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Non-linear regularized phase retrieval for unidirectional X-ray differential phase contrast radiographyThomas Thüring, Peter Modregger, Bernd R. Pinzer, Zhentian Wang, and Marco Stampanoni »View Author Affiliations
Thomas Thüring,1,2,*
Peter Modregger,1,3
Bernd R. Pinzer,1
Zhentian Wang,1
and Marco Stampanoni1,2
1Swiss Light Source, Paul Scherrer Institut, Villigen PSI, Switzerland 2Institute for Biomedical Engineering, Swiss Federal Institute of Technology, Zurich, Switzerland 3School of Biology and Medicine, University of Lausanne, Lausanne, Switzerland *Corresponding author: thomas.thuering@psi.ch |
Optics Express, Vol. 19, Issue 25, pp. 25545-25558 (2011)
http://dx.doi.org/10.1364/OE.19.025545
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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
OCIS Codes
(100.5070) Image processing : Phase retrieval
(110.7440) Imaging systems : X-ray imaging
(110.3010) Imaging systems : Image reconstruction techniques
ToC Category:
Image Processing
Virtual Issues
Vol. 7, Iss. 2 Virtual Journal for Biomedical Optics
Citation
Thomas Thüring, Peter Modregger, Bernd R. Pinzer, Zhentian Wang, and Marco Stampanoni, "Non-linear regularized phase retrieval for unidirectional X-ray differential phase contrast radiography," Opt. Express 19, 25545-25558 (2011)
http://www.opticsinfobase.org/vjbo/abstract.cfm?URI=oe-19-25-25545
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References
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- 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. Express15, 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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- 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. Express12, 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]
- T. Davis, D. Gao, T. Gureyev, A. Stevenson, and S. Wilkins, “Phase-contrast imaging of weakly absorbing materials using hard X-rays,” Nature373, 595–598 (1995). [CrossRef]
- 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]
- 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]
- T. Weitkamp, A. Diaz, C. David, F. Pfeiffer, M. Stampanoni, P. Cloetens, and E. Ziegler, “X-ray phase imaging with a grating interferometer,” Opt. Express12, 6296–6304 (2005). [CrossRef]
- P. Thibault, M. Dierolf, A. Menzel, O. Bunk, C. David, and F. Pfeiffer, “High-resolution scanning X-ray diffraction microscopy” Science321, 379–382 (2008). [CrossRef] [PubMed]
- 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, 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]
- 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 A648, 202–207 (2010). [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]
- L. Rudin, S. Osher, and E. Fatemi, “Nonlinear total variation based noise removal algorithms,” Phys. D Nonlinear Phenom.60, 259–268 (1992). [CrossRef]
- 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]
- B. Hornberger, M. Feser, and C. Jacobsen, “Quantitative amplitude and phase contrast imaging in a scanning transmission X-ray microscope” Ultramicroscopy107, 644–655 (2007). [CrossRef] [PubMed]
- R. Fitzgerald, “Phase-sensitive X-ray imaging,” Phys. Today53, 23–26 (2000). [CrossRef]
- 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. SPIE6318, 63180M (2006). [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 A648, 202–207 (2010). [CrossRef]
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- 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. Express12, 6296–6304 (2005). [CrossRef]
- 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. 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,” Nature384, 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,” Nature373, 595–598 (1995). [CrossRef]
- J. Nocedal and S. Wright, Numerical optimization (Springer verlag, 1999). [CrossRef]
- 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]
- I. Zanette, T. Weitkamp, T. Donath, S. Rutishauser, and C. David, “Two-Dimensional X-Ray Grating Interferometer,” Phys. Rev. Lett.105 (2010). [CrossRef]
- 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]
- T. Weitkamp, A. Diaz, C. David, F. Pfeiffer, M. Stampanoni, P. Cloetens, and E. Ziegler, “X-ray phase imaging with a grating interferometer,” Opt. Express12, 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.
- 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.
- L. Shepp and B. Logan, “The Fourier reconstruction of a head section,” IEEE Trans. Nucl. Sci.21, 21–43 (1974).
IEEE Trans. Signal Process.
- 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.
- 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.
- 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.
- 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.
- 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.
- A. Momose and J. Fukuda, “Phase-contrast radiographs of nonstained rat cerebellar specimen,” Med. Phys.22, 375–379 (1995). [CrossRef] [PubMed]
Microelectron. Eng.
- 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.
- 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
- T. Davis, D. Gao, T. Gureyev, A. Stevenson, and S. Wilkins, “Phase-contrast imaging of weakly absorbing materials using hard X-rays,” Nature373, 595–598 (1995). [CrossRef]
- S. Wilkins, T. Gureyev, D. Gao, A. Pogany, and A. Stevenson, “Phase-contrast imaging using polychromatic hard X-rays,” Nature384, 335–338 (1996). [CrossRef]
Nature Mater.
- 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.
- 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
- 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 A648, 202–207 (2010). [CrossRef]
Opt. Commun.
- D. Paganin, “Phase retrieval using coherent imaging systems with linear transfer functions,” Opt. Commun.234, 87–105 (2004). [CrossRef]
Opt. Express
- T. Weitkamp, A. Diaz, C. David, F. Pfeiffer, M. Stampanoni, P. Cloetens, and E. Ziegler, “X-ray phase imaging with a grating interferometer,” Opt. Express12, 6296–6304 (2005). [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. Express15, 1175–1181 (2007). [CrossRef] [PubMed]
- A. Groso, R. Abela, and M. Stampanoni, “Implementation of a fast method for high resolution phase contrast tomography” Opt. Express14, 8103–8110 (2006). [CrossRef] [PubMed]
Phys. D Nonlinear Phenom.
- 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.
- 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.
- 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
- R. Fitzgerald, “Phase-sensitive X-ray imaging,” Phys. Today53, 23–26 (2000). [CrossRef]
Proc. SPIE
- 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. SPIE6318, 63180M (2006). [CrossRef]
Rad. Phys. Chem.
- 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.
- 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
- P. Thibault, M. Dierolf, A. Menzel, O. Bunk, C. David, and F. Pfeiffer, “High-resolution scanning X-ray diffraction microscopy” Science321, 379–382 (2008). [CrossRef] [PubMed]
SIAM Rev.
- P. Hansen, “Analysis of Discrete Ill-Posed Problems by Means of the L-Curve,” SIAM Rev.34, 561–580 (1992). [CrossRef]
Ultramicroscopy
- B. Hornberger, M. Feser, and C. Jacobsen, “Quantitative amplitude and phase contrast imaging in a scanning transmission X-ray microscope” Ultramicroscopy107, 644–655 (2007). [CrossRef] [PubMed]
Other
- J. Nocedal and S. Wright, Numerical optimization (Springer verlag, 1999). [CrossRef]
2010, Zanette, Phys. Rev. Lett.
- 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 A648, 202–207 (2010). [CrossRef]
- 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]
- 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” Science321, 379–382 (2008). [CrossRef] [PubMed]
- B. Hornberger, M. Feser, and C. Jacobsen, “Quantitative amplitude and phase contrast imaging in a scanning transmission X-ray microscope” Ultramicroscopy107, 644–655 (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]
- 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]
- M. Stampanoni, G. Borchert, and R. Abela, “Progress in microtomography with the Bragg Magnifier at SLS,” Rad. Phys. Chem.75, 1956–1961 (2006). [CrossRef]
- 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. SPIE6318, 63180M (2006). [CrossRef]
- 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]
- T. Weitkamp, A. Diaz, C. David, F. Pfeiffer, M. Stampanoni, P. Cloetens, and E. Ziegler, “X-ray phase imaging with a grating interferometer,” Opt. Express12, 6296–6304 (2005). [CrossRef]
- 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]
- 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]
- 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]
- R. Fitzgerald, “Phase-sensitive X-ray imaging,” Phys. Today53, 23–26 (2000). [CrossRef]
- 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]
- 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,” Nature384, 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]
- 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,” Nature373, 595–598 (1995). [CrossRef]
- K. Sauer and C. Bouman, “A local update strategy for iterative reconstruction from projections,” IEEE Trans. Signal Process.41, 534–548 (1993). [CrossRef]
- 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]
- W. Boettinger, H. Burdette, and M. Kuriyama, “X-ray magnifier,” Rev. Sci. Instrum.50, 26–30 (1979). [CrossRef] [PubMed]
- L. Shepp and B. Logan, “The Fourier reconstruction of a head section,” IEEE Trans. Nucl. Sci.21, 21–43 (1974).
- 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).
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