Abstract

State-of-the-art techniques for phase retrieval in propagation based X-ray phase-contrast imaging are aiming to solve an underdetermined linear system of equations. They commonly employ Tikhonov regularization – an L2-norm regularized deconvolution scheme – despite some of its limitations. We present a novel approach to phase retrieval based on Total Variation (TV) minimization. We incorporated TV minimization for deconvolution in phase retrieval using a variety of the most common linear phase-contrast models. The results of our TV minimization was compared with Tikhonov regularized deconvolution on simulated as well as experimental data. The presented method was shown to deliver improved accuracy in reconstructions based on a single distance as well as multiple distance phase-contrast images corrupted by noise and hampered by errors due to nonlinear imaging effects.

© 2012 OSA

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    [CrossRef]
  6. J. Xu, A. W. Stevenson, D. Gao, M. Tykocinski, D. Lawrence, S. W. Wilkins, G. M. Clark, E. Saunders, and R. S. Cowan, “The role of radiographic phase-contrast imaging in the development of intracochlear electrode arrays,” Otology and Neur.22, 862–868 (2001).
    [CrossRef]
  7. P. Weiss, L. Obadia, D. Magne, X. Bourges, C. Rau, T. Weitkamp, I. Khairoun, J.M. Bouler, D. Chappard, O. Gauthier, and G. Daculsi, “Synchrotron x-ray microtomography (on a micron scale) provides three dimensional imaging representation of bone ingrowth in calcium phosphate biomaterials,” Biomat.24, 4591 (2003).
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  8. P. Coan, F. Gruener, C. Glaser, T. Schneider, A. Bravin, M. Reiser, and D. Habs, “Phase contrast medical imaging with compact x-ray sources at the Munich-centre for advance photonics,” Nucl. Instr. and Meth. Phys. Res. A608, S44–S46 (2009).
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    [CrossRef]
  21. M. Langer, P. Cloetens, and F. Peyrin, “Fourier-wavelet regularization of phase retrieval in x-ray in-line phase tomography,” J. Opt. Soc. Am. A26, 1876–1881 (2009).
    [CrossRef]
  22. L.I. Rudin, S. Osher, and E. Fatemi, “Nonlinear total variation based noise removal algorithms,” Physica D.60, 259–268 (1992).
    [CrossRef]
  23. A. Chambolle, “An algorithm for total variation minimization and applications,” J. Math. Imaging Vision20, 89–97 (2004).
    [CrossRef]
  24. E. Y. Sidky, M. A. Anastasio, and X. Pan, “Image reconstruction exploiting object sparsity in boundary-enhanced x-ray phase-contrast tomography,” Opt. Express18, 10404 (2010).
    [CrossRef] [PubMed]
  25. J. Dahl, P. C. Hansen, S. H. Jensen, and T. L. Jensen, “Algorithms and software for total variation image reconstruction via first-order methods,” Num. Alg.53, 67–92 (2010).
    [CrossRef]
  26. A. W. M. van Eekeren, K. Schutte, and L. J. van Vliet, “Multiframe super-resolution reconstruction of small moving objects,” IEEE Trans. Im. Proc.19, 2901–2912 (2010).
    [CrossRef]
  27. E. Candes, J. Romberg, and T. Tao, “Robust uncertainty principles: exact signal reconstruction from highly incomplete frequency information,” IEEE Trans. Inform. Theory52, 489–509 (2006).
    [CrossRef]
  28. X. Bresson and T. F. Chan, “Fast dual minimization of the vectorial total variation norm and applications to color image processing,” Inverse Probl. and Imag.2, 455–484 (2008).
    [CrossRef]
  29. A. Beck and M. Teboulle, “A fast iterative shrinkage-thresholding algorithm for linear inverse problems,” SIAM J. Im. Sci.2, 183–202 (2009).
    [CrossRef]
  30. A. Beck and M. Teboulle, “Fast gradient-based algorithms for constrained total variation image denoising and deblurring problems,” IEEE Trans. Im. Proc.18, 2419–2434 (2009).
    [CrossRef]
  31. T. Q. Pham, L. J. van Vliet, and K. Schutte, “Robust super-resolution by minimizing a gaussian-weighted l2 error norm,” J. Phys.: Conf.124, 012037 (2008).
    [CrossRef]
  32. R. Barrett, R. Baker, P. Cloetens, Y. Dabin, C. Morawe, H. Suhonen, R. Tucoulou, A. Vivo, and L. Zhang. “Dynamically-figured mirror system for high-energy nanofocusing at the ESRF, Proc. SPIE12, 813904 (2011).
    [CrossRef]
  33. R. Mokso, P. Cloetens, E. Maire, W. Ludwig, and J.-Y. Buffire, “Nanoscale zoom tomography with hard x-rays using Kirkpatrick-Baez optics, Appl. Phys. Lett.90, 144104 (2007).
    [CrossRef]

2012

A. Kostenko, H. Sharma, E. G. Dere, A. King, W. Ludwig, W. van Oel, S. Stallinga, L. J. van Vliet, and S. E. Offerman, “Three-dimensional morphology of cementite in steel studied by x-ray phase-contrast tomography,” Scr. Mater.67, 261–264 (2012).
[CrossRef]

Q. Tao, D. li, L. Zhang, and S. Luo, “Using x-ray in-line phase-contrast imaging for the investigation of nude mouse hepatic tumors,” PLoS ONE7, e39936, (2012).
[CrossRef] [PubMed]

A. A. Appel, J. C. Larson, S. Somo, Z. Zhong, P. P. Spicer, F. K. Kasper, A. B. Garson, A. M. Zysk, A. G. Mikos, M. A. Anastasio, and E. M. Brey, “Imaging of poly(α-hydroxy-ester) scaffolds with x-ray phase-contrast microcomputed tomography,” Tissue Eng. Part C18, (2012).

2011

T. Argunova, M. Gutkin, J. H. Je, E. Mokhov, S. Nagalyuk, S. Sergey, and Y. Hwu, “SR phase-contrast imaging to address the evolution of defects during SiC growth,” Phys. Status Solid. A Appl. Mat.208, 819–824 (2011).
[CrossRef]

O. Coindreau, C. Mulat, C. Germain, J. Lachaud, and G. L. Vignoles, “Benefits of x-ray CMT for the modeling of C/C composites,” Adv. Eng. Mat.13, 178–185 (2011).
[CrossRef]

F. Cosmi, A. Bernasconi, and N. Sodini, “Phase contrast micro-tomography and morphological analysis of a short carbon fibre reinforced polyamide,” Compos. Sci. Technol.71, 23–30 (2011).
[CrossRef]

M. Herbig, A. King, P. Reischig, H. Proudhon, E. M. Lauridsen, J. Marrow, J. -Y. Buffire, and W. Ludwig, “3-D growth of a short fatigue crack within a polycrystalline microstructure studied using combined diffraction and phase-contrast x-ray tomography,” Acta Mater.59, 590–601 (2011).
[CrossRef]

R. Barrett, R. Baker, P. Cloetens, Y. Dabin, C. Morawe, H. Suhonen, R. Tucoulou, A. Vivo, and L. Zhang. “Dynamically-figured mirror system for high-energy nanofocusing at the ESRF, Proc. SPIE12, 813904 (2011).
[CrossRef]

R. Hofmann, J. Moosmann, and T. Baumbach, “Criticality in single-distance phase retrieval,” Opt. Express19, 25881–25890 (2011).
[CrossRef]

2010

E. Y. Sidky, M. A. Anastasio, and X. Pan, “Image reconstruction exploiting object sparsity in boundary-enhanced x-ray phase-contrast tomography,” Opt. Express18, 10404 (2010).
[CrossRef] [PubMed]

E. C. Ismaila, W. Kaabara, D. Garritya, O. Gundogdua, O. Bunkb, F. Pfeifferb, M. J. Farquharsond, and D. A. Bradleya, “X-ray phase contrast imaging of the bone-cartilage interface,” Appl. Rad. and Isot.68, 767–771 (2010).
[CrossRef]

J. Dahl, P. C. Hansen, S. H. Jensen, and T. L. Jensen, “Algorithms and software for total variation image reconstruction via first-order methods,” Num. Alg.53, 67–92 (2010).
[CrossRef]

A. W. M. van Eekeren, K. Schutte, and L. J. van Vliet, “Multiframe super-resolution reconstruction of small moving objects,” IEEE Trans. Im. Proc.19, 2901–2912 (2010).
[CrossRef]

2009

P. Coan, F. Gruener, C. Glaser, T. Schneider, A. Bravin, M. Reiser, and D. Habs, “Phase contrast medical imaging with compact x-ray sources at the Munich-centre for advance photonics,” Nucl. Instr. and Meth. Phys. Res. A608, S44–S46 (2009).
[CrossRef]

X. Wu and A. Yan, “Phase retrieval from one single phase contrast x-ray image,” Opt. Express17, 11187 (2009).
[CrossRef] [PubMed]

M. Langer, P. Cloetens, and F. Peyrin, “Fourier-wavelet regularization of phase retrieval in x-ray in-line phase tomography,” J. Opt. Soc. Am. A26, 1876–1881 (2009).
[CrossRef]

A. Beck and M. Teboulle, “A fast iterative shrinkage-thresholding algorithm for linear inverse problems,” SIAM J. Im. Sci.2, 183–202 (2009).
[CrossRef]

A. Beck and M. Teboulle, “Fast gradient-based algorithms for constrained total variation image denoising and deblurring problems,” IEEE Trans. Im. Proc.18, 2419–2434 (2009).
[CrossRef]

2008

T. Q. Pham, L. J. van Vliet, and K. Schutte, “Robust super-resolution by minimizing a gaussian-weighted l2 error norm,” J. Phys.: Conf.124, 012037 (2008).
[CrossRef]

X. Bresson and T. F. Chan, “Fast dual minimization of the vectorial total variation norm and applications to color image processing,” Inverse Probl. and Imag.2, 455–484 (2008).
[CrossRef]

M. Langer, F. Peyrin, P. Cloetens, and J. -P. Guigay, “Quantitative comparison of direct phase retrieval algorithms in in-line phase tomography,” Med. Phys.35, 4556 (2008).
[CrossRef] [PubMed]

2007

J. -P. Guigay, M. Langer, R. Boistel, and P. Cloetens, “A mixed contrast transfer and transport of intensity approach for phase retrieval in the Fresnel region,” Opt. Lett.32, 1617–1619 (2007).
[CrossRef] [PubMed]

R. Mokso, P. Cloetens, E. Maire, W. Ludwig, and J.-Y. Buffire, “Nanoscale zoom tomography with hard x-rays using Kirkpatrick-Baez optics, Appl. Phys. Lett.90, 144104 (2007).
[CrossRef]

2006

E. Candes, J. Romberg, and T. Tao, “Robust uncertainty principles: exact signal reconstruction from highly incomplete frequency information,” IEEE Trans. Inform. Theory52, 489–509 (2006).
[CrossRef]

2004

2003

X. Wu and H. Liu, “A general theoretical formalism for x-ray phase contrast imaging,” J. of X-ray Sci. and Techn.11, 33–42 (2003).

P. Weiss, L. Obadia, D. Magne, X. Bourges, C. Rau, T. Weitkamp, I. Khairoun, J.M. Bouler, D. Chappard, O. Gauthier, and G. Daculsi, “Synchrotron x-ray microtomography (on a micron scale) provides three dimensional imaging representation of bone ingrowth in calcium phosphate biomaterials,” Biomat.24, 4591 (2003).
[CrossRef]

2002

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

2001

J. Xu, A. W. Stevenson, D. Gao, M. Tykocinski, D. Lawrence, S. W. Wilkins, G. M. Clark, E. Saunders, and R. S. Cowan, “The role of radiographic phase-contrast imaging in the development of intracochlear electrode arrays,” Otology and Neur.22, 862–868 (2001).
[CrossRef]

1999

P. Cloetens, W. Ludwig, J. Baruchel, D. Van Dyck, J. Van Landuyt, J. P. Guigay, and M. Schlenker, “Holotomography: quantitative phase tomography with micrometer resolution using hard synchrotron radiation x-rays,” Appl. Phys. Lett.75, 2912–2914 (1999).
[CrossRef]

1992

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

Anastasio, M. A.

A. A. Appel, J. C. Larson, S. Somo, Z. Zhong, P. P. Spicer, F. K. Kasper, A. B. Garson, A. M. Zysk, A. G. Mikos, M. A. Anastasio, and E. M. Brey, “Imaging of poly(α-hydroxy-ester) scaffolds with x-ray phase-contrast microcomputed tomography,” Tissue Eng. Part C18, (2012).

E. Y. Sidky, M. A. Anastasio, and X. Pan, “Image reconstruction exploiting object sparsity in boundary-enhanced x-ray phase-contrast tomography,” Opt. Express18, 10404 (2010).
[CrossRef] [PubMed]

Appel, A. A.

A. A. Appel, J. C. Larson, S. Somo, Z. Zhong, P. P. Spicer, F. K. Kasper, A. B. Garson, A. M. Zysk, A. G. Mikos, M. A. Anastasio, and E. M. Brey, “Imaging of poly(α-hydroxy-ester) scaffolds with x-ray phase-contrast microcomputed tomography,” Tissue Eng. Part C18, (2012).

Argunova, T.

T. Argunova, M. Gutkin, J. H. Je, E. Mokhov, S. Nagalyuk, S. Sergey, and Y. Hwu, “SR phase-contrast imaging to address the evolution of defects during SiC growth,” Phys. Status Solid. A Appl. Mat.208, 819–824 (2011).
[CrossRef]

Baker, R.

R. Barrett, R. Baker, P. Cloetens, Y. Dabin, C. Morawe, H. Suhonen, R. Tucoulou, A. Vivo, and L. Zhang. “Dynamically-figured mirror system for high-energy nanofocusing at the ESRF, Proc. SPIE12, 813904 (2011).
[CrossRef]

Barrett, R.

R. Barrett, R. Baker, P. Cloetens, Y. Dabin, C. Morawe, H. Suhonen, R. Tucoulou, A. Vivo, and L. Zhang. “Dynamically-figured mirror system for high-energy nanofocusing at the ESRF, Proc. SPIE12, 813904 (2011).
[CrossRef]

Baruchel, J.

P. Cloetens, W. Ludwig, J. Baruchel, D. Van Dyck, J. Van Landuyt, J. P. Guigay, and M. Schlenker, “Holotomography: quantitative phase tomography with micrometer resolution using hard synchrotron radiation x-rays,” Appl. Phys. Lett.75, 2912–2914 (1999).
[CrossRef]

Baumbach, T.

Beck, A.

A. Beck and M. Teboulle, “A fast iterative shrinkage-thresholding algorithm for linear inverse problems,” SIAM J. Im. Sci.2, 183–202 (2009).
[CrossRef]

A. Beck and M. Teboulle, “Fast gradient-based algorithms for constrained total variation image denoising and deblurring problems,” IEEE Trans. Im. Proc.18, 2419–2434 (2009).
[CrossRef]

Bernasconi, A.

F. Cosmi, A. Bernasconi, and N. Sodini, “Phase contrast micro-tomography and morphological analysis of a short carbon fibre reinforced polyamide,” Compos. Sci. Technol.71, 23–30 (2011).
[CrossRef]

Boistel, R.

Bouler, J.M.

P. Weiss, L. Obadia, D. Magne, X. Bourges, C. Rau, T. Weitkamp, I. Khairoun, J.M. Bouler, D. Chappard, O. Gauthier, and G. Daculsi, “Synchrotron x-ray microtomography (on a micron scale) provides three dimensional imaging representation of bone ingrowth in calcium phosphate biomaterials,” Biomat.24, 4591 (2003).
[CrossRef]

Bourges, X.

P. Weiss, L. Obadia, D. Magne, X. Bourges, C. Rau, T. Weitkamp, I. Khairoun, J.M. Bouler, D. Chappard, O. Gauthier, and G. Daculsi, “Synchrotron x-ray microtomography (on a micron scale) provides three dimensional imaging representation of bone ingrowth in calcium phosphate biomaterials,” Biomat.24, 4591 (2003).
[CrossRef]

Bradleya, D. A.

E. C. Ismaila, W. Kaabara, D. Garritya, O. Gundogdua, O. Bunkb, F. Pfeifferb, M. J. Farquharsond, and D. A. Bradleya, “X-ray phase contrast imaging of the bone-cartilage interface,” Appl. Rad. and Isot.68, 767–771 (2010).
[CrossRef]

Bravin, A.

P. Coan, F. Gruener, C. Glaser, T. Schneider, A. Bravin, M. Reiser, and D. Habs, “Phase contrast medical imaging with compact x-ray sources at the Munich-centre for advance photonics,” Nucl. Instr. and Meth. Phys. Res. A608, S44–S46 (2009).
[CrossRef]

Bresson, X.

X. Bresson and T. F. Chan, “Fast dual minimization of the vectorial total variation norm and applications to color image processing,” Inverse Probl. and Imag.2, 455–484 (2008).
[CrossRef]

Brey, E. M.

A. A. Appel, J. C. Larson, S. Somo, Z. Zhong, P. P. Spicer, F. K. Kasper, A. B. Garson, A. M. Zysk, A. G. Mikos, M. A. Anastasio, and E. M. Brey, “Imaging of poly(α-hydroxy-ester) scaffolds with x-ray phase-contrast microcomputed tomography,” Tissue Eng. Part C18, (2012).

Buffire, J. -Y.

M. Herbig, A. King, P. Reischig, H. Proudhon, E. M. Lauridsen, J. Marrow, J. -Y. Buffire, and W. Ludwig, “3-D growth of a short fatigue crack within a polycrystalline microstructure studied using combined diffraction and phase-contrast x-ray tomography,” Acta Mater.59, 590–601 (2011).
[CrossRef]

Buffire, J.-Y.

R. Mokso, P. Cloetens, E. Maire, W. Ludwig, and J.-Y. Buffire, “Nanoscale zoom tomography with hard x-rays using Kirkpatrick-Baez optics, Appl. Phys. Lett.90, 144104 (2007).
[CrossRef]

Bunkb, O.

E. C. Ismaila, W. Kaabara, D. Garritya, O. Gundogdua, O. Bunkb, F. Pfeifferb, M. J. Farquharsond, and D. A. Bradleya, “X-ray phase contrast imaging of the bone-cartilage interface,” Appl. Rad. and Isot.68, 767–771 (2010).
[CrossRef]

Candes, E.

E. Candes, J. Romberg, and T. Tao, “Robust uncertainty principles: exact signal reconstruction from highly incomplete frequency information,” IEEE Trans. Inform. Theory52, 489–509 (2006).
[CrossRef]

Chambolle, A.

A. Chambolle, “An algorithm for total variation minimization and applications,” J. Math. Imaging Vision20, 89–97 (2004).
[CrossRef]

Chan, T. F.

X. Bresson and T. F. Chan, “Fast dual minimization of the vectorial total variation norm and applications to color image processing,” Inverse Probl. and Imag.2, 455–484 (2008).
[CrossRef]

Chappard, D.

P. Weiss, L. Obadia, D. Magne, X. Bourges, C. Rau, T. Weitkamp, I. Khairoun, J.M. Bouler, D. Chappard, O. Gauthier, and G. Daculsi, “Synchrotron x-ray microtomography (on a micron scale) provides three dimensional imaging representation of bone ingrowth in calcium phosphate biomaterials,” Biomat.24, 4591 (2003).
[CrossRef]

Clark, G. M.

J. Xu, A. W. Stevenson, D. Gao, M. Tykocinski, D. Lawrence, S. W. Wilkins, G. M. Clark, E. Saunders, and R. S. Cowan, “The role of radiographic phase-contrast imaging in the development of intracochlear electrode arrays,” Otology and Neur.22, 862–868 (2001).
[CrossRef]

Cloetens, P.

R. Barrett, R. Baker, P. Cloetens, Y. Dabin, C. Morawe, H. Suhonen, R. Tucoulou, A. Vivo, and L. Zhang. “Dynamically-figured mirror system for high-energy nanofocusing at the ESRF, Proc. SPIE12, 813904 (2011).
[CrossRef]

M. Langer, P. Cloetens, and F. Peyrin, “Fourier-wavelet regularization of phase retrieval in x-ray in-line phase tomography,” J. Opt. Soc. Am. A26, 1876–1881 (2009).
[CrossRef]

M. Langer, F. Peyrin, P. Cloetens, and J. -P. Guigay, “Quantitative comparison of direct phase retrieval algorithms in in-line phase tomography,” Med. Phys.35, 4556 (2008).
[CrossRef] [PubMed]

R. Mokso, P. Cloetens, E. Maire, W. Ludwig, and J.-Y. Buffire, “Nanoscale zoom tomography with hard x-rays using Kirkpatrick-Baez optics, Appl. Phys. Lett.90, 144104 (2007).
[CrossRef]

J. -P. Guigay, M. Langer, R. Boistel, and P. Cloetens, “A mixed contrast transfer and transport of intensity approach for phase retrieval in the Fresnel region,” Opt. Lett.32, 1617–1619 (2007).
[CrossRef] [PubMed]

P. Cloetens, W. Ludwig, J. Baruchel, D. Van Dyck, J. Van Landuyt, J. P. Guigay, and M. Schlenker, “Holotomography: quantitative phase tomography with micrometer resolution using hard synchrotron radiation x-rays,” Appl. Phys. Lett.75, 2912–2914 (1999).
[CrossRef]

Coan, P.

P. Coan, F. Gruener, C. Glaser, T. Schneider, A. Bravin, M. Reiser, and D. Habs, “Phase contrast medical imaging with compact x-ray sources at the Munich-centre for advance photonics,” Nucl. Instr. and Meth. Phys. Res. A608, S44–S46 (2009).
[CrossRef]

Coindreau, O.

O. Coindreau, C. Mulat, C. Germain, J. Lachaud, and G. L. Vignoles, “Benefits of x-ray CMT for the modeling of C/C composites,” Adv. Eng. Mat.13, 178–185 (2011).
[CrossRef]

Cosmi, F.

F. Cosmi, A. Bernasconi, and N. Sodini, “Phase contrast micro-tomography and morphological analysis of a short carbon fibre reinforced polyamide,” Compos. Sci. Technol.71, 23–30 (2011).
[CrossRef]

Cowan, R. S.

J. Xu, A. W. Stevenson, D. Gao, M. Tykocinski, D. Lawrence, S. W. Wilkins, G. M. Clark, E. Saunders, and R. S. Cowan, “The role of radiographic phase-contrast imaging in the development of intracochlear electrode arrays,” Otology and Neur.22, 862–868 (2001).
[CrossRef]

Dabin, Y.

R. Barrett, R. Baker, P. Cloetens, Y. Dabin, C. Morawe, H. Suhonen, R. Tucoulou, A. Vivo, and L. Zhang. “Dynamically-figured mirror system for high-energy nanofocusing at the ESRF, Proc. SPIE12, 813904 (2011).
[CrossRef]

Daculsi, G.

P. Weiss, L. Obadia, D. Magne, X. Bourges, C. Rau, T. Weitkamp, I. Khairoun, J.M. Bouler, D. Chappard, O. Gauthier, and G. Daculsi, “Synchrotron x-ray microtomography (on a micron scale) provides three dimensional imaging representation of bone ingrowth in calcium phosphate biomaterials,” Biomat.24, 4591 (2003).
[CrossRef]

Dahl, J.

J. Dahl, P. C. Hansen, S. H. Jensen, and T. L. Jensen, “Algorithms and software for total variation image reconstruction via first-order methods,” Num. Alg.53, 67–92 (2010).
[CrossRef]

Davis, T.J.

Dere, E. G.

A. Kostenko, H. Sharma, E. G. Dere, A. King, W. Ludwig, W. van Oel, S. Stallinga, L. J. van Vliet, and S. E. Offerman, “Three-dimensional morphology of cementite in steel studied by x-ray phase-contrast tomography,” Scr. Mater.67, 261–264 (2012).
[CrossRef]

Dhal, B.

Farquharsond, M. J.

E. C. Ismaila, W. Kaabara, D. Garritya, O. Gundogdua, O. Bunkb, F. Pfeifferb, M. J. Farquharsond, and D. A. Bradleya, “X-ray phase contrast imaging of the bone-cartilage interface,” Appl. Rad. and Isot.68, 767–771 (2010).
[CrossRef]

Fatemi, E.

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

Gao, D.

J. Xu, A. W. Stevenson, D. Gao, M. Tykocinski, D. Lawrence, S. W. Wilkins, G. M. Clark, E. Saunders, and R. S. Cowan, “The role of radiographic phase-contrast imaging in the development of intracochlear electrode arrays,” Otology and Neur.22, 862–868 (2001).
[CrossRef]

Garritya, D.

E. C. Ismaila, W. Kaabara, D. Garritya, O. Gundogdua, O. Bunkb, F. Pfeifferb, M. J. Farquharsond, and D. A. Bradleya, “X-ray phase contrast imaging of the bone-cartilage interface,” Appl. Rad. and Isot.68, 767–771 (2010).
[CrossRef]

Garson, A. B.

A. A. Appel, J. C. Larson, S. Somo, Z. Zhong, P. P. Spicer, F. K. Kasper, A. B. Garson, A. M. Zysk, A. G. Mikos, M. A. Anastasio, and E. M. Brey, “Imaging of poly(α-hydroxy-ester) scaffolds with x-ray phase-contrast microcomputed tomography,” Tissue Eng. Part C18, (2012).

Gauthier, O.

P. Weiss, L. Obadia, D. Magne, X. Bourges, C. Rau, T. Weitkamp, I. Khairoun, J.M. Bouler, D. Chappard, O. Gauthier, and G. Daculsi, “Synchrotron x-ray microtomography (on a micron scale) provides three dimensional imaging representation of bone ingrowth in calcium phosphate biomaterials,” Biomat.24, 4591 (2003).
[CrossRef]

Germain, C.

O. Coindreau, C. Mulat, C. Germain, J. Lachaud, and G. L. Vignoles, “Benefits of x-ray CMT for the modeling of C/C composites,” Adv. Eng. Mat.13, 178–185 (2011).
[CrossRef]

Glaser, C.

P. Coan, F. Gruener, C. Glaser, T. Schneider, A. Bravin, M. Reiser, and D. Habs, “Phase contrast medical imaging with compact x-ray sources at the Munich-centre for advance photonics,” Nucl. Instr. and Meth. Phys. Res. A608, S44–S46 (2009).
[CrossRef]

Gruener, F.

P. Coan, F. Gruener, C. Glaser, T. Schneider, A. Bravin, M. Reiser, and D. Habs, “Phase contrast medical imaging with compact x-ray sources at the Munich-centre for advance photonics,” Nucl. Instr. and Meth. Phys. Res. A608, S44–S46 (2009).
[CrossRef]

Guigay, J. P.

P. Cloetens, W. Ludwig, J. Baruchel, D. Van Dyck, J. Van Landuyt, J. P. Guigay, and M. Schlenker, “Holotomography: quantitative phase tomography with micrometer resolution using hard synchrotron radiation x-rays,” Appl. Phys. Lett.75, 2912–2914 (1999).
[CrossRef]

Guigay, J. -P.

M. Langer, F. Peyrin, P. Cloetens, and J. -P. Guigay, “Quantitative comparison of direct phase retrieval algorithms in in-line phase tomography,” Med. Phys.35, 4556 (2008).
[CrossRef] [PubMed]

J. -P. Guigay, M. Langer, R. Boistel, and P. Cloetens, “A mixed contrast transfer and transport of intensity approach for phase retrieval in the Fresnel region,” Opt. Lett.32, 1617–1619 (2007).
[CrossRef] [PubMed]

Gundogdua, O.

E. C. Ismaila, W. Kaabara, D. Garritya, O. Gundogdua, O. Bunkb, F. Pfeifferb, M. J. Farquharsond, and D. A. Bradleya, “X-ray phase contrast imaging of the bone-cartilage interface,” Appl. Rad. and Isot.68, 767–771 (2010).
[CrossRef]

Gureyev, T.

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

Gureyev, T.E.

Gutkin, M.

T. Argunova, M. Gutkin, J. H. Je, E. Mokhov, S. Nagalyuk, S. Sergey, and Y. Hwu, “SR phase-contrast imaging to address the evolution of defects during SiC growth,” Phys. Status Solid. A Appl. Mat.208, 819–824 (2011).
[CrossRef]

Habs, D.

P. Coan, F. Gruener, C. Glaser, T. Schneider, A. Bravin, M. Reiser, and D. Habs, “Phase contrast medical imaging with compact x-ray sources at the Munich-centre for advance photonics,” Nucl. Instr. and Meth. Phys. Res. A608, S44–S46 (2009).
[CrossRef]

Hansen, P. C.

J. Dahl, P. C. Hansen, S. H. Jensen, and T. L. Jensen, “Algorithms and software for total variation image reconstruction via first-order methods,” Num. Alg.53, 67–92 (2010).
[CrossRef]

Hayes, J.

Herbig, M.

M. Herbig, A. King, P. Reischig, H. Proudhon, E. M. Lauridsen, J. Marrow, J. -Y. Buffire, and W. Ludwig, “3-D growth of a short fatigue crack within a polycrystalline microstructure studied using combined diffraction and phase-contrast x-ray tomography,” Acta Mater.59, 590–601 (2011).
[CrossRef]

Hofmann, R.

Hwu, Y.

T. Argunova, M. Gutkin, J. H. Je, E. Mokhov, S. Nagalyuk, S. Sergey, and Y. Hwu, “SR phase-contrast imaging to address the evolution of defects during SiC growth,” Phys. Status Solid. A Appl. Mat.208, 819–824 (2011).
[CrossRef]

Ismaila, E. C.

E. C. Ismaila, W. Kaabara, D. Garritya, O. Gundogdua, O. Bunkb, F. Pfeifferb, M. J. Farquharsond, and D. A. Bradleya, “X-ray phase contrast imaging of the bone-cartilage interface,” Appl. Rad. and Isot.68, 767–771 (2010).
[CrossRef]

Je, J. H.

T. Argunova, M. Gutkin, J. H. Je, E. Mokhov, S. Nagalyuk, S. Sergey, and Y. Hwu, “SR phase-contrast imaging to address the evolution of defects during SiC growth,” Phys. Status Solid. A Appl. Mat.208, 819–824 (2011).
[CrossRef]

Jensen, S. H.

J. Dahl, P. C. Hansen, S. H. Jensen, and T. L. Jensen, “Algorithms and software for total variation image reconstruction via first-order methods,” Num. Alg.53, 67–92 (2010).
[CrossRef]

Jensen, T. L.

J. Dahl, P. C. Hansen, S. H. Jensen, and T. L. Jensen, “Algorithms and software for total variation image reconstruction via first-order methods,” Num. Alg.53, 67–92 (2010).
[CrossRef]

Kaabara, W.

E. C. Ismaila, W. Kaabara, D. Garritya, O. Gundogdua, O. Bunkb, F. Pfeifferb, M. J. Farquharsond, and D. A. Bradleya, “X-ray phase contrast imaging of the bone-cartilage interface,” Appl. Rad. and Isot.68, 767–771 (2010).
[CrossRef]

Kasper, F. K.

A. A. Appel, J. C. Larson, S. Somo, Z. Zhong, P. P. Spicer, F. K. Kasper, A. B. Garson, A. M. Zysk, A. G. Mikos, M. A. Anastasio, and E. M. Brey, “Imaging of poly(α-hydroxy-ester) scaffolds with x-ray phase-contrast microcomputed tomography,” Tissue Eng. Part C18, (2012).

Khairoun, I.

P. Weiss, L. Obadia, D. Magne, X. Bourges, C. Rau, T. Weitkamp, I. Khairoun, J.M. Bouler, D. Chappard, O. Gauthier, and G. Daculsi, “Synchrotron x-ray microtomography (on a micron scale) provides three dimensional imaging representation of bone ingrowth in calcium phosphate biomaterials,” Biomat.24, 4591 (2003).
[CrossRef]

King, A.

A. Kostenko, H. Sharma, E. G. Dere, A. King, W. Ludwig, W. van Oel, S. Stallinga, L. J. van Vliet, and S. E. Offerman, “Three-dimensional morphology of cementite in steel studied by x-ray phase-contrast tomography,” Scr. Mater.67, 261–264 (2012).
[CrossRef]

M. Herbig, A. King, P. Reischig, H. Proudhon, E. M. Lauridsen, J. Marrow, J. -Y. Buffire, and W. Ludwig, “3-D growth of a short fatigue crack within a polycrystalline microstructure studied using combined diffraction and phase-contrast x-ray tomography,” Acta Mater.59, 590–601 (2011).
[CrossRef]

Kostenko, A.

A. Kostenko, H. Sharma, E. G. Dere, A. King, W. Ludwig, W. van Oel, S. Stallinga, L. J. van Vliet, and S. E. Offerman, “Three-dimensional morphology of cementite in steel studied by x-ray phase-contrast tomography,” Scr. Mater.67, 261–264 (2012).
[CrossRef]

Lachaud, J.

O. Coindreau, C. Mulat, C. Germain, J. Lachaud, and G. L. Vignoles, “Benefits of x-ray CMT for the modeling of C/C composites,” Adv. Eng. Mat.13, 178–185 (2011).
[CrossRef]

Langer, M.

Larson, J. C.

A. A. Appel, J. C. Larson, S. Somo, Z. Zhong, P. P. Spicer, F. K. Kasper, A. B. Garson, A. M. Zysk, A. G. Mikos, M. A. Anastasio, and E. M. Brey, “Imaging of poly(α-hydroxy-ester) scaffolds with x-ray phase-contrast microcomputed tomography,” Tissue Eng. Part C18, (2012).

Lauridsen, E. M.

M. Herbig, A. King, P. Reischig, H. Proudhon, E. M. Lauridsen, J. Marrow, J. -Y. Buffire, and W. Ludwig, “3-D growth of a short fatigue crack within a polycrystalline microstructure studied using combined diffraction and phase-contrast x-ray tomography,” Acta Mater.59, 590–601 (2011).
[CrossRef]

Lawrence, D.

J. Xu, A. W. Stevenson, D. Gao, M. Tykocinski, D. Lawrence, S. W. Wilkins, G. M. Clark, E. Saunders, and R. S. Cowan, “The role of radiographic phase-contrast imaging in the development of intracochlear electrode arrays,” Otology and Neur.22, 862–868 (2001).
[CrossRef]

li, D.

Q. Tao, D. li, L. Zhang, and S. Luo, “Using x-ray in-line phase-contrast imaging for the investigation of nude mouse hepatic tumors,” PLoS ONE7, e39936, (2012).
[CrossRef] [PubMed]

Liu, H.

X. Wu and H. Liu, “A general theoretical formalism for x-ray phase contrast imaging,” J. of X-ray Sci. and Techn.11, 33–42 (2003).

Ludwig, W.

A. Kostenko, H. Sharma, E. G. Dere, A. King, W. Ludwig, W. van Oel, S. Stallinga, L. J. van Vliet, and S. E. Offerman, “Three-dimensional morphology of cementite in steel studied by x-ray phase-contrast tomography,” Scr. Mater.67, 261–264 (2012).
[CrossRef]

M. Herbig, A. King, P. Reischig, H. Proudhon, E. M. Lauridsen, J. Marrow, J. -Y. Buffire, and W. Ludwig, “3-D growth of a short fatigue crack within a polycrystalline microstructure studied using combined diffraction and phase-contrast x-ray tomography,” Acta Mater.59, 590–601 (2011).
[CrossRef]

R. Mokso, P. Cloetens, E. Maire, W. Ludwig, and J.-Y. Buffire, “Nanoscale zoom tomography with hard x-rays using Kirkpatrick-Baez optics, Appl. Phys. Lett.90, 144104 (2007).
[CrossRef]

P. Cloetens, W. Ludwig, J. Baruchel, D. Van Dyck, J. Van Landuyt, J. P. Guigay, and M. Schlenker, “Holotomography: quantitative phase tomography with micrometer resolution using hard synchrotron radiation x-rays,” Appl. Phys. Lett.75, 2912–2914 (1999).
[CrossRef]

Luo, S.

Q. Tao, D. li, L. Zhang, and S. Luo, “Using x-ray in-line phase-contrast imaging for the investigation of nude mouse hepatic tumors,” PLoS ONE7, e39936, (2012).
[CrossRef] [PubMed]

Magne, D.

P. Weiss, L. Obadia, D. Magne, X. Bourges, C. Rau, T. Weitkamp, I. Khairoun, J.M. Bouler, D. Chappard, O. Gauthier, and G. Daculsi, “Synchrotron x-ray microtomography (on a micron scale) provides three dimensional imaging representation of bone ingrowth in calcium phosphate biomaterials,” Biomat.24, 4591 (2003).
[CrossRef]

Maire, E.

R. Mokso, P. Cloetens, E. Maire, W. Ludwig, and J.-Y. Buffire, “Nanoscale zoom tomography with hard x-rays using Kirkpatrick-Baez optics, Appl. Phys. Lett.90, 144104 (2007).
[CrossRef]

Mancuso, A.

Marrow, J.

M. Herbig, A. King, P. Reischig, H. Proudhon, E. M. Lauridsen, J. Marrow, J. -Y. Buffire, and W. Ludwig, “3-D growth of a short fatigue crack within a polycrystalline microstructure studied using combined diffraction and phase-contrast x-ray tomography,” Acta Mater.59, 590–601 (2011).
[CrossRef]

Mayo, S.

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

Mayo, S.C.

Mikos, A. G.

A. A. Appel, J. C. Larson, S. Somo, Z. Zhong, P. P. Spicer, F. K. Kasper, A. B. Garson, A. M. Zysk, A. G. Mikos, M. A. Anastasio, and E. M. Brey, “Imaging of poly(α-hydroxy-ester) scaffolds with x-ray phase-contrast microcomputed tomography,” Tissue Eng. Part C18, (2012).

Miller, P.

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

Mokhov, E.

T. Argunova, M. Gutkin, J. H. Je, E. Mokhov, S. Nagalyuk, S. Sergey, and Y. Hwu, “SR phase-contrast imaging to address the evolution of defects during SiC growth,” Phys. Status Solid. A Appl. Mat.208, 819–824 (2011).
[CrossRef]

Mokso, R.

R. Mokso, P. Cloetens, E. Maire, W. Ludwig, and J.-Y. Buffire, “Nanoscale zoom tomography with hard x-rays using Kirkpatrick-Baez optics, Appl. Phys. Lett.90, 144104 (2007).
[CrossRef]

Moosmann, J.

Morawe, C.

R. Barrett, R. Baker, P. Cloetens, Y. Dabin, C. Morawe, H. Suhonen, R. Tucoulou, A. Vivo, and L. Zhang. “Dynamically-figured mirror system for high-energy nanofocusing at the ESRF, Proc. SPIE12, 813904 (2011).
[CrossRef]

Mulat, C.

O. Coindreau, C. Mulat, C. Germain, J. Lachaud, and G. L. Vignoles, “Benefits of x-ray CMT for the modeling of C/C composites,” Adv. Eng. Mat.13, 178–185 (2011).
[CrossRef]

Nagalyuk, S.

T. Argunova, M. Gutkin, J. H. Je, E. Mokhov, S. Nagalyuk, S. Sergey, and Y. Hwu, “SR phase-contrast imaging to address the evolution of defects during SiC growth,” Phys. Status Solid. A Appl. Mat.208, 819–824 (2011).
[CrossRef]

Nugent, K.

Obadia, L.

P. Weiss, L. Obadia, D. Magne, X. Bourges, C. Rau, T. Weitkamp, I. Khairoun, J.M. Bouler, D. Chappard, O. Gauthier, and G. Daculsi, “Synchrotron x-ray microtomography (on a micron scale) provides three dimensional imaging representation of bone ingrowth in calcium phosphate biomaterials,” Biomat.24, 4591 (2003).
[CrossRef]

Offerman, S. E.

A. Kostenko, H. Sharma, E. G. Dere, A. King, W. Ludwig, W. van Oel, S. Stallinga, L. J. van Vliet, and S. E. Offerman, “Three-dimensional morphology of cementite in steel studied by x-ray phase-contrast tomography,” Scr. Mater.67, 261–264 (2012).
[CrossRef]

Osher, S.

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

Paganin, D.

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

Pan, X.

Paterson, D.

Peele, A.

Peyrin, F.

M. Langer, P. Cloetens, and F. Peyrin, “Fourier-wavelet regularization of phase retrieval in x-ray in-line phase tomography,” J. Opt. Soc. Am. A26, 1876–1881 (2009).
[CrossRef]

M. Langer, F. Peyrin, P. Cloetens, and J. -P. Guigay, “Quantitative comparison of direct phase retrieval algorithms in in-line phase tomography,” Med. Phys.35, 4556 (2008).
[CrossRef] [PubMed]

Pfeifferb, F.

E. C. Ismaila, W. Kaabara, D. Garritya, O. Gundogdua, O. Bunkb, F. Pfeifferb, M. J. Farquharsond, and D. A. Bradleya, “X-ray phase contrast imaging of the bone-cartilage interface,” Appl. Rad. and Isot.68, 767–771 (2010).
[CrossRef]

Pham, T. Q.

T. Q. Pham, L. J. van Vliet, and K. Schutte, “Robust super-resolution by minimizing a gaussian-weighted l2 error norm,” J. Phys.: Conf.124, 012037 (2008).
[CrossRef]

Pogany, A.

Proudhon, H.

M. Herbig, A. King, P. Reischig, H. Proudhon, E. M. Lauridsen, J. Marrow, J. -Y. Buffire, and W. Ludwig, “3-D growth of a short fatigue crack within a polycrystalline microstructure studied using combined diffraction and phase-contrast x-ray tomography,” Acta Mater.59, 590–601 (2011).
[CrossRef]

Rau, C.

P. Weiss, L. Obadia, D. Magne, X. Bourges, C. Rau, T. Weitkamp, I. Khairoun, J.M. Bouler, D. Chappard, O. Gauthier, and G. Daculsi, “Synchrotron x-ray microtomography (on a micron scale) provides three dimensional imaging representation of bone ingrowth in calcium phosphate biomaterials,” Biomat.24, 4591 (2003).
[CrossRef]

Reischig, P.

M. Herbig, A. King, P. Reischig, H. Proudhon, E. M. Lauridsen, J. Marrow, J. -Y. Buffire, and W. Ludwig, “3-D growth of a short fatigue crack within a polycrystalline microstructure studied using combined diffraction and phase-contrast x-ray tomography,” Acta Mater.59, 590–601 (2011).
[CrossRef]

Reiser, M.

P. Coan, F. Gruener, C. Glaser, T. Schneider, A. Bravin, M. Reiser, and D. Habs, “Phase contrast medical imaging with compact x-ray sources at the Munich-centre for advance photonics,” Nucl. Instr. and Meth. Phys. Res. A608, S44–S46 (2009).
[CrossRef]

Romberg, J.

E. Candes, J. Romberg, and T. Tao, “Robust uncertainty principles: exact signal reconstruction from highly incomplete frequency information,” IEEE Trans. Inform. Theory52, 489–509 (2006).
[CrossRef]

Rudin, L.I.

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

Saunders, E.

J. Xu, A. W. Stevenson, D. Gao, M. Tykocinski, D. Lawrence, S. W. Wilkins, G. M. Clark, E. Saunders, and R. S. Cowan, “The role of radiographic phase-contrast imaging in the development of intracochlear electrode arrays,” Otology and Neur.22, 862–868 (2001).
[CrossRef]

Schlenker, M.

P. Cloetens, W. Ludwig, J. Baruchel, D. Van Dyck, J. Van Landuyt, J. P. Guigay, and M. Schlenker, “Holotomography: quantitative phase tomography with micrometer resolution using hard synchrotron radiation x-rays,” Appl. Phys. Lett.75, 2912–2914 (1999).
[CrossRef]

Schneider, T.

P. Coan, F. Gruener, C. Glaser, T. Schneider, A. Bravin, M. Reiser, and D. Habs, “Phase contrast medical imaging with compact x-ray sources at the Munich-centre for advance photonics,” Nucl. Instr. and Meth. Phys. Res. A608, S44–S46 (2009).
[CrossRef]

Scholten, R.

Schutte, K.

A. W. M. van Eekeren, K. Schutte, and L. J. van Vliet, “Multiframe super-resolution reconstruction of small moving objects,” IEEE Trans. Im. Proc.19, 2901–2912 (2010).
[CrossRef]

T. Q. Pham, L. J. van Vliet, and K. Schutte, “Robust super-resolution by minimizing a gaussian-weighted l2 error norm,” J. Phys.: Conf.124, 012037 (2008).
[CrossRef]

Sergey, S.

T. Argunova, M. Gutkin, J. H. Je, E. Mokhov, S. Nagalyuk, S. Sergey, and Y. Hwu, “SR phase-contrast imaging to address the evolution of defects during SiC growth,” Phys. Status Solid. A Appl. Mat.208, 819–824 (2011).
[CrossRef]

Sharma, H.

A. Kostenko, H. Sharma, E. G. Dere, A. King, W. Ludwig, W. van Oel, S. Stallinga, L. J. van Vliet, and S. E. Offerman, “Three-dimensional morphology of cementite in steel studied by x-ray phase-contrast tomography,” Scr. Mater.67, 261–264 (2012).
[CrossRef]

Sidky, E. Y.

Sodini, N.

F. Cosmi, A. Bernasconi, and N. Sodini, “Phase contrast micro-tomography and morphological analysis of a short carbon fibre reinforced polyamide,” Compos. Sci. Technol.71, 23–30 (2011).
[CrossRef]

Somo, S.

A. A. Appel, J. C. Larson, S. Somo, Z. Zhong, P. P. Spicer, F. K. Kasper, A. B. Garson, A. M. Zysk, A. G. Mikos, M. A. Anastasio, and E. M. Brey, “Imaging of poly(α-hydroxy-ester) scaffolds with x-ray phase-contrast microcomputed tomography,” Tissue Eng. Part C18, (2012).

Spicer, P. P.

A. A. Appel, J. C. Larson, S. Somo, Z. Zhong, P. P. Spicer, F. K. Kasper, A. B. Garson, A. M. Zysk, A. G. Mikos, M. A. Anastasio, and E. M. Brey, “Imaging of poly(α-hydroxy-ester) scaffolds with x-ray phase-contrast microcomputed tomography,” Tissue Eng. Part C18, (2012).

Stallinga, S.

A. Kostenko, H. Sharma, E. G. Dere, A. King, W. Ludwig, W. van Oel, S. Stallinga, L. J. van Vliet, and S. E. Offerman, “Three-dimensional morphology of cementite in steel studied by x-ray phase-contrast tomography,” Scr. Mater.67, 261–264 (2012).
[CrossRef]

Stevenson, A. W.

J. Xu, A. W. Stevenson, D. Gao, M. Tykocinski, D. Lawrence, S. W. Wilkins, G. M. Clark, E. Saunders, and R. S. Cowan, “The role of radiographic phase-contrast imaging in the development of intracochlear electrode arrays,” Otology and Neur.22, 862–868 (2001).
[CrossRef]

Suhonen, H.

R. Barrett, R. Baker, P. Cloetens, Y. Dabin, C. Morawe, H. Suhonen, R. Tucoulou, A. Vivo, and L. Zhang. “Dynamically-figured mirror system for high-energy nanofocusing at the ESRF, Proc. SPIE12, 813904 (2011).
[CrossRef]

Tao, Q.

Q. Tao, D. li, L. Zhang, and S. Luo, “Using x-ray in-line phase-contrast imaging for the investigation of nude mouse hepatic tumors,” PLoS ONE7, e39936, (2012).
[CrossRef] [PubMed]

Tao, T.

E. Candes, J. Romberg, and T. Tao, “Robust uncertainty principles: exact signal reconstruction from highly incomplete frequency information,” IEEE Trans. Inform. Theory52, 489–509 (2006).
[CrossRef]

Teboulle, M.

A. Beck and M. Teboulle, “Fast gradient-based algorithms for constrained total variation image denoising and deblurring problems,” IEEE Trans. Im. Proc.18, 2419–2434 (2009).
[CrossRef]

A. Beck and M. Teboulle, “A fast iterative shrinkage-thresholding algorithm for linear inverse problems,” SIAM J. Im. Sci.2, 183–202 (2009).
[CrossRef]

Tran, C.

Tucoulou, R.

R. Barrett, R. Baker, P. Cloetens, Y. Dabin, C. Morawe, H. Suhonen, R. Tucoulou, A. Vivo, and L. Zhang. “Dynamically-figured mirror system for high-energy nanofocusing at the ESRF, Proc. SPIE12, 813904 (2011).
[CrossRef]

Turner, L.

Tykocinski, M.

J. Xu, A. W. Stevenson, D. Gao, M. Tykocinski, D. Lawrence, S. W. Wilkins, G. M. Clark, E. Saunders, and R. S. Cowan, “The role of radiographic phase-contrast imaging in the development of intracochlear electrode arrays,” Otology and Neur.22, 862–868 (2001).
[CrossRef]

Van Dyck, D.

P. Cloetens, W. Ludwig, J. Baruchel, D. Van Dyck, J. Van Landuyt, J. P. Guigay, and M. Schlenker, “Holotomography: quantitative phase tomography with micrometer resolution using hard synchrotron radiation x-rays,” Appl. Phys. Lett.75, 2912–2914 (1999).
[CrossRef]

van Eekeren, A. W. M.

A. W. M. van Eekeren, K. Schutte, and L. J. van Vliet, “Multiframe super-resolution reconstruction of small moving objects,” IEEE Trans. Im. Proc.19, 2901–2912 (2010).
[CrossRef]

Van Landuyt, J.

P. Cloetens, W. Ludwig, J. Baruchel, D. Van Dyck, J. Van Landuyt, J. P. Guigay, and M. Schlenker, “Holotomography: quantitative phase tomography with micrometer resolution using hard synchrotron radiation x-rays,” Appl. Phys. Lett.75, 2912–2914 (1999).
[CrossRef]

van Oel, W.

A. Kostenko, H. Sharma, E. G. Dere, A. King, W. Ludwig, W. van Oel, S. Stallinga, L. J. van Vliet, and S. E. Offerman, “Three-dimensional morphology of cementite in steel studied by x-ray phase-contrast tomography,” Scr. Mater.67, 261–264 (2012).
[CrossRef]

van Vliet, L. J.

A. Kostenko, H. Sharma, E. G. Dere, A. King, W. Ludwig, W. van Oel, S. Stallinga, L. J. van Vliet, and S. E. Offerman, “Three-dimensional morphology of cementite in steel studied by x-ray phase-contrast tomography,” Scr. Mater.67, 261–264 (2012).
[CrossRef]

A. W. M. van Eekeren, K. Schutte, and L. J. van Vliet, “Multiframe super-resolution reconstruction of small moving objects,” IEEE Trans. Im. Proc.19, 2901–2912 (2010).
[CrossRef]

T. Q. Pham, L. J. van Vliet, and K. Schutte, “Robust super-resolution by minimizing a gaussian-weighted l2 error norm,” J. Phys.: Conf.124, 012037 (2008).
[CrossRef]

Vignoles, G. L.

O. Coindreau, C. Mulat, C. Germain, J. Lachaud, and G. L. Vignoles, “Benefits of x-ray CMT for the modeling of C/C composites,” Adv. Eng. Mat.13, 178–185 (2011).
[CrossRef]

Vivo, A.

R. Barrett, R. Baker, P. Cloetens, Y. Dabin, C. Morawe, H. Suhonen, R. Tucoulou, A. Vivo, and L. Zhang. “Dynamically-figured mirror system for high-energy nanofocusing at the ESRF, Proc. SPIE12, 813904 (2011).
[CrossRef]

Weiss, P.

P. Weiss, L. Obadia, D. Magne, X. Bourges, C. Rau, T. Weitkamp, I. Khairoun, J.M. Bouler, D. Chappard, O. Gauthier, and G. Daculsi, “Synchrotron x-ray microtomography (on a micron scale) provides three dimensional imaging representation of bone ingrowth in calcium phosphate biomaterials,” Biomat.24, 4591 (2003).
[CrossRef]

Weitkamp, T.

P. Weiss, L. Obadia, D. Magne, X. Bourges, C. Rau, T. Weitkamp, I. Khairoun, J.M. Bouler, D. Chappard, O. Gauthier, and G. Daculsi, “Synchrotron x-ray microtomography (on a micron scale) provides three dimensional imaging representation of bone ingrowth in calcium phosphate biomaterials,” Biomat.24, 4591 (2003).
[CrossRef]

Wilkins, S.

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

Wilkins, S. W.

J. Xu, A. W. Stevenson, D. Gao, M. Tykocinski, D. Lawrence, S. W. Wilkins, G. M. Clark, E. Saunders, and R. S. Cowan, “The role of radiographic phase-contrast imaging in the development of intracochlear electrode arrays,” Otology and Neur.22, 862–868 (2001).
[CrossRef]

Wilkins, S.W.

Wu, X.

X. Wu and A. Yan, “Phase retrieval from one single phase contrast x-ray image,” Opt. Express17, 11187 (2009).
[CrossRef] [PubMed]

X. Wu and H. Liu, “A general theoretical formalism for x-ray phase contrast imaging,” J. of X-ray Sci. and Techn.11, 33–42 (2003).

Xu, J.

J. Xu, A. W. Stevenson, D. Gao, M. Tykocinski, D. Lawrence, S. W. Wilkins, G. M. Clark, E. Saunders, and R. S. Cowan, “The role of radiographic phase-contrast imaging in the development of intracochlear electrode arrays,” Otology and Neur.22, 862–868 (2001).
[CrossRef]

Yan, A.

Zhang, L.

Q. Tao, D. li, L. Zhang, and S. Luo, “Using x-ray in-line phase-contrast imaging for the investigation of nude mouse hepatic tumors,” PLoS ONE7, e39936, (2012).
[CrossRef] [PubMed]

R. Barrett, R. Baker, P. Cloetens, Y. Dabin, C. Morawe, H. Suhonen, R. Tucoulou, A. Vivo, and L. Zhang. “Dynamically-figured mirror system for high-energy nanofocusing at the ESRF, Proc. SPIE12, 813904 (2011).
[CrossRef]

Zhong, Z.

A. A. Appel, J. C. Larson, S. Somo, Z. Zhong, P. P. Spicer, F. K. Kasper, A. B. Garson, A. M. Zysk, A. G. Mikos, M. A. Anastasio, and E. M. Brey, “Imaging of poly(α-hydroxy-ester) scaffolds with x-ray phase-contrast microcomputed tomography,” Tissue Eng. Part C18, (2012).

Zysk, A. M.

A. A. Appel, J. C. Larson, S. Somo, Z. Zhong, P. P. Spicer, F. K. Kasper, A. B. Garson, A. M. Zysk, A. G. Mikos, M. A. Anastasio, and E. M. Brey, “Imaging of poly(α-hydroxy-ester) scaffolds with x-ray phase-contrast microcomputed tomography,” Tissue Eng. Part C18, (2012).

Acta Mater.

M. Herbig, A. King, P. Reischig, H. Proudhon, E. M. Lauridsen, J. Marrow, J. -Y. Buffire, and W. Ludwig, “3-D growth of a short fatigue crack within a polycrystalline microstructure studied using combined diffraction and phase-contrast x-ray tomography,” Acta Mater.59, 590–601 (2011).
[CrossRef]

Adv. Eng. Mat.

O. Coindreau, C. Mulat, C. Germain, J. Lachaud, and G. L. Vignoles, “Benefits of x-ray CMT for the modeling of C/C composites,” Adv. Eng. Mat.13, 178–185 (2011).
[CrossRef]

Appl. Opt.

Appl. Phys. Lett.

P. Cloetens, W. Ludwig, J. Baruchel, D. Van Dyck, J. Van Landuyt, J. P. Guigay, and M. Schlenker, “Holotomography: quantitative phase tomography with micrometer resolution using hard synchrotron radiation x-rays,” Appl. Phys. Lett.75, 2912–2914 (1999).
[CrossRef]

R. Mokso, P. Cloetens, E. Maire, W. Ludwig, and J.-Y. Buffire, “Nanoscale zoom tomography with hard x-rays using Kirkpatrick-Baez optics, Appl. Phys. Lett.90, 144104 (2007).
[CrossRef]

Appl. Rad. and Isot.

E. C. Ismaila, W. Kaabara, D. Garritya, O. Gundogdua, O. Bunkb, F. Pfeifferb, M. J. Farquharsond, and D. A. Bradleya, “X-ray phase contrast imaging of the bone-cartilage interface,” Appl. Rad. and Isot.68, 767–771 (2010).
[CrossRef]

Biomat.

P. Weiss, L. Obadia, D. Magne, X. Bourges, C. Rau, T. Weitkamp, I. Khairoun, J.M. Bouler, D. Chappard, O. Gauthier, and G. Daculsi, “Synchrotron x-ray microtomography (on a micron scale) provides three dimensional imaging representation of bone ingrowth in calcium phosphate biomaterials,” Biomat.24, 4591 (2003).
[CrossRef]

Compos. Sci. Technol.

F. Cosmi, A. Bernasconi, and N. Sodini, “Phase contrast micro-tomography and morphological analysis of a short carbon fibre reinforced polyamide,” Compos. Sci. Technol.71, 23–30 (2011).
[CrossRef]

IEEE Trans. Im. Proc.

A. Beck and M. Teboulle, “Fast gradient-based algorithms for constrained total variation image denoising and deblurring problems,” IEEE Trans. Im. Proc.18, 2419–2434 (2009).
[CrossRef]

A. W. M. van Eekeren, K. Schutte, and L. J. van Vliet, “Multiframe super-resolution reconstruction of small moving objects,” IEEE Trans. Im. Proc.19, 2901–2912 (2010).
[CrossRef]

IEEE Trans. Inform. Theory

E. Candes, J. Romberg, and T. Tao, “Robust uncertainty principles: exact signal reconstruction from highly incomplete frequency information,” IEEE Trans. Inform. Theory52, 489–509 (2006).
[CrossRef]

Inverse Probl. and Imag.

X. Bresson and T. F. Chan, “Fast dual minimization of the vectorial total variation norm and applications to color image processing,” Inverse Probl. and Imag.2, 455–484 (2008).
[CrossRef]

J. Math. Imaging Vision

A. Chambolle, “An algorithm for total variation minimization and applications,” J. Math. Imaging Vision20, 89–97 (2004).
[CrossRef]

J. Microsc.

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

J. of X-ray Sci. and Techn.

X. Wu and H. Liu, “A general theoretical formalism for x-ray phase contrast imaging,” J. of X-ray Sci. and Techn.11, 33–42 (2003).

J. Opt. Soc. Am. A

J. Phys.: Conf.

T. Q. Pham, L. J. van Vliet, and K. Schutte, “Robust super-resolution by minimizing a gaussian-weighted l2 error norm,” J. Phys.: Conf.124, 012037 (2008).
[CrossRef]

Med. Phys.

M. Langer, F. Peyrin, P. Cloetens, and J. -P. Guigay, “Quantitative comparison of direct phase retrieval algorithms in in-line phase tomography,” Med. Phys.35, 4556 (2008).
[CrossRef] [PubMed]

Nucl. Instr. and Meth. Phys. Res. A

P. Coan, F. Gruener, C. Glaser, T. Schneider, A. Bravin, M. Reiser, and D. Habs, “Phase contrast medical imaging with compact x-ray sources at the Munich-centre for advance photonics,” Nucl. Instr. and Meth. Phys. Res. A608, S44–S46 (2009).
[CrossRef]

Num. Alg.

J. Dahl, P. C. Hansen, S. H. Jensen, and T. L. Jensen, “Algorithms and software for total variation image reconstruction via first-order methods,” Num. Alg.53, 67–92 (2010).
[CrossRef]

Opt. Express

Opt. Lett.

Otology and Neur.

J. Xu, A. W. Stevenson, D. Gao, M. Tykocinski, D. Lawrence, S. W. Wilkins, G. M. Clark, E. Saunders, and R. S. Cowan, “The role of radiographic phase-contrast imaging in the development of intracochlear electrode arrays,” Otology and Neur.22, 862–868 (2001).
[CrossRef]

Phys. Status Solid. A Appl. Mat.

T. Argunova, M. Gutkin, J. H. Je, E. Mokhov, S. Nagalyuk, S. Sergey, and Y. Hwu, “SR phase-contrast imaging to address the evolution of defects during SiC growth,” Phys. Status Solid. A Appl. Mat.208, 819–824 (2011).
[CrossRef]

Physica D.

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

PLoS ONE

Q. Tao, D. li, L. Zhang, and S. Luo, “Using x-ray in-line phase-contrast imaging for the investigation of nude mouse hepatic tumors,” PLoS ONE7, e39936, (2012).
[CrossRef] [PubMed]

Proc. SPIE

R. Barrett, R. Baker, P. Cloetens, Y. Dabin, C. Morawe, H. Suhonen, R. Tucoulou, A. Vivo, and L. Zhang. “Dynamically-figured mirror system for high-energy nanofocusing at the ESRF, Proc. SPIE12, 813904 (2011).
[CrossRef]

Scr. Mater.

A. Kostenko, H. Sharma, E. G. Dere, A. King, W. Ludwig, W. van Oel, S. Stallinga, L. J. van Vliet, and S. E. Offerman, “Three-dimensional morphology of cementite in steel studied by x-ray phase-contrast tomography,” Scr. Mater.67, 261–264 (2012).
[CrossRef]

SIAM J. Im. Sci.

A. Beck and M. Teboulle, “A fast iterative shrinkage-thresholding algorithm for linear inverse problems,” SIAM J. Im. Sci.2, 183–202 (2009).
[CrossRef]

Tissue Eng. Part C

A. A. Appel, J. C. Larson, S. Somo, Z. Zhong, P. P. Spicer, F. K. Kasper, A. B. Garson, A. M. Zysk, A. G. Mikos, M. A. Anastasio, and E. M. Brey, “Imaging of poly(α-hydroxy-ester) scaffolds with x-ray phase-contrast microcomputed tomography,” Tissue Eng. Part C18, (2012).

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Figures (7)

Fig. 1
Fig. 1

Phase reconstructions based on the simulated data of the ’flat’ phantom for TV and L2 regularization with and without frequency weighting (see Sec. 3.1 for simulation parameters). (a) Ground truth. (b) Intensity image at zero distance with Gaussian noise. (c) Propagated phase-contrast image at 1m with Gaussian noise. (d) L2-regularized solution, no frequency weighting. (e) TV-regularized solution, no frequency weighting. (f) L2-regularized solution, with frequency weighting. (g) TV-regularized solution with frequency weighting.

Fig. 2
Fig. 2

The radial frequency spectrum (angular averaged) of the reconstruction. Dotted line shows |Ãj,j| on both graphs. Left: normalized difference between the solutions with and without TV-regularization, i.e. (εTV = 0.02) vs. (εTV = 0). Right: normalized error-magnitude of the TV-regularized solution (solid line) and the L2-regularized solution (dashed line). The graphs are normalized against the radial frequency spectrum of the ground truth image.

Fig. 3
Fig. 3

Radial frequency spectrum (angular averaged) of the retrieval results using the frequency dependent regularization weights (i.e. frequency weighting). Dotted line shows |Ãj,j| on both graphs. Left: estimated frequency spectrum of the specimen (solid line). Right: normalized error-magnitude of the TV-regularized solution (solid line) and L2-regularized solution (dashed line) with frequency weighting.

Fig. 4
Fig. 4

Reconstructions based on the simulated data of the ’spheres’ phantom for different propagation models. All results are obtained using frequency weighting. Top row of images (a, c, e, g) show L2-regularized solutions. The bottom row of images (b, d, f, h) show TV-regularized solutions. Solutions (a, b) are based on the CTF model, (c, d) on the Mixed model, (e, f) on the dual-CTF model and (g, h) on the dual-Mixed model.

Fig. 5
Fig. 5

The radial frequency spectrum (angular averaged) of the reconstructions for different propagation models. Dotted line shows |Ãj,j| on all graphs. Normalized error magnitude of the TV-regularized solution (solid line) and L2-regularized solution (dashed line).

Fig. 6
Fig. 6

Influence of the regularization weight on samples with increasing noise levels and on samples of increasing thickness. RMSE of the solution is shown against the magnitude of regularization parameter for L2 regularization (crosses) against TV regularization (plus signs). Left: each curve shows errors for different noise levels, STD = 0.01, 0.02, 0.05, 0.1. Right: each curve shows errors for different specimen thickness, 0.1 mm, 0.2 mm, 0.5 mm, 1 mm.

Fig. 7
Fig. 7

Phase retrieval from experimental data using the dual-CTF model. (a–d) Phase retrieval of a ’dots and lines’ pattern. (e–h) Phase retrieval of a ’star’ pattern. (a, e) L2-regularized solution based on 4 images recorded at different propagation distances. (b, f) TV-regularized solution based on 4 images recorded at different propagation distances. (c, g) L2-regularized solution based on a single recorded image. (d, h) TV-regularized solution based on a single recorded image.

Equations (25)

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H D = P D T .
H ˜ D ( f ) = P ˜ D ( f ) T ˜ ( f ) ,
t ˜ = ( T ˜ ( f 1 ) T ˜ ( f k ) ) and P ˜ D = ( P ˜ D ( f 1 ) P ˜ D ( f 2 ) P ˜ D ( f k ) ) .
h ˜ D = P ˜ D t ˜ ,
h D = P D t
b = A x
argmin x : A x b 2 2 + ε L 2 x 2 2 ,
x ˜ j = A ˜ j , j * b ˜ j | A ˜ j , j | 2 + ε L 2 ,
x ˜ j A ˜ j , j * b ˜ j ε L 2 .
argmin x : A x b 2 2 + ε T V x T V ,
x T V = ( h x ) 2 + ( v x ) 2 ,
argmin x : x 0 x 2 2 + ε T V x T V .
1 : y n 1 = x n 1 + ( t n 1 1 t n ) ( x n 1 x n 2 ) ; 2 : y n = y n 1 2 L A T ( A y n 1 b ) ; 3 : x n = D L , ε ( y n ) .
y ˜ j n = y ˜ j n 1 2 ω j L A ˜ j , j * ( A ˜ j , j y ˜ j n 1 b ˜ j ) .
C ˜ D = ( cos ( α 1 ) cos ( α k ) ) , S ˜ D = ( sin ( α 1 ) sin ( α k ) )
I ˜ D ( f ) = δ ( f ) 2 cos ( α ) μ ˜ ( f ) + 2 sin ( α ) φ ˜ ( f ) .
A ˜ = ( 2 C ˜ D ( 1 ) 2 S ˜ D ( 1 ) 2 C ˜ D ( m ) 2 S ˜ D ( m ) ) , x ˜ = ( μ ˜ φ ˜ ) , b ˜ = ( I ˜ D ( 1 ) I ˜ D ( m ) ) .
μ ˜ j n = μ ˜ j n 1 + 4 ω j L m i = 1 m C ˜ D ( i ) ( j , j ) ( 2 C ˜ D ( i ) ( j , j ) μ ˜ j n 1 + 2 S ˜ D ( i ) ( j , j ) φ ˜ j n 1 I ˜ D ( i ) j ) , φ ˜ j n = φ ˜ j n 1 4 ω j L m i = 1 m S ˜ D ( i ) ( j , j ) ( 2 C ˜ D ( i ) ( j , j ) μ ˜ j n 1 + 2 S ˜ D ( i ) ( j , j ) φ ˜ j n 1 I ˜ D ( i ) j ) .
I ˜ D ( f ) = cos ( α ) I ˜ 0 ( f ) + 2 sin ( α ) ( I 0 φ ˜ ) ( f ) .
A = ( C ˜ D ( 1 ) 2 S ˜ D ( 1 ) C ˜ D ( m ) 2 S ˜ D ( m ) ) , x ˜ = ( I ˜ 0 I 0 φ ˜ ) , b ˜ = ( I ˜ D ( 1 ) I ˜ D ( m ) ) .
I ˜ 0 , j n = I ˜ 0 , j n 1 2 ω j L m i = 1 m C ˜ D ( i ) ( j , j ) ( C ˜ D ( i ) ( j , j ) I ˜ 0 , j n 1 + 2 S ˜ D ( i ) ( j , j ) ( I 0 φ ˜ ) j n 1 I ˜ D ( i ) j ) ( I 0 φ ˜ ) j n = ( I 0 φ ˜ ) j n 1 4 ω j L m i = 1 m S ˜ D ( i ) ( j , j ) ( C ˜ D ( i ) ( j , j ) I 0 , j n 1 + 2 S ˜ D ( i ) ( j , j ) ( I 0 φ ˜ ) j n 1 I ˜ D ( i ) j ) .
I ˜ D ( f ) = 2 ( σ sin ( α ) cos ( α ) ) μ ˜ ( f ) ,
I ˜ D ( f ) = ( cos ( α ) + ( α σ ) sin ( α ) ) I ˜ 0 ( f ) ,
μ ˜ j n = μ ˜ j n 1 4 ω j L m i = 1 m B ˜ i ( 2 B ˜ i μ ˜ j n 1 I ˜ D ( i ) j ) ,
I ˜ 0 , j n = I ˜ 0 , j n 1 2 ω j L m i 1 m B ˜ i ( B ˜ i I ˜ 0 , j n 1 I ˜ D ( i ) j ) ,

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