Abstract

We investigated effects of unresolvable sharp edges on images obtained in a grating-based X-ray differential phase imaging technique. Results of numerical calculations for monochromatic X-rays show that an unresolvable sharp edge generates not only differential-phase contrast but also visibility contrast. The latter shows that the visibility contrast has another major origin other than ultra-small-angle X-ray scattering (USAXS) from randomly distributed unresolvable microstructures, which has been considered the main origin for the contrast. The effects were experimentally confirmed using a synchrotron X-ray source.

© 2015 Optical Society of America

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2014 (9)

M. Endrizzi, P. C. Diemoz, T. P. Millard, J. L. Jones, R. D. Speller, I. K. Robinson, and A. Olivo, “Hard x-ray dark-field imaging with incoherent sample illumination,” Appl. Phys. Lett. 104, 024106 (2014).
[Crossref]

F. Schaff, A. Malecki, G. Potdevin, E. Eggl, P. B. Noël, T. Baum, E. G. Garcia, J. S. Bauer, and F. Pfeiffer, “Correlation of x-ray vector radiography to bone micro-architecture,” Sci. Rep. 4, 3695 (2014).
[Crossref] [PubMed]

A. Malecki, G. Potdevin, T. Biernath, E. Eggl, K. Willer, T. Lasser, J. Maisenbacher, J. Gibmeier, A. Wanner, and F. Pfeiffer, “X-ray tensor tomography,” Europhys. Lett. 105, 38002 (2014).
[Crossref]

T. Lauridsen, E. M. Lauridsen, and R. Feidenhans’l, “Mapping misoriented fibers using x-ray dark field tomography,” Appl. Phys. A 115, 741–745 (2014).
[Crossref]

P. Modregger, M. Kagias, S. Peter, M. Abis, V. A. Guzenko, C. David, and M. Stampanoni, “Multiple scattering tomography,” Phys. Rev. Lett. 113, 020801 (2014).
[Crossref] [PubMed]

P. Modregger, S. Rutishauser, J. Meiser, C. David, and M. Stampanoni, “Two-dimensional ultra-small angle X-ray scattering with grating interferometry,” Appl. Phys. Lett. 105, 024102 (2014).
[Crossref]

F. L. Bayer, S. Hu, A. Maier, T. Weber, G. Anton, T. Michel, and C. P. Riess, “Reconstruction of scalar and vectorial components in x-ray dark-field tomography,” Proc. Natl. Acad. Sci. U. S. A. 111, 12699–12704 (2014).
[Crossref] [PubMed]

S. Peter, P. Modregger, M. K. Fix, W. Volken, D. Frei, P. Manserd, and M. Stampanoni, “Combining Monte Carlo methods with coherent wave optics for the simulation of phase-sensitive x-ray imaging,” J. Synchro. Rad. 21, 613–622 (2014).
[Crossref]

J. Kim, S. W. Lee, and G. Cho, “Visibility evaluation of a neutron grating interferometer operated with a poly-chromatic thermal neutron beam,” Nucl. Instrum. Methods A 746, 26–32 (2014).
[Crossref]

2013 (10)

C. Grünzweig, J. Kopecek, B. Betz, A. Kaestner, K. Jefimovs, J. Kohlbrecher, U. Gasser, O. Bunk, C. David, E. Lehmann, T. Donath, and F. Pfeiffer, “Quantification of the neutron dark-field imaging signal in grating interferometry,” Phys. Rev. B 88, 125104 (2013).
[Crossref]

S. W. Lee, Y. Zhou, T. Zhou, M. Jiang, J. Kim, C. W. Ahn, and A. K. Louis, “Visibility studies of grating-based neutron phase contrast and dark-field imaging by using partial coherence theory,” J. Kor. Phys. Soc. 63, 2093–2097 (2013).
[Crossref]

F. Schwab, S. Schleede, D. Hahn, M. Bech, J. Herzen, S. Auweter, F. Bamberg, K. Achterhold, A. Ö. Yildirim, A. Bohla, O. Eickelberg, R. Loewen, M. Gifford, R. Ruth, M. F. Reiser, K. Nikolaou, F. Pfeiffer, and F. G. Meinel, “Comparison of contrast-to-noise ratios of transmission and dark-field signal in grating-based X-ray imaging for healthy murine lung tissue,” Z. Med. Phys. 33, 236–242 (2013).
[Crossref]

T. Michel, J. Rieger, G. Anton, F. Bayer, M. W. Beckmann, J. Dürst, P. A. Fasching, W. Haas, A. Hartmann, G. Pelzer, M. Radicke, C. Rauh, A. Ritter, P. Sievers, R. Schulz-Wendtland, M. Uder, D. L. Wachter, T. Weber, E. Wenkel, and A. Zang, “On a dark-field signal generated by micrometer-sized calcifications in phase-contrast mammography,” Phys. Med. Biol. 58, 2713–2732 (2013).
[Crossref] [PubMed]

A. Malecki, G. Potdevin, T. Biernath, E. Eggl, E. G. Garcia, T. Baum, P. B. Noël, J. S. Bauer, and F. Pfeiffer, “Coherent superposition in grating-based directional dark-field imaging,” PLOSo One 8, e61268 (2013).
[Crossref]

T. Weber, G. Pelzer, F. Bayer, F. Horn, J. Rieger, A. Ritter, A. Zang, J. Durst, G. Anton, and T. Michel, “Increasing the darkfield contrast-to-noise ratio using a deconvolution-based information retrieval algorithm in X-ray grating-based phase-contrast imaging,” Opt. Express 21, 18011–18020 (2013).
[Crossref] [PubMed]

F. Bayer, S. Zabler, C. Brendel, G. Pelzer, J. Rieger, A. Ritter, T. Weber, T. Michel, and G. Anton, “Projection angle dependence in grating-based X-ray dark-field imaging of ordered structures,” Opt. Express 21, 19923–19933 (2013).
[Crossref]

V. Revol, B. Plank, R. Kaufmann, J. Kastner, C. Kottler, and A. Neels, “Laminate fibre structure characterisation of carbon fibre-reinforced polymers by x-ray scatter dark field imaging with a grating interferometer,” NDT&E Int. 58, 64–71 (2013).
[Crossref]

F. Scattarella, S. Tangaro, P. Modregger, M. Stampanoni, L. De Caro, and R. Bellotti, “Post-detection analysis for grating-based ultra-small angle x-ray scattering,” Phys. Med. 29, 478–486 (2013).
[Crossref] [PubMed]

Z. Wang and Marco Stampanoni, “Quantitative x-ray radiography using grating interferometry: a feasibility study,” Phys. Med. Biol. 58, 6815–6826 (2013).
[Crossref] [PubMed]

2012 (6)

P. Modregger, F. Scattarella, B. R. Pinzer, C. David, R. Bellotti, and M. Stampanoni, “Imaging the ultrasmall-angle x-ray scattering distribution with grating interferometry,” Phys. Rev. Lett. 108, 048101 (2012).
[Crossref] [PubMed]

W. Cong, F. Pfeiffer, M. Bech, and G. Wang, “X-ray dark-field imaging modeling,” J. Opt. Soc. Am. A 29, 908–912 (2012).
[Crossref]

G. Potdevin, A. Malecki, T. Biernath, M. Bech, T. H. Jensen, R. Feidenhans’l, I. Zanette, T. Weitkamp, J. Kenntner, J. Mohr, P. Roschger, M. Kerschnitzki, W. Wagermaier, K. Klaushofer, P. Fratzl, and F. Pfeiffer, “X-ray vector radiography for bone micro-architecture diagnostics,” Phys. Med. Biol. 57, 3451–3461 (2012).
[Crossref] [PubMed]

A. Malecki, G. Potdevin, and F. Pfeiffer, “Quantitative wave-optical numerical analysis of the dark-field signal in grating-based x-ray interferometry,” Europhys. Lett. 99, 48001 (2012).
[Crossref]

Y. Yang and X. Tang, “The second-order differential phase contrast and its retrieval for imaging with x-ray Talbot interferometry,” Med. Phys. 39, 7237–7253 (2012).
[Crossref] [PubMed]

V. Revol, C. Kottler, R. Kaufmann, A. Neels, and A. Dommann, “Orientation-selective x-ray dark field imaging of ordered systems,” J. Appl. Phys. 112, 114903 (2012).
[Crossref]

2011 (7)

E. Hack and J. Burke, “Invited review article: Measurement uncertainty of linear phase-stepping algorithms,” Rev. Sci. Instrum. 82, 061101 (2011).
[Crossref] [PubMed]

S. W. Lee, Y. K. Jun, and O. Y. Kwon, “A neutron dark-field imaging experiment with a neutron grating Iinterferometer at a thermal neutron beam line at HANARO,” J. Korean Phys. Soc. 58, 730–734 (2011).
[Crossref]

S. K. Lynch, V. Pai, J. Auxier, A. F. Stein, E. E. Bennett, C. K. Kemble, X. Xiao, W.-K. Lee, N. Y. Morgan, and H. H. Wen, “Interpretation of dark-field contrast and particle-size selectivity in grating interferometers,” Appl. Opt. 50, 4310–4319 (2011).
[Crossref] [PubMed]

W. Han, J. A. Eichholz, X. Cheng, and G. Wang, “A theoretical framework of x-ray dark-field tomograpy,” Siam. J. Appl. Math. 71, 1557–1577 (2011).
[Crossref]

V. Revol, I. Jerjen, C. Kottler, P. Schütz, R. Kaufmann, T. Lthi, U. Sennhauser, U. Straumann, and C. Urban, “Sub-pixel porosity revealed by x-ray scatter dark field imaging,” J. Appl. Phys. 110, 044912 (2011).
[Crossref]

M. Chabior, T. Donath, C. David, M. Schuster, C. Schroer, and F. Pfeiffer, “Signal-to-noise ratio in x ray dark-field imaging using a grating interferometer,” J. Appl. Phys. 110, 053105 (2011).
[Crossref]

W. Yashiro, S. Harasse, K. Kawabata, H. Kuwabara, T. Yamazaki, and A. Momose, “Distribution of unresolvable anisotropic microstructures revealed in visibility-contrast images using x-ray Talbot interferometry,” Phys. Rev. B 84, 094106 (2011).
[Crossref]

2010 (9)

M. Bech, T. H. Jensen, O. Bunk, T. Donath, C. David, T. Weitkamp, G. Le Duc, A. Bravin, and P. Cloetens, “Advanced contrast modalities for x-ray radiology: phase-contrast and dark-field imaging using a grating interferometer,” Z. Med. Phys. 20, 7–16 (2010).
[Crossref] [PubMed]

A. F. Stein, J. Ilavsky, R. Kopace, E. E. Bennett, and H. Wen, “Selective imaging of nano-particle contrast agents by a single-shot x-ray diffraction technique,” Opt. Express 18, 13271–13278 (2010).
[Crossref] [PubMed]

G.-H. Chen, N. Bevins, J. Zambelli, and Z. Qi, “Small-angle scattering computed tomography (SAS-CT) using a Talbot–Lau interferometer and a rotating anode x-ray tube: theory and experiments,” Opt. Express 18, 12960–12970 (2010).
[Crossref] [PubMed]

M. Bech, O. Bunk, T. Donath, R. Feidenhans’l, C David, and F. Pfeiffer, “Quantitative x-ray dark-field computed tomography,” Phys. Med. Biol. 55, 5529–5539 (2010).
[Crossref] [PubMed]

T. H. Jensen, M. Bech, I. Zanette, T. Weitkamp, C. David, H. Deyhle, S. Rutishauser, E. Reznikova, J. Mohr, R. Feidenhans’l, and F. Pfeiffer, “Directional x-ray dark-field imaging of strongly ordered systems,” Phys. Rev. B 82, 214103 (2010).
[Crossref]

W. Yashiro, Y. Terui, K. Kawabara, and A. Momose, “On the origin of visibility contrast in x-ray Talbot interferometry,” Opt. Express 18, 16890–16901 (2010).
[Crossref] [PubMed]

I. Manke, N. Kardjilov, R. Schäfer, A. Hilger, M. Strobl, M. Dawson, C. Grünzweig, G. Behr, M. Hentschel, C. David, A. Kupsch, A. Lange, and J. Banhart, “Three-dimensional imaging of magnetic domains,” Nat. Commun. 1, 125 (2010).
[Crossref] [PubMed]

V. Revol, C. Kottler, R. Kaufmann, U. Straumann, and C. Urban, “Noise analysis of grating-based x-ray differential phase contrast imaging,” Rev. Sci. Instr. 81, 073709 (2010).
[Crossref]

K. S. Morgan, K. K. W. Siu, and D. Paganin, “The projection approximation and edge contrast for x-ray propagation-based phase contrast imaging of a cylindrical edge,” Opt. Express 18, 9865–9878 (2010).
[Crossref] [PubMed]

2009 (3)

S. W. Lee, D. S. Hussey, D. L. Jacobson, C. M. Sim, and M. Arif, “Development of the grating phase neutron interferometer at a monochromatic beam line,” Nucl. Instrum. Methods A 605, 16–20 (2009).
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M. Strobl, A. Hilger, N. Kardjilov, O. Ebrahimi, S. Keil, and I. Manke, “Differential phase contrast and dark field neutron imaging,” Nucl. Instrum. Methods A 605, 9–12 (2009).
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Z.-T. Wang, K.-J. Kang, Z.-F. Huang, and Z.-Q. Chen, “Quantitative grating-based x-ray dark-field computed tomography,” Appl. Phys. Lett. 95, 094105 (2009).
[Crossref]

2008 (6)

F. Pfeiffer, T. Weitkamp, O. Bunk, and C. David, “Hard-x-ray dark-field imaging using a grating interferometer,” Nat. Mat. 7, 134–137 (2008).
[Crossref]

W. Yashiro, Y. Takeda, and A. Momose, “Efficiency of capturing a phase image using cone-beam x-ray Talbot interferometry,” J. Opt. Soc. Am. A 25, 2025–2039 (2008).
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C. Grünzweig, C. David, O. Bunk, M. Dierolf, G. Frei, G. Kühne, J. Kohlbrecher, R. Schäfer, P. Lejcek, H. M. R. Ronnow, and F. Pfeiffer, “Neutron decoherence imaging for visualizing bulk magnetic domain structures,” Phys. Rev. Lett. 101, 025504 (2008).
[Crossref] [PubMed]

C. Grünzweig, C. David, O. Bunk, M. Dierolf, G. Frei, G. Kuhne, R. Schafer, S. Pofahl, H. M. R. Ronnow, and F. Pfeiffer, “Bulk magnetic domain structures visualized by neutron dark-field imaging,” Appl. Phys. Lett. 93, 112504 (2008).
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M. Strobl, C. Grünzweig, A. Hilger, I. Manke, N. Kardjilov, C. David, and F. Pfeiffer, “Neutron dark-field tomography,” Phys. Rev. Lett. 101, 123902 (2008).
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Ya. I. Nesterets, “On the origins of decoherence and extinction contrast in phase-contrast imaging,” Opt. Commun. 281, 533–542 (2008).
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2006 (2)

A. Momose, W. Yashiro, Y. Takeda, Y. Suzuki, and T. Hattori, “Phase tomography by x-ray Talbot interferometry for biological imaging,” Jpn. J. Appl. Phys. 45, 5254–5262 (2006).
[Crossref]

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)

T. Weitkamp, B. Nöhammer, A. Diaz, C. David, and E. Ziegler, “X-ray wavefront analysis and optics characterization with a grating interferometer,” Appl. Phys. Lett. 86, 054101 (2005).
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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).
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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).
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1992 (1)

K. Ohmi, T. Nogami, Y. Fukushima, M. Katoh, and T. Yamakawa, “Characteristics of the five-pole superconducting vertical wiggler at the Photon Factory,” Rev. Sci. Instr. 63, 301–304 (1992).
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1990 (1)

1982 (1)

1974 (1)

Abis, M.

P. Modregger, M. Kagias, S. Peter, M. Abis, V. A. Guzenko, C. David, and M. Stampanoni, “Multiple scattering tomography,” Phys. Rev. Lett. 113, 020801 (2014).
[Crossref] [PubMed]

Achterhold, K.

F. Schwab, S. Schleede, D. Hahn, M. Bech, J. Herzen, S. Auweter, F. Bamberg, K. Achterhold, A. Ö. Yildirim, A. Bohla, O. Eickelberg, R. Loewen, M. Gifford, R. Ruth, M. F. Reiser, K. Nikolaou, F. Pfeiffer, and F. G. Meinel, “Comparison of contrast-to-noise ratios of transmission and dark-field signal in grating-based X-ray imaging for healthy murine lung tissue,” Z. Med. Phys. 33, 236–242 (2013).
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Ahn, C. W.

S. W. Lee, Y. Zhou, T. Zhou, M. Jiang, J. Kim, C. W. Ahn, and A. K. Louis, “Visibility studies of grating-based neutron phase contrast and dark-field imaging by using partial coherence theory,” J. Kor. Phys. Soc. 63, 2093–2097 (2013).
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Anton, G.

F. L. Bayer, S. Hu, A. Maier, T. Weber, G. Anton, T. Michel, and C. P. Riess, “Reconstruction of scalar and vectorial components in x-ray dark-field tomography,” Proc. Natl. Acad. Sci. U. S. A. 111, 12699–12704 (2014).
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F. Bayer, S. Zabler, C. Brendel, G. Pelzer, J. Rieger, A. Ritter, T. Weber, T. Michel, and G. Anton, “Projection angle dependence in grating-based X-ray dark-field imaging of ordered structures,” Opt. Express 21, 19923–19933 (2013).
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T. Michel, J. Rieger, G. Anton, F. Bayer, M. W. Beckmann, J. Dürst, P. A. Fasching, W. Haas, A. Hartmann, G. Pelzer, M. Radicke, C. Rauh, A. Ritter, P. Sievers, R. Schulz-Wendtland, M. Uder, D. L. Wachter, T. Weber, E. Wenkel, and A. Zang, “On a dark-field signal generated by micrometer-sized calcifications in phase-contrast mammography,” Phys. Med. Biol. 58, 2713–2732 (2013).
[Crossref] [PubMed]

T. Weber, G. Pelzer, F. Bayer, F. Horn, J. Rieger, A. Ritter, A. Zang, J. Durst, G. Anton, and T. Michel, “Increasing the darkfield contrast-to-noise ratio using a deconvolution-based information retrieval algorithm in X-ray grating-based phase-contrast imaging,” Opt. Express 21, 18011–18020 (2013).
[Crossref] [PubMed]

Arif, M.

S. W. Lee, D. S. Hussey, D. L. Jacobson, C. M. Sim, and M. Arif, “Development of the grating phase neutron interferometer at a monochromatic beam line,” Nucl. Instrum. Methods A 605, 16–20 (2009).
[Crossref]

Auweter, S.

F. Schwab, S. Schleede, D. Hahn, M. Bech, J. Herzen, S. Auweter, F. Bamberg, K. Achterhold, A. Ö. Yildirim, A. Bohla, O. Eickelberg, R. Loewen, M. Gifford, R. Ruth, M. F. Reiser, K. Nikolaou, F. Pfeiffer, and F. G. Meinel, “Comparison of contrast-to-noise ratios of transmission and dark-field signal in grating-based X-ray imaging for healthy murine lung tissue,” Z. Med. Phys. 33, 236–242 (2013).
[Crossref]

Auxier, J.

Bamberg, F.

F. Schwab, S. Schleede, D. Hahn, M. Bech, J. Herzen, S. Auweter, F. Bamberg, K. Achterhold, A. Ö. Yildirim, A. Bohla, O. Eickelberg, R. Loewen, M. Gifford, R. Ruth, M. F. Reiser, K. Nikolaou, F. Pfeiffer, and F. G. Meinel, “Comparison of contrast-to-noise ratios of transmission and dark-field signal in grating-based X-ray imaging for healthy murine lung tissue,” Z. Med. Phys. 33, 236–242 (2013).
[Crossref]

Banhart, J.

I. Manke, N. Kardjilov, R. Schäfer, A. Hilger, M. Strobl, M. Dawson, C. Grünzweig, G. Behr, M. Hentschel, C. David, A. Kupsch, A. Lange, and J. Banhart, “Three-dimensional imaging of magnetic domains,” Nat. Commun. 1, 125 (2010).
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Bauer, J. S.

F. Schaff, A. Malecki, G. Potdevin, E. Eggl, P. B. Noël, T. Baum, E. G. Garcia, J. S. Bauer, and F. Pfeiffer, “Correlation of x-ray vector radiography to bone micro-architecture,” Sci. Rep. 4, 3695 (2014).
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A. Malecki, G. Potdevin, T. Biernath, E. Eggl, E. G. Garcia, T. Baum, P. B. Noël, J. S. Bauer, and F. Pfeiffer, “Coherent superposition in grating-based directional dark-field imaging,” PLOSo One 8, e61268 (2013).
[Crossref]

Baum, T.

F. Schaff, A. Malecki, G. Potdevin, E. Eggl, P. B. Noël, T. Baum, E. G. Garcia, J. S. Bauer, and F. Pfeiffer, “Correlation of x-ray vector radiography to bone micro-architecture,” Sci. Rep. 4, 3695 (2014).
[Crossref] [PubMed]

A. Malecki, G. Potdevin, T. Biernath, E. Eggl, E. G. Garcia, T. Baum, P. B. Noël, J. S. Bauer, and F. Pfeiffer, “Coherent superposition in grating-based directional dark-field imaging,” PLOSo One 8, e61268 (2013).
[Crossref]

Bayer, F.

F. Bayer, S. Zabler, C. Brendel, G. Pelzer, J. Rieger, A. Ritter, T. Weber, T. Michel, and G. Anton, “Projection angle dependence in grating-based X-ray dark-field imaging of ordered structures,” Opt. Express 21, 19923–19933 (2013).
[Crossref]

T. Michel, J. Rieger, G. Anton, F. Bayer, M. W. Beckmann, J. Dürst, P. A. Fasching, W. Haas, A. Hartmann, G. Pelzer, M. Radicke, C. Rauh, A. Ritter, P. Sievers, R. Schulz-Wendtland, M. Uder, D. L. Wachter, T. Weber, E. Wenkel, and A. Zang, “On a dark-field signal generated by micrometer-sized calcifications in phase-contrast mammography,” Phys. Med. Biol. 58, 2713–2732 (2013).
[Crossref] [PubMed]

T. Weber, G. Pelzer, F. Bayer, F. Horn, J. Rieger, A. Ritter, A. Zang, J. Durst, G. Anton, and T. Michel, “Increasing the darkfield contrast-to-noise ratio using a deconvolution-based information retrieval algorithm in X-ray grating-based phase-contrast imaging,” Opt. Express 21, 18011–18020 (2013).
[Crossref] [PubMed]

Bayer, F. L.

F. L. Bayer, S. Hu, A. Maier, T. Weber, G. Anton, T. Michel, and C. P. Riess, “Reconstruction of scalar and vectorial components in x-ray dark-field tomography,” Proc. Natl. Acad. Sci. U. S. A. 111, 12699–12704 (2014).
[Crossref] [PubMed]

Bech, M.

F. Schwab, S. Schleede, D. Hahn, M. Bech, J. Herzen, S. Auweter, F. Bamberg, K. Achterhold, A. Ö. Yildirim, A. Bohla, O. Eickelberg, R. Loewen, M. Gifford, R. Ruth, M. F. Reiser, K. Nikolaou, F. Pfeiffer, and F. G. Meinel, “Comparison of contrast-to-noise ratios of transmission and dark-field signal in grating-based X-ray imaging for healthy murine lung tissue,” Z. Med. Phys. 33, 236–242 (2013).
[Crossref]

G. Potdevin, A. Malecki, T. Biernath, M. Bech, T. H. Jensen, R. Feidenhans’l, I. Zanette, T. Weitkamp, J. Kenntner, J. Mohr, P. Roschger, M. Kerschnitzki, W. Wagermaier, K. Klaushofer, P. Fratzl, and F. Pfeiffer, “X-ray vector radiography for bone micro-architecture diagnostics,” Phys. Med. Biol. 57, 3451–3461 (2012).
[Crossref] [PubMed]

W. Cong, F. Pfeiffer, M. Bech, and G. Wang, “X-ray dark-field imaging modeling,” J. Opt. Soc. Am. A 29, 908–912 (2012).
[Crossref]

M. Bech, T. H. Jensen, O. Bunk, T. Donath, C. David, T. Weitkamp, G. Le Duc, A. Bravin, and P. Cloetens, “Advanced contrast modalities for x-ray radiology: phase-contrast and dark-field imaging using a grating interferometer,” Z. Med. Phys. 20, 7–16 (2010).
[Crossref] [PubMed]

M. Bech, O. Bunk, T. Donath, R. Feidenhans’l, C David, and F. Pfeiffer, “Quantitative x-ray dark-field computed tomography,” Phys. Med. Biol. 55, 5529–5539 (2010).
[Crossref] [PubMed]

T. H. Jensen, M. Bech, I. Zanette, T. Weitkamp, C. David, H. Deyhle, S. Rutishauser, E. Reznikova, J. Mohr, R. Feidenhans’l, and F. Pfeiffer, “Directional x-ray dark-field imaging of strongly ordered systems,” Phys. Rev. B 82, 214103 (2010).
[Crossref]

Beckmann, M. W.

T. Michel, J. Rieger, G. Anton, F. Bayer, M. W. Beckmann, J. Dürst, P. A. Fasching, W. Haas, A. Hartmann, G. Pelzer, M. Radicke, C. Rauh, A. Ritter, P. Sievers, R. Schulz-Wendtland, M. Uder, D. L. Wachter, T. Weber, E. Wenkel, and A. Zang, “On a dark-field signal generated by micrometer-sized calcifications in phase-contrast mammography,” Phys. Med. Biol. 58, 2713–2732 (2013).
[Crossref] [PubMed]

Behr, G.

I. Manke, N. Kardjilov, R. Schäfer, A. Hilger, M. Strobl, M. Dawson, C. Grünzweig, G. Behr, M. Hentschel, C. David, A. Kupsch, A. Lange, and J. Banhart, “Three-dimensional imaging of magnetic domains,” Nat. Commun. 1, 125 (2010).
[Crossref] [PubMed]

Bellotti, R.

F. Scattarella, S. Tangaro, P. Modregger, M. Stampanoni, L. De Caro, and R. Bellotti, “Post-detection analysis for grating-based ultra-small angle x-ray scattering,” Phys. Med. 29, 478–486 (2013).
[Crossref] [PubMed]

P. Modregger, F. Scattarella, B. R. Pinzer, C. David, R. Bellotti, and M. Stampanoni, “Imaging the ultrasmall-angle x-ray scattering distribution with grating interferometry,” Phys. Rev. Lett. 108, 048101 (2012).
[Crossref] [PubMed]

Bennett, E. E.

Betz, B.

C. Grünzweig, J. Kopecek, B. Betz, A. Kaestner, K. Jefimovs, J. Kohlbrecher, U. Gasser, O. Bunk, C. David, E. Lehmann, T. Donath, and F. Pfeiffer, “Quantification of the neutron dark-field imaging signal in grating interferometry,” Phys. Rev. B 88, 125104 (2013).
[Crossref]

Bevins, N.

Biernath, T.

A. Malecki, G. Potdevin, T. Biernath, E. Eggl, K. Willer, T. Lasser, J. Maisenbacher, J. Gibmeier, A. Wanner, and F. Pfeiffer, “X-ray tensor tomography,” Europhys. Lett. 105, 38002 (2014).
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A. Malecki, G. Potdevin, T. Biernath, E. Eggl, E. G. Garcia, T. Baum, P. B. Noël, J. S. Bauer, and F. Pfeiffer, “Coherent superposition in grating-based directional dark-field imaging,” PLOSo One 8, e61268 (2013).
[Crossref]

G. Potdevin, A. Malecki, T. Biernath, M. Bech, T. H. Jensen, R. Feidenhans’l, I. Zanette, T. Weitkamp, J. Kenntner, J. Mohr, P. Roschger, M. Kerschnitzki, W. Wagermaier, K. Klaushofer, P. Fratzl, and F. Pfeiffer, “X-ray vector radiography for bone micro-architecture diagnostics,” Phys. Med. Biol. 57, 3451–3461 (2012).
[Crossref] [PubMed]

Bohla, A.

F. Schwab, S. Schleede, D. Hahn, M. Bech, J. Herzen, S. Auweter, F. Bamberg, K. Achterhold, A. Ö. Yildirim, A. Bohla, O. Eickelberg, R. Loewen, M. Gifford, R. Ruth, M. F. Reiser, K. Nikolaou, F. Pfeiffer, and F. G. Meinel, “Comparison of contrast-to-noise ratios of transmission and dark-field signal in grating-based X-ray imaging for healthy murine lung tissue,” Z. Med. Phys. 33, 236–242 (2013).
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Born, M.

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge University Press, 1999).
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Brangaccio, D. J.

Bravin, A.

M. Bech, T. H. Jensen, O. Bunk, T. Donath, C. David, T. Weitkamp, G. Le Duc, A. Bravin, and P. Cloetens, “Advanced contrast modalities for x-ray radiology: phase-contrast and dark-field imaging using a grating interferometer,” Z. Med. Phys. 20, 7–16 (2010).
[Crossref] [PubMed]

Brendel, C.

F. Bayer, S. Zabler, C. Brendel, G. Pelzer, J. Rieger, A. Ritter, T. Weber, T. Michel, and G. Anton, “Projection angle dependence in grating-based X-ray dark-field imaging of ordered structures,” Opt. Express 21, 19923–19933 (2013).
[Crossref]

Bruning, J. H.

Bunk, O.

C. Grünzweig, J. Kopecek, B. Betz, A. Kaestner, K. Jefimovs, J. Kohlbrecher, U. Gasser, O. Bunk, C. David, E. Lehmann, T. Donath, and F. Pfeiffer, “Quantification of the neutron dark-field imaging signal in grating interferometry,” Phys. Rev. B 88, 125104 (2013).
[Crossref]

M. Bech, T. H. Jensen, O. Bunk, T. Donath, C. David, T. Weitkamp, G. Le Duc, A. Bravin, and P. Cloetens, “Advanced contrast modalities for x-ray radiology: phase-contrast and dark-field imaging using a grating interferometer,” Z. Med. Phys. 20, 7–16 (2010).
[Crossref] [PubMed]

M. Bech, O. Bunk, T. Donath, R. Feidenhans’l, C David, and F. Pfeiffer, “Quantitative x-ray dark-field computed tomography,” Phys. Med. Biol. 55, 5529–5539 (2010).
[Crossref] [PubMed]

F. Pfeiffer, T. Weitkamp, O. Bunk, and C. David, “Hard-x-ray dark-field imaging using a grating interferometer,” Nat. Mat. 7, 134–137 (2008).
[Crossref]

C. Grünzweig, C. David, O. Bunk, M. Dierolf, G. Frei, G. Kühne, J. Kohlbrecher, R. Schäfer, P. Lejcek, H. M. R. Ronnow, and F. Pfeiffer, “Neutron decoherence imaging for visualizing bulk magnetic domain structures,” Phys. Rev. Lett. 101, 025504 (2008).
[Crossref] [PubMed]

C. Grünzweig, C. David, O. Bunk, M. Dierolf, G. Frei, G. Kuhne, R. Schafer, S. Pofahl, H. M. R. Ronnow, and F. Pfeiffer, “Bulk magnetic domain structures visualized by neutron dark-field imaging,” Appl. Phys. Lett. 93, 112504 (2008).
[Crossref]

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).
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E. Hack and J. Burke, “Invited review article: Measurement uncertainty of linear phase-stepping algorithms,” Rev. Sci. Instrum. 82, 061101 (2011).
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M. Chabior, T. Donath, C. David, M. Schuster, C. Schroer, and F. Pfeiffer, “Signal-to-noise ratio in x ray dark-field imaging using a grating interferometer,” J. Appl. Phys. 110, 053105 (2011).
[Crossref]

Chen, G.-H.

Chen, Z.-Q.

Z.-T. Wang, K.-J. Kang, Z.-F. Huang, and Z.-Q. Chen, “Quantitative grating-based x-ray dark-field computed tomography,” Appl. Phys. Lett. 95, 094105 (2009).
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Cheng, X.

W. Han, J. A. Eichholz, X. Cheng, and G. Wang, “A theoretical framework of x-ray dark-field tomograpy,” Siam. J. Appl. Math. 71, 1557–1577 (2011).
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Cho, G.

J. Kim, S. W. Lee, and G. Cho, “Visibility evaluation of a neutron grating interferometer operated with a poly-chromatic thermal neutron beam,” Nucl. Instrum. Methods A 746, 26–32 (2014).
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Cloetens, P.

M. Bech, T. H. Jensen, O. Bunk, T. Donath, C. David, T. Weitkamp, G. Le Duc, A. Bravin, and P. Cloetens, “Advanced contrast modalities for x-ray radiology: phase-contrast and dark-field imaging using a grating interferometer,” Z. Med. Phys. 20, 7–16 (2010).
[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]

Cong, W.

David, C

M. Bech, O. Bunk, T. Donath, R. Feidenhans’l, C David, and F. Pfeiffer, “Quantitative x-ray dark-field computed tomography,” Phys. Med. Biol. 55, 5529–5539 (2010).
[Crossref] [PubMed]

David, C.

P. Modregger, S. Rutishauser, J. Meiser, C. David, and M. Stampanoni, “Two-dimensional ultra-small angle X-ray scattering with grating interferometry,” Appl. Phys. Lett. 105, 024102 (2014).
[Crossref]

P. Modregger, M. Kagias, S. Peter, M. Abis, V. A. Guzenko, C. David, and M. Stampanoni, “Multiple scattering tomography,” Phys. Rev. Lett. 113, 020801 (2014).
[Crossref] [PubMed]

C. Grünzweig, J. Kopecek, B. Betz, A. Kaestner, K. Jefimovs, J. Kohlbrecher, U. Gasser, O. Bunk, C. David, E. Lehmann, T. Donath, and F. Pfeiffer, “Quantification of the neutron dark-field imaging signal in grating interferometry,” Phys. Rev. B 88, 125104 (2013).
[Crossref]

P. Modregger, F. Scattarella, B. R. Pinzer, C. David, R. Bellotti, and M. Stampanoni, “Imaging the ultrasmall-angle x-ray scattering distribution with grating interferometry,” Phys. Rev. Lett. 108, 048101 (2012).
[Crossref] [PubMed]

M. Chabior, T. Donath, C. David, M. Schuster, C. Schroer, and F. Pfeiffer, “Signal-to-noise ratio in x ray dark-field imaging using a grating interferometer,” J. Appl. Phys. 110, 053105 (2011).
[Crossref]

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P. Modregger, M. Kagias, S. Peter, M. Abis, V. A. Guzenko, C. David, and M. Stampanoni, “Multiple scattering tomography,” Phys. Rev. Lett. 113, 020801 (2014).
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Z.-T. Wang, K.-J. Kang, Z.-F. Huang, and Z.-Q. Chen, “Quantitative grating-based x-ray dark-field computed tomography,” Appl. Phys. Lett. 95, 094105 (2009).
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I. Manke, N. Kardjilov, R. Schäfer, A. Hilger, M. Strobl, M. Dawson, C. Grünzweig, G. Behr, M. Hentschel, C. David, A. Kupsch, A. Lange, and J. Banhart, “Three-dimensional imaging of magnetic domains,” Nat. Commun. 1, 125 (2010).
[Crossref] [PubMed]

M. Strobl, A. Hilger, N. Kardjilov, O. Ebrahimi, S. Keil, and I. Manke, “Differential phase contrast and dark field neutron imaging,” Nucl. Instrum. Methods A 605, 9–12 (2009).
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M. Strobl, C. Grünzweig, A. Hilger, I. Manke, N. Kardjilov, C. David, and F. Pfeiffer, “Neutron dark-field tomography,” Phys. Rev. Lett. 101, 123902 (2008).
[Crossref] [PubMed]

Kastner, J.

V. Revol, B. Plank, R. Kaufmann, J. Kastner, C. Kottler, and A. Neels, “Laminate fibre structure characterisation of carbon fibre-reinforced polymers by x-ray scatter dark field imaging with a grating interferometer,” NDT&E Int. 58, 64–71 (2013).
[Crossref]

Katoh, M.

K. Ohmi, T. Nogami, Y. Fukushima, M. Katoh, and T. Yamakawa, “Characteristics of the five-pole superconducting vertical wiggler at the Photon Factory,” Rev. Sci. Instr. 63, 301–304 (1992).
[Crossref]

Kaufmann, R.

V. Revol, B. Plank, R. Kaufmann, J. Kastner, C. Kottler, and A. Neels, “Laminate fibre structure characterisation of carbon fibre-reinforced polymers by x-ray scatter dark field imaging with a grating interferometer,” NDT&E Int. 58, 64–71 (2013).
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V. Revol, C. Kottler, R. Kaufmann, A. Neels, and A. Dommann, “Orientation-selective x-ray dark field imaging of ordered systems,” J. Appl. Phys. 112, 114903 (2012).
[Crossref]

V. Revol, I. Jerjen, C. Kottler, P. Schütz, R. Kaufmann, T. Lthi, U. Sennhauser, U. Straumann, and C. Urban, “Sub-pixel porosity revealed by x-ray scatter dark field imaging,” J. Appl. Phys. 110, 044912 (2011).
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V. Revol, C. Kottler, R. Kaufmann, U. Straumann, and C. Urban, “Noise analysis of grating-based x-ray differential phase contrast imaging,” Rev. Sci. Instr. 81, 073709 (2010).
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Kawabara, K.

Kawabata, K.

W. Yashiro, S. Harasse, K. Kawabata, H. Kuwabara, T. Yamazaki, and A. Momose, “Distribution of unresolvable anisotropic microstructures revealed in visibility-contrast images using x-ray Talbot interferometry,” Phys. Rev. B 84, 094106 (2011).
[Crossref]

Kawamoto, S.

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]

Keil, S.

M. Strobl, A. Hilger, N. Kardjilov, O. Ebrahimi, S. Keil, and I. Manke, “Differential phase contrast and dark field neutron imaging,” Nucl. Instrum. Methods A 605, 9–12 (2009).
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Kemble, C. K.

Kenntner, J.

G. Potdevin, A. Malecki, T. Biernath, M. Bech, T. H. Jensen, R. Feidenhans’l, I. Zanette, T. Weitkamp, J. Kenntner, J. Mohr, P. Roschger, M. Kerschnitzki, W. Wagermaier, K. Klaushofer, P. Fratzl, and F. Pfeiffer, “X-ray vector radiography for bone micro-architecture diagnostics,” Phys. Med. Biol. 57, 3451–3461 (2012).
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Kerschnitzki, M.

G. Potdevin, A. Malecki, T. Biernath, M. Bech, T. H. Jensen, R. Feidenhans’l, I. Zanette, T. Weitkamp, J. Kenntner, J. Mohr, P. Roschger, M. Kerschnitzki, W. Wagermaier, K. Klaushofer, P. Fratzl, and F. Pfeiffer, “X-ray vector radiography for bone micro-architecture diagnostics,” Phys. Med. Biol. 57, 3451–3461 (2012).
[Crossref] [PubMed]

Kim, J.

J. Kim, S. W. Lee, and G. Cho, “Visibility evaluation of a neutron grating interferometer operated with a poly-chromatic thermal neutron beam,” Nucl. Instrum. Methods A 746, 26–32 (2014).
[Crossref]

S. W. Lee, Y. Zhou, T. Zhou, M. Jiang, J. Kim, C. W. Ahn, and A. K. Louis, “Visibility studies of grating-based neutron phase contrast and dark-field imaging by using partial coherence theory,” J. Kor. Phys. Soc. 63, 2093–2097 (2013).
[Crossref]

Klaushofer, K.

G. Potdevin, A. Malecki, T. Biernath, M. Bech, T. H. Jensen, R. Feidenhans’l, I. Zanette, T. Weitkamp, J. Kenntner, J. Mohr, P. Roschger, M. Kerschnitzki, W. Wagermaier, K. Klaushofer, P. Fratzl, and F. Pfeiffer, “X-ray vector radiography for bone micro-architecture diagnostics,” Phys. Med. Biol. 57, 3451–3461 (2012).
[Crossref] [PubMed]

Kobayashi, S.

Kohlbrecher, J.

C. Grünzweig, J. Kopecek, B. Betz, A. Kaestner, K. Jefimovs, J. Kohlbrecher, U. Gasser, O. Bunk, C. David, E. Lehmann, T. Donath, and F. Pfeiffer, “Quantification of the neutron dark-field imaging signal in grating interferometry,” Phys. Rev. B 88, 125104 (2013).
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C. Grünzweig, C. David, O. Bunk, M. Dierolf, G. Frei, G. Kühne, J. Kohlbrecher, R. Schäfer, P. Lejcek, H. M. R. Ronnow, and F. Pfeiffer, “Neutron decoherence imaging for visualizing bulk magnetic domain structures,” Phys. Rev. Lett. 101, 025504 (2008).
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Kopecek, J.

C. Grünzweig, J. Kopecek, B. Betz, A. Kaestner, K. Jefimovs, J. Kohlbrecher, U. Gasser, O. Bunk, C. David, E. Lehmann, T. Donath, and F. Pfeiffer, “Quantification of the neutron dark-field imaging signal in grating interferometry,” Phys. Rev. B 88, 125104 (2013).
[Crossref]

Kottler, C.

V. Revol, B. Plank, R. Kaufmann, J. Kastner, C. Kottler, and A. Neels, “Laminate fibre structure characterisation of carbon fibre-reinforced polymers by x-ray scatter dark field imaging with a grating interferometer,” NDT&E Int. 58, 64–71 (2013).
[Crossref]

V. Revol, C. Kottler, R. Kaufmann, A. Neels, and A. Dommann, “Orientation-selective x-ray dark field imaging of ordered systems,” J. Appl. Phys. 112, 114903 (2012).
[Crossref]

V. Revol, I. Jerjen, C. Kottler, P. Schütz, R. Kaufmann, T. Lthi, U. Sennhauser, U. Straumann, and C. Urban, “Sub-pixel porosity revealed by x-ray scatter dark field imaging,” J. Appl. Phys. 110, 044912 (2011).
[Crossref]

V. Revol, C. Kottler, R. Kaufmann, U. Straumann, and C. Urban, “Noise analysis of grating-based x-ray differential phase contrast imaging,” Rev. Sci. Instr. 81, 073709 (2010).
[Crossref]

Koyama, I.

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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C. Grünzweig, C. David, O. Bunk, M. Dierolf, G. Frei, G. Kuhne, R. Schafer, S. Pofahl, H. M. R. Ronnow, and F. Pfeiffer, “Bulk magnetic domain structures visualized by neutron dark-field imaging,” Appl. Phys. Lett. 93, 112504 (2008).
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C. Grünzweig, C. David, O. Bunk, M. Dierolf, G. Frei, G. Kühne, J. Kohlbrecher, R. Schäfer, P. Lejcek, H. M. R. Ronnow, and F. Pfeiffer, “Neutron decoherence imaging for visualizing bulk magnetic domain structures,” Phys. Rev. Lett. 101, 025504 (2008).
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Kupsch, A.

I. Manke, N. Kardjilov, R. Schäfer, A. Hilger, M. Strobl, M. Dawson, C. Grünzweig, G. Behr, M. Hentschel, C. David, A. Kupsch, A. Lange, and J. Banhart, “Three-dimensional imaging of magnetic domains,” Nat. Commun. 1, 125 (2010).
[Crossref] [PubMed]

Kuwabara, H.

W. Yashiro, S. Harasse, K. Kawabata, H. Kuwabara, T. Yamazaki, and A. Momose, “Distribution of unresolvable anisotropic microstructures revealed in visibility-contrast images using x-ray Talbot interferometry,” Phys. Rev. B 84, 094106 (2011).
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S. W. Lee, Y. K. Jun, and O. Y. Kwon, “A neutron dark-field imaging experiment with a neutron grating Iinterferometer at a thermal neutron beam line at HANARO,” J. Korean Phys. Soc. 58, 730–734 (2011).
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Lange, A.

I. Manke, N. Kardjilov, R. Schäfer, A. Hilger, M. Strobl, M. Dawson, C. Grünzweig, G. Behr, M. Hentschel, C. David, A. Kupsch, A. Lange, and J. Banhart, “Three-dimensional imaging of magnetic domains,” Nat. Commun. 1, 125 (2010).
[Crossref] [PubMed]

Lasser, T.

A. Malecki, G. Potdevin, T. Biernath, E. Eggl, K. Willer, T. Lasser, J. Maisenbacher, J. Gibmeier, A. Wanner, and F. Pfeiffer, “X-ray tensor tomography,” Europhys. Lett. 105, 38002 (2014).
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T. Lauridsen, E. M. Lauridsen, and R. Feidenhans’l, “Mapping misoriented fibers using x-ray dark field tomography,” Appl. Phys. A 115, 741–745 (2014).
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T. Lauridsen, E. M. Lauridsen, and R. Feidenhans’l, “Mapping misoriented fibers using x-ray dark field tomography,” Appl. Phys. A 115, 741–745 (2014).
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M. Bech, T. H. Jensen, O. Bunk, T. Donath, C. David, T. Weitkamp, G. Le Duc, A. Bravin, and P. Cloetens, “Advanced contrast modalities for x-ray radiology: phase-contrast and dark-field imaging using a grating interferometer,” Z. Med. Phys. 20, 7–16 (2010).
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Lee, S. W.

J. Kim, S. W. Lee, and G. Cho, “Visibility evaluation of a neutron grating interferometer operated with a poly-chromatic thermal neutron beam,” Nucl. Instrum. Methods A 746, 26–32 (2014).
[Crossref]

S. W. Lee, Y. Zhou, T. Zhou, M. Jiang, J. Kim, C. W. Ahn, and A. K. Louis, “Visibility studies of grating-based neutron phase contrast and dark-field imaging by using partial coherence theory,” J. Kor. Phys. Soc. 63, 2093–2097 (2013).
[Crossref]

S. W. Lee, Y. K. Jun, and O. Y. Kwon, “A neutron dark-field imaging experiment with a neutron grating Iinterferometer at a thermal neutron beam line at HANARO,” J. Korean Phys. Soc. 58, 730–734 (2011).
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S. W. Lee, D. S. Hussey, D. L. Jacobson, C. M. Sim, and M. Arif, “Development of the grating phase neutron interferometer at a monochromatic beam line,” Nucl. Instrum. Methods A 605, 16–20 (2009).
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Lehmann, E.

C. Grünzweig, J. Kopecek, B. Betz, A. Kaestner, K. Jefimovs, J. Kohlbrecher, U. Gasser, O. Bunk, C. David, E. Lehmann, T. Donath, and F. Pfeiffer, “Quantification of the neutron dark-field imaging signal in grating interferometry,” Phys. Rev. B 88, 125104 (2013).
[Crossref]

Lejcek, P.

C. Grünzweig, C. David, O. Bunk, M. Dierolf, G. Frei, G. Kühne, J. Kohlbrecher, R. Schäfer, P. Lejcek, H. M. R. Ronnow, and F. Pfeiffer, “Neutron decoherence imaging for visualizing bulk magnetic domain structures,” Phys. Rev. Lett. 101, 025504 (2008).
[Crossref] [PubMed]

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F. Schwab, S. Schleede, D. Hahn, M. Bech, J. Herzen, S. Auweter, F. Bamberg, K. Achterhold, A. Ö. Yildirim, A. Bohla, O. Eickelberg, R. Loewen, M. Gifford, R. Ruth, M. F. Reiser, K. Nikolaou, F. Pfeiffer, and F. G. Meinel, “Comparison of contrast-to-noise ratios of transmission and dark-field signal in grating-based X-ray imaging for healthy murine lung tissue,” Z. Med. Phys. 33, 236–242 (2013).
[Crossref]

Lohmann, A. W.

Louis, A. K.

S. W. Lee, Y. Zhou, T. Zhou, M. Jiang, J. Kim, C. W. Ahn, and A. K. Louis, “Visibility studies of grating-based neutron phase contrast and dark-field imaging by using partial coherence theory,” J. Kor. Phys. Soc. 63, 2093–2097 (2013).
[Crossref]

Lthi, T.

V. Revol, I. Jerjen, C. Kottler, P. Schütz, R. Kaufmann, T. Lthi, U. Sennhauser, U. Straumann, and C. Urban, “Sub-pixel porosity revealed by x-ray scatter dark field imaging,” J. Appl. Phys. 110, 044912 (2011).
[Crossref]

Lynch, S. K.

Maier, A.

F. L. Bayer, S. Hu, A. Maier, T. Weber, G. Anton, T. Michel, and C. P. Riess, “Reconstruction of scalar and vectorial components in x-ray dark-field tomography,” Proc. Natl. Acad. Sci. U. S. A. 111, 12699–12704 (2014).
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Maisenbacher, J.

A. Malecki, G. Potdevin, T. Biernath, E. Eggl, K. Willer, T. Lasser, J. Maisenbacher, J. Gibmeier, A. Wanner, and F. Pfeiffer, “X-ray tensor tomography,” Europhys. Lett. 105, 38002 (2014).
[Crossref]

Malecki, A.

A. Malecki, G. Potdevin, T. Biernath, E. Eggl, K. Willer, T. Lasser, J. Maisenbacher, J. Gibmeier, A. Wanner, and F. Pfeiffer, “X-ray tensor tomography,” Europhys. Lett. 105, 38002 (2014).
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F. Schaff, A. Malecki, G. Potdevin, E. Eggl, P. B. Noël, T. Baum, E. G. Garcia, J. S. Bauer, and F. Pfeiffer, “Correlation of x-ray vector radiography to bone micro-architecture,” Sci. Rep. 4, 3695 (2014).
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A. Malecki, G. Potdevin, T. Biernath, E. Eggl, E. G. Garcia, T. Baum, P. B. Noël, J. S. Bauer, and F. Pfeiffer, “Coherent superposition in grating-based directional dark-field imaging,” PLOSo One 8, e61268 (2013).
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A. Malecki, G. Potdevin, and F. Pfeiffer, “Quantitative wave-optical numerical analysis of the dark-field signal in grating-based x-ray interferometry,” Europhys. Lett. 99, 48001 (2012).
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G. Potdevin, A. Malecki, T. Biernath, M. Bech, T. H. Jensen, R. Feidenhans’l, I. Zanette, T. Weitkamp, J. Kenntner, J. Mohr, P. Roschger, M. Kerschnitzki, W. Wagermaier, K. Klaushofer, P. Fratzl, and F. Pfeiffer, “X-ray vector radiography for bone micro-architecture diagnostics,” Phys. Med. Biol. 57, 3451–3461 (2012).
[Crossref] [PubMed]

Manke, I.

I. Manke, N. Kardjilov, R. Schäfer, A. Hilger, M. Strobl, M. Dawson, C. Grünzweig, G. Behr, M. Hentschel, C. David, A. Kupsch, A. Lange, and J. Banhart, “Three-dimensional imaging of magnetic domains,” Nat. Commun. 1, 125 (2010).
[Crossref] [PubMed]

M. Strobl, A. Hilger, N. Kardjilov, O. Ebrahimi, S. Keil, and I. Manke, “Differential phase contrast and dark field neutron imaging,” Nucl. Instrum. Methods A 605, 9–12 (2009).
[Crossref]

M. Strobl, C. Grünzweig, A. Hilger, I. Manke, N. Kardjilov, C. David, and F. Pfeiffer, “Neutron dark-field tomography,” Phys. Rev. Lett. 101, 123902 (2008).
[Crossref] [PubMed]

Manserd, P.

S. Peter, P. Modregger, M. K. Fix, W. Volken, D. Frei, P. Manserd, and M. Stampanoni, “Combining Monte Carlo methods with coherent wave optics for the simulation of phase-sensitive x-ray imaging,” J. Synchro. Rad. 21, 613–622 (2014).
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Meinel, F. G.

F. Schwab, S. Schleede, D. Hahn, M. Bech, J. Herzen, S. Auweter, F. Bamberg, K. Achterhold, A. Ö. Yildirim, A. Bohla, O. Eickelberg, R. Loewen, M. Gifford, R. Ruth, M. F. Reiser, K. Nikolaou, F. Pfeiffer, and F. G. Meinel, “Comparison of contrast-to-noise ratios of transmission and dark-field signal in grating-based X-ray imaging for healthy murine lung tissue,” Z. Med. Phys. 33, 236–242 (2013).
[Crossref]

Meiser, J.

P. Modregger, S. Rutishauser, J. Meiser, C. David, and M. Stampanoni, “Two-dimensional ultra-small angle X-ray scattering with grating interferometry,” Appl. Phys. Lett. 105, 024102 (2014).
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Michel, T.

F. L. Bayer, S. Hu, A. Maier, T. Weber, G. Anton, T. Michel, and C. P. Riess, “Reconstruction of scalar and vectorial components in x-ray dark-field tomography,” Proc. Natl. Acad. Sci. U. S. A. 111, 12699–12704 (2014).
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F. Bayer, S. Zabler, C. Brendel, G. Pelzer, J. Rieger, A. Ritter, T. Weber, T. Michel, and G. Anton, “Projection angle dependence in grating-based X-ray dark-field imaging of ordered structures,” Opt. Express 21, 19923–19933 (2013).
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T. Michel, J. Rieger, G. Anton, F. Bayer, M. W. Beckmann, J. Dürst, P. A. Fasching, W. Haas, A. Hartmann, G. Pelzer, M. Radicke, C. Rauh, A. Ritter, P. Sievers, R. Schulz-Wendtland, M. Uder, D. L. Wachter, T. Weber, E. Wenkel, and A. Zang, “On a dark-field signal generated by micrometer-sized calcifications in phase-contrast mammography,” Phys. Med. Biol. 58, 2713–2732 (2013).
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T. Weber, G. Pelzer, F. Bayer, F. Horn, J. Rieger, A. Ritter, A. Zang, J. Durst, G. Anton, and T. Michel, “Increasing the darkfield contrast-to-noise ratio using a deconvolution-based information retrieval algorithm in X-ray grating-based phase-contrast imaging,” Opt. Express 21, 18011–18020 (2013).
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M. Endrizzi, P. C. Diemoz, T. P. Millard, J. L. Jones, R. D. Speller, I. K. Robinson, and A. Olivo, “Hard x-ray dark-field imaging with incoherent sample illumination,” Appl. Phys. Lett. 104, 024106 (2014).
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P. Modregger, M. Kagias, S. Peter, M. Abis, V. A. Guzenko, C. David, and M. Stampanoni, “Multiple scattering tomography,” Phys. Rev. Lett. 113, 020801 (2014).
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P. Modregger, S. Rutishauser, J. Meiser, C. David, and M. Stampanoni, “Two-dimensional ultra-small angle X-ray scattering with grating interferometry,” Appl. Phys. Lett. 105, 024102 (2014).
[Crossref]

S. Peter, P. Modregger, M. K. Fix, W. Volken, D. Frei, P. Manserd, and M. Stampanoni, “Combining Monte Carlo methods with coherent wave optics for the simulation of phase-sensitive x-ray imaging,” J. Synchro. Rad. 21, 613–622 (2014).
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F. Scattarella, S. Tangaro, P. Modregger, M. Stampanoni, L. De Caro, and R. Bellotti, “Post-detection analysis for grating-based ultra-small angle x-ray scattering,” Phys. Med. 29, 478–486 (2013).
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P. Modregger, F. Scattarella, B. R. Pinzer, C. David, R. Bellotti, and M. Stampanoni, “Imaging the ultrasmall-angle x-ray scattering distribution with grating interferometry,” Phys. Rev. Lett. 108, 048101 (2012).
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G. Potdevin, A. Malecki, T. Biernath, M. Bech, T. H. Jensen, R. Feidenhans’l, I. Zanette, T. Weitkamp, J. Kenntner, J. Mohr, P. Roschger, M. Kerschnitzki, W. Wagermaier, K. Klaushofer, P. Fratzl, and F. Pfeiffer, “X-ray vector radiography for bone micro-architecture diagnostics,” Phys. Med. Biol. 57, 3451–3461 (2012).
[Crossref] [PubMed]

T. H. Jensen, M. Bech, I. Zanette, T. Weitkamp, C. David, H. Deyhle, S. Rutishauser, E. Reznikova, J. Mohr, R. Feidenhans’l, and F. Pfeiffer, “Directional x-ray dark-field imaging of strongly ordered systems,” Phys. Rev. B 82, 214103 (2010).
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Momose, A.

W. Yashiro, S. Harasse, K. Kawabata, H. Kuwabara, T. Yamazaki, and A. Momose, “Distribution of unresolvable anisotropic microstructures revealed in visibility-contrast images using x-ray Talbot interferometry,” Phys. Rev. B 84, 094106 (2011).
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W. Yashiro, Y. Terui, K. Kawabara, and A. Momose, “On the origin of visibility contrast in x-ray Talbot interferometry,” Opt. Express 18, 16890–16901 (2010).
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A. Momose, W. Yashiro, Y. Takeda, Y. Suzuki, and T. Hattori, “Phase tomography by x-ray Talbot interferometry for biological imaging,” Jpn. J. Appl. Phys. 45, 5254–5262 (2006).
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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, W. Yashiro, and Y. Takeda, Biomedical Mathematics: Promising Directions in Imaging, Therapy Planning and Inverse Problems, Y. Censor, M. Jiang, and G. Wang, eds. (Medical Physics Publishing, Madison, 2010).

Morgan, K. S.

Morgan, N. Y.

Neels, A.

V. Revol, B. Plank, R. Kaufmann, J. Kastner, C. Kottler, and A. Neels, “Laminate fibre structure characterisation of carbon fibre-reinforced polymers by x-ray scatter dark field imaging with a grating interferometer,” NDT&E Int. 58, 64–71 (2013).
[Crossref]

V. Revol, C. Kottler, R. Kaufmann, A. Neels, and A. Dommann, “Orientation-selective x-ray dark field imaging of ordered systems,” J. Appl. Phys. 112, 114903 (2012).
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Ya. I. Nesterets, “On the origins of decoherence and extinction contrast in phase-contrast imaging,” Opt. Commun. 281, 533–542 (2008).
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Noël, P. B.

F. Schaff, A. Malecki, G. Potdevin, E. Eggl, P. B. Noël, T. Baum, E. G. Garcia, J. S. Bauer, and F. Pfeiffer, “Correlation of x-ray vector radiography to bone micro-architecture,” Sci. Rep. 4, 3695 (2014).
[Crossref] [PubMed]

A. Malecki, G. Potdevin, T. Biernath, E. Eggl, E. G. Garcia, T. Baum, P. B. Noël, J. S. Bauer, and F. Pfeiffer, “Coherent superposition in grating-based directional dark-field imaging,” PLOSo One 8, e61268 (2013).
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Nogami, T.

K. Ohmi, T. Nogami, Y. Fukushima, M. Katoh, and T. Yamakawa, “Characteristics of the five-pole superconducting vertical wiggler at the Photon Factory,” Rev. Sci. Instr. 63, 301–304 (1992).
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T. Weitkamp, B. Nöhammer, A. Diaz, C. David, and E. Ziegler, “X-ray wavefront analysis and optics characterization with a grating interferometer,” Appl. Phys. Lett. 86, 054101 (2005).
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Pai, V.

Pelzer, G.

T. Weber, G. Pelzer, F. Bayer, F. Horn, J. Rieger, A. Ritter, A. Zang, J. Durst, G. Anton, and T. Michel, “Increasing the darkfield contrast-to-noise ratio using a deconvolution-based information retrieval algorithm in X-ray grating-based phase-contrast imaging,” Opt. Express 21, 18011–18020 (2013).
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F. Bayer, S. Zabler, C. Brendel, G. Pelzer, J. Rieger, A. Ritter, T. Weber, T. Michel, and G. Anton, “Projection angle dependence in grating-based X-ray dark-field imaging of ordered structures,” Opt. Express 21, 19923–19933 (2013).
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T. Michel, J. Rieger, G. Anton, F. Bayer, M. W. Beckmann, J. Dürst, P. A. Fasching, W. Haas, A. Hartmann, G. Pelzer, M. Radicke, C. Rauh, A. Ritter, P. Sievers, R. Schulz-Wendtland, M. Uder, D. L. Wachter, T. Weber, E. Wenkel, and A. Zang, “On a dark-field signal generated by micrometer-sized calcifications in phase-contrast mammography,” Phys. Med. Biol. 58, 2713–2732 (2013).
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Peter, S.

P. Modregger, M. Kagias, S. Peter, M. Abis, V. A. Guzenko, C. David, and M. Stampanoni, “Multiple scattering tomography,” Phys. Rev. Lett. 113, 020801 (2014).
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S. Peter, P. Modregger, M. K. Fix, W. Volken, D. Frei, P. Manserd, and M. Stampanoni, “Combining Monte Carlo methods with coherent wave optics for the simulation of phase-sensitive x-ray imaging,” J. Synchro. Rad. 21, 613–622 (2014).
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Pfeiffer, F.

A. Malecki, G. Potdevin, T. Biernath, E. Eggl, K. Willer, T. Lasser, J. Maisenbacher, J. Gibmeier, A. Wanner, and F. Pfeiffer, “X-ray tensor tomography,” Europhys. Lett. 105, 38002 (2014).
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F. Schaff, A. Malecki, G. Potdevin, E. Eggl, P. B. Noël, T. Baum, E. G. Garcia, J. S. Bauer, and F. Pfeiffer, “Correlation of x-ray vector radiography to bone micro-architecture,” Sci. Rep. 4, 3695 (2014).
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A. Malecki, G. Potdevin, T. Biernath, E. Eggl, E. G. Garcia, T. Baum, P. B. Noël, J. S. Bauer, and F. Pfeiffer, “Coherent superposition in grating-based directional dark-field imaging,” PLOSo One 8, e61268 (2013).
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F. Schwab, S. Schleede, D. Hahn, M. Bech, J. Herzen, S. Auweter, F. Bamberg, K. Achterhold, A. Ö. Yildirim, A. Bohla, O. Eickelberg, R. Loewen, M. Gifford, R. Ruth, M. F. Reiser, K. Nikolaou, F. Pfeiffer, and F. G. Meinel, “Comparison of contrast-to-noise ratios of transmission and dark-field signal in grating-based X-ray imaging for healthy murine lung tissue,” Z. Med. Phys. 33, 236–242 (2013).
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C. Grünzweig, J. Kopecek, B. Betz, A. Kaestner, K. Jefimovs, J. Kohlbrecher, U. Gasser, O. Bunk, C. David, E. Lehmann, T. Donath, and F. Pfeiffer, “Quantification of the neutron dark-field imaging signal in grating interferometry,” Phys. Rev. B 88, 125104 (2013).
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W. Cong, F. Pfeiffer, M. Bech, and G. Wang, “X-ray dark-field imaging modeling,” J. Opt. Soc. Am. A 29, 908–912 (2012).
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A. Malecki, G. Potdevin, and F. Pfeiffer, “Quantitative wave-optical numerical analysis of the dark-field signal in grating-based x-ray interferometry,” Europhys. Lett. 99, 48001 (2012).
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G. Potdevin, A. Malecki, T. Biernath, M. Bech, T. H. Jensen, R. Feidenhans’l, I. Zanette, T. Weitkamp, J. Kenntner, J. Mohr, P. Roschger, M. Kerschnitzki, W. Wagermaier, K. Klaushofer, P. Fratzl, and F. Pfeiffer, “X-ray vector radiography for bone micro-architecture diagnostics,” Phys. Med. Biol. 57, 3451–3461 (2012).
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M. Chabior, T. Donath, C. David, M. Schuster, C. Schroer, and F. Pfeiffer, “Signal-to-noise ratio in x ray dark-field imaging using a grating interferometer,” J. Appl. Phys. 110, 053105 (2011).
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T. H. Jensen, M. Bech, I. Zanette, T. Weitkamp, C. David, H. Deyhle, S. Rutishauser, E. Reznikova, J. Mohr, R. Feidenhans’l, and F. Pfeiffer, “Directional x-ray dark-field imaging of strongly ordered systems,” Phys. Rev. B 82, 214103 (2010).
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F. Pfeiffer, T. Weitkamp, O. Bunk, and C. David, “Hard-x-ray dark-field imaging using a grating interferometer,” Nat. Mat. 7, 134–137 (2008).
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C. Grünzweig, C. David, O. Bunk, M. Dierolf, G. Frei, G. Kühne, J. Kohlbrecher, R. Schäfer, P. Lejcek, H. M. R. Ronnow, and F. Pfeiffer, “Neutron decoherence imaging for visualizing bulk magnetic domain structures,” Phys. Rev. Lett. 101, 025504 (2008).
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M. Strobl, C. Grünzweig, A. Hilger, I. Manke, N. Kardjilov, C. David, and F. Pfeiffer, “Neutron dark-field tomography,” Phys. Rev. Lett. 101, 123902 (2008).
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C. Grünzweig, C. David, O. Bunk, M. Dierolf, G. Frei, G. Kuhne, R. Schafer, S. Pofahl, H. M. R. Ronnow, and F. Pfeiffer, “Bulk magnetic domain structures visualized by neutron dark-field imaging,” Appl. Phys. Lett. 93, 112504 (2008).
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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,” 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]

Pinzer, B. R.

P. Modregger, F. Scattarella, B. R. Pinzer, C. David, R. Bellotti, and M. Stampanoni, “Imaging the ultrasmall-angle x-ray scattering distribution with grating interferometry,” Phys. Rev. Lett. 108, 048101 (2012).
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Plank, B.

V. Revol, B. Plank, R. Kaufmann, J. Kastner, C. Kottler, and A. Neels, “Laminate fibre structure characterisation of carbon fibre-reinforced polymers by x-ray scatter dark field imaging with a grating interferometer,” NDT&E Int. 58, 64–71 (2013).
[Crossref]

Pofahl, S.

C. Grünzweig, C. David, O. Bunk, M. Dierolf, G. Frei, G. Kuhne, R. Schafer, S. Pofahl, H. M. R. Ronnow, and F. Pfeiffer, “Bulk magnetic domain structures visualized by neutron dark-field imaging,” Appl. Phys. Lett. 93, 112504 (2008).
[Crossref]

Potdevin, G.

A. Malecki, G. Potdevin, T. Biernath, E. Eggl, K. Willer, T. Lasser, J. Maisenbacher, J. Gibmeier, A. Wanner, and F. Pfeiffer, “X-ray tensor tomography,” Europhys. Lett. 105, 38002 (2014).
[Crossref]

F. Schaff, A. Malecki, G. Potdevin, E. Eggl, P. B. Noël, T. Baum, E. G. Garcia, J. S. Bauer, and F. Pfeiffer, “Correlation of x-ray vector radiography to bone micro-architecture,” Sci. Rep. 4, 3695 (2014).
[Crossref] [PubMed]

A. Malecki, G. Potdevin, T. Biernath, E. Eggl, E. G. Garcia, T. Baum, P. B. Noël, J. S. Bauer, and F. Pfeiffer, “Coherent superposition in grating-based directional dark-field imaging,” PLOSo One 8, e61268 (2013).
[Crossref]

G. Potdevin, A. Malecki, T. Biernath, M. Bech, T. H. Jensen, R. Feidenhans’l, I. Zanette, T. Weitkamp, J. Kenntner, J. Mohr, P. Roschger, M. Kerschnitzki, W. Wagermaier, K. Klaushofer, P. Fratzl, and F. Pfeiffer, “X-ray vector radiography for bone micro-architecture diagnostics,” Phys. Med. Biol. 57, 3451–3461 (2012).
[Crossref] [PubMed]

A. Malecki, G. Potdevin, and F. Pfeiffer, “Quantitative wave-optical numerical analysis of the dark-field signal in grating-based x-ray interferometry,” Europhys. Lett. 99, 48001 (2012).
[Crossref]

Qi, Z.

Radicke, M.

T. Michel, J. Rieger, G. Anton, F. Bayer, M. W. Beckmann, J. Dürst, P. A. Fasching, W. Haas, A. Hartmann, G. Pelzer, M. Radicke, C. Rauh, A. Ritter, P. Sievers, R. Schulz-Wendtland, M. Uder, D. L. Wachter, T. Weber, E. Wenkel, and A. Zang, “On a dark-field signal generated by micrometer-sized calcifications in phase-contrast mammography,” Phys. Med. Biol. 58, 2713–2732 (2013).
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Rauh, C.

T. Michel, J. Rieger, G. Anton, F. Bayer, M. W. Beckmann, J. Dürst, P. A. Fasching, W. Haas, A. Hartmann, G. Pelzer, M. Radicke, C. Rauh, A. Ritter, P. Sievers, R. Schulz-Wendtland, M. Uder, D. L. Wachter, T. Weber, E. Wenkel, and A. Zang, “On a dark-field signal generated by micrometer-sized calcifications in phase-contrast mammography,” Phys. Med. Biol. 58, 2713–2732 (2013).
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Reiser, M. F.

F. Schwab, S. Schleede, D. Hahn, M. Bech, J. Herzen, S. Auweter, F. Bamberg, K. Achterhold, A. Ö. Yildirim, A. Bohla, O. Eickelberg, R. Loewen, M. Gifford, R. Ruth, M. F. Reiser, K. Nikolaou, F. Pfeiffer, and F. G. Meinel, “Comparison of contrast-to-noise ratios of transmission and dark-field signal in grating-based X-ray imaging for healthy murine lung tissue,” Z. Med. Phys. 33, 236–242 (2013).
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Revol, V.

V. Revol, B. Plank, R. Kaufmann, J. Kastner, C. Kottler, and A. Neels, “Laminate fibre structure characterisation of carbon fibre-reinforced polymers by x-ray scatter dark field imaging with a grating interferometer,” NDT&E Int. 58, 64–71 (2013).
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V. Revol, C. Kottler, R. Kaufmann, A. Neels, and A. Dommann, “Orientation-selective x-ray dark field imaging of ordered systems,” J. Appl. Phys. 112, 114903 (2012).
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V. Revol, I. Jerjen, C. Kottler, P. Schütz, R. Kaufmann, T. Lthi, U. Sennhauser, U. Straumann, and C. Urban, “Sub-pixel porosity revealed by x-ray scatter dark field imaging,” J. Appl. Phys. 110, 044912 (2011).
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V. Revol, C. Kottler, R. Kaufmann, U. Straumann, and C. Urban, “Noise analysis of grating-based x-ray differential phase contrast imaging,” Rev. Sci. Instr. 81, 073709 (2010).
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Reznikova, E.

T. H. Jensen, M. Bech, I. Zanette, T. Weitkamp, C. David, H. Deyhle, S. Rutishauser, E. Reznikova, J. Mohr, R. Feidenhans’l, and F. Pfeiffer, “Directional x-ray dark-field imaging of strongly ordered systems,” Phys. Rev. B 82, 214103 (2010).
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Rieger, J.

T. Michel, J. Rieger, G. Anton, F. Bayer, M. W. Beckmann, J. Dürst, P. A. Fasching, W. Haas, A. Hartmann, G. Pelzer, M. Radicke, C. Rauh, A. Ritter, P. Sievers, R. Schulz-Wendtland, M. Uder, D. L. Wachter, T. Weber, E. Wenkel, and A. Zang, “On a dark-field signal generated by micrometer-sized calcifications in phase-contrast mammography,” Phys. Med. Biol. 58, 2713–2732 (2013).
[Crossref] [PubMed]

F. Bayer, S. Zabler, C. Brendel, G. Pelzer, J. Rieger, A. Ritter, T. Weber, T. Michel, and G. Anton, “Projection angle dependence in grating-based X-ray dark-field imaging of ordered structures,” Opt. Express 21, 19923–19933 (2013).
[Crossref]

T. Weber, G. Pelzer, F. Bayer, F. Horn, J. Rieger, A. Ritter, A. Zang, J. Durst, G. Anton, and T. Michel, “Increasing the darkfield contrast-to-noise ratio using a deconvolution-based information retrieval algorithm in X-ray grating-based phase-contrast imaging,” Opt. Express 21, 18011–18020 (2013).
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Riess, C. P.

F. L. Bayer, S. Hu, A. Maier, T. Weber, G. Anton, T. Michel, and C. P. Riess, “Reconstruction of scalar and vectorial components in x-ray dark-field tomography,” Proc. Natl. Acad. Sci. U. S. A. 111, 12699–12704 (2014).
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Ritter, A.

F. Bayer, S. Zabler, C. Brendel, G. Pelzer, J. Rieger, A. Ritter, T. Weber, T. Michel, and G. Anton, “Projection angle dependence in grating-based X-ray dark-field imaging of ordered structures,” Opt. Express 21, 19923–19933 (2013).
[Crossref]

T. Michel, J. Rieger, G. Anton, F. Bayer, M. W. Beckmann, J. Dürst, P. A. Fasching, W. Haas, A. Hartmann, G. Pelzer, M. Radicke, C. Rauh, A. Ritter, P. Sievers, R. Schulz-Wendtland, M. Uder, D. L. Wachter, T. Weber, E. Wenkel, and A. Zang, “On a dark-field signal generated by micrometer-sized calcifications in phase-contrast mammography,” Phys. Med. Biol. 58, 2713–2732 (2013).
[Crossref] [PubMed]

T. Weber, G. Pelzer, F. Bayer, F. Horn, J. Rieger, A. Ritter, A. Zang, J. Durst, G. Anton, and T. Michel, “Increasing the darkfield contrast-to-noise ratio using a deconvolution-based information retrieval algorithm in X-ray grating-based phase-contrast imaging,” Opt. Express 21, 18011–18020 (2013).
[Crossref] [PubMed]

Robinson, I. K.

M. Endrizzi, P. C. Diemoz, T. P. Millard, J. L. Jones, R. D. Speller, I. K. Robinson, and A. Olivo, “Hard x-ray dark-field imaging with incoherent sample illumination,” Appl. Phys. Lett. 104, 024106 (2014).
[Crossref]

Ronnow, H. M. R.

C. Grünzweig, C. David, O. Bunk, M. Dierolf, G. Frei, G. Kühne, J. Kohlbrecher, R. Schäfer, P. Lejcek, H. M. R. Ronnow, and F. Pfeiffer, “Neutron decoherence imaging for visualizing bulk magnetic domain structures,” Phys. Rev. Lett. 101, 025504 (2008).
[Crossref] [PubMed]

C. Grünzweig, C. David, O. Bunk, M. Dierolf, G. Frei, G. Kuhne, R. Schafer, S. Pofahl, H. M. R. Ronnow, and F. Pfeiffer, “Bulk magnetic domain structures visualized by neutron dark-field imaging,” Appl. Phys. Lett. 93, 112504 (2008).
[Crossref]

Roschger, P.

G. Potdevin, A. Malecki, T. Biernath, M. Bech, T. H. Jensen, R. Feidenhans’l, I. Zanette, T. Weitkamp, J. Kenntner, J. Mohr, P. Roschger, M. Kerschnitzki, W. Wagermaier, K. Klaushofer, P. Fratzl, and F. Pfeiffer, “X-ray vector radiography for bone micro-architecture diagnostics,” Phys. Med. Biol. 57, 3451–3461 (2012).
[Crossref] [PubMed]

Rosenfeld, D. P.

Ruth, R.

F. Schwab, S. Schleede, D. Hahn, M. Bech, J. Herzen, S. Auweter, F. Bamberg, K. Achterhold, A. Ö. Yildirim, A. Bohla, O. Eickelberg, R. Loewen, M. Gifford, R. Ruth, M. F. Reiser, K. Nikolaou, F. Pfeiffer, and F. G. Meinel, “Comparison of contrast-to-noise ratios of transmission and dark-field signal in grating-based X-ray imaging for healthy murine lung tissue,” Z. Med. Phys. 33, 236–242 (2013).
[Crossref]

Rutishauser, S.

P. Modregger, S. Rutishauser, J. Meiser, C. David, and M. Stampanoni, “Two-dimensional ultra-small angle X-ray scattering with grating interferometry,” Appl. Phys. Lett. 105, 024102 (2014).
[Crossref]

T. H. Jensen, M. Bech, I. Zanette, T. Weitkamp, C. David, H. Deyhle, S. Rutishauser, E. Reznikova, J. Mohr, R. Feidenhans’l, and F. Pfeiffer, “Directional x-ray dark-field imaging of strongly ordered systems,” Phys. Rev. B 82, 214103 (2010).
[Crossref]

Scattarella, F.

F. Scattarella, S. Tangaro, P. Modregger, M. Stampanoni, L. De Caro, and R. Bellotti, “Post-detection analysis for grating-based ultra-small angle x-ray scattering,” Phys. Med. 29, 478–486 (2013).
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P. Modregger, F. Scattarella, B. R. Pinzer, C. David, R. Bellotti, and M. Stampanoni, “Imaging the ultrasmall-angle x-ray scattering distribution with grating interferometry,” Phys. Rev. Lett. 108, 048101 (2012).
[Crossref] [PubMed]

Schafer, R.

C. Grünzweig, C. David, O. Bunk, M. Dierolf, G. Frei, G. Kuhne, R. Schafer, S. Pofahl, H. M. R. Ronnow, and F. Pfeiffer, “Bulk magnetic domain structures visualized by neutron dark-field imaging,” Appl. Phys. Lett. 93, 112504 (2008).
[Crossref]

Schäfer, R.

I. Manke, N. Kardjilov, R. Schäfer, A. Hilger, M. Strobl, M. Dawson, C. Grünzweig, G. Behr, M. Hentschel, C. David, A. Kupsch, A. Lange, and J. Banhart, “Three-dimensional imaging of magnetic domains,” Nat. Commun. 1, 125 (2010).
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C. Grünzweig, C. David, O. Bunk, M. Dierolf, G. Frei, G. Kühne, J. Kohlbrecher, R. Schäfer, P. Lejcek, H. M. R. Ronnow, and F. Pfeiffer, “Neutron decoherence imaging for visualizing bulk magnetic domain structures,” Phys. Rev. Lett. 101, 025504 (2008).
[Crossref] [PubMed]

Schaff, F.

F. Schaff, A. Malecki, G. Potdevin, E. Eggl, P. B. Noël, T. Baum, E. G. Garcia, J. S. Bauer, and F. Pfeiffer, “Correlation of x-ray vector radiography to bone micro-architecture,” Sci. Rep. 4, 3695 (2014).
[Crossref] [PubMed]

Schleede, S.

F. Schwab, S. Schleede, D. Hahn, M. Bech, J. Herzen, S. Auweter, F. Bamberg, K. Achterhold, A. Ö. Yildirim, A. Bohla, O. Eickelberg, R. Loewen, M. Gifford, R. Ruth, M. F. Reiser, K. Nikolaou, F. Pfeiffer, and F. G. Meinel, “Comparison of contrast-to-noise ratios of transmission and dark-field signal in grating-based X-ray imaging for healthy murine lung tissue,” Z. Med. Phys. 33, 236–242 (2013).
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Schreiber, H.

H. Schreiber and J. H. Bruning, Optical Shop Testing, D. Malacara, ed. (Wiley Interscience, 2007), Chap. 14.

Schroer, C.

M. Chabior, T. Donath, C. David, M. Schuster, C. Schroer, and F. Pfeiffer, “Signal-to-noise ratio in x ray dark-field imaging using a grating interferometer,” J. Appl. Phys. 110, 053105 (2011).
[Crossref]

Schulz-Wendtland, R.

T. Michel, J. Rieger, G. Anton, F. Bayer, M. W. Beckmann, J. Dürst, P. A. Fasching, W. Haas, A. Hartmann, G. Pelzer, M. Radicke, C. Rauh, A. Ritter, P. Sievers, R. Schulz-Wendtland, M. Uder, D. L. Wachter, T. Weber, E. Wenkel, and A. Zang, “On a dark-field signal generated by micrometer-sized calcifications in phase-contrast mammography,” Phys. Med. Biol. 58, 2713–2732 (2013).
[Crossref] [PubMed]

Schuster, M.

M. Chabior, T. Donath, C. David, M. Schuster, C. Schroer, and F. Pfeiffer, “Signal-to-noise ratio in x ray dark-field imaging using a grating interferometer,” J. Appl. Phys. 110, 053105 (2011).
[Crossref]

Schütz, P.

V. Revol, I. Jerjen, C. Kottler, P. Schütz, R. Kaufmann, T. Lthi, U. Sennhauser, U. Straumann, and C. Urban, “Sub-pixel porosity revealed by x-ray scatter dark field imaging,” J. Appl. Phys. 110, 044912 (2011).
[Crossref]

Schwab, F.

F. Schwab, S. Schleede, D. Hahn, M. Bech, J. Herzen, S. Auweter, F. Bamberg, K. Achterhold, A. Ö. Yildirim, A. Bohla, O. Eickelberg, R. Loewen, M. Gifford, R. Ruth, M. F. Reiser, K. Nikolaou, F. Pfeiffer, and F. G. Meinel, “Comparison of contrast-to-noise ratios of transmission and dark-field signal in grating-based X-ray imaging for healthy murine lung tissue,” Z. Med. Phys. 33, 236–242 (2013).
[Crossref]

Sennhauser, U.

V. Revol, I. Jerjen, C. Kottler, P. Schütz, R. Kaufmann, T. Lthi, U. Sennhauser, U. Straumann, and C. Urban, “Sub-pixel porosity revealed by x-ray scatter dark field imaging,” J. Appl. Phys. 110, 044912 (2011).
[Crossref]

Sievers, P.

T. Michel, J. Rieger, G. Anton, F. Bayer, M. W. Beckmann, J. Dürst, P. A. Fasching, W. Haas, A. Hartmann, G. Pelzer, M. Radicke, C. Rauh, A. Ritter, P. Sievers, R. Schulz-Wendtland, M. Uder, D. L. Wachter, T. Weber, E. Wenkel, and A. Zang, “On a dark-field signal generated by micrometer-sized calcifications in phase-contrast mammography,” Phys. Med. Biol. 58, 2713–2732 (2013).
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S. W. Lee, D. S. Hussey, D. L. Jacobson, C. M. Sim, and M. Arif, “Development of the grating phase neutron interferometer at a monochromatic beam line,” Nucl. Instrum. Methods A 605, 16–20 (2009).
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Siu, K. K. W.

Speller, R. D.

M. Endrizzi, P. C. Diemoz, T. P. Millard, J. L. Jones, R. D. Speller, I. K. Robinson, and A. Olivo, “Hard x-ray dark-field imaging with incoherent sample illumination,” Appl. Phys. Lett. 104, 024106 (2014).
[Crossref]

Stampanoni, M.

P. Modregger, M. Kagias, S. Peter, M. Abis, V. A. Guzenko, C. David, and M. Stampanoni, “Multiple scattering tomography,” Phys. Rev. Lett. 113, 020801 (2014).
[Crossref] [PubMed]

P. Modregger, S. Rutishauser, J. Meiser, C. David, and M. Stampanoni, “Two-dimensional ultra-small angle X-ray scattering with grating interferometry,” Appl. Phys. Lett. 105, 024102 (2014).
[Crossref]

S. Peter, P. Modregger, M. K. Fix, W. Volken, D. Frei, P. Manserd, and M. Stampanoni, “Combining Monte Carlo methods with coherent wave optics for the simulation of phase-sensitive x-ray imaging,” J. Synchro. Rad. 21, 613–622 (2014).
[Crossref]

F. Scattarella, S. Tangaro, P. Modregger, M. Stampanoni, L. De Caro, and R. Bellotti, “Post-detection analysis for grating-based ultra-small angle x-ray scattering,” Phys. Med. 29, 478–486 (2013).
[Crossref] [PubMed]

P. Modregger, F. Scattarella, B. R. Pinzer, C. David, R. Bellotti, and M. Stampanoni, “Imaging the ultrasmall-angle x-ray scattering distribution with grating interferometry,” Phys. Rev. Lett. 108, 048101 (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]

Stampanoni, Marco

Z. Wang and Marco Stampanoni, “Quantitative x-ray radiography using grating interferometry: a feasibility study,” Phys. Med. Biol. 58, 6815–6826 (2013).
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Straumann, U.

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W. Yashiro, Y. Takeda, and A. Momose, “Efficiency of capturing a phase image using cone-beam x-ray Talbot interferometry,” J. Opt. Soc. Am. A 25, 2025–2039 (2008).
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T. Weber, G. Pelzer, F. Bayer, F. Horn, J. Rieger, A. Ritter, A. Zang, J. Durst, G. Anton, and T. Michel, “Increasing the darkfield contrast-to-noise ratio using a deconvolution-based information retrieval algorithm in X-ray grating-based phase-contrast imaging,” Opt. Express 21, 18011–18020 (2013).
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Phys. Med. (1)

F. Scattarella, S. Tangaro, P. Modregger, M. Stampanoni, L. De Caro, and R. Bellotti, “Post-detection analysis for grating-based ultra-small angle x-ray scattering,” Phys. Med. 29, 478–486 (2013).
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T. Michel, J. Rieger, G. Anton, F. Bayer, M. W. Beckmann, J. Dürst, P. A. Fasching, W. Haas, A. Hartmann, G. Pelzer, M. Radicke, C. Rauh, A. Ritter, P. Sievers, R. Schulz-Wendtland, M. Uder, D. L. Wachter, T. Weber, E. Wenkel, and A. Zang, “On a dark-field signal generated by micrometer-sized calcifications in phase-contrast mammography,” Phys. Med. Biol. 58, 2713–2732 (2013).
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G. Potdevin, A. Malecki, T. Biernath, M. Bech, T. H. Jensen, R. Feidenhans’l, I. Zanette, T. Weitkamp, J. Kenntner, J. Mohr, P. Roschger, M. Kerschnitzki, W. Wagermaier, K. Klaushofer, P. Fratzl, and F. Pfeiffer, “X-ray vector radiography for bone micro-architecture diagnostics,” Phys. Med. Biol. 57, 3451–3461 (2012).
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M. Bech, O. Bunk, T. Donath, R. Feidenhans’l, C David, and F. Pfeiffer, “Quantitative x-ray dark-field computed tomography,” Phys. Med. Biol. 55, 5529–5539 (2010).
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Phys. Rev. B (3)

T. H. Jensen, M. Bech, I. Zanette, T. Weitkamp, C. David, H. Deyhle, S. Rutishauser, E. Reznikova, J. Mohr, R. Feidenhans’l, and F. Pfeiffer, “Directional x-ray dark-field imaging of strongly ordered systems,” Phys. Rev. B 82, 214103 (2010).
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W. Yashiro, S. Harasse, K. Kawabata, H. Kuwabara, T. Yamazaki, and A. Momose, “Distribution of unresolvable anisotropic microstructures revealed in visibility-contrast images using x-ray Talbot interferometry,” Phys. Rev. B 84, 094106 (2011).
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C. Grünzweig, J. Kopecek, B. Betz, A. Kaestner, K. Jefimovs, J. Kohlbrecher, U. Gasser, O. Bunk, C. David, E. Lehmann, T. Donath, and F. Pfeiffer, “Quantification of the neutron dark-field imaging signal in grating interferometry,” Phys. Rev. B 88, 125104 (2013).
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Phys. Rev. Lett. (4)

M. Strobl, C. Grünzweig, A. Hilger, I. Manke, N. Kardjilov, C. David, and F. Pfeiffer, “Neutron dark-field tomography,” Phys. Rev. Lett. 101, 123902 (2008).
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P. Modregger, M. Kagias, S. Peter, M. Abis, V. A. Guzenko, C. David, and M. Stampanoni, “Multiple scattering tomography,” Phys. Rev. Lett. 113, 020801 (2014).
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C. Grünzweig, C. David, O. Bunk, M. Dierolf, G. Frei, G. Kühne, J. Kohlbrecher, R. Schäfer, P. Lejcek, H. M. R. Ronnow, and F. Pfeiffer, “Neutron decoherence imaging for visualizing bulk magnetic domain structures,” Phys. Rev. Lett. 101, 025504 (2008).
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P. Modregger, F. Scattarella, B. R. Pinzer, C. David, R. Bellotti, and M. Stampanoni, “Imaging the ultrasmall-angle x-ray scattering distribution with grating interferometry,” Phys. Rev. Lett. 108, 048101 (2012).
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PLOSo One (1)

A. Malecki, G. Potdevin, T. Biernath, E. Eggl, E. G. Garcia, T. Baum, P. B. Noël, J. S. Bauer, and F. Pfeiffer, “Coherent superposition in grating-based directional dark-field imaging,” PLOSo One 8, e61268 (2013).
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Proc. Natl. Acad. Sci. U. S. A. (1)

F. L. Bayer, S. Hu, A. Maier, T. Weber, G. Anton, T. Michel, and C. P. Riess, “Reconstruction of scalar and vectorial components in x-ray dark-field tomography,” Proc. Natl. Acad. Sci. U. S. A. 111, 12699–12704 (2014).
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Rev. Sci. Instr. (2)

K. Ohmi, T. Nogami, Y. Fukushima, M. Katoh, and T. Yamakawa, “Characteristics of the five-pole superconducting vertical wiggler at the Photon Factory,” Rev. Sci. Instr. 63, 301–304 (1992).
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V. Revol, C. Kottler, R. Kaufmann, U. Straumann, and C. Urban, “Noise analysis of grating-based x-ray differential phase contrast imaging,” Rev. Sci. Instr. 81, 073709 (2010).
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Rev. Sci. Instrum. (1)

E. Hack and J. Burke, “Invited review article: Measurement uncertainty of linear phase-stepping algorithms,” Rev. Sci. Instrum. 82, 061101 (2011).
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Sci. Rep. (1)

F. Schaff, A. Malecki, G. Potdevin, E. Eggl, P. B. Noël, T. Baum, E. G. Garcia, J. S. Bauer, and F. Pfeiffer, “Correlation of x-ray vector radiography to bone micro-architecture,” Sci. Rep. 4, 3695 (2014).
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Siam. J. Appl. Math. (1)

W. Han, J. A. Eichholz, X. Cheng, and G. Wang, “A theoretical framework of x-ray dark-field tomograpy,” Siam. J. Appl. Math. 71, 1557–1577 (2011).
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Z. Med. Phys. (2)

M. Bech, T. H. Jensen, O. Bunk, T. Donath, C. David, T. Weitkamp, G. Le Duc, A. Bravin, and P. Cloetens, “Advanced contrast modalities for x-ray radiology: phase-contrast and dark-field imaging using a grating interferometer,” Z. Med. Phys. 20, 7–16 (2010).
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F. Schwab, S. Schleede, D. Hahn, M. Bech, J. Herzen, S. Auweter, F. Bamberg, K. Achterhold, A. Ö. Yildirim, A. Bohla, O. Eickelberg, R. Loewen, M. Gifford, R. Ruth, M. F. Reiser, K. Nikolaou, F. Pfeiffer, and F. G. Meinel, “Comparison of contrast-to-noise ratios of transmission and dark-field signal in grating-based X-ray imaging for healthy murine lung tissue,” Z. Med. Phys. 33, 236–242 (2013).
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Other (3)

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H. Schreiber and J. H. Bruning, Optical Shop Testing, D. Malacara, ed. (Wiley Interscience, 2007), Chap. 14.

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

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Figures (8)

Fig. 1
Fig. 1 Setup of a spherical-wave X-ray Talbot interferometer.
Fig. 2
Fig. 2 Effect of finite size of X-ray source in X-ray Talbot interferometer.
Fig. 3
Fig. 3 Results of numerical calculations for line profiles across the edges of PMMA cuboid plates with several thicknesses (T) in (a) transmittance (��), (b) moiré-phase (��), and (c) normalized-visibility (��) images, and dependences of (d) the maximum value of |��| and (e) the minimum value of �� on the thickness of the plates. Solid and broken lines are the results with and without the Fresnel diffraction by the samples taken into account.
Fig. 4
Fig. 4 Results of numerical calculations for line profiles across the edges of PMMA cylinders with several diameters (D) in transmittance (��; (a) and (d)), moiré-phase (��; (b) and (e)), and normalized-visibility (��; (c) and (f)) images. Solid and broken lines are the results with and without the Fresnel diffraction by the samples taken into account.
Fig. 5
Fig. 5 Minimum value of the normalized visibility (��) for PMMA cuboid plates is plotted with respect to a universal parameter (Uplate). Here w D 2 2 ln 2 σ D.
Fig. 6
Fig. 6 Minimum value of the normalized visibility (��) for PMMA cylinders is plotted with v respect to a universal parameter (Ucylinder). Here w D 2 2 ln 2 σ D. Red lines: wD changed; blue lines: distance between G1 and G2 (corresponding to p12) changed; green lines: mass density of PMMA changed (but kept uniform). Solid, broken, and dotted lines correspond to diameters (D) of 1, 2, and 5 mm.
Fig. 7
Fig. 7 Experimental results (open circles) of line profiles across edges of three PMMA cylinders with diameters of 3, 6, and 10 mm in (a) transmittance, (b) moiré-phase, and (c) normalized-visibility images. Solid lines were results of numerical calculations based on Eqs. (53), (54), and (55) with the Fresnel diffraction by the samples taken into account.
Fig. 8
Fig. 8 Experimental results (open circles) of |��|max and ��min near the edges of three PMMA cylinders with diameters of 3, 6, and 10 mm. Solid lines were results of numerical calculations based on Eqs. (53), (54), and (55) with the Fresnel diffraction by the samples taken into account. (a) and (b): Talbot-order (ps)-dependence at 25 keV; (c) and (d): X-ray-wavelength-dependence for ps = 0.5

Equations (69)

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d i M i 1 d 1 ,
z eff , i j ( 1 R i + 1 R j R i ) 1
p i j z eff , i j λ d i 2
= λ d i d j ( R j R i ) .
E 2 ( x 20 , y 20 ) E 0 exp [ 2 π i λ R 2 ] R 2 n = a n T s 2 , n F ( x s 0 , y s 0 ) exp [ 2 π i n x 10 d 1 ] ,
a n a n exp [ π i p 12 n 2 ] ,
T s 2 , n F ( x s 0 , y s 0 ) T s 2 F ( x s 0 n p s d s , y s 0 ) ,
T s 2 F ( x s 0 , y s 0 ) T s * P s 2 ( x s 0 , y s 0 ) ,
P s 2 ( x s 0 , y s 0 ) i λ z eff , s 2 exp [ π i λ z eff , s 2 ( x s 0 2 + y s 0 2 ) ] .
p s { p 12 ( R s R 1 ) p s 2 ( R s R 1 ) ,
T s ( x s 0 , y s 0 ) = exp [ i Φ s ( x s 0 , y s 0 ) ] ,
E s + ( x s 0 , y s 0 ) = T s ( x s 0 , y s 0 ) E 0 exp [ 2 π i λ R s ] R s exp [ 2 π i ( x s 0 2 + y s 0 2 ) ] .
T s 2 , n F ( x s 0 , y s 0 ) T s , n ( x s 0 , y s 0 ) ,
T s , n ( x s 0 , y s 0 ) exp [ Φ s i ( x s 0 , y s 0 ) ] exp [ i ( Φ s r ( x s 0 , y s 0 ) n p s d s Φ s r x s 0 ( x s 0 , y s 0 ) ) ] .
I 2 ( x 20 , y 20 ) I 0 R 2 2 n , n a n a n n 1 ¯ T s 2 , n F ( x s 0 , y s 0 ) T s 2 , n n 1 F ( x s 0 , y s 0 ) ¯ exp [ 2 π i n 1 x 10 d 1 ] ,
I 0 exp [ 2 Φ s i ] R 2 2 n 1 b n 1 exp [ i n 1 p s d s Φ s r x s 0 ] exp [ 2 π i n 1 x 10 d 1 ] ,
I 2 + ( x 20 , y 20 ) = I 2 ( x 20 , y 20 ) | T 2 ( x 20 , y 20 ) | 2 ,
I 2 + ( x 20 , y 20 ) I 0 exp [ 2 Φ s i ] R 2 2 × n 1 , n 2 b n 1 c n 2 exp [ i n 1 p s d s Φ s r x s 0 ] exp [ 2 π i ( n 1 + n 2 ) x 20 + n 2 χ 2 d 2 ] ,
I D ( x D 0 , y D 0 ) = ( M 2 D ) 2 I 2 + ( x 20 , y 20 ) .
I D ( x D 0 , y D 0 Δ S ) [ cps ] = Δ S ( M 2 D ) 2 { I 2 + * W ^ D ( x D 0 , y D 0 ) } .
I D ( x D 0 , y D 0 ) I 0 exp [ 2 Φ s i ] R D 2 n b n c n exp [ 2 π i n p s d s λ φ x s 0 ] exp [ 2 π i n χ 2 d 2 ] ,
I D [ x D 0 , y D 0 ; T s ( x s 0 , y s 0 ) , T 1 ( x 10 , y 10 ) , T 2 ( x 20 , y 20 ) ; W ^ D ( x D 0 , y D 0 ) ] .
I D [ x D 0 , y D 0 ; T s ( x s 0 + Δ x s + y s 0 + Δ y s ) , T 1 ( x 10 + Δ x 1 , y 10 + Δ y 1 ) , T 2 ( x 20 + Δ x 2 , y 20 + Δ y 2 ) ; W ^ D ( x D 0 , y D 0 ) ] ,
( Δ x i , Δ y i ) = R D R i R D ( X 0 , Y 0 ) .
I D , σ ( x D 0 , y D 0 ) I 0 I ^ 0 ( X 0 , Y 0 ) I D ( x D 0 , y D 0 ; X 0 , Y 0 ) d X 0 d Y 0 .
I D , σ ( x D 0 , y D 0 ) I 0 exp [ 2 Φ s i ] R D 2 × I ^ 0 ( X 0 , Y 0 ) n 1 , n 2 b n 1 c n 2 exp [ i n 1 p s d s Φ s r x s 0 ( x s 0 + Δ x s , y s 0 + Δ y s ) ] × exp [ 2 π i ( n 1 + n 2 ) x 20 + n 2 χ 2 d 2 ] exp [ 2 π i ( n 1 Δ x 1 d 1 + n 2 Δ x 2 d 2 ) ] d X 0 d Y 0 .
I D , σ ( x D 0 , y D 0 ) I 0 exp [ 2 Φ s i ] R D 2 n μ ( n p 12 d 1 λ R 1 , 0 ) b n c n × exp [ 2 π i n p s d s λ φ x s 0 ] D exp [ 2 π i n χ 2 d 2 ] ,
μ ( k x , k y ) = I ^ 0 ( X 0 , Y 0 ) exp [ 2 π i ( k x X 0 + k y Y 0 ) ] d X 0 d Y 0 ,
( k x , k y ) = ( n d 2 R 2 R 1 R 1 , 0 ) ,
= ( n p 12 d 1 λ R 1 , 0 ) .
q 0 I 0 exp [ 2 Φ s i ] R D 2 b 0 c 0 ,
𝒯 ( x D 0 , y D 0 ) q 0 q 0 ref ,
exp [ 2 Φ s i ( x s 0 , y s 0 ) ] .
𝒫 ( x D 0 , y D 0 ) arg [ q 1 q 1 ref ] ,
arg exp [ i p s d s Φ s r x s 0 ( x s 0 , y s 0 ) ] D .
1 p s d s 𝒫 ( x D 0 , y D 0 ) Φ s r x s 0 ( x s 0 , y s 0 ) ,
𝒱 ( x D 0 , y D 0 ) | q 1 | / q 0 | q 1 ref | / q 0 ref .
I D , σ ( x D 0 , y D 0 ) I ^ 0 ( X 0 , Y 0 ) × I D [ x D 0 , y D 0 ; T sD F ( x S 0 + Δ x s , y s 0 + Δ y s ) , T 1 ( x 10 + Δ x 1 , y 10 + Δ y 1 ) , T 2 ( x 20 + Δ x 2 , y 20 + Δ y 2 ) ; W ^ D ( x D 0 , y D 0 ) ] d X 0 d Y 0 ,
( x i 0 , y i 0 ) = R i R D ( x D 0 , y D 0 ) ,
( Δ x i , Δ y i ) R D R i R D ( X 0 , Y 0 )
I D , σ ( x D 0 , y D 0 ) Δ S [ cps ] Δ S 𝒫 I ^ 0 ( X 0 , Y 0 ) [ I D * W ^ D ( x D 0 , y D 0 ) ] d X 0 d Y 0 ,
I D [ x D 0 , y D 0 ; T sD F ( x s 0 + Δ x s , y s 0 + Δ y s ) , T 1 ( x 10 + Δ x 1 , y 10 + Δ y 1 ) , T 2 ( x 20 + Δ x 2 , y 20 + Δ y 2 ) ] I 0 R D 2 n , n 1 , n 2 a n a n n 1 ¯ T sD , n F ( x s 0 + Δ x s , y s 0 + Δ y s ) T sD , n n 1 F ( x s 0 + Δ x s , y s 0 + Δ y s ) ¯ × c n 2 exp [ 2 π i ( n 1 + n 2 ) x 20 d 2 ] exp [ 2 π i ( n 1 Δ x 1 d 1 + n 2 Δ x 2 d 2 ) ] exp [ 2 π i n 2 χ 2 d 2 ] ,
I D * W ^ D ( x D 0 , y D 0 ) = I D ( x D 0 x , y D 0 y ) W ^ D ( x , y ) d x d y .
( x , y ) ( x , y ) ( Δ X 0 , Y 0 ) ,
( Δ X 0 , Y 0 ) R D R s R s ( X 0 , Y 0 ) .
I D * W ^ D ( x D 0 , y D 0 ) = I D ( x D 0 x , y D 0 y ) W ^ D ( x + Δ X 0 , y + Δ Y 0 ) d x d y ,
I D ( x D 0 , y D 0 ) I D [ x D 0 , y D 0 ; T s ( x s 0 , y s 0 ) , T 1 ( x 10 + Δ x 1 , y 10 + Δ y 1 ) T 2 ( x 20 + Δ x 2 , y 20 + Δ y 2 ) ] ,
( Δ x i , Δ y i ) R i R s R s ( X 0 , Y 0 )
I D , σ ( x D 0 , y D 0 ) I D , σ * W ^ D ( x D 0 , y D 0 ) ,
I D , σ ( x D 0 , y D 0 ) I 0 R D 2 n , n 1 , n 2 μ ( ( n 1 p s 1 + n 2 p s 2 ) d s λ R s , 0 ) a n a n n 1 ¯ × T sD , n F ( x s 0 , y s 0 ) T sD , n n 1 F ( x s 0 , y s 0 ) ¯ × c n 2 exp [ 2 π i ( n 1 + n 2 ) x 20 d 2 ] exp [ 2 π i n 2 χ 2 d 2 ] .
I D , σ ( x D 0 , y D 0 ) I 0 R D 2 n , n 1 , n 2 μ ( ( n 1 p s 1 + n 2 p s 2 ) d s λ R s , 0 ) a n a n n 1 ¯ c n 2 exp [ 2 π i n 2 χ 2 d 2 ] × T sD , n F ( x s 0 , y s 0 ) T sD , n n 1 F ( x s 0 , y s 0 ) ¯ exp [ 2 π i ( n 1 + n 2 ) x 20 d 2 ] D .
I D , σ ( x D 0 , y D 0 ) I 0 R D 2 n 2 μ ( n 2 p 12 d 1 λ R 1 , 0 ) c n 2 exp [ 2 π i n 2 χ 2 d 2 ] × n a n a n + n 2 ¯ T sD , n F ( x s 0 , y s 0 ) T sD , n + n 2 F ( x s 0 , y s 0 ) ¯ D .
𝒯 ( x D 0 , y D 0 ) n | a n | 2 | T sD , n F ( x s 0 , y s 0 ) | 2 D n | a n | 2 ,
𝒫 ( x D 0 , y D 0 ) arg [ n a n a n + 1 ¯ T sD , n F ( x s 0 , y s 0 ) T sD , n + 1 F ( x s 0 , y s 0 ¯ D n a n a n + 1 ¯ ] ,
𝒱 ( x D 0 , y D 0 ) | n a n a n + 1 ¯ T sD , n F ( x s 0 , y s 0 ) T sD , n + 1 F ( x s 0 , y s 0 ¯ D | / | n a n a n + 1 ¯ | | n a n | 2 | T sD , n F ( x s 0 , y s 0 ) | 2 D / | n a n | 2 .
𝒯 ( x D 0 , y D 0 ) | T sD , 0 F ( x s 0 , y s 0 ) | 2 D ,
𝒫 ( x D 0 , y D 0 ) arg [ T sD , 0 F ( x s 0 , y s 0 ) T sD , 1 F ( x s 0 , y s 0 ) ¯ D ] ,
𝒱 ( x D 0 , y D 0 ) | T sD , 0 F ( x s 0 , y s 0 ) T sD , 1 F ( x s 0 , y s 0 ) ¯ D | | T sD , 0 F ( x s 0 , y s 0 ) | 2 D .
T sD F ( x s 0 , y s 0 ) = exp [ i Φ sD F ( x s 0 , y s 0 ) ] ,
= exp [ i { Φ sD , sm F ( x s 0 , y s 0 ) + Δ Φ sD F ( x s 0 , y s 0 ) } ] ,
Φ sD , sm F ( x s 0 , y s 0 ) = C r x s 0 + i C i ,
( exp [ i Φ sD , sm F ( x s 0 , y s 0 ) ] exp [ i Φ sD , sm F ( x s 0 + p s d s , y s 0 ) ] ¯ ) x s 0 = 0
𝒯 ( x D 0 , y D 0 ) exp [ 2 ( Φ sD , sm F ( x s 0 , y s 0 ) ) i ] ,
𝒫 ( x D 0 , y D 0 ) p s d s [ ( Φ sD , sm F ( x s 0 , y s 0 ) x s 0 ) r ] ,
𝒱 ( x D 0 , y D 0 ) | exp [ i Δ Φ sD F ( x s 0 , y s 0 ) ] exp [ i Δ Φ sD F ( x s 0 + p s d s , y s 0 ) ] ¯ D | .
| exp [ i δ PMMA T ] 1 | 2 ( 2 π k x ) 2 .
U plate = p s d s σ D ( 1 cos 2 π δ PMMA T λ )
U cylinder = δ PMMA λ p s d s D σ D
U cylinder ρ electron λ p s d s D σ D ,

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