Abstract

We present a spectral phase unwrapping approach for grating-based differential phase-contrast data where the unwrapped interferometer phase shift is estimated from energy discriminated measurements using maximum likelihood principles. We demonstrate the method on tomographic data sets of a test specimen taken at different x-ray energies using synchrotron radiation. The proposed unwrapping technique was demonstrated to successfully correct the data set for phase wrapping.

© 2013 Optical Society of America

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  1. A. Baldi, “Phase unwrapping by region growing,” Appl. Opt.42, 2498–2505 (2003).
    [CrossRef] [PubMed]
  2. C. W. Chen and H. A. Zebker, “Two-dimensional phase unwrapping with use of statistical models for cost functions in nonlinear optimizatio,” J.Opt. Soc. Am. A18, 338–351 (2001).
    [CrossRef]
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    [CrossRef]
  6. C. David, J. Bruder, T. Rohbeck, C. Grnzweig, 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]
  7. I. Zanette, T. Weitkamp, S. Lang, M. Langer, J. Mohr, C. David, and J. Baruchel, “Quantitative phase and absorption tomography with an X-ray interferometer and synchrotron radiation,” Phys. Status Solidi208, 2526–2532 (2011).
    [CrossRef]
  8. K. Li, N. B. Bevins, J. N. Zambelli, and G. Chen, “Feasibility of differential phase contrast CT for whole body imaging,” AIP Conference Proceedings. 1466, 175–180 (2012).
    [CrossRef]
  9. I. Jerjen, V. Revol, P. Schuetz, C. Kottler, R. Kaufmann, T. Luethi, K. Jefimovs, C. Urban, and U. Sennhauser, “Reduction of phase artifacts in differential phase contrast computed tomography,” Opt. Express19, 13604–13611 (2011).
    [CrossRef] [PubMed]
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  11. W. Xu, E. Chang, L. Kwoh, H. Lim, W. Cheng, and A. Heng, “Phase-unwrapping of SAR Interferogram with Multi-frequency or Multi-baseline,” in Proceedings of IEEE Conference on Geoscience and Remote Sensing Symposium, (IEEE1994), pp. 730–732.
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    [CrossRef]
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    [CrossRef] [PubMed]
  23. F. Pfeiffer, T. Weitkamp, O. Bunk, and C. David, “Phase retrieval and differential phase-contrast imaging with low-brilliance X-ray sources,” Nature Phys. Lett.2, 258–261 (2006).
    [CrossRef]
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    [CrossRef]
  26. F. Pfeiffer, M. Bech, O. Bunk, T. Donath, B. Henrich, P. Kraft, and C. David, “X-ray dark-field and phase-contrast imaging using a grating interferometer,” J. Appl. Phys.105, 102006 (2009).
    [CrossRef]
  27. M. Born and E. Wolf, Principles of Optics (Academic, 1993).
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    [CrossRef]
  29. A. N. Tikhonov, “On the stability of inverse problems,” Doklady Akademii Nauk SSSR39, 195–198 (1943).
  30. J. Herzen, T. Donath, F. Beckmann, M. Ogurreck, C. David, J. Mohr, F. Pfeiffer, and A. Schreyer, “X-ray grating interferometer for materials-science imaging at a low-coherent wiggler source,” Rev. Sci. Instrum.82, 113711 (2011).
    [CrossRef] [PubMed]
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2012

J. Kenntner, V. Altapova, T. Grund, F. J. Pantenburg, J. Meiser, T. Baumbach, and J. Mohr, “Fabrication and characterization of analyzer gratings with high aspect ratios for phase contrast imaging using a Talbot interferometer,” AIP Conference Proceedings. 1437, 89–93 (2012).
[CrossRef]

K. Li, N. B. Bevins, J. N. Zambelli, and G. Chen, “Feasibility of differential phase contrast CT for whole body imaging,” AIP Conference Proceedings. 1466, 175–180 (2012).
[CrossRef]

D. Pennicard, S. Lange, S. Smoljanin, H. Hirsemann, and H. Graafsma, “LAMBDA - Large Area Medipix3-Based Detector Array,” JINST7, C11009 (2012).
[CrossRef]

2011

R. Ballabriga, M. Campbell, E. Heijne, X. Llopart, L. Tlustos, and W. Wong, “Medipix3: A 64k pixel detector readout chip working in single photon counting mode with improved spectrometric performance,” Nucl. Instrum. Meth. A633, 15–18 (2011).
[CrossRef]

I. Zanette, T. Weitkamp, S. Lang, M. Langer, J. Mohr, C. David, and J. Baruchel, “Quantitative phase and absorption tomography with an X-ray interferometer and synchrotron radiation,” Phys. Status Solidi208, 2526–2532 (2011).
[CrossRef]

J. Herzen, T. Donath, F. Beckmann, M. Ogurreck, C. David, J. Mohr, F. Pfeiffer, and A. Schreyer, “X-ray grating interferometer for materials-science imaging at a low-coherent wiggler source,” Rev. Sci. Instrum.82, 113711 (2011).
[CrossRef] [PubMed]

I. Jerjen, V. Revol, P. Schuetz, C. Kottler, R. Kaufmann, T. Luethi, K. Jefimovs, C. Urban, and U. Sennhauser, “Reduction of phase artifacts in differential phase contrast computed tomography,” Opt. Express19, 13604–13611 (2011).
[CrossRef] [PubMed]

2010

R. Steadman, C. Herrmann, O. Mlhens, D. G. Maeding, J. Colley, T. Firlit, R. Luhta, M. Chappo, B. Harwood, and D. Kosty, “ChromAIX: A high-rate energy-resolving photon-counting ASIC for Spectral Computed Tomography,” Proc. of SPIE7622, 762220 (2010).

2009

F. Pfeiffer, M. Bech, O. Bunk, T. Donath, B. Henrich, P. Kraft, and C. David, “X-ray dark-field and phase-contrast imaging using a grating interferometer,” J. Appl. Phys.105, 102006 (2009).
[CrossRef]

2008

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

2007

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

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

2006

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 (2006).
[CrossRef]

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

2005

2003

A. Baldi, “Phase unwrapping by region growing,” Appl. Opt.42, 2498–2505 (2003).
[CrossRef] [PubMed]

A. Momose, S. Kawamoto, I. Koyama, Y. Hamaishi, K. Takai, and Y. Suzuki, “Demonstration of X-ray Talbot Interferometry,” Jpn. J. Appl. Phys., 42, 866–868 (2003).
[CrossRef]

2001

C. W. Chen and H. A. Zebker, “Two-dimensional phase unwrapping with use of statistical models for cost functions in nonlinear optimizatio,” J.Opt. Soc. Am. A18, 338–351 (2001).
[CrossRef]

2000

G. Nico, “Bayesian approaches to phase unwrapping: theoretical study,” IEEE Trans. Sig. Process.48, 2454–2556 (2000).
[CrossRef]

1943

A. N. Tikhonov, “On the stability of inverse problems,” Doklady Akademii Nauk SSSR39, 195–198 (1943).

Achterhold, K.

W. Haas, M. Bech, P. Bartl, F. Bayer, A. Ritter, T. Weber, G. Pelzer, M. Willner, K. Achterhold, J. Durst, T. Michel, M. Prmmer, F. Pfeiffer, G. Anton, and J. Hornegger, “Phase-Unwrapping of Differential Phase-Contrast Data using Attenuation Information,” Proc. SPIE7962, (2011).
[CrossRef]

Altapova, V.

J. Kenntner, V. Altapova, T. Grund, F. J. Pantenburg, J. Meiser, T. Baumbach, and J. Mohr, “Fabrication and characterization of analyzer gratings with high aspect ratios for phase contrast imaging using a Talbot interferometer,” AIP Conference Proceedings. 1437, 89–93 (2012).
[CrossRef]

Anton, G.

W. Haas, M. Bech, P. Bartl, F. Bayer, A. Ritter, T. Weber, G. Pelzer, M. Willner, K. Achterhold, J. Durst, T. Michel, M. Prmmer, F. Pfeiffer, G. Anton, and J. Hornegger, “Phase-Unwrapping of Differential Phase-Contrast Data using Attenuation Information,” Proc. SPIE7962, (2011).
[CrossRef]

Astola, J.

J. Bioucas-Dias, V. Katkovnik, J. Astola, and K. Egiazarian, Multi-frequency Phase Unwrapping from Noisy Data: Adaptive Local Maximum Likelihood Approach, (Academic, 2009), pp. 310–320.

Attwood, D.

D. Attwood, Soft X-rays and Extreme Ultraviolet Radiation (Academic, 2005).

Baldi, A.

Ballabriga, R.

R. Ballabriga, M. Campbell, E. Heijne, X. Llopart, L. Tlustos, and W. Wong, “Medipix3: A 64k pixel detector readout chip working in single photon counting mode with improved spectrometric performance,” Nucl. Instrum. Meth. A633, 15–18 (2011).
[CrossRef]

Barber, W. C.

W. C. Barber, E. Nygard, J. C. Wessel, N. Malakhov, N. E. Hartsough, T. Gandhi, G. Wawrzyniak, and J. S. Iwanczyk, “Photon-counting energy-resolving CdTe detectors for high-flux X-ray imaging,” in Proceedings of IEEE Conference on Nuclear Science Symposium (IEEE, 2010), 3953–3955.
[CrossRef]

Bartl, P.

W. Haas, M. Bech, P. Bartl, F. Bayer, A. Ritter, T. Weber, G. Pelzer, M. Willner, K. Achterhold, J. Durst, T. Michel, M. Prmmer, F. Pfeiffer, G. Anton, and J. Hornegger, “Phase-Unwrapping of Differential Phase-Contrast Data using Attenuation Information,” Proc. SPIE7962, (2011).
[CrossRef]

Baruchel, J.

I. Zanette, T. Weitkamp, S. Lang, M. Langer, J. Mohr, C. David, and J. Baruchel, “Quantitative phase and absorption tomography with an X-ray interferometer and synchrotron radiation,” Phys. Status Solidi208, 2526–2532 (2011).
[CrossRef]

Baumann, J.

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

Baumbach, T.

J. Kenntner, V. Altapova, T. Grund, F. J. Pantenburg, J. Meiser, T. Baumbach, and J. Mohr, “Fabrication and characterization of analyzer gratings with high aspect ratios for phase contrast imaging using a Talbot interferometer,” AIP Conference Proceedings. 1437, 89–93 (2012).
[CrossRef]

Bayer, F.

W. Haas, M. Bech, P. Bartl, F. Bayer, A. Ritter, T. Weber, G. Pelzer, M. Willner, K. Achterhold, J. Durst, T. Michel, M. Prmmer, F. Pfeiffer, G. Anton, and J. Hornegger, “Phase-Unwrapping of Differential Phase-Contrast Data using Attenuation Information,” Proc. SPIE7962, (2011).
[CrossRef]

Bech, M.

F. Pfeiffer, M. Bech, O. Bunk, T. Donath, B. Henrich, P. Kraft, and C. David, “X-ray dark-field and phase-contrast imaging using a grating interferometer,” J. Appl. Phys.105, 102006 (2009).
[CrossRef]

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

W. Haas, M. Bech, P. Bartl, F. Bayer, A. Ritter, T. Weber, G. Pelzer, M. Willner, K. Achterhold, J. Durst, T. Michel, M. Prmmer, F. Pfeiffer, G. Anton, and J. Hornegger, “Phase-Unwrapping of Differential Phase-Contrast Data using Attenuation Information,” Proc. SPIE7962, (2011).
[CrossRef]

Beckmann, F.

J. Herzen, T. Donath, F. Beckmann, M. Ogurreck, C. David, J. Mohr, F. Pfeiffer, and A. Schreyer, “X-ray grating interferometer for materials-science imaging at a low-coherent wiggler source,” Rev. Sci. Instrum.82, 113711 (2011).
[CrossRef] [PubMed]

Bevins, N. B.

K. Li, N. B. Bevins, J. N. Zambelli, and G. Chen, “Feasibility of differential phase contrast CT for whole body imaging,” AIP Conference Proceedings. 1466, 175–180 (2012).
[CrossRef]

Bioucas-Dias, J.

J. Bioucas-Dias, V. Katkovnik, J. Astola, and K. Egiazarian, Multi-frequency Phase Unwrapping from Noisy Data: Adaptive Local Maximum Likelihood Approach, (Academic, 2009), pp. 310–320.

Born, M.

M. Born and E. Wolf, Principles of Optics (Academic, 1993).

Brnnimann, C.

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

Bruder, J.

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

Bunk, O.

F. Pfeiffer, M. Bech, O. Bunk, T. Donath, B. Henrich, P. Kraft, and C. David, “X-ray dark-field and phase-contrast imaging using a grating interferometer,” J. Appl. Phys.105, 102006 (2009).
[CrossRef]

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

C. David, J. Bruder, T. Rohbeck, C. Grnzweig, 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. Lett.2, 258–261 (2006).
[CrossRef]

Campbell, M.

R. Ballabriga, M. Campbell, E. Heijne, X. Llopart, L. Tlustos, and W. Wong, “Medipix3: A 64k pixel detector readout chip working in single photon counting mode with improved spectrometric performance,” Nucl. Instrum. Meth. A633, 15–18 (2011).
[CrossRef]

Chang, E.

W. Xu, E. Chang, L. Kwoh, H. Lim, W. Cheng, and A. Heng, “Phase-unwrapping of SAR Interferogram with Multi-frequency or Multi-baseline,” in Proceedings of IEEE Conference on Geoscience and Remote Sensing Symposium, (IEEE1994), pp. 730–732.

Chappo, M.

R. Steadman, C. Herrmann, O. Mlhens, D. G. Maeding, J. Colley, T. Firlit, R. Luhta, M. Chappo, B. Harwood, and D. Kosty, “ChromAIX: A high-rate energy-resolving photon-counting ASIC for Spectral Computed Tomography,” Proc. of SPIE7622, 762220 (2010).

Chen, C. W.

C. W. Chen and H. A. Zebker, “Two-dimensional phase unwrapping with use of statistical models for cost functions in nonlinear optimizatio,” J.Opt. Soc. Am. A18, 338–351 (2001).
[CrossRef]

Chen, G.

K. Li, N. B. Bevins, J. N. Zambelli, and G. Chen, “Feasibility of differential phase contrast CT for whole body imaging,” AIP Conference Proceedings. 1466, 175–180 (2012).
[CrossRef]

Cheng, W.

W. Xu, E. Chang, L. Kwoh, H. Lim, W. Cheng, and A. Heng, “Phase-unwrapping of SAR Interferogram with Multi-frequency or Multi-baseline,” in Proceedings of IEEE Conference on Geoscience and Remote Sensing Symposium, (IEEE1994), pp. 730–732.

Cloetens, P.

Colley, J.

R. Steadman, C. Herrmann, O. Mlhens, D. G. Maeding, J. Colley, T. Firlit, R. Luhta, M. Chappo, B. Harwood, and D. Kosty, “ChromAIX: A high-rate energy-resolving photon-counting ASIC for Spectral Computed Tomography,” Proc. of SPIE7622, 762220 (2010).

David, C.

J. Herzen, T. Donath, F. Beckmann, M. Ogurreck, C. David, J. Mohr, F. Pfeiffer, and A. Schreyer, “X-ray grating interferometer for materials-science imaging at a low-coherent wiggler source,” Rev. Sci. Instrum.82, 113711 (2011).
[CrossRef] [PubMed]

I. Zanette, T. Weitkamp, S. Lang, M. Langer, J. Mohr, C. David, and J. Baruchel, “Quantitative phase and absorption tomography with an X-ray interferometer and synchrotron radiation,” Phys. Status Solidi208, 2526–2532 (2011).
[CrossRef]

F. Pfeiffer, M. Bech, O. Bunk, T. Donath, B. Henrich, P. Kraft, and C. David, “X-ray dark-field and phase-contrast imaging using a grating interferometer,” J. Appl. Phys.105, 102006 (2009).
[CrossRef]

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

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

C. David, J. Bruder, T. Rohbeck, C. Grnzweig, 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]

F. Pfeiffer, T. Weitkamp, O. Bunk, and C. David, “Phase retrieval and differential phase-contrast imaging with low-brilliance X-ray sources,” Nature Phys. Lett.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. Express13, 6296–6304 (2005).
[CrossRef] [PubMed]

Diaz, A.

C. David, J. Bruder, T. Rohbeck, C. Grnzweig, 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. Express13, 6296–6304 (2005).
[CrossRef] [PubMed]

Donath, T.

J. Herzen, T. Donath, F. Beckmann, M. Ogurreck, C. David, J. Mohr, F. Pfeiffer, and A. Schreyer, “X-ray grating interferometer for materials-science imaging at a low-coherent wiggler source,” Rev. Sci. Instrum.82, 113711 (2011).
[CrossRef] [PubMed]

F. Pfeiffer, M. Bech, O. Bunk, T. Donath, B. Henrich, P. Kraft, and C. David, “X-ray dark-field and phase-contrast imaging using a grating interferometer,” J. Appl. Phys.105, 102006 (2009).
[CrossRef]

Durst, J.

W. Haas, M. Bech, P. Bartl, F. Bayer, A. Ritter, T. Weber, G. Pelzer, M. Willner, K. Achterhold, J. Durst, T. Michel, M. Prmmer, F. Pfeiffer, G. Anton, and J. Hornegger, “Phase-Unwrapping of Differential Phase-Contrast Data using Attenuation Information,” Proc. SPIE7962, (2011).
[CrossRef]

Egiazarian, K.

J. Bioucas-Dias, V. Katkovnik, J. Astola, and K. Egiazarian, Multi-frequency Phase Unwrapping from Noisy Data: Adaptive Local Maximum Likelihood Approach, (Academic, 2009), pp. 310–320.

Eikenberry, E. F.

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

Engelhardt, M.

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

Firlit, T.

R. Steadman, C. Herrmann, O. Mlhens, D. G. Maeding, J. Colley, T. Firlit, R. Luhta, M. Chappo, B. Harwood, and D. Kosty, “ChromAIX: A high-rate energy-resolving photon-counting ASIC for Spectral Computed Tomography,” Proc. of SPIE7622, 762220 (2010).

Gandhi, T.

W. C. Barber, E. Nygard, J. C. Wessel, N. Malakhov, N. E. Hartsough, T. Gandhi, G. Wawrzyniak, and J. S. Iwanczyk, “Photon-counting energy-resolving CdTe detectors for high-flux X-ray imaging,” in Proceedings of IEEE Conference on Nuclear Science Symposium (IEEE, 2010), 3953–3955.
[CrossRef]

Ghiglia, D.

D. Ghiglia and M. Pritt, Two-Dimensional Phase Unwrapping: Theory, Algorithms, and Software (Academic, 1998).

Graafsma, H.

D. Pennicard, S. Lange, S. Smoljanin, H. Hirsemann, and H. Graafsma, “LAMBDA - Large Area Medipix3-Based Detector Array,” JINST7, C11009 (2012).
[CrossRef]

Grnzweig, C.

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

C. David, J. Bruder, T. Rohbeck, C. Grnzweig, 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]

Grund, T.

J. Kenntner, V. Altapova, T. Grund, F. J. Pantenburg, J. Meiser, T. Baumbach, and J. Mohr, “Fabrication and characterization of analyzer gratings with high aspect ratios for phase contrast imaging using a Talbot interferometer,” AIP Conference Proceedings. 1437, 89–93 (2012).
[CrossRef]

Haas, W.

W. Haas, M. Bech, P. Bartl, F. Bayer, A. Ritter, T. Weber, G. Pelzer, M. Willner, K. Achterhold, J. Durst, T. Michel, M. Prmmer, F. Pfeiffer, G. Anton, and J. Hornegger, “Phase-Unwrapping of Differential Phase-Contrast Data using Attenuation Information,” Proc. SPIE7962, (2011).
[CrossRef]

Hamaishi, Y.

A. Momose, S. Kawamoto, I. Koyama, Y. Hamaishi, K. Takai, and Y. Suzuki, “Demonstration of X-ray Talbot Interferometry,” Jpn. J. Appl. Phys., 42, 866–868 (2003).
[CrossRef]

Hartsough, N. E.

W. C. Barber, E. Nygard, J. C. Wessel, N. Malakhov, N. E. Hartsough, T. Gandhi, G. Wawrzyniak, and J. S. Iwanczyk, “Photon-counting energy-resolving CdTe detectors for high-flux X-ray imaging,” in Proceedings of IEEE Conference on Nuclear Science Symposium (IEEE, 2010), 3953–3955.
[CrossRef]

Harwood, B.

R. Steadman, C. Herrmann, O. Mlhens, D. G. Maeding, J. Colley, T. Firlit, R. Luhta, M. Chappo, B. Harwood, and D. Kosty, “ChromAIX: A high-rate energy-resolving photon-counting ASIC for Spectral Computed Tomography,” Proc. of SPIE7622, 762220 (2010).

Hattori, T.

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 (2006).
[CrossRef]

Heijne, E.

R. Ballabriga, M. Campbell, E. Heijne, X. Llopart, L. Tlustos, and W. Wong, “Medipix3: A 64k pixel detector readout chip working in single photon counting mode with improved spectrometric performance,” Nucl. Instrum. Meth. A633, 15–18 (2011).
[CrossRef]

Heng, A.

W. Xu, E. Chang, L. Kwoh, H. Lim, W. Cheng, and A. Heng, “Phase-unwrapping of SAR Interferogram with Multi-frequency or Multi-baseline,” in Proceedings of IEEE Conference on Geoscience and Remote Sensing Symposium, (IEEE1994), pp. 730–732.

Henrich, B.

F. Pfeiffer, M. Bech, O. Bunk, T. Donath, B. Henrich, P. Kraft, and C. David, “X-ray dark-field and phase-contrast imaging using a grating interferometer,” J. Appl. Phys.105, 102006 (2009).
[CrossRef]

Herrmann, C.

R. Steadman, C. Herrmann, O. Mlhens, D. G. Maeding, J. Colley, T. Firlit, R. Luhta, M. Chappo, B. Harwood, and D. Kosty, “ChromAIX: A high-rate energy-resolving photon-counting ASIC for Spectral Computed Tomography,” Proc. of SPIE7622, 762220 (2010).

C. Herrmann, R. Steadman, and O. Mulhens, “ChromAIX: Fast energy resolved photon-counting readout electronics for Future Human Computed Tomography,” in Proceedings of IEEE Conference on Nuclear Science Symposium (IEEE, 2010), pp. 1996–1999.
[CrossRef]

Herzen, J.

J. Herzen, T. Donath, F. Beckmann, M. Ogurreck, C. David, J. Mohr, F. Pfeiffer, and A. Schreyer, “X-ray grating interferometer for materials-science imaging at a low-coherent wiggler source,” Rev. Sci. Instrum.82, 113711 (2011).
[CrossRef] [PubMed]

Hirsemann, H.

D. Pennicard, S. Lange, S. Smoljanin, H. Hirsemann, and H. Graafsma, “LAMBDA - Large Area Medipix3-Based Detector Array,” JINST7, C11009 (2012).
[CrossRef]

Hornegger, J.

W. Haas, M. Bech, P. Bartl, F. Bayer, A. Ritter, T. Weber, G. Pelzer, M. Willner, K. Achterhold, J. Durst, T. Michel, M. Prmmer, F. Pfeiffer, G. Anton, and J. Hornegger, “Phase-Unwrapping of Differential Phase-Contrast Data using Attenuation Information,” Proc. SPIE7962, (2011).
[CrossRef]

Iwanczyk, J. S.

W. C. Barber, E. Nygard, J. C. Wessel, N. Malakhov, N. E. Hartsough, T. Gandhi, G. Wawrzyniak, and J. S. Iwanczyk, “Photon-counting energy-resolving CdTe detectors for high-flux X-ray imaging,” in Proceedings of IEEE Conference on Nuclear Science Symposium (IEEE, 2010), 3953–3955.
[CrossRef]

Jefimovs, K.

Jerjen, I.

Katkovnik, V.

J. Bioucas-Dias, V. Katkovnik, J. Astola, and K. Egiazarian, Multi-frequency Phase Unwrapping from Noisy Data: Adaptive Local Maximum Likelihood Approach, (Academic, 2009), pp. 310–320.

Kaufmann, R.

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, 866–868 (2003).
[CrossRef]

Kenntner, J.

J. Kenntner, V. Altapova, T. Grund, F. J. Pantenburg, J. Meiser, T. Baumbach, and J. Mohr, “Fabrication and characterization of analyzer gratings with high aspect ratios for phase contrast imaging using a Talbot interferometer,” AIP Conference Proceedings. 1437, 89–93 (2012).
[CrossRef]

Kosty, D.

R. Steadman, C. Herrmann, O. Mlhens, D. G. Maeding, J. Colley, T. Firlit, R. Luhta, M. Chappo, B. Harwood, and D. Kosty, “ChromAIX: A high-rate energy-resolving photon-counting ASIC for Spectral Computed Tomography,” Proc. of SPIE7622, 762220 (2010).

Kottler, C.

I. Jerjen, V. Revol, P. Schuetz, C. Kottler, R. Kaufmann, T. Luethi, K. Jefimovs, C. Urban, and U. Sennhauser, “Reduction of phase artifacts in differential phase contrast computed tomography,” Opt. Express19, 13604–13611 (2011).
[CrossRef] [PubMed]

C. David, J. Bruder, T. Rohbeck, C. Grnzweig, 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]

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, 866–868 (2003).
[CrossRef]

Kraft, P.

F. Pfeiffer, M. Bech, O. Bunk, T. Donath, B. Henrich, P. Kraft, and C. David, “X-ray dark-field and phase-contrast imaging using a grating interferometer,” J. Appl. Phys.105, 102006 (2009).
[CrossRef]

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

Kwoh, L.

W. Xu, E. Chang, L. Kwoh, H. Lim, W. Cheng, and A. Heng, “Phase-unwrapping of SAR Interferogram with Multi-frequency or Multi-baseline,” in Proceedings of IEEE Conference on Geoscience and Remote Sensing Symposium, (IEEE1994), pp. 730–732.

Lang, S.

I. Zanette, T. Weitkamp, S. Lang, M. Langer, J. Mohr, C. David, and J. Baruchel, “Quantitative phase and absorption tomography with an X-ray interferometer and synchrotron radiation,” Phys. Status Solidi208, 2526–2532 (2011).
[CrossRef]

Lange, S.

D. Pennicard, S. Lange, S. Smoljanin, H. Hirsemann, and H. Graafsma, “LAMBDA - Large Area Medipix3-Based Detector Array,” JINST7, C11009 (2012).
[CrossRef]

Langer, M.

I. Zanette, T. Weitkamp, S. Lang, M. Langer, J. Mohr, C. David, and J. Baruchel, “Quantitative phase and absorption tomography with an X-ray interferometer and synchrotron radiation,” Phys. Status Solidi208, 2526–2532 (2011).
[CrossRef]

Li, K.

K. Li, N. B. Bevins, J. N. Zambelli, and G. Chen, “Feasibility of differential phase contrast CT for whole body imaging,” AIP Conference Proceedings. 1466, 175–180 (2012).
[CrossRef]

Lim, H.

W. Xu, E. Chang, L. Kwoh, H. Lim, W. Cheng, and A. Heng, “Phase-unwrapping of SAR Interferogram with Multi-frequency or Multi-baseline,” in Proceedings of IEEE Conference on Geoscience and Remote Sensing Symposium, (IEEE1994), pp. 730–732.

Llopart, X.

R. Ballabriga, M. Campbell, E. Heijne, X. Llopart, L. Tlustos, and W. Wong, “Medipix3: A 64k pixel detector readout chip working in single photon counting mode with improved spectrometric performance,” Nucl. Instrum. Meth. A633, 15–18 (2011).
[CrossRef]

Luethi, T.

Luhta, R.

R. Steadman, C. Herrmann, O. Mlhens, D. G. Maeding, J. Colley, T. Firlit, R. Luhta, M. Chappo, B. Harwood, and D. Kosty, “ChromAIX: A high-rate energy-resolving photon-counting ASIC for Spectral Computed Tomography,” Proc. of SPIE7622, 762220 (2010).

Maeding, D. G.

R. Steadman, C. Herrmann, O. Mlhens, D. G. Maeding, J. Colley, T. Firlit, R. Luhta, M. Chappo, B. Harwood, and D. Kosty, “ChromAIX: A high-rate energy-resolving photon-counting ASIC for Spectral Computed Tomography,” Proc. of SPIE7622, 762220 (2010).

Malakhov, N.

W. C. Barber, E. Nygard, J. C. Wessel, N. Malakhov, N. E. Hartsough, T. Gandhi, G. Wawrzyniak, and J. S. Iwanczyk, “Photon-counting energy-resolving CdTe detectors for high-flux X-ray imaging,” in Proceedings of IEEE Conference on Nuclear Science Symposium (IEEE, 2010), 3953–3955.
[CrossRef]

Meiser, J.

J. Kenntner, V. Altapova, T. Grund, F. J. Pantenburg, J. Meiser, T. Baumbach, and J. Mohr, “Fabrication and characterization of analyzer gratings with high aspect ratios for phase contrast imaging using a Talbot interferometer,” AIP Conference Proceedings. 1437, 89–93 (2012).
[CrossRef]

Michel, T.

W. Haas, M. Bech, P. Bartl, F. Bayer, A. Ritter, T. Weber, G. Pelzer, M. Willner, K. Achterhold, J. Durst, T. Michel, M. Prmmer, F. Pfeiffer, G. Anton, and J. Hornegger, “Phase-Unwrapping of Differential Phase-Contrast Data using Attenuation Information,” Proc. SPIE7962, (2011).
[CrossRef]

Mlhens, O.

R. Steadman, C. Herrmann, O. Mlhens, D. G. Maeding, J. Colley, T. Firlit, R. Luhta, M. Chappo, B. Harwood, and D. Kosty, “ChromAIX: A high-rate energy-resolving photon-counting ASIC for Spectral Computed Tomography,” Proc. of SPIE7622, 762220 (2010).

Mohr, J.

J. Kenntner, V. Altapova, T. Grund, F. J. Pantenburg, J. Meiser, T. Baumbach, and J. Mohr, “Fabrication and characterization of analyzer gratings with high aspect ratios for phase contrast imaging using a Talbot interferometer,” AIP Conference Proceedings. 1437, 89–93 (2012).
[CrossRef]

I. Zanette, T. Weitkamp, S. Lang, M. Langer, J. Mohr, C. David, and J. Baruchel, “Quantitative phase and absorption tomography with an X-ray interferometer and synchrotron radiation,” Phys. Status Solidi208, 2526–2532 (2011).
[CrossRef]

J. Herzen, T. Donath, F. Beckmann, M. Ogurreck, C. David, J. Mohr, F. Pfeiffer, and A. Schreyer, “X-ray grating interferometer for materials-science imaging at a low-coherent wiggler source,” Rev. Sci. Instrum.82, 113711 (2011).
[CrossRef] [PubMed]

Momose, A.

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 (2006).
[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, 866–868 (2003).
[CrossRef]

Mulhens, O.

C. Herrmann, R. Steadman, and O. Mulhens, “ChromAIX: Fast energy resolved photon-counting readout electronics for Future Human Computed Tomography,” in Proceedings of IEEE Conference on Nuclear Science Symposium (IEEE, 2010), pp. 1996–1999.
[CrossRef]

Nico, G.

G. Nico, “Bayesian approaches to phase unwrapping: theoretical study,” IEEE Trans. Sig. Process.48, 2454–2556 (2000).
[CrossRef]

Nygard, E.

W. C. Barber, E. Nygard, J. C. Wessel, N. Malakhov, N. E. Hartsough, T. Gandhi, G. Wawrzyniak, and J. S. Iwanczyk, “Photon-counting energy-resolving CdTe detectors for high-flux X-ray imaging,” in Proceedings of IEEE Conference on Nuclear Science Symposium (IEEE, 2010), 3953–3955.
[CrossRef]

Ogurreck, M.

J. Herzen, T. Donath, F. Beckmann, M. Ogurreck, C. David, J. Mohr, F. Pfeiffer, and A. Schreyer, “X-ray grating interferometer for materials-science imaging at a low-coherent wiggler source,” Rev. Sci. Instrum.82, 113711 (2011).
[CrossRef] [PubMed]

Pantenburg, F. J.

J. Kenntner, V. Altapova, T. Grund, F. J. Pantenburg, J. Meiser, T. Baumbach, and J. Mohr, “Fabrication and characterization of analyzer gratings with high aspect ratios for phase contrast imaging using a Talbot interferometer,” AIP Conference Proceedings. 1437, 89–93 (2012).
[CrossRef]

Pascazio, V.

V. Pascazio and G. Schirinzi, “Multifrequency InSAR Height Reconstruction Through Maximum Likelihood Estimation of Local Planes Parameters,” IEEE transaction on image processing, 11,(2002).
[CrossRef]

Pelzer, G.

W. Haas, M. Bech, P. Bartl, F. Bayer, A. Ritter, T. Weber, G. Pelzer, M. Willner, K. Achterhold, J. Durst, T. Michel, M. Prmmer, F. Pfeiffer, G. Anton, and J. Hornegger, “Phase-Unwrapping of Differential Phase-Contrast Data using Attenuation Information,” Proc. SPIE7962, (2011).
[CrossRef]

Pennicard, D.

D. Pennicard, S. Lange, S. Smoljanin, H. Hirsemann, and H. Graafsma, “LAMBDA - Large Area Medipix3-Based Detector Array,” JINST7, C11009 (2012).
[CrossRef]

Pfeiffer, F.

J. Herzen, T. Donath, F. Beckmann, M. Ogurreck, C. David, J. Mohr, F. Pfeiffer, and A. Schreyer, “X-ray grating interferometer for materials-science imaging at a low-coherent wiggler source,” Rev. Sci. Instrum.82, 113711 (2011).
[CrossRef] [PubMed]

F. Pfeiffer, M. Bech, O. Bunk, T. Donath, B. Henrich, P. Kraft, and C. David, “X-ray dark-field and phase-contrast imaging using a grating interferometer,” J. Appl. Phys.105, 102006 (2009).
[CrossRef]

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

C. David, J. Bruder, T. Rohbeck, C. Grnzweig, 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. Lett.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. Express13, 6296–6304 (2005).
[CrossRef] [PubMed]

W. Haas, M. Bech, P. Bartl, F. Bayer, A. Ritter, T. Weber, G. Pelzer, M. Willner, K. Achterhold, J. Durst, T. Michel, M. Prmmer, F. Pfeiffer, G. Anton, and J. Hornegger, “Phase-Unwrapping of Differential Phase-Contrast Data using Attenuation Information,” Proc. SPIE7962, (2011).
[CrossRef]

Pritt, M.

D. Ghiglia and M. Pritt, Two-Dimensional Phase Unwrapping: Theory, Algorithms, and Software (Academic, 1998).

Prmmer, M.

W. Haas, M. Bech, P. Bartl, F. Bayer, A. Ritter, T. Weber, G. Pelzer, M. Willner, K. Achterhold, J. Durst, T. Michel, M. Prmmer, F. Pfeiffer, G. Anton, and J. Hornegger, “Phase-Unwrapping of Differential Phase-Contrast Data using Attenuation Information,” Proc. SPIE7962, (2011).
[CrossRef]

Revol, V.

Ritter, A.

W. Haas, M. Bech, P. Bartl, F. Bayer, A. Ritter, T. Weber, G. Pelzer, M. Willner, K. Achterhold, J. Durst, T. Michel, M. Prmmer, F. Pfeiffer, G. Anton, and J. Hornegger, “Phase-Unwrapping of Differential Phase-Contrast Data using Attenuation Information,” Proc. SPIE7962, (2011).
[CrossRef]

Rohbeck, T.

C. David, J. Bruder, T. Rohbeck, C. Grnzweig, 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]

Schirinzi, G.

V. Pascazio and G. Schirinzi, “Multifrequency InSAR Height Reconstruction Through Maximum Likelihood Estimation of Local Planes Parameters,” IEEE transaction on image processing, 11,(2002).
[CrossRef]

Schreyer, A.

J. Herzen, T. Donath, F. Beckmann, M. Ogurreck, C. David, J. Mohr, F. Pfeiffer, and A. Schreyer, “X-ray grating interferometer for materials-science imaging at a low-coherent wiggler source,” Rev. Sci. Instrum.82, 113711 (2011).
[CrossRef] [PubMed]

Schuetz, P.

Schuster, M.

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

Sennhauser, U.

Smoljanin, S.

D. Pennicard, S. Lange, S. Smoljanin, H. Hirsemann, and H. Graafsma, “LAMBDA - Large Area Medipix3-Based Detector Array,” JINST7, C11009 (2012).
[CrossRef]

Stampanoni, M.

Steadman, R.

R. Steadman, C. Herrmann, O. Mlhens, D. G. Maeding, J. Colley, T. Firlit, R. Luhta, M. Chappo, B. Harwood, and D. Kosty, “ChromAIX: A high-rate energy-resolving photon-counting ASIC for Spectral Computed Tomography,” Proc. of SPIE7622, 762220 (2010).

C. Herrmann, R. Steadman, and O. Mulhens, “ChromAIX: Fast energy resolved photon-counting readout electronics for Future Human Computed Tomography,” in Proceedings of IEEE Conference on Nuclear Science Symposium (IEEE, 2010), pp. 1996–1999.
[CrossRef]

Suzuki, Y.

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 (2006).
[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, 866–868 (2003).
[CrossRef]

Takai, K.

A. Momose, S. Kawamoto, I. Koyama, Y. Hamaishi, K. Takai, and Y. Suzuki, “Demonstration of X-ray Talbot Interferometry,” Jpn. J. Appl. Phys., 42, 866–868 (2003).
[CrossRef]

Takeda, Y.

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 (2006).
[CrossRef]

Tikhonov, A. N.

A. N. Tikhonov, “On the stability of inverse problems,” Doklady Akademii Nauk SSSR39, 195–198 (1943).

Tlustos, L.

R. Ballabriga, M. Campbell, E. Heijne, X. Llopart, L. Tlustos, and W. Wong, “Medipix3: A 64k pixel detector readout chip working in single photon counting mode with improved spectrometric performance,” Nucl. Instrum. Meth. A633, 15–18 (2011).
[CrossRef]

Urban, C.

Wawrzyniak, G.

W. C. Barber, E. Nygard, J. C. Wessel, N. Malakhov, N. E. Hartsough, T. Gandhi, G. Wawrzyniak, and J. S. Iwanczyk, “Photon-counting energy-resolving CdTe detectors for high-flux X-ray imaging,” in Proceedings of IEEE Conference on Nuclear Science Symposium (IEEE, 2010), 3953–3955.
[CrossRef]

Weber, T.

W. Haas, M. Bech, P. Bartl, F. Bayer, A. Ritter, T. Weber, G. Pelzer, M. Willner, K. Achterhold, J. Durst, T. Michel, M. Prmmer, F. Pfeiffer, G. Anton, and J. Hornegger, “Phase-Unwrapping of Differential Phase-Contrast Data using Attenuation Information,” Proc. SPIE7962, (2011).
[CrossRef]

Weitkamp, T.

I. Zanette, T. Weitkamp, S. Lang, M. Langer, J. Mohr, C. David, and J. Baruchel, “Quantitative phase and absorption tomography with an X-ray interferometer and synchrotron radiation,” Phys. Status Solidi208, 2526–2532 (2011).
[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. Lett.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. Express13, 6296–6304 (2005).
[CrossRef] [PubMed]

Wessel, J. C.

W. C. Barber, E. Nygard, J. C. Wessel, N. Malakhov, N. E. Hartsough, T. Gandhi, G. Wawrzyniak, and J. S. Iwanczyk, “Photon-counting energy-resolving CdTe detectors for high-flux X-ray imaging,” in Proceedings of IEEE Conference on Nuclear Science Symposium (IEEE, 2010), 3953–3955.
[CrossRef]

Willner, M.

W. Haas, M. Bech, P. Bartl, F. Bayer, A. Ritter, T. Weber, G. Pelzer, M. Willner, K. Achterhold, J. Durst, T. Michel, M. Prmmer, F. Pfeiffer, G. Anton, and J. Hornegger, “Phase-Unwrapping of Differential Phase-Contrast Data using Attenuation Information,” Proc. SPIE7962, (2011).
[CrossRef]

Wolf, E.

M. Born and E. Wolf, Principles of Optics (Academic, 1993).

Wong, W.

R. Ballabriga, M. Campbell, E. Heijne, X. Llopart, L. Tlustos, and W. Wong, “Medipix3: A 64k pixel detector readout chip working in single photon counting mode with improved spectrometric performance,” Nucl. Instrum. Meth. A633, 15–18 (2011).
[CrossRef]

Xu, W.

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Yashiro, W.

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Figures (5)

Fig. 1
Fig. 1

Sketch of the Talbot-Lau interferometer. It consists of the source grating G0 with period p0, the phase grating G1 with period p1 and the analyzer grating G2 with period p2. In this case, the G1 grating was used for the stepping process. The doted black curve and the continuous red curve illustrate the interference pattern without and with a sample in the x-ray beam, respectively.

Fig. 2
Fig. 2

The top row shows a sketch of the plastic (PTFE) cube test specimen at rotation angle α. The x-ray propagation direction is illustrated by the dashed arrows. The middle and bottom rows show the measured projections of the interference pattern phase shift φ and the corresponding line plots along the dashed lines, respectively. The projections where taken at an x-ray energy of 24keV. Since phase wrapping is present in c) and d), the measured interference pattern phase shift is not anymore proportional to the actual differential phase shift of the object as expressed by Eq. (3).

Fig. 3
Fig. 3

a) The interference pattern phase shifts φ for λ1 = 0.52Å (thick black), λ2 = 0.42Å (thick green) and λ3 = 0.26Å (thick blue) together with the estimated interference pattern phase shift 2 (thin lines) for all x-ray wavelength from a single pixel over a 90° rotation of the cube. b) Zoom into the area where phase wrapping, indicated by arrows, occurs.

Fig. 4
Fig. 4

Measured (dashed) and estimated (solid) interference pattern phase shift φ and 2 respectively at an x-ray energy of 24keV for 4 different rotation angles of the cube. The arrows in d) indicate the artifacts which derive from the total external reflection of the x-rays at the surface of the cube.

Fig. 5
Fig. 5

Slices through the reconstructed electron density from a) the measured interference pattern phase shift at 24keV and b) from the estimated energy independent interference pattern phase shift M. The dashed squares indicate the area from which we extracted the mean electron densities of ρm = (4.79±0.93)×1029m−3 and ρe = (5.59±0.61)×1029m−3 from the measured and estimated interference pattern phase shift, respectively. c) Line plots along the horizontal lines of a) (continuous) and b) (dashed).

Equations (9)

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

f 0 = 1 N s = 0 N 1 I s = I ,
f 1 = 1 N s = 0 N 1 I s e 2 π i N s = 1 2 A 1 e i φ .
φ ( x , y ) = l λ p 2 Φ x
= l p 2 r e λ 2 x Sample ρ ( x , y , z ) d z
= λ 2 M .
φ = Ψ modulo 2 π ,
L ( φ j , M ) j exp ( 1 ( 2 σ j 2 ) | f 1 j 1 2 A 1 j e i λ j 2 M | 2 )
j exp ( A 1 j 2 8 I j | e i φ j e i λ j 2 M | 2 ) ,
= log ( L ) j A 1 j 2 I j ( 1 cos ( φ j λ j 2 M ) ) + β R 2 .

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