st, 2nd, and 3rd minimum of z as a function of S and the duality parameter ε for 0 ≤ ε ≤ 10−2 in simulated forward propagation using Lena as a phase pattern (δϕmax = 0.01 at S = 1). Note stability of the critical behavior in S under variations of ε. (b) gz for a fixed projection angle (left) and modulus of Fourier transform, |ℱgz|, (right). The visibility of several rings demonstrates the presence of information at high frequencies which would be suppressed by Paganin phase retrieval. (c) gz in blue is obtained from experiment; gz in green results from forward-propagation after upscaling by a factor of two of the quasi-particle retrieved phase δϕ (ε = 10−2.5, δ = 0.1, see Appendix B), (d), (e), and (f) equal slice through tomographic reconstruction after Paganin phase retrieval, phase retrieval using Eq. (10), and its quasi-particle version, respectively.

Fig. 5
Fig. 5

In vivo XPCμT of Xenopus laevis development within stage-19 embryo (late neural stage) as imaged at beamline 32-ID of APS. (a) intensity contrast gz of posterior part at fixed projection angle (left), and modulus of Fourier transform, |ℱgz|, (right). Ring orders higher than second are washed out by shot noise. (b) z in blue is obtained from experiment; z in green results from simulated forward propagation after upscaling by a factor of two of the quasi-particle retrieved phase δϕ (ε = 0, δ = 0.1). The vertical dashed line indicates the frequency cutoff ξnoise, chosen such as to discriminate signal-dominating noise, see text. (c) reconstructed dorsal-ventral slice using quasi-particle phase retrieval. (d) reconstruction of same slice using Paganin (left) and modified quasi-particle (right) phase retrieval (Eq. 12). The former sacrifices resolution by neglecting frequencies higher than the first maximum of z, the latter cuts off shot-noise dominated frequencies beyond the second maximum (b). (e) reconstruction of more posterior dorsal-ventral slice subject to modified quasi-particle phase retrieval. (f) same reconstruction of same slice as in (e) but now after 14 min time lapse. Comparing (e) and (f), note closing of neural tube.

Fig. 6
Fig. 6

In vivo XPCμT of Xenopus embryo at stage-19 (late neurula) as imaged at beamlines 32-ID of APS ((a) and (c), z = 0.7m, Δx = 1.3μm, and event number per pixel of N ∼ 10.000) and ID19 of ESRF ((b) and (d), z = 3.6m, Δx = 1.6μm, and event number of N ∼ 4.000). In both cases, the first member of the time-lapse series is visualized. Compared is a central slice, reconstructed with 499 projections, and acquired at similar X-ray energies E = 30keV (APS) and E = 26.2keV (ESRF). Reconstructions are based on Paganin phase retrieval ((a) and (b)) and modified quasi-particle phase retrieval ((c) and (d)).

Equations (18)

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( I z ) ( ξ ) = d 2 x ψ ( x π z k ξ ) ψ * ( x + π z k ξ ) × exp ( 2 π i ξ x ) ,
( g z ) = 2 sin ( σ ) ( ϕ ) ( ξ ) cos ( σ ) d 2 ξ ( ϕ ) ( ξ ) ( ϕ ) ( ξ ξ ) + exp ( i σ ) d 2 ξ exp ( i 4 π 2 ξ ξ z k ) × ( ϕ ) ( ξ ) ( ϕ ) ( ξ ξ ) + 𝒪 [ ( ϕ ) 3 ] ,
| ϕ ( x π z k ξ ) ϕ ( x + π z k ξ ) | 1
Q ˜ ( | ξ | ) 1 2 π FoV d 2 x 0 2 π d φ | Q | ( ξ ) .
Q ¯ FoV d 2 x | Q | FoV d 2 x .
R g z 𝒩 g z ¯ g z ¯ ,
M i d 2 ξ ( | ξ | / ξ c ) i | I z / I 0 | ( ξ ) d 2 ξ | I z / I 0 | ( ξ ) , ( i = 1 , 2 ) ,
ψ exp ( i α ) ψ = I inc exp [ i ( 1 + α ϕ ) ϕ ] , ( α real ) .
g ˜ z ( | ξ | m , m 1 ) ( S ) / g ˜ z ( | ξ | 1 ) ( S = 1 ) ,
g ˜ z ( | ξ | m ) ( S ) / g ˜ z ( | ξ | 1 ) ( S = 1 )
( g z ) ( ξ ) ( g z ^ ) ( ξ ) Θ ( | sin ( σ ) | δ ^ ) ( g z ) ( ξ ) ,
( g z ^ ) ( ξ ) 1 δ ^ sin ( σ ) ( g z ) ( ξ )
A R = m = 1 2 π z ξ c 2 / k 2 π | ξ | m 2 δ ^ k 4 π 2 z | ξ | m 1 = 2 δ ^ ξ c 2 ,
A R A D = 2 δ ^ / π .
( g z ) ( ξ ) = 2 ( sin ( σ ) + ε cos ( σ ) ) ( ϕ ) ( ξ ) = 2 1 + ε 2 sin ( σ + arctan ε ) ( ϕ ) ( ξ ) .
| ξ | m m k 2 π z ( 1 ε 2 π m ) + 𝒪 ( ε 2 ) , ( m 1 , ε 1 ) .
( g z ^ ( ξ ) ) Θ ( ξ noise | ξ | ) ( g z ^ ( ξ ) ) .
z z M = ( 1 1 M ) R

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