Abstract

For medical applications of the in-line phase-contrast x-ray imaging, phase retrieval is a crucial step for quantitative imaging such as reconstructing the 3-D distribution of tissue linear attenuation coefficients and refraction indices. The conventional phase retrieval algorithms, such as the transport of intensity equation (TIE) based algorithms, are not robust against the quantum noise that appears in acquired images due to the radiation dose constraints in medical imaging. In this work a new attenuation-partition based iterative phase retrieval algorithm is proposed. This new algorithm takes advantage of the correlation between the attenuation and phase-shift, and is much robust against noise in acquired images. Phase retrieval results from experimental images show that this new iterative algorithm is fast and robust, and it has good potential for medical x-ray imaging applications.

© 2008 Optical Society of America

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  1. 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]
  2. 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]
  3. A. Pogany, D. Gao, and S. Wilkins, "Contrast and resolution in imaging with a microfocus x-ray source," Rev. Sci. Instrum. 68, 2774 (1997).
    [CrossRef]
  4. D. Paganin and K. Nugent, "Noninterferometric Phase Imaging with Partially Coherent Light," Phy. Rev. Lett. 80, 2586 - 2589 (1998).
    [CrossRef]
  5. X. Wu and H. Liu, "A general theoretical formalism for X-ray phase contrast imaging," J. X-ray Sci. Technol. 11, 33 - 42 (2003).
  6. X.  Wu and H. Liu, "Clinical implementation of x-ray phase-contrast imaging: Theoretical foundations and design considerations," Med. Phys. 30, 2169 - 2179 (2003).
    [CrossRef] [PubMed]
  7. E. Donnelly, R. Price, and D. Pickens, "Dual focal-spot imaging for phase extraction in phase-contrast radiography," Med. Phys. 30, 2292 - 2296 (2003).
    [CrossRef] [PubMed]
  8. X. Wu and H. Liu, "A dual detector approach for X-ray attenuation and phase imaging," J. X-ray Sci. Technol. 12, 35-42 (2004).
  9. X. Wu and H. Liu, "An experimental method of determining relative phase-contrast factor for x-ray imaging systems," Med. Phys. 31, 997 - 1002 (2004).
    [CrossRef]
  10. 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]
  11. X. Wu and H. Liu, "A reconstruction formula for soft tissue X-ray phase tomography," J. X-ray Sci. Technol. 12, 273 - 279 (2004).
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    [CrossRef] [PubMed]
  13. X. Wu, H. Liu, and A. Yan, "Phase-Contrast X-Ray Tomography: Contrast Mechanism and Roles of Phase Retrieval," Eur. J. Radiol. (to be published) (2008).
    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef] [PubMed]
  16. T. Gureyev, D. Paganin, G. Myers, Y. Nesterets, and S. Wilkins, "Phase-and-amplitude computer tomography," Appl. Phys. Lett. 89, 034,102 (2006).
    [CrossRef]
  17. R. M. P. Cloetens, M. Schlenker, and S. Lerbs-Mache, "Quantitative phase tomography of Arabidopsis seeds reveals intercellular void network," PNAS 103, 14,626 - 14,630 (2006).
    [CrossRef]
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    [CrossRef]
  19. L. Allen and M. Oxley, "Phase retrieval from series of images obtainedbe defocus variation," Opt. Commun. 199, 65 - 75 (2001).
    [CrossRef]
  20. X. Wu and H. Liu, "Phase-space evolution of x-ray coherence in phase-sensitive imaging," Appl. Opt. 47, E44 - E52 (2008).
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    [CrossRef]
  25. T. Gureyev, A. Pogany, D. Paganin, and S. Wilkins, "Linear algorithms for phase retrieval in the Fresnel region," Opt. Commun. 231, 53 - 70 (2004).
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2008

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

X. Wu, H. Liu, and A. Yan, "Phase-Contrast X-Ray Tomography: Contrast Mechanism and Roles of Phase Retrieval," Eur. J. Radiol. (to be published) (2008).
[CrossRef] [PubMed]

X. Wu and H. Liu, "Phase-space evolution of x-ray coherence in phase-sensitive imaging," Appl. Opt. 47, E44 - E52 (2008).
[CrossRef] [PubMed]

2007

X. Wu and H. Liu, "Robustness of a phase-retrieval approach based on phase-attenuation duality," J. X-Ray Sci. Technol. 15, 85 - 95 (2007).

2006

T. Gureyev, D. Paganin, G. Myers, Y. Nesterets, and S. Wilkins, "Phase-and-amplitude computer tomography," Appl. Phys. Lett. 89, 034,102 (2006).
[CrossRef]

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

2005

2004

T. Gureyev, A. Pogany, D. Paganin, and S. Wilkins, "Linear algorithms for phase retrieval in the Fresnel region," Opt. Commun. 231, 53 - 70 (2004).
[CrossRef]

X. Wu and H. Liu, "A dual detector approach for X-ray attenuation and phase imaging," J. X-ray Sci. Technol. 12, 35-42 (2004).

X. Wu and H. Liu, "An experimental method of determining relative phase-contrast factor for x-ray imaging systems," Med. Phys. 31, 997 - 1002 (2004).
[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, "A reconstruction formula for soft tissue X-ray phase tomography," J. X-ray Sci. Technol. 12, 273 - 279 (2004).

2003

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

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

E. Donnelly, R. Price, and D. Pickens, "Dual focal-spot imaging for phase extraction in phase-contrast radiography," Med. Phys. 30, 2292 - 2296 (2003).
[CrossRef] [PubMed]

2002

I. Schelokov, T. Weitkamp, and A. Snigirev, "Reconstruction of an object phase transmission function from inline X-ray holograms," Opt. Commun. 213, 247 - 258 (2002).
[CrossRef]

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

2001

L. Allen and M. Oxley, "Phase retrieval from series of images obtainedbe defocus variation," Opt. Commun. 199, 65 - 75 (2001).
[CrossRef]

1998

D. Paganin and K. Nugent, "Noninterferometric Phase Imaging with Partially Coherent Light," Phy. Rev. Lett. 80, 2586 - 2589 (1998).
[CrossRef]

1997

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

1996

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 - 2964 (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]

1983

Allen, L.

L. Allen and M. Oxley, "Phase retrieval from series of images obtainedbe defocus variation," Opt. Commun. 199, 65 - 75 (2001).
[CrossRef]

Archet, A.

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

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 - 2964 (1996).
[CrossRef]

Bronnikov, A.

Cloetens, R. M. P.

R. M. P. Cloetens, 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 - 2964 (1996).
[CrossRef]

Donnelly, E.

E. Donnelly, R. Price, and D. Pickens, "Dual focal-spot imaging for phase extraction in phase-contrast radiography," Med. Phys. 30, 2292 - 2296 (2003).
[CrossRef] [PubMed]

Donvan, M.

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

Fajardo, L.

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

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 (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]

Gureyev, T.

T. Gureyev, D. Paganin, G. Myers, Y. Nesterets, and S. Wilkins, "Phase-and-amplitude computer tomography," Appl. Phys. Lett. 89, 034,102 (2006).
[CrossRef]

T. Gureyev, A. Pogany, D. Paganin, and S. Wilkins, "Linear algorithms for phase retrieval in the Fresnel region," Opt. Commun. 231, 53 - 70 (2004).
[CrossRef]

K. Nugent, T. Gureyev, D. Cookson, D. Paganin, and Z. Barnea, "Quantitative Phase Imaging Using Hard X Rays," Phy. Rev. Lett. 77, 2961 - 2964 (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]

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]

Lerbs-Mache, S.

R. M. P. Cloetens, 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. Radiol. (to be published) (2008).
[CrossRef] [PubMed]

X. Wu and H. Liu, "Phase-space evolution of x-ray coherence in phase-sensitive imaging," Appl. Opt. 47, E44 - E52 (2008).
[CrossRef] [PubMed]

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

X. Wu and H. Liu, "Robustness of a phase-retrieval approach based on phase-attenuation duality," J. X-Ray Sci. Technol. 15, 85 - 95 (2007).

X. Wu, H. Liu, and A. Yan, "X-ray phase-attenuation duality and phase retrieval," Opt. Lett. 30(4), 379 - 381 (2005).
[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 and H. Liu, "A dual detector approach for X-ray attenuation and phase imaging," J. X-ray Sci. Technol. 12, 35-42 (2004).

X. Wu and H. Liu, "An experimental method of determining relative phase-contrast factor for x-ray imaging systems," Med. Phys. 31, 997 - 1002 (2004).
[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, "A reconstruction formula for soft tissue X-ray phase tomography," J. X-ray Sci. Technol. 12, 273 - 279 (2004).

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

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

Myers, G.

T. Gureyev, D. Paganin, G. Myers, Y. Nesterets, and S. Wilkins, "Phase-and-amplitude computer tomography," Appl. Phys. Lett. 89, 034,102 (2006).
[CrossRef]

Nesterets, Y.

T. Gureyev, D. Paganin, G. Myers, Y. Nesterets, and S. Wilkins, "Phase-and-amplitude computer tomography," Appl. Phys. Lett. 89, 034,102 (2006).
[CrossRef]

Nugent, K.

D. Paganin and K. Nugent, "Noninterferometric Phase Imaging with Partially Coherent Light," Phy. Rev. Lett. 80, 2586 - 2589 (1998).
[CrossRef]

K. Nugent, T. Gureyev, D. Cookson, D. Paganin, and Z. Barnea, "Quantitative Phase Imaging Using Hard X Rays," Phy. Rev. Lett. 77, 2961 - 2964 (1996).
[CrossRef]

Oxley, M.

L. Allen and M. Oxley, "Phase retrieval from series of images obtainedbe defocus variation," Opt. Commun. 199, 65 - 75 (2001).
[CrossRef]

Paganin, D.

T. Gureyev, D. Paganin, G. Myers, Y. Nesterets, and S. Wilkins, "Phase-and-amplitude computer tomography," Appl. Phys. Lett. 89, 034,102 (2006).
[CrossRef]

T. Gureyev, A. Pogany, D. Paganin, and S. Wilkins, "Linear algorithms for phase retrieval in the Fresnel region," Opt. Commun. 231, 53 - 70 (2004).
[CrossRef]

D. Paganin and K. Nugent, "Noninterferometric Phase Imaging with Partially Coherent Light," Phy. Rev. Lett. 80, 2586 - 2589 (1998).
[CrossRef]

K. Nugent, T. Gureyev, D. Cookson, D. Paganin, and Z. Barnea, "Quantitative Phase Imaging Using Hard X Rays," Phy. Rev. Lett. 77, 2961 - 2964 (1996).
[CrossRef]

Pickens, D.

E. Donnelly, R. Price, and D. Pickens, "Dual focal-spot imaging for phase extraction in phase-contrast radiography," Med. Phys. 30, 2292 - 2296 (2003).
[CrossRef] [PubMed]

Pogany, A.

T. Gureyev, A. Pogany, D. Paganin, and S. Wilkins, "Linear algorithms for phase retrieval in the Fresnel region," Opt. Commun. 231, 53 - 70 (2004).
[CrossRef]

A. Pogany, D. Gao, and S. Wilkins, "Contrast and resolution in imaging with a microfocus x-ray source," Rev. Sci. Instrum. 68, 2774 (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, "Dual focal-spot imaging for phase extraction in phase-contrast radiography," Med. Phys. 30, 2292 - 2296 (2003).
[CrossRef] [PubMed]

Schelokov, I.

I. Schelokov, T. Weitkamp, and A. Snigirev, "Reconstruction of an object phase transmission function from inline X-ray holograms," Opt. Commun. 213, 247 - 258 (2002).
[CrossRef]

Schlenker, M.

R. M. P. Cloetens, 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.

I. Schelokov, T. Weitkamp, and A. Snigirev, "Reconstruction of an object phase transmission function from inline X-ray holograms," Opt. Commun. 213, 247 - 258 (2002).
[CrossRef]

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

Teague, M.

Weitkamp, T.

I. Schelokov, T. Weitkamp, and A. Snigirev, "Reconstruction of an object phase transmission function from inline X-ray holograms," Opt. Commun. 213, 247 - 258 (2002).
[CrossRef]

Wilkins, S.

T. Gureyev, D. Paganin, G. Myers, Y. Nesterets, and S. Wilkins, "Phase-and-amplitude computer tomography," Appl. Phys. Lett. 89, 034,102 (2006).
[CrossRef]

T. Gureyev, A. Pogany, D. Paganin, and S. Wilkins, "Linear algorithms for phase retrieval in the Fresnel region," Opt. Commun. 231, 53 - 70 (2004).
[CrossRef]

A. Pogany, D. Gao, and S. Wilkins, "Contrast and resolution in imaging with a microfocus x-ray source," Rev. Sci. Instrum. 68, 2774 (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. Radiol. (to be published) (2008).
[CrossRef] [PubMed]

X. Wu and H. Liu, "Phase-space evolution of x-ray coherence in phase-sensitive imaging," Appl. Opt. 47, E44 - E52 (2008).
[CrossRef] [PubMed]

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

X. Wu and H. Liu, "Robustness of a phase-retrieval approach based on phase-attenuation duality," J. X-Ray Sci. Technol. 15, 85 - 95 (2007).

X. Wu, H. Liu, and A. Yan, "X-ray phase-attenuation duality and phase retrieval," Opt. Lett. 30(4), 379 - 381 (2005).
[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 and H. Liu, "A dual detector approach for X-ray attenuation and phase imaging," J. X-ray Sci. Technol. 12, 35-42 (2004).

X. Wu and H. Liu, "An experimental method of determining relative phase-contrast factor for x-ray imaging systems," Med. Phys. 31, 997 - 1002 (2004).
[CrossRef]

X. Wu and H. Liu, "A reconstruction formula for soft tissue X-ray phase tomography," J. X-ray Sci. Technol. 12, 273 - 279 (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]

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

Wu, X.

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

Yan, A.

X. Wu, H. Liu, and A. Yan, "Phase-Contrast X-Ray Tomography: Contrast Mechanism and Roles of Phase Retrieval," Eur. J. Radiol. (to be published) (2008).
[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] [PubMed]

Zhang, D.

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

Appl. Opt.

Appl. Phys. Lett.

T. Gureyev, D. Paganin, G. Myers, Y. Nesterets, and S. Wilkins, "Phase-and-amplitude computer tomography," Appl. Phys. Lett. 89, 034,102 (2006).
[CrossRef]

Eur. J. Radiol.

X. Wu, H. Liu, and A. Yan, "Phase-Contrast X-Ray Tomography: Contrast Mechanism and Roles of Phase Retrieval," Eur. J. Radiol. (to be published) (2008).
[CrossRef] [PubMed]

IEEE Trans. Biomed. Eng.

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

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

J. X-ray Sci. Technol.

X. Wu and H. Liu, "A reconstruction formula for soft tissue X-ray phase tomography," J. X-ray Sci. Technol. 12, 273 - 279 (2004).

X. Wu and H. Liu, "Robustness of a phase-retrieval approach based on phase-attenuation duality," J. X-Ray Sci. Technol. 15, 85 - 95 (2007).

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

X. Wu and H. Liu, "A dual detector approach for X-ray attenuation and phase imaging," J. X-ray Sci. Technol. 12, 35-42 (2004).

Med. Phys.

X. Wu and H. Liu, "An experimental method of determining relative phase-contrast factor for x-ray imaging systems," Med. Phys. 31, 997 - 1002 (2004).
[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 x-ray phase-contrast imaging: Theoretical foundations and design considerations," Med. Phys. 30, 2169 - 2179 (2003).
[CrossRef] [PubMed]

E. Donnelly, R. Price, and D. Pickens, "Dual focal-spot imaging for phase extraction in phase-contrast radiography," Med. Phys. 30, 2292 - 2296 (2003).
[CrossRef] [PubMed]

Nature

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]

Opt. Commun.

I. Schelokov, T. Weitkamp, and A. Snigirev, "Reconstruction of an object phase transmission function from inline X-ray holograms," Opt. Commun. 213, 247 - 258 (2002).
[CrossRef]

T. Gureyev, A. Pogany, D. Paganin, and S. Wilkins, "Linear algorithms for phase retrieval in the Fresnel region," Opt. Commun. 231, 53 - 70 (2004).
[CrossRef]

L. Allen and M. Oxley, "Phase retrieval from series of images obtainedbe defocus variation," Opt. Commun. 199, 65 - 75 (2001).
[CrossRef]

Opt. Express

Opt. Lett.

Phy. Rev. Lett.

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[CrossRef]

D. Paganin and K. Nugent, "Noninterferometric Phase Imaging with Partially Coherent Light," Phy. Rev. Lett. 80, 2586 - 2589 (1998).
[CrossRef]

PNAS

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

Rev. Sci. Instrum.

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]

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[CrossRef]

Other

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[CrossRef]

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Figures (5)

Fig. 1.
Fig. 1.

Flow chart of APBA

Fig. 2.
Fig. 2.

x-ray edge images and the recovered phase map. (a) Attenuation image A 2 0 ; (b) Phase-contrast image I; (c) phase map recovered using APBA (with 10 iteration steps)

Fig. 3.
Fig. 3.

Recovered phase maps using TIE algorithm, with different Tikhonov regularization parameters α: (a) α=0; (b) α=10Δ2; (c) α=50Δ2. Δ is the sampling step-size in FT space.

Fig. 4.
Fig. 4.

x-ray lumpectomy images and the recovered phase map. (a) Attenuation image A 2 0 ; (©[2008] IEEE);(b) Phase-contrast image I; (©[2008] IEEE);(c) phase map recovered using APBA with 10 iteration steps.

Fig. 5.
Fig. 5.

Recovered lumpectomy phase maps using TIE algorithm, with different Tikhonov regularization parameters α: (a) α=0; (b) α=10Δ2; (c) α=50Δ2. Δ is the sampling step-size in FT space.

Equations (25)

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n = 1 δ + i β
δ = ( r e λ 2 2 π ) k N k ( Z k + f k r )
I ˜ ( u M ; R 1 + R 2 ) = I in { cos ( π λ R 2 M u 2 ) · 𝓕 ̂ ( A 0 2 ) +
+ [ 2 sin ( π λ R 2 M u 2 ) ( 2 π λ R 2 M u 2 ) · cos ( π λ R 2 M u 2 ) ] · 𝓕 ̂ ( A 0 2 ϕ )
cos ( π λ R 2 M u 2 ) · λ R 2 2 π M · 𝓕 ̂ ( · ( A 0 2 ϕ ) ) } ,
I ˜ ( u ) = 2 e i 2 π x · u I ( x ) d x ,
π λ R 2 u max 2 1 ,
I ( r ; R 1 + R 2 ) = I in M 2 { A 0 2 ( r M ) λ R 2 2 π M · ( A 0 2 ϕ ) ( r M ) } .
ϕ ( r ) = 2 π M λ R 2 2 { · [ 1 A 0 2 ( 2 ( M 2 I in I A 0 2 ) ) ] } ,
2 = 2 ( 2 ) 2 + α 2 ,
A 0 ( r ) = A incoh ( r ) · A pe , coh ( r ) .
A KN 2 ( r ) = exp ( μ KN ( r , s ) ds ) = exp [ σ KN λ r e ϕ ( r ) ] ,
σ KN = 2 π r e 2 { 1 + η η 2 [ 2 ( 1 + η ) 1 + 2 η 1 η log ( 1 + 2 η ) ] + 1 2 η log ( 1 + 2 η ) 1 + 3 η ( 1 + 2 η ) 2 } .
A KN 2 ( r ) = 𝓕 ˇ ( 𝓕 ̂ ( M 2 I in I ) 1 + 4 π 2 k u 2 ) ,
k = λ R 2 2 π M λ r e σ KN ,
A 0 ( r ) = A ~ ( r ) δ A ( r ) ,
A ˜ ( r ) = A KN ( r ) = A incoh ( r ), δ A ( r ) = A KN ( r ) · ( 1 A pe ,coh( r )).
E ( r ; R 1 , R 2 ) = I in λ R 2 · exp ( 2 π i R 1 + R 2 λ ) 2 exp [ i π λ ( ξ 2 R 1 + ( ξ r ) 2 R 2 ) ] A 0 ( ξ ) e i ϕ ( ξ ) d ξ
h ( r ) = ( M 1 ) r 2 2 R 1 R 2 , ω = λ R 2 2 π M
E 1 ( r ) = M I in E ( M r ; R 1 , R 2 ) · e i 2 ω h ( r ) ,
E 1 ( r ) = 1 2 π ω · 2 e i ( r ξ ) 2 2 ω T 0 ( ξ ) d ξ = 1 2 π ω . G * T 0 ( r ) ,
ψ ( r ) = π 2 + ϕ ( r ) + ω 2 [ 2 A 0 ( r ) A 0 ( r ) ϕ ( r ) 2 ] .
I ( r ) = I ~ ( r ) + δ I ( r ) 2 I ~ ( r ) · δ I ( r ) · cos ( ψ ~ ( r ) δ ψ ( r ) ) .
I ( r ) I ~ ( r ) δI ( r ) ,
δ = δ δ A old δ A ,

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