Abstract

We report the implementation of a method which can yield the 3D distribution of the phase (refractive index) of a weakly absorbing object from a single tomographic data set. In order to reduce the residual absorption artifact (due to the fact that only one data set is used) the original algorithm presented by A. V. Bronnikov is amended by adding in the filter a factor found by using a semi empirical approach. The quality of the reconstruction is largely sufficient for optimal segmentation and further postprocessing even though the filter correction is based on assumption of constant absorption. This one step approach allows keeping radiation dose to the minimum. Spatial resolution is comparable to the conventional absorption based technique. The performance of the method is validated by using an established phase contrast technique.

© 2006 Optical Society of America

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  1. U. Bonse and F. Bush , “ X-ray computed microtomography using synchrotron radiation ,” Prog. Biophys. Mol. Biol. ,   65 , 133 – 169 ( 1996 ).
    [CrossRef] [PubMed]
  2. B. A. Dowd , G. H. Campbell , R. B. Marr , V. V. Nagarkar , S. V. Tipnis , L. Axe , and D. P. Siddons , “ Developments in synchrotron x-ray computed microtomography at the National Synchrotron Light Source ,” in Developments in X-Ray Tomography II , U. Bonse , ed., Proc. SPIE   3772 , 224 – 236 ( 1999 ).
    [CrossRef]
  3. M. Stampanoni , G. Borchert , P. Wyss , R. Abela , B. Patterson , S. Hunt , D. Vermeulen , and P. Rüegsegger , “ High resolution X-ray detector for synchrotron-based microtomography ,” Nucl Instrum Methods A   491 , 291 – 301 ( 2002 ).
    [CrossRef]
  4. T. Weitkamp , C. Raven , and A. Snigirev , “ An imaging and microtomography facility at the ESRF beamline ID 22 ,” in Developments in X-Ray Tomography II , U. Bonse , ed., Proc. SPIE   3772 , 311 – 317 ( 1999 ).
    [CrossRef]
  5. U. Bonse and M. Hart , “ An x-ray interferometer ,” Appl. Phys. Lett.   6 , 155 – 156 ( 1965 ).
    [CrossRef]
  6. U. Bonse and F. Beckmann , “ Multiple-beam X-ray interferometry for phase-contrast microtomography ,” Journal of Synchrotron Radiat.   8 , 1 – 5 ( 2001 ).
    [CrossRef]
  7. A. Momose , “ Demonstration of phase-contrast x-ray computed-tomography using an x-ray interferometer ,” Nuclear Instruments and Methods A   352 , 622 – 628 ( 1995 ).
    [CrossRef]
  8. T. Weitkamp , A. Diaz , C. David , F. Pfeiffer , M. Stampanoni , P. Cloetens , and E. Ziegler , “ X-ray phase imaging with a grating interferometer ,” Opt. Express   13 , 6296 – 6304 ( 2005 ).
    [CrossRef] [PubMed]
  9. M. R. Teague , “ Deterministic phase retrieval - a Green-function solution ,” J. of the Opt. Soc. Am.   73 , 1434 – 1441 ( 1983 ).
    [CrossRef]
  10. K. A. Nugent , T. E. Gureyev , D. F. Cookson , D. Paganin , and Z. Barnea , “ Quantitative phase imaging using hard x rays ,” Phys. Rev. Lett.   77 , 2961 – 2964 ( 1996 ).
    [CrossRef] [PubMed]
  11. L. J. Allen , W. McBride , N. L. O’Leary , and M. P. Oxley , “ Exit wave reconstruction at atomic resolution ,” Ultramicroscopy   100 , 91 – 104 ( 2004 ).
    [CrossRef] [PubMed]
  12. W. K. Hsieh , F. R. Chen , J. J. Kai , and A. I. Kirkland , “ Resolution extension and exit wave reconstruction in complex HREM ,” Ultramicroscopy   98 , 99 – 114 ( 2004 ).
    [CrossRef] [PubMed]
  13. S. C. Mayo , P. R. Miller , S. W. Wilkins , T. J. Davis , D. Gao , T. E. Gureyev , D. Paganin , D. J. Parry , A. Pogany , and A. W. Stevenson , “ Quantitative X-ray projection microscopy: phase-contrast and multi-spectral imaging ,” J. Microsc.   207 , 79 – 96 ( 2002 ).
    [CrossRef] [PubMed]
  14. T. E. Gureyev , A. Pogany , D. M. Paganin , and S. W. Wilkins , “ Linear algorithms for phase retrieval in the Fresnel region ,” Optics Commun.   231 , 53 – 70 ( 2004 ).
    [CrossRef]
  15. A. G. Peele , F. DeCarlo , P. J. Mahon , B. B. Dhal , and K. A. Nugent , “ X-ray phase contrast tomography with a bending magnet source ,” Rev. Sci. Instrum.   76 , 083707 ( 2005 ).
    [CrossRef]
  16. T. E. Gureyev and K. A. Nugent , “ Phase retrieval with the transport-of-intensity equation II. Orthogonal series solution for nonuniform illumination ,” J. Opt. Soc. Am. A   13 , 1670 – 1682 ( 1996 ).
    [CrossRef]
  17. 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]
  18. U. Bonse , F. Beckmann , M. Bartscher , T. Biermann , F. Busch , and O. Günnewig , “ Phase contrast x-ray tomography using synchrotron radiation ” in Developments in X-Ray Tomography , U. Bonse , ed., Proc. SPIE   3149 , 108 – 119 ( 1997 ).
    [CrossRef]
  19. A. V. Bronnikov , “ Reconstruction formulas in phase-contrast tomography ,” Optics Communications   171 , 239 – 244 ( 1999 ).
    [CrossRef]
  20. A. V. Bronnikov , “ Theory of quantitative phase-contrast computed tomography ,” J. of the Opt. Soc. of Am. A   19 , 472 – 480 ( 2002 ).
    [CrossRef]
  21. A. Groso , M. Stampanoni , R. Abela , P. Schneider , S. Linga , and R. Müller , “ Phase contrast tomography: an alternative approach ,” Appl. Phys. Lett.   88 , 214104 ( 2006 ).
    [CrossRef]
  22. The computation of the two-dimensional Radon transform and backprojection over the angle ω are combined in a single step of the algorithm [20].
  23. J. M. Cowley , Diffraction Physics ( North-Holland , 1995 ), Chap. 3.4.
  24. A. Pogany , D. Gao , and S. W. Wilkins , “ Contrast and resolution in imaging with a microfocus x-ray source ,” Rev. Sci. Instr.   68 , 2774 – 2782 ( 1997 ).
    [CrossRef]

2006 (1)

A. Groso , M. Stampanoni , R. Abela , P. Schneider , S. Linga , and R. Müller , “ Phase contrast tomography: an alternative approach ,” Appl. Phys. Lett.   88 , 214104 ( 2006 ).
[CrossRef]

2005 (2)

T. Weitkamp , A. Diaz , C. David , F. Pfeiffer , M. Stampanoni , P. Cloetens , and E. Ziegler , “ X-ray phase imaging with a grating interferometer ,” Opt. Express   13 , 6296 – 6304 ( 2005 ).
[CrossRef] [PubMed]

A. G. Peele , F. DeCarlo , P. J. Mahon , B. B. Dhal , and K. A. Nugent , “ X-ray phase contrast tomography with a bending magnet source ,” Rev. Sci. Instrum.   76 , 083707 ( 2005 ).
[CrossRef]

2004 (3)

L. J. Allen , W. McBride , N. L. O’Leary , and M. P. Oxley , “ Exit wave reconstruction at atomic resolution ,” Ultramicroscopy   100 , 91 – 104 ( 2004 ).
[CrossRef] [PubMed]

W. K. Hsieh , F. R. Chen , J. J. Kai , and A. I. Kirkland , “ Resolution extension and exit wave reconstruction in complex HREM ,” Ultramicroscopy   98 , 99 – 114 ( 2004 ).
[CrossRef] [PubMed]

T. E. Gureyev , A. Pogany , D. M. Paganin , and S. W. Wilkins , “ Linear algorithms for phase retrieval in the Fresnel region ,” Optics Commun.   231 , 53 – 70 ( 2004 ).
[CrossRef]

2002 (3)

S. C. Mayo , P. R. Miller , S. W. Wilkins , T. J. Davis , D. Gao , T. E. Gureyev , D. Paganin , D. J. Parry , A. Pogany , and A. W. Stevenson , “ Quantitative X-ray projection microscopy: phase-contrast and multi-spectral imaging ,” J. Microsc.   207 , 79 – 96 ( 2002 ).
[CrossRef] [PubMed]

M. Stampanoni , G. Borchert , P. Wyss , R. Abela , B. Patterson , S. Hunt , D. Vermeulen , and P. Rüegsegger , “ High resolution X-ray detector for synchrotron-based microtomography ,” Nucl Instrum Methods A   491 , 291 – 301 ( 2002 ).
[CrossRef]

A. V. Bronnikov , “ Theory of quantitative phase-contrast computed tomography ,” J. of the Opt. Soc. of Am. A   19 , 472 – 480 ( 2002 ).
[CrossRef]

2001 (1)

U. Bonse and F. Beckmann , “ Multiple-beam X-ray interferometry for phase-contrast microtomography ,” Journal of Synchrotron Radiat.   8 , 1 – 5 ( 2001 ).
[CrossRef]

1999 (4)

T. Weitkamp , C. Raven , and A. Snigirev , “ An imaging and microtomography facility at the ESRF beamline ID 22 ,” in Developments in X-Ray Tomography II , U. Bonse , ed., Proc. SPIE   3772 , 311 – 317 ( 1999 ).
[CrossRef]

B. A. Dowd , G. H. Campbell , R. B. Marr , V. V. Nagarkar , S. V. Tipnis , L. Axe , and D. P. Siddons , “ Developments in synchrotron x-ray computed microtomography at the National Synchrotron Light Source ,” in Developments in X-Ray Tomography II , U. Bonse , ed., Proc. SPIE   3772 , 224 – 236 ( 1999 ).
[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]

A. V. Bronnikov , “ Reconstruction formulas in phase-contrast tomography ,” Optics Communications   171 , 239 – 244 ( 1999 ).
[CrossRef]

1997 (2)

A. Pogany , D. Gao , and S. W. Wilkins , “ Contrast and resolution in imaging with a microfocus x-ray source ,” Rev. Sci. Instr.   68 , 2774 – 2782 ( 1997 ).
[CrossRef]

U. Bonse , F. Beckmann , M. Bartscher , T. Biermann , F. Busch , and O. Günnewig , “ Phase contrast x-ray tomography using synchrotron radiation ” in Developments in X-Ray Tomography , U. Bonse , ed., Proc. SPIE   3149 , 108 – 119 ( 1997 ).
[CrossRef]

1996 (3)

T. E. Gureyev and K. A. Nugent , “ Phase retrieval with the transport-of-intensity equation II. Orthogonal series solution for nonuniform illumination ,” J. Opt. Soc. Am. A   13 , 1670 – 1682 ( 1996 ).
[CrossRef]

K. A. Nugent , T. E. Gureyev , D. F. Cookson , D. Paganin , and Z. Barnea , “ Quantitative phase imaging using hard x rays ,” Phys. Rev. Lett.   77 , 2961 – 2964 ( 1996 ).
[CrossRef] [PubMed]

U. Bonse and F. Bush , “ X-ray computed microtomography using synchrotron radiation ,” Prog. Biophys. Mol. Biol. ,   65 , 133 – 169 ( 1996 ).
[CrossRef] [PubMed]

1995 (1)

A. Momose , “ Demonstration of phase-contrast x-ray computed-tomography using an x-ray interferometer ,” Nuclear Instruments and Methods A   352 , 622 – 628 ( 1995 ).
[CrossRef]

1983 (1)

M. R. Teague , “ Deterministic phase retrieval - a Green-function solution ,” J. of the Opt. Soc. Am.   73 , 1434 – 1441 ( 1983 ).
[CrossRef]

1965 (1)

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

Abela, R.

A. Groso , M. Stampanoni , R. Abela , P. Schneider , S. Linga , and R. Müller , “ Phase contrast tomography: an alternative approach ,” Appl. Phys. Lett.   88 , 214104 ( 2006 ).
[CrossRef]

M. Stampanoni , G. Borchert , P. Wyss , R. Abela , B. Patterson , S. Hunt , D. Vermeulen , and P. Rüegsegger , “ High resolution X-ray detector for synchrotron-based microtomography ,” Nucl Instrum Methods A   491 , 291 – 301 ( 2002 ).
[CrossRef]

Allen, L. J.

L. J. Allen , W. McBride , N. L. O’Leary , and M. P. Oxley , “ Exit wave reconstruction at atomic resolution ,” Ultramicroscopy   100 , 91 – 104 ( 2004 ).
[CrossRef] [PubMed]

Axe, L.

B. A. Dowd , G. H. Campbell , R. B. Marr , V. V. Nagarkar , S. V. Tipnis , L. Axe , and D. P. Siddons , “ Developments in synchrotron x-ray computed microtomography at the National Synchrotron Light Source ,” in Developments in X-Ray Tomography II , U. Bonse , ed., Proc. SPIE   3772 , 224 – 236 ( 1999 ).
[CrossRef]

Barnea, Z.

K. A. Nugent , T. E. Gureyev , D. F. Cookson , D. Paganin , and Z. Barnea , “ Quantitative phase imaging using hard x rays ,” Phys. Rev. Lett.   77 , 2961 – 2964 ( 1996 ).
[CrossRef] [PubMed]

Bartscher, M.

U. Bonse , F. Beckmann , M. Bartscher , T. Biermann , F. Busch , and O. Günnewig , “ Phase contrast x-ray tomography using synchrotron radiation ” in Developments in X-Ray Tomography , U. Bonse , ed., Proc. SPIE   3149 , 108 – 119 ( 1997 ).
[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]

Beckmann, F.

U. Bonse and F. Beckmann , “ Multiple-beam X-ray interferometry for phase-contrast microtomography ,” Journal of Synchrotron Radiat.   8 , 1 – 5 ( 2001 ).
[CrossRef]

U. Bonse , F. Beckmann , M. Bartscher , T. Biermann , F. Busch , and O. Günnewig , “ Phase contrast x-ray tomography using synchrotron radiation ” in Developments in X-Ray Tomography , U. Bonse , ed., Proc. SPIE   3149 , 108 – 119 ( 1997 ).
[CrossRef]

Biermann, T.

U. Bonse , F. Beckmann , M. Bartscher , T. Biermann , F. Busch , and O. Günnewig , “ Phase contrast x-ray tomography using synchrotron radiation ” in Developments in X-Ray Tomography , U. Bonse , ed., Proc. SPIE   3149 , 108 – 119 ( 1997 ).
[CrossRef]

Bonse, U.

U. Bonse and F. Beckmann , “ Multiple-beam X-ray interferometry for phase-contrast microtomography ,” Journal of Synchrotron Radiat.   8 , 1 – 5 ( 2001 ).
[CrossRef]

U. Bonse , F. Beckmann , M. Bartscher , T. Biermann , F. Busch , and O. Günnewig , “ Phase contrast x-ray tomography using synchrotron radiation ” in Developments in X-Ray Tomography , U. Bonse , ed., Proc. SPIE   3149 , 108 – 119 ( 1997 ).
[CrossRef]

U. Bonse and F. Bush , “ X-ray computed microtomography using synchrotron radiation ,” Prog. Biophys. Mol. Biol. ,   65 , 133 – 169 ( 1996 ).
[CrossRef] [PubMed]

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

Borchert, G.

M. Stampanoni , G. Borchert , P. Wyss , R. Abela , B. Patterson , S. Hunt , D. Vermeulen , and P. Rüegsegger , “ High resolution X-ray detector for synchrotron-based microtomography ,” Nucl Instrum Methods A   491 , 291 – 301 ( 2002 ).
[CrossRef]

Bronnikov, A. V.

A. V. Bronnikov , “ Theory of quantitative phase-contrast computed tomography ,” J. of the Opt. Soc. of Am. A   19 , 472 – 480 ( 2002 ).
[CrossRef]

A. V. Bronnikov , “ Reconstruction formulas in phase-contrast tomography ,” Optics Communications   171 , 239 – 244 ( 1999 ).
[CrossRef]

Busch, F.

U. Bonse , F. Beckmann , M. Bartscher , T. Biermann , F. Busch , and O. Günnewig , “ Phase contrast x-ray tomography using synchrotron radiation ” in Developments in X-Ray Tomography , U. Bonse , ed., Proc. SPIE   3149 , 108 – 119 ( 1997 ).
[CrossRef]

Bush, F.

U. Bonse and F. Bush , “ X-ray computed microtomography using synchrotron radiation ,” Prog. Biophys. Mol. Biol. ,   65 , 133 – 169 ( 1996 ).
[CrossRef] [PubMed]

Campbell, G. H.

B. A. Dowd , G. H. Campbell , R. B. Marr , V. V. Nagarkar , S. V. Tipnis , L. Axe , and D. P. Siddons , “ Developments in synchrotron x-ray computed microtomography at the National Synchrotron Light Source ,” in Developments in X-Ray Tomography II , U. Bonse , ed., Proc. SPIE   3772 , 224 – 236 ( 1999 ).
[CrossRef]

Chen, F. R.

W. K. Hsieh , F. R. Chen , J. J. Kai , and A. I. Kirkland , “ Resolution extension and exit wave reconstruction in complex HREM ,” Ultramicroscopy   98 , 99 – 114 ( 2004 ).
[CrossRef] [PubMed]

Cloetens, P.

T. Weitkamp , A. Diaz , C. David , F. Pfeiffer , M. Stampanoni , P. Cloetens , and E. Ziegler , “ X-ray phase imaging with a grating interferometer ,” Opt. Express   13 , 6296 – 6304 ( 2005 ).
[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]

Cookson, D. F.

K. A. Nugent , T. E. Gureyev , D. F. Cookson , D. Paganin , and Z. Barnea , “ Quantitative phase imaging using hard x rays ,” Phys. Rev. Lett.   77 , 2961 – 2964 ( 1996 ).
[CrossRef] [PubMed]

Cowley, J. M.

J. M. Cowley , Diffraction Physics ( North-Holland , 1995 ), Chap. 3.4.

David, C.

Davis, T. J.

S. C. Mayo , P. R. Miller , S. W. Wilkins , T. J. Davis , D. Gao , T. E. Gureyev , D. Paganin , D. J. Parry , A. Pogany , and A. W. Stevenson , “ Quantitative X-ray projection microscopy: phase-contrast and multi-spectral imaging ,” J. Microsc.   207 , 79 – 96 ( 2002 ).
[CrossRef] [PubMed]

DeCarlo, F.

A. G. Peele , F. DeCarlo , P. J. Mahon , B. B. Dhal , and K. A. Nugent , “ X-ray phase contrast tomography with a bending magnet source ,” Rev. Sci. Instrum.   76 , 083707 ( 2005 ).
[CrossRef]

Dhal, B. B.

A. G. Peele , F. DeCarlo , P. J. Mahon , B. B. Dhal , and K. A. Nugent , “ X-ray phase contrast tomography with a bending magnet source ,” Rev. Sci. Instrum.   76 , 083707 ( 2005 ).
[CrossRef]

Diaz, A.

Dowd, B. A.

B. A. Dowd , G. H. Campbell , R. B. Marr , V. V. Nagarkar , S. V. Tipnis , L. Axe , and D. P. Siddons , “ Developments in synchrotron x-ray computed microtomography at the National Synchrotron Light Source ,” in Developments in X-Ray Tomography II , U. Bonse , ed., Proc. SPIE   3772 , 224 – 236 ( 1999 ).
[CrossRef]

Gao, D.

S. C. Mayo , P. R. Miller , S. W. Wilkins , T. J. Davis , D. Gao , T. E. Gureyev , D. Paganin , D. J. Parry , A. Pogany , and A. W. Stevenson , “ Quantitative X-ray projection microscopy: phase-contrast and multi-spectral imaging ,” J. Microsc.   207 , 79 – 96 ( 2002 ).
[CrossRef] [PubMed]

A. Pogany , D. Gao , and S. W. Wilkins , “ Contrast and resolution in imaging with a microfocus x-ray source ,” Rev. Sci. Instr.   68 , 2774 – 2782 ( 1997 ).
[CrossRef]

Groso, A.

A. Groso , M. Stampanoni , R. Abela , P. Schneider , S. Linga , and R. Müller , “ Phase contrast tomography: an alternative approach ,” Appl. Phys. Lett.   88 , 214104 ( 2006 ).
[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]

Günnewig, O.

U. Bonse , F. Beckmann , M. Bartscher , T. Biermann , F. Busch , and O. Günnewig , “ Phase contrast x-ray tomography using synchrotron radiation ” in Developments in X-Ray Tomography , U. Bonse , ed., Proc. SPIE   3149 , 108 – 119 ( 1997 ).
[CrossRef]

Gureyev, T. E.

T. E. Gureyev , A. Pogany , D. M. Paganin , and S. W. Wilkins , “ Linear algorithms for phase retrieval in the Fresnel region ,” Optics Commun.   231 , 53 – 70 ( 2004 ).
[CrossRef]

S. C. Mayo , P. R. Miller , S. W. Wilkins , T. J. Davis , D. Gao , T. E. Gureyev , D. Paganin , D. J. Parry , A. Pogany , and A. W. Stevenson , “ Quantitative X-ray projection microscopy: phase-contrast and multi-spectral imaging ,” J. Microsc.   207 , 79 – 96 ( 2002 ).
[CrossRef] [PubMed]

K. A. Nugent , T. E. Gureyev , D. F. Cookson , D. Paganin , and Z. Barnea , “ Quantitative phase imaging using hard x rays ,” Phys. Rev. Lett.   77 , 2961 – 2964 ( 1996 ).
[CrossRef] [PubMed]

T. E. Gureyev and K. A. Nugent , “ Phase retrieval with the transport-of-intensity equation II. Orthogonal series solution for nonuniform illumination ,” J. Opt. Soc. Am. A   13 , 1670 – 1682 ( 1996 ).
[CrossRef]

Hart, M.

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

Hsieh, W. K.

W. K. Hsieh , F. R. Chen , J. J. Kai , and A. I. Kirkland , “ Resolution extension and exit wave reconstruction in complex HREM ,” Ultramicroscopy   98 , 99 – 114 ( 2004 ).
[CrossRef] [PubMed]

Hunt, S.

M. Stampanoni , G. Borchert , P. Wyss , R. Abela , B. Patterson , S. Hunt , D. Vermeulen , and P. Rüegsegger , “ High resolution X-ray detector for synchrotron-based microtomography ,” Nucl Instrum Methods A   491 , 291 – 301 ( 2002 ).
[CrossRef]

Kai, J. J.

W. K. Hsieh , F. R. Chen , J. J. Kai , and A. I. Kirkland , “ Resolution extension and exit wave reconstruction in complex HREM ,” Ultramicroscopy   98 , 99 – 114 ( 2004 ).
[CrossRef] [PubMed]

Kirkland, A. I.

W. K. Hsieh , F. R. Chen , J. J. Kai , and A. I. Kirkland , “ Resolution extension and exit wave reconstruction in complex HREM ,” Ultramicroscopy   98 , 99 – 114 ( 2004 ).
[CrossRef] [PubMed]

Linga, S.

A. Groso , M. Stampanoni , R. Abela , P. Schneider , S. Linga , and R. Müller , “ Phase contrast tomography: an alternative approach ,” Appl. Phys. Lett.   88 , 214104 ( 2006 ).
[CrossRef]

Ludwig, W.

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]

Mahon, P. J.

A. G. Peele , F. DeCarlo , P. J. Mahon , B. B. Dhal , and K. A. Nugent , “ X-ray phase contrast tomography with a bending magnet source ,” Rev. Sci. Instrum.   76 , 083707 ( 2005 ).
[CrossRef]

Marr, R. B.

B. A. Dowd , G. H. Campbell , R. B. Marr , V. V. Nagarkar , S. V. Tipnis , L. Axe , and D. P. Siddons , “ Developments in synchrotron x-ray computed microtomography at the National Synchrotron Light Source ,” in Developments in X-Ray Tomography II , U. Bonse , ed., Proc. SPIE   3772 , 224 – 236 ( 1999 ).
[CrossRef]

Mayo, S. C.

S. C. Mayo , P. R. Miller , S. W. Wilkins , T. J. Davis , D. Gao , T. E. Gureyev , D. Paganin , D. J. Parry , A. Pogany , and A. W. Stevenson , “ Quantitative X-ray projection microscopy: phase-contrast and multi-spectral imaging ,” J. Microsc.   207 , 79 – 96 ( 2002 ).
[CrossRef] [PubMed]

McBride, W.

L. J. Allen , W. McBride , N. L. O’Leary , and M. P. Oxley , “ Exit wave reconstruction at atomic resolution ,” Ultramicroscopy   100 , 91 – 104 ( 2004 ).
[CrossRef] [PubMed]

Miller, P. R.

S. C. Mayo , P. R. Miller , S. W. Wilkins , T. J. Davis , D. Gao , T. E. Gureyev , D. Paganin , D. J. Parry , A. Pogany , and A. W. Stevenson , “ Quantitative X-ray projection microscopy: phase-contrast and multi-spectral imaging ,” J. Microsc.   207 , 79 – 96 ( 2002 ).
[CrossRef] [PubMed]

Momose, A.

A. Momose , “ Demonstration of phase-contrast x-ray computed-tomography using an x-ray interferometer ,” Nuclear Instruments and Methods A   352 , 622 – 628 ( 1995 ).
[CrossRef]

Müller, R.

A. Groso , M. Stampanoni , R. Abela , P. Schneider , S. Linga , and R. Müller , “ Phase contrast tomography: an alternative approach ,” Appl. Phys. Lett.   88 , 214104 ( 2006 ).
[CrossRef]

Nagarkar, V. V.

B. A. Dowd , G. H. Campbell , R. B. Marr , V. V. Nagarkar , S. V. Tipnis , L. Axe , and D. P. Siddons , “ Developments in synchrotron x-ray computed microtomography at the National Synchrotron Light Source ,” in Developments in X-Ray Tomography II , U. Bonse , ed., Proc. SPIE   3772 , 224 – 236 ( 1999 ).
[CrossRef]

Nugent, K. A.

A. G. Peele , F. DeCarlo , P. J. Mahon , B. B. Dhal , and K. A. Nugent , “ X-ray phase contrast tomography with a bending magnet source ,” Rev. Sci. Instrum.   76 , 083707 ( 2005 ).
[CrossRef]

K. A. Nugent , T. E. Gureyev , D. F. Cookson , D. Paganin , and Z. Barnea , “ Quantitative phase imaging using hard x rays ,” Phys. Rev. Lett.   77 , 2961 – 2964 ( 1996 ).
[CrossRef] [PubMed]

T. E. Gureyev and K. A. Nugent , “ Phase retrieval with the transport-of-intensity equation II. Orthogonal series solution for nonuniform illumination ,” J. Opt. Soc. Am. A   13 , 1670 – 1682 ( 1996 ).
[CrossRef]

O’Leary, N. L.

L. J. Allen , W. McBride , N. L. O’Leary , and M. P. Oxley , “ Exit wave reconstruction at atomic resolution ,” Ultramicroscopy   100 , 91 – 104 ( 2004 ).
[CrossRef] [PubMed]

Oxley, M. P.

L. J. Allen , W. McBride , N. L. O’Leary , and M. P. Oxley , “ Exit wave reconstruction at atomic resolution ,” Ultramicroscopy   100 , 91 – 104 ( 2004 ).
[CrossRef] [PubMed]

Paganin, D.

S. C. Mayo , P. R. Miller , S. W. Wilkins , T. J. Davis , D. Gao , T. E. Gureyev , D. Paganin , D. J. Parry , A. Pogany , and A. W. Stevenson , “ Quantitative X-ray projection microscopy: phase-contrast and multi-spectral imaging ,” J. Microsc.   207 , 79 – 96 ( 2002 ).
[CrossRef] [PubMed]

K. A. Nugent , T. E. Gureyev , D. F. Cookson , D. Paganin , and Z. Barnea , “ Quantitative phase imaging using hard x rays ,” Phys. Rev. Lett.   77 , 2961 – 2964 ( 1996 ).
[CrossRef] [PubMed]

Paganin, D. M.

T. E. Gureyev , A. Pogany , D. M. Paganin , and S. W. Wilkins , “ Linear algorithms for phase retrieval in the Fresnel region ,” Optics Commun.   231 , 53 – 70 ( 2004 ).
[CrossRef]

Parry, D. J.

S. C. Mayo , P. R. Miller , S. W. Wilkins , T. J. Davis , D. Gao , T. E. Gureyev , D. Paganin , D. J. Parry , A. Pogany , and A. W. Stevenson , “ Quantitative X-ray projection microscopy: phase-contrast and multi-spectral imaging ,” J. Microsc.   207 , 79 – 96 ( 2002 ).
[CrossRef] [PubMed]

Patterson, B.

M. Stampanoni , G. Borchert , P. Wyss , R. Abela , B. Patterson , S. Hunt , D. Vermeulen , and P. Rüegsegger , “ High resolution X-ray detector for synchrotron-based microtomography ,” Nucl Instrum Methods A   491 , 291 – 301 ( 2002 ).
[CrossRef]

Peele, A. G.

A. G. Peele , F. DeCarlo , P. J. Mahon , B. B. Dhal , and K. A. Nugent , “ X-ray phase contrast tomography with a bending magnet source ,” Rev. Sci. Instrum.   76 , 083707 ( 2005 ).
[CrossRef]

Pfeiffer, F.

Pogany, A.

T. E. Gureyev , A. Pogany , D. M. Paganin , and S. W. Wilkins , “ Linear algorithms for phase retrieval in the Fresnel region ,” Optics Commun.   231 , 53 – 70 ( 2004 ).
[CrossRef]

S. C. Mayo , P. R. Miller , S. W. Wilkins , T. J. Davis , D. Gao , T. E. Gureyev , D. Paganin , D. J. Parry , A. Pogany , and A. W. Stevenson , “ Quantitative X-ray projection microscopy: phase-contrast and multi-spectral imaging ,” J. Microsc.   207 , 79 – 96 ( 2002 ).
[CrossRef] [PubMed]

A. Pogany , D. Gao , and S. W. Wilkins , “ Contrast and resolution in imaging with a microfocus x-ray source ,” Rev. Sci. Instr.   68 , 2774 – 2782 ( 1997 ).
[CrossRef]

Raven, C.

T. Weitkamp , C. Raven , and A. Snigirev , “ An imaging and microtomography facility at the ESRF beamline ID 22 ,” in Developments in X-Ray Tomography II , U. Bonse , ed., Proc. SPIE   3772 , 311 – 317 ( 1999 ).
[CrossRef]

Rüegsegger, P.

M. Stampanoni , G. Borchert , P. Wyss , R. Abela , B. Patterson , S. Hunt , D. Vermeulen , and P. Rüegsegger , “ High resolution X-ray detector for synchrotron-based microtomography ,” Nucl Instrum Methods A   491 , 291 – 301 ( 2002 ).
[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, P.

A. Groso , M. Stampanoni , R. Abela , P. Schneider , S. Linga , and R. Müller , “ Phase contrast tomography: an alternative approach ,” Appl. Phys. Lett.   88 , 214104 ( 2006 ).
[CrossRef]

Siddons, D. P.

B. A. Dowd , G. H. Campbell , R. B. Marr , V. V. Nagarkar , S. V. Tipnis , L. Axe , and D. P. Siddons , “ Developments in synchrotron x-ray computed microtomography at the National Synchrotron Light Source ,” in Developments in X-Ray Tomography II , U. Bonse , ed., Proc. SPIE   3772 , 224 – 236 ( 1999 ).
[CrossRef]

Snigirev, A.

T. Weitkamp , C. Raven , and A. Snigirev , “ An imaging and microtomography facility at the ESRF beamline ID 22 ,” in Developments in X-Ray Tomography II , U. Bonse , ed., Proc. SPIE   3772 , 311 – 317 ( 1999 ).
[CrossRef]

Stampanoni, M.

A. Groso , M. Stampanoni , R. Abela , P. Schneider , S. Linga , and R. Müller , “ Phase contrast tomography: an alternative approach ,” Appl. Phys. Lett.   88 , 214104 ( 2006 ).
[CrossRef]

T. Weitkamp , A. Diaz , C. David , F. Pfeiffer , M. Stampanoni , P. Cloetens , and E. Ziegler , “ X-ray phase imaging with a grating interferometer ,” Opt. Express   13 , 6296 – 6304 ( 2005 ).
[CrossRef] [PubMed]

M. Stampanoni , G. Borchert , P. Wyss , R. Abela , B. Patterson , S. Hunt , D. Vermeulen , and P. Rüegsegger , “ High resolution X-ray detector for synchrotron-based microtomography ,” Nucl Instrum Methods A   491 , 291 – 301 ( 2002 ).
[CrossRef]

Stevenson, A. W.

S. C. Mayo , P. R. Miller , S. W. Wilkins , T. J. Davis , D. Gao , T. E. Gureyev , D. Paganin , D. J. Parry , A. Pogany , and A. W. Stevenson , “ Quantitative X-ray projection microscopy: phase-contrast and multi-spectral imaging ,” J. Microsc.   207 , 79 – 96 ( 2002 ).
[CrossRef] [PubMed]

Teague, M. R.

M. R. Teague , “ Deterministic phase retrieval - a Green-function solution ,” J. of the Opt. Soc. Am.   73 , 1434 – 1441 ( 1983 ).
[CrossRef]

Tipnis, S. V.

B. A. Dowd , G. H. Campbell , R. B. Marr , V. V. Nagarkar , S. V. Tipnis , L. Axe , and D. P. Siddons , “ Developments in synchrotron x-ray computed microtomography at the National Synchrotron Light Source ,” in Developments in X-Ray Tomography II , U. Bonse , ed., Proc. SPIE   3772 , 224 – 236 ( 1999 ).
[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 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]

Vermeulen, D.

M. Stampanoni , G. Borchert , P. Wyss , R. Abela , B. Patterson , S. Hunt , D. Vermeulen , and P. Rüegsegger , “ High resolution X-ray detector for synchrotron-based microtomography ,” Nucl Instrum Methods A   491 , 291 – 301 ( 2002 ).
[CrossRef]

Weitkamp, T.

T. Weitkamp , A. Diaz , C. David , F. Pfeiffer , M. Stampanoni , P. Cloetens , and E. Ziegler , “ X-ray phase imaging with a grating interferometer ,” Opt. Express   13 , 6296 – 6304 ( 2005 ).
[CrossRef] [PubMed]

T. Weitkamp , C. Raven , and A. Snigirev , “ An imaging and microtomography facility at the ESRF beamline ID 22 ,” in Developments in X-Ray Tomography II , U. Bonse , ed., Proc. SPIE   3772 , 311 – 317 ( 1999 ).
[CrossRef]

Wilkins, S. W.

T. E. Gureyev , A. Pogany , D. M. Paganin , and S. W. Wilkins , “ Linear algorithms for phase retrieval in the Fresnel region ,” Optics Commun.   231 , 53 – 70 ( 2004 ).
[CrossRef]

S. C. Mayo , P. R. Miller , S. W. Wilkins , T. J. Davis , D. Gao , T. E. Gureyev , D. Paganin , D. J. Parry , A. Pogany , and A. W. Stevenson , “ Quantitative X-ray projection microscopy: phase-contrast and multi-spectral imaging ,” J. Microsc.   207 , 79 – 96 ( 2002 ).
[CrossRef] [PubMed]

A. Pogany , D. Gao , and S. W. Wilkins , “ Contrast and resolution in imaging with a microfocus x-ray source ,” Rev. Sci. Instr.   68 , 2774 – 2782 ( 1997 ).
[CrossRef]

Wyss, P.

M. Stampanoni , G. Borchert , P. Wyss , R. Abela , B. Patterson , S. Hunt , D. Vermeulen , and P. Rüegsegger , “ High resolution X-ray detector for synchrotron-based microtomography ,” Nucl Instrum Methods A   491 , 291 – 301 ( 2002 ).
[CrossRef]

Ziegler, E.

Appl. Phys. Lett. (3)

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

A. Groso , M. Stampanoni , R. Abela , P. Schneider , S. Linga , and R. Müller , “ Phase contrast tomography: an alternative approach ,” Appl. Phys. Lett.   88 , 214104 ( 2006 ).
[CrossRef]

J. Microsc. (1)

S. C. Mayo , P. R. Miller , S. W. Wilkins , T. J. Davis , D. Gao , T. E. Gureyev , D. Paganin , D. J. Parry , A. Pogany , and A. W. Stevenson , “ Quantitative X-ray projection microscopy: phase-contrast and multi-spectral imaging ,” J. Microsc.   207 , 79 – 96 ( 2002 ).
[CrossRef] [PubMed]

J. of the Opt. Soc. Am. (1)

M. R. Teague , “ Deterministic phase retrieval - a Green-function solution ,” J. of the Opt. Soc. Am.   73 , 1434 – 1441 ( 1983 ).
[CrossRef]

J. of the Opt. Soc. of Am. A (1)

A. V. Bronnikov , “ Theory of quantitative phase-contrast computed tomography ,” J. of the Opt. Soc. of Am. A   19 , 472 – 480 ( 2002 ).
[CrossRef]

J. Opt. Soc. Am. A (1)

Journal of Synchrotron Radiat. (1)

U. Bonse and F. Beckmann , “ Multiple-beam X-ray interferometry for phase-contrast microtomography ,” Journal of Synchrotron Radiat.   8 , 1 – 5 ( 2001 ).
[CrossRef]

Nucl Instrum Methods A (1)

M. Stampanoni , G. Borchert , P. Wyss , R. Abela , B. Patterson , S. Hunt , D. Vermeulen , and P. Rüegsegger , “ High resolution X-ray detector for synchrotron-based microtomography ,” Nucl Instrum Methods A   491 , 291 – 301 ( 2002 ).
[CrossRef]

Nuclear Instruments and Methods A (1)

A. Momose , “ Demonstration of phase-contrast x-ray computed-tomography using an x-ray interferometer ,” Nuclear Instruments and Methods A   352 , 622 – 628 ( 1995 ).
[CrossRef]

Opt. Express (1)

Optics Commun. (1)

T. E. Gureyev , A. Pogany , D. M. Paganin , and S. W. Wilkins , “ Linear algorithms for phase retrieval in the Fresnel region ,” Optics Commun.   231 , 53 – 70 ( 2004 ).
[CrossRef]

Optics Communications (1)

A. V. Bronnikov , “ Reconstruction formulas in phase-contrast tomography ,” Optics Communications   171 , 239 – 244 ( 1999 ).
[CrossRef]

Phys. Rev. Lett. (1)

K. A. Nugent , T. E. Gureyev , D. F. Cookson , D. Paganin , and Z. Barnea , “ Quantitative phase imaging using hard x rays ,” Phys. Rev. Lett.   77 , 2961 – 2964 ( 1996 ).
[CrossRef] [PubMed]

Proc. SPIE (3)

U. Bonse , F. Beckmann , M. Bartscher , T. Biermann , F. Busch , and O. Günnewig , “ Phase contrast x-ray tomography using synchrotron radiation ” in Developments in X-Ray Tomography , U. Bonse , ed., Proc. SPIE   3149 , 108 – 119 ( 1997 ).
[CrossRef]

T. Weitkamp , C. Raven , and A. Snigirev , “ An imaging and microtomography facility at the ESRF beamline ID 22 ,” in Developments in X-Ray Tomography II , U. Bonse , ed., Proc. SPIE   3772 , 311 – 317 ( 1999 ).
[CrossRef]

B. A. Dowd , G. H. Campbell , R. B. Marr , V. V. Nagarkar , S. V. Tipnis , L. Axe , and D. P. Siddons , “ Developments in synchrotron x-ray computed microtomography at the National Synchrotron Light Source ,” in Developments in X-Ray Tomography II , U. Bonse , ed., Proc. SPIE   3772 , 224 – 236 ( 1999 ).
[CrossRef]

Prog. Biophys. Mol. Biol. (1)

U. Bonse and F. Bush , “ X-ray computed microtomography using synchrotron radiation ,” Prog. Biophys. Mol. Biol. ,   65 , 133 – 169 ( 1996 ).
[CrossRef] [PubMed]

Rev. Sci. Instr. (1)

A. Pogany , D. Gao , and S. W. Wilkins , “ Contrast and resolution in imaging with a microfocus x-ray source ,” Rev. Sci. Instr.   68 , 2774 – 2782 ( 1997 ).
[CrossRef]

Rev. Sci. Instrum. (1)

A. G. Peele , F. DeCarlo , P. J. Mahon , B. B. Dhal , and K. A. Nugent , “ X-ray phase contrast tomography with a bending magnet source ,” Rev. Sci. Instrum.   76 , 083707 ( 2005 ).
[CrossRef]

Ultramicroscopy (2)

L. J. Allen , W. McBride , N. L. O’Leary , and M. P. Oxley , “ Exit wave reconstruction at atomic resolution ,” Ultramicroscopy   100 , 91 – 104 ( 2004 ).
[CrossRef] [PubMed]

W. K. Hsieh , F. R. Chen , J. J. Kai , and A. I. Kirkland , “ Resolution extension and exit wave reconstruction in complex HREM ,” Ultramicroscopy   98 , 99 – 114 ( 2004 ).
[CrossRef] [PubMed]

Other (2)

The computation of the two-dimensional Radon transform and backprojection over the angle ω are combined in a single step of the algorithm [20].

J. M. Cowley , Diffraction Physics ( North-Holland , 1995 ), Chap. 3.4.

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Figures (4)

Fig. 1.
Fig. 1.

Tomographic reconstruction of a 350 microns thin wood sample: (a) absorption contrast obtained with 10 keV photon energy, (b) phase contrast, obtained with 25 keV. Effective pixel size is 0.7 microns. The length of the scale bar is 50 μm. Sample by courtesy of P.Trtik, ETHZ

Fig. 2.
Fig. 2.

Values for the correction factor αsimul as function of transmission found by simulations.

Fig. 3.
Fig. 3.

Phase tomography reconstruction (a) and the 3D rendering (b) of a 350 microns thin wood sample using modified filter given in the Eq. (8). The length of the scale bar is 50 μm.

Fig. 4.
Fig. 4.

Validation of the MBA method: (a) Phase tomographic reconstruction of sample consisting of polyacrylate, starch and cross-linked rubber matrix obtained using DPC and (b) using MBA. The length of the scale bar is 100 μm.

Equations (10)

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

I θ , z = d ( x , y ) = I θ , z = 0 ( x , y ) λd 2 π [ I θ , z = 0 ( x , y ) ϕ θ ( x , y ) ]
I θ , z = d ( x , y ) = I θ , z = 0 ( x , y ) [ 1 λd 2 π 2 ϕ θ ( x , y ) ]
2 s 2 δ ̂ s θ ω = 1 d g ̂ θ ( s , ω )
δ ( x 1 , x 2 , x 3 ) = 1 4 π 2 d 0 π ( q * * g θ )
q ( x , y ) y x 2 + y 2
q ( ξ , η ) = ξ ξ 2 + η 2
g θ ( x , y ) = I θ , z = d ( x , y ) 1 , θ [ 0 , π ]
q ( ξ , η ) = ξ ξ 2 + η 2 + α exp
SSE = ( δ reconstructed δ simulated ) 2 ( δ simulated ) 2
α exp = α simul pixel simul array simul pixel exp array exp

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