Abstract

Phase retrieval from unidirectional radiographic differential phase contrast images requires integration of noisy data. A method is presented, which aims to suppress stripe artifacts arising from direct image integration. It is purely algorithmic and therefore, compared to alternative approaches, neither additional alignment nor an increased scan time is required. We report on the theory of this method and present results using numerical as well as experimental data. The method shows significant improvements on the phase retrieval accuracy and enhances contrast in the phase image. Due to its general applicability, the proposed method provides a valuable tool for various 2D imaging applications using differential data.

© 2011 OSA

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    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef] [PubMed]
  28. M. Wernick, Y. Yang, I. Mondal, D. Chapman, M. Hasnah, C. Parham, E. Pisano, and Z. Zhong, “Computation of mass-density images from X-ray refraction-angle images” Phys. Med. Biol. 51, 1769–1778 (2006).
    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]

2010 (2)

I. Zanette, T. Weitkamp, T. Donath, S. Rutishauser, and C. David, “Two-Dimensional X-Ray Grating Interferometer,” Phys. Rev. Lett. 105 (2010).
[CrossRef]

K. Engel, D. Geller, T. Köhler, G. Martens, S. Schusser, G. Vogtmeier, and E. Rössl, “Contrast-to-noise in X-ray differential phase contrast imaging,” Nucl. Instrum. Methods A 648, 202–207 (2010).
[CrossRef]

2008 (4)

M. de Jonge, B. Hornberger, C. Holzner, D. Legnini, D. Paterson, I. McNulty, C. Jacobsen, and S. Vogt, “Quantitative Phase Imaging with a Scanning Transmission X-Ray Microscope,” Phys. Rev. Lett. 100, 163902 (2008).
[CrossRef] [PubMed]

B. Hornberger, M. de Jonge, M. Feser, P. Holl, C. Holzner, C. Jacobsen, D. Legnini, D. Paterson, P. Rehak, L. Strüder, and S. Vogt, “Differential phase contrast with a segmented detector in a scanning X-ray microprobe” J. Synchrotron Radiat. 15, 355–362 (2008).
[CrossRef] [PubMed]

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

F. Pfeiffer, M. Bech, O. Bunk, P. Kraft, E. Eikenberry, C. Brönnimann, C. Grünzweig, and C. David, “Hard-X-ray dark-field imaging using a grating interferometer” Nature Mater. 7, 134–137 (2008).
[CrossRef]

2007 (4)

M. Engelhardt, J. Baumann, M. Schuster, C. Kottler, F. Pfeiffer, O. Bunk, and C. David, “High-resolution differential phase contrast imaging using a magnifying projection geometry with a microfocus X-ray source,” Appl. Phys. Lett. 90, 224101 (2007).
[CrossRef]

B. Hornberger, M. Feser, and C. Jacobsen, “Quantitative amplitude and phase contrast imaging in a scanning transmission X-ray microscope” Ultramicroscopy 107, 644–655 (2007).
[CrossRef] [PubMed]

C. Kottler, C. David, F. Pfeiffer, and O. Bunk, “A two-directional approach for grating based differential phase contrast imaging using hard X-rays,” Opt. Express 15, 1175–1181 (2007).
[CrossRef] [PubMed]

C. David, J. Bruder, T. Rohbeck, C. Grünzweig, C. Kottler, A. Diaz, O. Bunk, and F. Pfeiffer, “Fabrication of diffraction gratings for hard X-ray phase contrast imaging,” Microelectron. Eng. 84, 1172–1177 (2007).
[CrossRef]

2006 (5)

M. Wernick, Y. Yang, I. Mondal, D. Chapman, M. Hasnah, C. Parham, E. Pisano, and Z. Zhong, “Computation of mass-density images from X-ray refraction-angle images” Phys. Med. Biol. 51, 1769–1778 (2006).
[CrossRef] [PubMed]

M. Stampanoni, A. Groso, A. Isenegger, G. Mikuljan, Q. Chen, A. Bertrand, S. Henein, R. Betemps, U. Frommherz, P. Böhler, D. Meister, M. Lange, and R. Abela, “Trends in synchrotron-based tomographic imaging: The SLS experience,” Proc. SPIE 6318, 63180M (2006).
[CrossRef]

A. Groso, R. Abela, and M. Stampanoni, “Implementation of a fast method for high resolution phase contrast tomography” Opt. Express 14, 8103–8110 (2006).
[CrossRef] [PubMed]

M. Stampanoni, G. Borchert, and R. Abela, “Progress in microtomography with the Bragg Magnifier at SLS,” Rad. Phys. Chem. 75, 1956–1961 (2006).
[CrossRef]

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

2005 (1)

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 12, 6296–6304 (2005).
[CrossRef]

2004 (2)

M. Arnison, K. Larkin, C. Sheppard, N. Smith, and C. Cogswell, “Linear phase imaging using differential interference contrast microscopy” J. microsc. 214, 7–12 (2004).
[CrossRef] [PubMed]

D. Paganin, “Phase retrieval using coherent imaging systems with linear transfer functions,” Opt. Commun. 234, 87–105 (2004).
[CrossRef]

2003 (1)

A. Momose, S. Kawamoto, I. Koyama, Y. Hamaishi, K. Takai, and Y. Suzuki, “Demonstration of X-Ray Talbot Interferometry,” Jpn. J. Appl. Phys. 42, L866–L868 (2003).
[CrossRef]

2002 (2)

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

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

2000 (1)

R. Fitzgerald, “Phase-sensitive X-ray imaging,” Phys. Today 53, 23–26 (2000).
[CrossRef]

1997 (1)

D. Chapman, W. Thomlinson, R. Johnston, D. Washburn, E. Pisano, N. Gmür, Z. Zhong, R. Menk, F. Arfelli, and D. Sayers, “Diffraction enhanced X-ray imaging,” Phys. Med. Biol. 42, 2015–2025 (1997).
[CrossRef] [PubMed]

1996 (3)

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

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

P. Cloetens, R. Barrett, J. Baruchel, J. Guigay, and M. Schlenker, “Phase objects in synchrotron radiation hard X-ray imaging,” J. Phys. D. 29, 133–146 (1996).
[CrossRef]

1995 (3)

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]

T. Davis, D. Gao, T. Gureyev, A. Stevenson, and S. Wilkins, “Phase-contrast imaging of weakly absorbing materials using hard X-rays,” Nature 373, 595–598 (1995).
[CrossRef]

1993 (1)

K. Sauer and C. Bouman, “A local update strategy for iterative reconstruction from projections,” IEEE Trans. Signal Process. 41, 534–548 (1993).
[CrossRef]

1992 (2)

L. Rudin, S. Osher, and E. Fatemi, “Nonlinear total variation based noise removal algorithms,” Phys. D Nonlinear Phenom. 60, 259–268 (1992).
[CrossRef]

P. Hansen, “Analysis of Discrete Ill-Posed Problems by Means of the L-Curve,” SIAM Rev. 34, 561–580 (1992).
[CrossRef]

1979 (1)

W. Boettinger, H. Burdette, and M. Kuriyama, “X-ray magnifier,” Rev. Sci. Instrum. 50, 26–30 (1979).
[CrossRef] [PubMed]

1974 (1)

L. Shepp and B. Logan, “The Fourier reconstruction of a head section,” IEEE Trans. Nucl. Sci. 21, 21–43 (1974).

1965 (1)

U. Bonse and M. Hart, “An X-ray interferometer,” Appl. Phys. Lett. 6, 155–156 (1965).
[CrossRef]

Abela, R.

M. Stampanoni, G. Borchert, and R. Abela, “Progress in microtomography with the Bragg Magnifier at SLS,” Rad. Phys. Chem. 75, 1956–1961 (2006).
[CrossRef]

A. Groso, R. Abela, and M. Stampanoni, “Implementation of a fast method for high resolution phase contrast tomography” Opt. Express 14, 8103–8110 (2006).
[CrossRef] [PubMed]

M. Stampanoni, A. Groso, A. Isenegger, G. Mikuljan, Q. Chen, A. Bertrand, S. Henein, R. Betemps, U. Frommherz, P. Böhler, D. Meister, M. Lange, and R. Abela, “Trends in synchrotron-based tomographic imaging: The SLS experience,” Proc. SPIE 6318, 63180M (2006).
[CrossRef]

Arfelli, F.

D. Chapman, W. Thomlinson, R. Johnston, D. Washburn, E. Pisano, N. Gmür, Z. Zhong, R. Menk, F. Arfelli, and D. Sayers, “Diffraction enhanced X-ray imaging,” Phys. Med. Biol. 42, 2015–2025 (1997).
[CrossRef] [PubMed]

Arnison, M.

M. Arnison, K. Larkin, C. Sheppard, N. Smith, and C. Cogswell, “Linear phase imaging using differential interference contrast microscopy” J. microsc. 214, 7–12 (2004).
[CrossRef] [PubMed]

Barrett, R.

P. Cloetens, R. Barrett, J. Baruchel, J. Guigay, and M. Schlenker, “Phase objects in synchrotron radiation hard X-ray imaging,” J. Phys. D. 29, 133–146 (1996).
[CrossRef]

Baruchel, J.

P. Cloetens, R. Barrett, J. Baruchel, J. Guigay, and M. Schlenker, “Phase objects in synchrotron radiation hard X-ray imaging,” J. Phys. D. 29, 133–146 (1996).
[CrossRef]

Baumann, J.

M. Engelhardt, J. Baumann, M. Schuster, C. Kottler, F. Pfeiffer, O. Bunk, and C. David, “High-resolution differential phase contrast imaging using a magnifying projection geometry with a microfocus X-ray source,” Appl. Phys. Lett. 90, 224101 (2007).
[CrossRef]

Bech, M.

F. Pfeiffer, M. Bech, O. Bunk, P. Kraft, E. Eikenberry, C. Brönnimann, C. Grünzweig, and C. David, “Hard-X-ray dark-field imaging using a grating interferometer” Nature Mater. 7, 134–137 (2008).
[CrossRef]

Bertrand, A.

M. Stampanoni, A. Groso, A. Isenegger, G. Mikuljan, Q. Chen, A. Bertrand, S. Henein, R. Betemps, U. Frommherz, P. Böhler, D. Meister, M. Lange, and R. Abela, “Trends in synchrotron-based tomographic imaging: The SLS experience,” Proc. SPIE 6318, 63180M (2006).
[CrossRef]

Betemps, R.

M. Stampanoni, A. Groso, A. Isenegger, G. Mikuljan, Q. Chen, A. Bertrand, S. Henein, R. Betemps, U. Frommherz, P. Böhler, D. Meister, M. Lange, and R. Abela, “Trends in synchrotron-based tomographic imaging: The SLS experience,” Proc. SPIE 6318, 63180M (2006).
[CrossRef]

Boettinger, W.

W. Boettinger, H. Burdette, and M. Kuriyama, “X-ray magnifier,” Rev. Sci. Instrum. 50, 26–30 (1979).
[CrossRef] [PubMed]

Böhler, P.

M. Stampanoni, A. Groso, A. Isenegger, G. Mikuljan, Q. Chen, A. Bertrand, S. Henein, R. Betemps, U. Frommherz, P. Böhler, D. Meister, M. Lange, and R. Abela, “Trends in synchrotron-based tomographic imaging: The SLS experience,” Proc. SPIE 6318, 63180M (2006).
[CrossRef]

Bonse, U.

U. Bonse and M. Hart, “An X-ray interferometer,” Appl. Phys. Lett. 6, 155–156 (1965).
[CrossRef]

Borchert, G.

M. Stampanoni, G. Borchert, and R. Abela, “Progress in microtomography with the Bragg Magnifier at SLS,” Rad. Phys. Chem. 75, 1956–1961 (2006).
[CrossRef]

Bouman, C.

K. Sauer and C. Bouman, “A local update strategy for iterative reconstruction from projections,” IEEE Trans. Signal Process. 41, 534–548 (1993).
[CrossRef]

Brönnimann, C.

F. Pfeiffer, M. Bech, O. Bunk, P. Kraft, E. Eikenberry, C. Brönnimann, C. Grünzweig, and C. David, “Hard-X-ray dark-field imaging using a grating interferometer” Nature Mater. 7, 134–137 (2008).
[CrossRef]

Bruder, J.

C. David, J. Bruder, T. Rohbeck, C. Grünzweig, C. Kottler, A. Diaz, O. Bunk, and F. Pfeiffer, “Fabrication of diffraction gratings for hard X-ray phase contrast imaging,” Microelectron. Eng. 84, 1172–1177 (2007).
[CrossRef]

Bunk, O.

F. Pfeiffer, M. Bech, O. Bunk, P. Kraft, E. Eikenberry, C. Brönnimann, C. Grünzweig, and C. David, “Hard-X-ray dark-field imaging using a grating interferometer” Nature Mater. 7, 134–137 (2008).
[CrossRef]

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

C. David, J. Bruder, T. Rohbeck, C. Grünzweig, C. Kottler, A. Diaz, O. Bunk, and F. Pfeiffer, “Fabrication of diffraction gratings for hard X-ray phase contrast imaging,” Microelectron. Eng. 84, 1172–1177 (2007).
[CrossRef]

M. Engelhardt, J. Baumann, M. Schuster, C. Kottler, F. Pfeiffer, O. Bunk, and C. David, “High-resolution differential phase contrast imaging using a magnifying projection geometry with a microfocus X-ray source,” Appl. Phys. Lett. 90, 224101 (2007).
[CrossRef]

C. Kottler, C. David, F. Pfeiffer, and O. Bunk, “A two-directional approach for grating based differential phase contrast imaging using hard X-rays,” Opt. Express 15, 1175–1181 (2007).
[CrossRef] [PubMed]

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

Burdette, H.

W. Boettinger, H. Burdette, and M. Kuriyama, “X-ray magnifier,” Rev. Sci. Instrum. 50, 26–30 (1979).
[CrossRef] [PubMed]

Chapman, D.

M. Wernick, Y. Yang, I. Mondal, D. Chapman, M. Hasnah, C. Parham, E. Pisano, and Z. Zhong, “Computation of mass-density images from X-ray refraction-angle images” Phys. Med. Biol. 51, 1769–1778 (2006).
[CrossRef] [PubMed]

D. Chapman, W. Thomlinson, R. Johnston, D. Washburn, E. Pisano, N. Gmür, Z. Zhong, R. Menk, F. Arfelli, and D. Sayers, “Diffraction enhanced X-ray imaging,” Phys. Med. Biol. 42, 2015–2025 (1997).
[CrossRef] [PubMed]

Chen, Q.

M. Stampanoni, A. Groso, A. Isenegger, G. Mikuljan, Q. Chen, A. Bertrand, S. Henein, R. Betemps, U. Frommherz, P. Böhler, D. Meister, M. Lange, and R. Abela, “Trends in synchrotron-based tomographic imaging: The SLS experience,” Proc. SPIE 6318, 63180M (2006).
[CrossRef]

Cloetens, P.

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 12, 6296–6304 (2005).
[CrossRef]

P. Cloetens, R. Barrett, J. Baruchel, J. Guigay, and M. Schlenker, “Phase objects in synchrotron radiation hard X-ray imaging,” J. Phys. D. 29, 133–146 (1996).
[CrossRef]

Cogswell, C.

M. Arnison, K. Larkin, C. Sheppard, N. Smith, and C. Cogswell, “Linear phase imaging using differential interference contrast microscopy” J. microsc. 214, 7–12 (2004).
[CrossRef] [PubMed]

David, C.

I. Zanette, T. Weitkamp, T. Donath, S. Rutishauser, and C. David, “Two-Dimensional X-Ray Grating Interferometer,” Phys. Rev. Lett. 105 (2010).
[CrossRef]

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

F. Pfeiffer, M. Bech, O. Bunk, P. Kraft, E. Eikenberry, C. Brönnimann, C. Grünzweig, and C. David, “Hard-X-ray dark-field imaging using a grating interferometer” Nature Mater. 7, 134–137 (2008).
[CrossRef]

C. David, J. Bruder, T. Rohbeck, C. Grünzweig, C. Kottler, A. Diaz, O. Bunk, and F. Pfeiffer, “Fabrication of diffraction gratings for hard X-ray phase contrast imaging,” Microelectron. Eng. 84, 1172–1177 (2007).
[CrossRef]

C. Kottler, C. David, F. Pfeiffer, and O. Bunk, “A two-directional approach for grating based differential phase contrast imaging using hard X-rays,” Opt. Express 15, 1175–1181 (2007).
[CrossRef] [PubMed]

M. Engelhardt, J. Baumann, M. Schuster, C. Kottler, F. Pfeiffer, O. Bunk, and C. David, “High-resolution differential phase contrast imaging using a magnifying projection geometry with a microfocus X-ray source,” Appl. Phys. Lett. 90, 224101 (2007).
[CrossRef]

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K. Engel, D. Geller, T. Köhler, G. Martens, S. Schusser, G. Vogtmeier, and E. Rössl, “Contrast-to-noise in X-ray differential phase contrast imaging,” Nucl. Instrum. Methods A 648, 202–207 (2010).
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D. Paganin, S. Mayo, T. Gureyev, P. Miller, and S. Wilkins, “Simultaneous phase and amplitude extraction from a single defocused image of a homogeneous object,” J. Microsc. 206, 33–40 (2002).
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M. de Jonge, B. Hornberger, C. Holzner, D. Legnini, D. Paterson, I. McNulty, C. Jacobsen, and S. Vogt, “Quantitative Phase Imaging with a Scanning Transmission X-Ray Microscope,” Phys. Rev. Lett. 100, 163902 (2008).
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M. Stampanoni, A. Groso, A. Isenegger, G. Mikuljan, Q. Chen, A. Bertrand, S. Henein, R. Betemps, U. Frommherz, P. Böhler, D. Meister, M. Lange, and R. Abela, “Trends in synchrotron-based tomographic imaging: The SLS experience,” Proc. SPIE 6318, 63180M (2006).
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D. Chapman, W. Thomlinson, R. Johnston, D. Washburn, E. Pisano, N. Gmür, Z. Zhong, R. Menk, F. Arfelli, and D. Sayers, “Diffraction enhanced X-ray imaging,” Phys. Med. Biol. 42, 2015–2025 (1997).
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P. Thibault, M. Dierolf, A. Menzel, O. Bunk, C. David, and F. Pfeiffer, “High-resolution scanning X-ray diffraction microscopy” Science 321, 379–382 (2008).
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M. Stampanoni, A. Groso, A. Isenegger, G. Mikuljan, Q. Chen, A. Bertrand, S. Henein, R. Betemps, U. Frommherz, P. Böhler, D. Meister, M. Lange, and R. Abela, “Trends in synchrotron-based tomographic imaging: The SLS experience,” Proc. SPIE 6318, 63180M (2006).
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D. Paganin, S. Mayo, T. Gureyev, P. Miller, and S. Wilkins, “Simultaneous phase and amplitude extraction from a single defocused image of a homogeneous object,” J. Microsc. 206, 33–40 (2002).
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P. Modregger, D. Lübbert, P. Schäfer, and R. Köhler, “Two dimensional diffraction enhanced imaging algorithm,” Appl. Phys. Lett. 90, 193501 (2007).

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A. Momose, S. Kawamoto, I. Koyama, Y. Hamaishi, K. Takai, and Y. Suzuki, “Demonstration of X-Ray Talbot Interferometry,” Jpn. J. Appl. Phys. 42, L866–L868 (2003).
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A. Momose, T. Takeda, Y. Itai, and K. Hirano, “Phase-contrast X-ray computed tomography for observing biological soft tissues,” Nat. Med. 2, 473–475 (1996).
[CrossRef] [PubMed]

A. Momose and J. Fukuda, “Phase-contrast radiographs of nonstained rat cerebellar specimen,” Med. Phys. 22, 375–379 (1995).
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M. Wernick, Y. Yang, I. Mondal, D. Chapman, M. Hasnah, C. Parham, E. Pisano, and Z. Zhong, “Computation of mass-density images from X-ray refraction-angle images” Phys. Med. Biol. 51, 1769–1778 (2006).
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L. Rudin, S. Osher, and E. Fatemi, “Nonlinear total variation based noise removal algorithms,” Phys. D Nonlinear Phenom. 60, 259–268 (1992).
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M. Wernick, Y. Yang, I. Mondal, D. Chapman, M. Hasnah, C. Parham, E. Pisano, and Z. Zhong, “Computation of mass-density images from X-ray refraction-angle images” Phys. Med. Biol. 51, 1769–1778 (2006).
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B. Hornberger, M. de Jonge, M. Feser, P. Holl, C. Holzner, C. Jacobsen, D. Legnini, D. Paterson, P. Rehak, L. Strüder, and S. Vogt, “Differential phase contrast with a segmented detector in a scanning X-ray microprobe” J. Synchrotron Radiat. 15, 355–362 (2008).
[CrossRef] [PubMed]

M. de Jonge, B. Hornberger, C. Holzner, D. Legnini, D. Paterson, I. McNulty, C. Jacobsen, and S. Vogt, “Quantitative Phase Imaging with a Scanning Transmission X-Ray Microscope,” Phys. Rev. Lett. 100, 163902 (2008).
[CrossRef] [PubMed]

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

F. Pfeiffer, M. Bech, O. Bunk, P. Kraft, E. Eikenberry, C. Brönnimann, C. Grünzweig, and C. David, “Hard-X-ray dark-field imaging using a grating interferometer” Nature Mater. 7, 134–137 (2008).
[CrossRef]

C. David, J. Bruder, T. Rohbeck, C. Grünzweig, C. Kottler, A. Diaz, O. Bunk, and F. Pfeiffer, “Fabrication of diffraction gratings for hard X-ray phase contrast imaging,” Microelectron. Eng. 84, 1172–1177 (2007).
[CrossRef]

C. Kottler, C. David, F. Pfeiffer, and O. Bunk, “A two-directional approach for grating based differential phase contrast imaging using hard X-rays,” Opt. Express 15, 1175–1181 (2007).
[CrossRef] [PubMed]

M. Engelhardt, J. Baumann, M. Schuster, C. Kottler, F. Pfeiffer, O. Bunk, and C. David, “High-resolution differential phase contrast imaging using a magnifying projection geometry with a microfocus X-ray source,” Appl. Phys. Lett. 90, 224101 (2007).
[CrossRef]

F. Pfeiffer, T. Weitkamp, O. Bunk, and C. David, “Phase retrieval and differential phase-contrast imaging with low-brilliance X-ray sources,” Nature 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 12, 6296–6304 (2005).
[CrossRef]

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M. Wernick, Y. Yang, I. Mondal, D. Chapman, M. Hasnah, C. Parham, E. Pisano, and Z. Zhong, “Computation of mass-density images from X-ray refraction-angle images” Phys. Med. Biol. 51, 1769–1778 (2006).
[CrossRef] [PubMed]

D. Chapman, W. Thomlinson, R. Johnston, D. Washburn, E. Pisano, N. Gmür, Z. Zhong, R. Menk, F. Arfelli, and D. Sayers, “Diffraction enhanced X-ray imaging,” Phys. Med. Biol. 42, 2015–2025 (1997).
[CrossRef] [PubMed]

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

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B. Hornberger, M. de Jonge, M. Feser, P. Holl, C. Holzner, C. Jacobsen, D. Legnini, D. Paterson, P. Rehak, L. Strüder, and S. Vogt, “Differential phase contrast with a segmented detector in a scanning X-ray microprobe” J. Synchrotron Radiat. 15, 355–362 (2008).
[CrossRef] [PubMed]

Rohbeck, T.

C. David, J. Bruder, T. Rohbeck, C. Grünzweig, C. Kottler, A. Diaz, O. Bunk, and F. Pfeiffer, “Fabrication of diffraction gratings for hard X-ray phase contrast imaging,” Microelectron. Eng. 84, 1172–1177 (2007).
[CrossRef]

Rössl, E.

K. Engel, D. Geller, T. Köhler, G. Martens, S. Schusser, G. Vogtmeier, and E. Rössl, “Contrast-to-noise in X-ray differential phase contrast imaging,” Nucl. Instrum. Methods A 648, 202–207 (2010).
[CrossRef]

Rudin, L.

L. Rudin, S. Osher, and E. Fatemi, “Nonlinear total variation based noise removal algorithms,” Phys. D Nonlinear Phenom. 60, 259–268 (1992).
[CrossRef]

Rutishauser, S.

I. Zanette, T. Weitkamp, T. Donath, S. Rutishauser, and C. David, “Two-Dimensional X-Ray Grating Interferometer,” Phys. Rev. Lett. 105 (2010).
[CrossRef]

Sauer, K.

K. Sauer and C. Bouman, “A local update strategy for iterative reconstruction from projections,” IEEE Trans. Signal Process. 41, 534–548 (1993).
[CrossRef]

Sayers, D.

D. Chapman, W. Thomlinson, R. Johnston, D. Washburn, E. Pisano, N. Gmür, Z. Zhong, R. Menk, F. Arfelli, and D. Sayers, “Diffraction enhanced X-ray imaging,” Phys. Med. Biol. 42, 2015–2025 (1997).
[CrossRef] [PubMed]

Schäfer, P.

P. Modregger, D. Lübbert, P. Schäfer, and R. Köhler, “Two dimensional diffraction enhanced imaging algorithm,” Appl. Phys. Lett. 90, 193501 (2007).

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.

P. Cloetens, R. Barrett, J. Baruchel, J. Guigay, and M. Schlenker, “Phase objects in synchrotron radiation hard X-ray imaging,” J. Phys. D. 29, 133–146 (1996).
[CrossRef]

Schusser, S.

K. Engel, D. Geller, T. Köhler, G. Martens, S. Schusser, G. Vogtmeier, and E. Rössl, “Contrast-to-noise in X-ray differential phase contrast imaging,” Nucl. Instrum. Methods A 648, 202–207 (2010).
[CrossRef]

Schuster, M.

M. Engelhardt, J. Baumann, M. Schuster, C. Kottler, F. Pfeiffer, O. Bunk, and C. David, “High-resolution differential phase contrast imaging using a magnifying projection geometry with a microfocus X-ray source,” Appl. Phys. Lett. 90, 224101 (2007).
[CrossRef]

Shepp, L.

L. Shepp and B. Logan, “The Fourier reconstruction of a head section,” IEEE Trans. Nucl. Sci. 21, 21–43 (1974).

Sheppard, C.

M. Arnison, K. Larkin, C. Sheppard, N. Smith, and C. Cogswell, “Linear phase imaging using differential interference contrast microscopy” J. microsc. 214, 7–12 (2004).
[CrossRef] [PubMed]

Smith, N.

M. Arnison, K. Larkin, C. Sheppard, N. Smith, and C. Cogswell, “Linear phase imaging using differential interference contrast microscopy” J. microsc. 214, 7–12 (2004).
[CrossRef] [PubMed]

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]

Solak, H.

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

Stampanoni, M.

M. Stampanoni, G. Borchert, and R. Abela, “Progress in microtomography with the Bragg Magnifier at SLS,” Rad. Phys. Chem. 75, 1956–1961 (2006).
[CrossRef]

A. Groso, R. Abela, and M. Stampanoni, “Implementation of a fast method for high resolution phase contrast tomography” Opt. Express 14, 8103–8110 (2006).
[CrossRef] [PubMed]

M. Stampanoni, A. Groso, A. Isenegger, G. Mikuljan, Q. Chen, A. Bertrand, S. Henein, R. Betemps, U. Frommherz, P. Böhler, D. Meister, M. Lange, and R. Abela, “Trends in synchrotron-based tomographic imaging: The SLS experience,” Proc. SPIE 6318, 63180M (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 12, 6296–6304 (2005).
[CrossRef]

Stevenson, A.

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

T. Davis, D. Gao, T. Gureyev, A. Stevenson, and S. Wilkins, “Phase-contrast imaging of weakly absorbing materials using hard X-rays,” Nature 373, 595–598 (1995).
[CrossRef]

Strüder, L.

B. Hornberger, M. de Jonge, M. Feser, P. Holl, C. Holzner, C. Jacobsen, D. Legnini, D. Paterson, P. Rehak, L. Strüder, and S. Vogt, “Differential phase contrast with a segmented detector in a scanning X-ray microprobe” J. Synchrotron Radiat. 15, 355–362 (2008).
[CrossRef] [PubMed]

Suzuki, Y.

A. Momose, S. Kawamoto, I. Koyama, Y. Hamaishi, K. Takai, and Y. Suzuki, “Demonstration of X-Ray Talbot Interferometry,” Jpn. J. Appl. Phys. 42, L866–L868 (2003).
[CrossRef]

Takai, K.

A. Momose, S. Kawamoto, I. Koyama, Y. Hamaishi, K. Takai, and Y. Suzuki, “Demonstration of X-Ray Talbot Interferometry,” Jpn. J. Appl. Phys. 42, L866–L868 (2003).
[CrossRef]

Takeda, T.

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

Thibault, P.

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

Thomlinson, W.

D. Chapman, W. Thomlinson, R. Johnston, D. Washburn, E. Pisano, N. Gmür, Z. Zhong, R. Menk, F. Arfelli, and D. Sayers, “Diffraction enhanced X-ray imaging,” Phys. Med. Biol. 42, 2015–2025 (1997).
[CrossRef] [PubMed]

Vogt, S.

M. de Jonge, B. Hornberger, C. Holzner, D. Legnini, D. Paterson, I. McNulty, C. Jacobsen, and S. Vogt, “Quantitative Phase Imaging with a Scanning Transmission X-Ray Microscope,” Phys. Rev. Lett. 100, 163902 (2008).
[CrossRef] [PubMed]

B. Hornberger, M. de Jonge, M. Feser, P. Holl, C. Holzner, C. Jacobsen, D. Legnini, D. Paterson, P. Rehak, L. Strüder, and S. Vogt, “Differential phase contrast with a segmented detector in a scanning X-ray microprobe” J. Synchrotron Radiat. 15, 355–362 (2008).
[CrossRef] [PubMed]

Vogtmeier, G.

K. Engel, D. Geller, T. Köhler, G. Martens, S. Schusser, G. Vogtmeier, and E. Rössl, “Contrast-to-noise in X-ray differential phase contrast imaging,” Nucl. Instrum. Methods A 648, 202–207 (2010).
[CrossRef]

Washburn, D.

D. Chapman, W. Thomlinson, R. Johnston, D. Washburn, E. Pisano, N. Gmür, Z. Zhong, R. Menk, F. Arfelli, and D. Sayers, “Diffraction enhanced X-ray imaging,” Phys. Med. Biol. 42, 2015–2025 (1997).
[CrossRef] [PubMed]

Weitkamp, T.

I. Zanette, T. Weitkamp, T. Donath, S. Rutishauser, and C. David, “Two-Dimensional X-Ray Grating Interferometer,” Phys. Rev. Lett. 105 (2010).
[CrossRef]

F. Pfeiffer, T. Weitkamp, O. Bunk, and C. David, “Phase retrieval and differential phase-contrast imaging with low-brilliance X-ray sources,” Nature 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 12, 6296–6304 (2005).
[CrossRef]

Wernick, M.

M. Wernick, Y. Yang, I. Mondal, D. Chapman, M. Hasnah, C. Parham, E. Pisano, and Z. Zhong, “Computation of mass-density images from X-ray refraction-angle images” Phys. Med. Biol. 51, 1769–1778 (2006).
[CrossRef] [PubMed]

Wilkins, S.

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

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

T. Davis, D. Gao, T. Gureyev, A. Stevenson, and S. Wilkins, “Phase-contrast imaging of weakly absorbing materials using hard X-rays,” Nature 373, 595–598 (1995).
[CrossRef]

Wright, S.

J. Nocedal and S. Wright, Numerical optimization (Springer verlag, 1999).
[CrossRef]

Yang, Y.

M. Wernick, Y. Yang, I. Mondal, D. Chapman, M. Hasnah, C. Parham, E. Pisano, and Z. Zhong, “Computation of mass-density images from X-ray refraction-angle images” Phys. Med. Biol. 51, 1769–1778 (2006).
[CrossRef] [PubMed]

Zanette, I.

I. Zanette, T. Weitkamp, T. Donath, S. Rutishauser, and C. David, “Two-Dimensional X-Ray Grating Interferometer,” Phys. Rev. Lett. 105 (2010).
[CrossRef]

Zhong, Z.

M. Wernick, Y. Yang, I. Mondal, D. Chapman, M. Hasnah, C. Parham, E. Pisano, and Z. Zhong, “Computation of mass-density images from X-ray refraction-angle images” Phys. Med. Biol. 51, 1769–1778 (2006).
[CrossRef] [PubMed]

D. Chapman, W. Thomlinson, R. Johnston, D. Washburn, E. Pisano, N. Gmür, Z. Zhong, R. Menk, F. Arfelli, and D. Sayers, “Diffraction enhanced X-ray imaging,” Phys. Med. Biol. 42, 2015–2025 (1997).
[CrossRef] [PubMed]

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 12, 6296–6304 (2005).
[CrossRef]

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

Appl. Phys. Lett. (4)

U. Bonse and M. Hart, “An X-ray interferometer,” Appl. Phys. Lett. 6, 155–156 (1965).
[CrossRef]

P. Modregger, D. Lübbert, P. Schäfer, and R. Köhler, “Two dimensional diffraction enhanced imaging algorithm,” Appl. Phys. Lett. 90, 193501 (2007).

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

M. Engelhardt, J. Baumann, M. Schuster, C. Kottler, F. Pfeiffer, O. Bunk, and C. David, “High-resolution differential phase contrast imaging using a magnifying projection geometry with a microfocus X-ray source,” Appl. Phys. Lett. 90, 224101 (2007).
[CrossRef]

IEEE Trans. Nucl. Sci. (1)

L. Shepp and B. Logan, “The Fourier reconstruction of a head section,” IEEE Trans. Nucl. Sci. 21, 21–43 (1974).

IEEE Trans. Signal Process. (1)

K. Sauer and C. Bouman, “A local update strategy for iterative reconstruction from projections,” IEEE Trans. Signal Process. 41, 534–548 (1993).
[CrossRef]

J. Microsc. (1)

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

M. Arnison, K. Larkin, C. Sheppard, N. Smith, and C. Cogswell, “Linear phase imaging using differential interference contrast microscopy” J. microsc. 214, 7–12 (2004).
[CrossRef] [PubMed]

J. Phys. D. (1)

P. Cloetens, R. Barrett, J. Baruchel, J. Guigay, and M. Schlenker, “Phase objects in synchrotron radiation hard X-ray imaging,” J. Phys. D. 29, 133–146 (1996).
[CrossRef]

J. Synchrotron Radiat. (1)

B. Hornberger, M. de Jonge, M. Feser, P. Holl, C. Holzner, C. Jacobsen, D. Legnini, D. Paterson, P. Rehak, L. Strüder, and S. Vogt, “Differential phase contrast with a segmented detector in a scanning X-ray microprobe” J. Synchrotron Radiat. 15, 355–362 (2008).
[CrossRef] [PubMed]

Jpn. J. Appl. Phys. (1)

A. Momose, S. Kawamoto, I. Koyama, Y. Hamaishi, K. Takai, and Y. Suzuki, “Demonstration of X-Ray Talbot Interferometry,” Jpn. J. Appl. Phys. 42, L866–L868 (2003).
[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]

Microelectron. Eng. (1)

C. David, J. Bruder, T. Rohbeck, C. Grünzweig, C. Kottler, A. Diaz, O. Bunk, and F. Pfeiffer, “Fabrication of diffraction gratings for hard X-ray phase contrast imaging,” Microelectron. Eng. 84, 1172–1177 (2007).
[CrossRef]

Nat. Med. (1)

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

Nature (2)

T. Davis, D. Gao, T. Gureyev, A. Stevenson, and S. Wilkins, “Phase-contrast imaging of weakly absorbing materials using hard X-rays,” Nature 373, 595–598 (1995).
[CrossRef]

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

Nature Mater. (1)

F. Pfeiffer, M. Bech, O. Bunk, P. Kraft, E. Eikenberry, C. Brönnimann, C. Grünzweig, and C. David, “Hard-X-ray dark-field imaging using a grating interferometer” Nature Mater. 7, 134–137 (2008).
[CrossRef]

Nature Phys. (1)

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

Nucl. Instrum. Methods A (1)

K. Engel, D. Geller, T. Köhler, G. Martens, S. Schusser, G. Vogtmeier, and E. Rössl, “Contrast-to-noise in X-ray differential phase contrast imaging,” Nucl. Instrum. Methods A 648, 202–207 (2010).
[CrossRef]

Opt. Commun. (1)

D. Paganin, “Phase retrieval using coherent imaging systems with linear transfer functions,” Opt. Commun. 234, 87–105 (2004).
[CrossRef]

Opt. Express (3)

Phys. D Nonlinear Phenom. (1)

L. Rudin, S. Osher, and E. Fatemi, “Nonlinear total variation based noise removal algorithms,” Phys. D Nonlinear Phenom. 60, 259–268 (1992).
[CrossRef]

Phys. Med. Biol. (2)

M. Wernick, Y. Yang, I. Mondal, D. Chapman, M. Hasnah, C. Parham, E. Pisano, and Z. Zhong, “Computation of mass-density images from X-ray refraction-angle images” Phys. Med. Biol. 51, 1769–1778 (2006).
[CrossRef] [PubMed]

D. Chapman, W. Thomlinson, R. Johnston, D. Washburn, E. Pisano, N. Gmür, Z. Zhong, R. Menk, F. Arfelli, and D. Sayers, “Diffraction enhanced X-ray imaging,” Phys. Med. Biol. 42, 2015–2025 (1997).
[CrossRef] [PubMed]

Phys. Rev. Lett. (2)

I. Zanette, T. Weitkamp, T. Donath, S. Rutishauser, and C. David, “Two-Dimensional X-Ray Grating Interferometer,” Phys. Rev. Lett. 105 (2010).
[CrossRef]

M. de Jonge, B. Hornberger, C. Holzner, D. Legnini, D. Paterson, I. McNulty, C. Jacobsen, and S. Vogt, “Quantitative Phase Imaging with a Scanning Transmission X-Ray Microscope,” Phys. Rev. Lett. 100, 163902 (2008).
[CrossRef] [PubMed]

Phys. Today (1)

R. Fitzgerald, “Phase-sensitive X-ray imaging,” Phys. Today 53, 23–26 (2000).
[CrossRef]

Proc. SPIE (1)

M. Stampanoni, A. Groso, A. Isenegger, G. Mikuljan, Q. Chen, A. Bertrand, S. Henein, R. Betemps, U. Frommherz, P. Böhler, D. Meister, M. Lange, and R. Abela, “Trends in synchrotron-based tomographic imaging: The SLS experience,” Proc. SPIE 6318, 63180M (2006).
[CrossRef]

Rad. Phys. Chem. (1)

M. Stampanoni, G. Borchert, and R. Abela, “Progress in microtomography with the Bragg Magnifier at SLS,” Rad. Phys. Chem. 75, 1956–1961 (2006).
[CrossRef]

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]

W. Boettinger, H. Burdette, and M. Kuriyama, “X-ray magnifier,” Rev. Sci. Instrum. 50, 26–30 (1979).
[CrossRef] [PubMed]

Science (1)

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

SIAM Rev. (1)

P. Hansen, “Analysis of Discrete Ill-Posed Problems by Means of the L-Curve,” SIAM Rev. 34, 561–580 (1992).
[CrossRef]

Ultramicroscopy (1)

B. Hornberger, M. Feser, and C. Jacobsen, “Quantitative amplitude and phase contrast imaging in a scanning transmission X-ray microscope” Ultramicroscopy 107, 644–655 (2007).
[CrossRef] [PubMed]

Other (1)

J. Nocedal and S. Wright, Numerical optimization (Springer verlag, 1999).
[CrossRef]

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Figures (6)

Fig. 1
Fig. 1

(a) DPC setup for a parallel beam configuration. The phase grating generates an interference pattern (beam splitter). A sample in front of the phase grating causes a lateral (x-direction) shift of the interference fringes at the position of the absorption grating. The absorption grating is used to analyze the interference fringes and determine this shift, which corresponds to the DPC signal. (b) Reference (blue) and object (light red) phase stepping curve (PSC). From these curves, the absorption, phase and a scattering signal can be calculated.

Fig. 2
Fig. 2

Simulation of the integration of noisy DPC data using the modified Shepp-Logan phantom. a) Original data (some low-pass filtering applied), b) noisy DPC measurement, c) phase retrieval by direct integration

Fig. 3
Fig. 3

Simulation of (a) the transmission image and (b) the visibility map of the Shepp-Logan phantom. (c) noise standard deviation map (normalized), obtained by combining (a) and (b) according to Eq. (7).

Fig. 4
Fig. 4

On the left side, simulation results on the Shepp-Logan phantom are shown. (a) Direct integration of the numerical phantom data of Fig. 2(b), causing the typical stripe artifacts. (b) Regularized image integration with p = 1 and λ = 5·10−3, containing no stripe artifacts. (c) noise pattern for the direct integration and (d) for the regularized integration, respectively, calculated by subtracting the output image from the ground-truth image. The noise standard deviations are displayed on the images. The right side of the figure shows the evaluation of (e) the RMSE and (f) the CNR as a function of the regularization parameter and for p ∈ {1, 2}. The red boxes in (a) show the regions used for the calculation of the CNR. The optimal regularization parameter for both metrics is indicated with a vertical line (λRMSE and λCNR). Figure (g) and (h) show vertical and horizontal line profiles, respectively, through the images in (a), (b) and the ground-truth image. The locations of the extracted line profiles are indicated with the dashed line in (b). In these profiles, the reduction of high vertical variations and the improved quantitative phase recovery in the x-direction compared to direct integration is clearly visible.

Fig. 5
Fig. 5

Projection images of a phantom, acquired at 25keV at the DPC setup of the TOM-CAT beamline at the Swiss Light Source. (a) DPC image, (b) direct integration, (c) regularized integration using p1 = 1, λ1 = 6 · 10−4, p2 = 2 and λ2 = 10−2 (d) ground-truth image. The colormaps of the images (b)–(d) all have the same minimum and maximum value. (e) shows the evaluation of RMSE and CNR for DI and RI and (f) a normalized horizontal profile after summing up the images in the vertical direction. CNR has been calculated using the ROIs marked with the red boxes in (b).

Fig. 6
Fig. 6

(a) DPC image of a mouse taken at an X-ray tube setup (40kV, 25mA), (b) integrated image, (c) regularized integration for λ = 4 · 10−3. In the zoomed area below, the ribs of the mouse can clearly be identified in the regularized integration, while these details vanished in the direct integration. The colormaps of the images in (b) and (c) with the same zoom level have the same minimum and maximum value.

Equations (17)

Equations on this page are rendered with MathJax. Learn more.

ϕ ( x , y ) = 2 π λ δ ( x , y , z ) d z .
α ( x , y ) = λ 2 π ϕ ( x , y ) x .
φ ( x , y ) = 2 π d g 2 α ( x , y ) = λ d g 2 ϕ ( x , y ) x .
a = a 0 , obj a 0 , ref
φ = φ 1 , obj φ 1 , ref
v = a 1 , obj a 0 , obj a 0 , ref a 1 , ref = V obj V ref ,
ϕ ( x , y ) = g 2 λ d 0 x φ ( x , y ) d x .
φ = D x ϕ + w .
σ DPC 1 V N ,
D x ϕ ( i , j ) = { ϕ ( i + 1 , j ) ϕ ( i , j ) if 1 i < N i 0 if x = N j ,
minimize D y f p subject to : W ( D x f φ ) 2 < ɛ .
a p = ( i = 1 N i j = 1 N j | a ( i , j ) | p ) 1 / p .
minimize F ( f ) = W ( D x f φ ) 2 2 + λ D y f p .
minimize F ( f ) = W ( D x f φ ) 2 2 + λ 1 T 1 f p 1 p 1 + + λ 2 T 2 f p 2 p 2 +
RMSE ( f 1 , f 2 ) = f 1 f 2 2 N i N j ,
CNR = 2 | S obj S bg | σ obj + σ bg ,
minimize F ( f ) = W ( D x f φ ) 2 2 + λ 1 D y f 1 + λ 2 M f 2 2 .

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