Abstract

Phase retrieval is required for achieving artifact-free x-ray phase-sensitive 3D imaging. A phase-retrieval approach based on the phase-attenuation duality with high energy x-rays can greatly facilitate for phase sensitive imaging by allowing phase retrieval from only one single projection image. The previously derived phase retrieval formula is valid only for small Fresnel propagator phases corresponding to common clinical imaging tasks. In this work we presented a new duality-based phase retrieval formula that can be applied for cases with large Fresnel propagator phases corresponding to high spatial resolution imaging. The computer simulation demonstrated superiority of this new formula over the previous phase retrieval formula in reconstructing the high frequency components of imaged objects. A modified Tikhonov regularization technique has been devised for phase retrieval in cases of very high resolution and large object-detector distance such that some Fresnel propagator phases may be close or greater than π. This new phase retrieval formula lays the foundation for implementing high-resolution phase-sensitive 3D imaging of soft tissue objects.

© 2009 Optical Society of America

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    [CrossRef]
  2. A. Snigirev, I. Snigireva, V. Kohn, S. Kuznetsov, and I. Shelokov, "On the possibilities of x-ray phase contrast microimaging by coherent high-energy synchrotron radiation," Rev. Sci. Instrum. 66, 5486 - 5492 (1995).
    [CrossRef]
  3. K. Nugent, T. Gureyev, D. Cookson, D. Paganin, and Z. Barnea, "Quantitative Phase Imaging Using Hard X Rays," Phy. Rev. Lett. 77, 2961 - 2965 (1996).
    [CrossRef]
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    [CrossRef]
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    [CrossRef] [PubMed]
  7. S. Mayo, T. Davis, T. Gureyev, P. Miller, D. Poganin, A. Pogany, A. Stevenson, and S. Wilkins, "X-ray phasecontrast microscopy and microtomography," Opt. Express 11, 2289 - 2302 (2003).
    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  24. A. Yan, X. Wu and H. Liu, "An attenuation-partition based iterative phase retrieval algorithm for in-line phasecontrast imaging" Opt. Express 16, 13330 - 13341 (2008)
    [CrossRef] [PubMed]

2008

D. Zhang, M. Donvan, L. Fajardo, A. Archer, X. Wu, and H. Liu, "Preliminary feasibility study of an in-line phase contrast x-ray imaging prototype," IEEE Trans. Biomed. Eng. 55, 2249 - 2257 (2008).
[CrossRef] [PubMed]

X. Wu, H. Liu, and A. Yan, "Phase-Contrast X-Ray Tomography: Contrast Mechanism and Roles of Phase Retrieval," Eur. J. Radiology 68, S8 - S12 (2008).
[CrossRef]

M. Langer, P. Cloetens, J.P. Guigay and F. Peyrin, "Quantitative comparison of direct phase retrieval in in-line phase tomography," Med. Physics 35, 4556 - 4566 (2008)
[CrossRef]

A. Yan, X. Wu and H. Liu, "An attenuation-partition based iterative phase retrieval algorithm for in-line phasecontrast imaging" Opt. Express 16, 13330 - 13341 (2008)
[CrossRef] [PubMed]

2007

2006

P. Cloetens, R. Mache, M. Schlenker, and S. Lerbs-Mache, "Quantitative phase tomography of Arabidopsis seeds reveals intercellular void network," PNAS 103, 14,626 - 14,630 (2006).
[CrossRef]

T. Gureyev, Y. Nesterets, D. Paganin, A. Pogany, and S. Wilkins, "Linear algorithms for phase retrieval in the Fresnel region. 2. Partially coherent illumination," Opt. Comm. 259, 569 - 580 (2006).
[CrossRef]

2005

2004

X. Wu and H. Liu, "A new theory of phase-contrast x-ray imaging based on Wigner distributions," Med. Phys. 31, 2378 - 2384 (2004).
[CrossRef] [PubMed]

2003

S. Mayo, T. Davis, T. Gureyev, P. Miller, D. Poganin, A. Pogany, A. Stevenson, and S. Wilkins, "X-ray phasecontrast microscopy and microtomography," Opt. Express 11, 2289 - 2302 (2003).
[CrossRef] [PubMed]

X. Wu and H. Liu, "Clinical implementation of phase-contrast x-ray imaging: Theoretical foundations and design considerations," Med. Phys. 30, 2169 - 2179 (2003).
[CrossRef] [PubMed]

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]

2000

F. Arfelli, V. Bonvicini,  and et al, "Mammography with synchrotron radiation: phase-detected Techniques," Radiology 215, 286 - 293 (2000).

1997

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

1996

K. Nugent, T. Gureyev, D. Cookson, D. Paganin, and Z. Barnea, "Quantitative Phase Imaging Using Hard X Rays," Phy. Rev. Lett. 77, 2961 - 2965 (1996).
[CrossRef]

S. Wilkins, T. Gureyev, D. Gao, A. Pogany, and A. Stevenson, "Phase-contrast imaging using polychromatic hard X-rays," Nature 384, 335 - 338 (1996).
[CrossRef]

1995

A. Snigirev, I. Snigireva, V. Kohn, S. Kuznetsov, and I. Shelokov, "On the possibilities of x-ray phase contrast microimaging by coherent high-energy synchrotron radiation," Rev. Sci. Instrum. 66, 5486 - 5492 (1995).
[CrossRef]

Archer, A.

D. Zhang, M. Donvan, L. Fajardo, A. Archer, X. Wu, and H. Liu, "Preliminary feasibility study of an in-line phase contrast x-ray imaging prototype," IEEE Trans. Biomed. Eng. 55, 2249 - 2257 (2008).
[CrossRef] [PubMed]

Arfelli, F.

F. Arfelli, V. Bonvicini,  and et al, "Mammography with synchrotron radiation: phase-detected Techniques," Radiology 215, 286 - 293 (2000).

Barnea, Z.

K. Nugent, T. Gureyev, D. Cookson, D. Paganin, and Z. Barnea, "Quantitative Phase Imaging Using Hard X Rays," Phy. Rev. Lett. 77, 2961 - 2965 (1996).
[CrossRef]

Boistel, R.

Bonvicini, V.

F. Arfelli, V. Bonvicini,  and et al, "Mammography with synchrotron radiation: phase-detected Techniques," Radiology 215, 286 - 293 (2000).

Cloetens, P.

M. Langer, P. Cloetens, J.P. Guigay and F. Peyrin, "Quantitative comparison of direct phase retrieval in in-line phase tomography," Med. Physics 35, 4556 - 4566 (2008)
[CrossRef]

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

P. Cloetens, R. Mache, M. Schlenker, and S. Lerbs-Mache, "Quantitative phase tomography of Arabidopsis seeds reveals intercellular void network," PNAS 103, 14,626 - 14,630 (2006).
[CrossRef]

Cookson, D.

K. Nugent, T. Gureyev, D. Cookson, D. Paganin, and Z. Barnea, "Quantitative Phase Imaging Using Hard X Rays," Phy. Rev. Lett. 77, 2961 - 2965 (1996).
[CrossRef]

Davis, T.

Donnelly, E.

E. Donnelly, R. Price, and D. Pickens, "Experimental validation of the Wigner distributions theory of phasecontrast imaging," Med. Phys. 32, 928 - 931 (2005).
[CrossRef] [PubMed]

Donvan, M.

D. Zhang, M. Donvan, L. Fajardo, A. Archer, X. Wu, and H. Liu, "Preliminary feasibility study of an in-line phase contrast x-ray imaging prototype," IEEE Trans. Biomed. Eng. 55, 2249 - 2257 (2008).
[CrossRef] [PubMed]

Fajardo, L.

D. Zhang, M. Donvan, L. Fajardo, A. Archer, X. Wu, and H. Liu, "Preliminary feasibility study of an in-line phase contrast x-ray imaging prototype," IEEE Trans. Biomed. Eng. 55, 2249 - 2257 (2008).
[CrossRef] [PubMed]

Gao, D.

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

S. Wilkins, T. Gureyev, D. Gao, A. Pogany, and A. Stevenson, "Phase-contrast imaging using polychromatic hard X-rays," Nature 384, 335 - 338 (1996).
[CrossRef]

Guigay, J.

Guigay, J.P.

M. Langer, P. Cloetens, J.P. Guigay and F. Peyrin, "Quantitative comparison of direct phase retrieval in in-line phase tomography," Med. Physics 35, 4556 - 4566 (2008)
[CrossRef]

Gureyev, T.

T. Gureyev, Y. Nesterets, D. Paganin, A. Pogany, and S. Wilkins, "Linear algorithms for phase retrieval in the Fresnel region. 2. Partially coherent illumination," Opt. Comm. 259, 569 - 580 (2006).
[CrossRef]

S. Mayo, T. Davis, T. Gureyev, P. Miller, D. Poganin, A. Pogany, A. Stevenson, and S. Wilkins, "X-ray phasecontrast microscopy and microtomography," Opt. Express 11, 2289 - 2302 (2003).
[CrossRef] [PubMed]

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

S. Wilkins, T. Gureyev, D. Gao, A. Pogany, and A. Stevenson, "Phase-contrast imaging using polychromatic hard X-rays," Nature 384, 335 - 338 (1996).
[CrossRef]

K. Nugent, T. Gureyev, D. Cookson, D. Paganin, and Z. Barnea, "Quantitative Phase Imaging Using Hard X Rays," Phy. Rev. Lett. 77, 2961 - 2965 (1996).
[CrossRef]

Kohn, V.

A. Snigirev, I. Snigireva, V. Kohn, S. Kuznetsov, and I. Shelokov, "On the possibilities of x-ray phase contrast microimaging by coherent high-energy synchrotron radiation," Rev. Sci. Instrum. 66, 5486 - 5492 (1995).
[CrossRef]

Kuznetsov, S.

A. Snigirev, I. Snigireva, V. Kohn, S. Kuznetsov, and I. Shelokov, "On the possibilities of x-ray phase contrast microimaging by coherent high-energy synchrotron radiation," Rev. Sci. Instrum. 66, 5486 - 5492 (1995).
[CrossRef]

Langer, M.

M. Langer, P. Cloetens, J.P. Guigay and F. Peyrin, "Quantitative comparison of direct phase retrieval in in-line phase tomography," Med. Physics 35, 4556 - 4566 (2008)
[CrossRef]

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

Lerbs-Mache, S.

P. Cloetens, R. Mache, M. Schlenker, and S. Lerbs-Mache, "Quantitative phase tomography of Arabidopsis seeds reveals intercellular void network," PNAS 103, 14,626 - 14,630 (2006).
[CrossRef]

Liu, H.

X. Wu, H. Liu, and A. Yan, "Phase-Contrast X-Ray Tomography: Contrast Mechanism and Roles of Phase Retrieval," Eur. J. Radiology 68, S8 - S12 (2008).
[CrossRef]

D. Zhang, M. Donvan, L. Fajardo, A. Archer, X. Wu, and H. Liu, "Preliminary feasibility study of an in-line phase contrast x-ray imaging prototype," IEEE Trans. Biomed. Eng. 55, 2249 - 2257 (2008).
[CrossRef] [PubMed]

A. Yan, X. Wu and H. Liu, "An attenuation-partition based iterative phase retrieval algorithm for in-line phasecontrast imaging" Opt. Express 16, 13330 - 13341 (2008)
[CrossRef] [PubMed]

F. Meng, H. Liu and X. Wu, "An iterative phase retrieval algorithm for in-line phase imaging," Opt. Express 15, 8383 - 8390 (2007)
[CrossRef] [PubMed]

X. Wu and H. Liu, "X-Ray cone-beam phase tomography formulas based on phase-attenuation duality," Opt. Express 13, 6000 - 6014 (2005).
[CrossRef] [PubMed]

X. Wu, H. Liu, and A. Yan, "X-ray phase-attenuation duality and phase retrieval," Opt. Lett. 30(4), 379 - 381 (2005).
[CrossRef]

X. Wu and H. Liu, "A new theory of phase-contrast x-ray imaging based on Wigner distributions," Med. Phys. 31, 2378 - 2384 (2004).
[CrossRef] [PubMed]

X. Wu and H. Liu, "Clinical implementation of phase-contrast x-ray imaging: Theoretical foundations and design considerations," Med. Phys. 30, 2169 - 2179 (2003).
[CrossRef] [PubMed]

X. Wu and H. Liu, "A general theoretical formalism for X-ray phase contrast imaging," J. X-ray Sci. and Tech. 11, 33 - 42 (2003).

Mache, R.

P. Cloetens, R. Mache, M. Schlenker, and S. Lerbs-Mache, "Quantitative phase tomography of Arabidopsis seeds reveals intercellular void network," PNAS 103, 14,626 - 14,630 (2006).
[CrossRef]

Mayo, S.

S. Mayo, T. Davis, T. Gureyev, P. Miller, D. Poganin, A. Pogany, A. Stevenson, and S. Wilkins, "X-ray phasecontrast microscopy and microtomography," Opt. Express 11, 2289 - 2302 (2003).
[CrossRef] [PubMed]

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

Meng, F.

Miller, P.

S. Mayo, T. Davis, T. Gureyev, P. Miller, D. Poganin, A. Pogany, A. Stevenson, and S. Wilkins, "X-ray phasecontrast microscopy and microtomography," Opt. Express 11, 2289 - 2302 (2003).
[CrossRef] [PubMed]

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

Nesterets, Y.

T. Gureyev, Y. Nesterets, D. Paganin, A. Pogany, and S. Wilkins, "Linear algorithms for phase retrieval in the Fresnel region. 2. Partially coherent illumination," Opt. Comm. 259, 569 - 580 (2006).
[CrossRef]

Nugent, K.

K. Nugent, T. Gureyev, D. Cookson, D. Paganin, and Z. Barnea, "Quantitative Phase Imaging Using Hard X Rays," Phy. Rev. Lett. 77, 2961 - 2965 (1996).
[CrossRef]

Paganin, D.

T. Gureyev, Y. Nesterets, D. Paganin, A. Pogany, and S. Wilkins, "Linear algorithms for phase retrieval in the Fresnel region. 2. Partially coherent illumination," Opt. Comm. 259, 569 - 580 (2006).
[CrossRef]

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

K. Nugent, T. Gureyev, D. Cookson, D. Paganin, and Z. Barnea, "Quantitative Phase Imaging Using Hard X Rays," Phy. Rev. Lett. 77, 2961 - 2965 (1996).
[CrossRef]

Peyrin, F.

M. Langer, P. Cloetens, J.P. Guigay and F. Peyrin, "Quantitative comparison of direct phase retrieval in in-line phase tomography," Med. Physics 35, 4556 - 4566 (2008)
[CrossRef]

Pickens, D.

E. Donnelly, R. Price, and D. Pickens, "Experimental validation of the Wigner distributions theory of phasecontrast imaging," Med. Phys. 32, 928 - 931 (2005).
[CrossRef] [PubMed]

Poganin, D.

Pogany, A.

T. Gureyev, Y. Nesterets, D. Paganin, A. Pogany, and S. Wilkins, "Linear algorithms for phase retrieval in the Fresnel region. 2. Partially coherent illumination," Opt. Comm. 259, 569 - 580 (2006).
[CrossRef]

S. Mayo, T. Davis, T. Gureyev, P. Miller, D. Poganin, A. Pogany, A. Stevenson, and S. Wilkins, "X-ray phasecontrast microscopy and microtomography," Opt. Express 11, 2289 - 2302 (2003).
[CrossRef] [PubMed]

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

S. Wilkins, T. Gureyev, D. Gao, A. Pogany, and A. Stevenson, "Phase-contrast imaging using polychromatic hard X-rays," Nature 384, 335 - 338 (1996).
[CrossRef]

Price, R.

E. Donnelly, R. Price, and D. Pickens, "Experimental validation of the Wigner distributions theory of phasecontrast imaging," Med. Phys. 32, 928 - 931 (2005).
[CrossRef] [PubMed]

Schlenker, M.

P. Cloetens, R. Mache, M. Schlenker, and S. Lerbs-Mache, "Quantitative phase tomography of Arabidopsis seeds reveals intercellular void network," PNAS 103, 14,626 - 14,630 (2006).
[CrossRef]

Shelokov, I.

A. Snigirev, I. Snigireva, V. Kohn, S. Kuznetsov, and I. Shelokov, "On the possibilities of x-ray phase contrast microimaging by coherent high-energy synchrotron radiation," Rev. Sci. Instrum. 66, 5486 - 5492 (1995).
[CrossRef]

Snigirev, A.

A. Snigirev, I. Snigireva, V. Kohn, S. Kuznetsov, and I. Shelokov, "On the possibilities of x-ray phase contrast microimaging by coherent high-energy synchrotron radiation," Rev. Sci. Instrum. 66, 5486 - 5492 (1995).
[CrossRef]

Snigireva, I.

A. Snigirev, I. Snigireva, V. Kohn, S. Kuznetsov, and I. Shelokov, "On the possibilities of x-ray phase contrast microimaging by coherent high-energy synchrotron radiation," Rev. Sci. Instrum. 66, 5486 - 5492 (1995).
[CrossRef]

Stevenson, A.

S. Mayo, T. Davis, T. Gureyev, P. Miller, D. Poganin, A. Pogany, A. Stevenson, and S. Wilkins, "X-ray phasecontrast microscopy and microtomography," Opt. Express 11, 2289 - 2302 (2003).
[CrossRef] [PubMed]

S. Wilkins, T. Gureyev, D. Gao, A. Pogany, and A. Stevenson, "Phase-contrast imaging using polychromatic hard X-rays," Nature 384, 335 - 338 (1996).
[CrossRef]

Wilkins, S.

T. Gureyev, Y. Nesterets, D. Paganin, A. Pogany, and S. Wilkins, "Linear algorithms for phase retrieval in the Fresnel region. 2. Partially coherent illumination," Opt. Comm. 259, 569 - 580 (2006).
[CrossRef]

S. Mayo, T. Davis, T. Gureyev, P. Miller, D. Poganin, A. Pogany, A. Stevenson, and S. Wilkins, "X-ray phasecontrast microscopy and microtomography," Opt. Express 11, 2289 - 2302 (2003).
[CrossRef] [PubMed]

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

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

S. Wilkins, T. Gureyev, D. Gao, A. Pogany, and A. Stevenson, "Phase-contrast imaging using polychromatic hard X-rays," Nature 384, 335 - 338 (1996).
[CrossRef]

Wu, X.

X. Wu, H. Liu, and A. Yan, "Phase-Contrast X-Ray Tomography: Contrast Mechanism and Roles of Phase Retrieval," Eur. J. Radiology 68, S8 - S12 (2008).
[CrossRef]

D. Zhang, M. Donvan, L. Fajardo, A. Archer, X. Wu, and H. Liu, "Preliminary feasibility study of an in-line phase contrast x-ray imaging prototype," IEEE Trans. Biomed. Eng. 55, 2249 - 2257 (2008).
[CrossRef] [PubMed]

A. Yan, X. Wu and H. Liu, "An attenuation-partition based iterative phase retrieval algorithm for in-line phasecontrast imaging" Opt. Express 16, 13330 - 13341 (2008)
[CrossRef] [PubMed]

F. Meng, H. Liu and X. Wu, "An iterative phase retrieval algorithm for in-line phase imaging," Opt. Express 15, 8383 - 8390 (2007)
[CrossRef] [PubMed]

X. Wu, H. Liu, and A. Yan, "X-ray phase-attenuation duality and phase retrieval," Opt. Lett. 30(4), 379 - 381 (2005).
[CrossRef]

X. Wu and H. Liu, "X-Ray cone-beam phase tomography formulas based on phase-attenuation duality," Opt. Express 13, 6000 - 6014 (2005).
[CrossRef] [PubMed]

X. Wu and H. Liu, "A new theory of phase-contrast x-ray imaging based on Wigner distributions," Med. Phys. 31, 2378 - 2384 (2004).
[CrossRef] [PubMed]

X. Wu and H. Liu, "Clinical implementation of phase-contrast x-ray imaging: Theoretical foundations and design considerations," Med. Phys. 30, 2169 - 2179 (2003).
[CrossRef] [PubMed]

X. Wu and H. Liu, "A general theoretical formalism for X-ray phase contrast imaging," J. X-ray Sci. and Tech. 11, 33 - 42 (2003).

Yan, A.

Zhang, D.

D. Zhang, M. Donvan, L. Fajardo, A. Archer, X. Wu, and H. Liu, "Preliminary feasibility study of an in-line phase contrast x-ray imaging prototype," IEEE Trans. Biomed. Eng. 55, 2249 - 2257 (2008).
[CrossRef] [PubMed]

Eur. J. Radiology

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[CrossRef]

IEEE Trans. Biomed. Eng.

D. Zhang, M. Donvan, L. Fajardo, A. Archer, X. Wu, and H. Liu, "Preliminary feasibility study of an in-line phase contrast x-ray imaging prototype," IEEE Trans. Biomed. Eng. 55, 2249 - 2257 (2008).
[CrossRef] [PubMed]

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[CrossRef] [PubMed]

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[CrossRef] [PubMed]

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[CrossRef] [PubMed]

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[CrossRef] [PubMed]

Med. Physics

M. Langer, P. Cloetens, J.P. Guigay and F. Peyrin, "Quantitative comparison of direct phase retrieval in in-line phase tomography," Med. Physics 35, 4556 - 4566 (2008)
[CrossRef]

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[CrossRef]

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[CrossRef]

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[CrossRef]

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Figures (4)

Fig. 1.
Fig. 1.

A shematic diagram for the measuring system

Fig. 2.
Fig. 2.

Plots of filters (13) and (12)

Fig. 3.
Fig. 3.

Comparation of the quality of the retrieved results A 2 0 using Eq. (11) and Eq. (10). A 2 0=exp[ϕ/γ], where phase shift ϕ=0.5sin(32px)-1, pixel pitch p=1µm, and object to detector distance R 2=49.55m. (a) is the true A 2 0 ; (b) is the phase-contrast intensity image I acquired from Fresnel diffraction; (c) is retrieved A 2 0 using Eq. (11); (d) is retrieved A 2 0 using Eq. (10).

Fig. 4.
Fig. 4.

Fig. 4. Comparation of the retrieved results using Eq. (11) and Eq. (10) with R2=5m. (a) is the true A20; (b) is the phase-contrast image I acquired from Fresnel diffraction; (c) is retrieved using Eq. (11); (d) is retrieved using Eq. (10).

Equations (18)

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ϕ(r)=(hcE)reρe(r,z)dz=(hcE)reρe,p(r),
μrecon(r)=μ (r)R2.reλ22π.(2x2+2y2+2z2)ρe+μmixed(r),
ϕ(r)=λreσKNln{(1(λ2reR2(2πMσKN))2)1(M2I(Mr;R1+R2)Iin)},
𝓖̂(I(rD))=IinM2{cos(πλR2u2M)(𝓖̂(A02)iλR2Mu.𝓖̂(ϕA02))+
+2 sin (πλR2u2M)(𝓖̂(A02ϕ)+iλR24Mu.𝓖̂(A02))} ,
I(r;R1+R2)=IinM2{A02(rM)λR22πM.(A02ϕ(rM))}.
ϕ(r)=λreρe,p(r),A02(r)=exp(σKNρe,p(r)),
σKN=2πre2{1+ηη2[(2(1+η)1+2η)1ηln(1+2η)]+12ηln(1+2η)1+3η(1+2η)2},
𝓕(I(rD))=IinM2{cos(πλR2u2M)[𝓕̂(A02)λR22πM𝓕(.(A02ϕ))]+
+2 [sin(πλR2u2M)πλR2u2Mcos(πλR2u2M)] 𝓕̂ (A02ϕ)
λR24πMsin(πλR2u2M)𝓕̂(2A02) } .
𝓕̂(.(A02ϕ))=(4π2u2)(λreσKN).𝓕̂(A02),
𝓕̂(A02ϕ)=λre𝓕̂(A02σKN)λreσKN𝓕̂(A011).
𝓕̂(M2I(rD)Iin)[cos(πλR2u2M)+(2λreσKN+πλR2u2M)sin(πλR2u2M)].𝓖̂(A02).
𝓕̂(M2I(rD)Iin)=(1+2(λreσKN)(πλR2u2M)).𝓕̂(A02),
1cos(α)+(2γ+α)sin(α)=1δ.cos(α1),
11+2γα .
cos(α1)cos(α2)2+κ2 ,

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