Abstract

The phase retrieval is an important task in x-ray phase contrast imaging. The robustness of phase retrieval is especially important for potential medical imaging applications such as phase contrast mammography. Recently the authors developed an iterative phase retrieval algorithm, the attenuation-partition based algorithm, for the phase retrieval in inline phase-contrast imaging [1]. Applied to experimental images, the algorithm was proven to be fast and robust. However, a quantitative analysis of the performance of this new algorithm is desirable. In this work, we systematically compared the performance of this algorithm with other two widely used phase retrieval algorithms, namely the Gerchberg-Saxton (GS) algorithm and the Transport of Intensity Equation (TIE) algorithm. The systematical comparison is conducted by analyzing phase retrieval performances with a digital breast specimen model. We show that the proposed algorithm converges faster than the GS algorithm in the Fresnel diffraction regime, and is more robust against image noise than the TIE algorithm. These results suggest the significance of the proposed algorithm for future medical applications with the x-ray phase contrast imaging technique.

© 2010 Optical Society of America

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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  4. 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] [PubMed]
  8. 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]
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    [CrossRef] [PubMed]
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    [PubMed]
  35. X. Wu, E. L. Gingold, G. T. Barnes and D. M. Tucker, "Normalized average glandular dose in Molybdenum target-Rhodium filter and Rhodium target-Rhodium filter mammography," Radiology 193, 83 - 89 (1994).
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    [CrossRef]
  40. T. Gureyev and K. 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]
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2009 (1)

2008 (4)

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

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]

2007 (1)

2006 (2)

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. Commun. 259, 569 - 580 (2006).
[CrossRef]

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]

2005 (3)

2004 (2)

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 dual detector approach for X-ray attenuation and phase imaging," J. X-ray Sci. Tech. 12, 35-42 (2004).

2003 (3)

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

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 (1)

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 (1)

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

2000 (1)

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

1998 (1)

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

1997 (1)

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 (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]

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]

T. Gureyev and K. 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]

1995 (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]

T. Gureyev, A. Roberts, and K. Nugent, "Partially coherent fields, the transport-of-intensity equation, and phase uniqueness," J. Opt. Soc. Am. A 12, 1942 - 1946 (1995).
[CrossRef]

1994 (1)

X. Wu, E. L. Gingold, G. T. Barnes and D. M. Tucker, "Normalized average glandular dose in Molybdenum target-Rhodium filter and Rhodium target-Rhodium filter mammography," Radiology 193, 83 - 89 (1994).
[PubMed]

1993 (1)

1992 (1)

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

1991 (2)

X. Wu, G. T. Barnes and D. M. Tucker, "Spectral dependence of glandular tissue dose in screen-film mammography," Radiology 179, 143 - 148 (1991).
[PubMed]

F. Roddier and C. Roddier, "Wavefront reconstruction using Iterative Fourier transforms," Appl. Opt. 30, 1325 -1327 (1991).
[CrossRef] [PubMed]

1990 (1)

1983 (1)

1982 (1)

1978 (1)

1975 (1)

J. H. Hubbell, W. I. Veigele, E. A. Briggs,  et al., "Atomic form factors, incohoerent scattering functions, and photon scattering cross sections," J. Phys. Chem. Ref. Data 4, 471-538 (1975).
[CrossRef]

1972 (1)

R. W. Gerchberg and W. O. Saxton, "A practical algorithm for the determination of the phase from image and diffraction plane pictures," Optik 35, 237 - 246 (1972).

Allen, L.

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

Barnes, G. T.

X. Wu, E. L. Gingold, G. T. Barnes and D. M. Tucker, "Normalized average glandular dose in Molybdenum target-Rhodium filter and Rhodium target-Rhodium filter mammography," Radiology 193, 83 - 89 (1994).
[PubMed]

X. Wu, G. T. Barnes and D. M. Tucker, "Spectral dependence of glandular tissue dose in screen-film mammography," Radiology 179, 143 - 148 (1991).
[PubMed]

Boistel, R.

Bonvicini, V.

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

Briggs, E. A.

J. H. Hubbell, W. I. Veigele, E. A. Briggs,  et al., "Atomic form factors, incohoerent scattering functions, and photon scattering cross sections," J. Phys. Chem. Ref. Data 4, 471-538 (1975).
[CrossRef]

Cloetens, P.

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]

Fatemi, E.

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

Fienup, J.

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]

Gerchberg, R. W.

R. W. Gerchberg and W. O. Saxton, "A practical algorithm for the determination of the phase from image and diffraction plane pictures," Optik 35, 237 - 246 (1972).

Gingold, E. L.

X. Wu, E. L. Gingold, G. T. Barnes and D. M. Tucker, "Normalized average glandular dose in Molybdenum target-Rhodium filter and Rhodium target-Rhodium filter mammography," Radiology 193, 83 - 89 (1994).
[PubMed]

Guigay, J.

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. Commun. 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]

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]

T. Gureyev and K. 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]

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]

T. Gureyev, A. Roberts, and K. Nugent, "Partially coherent fields, the transport-of-intensity equation, and phase uniqueness," J. Opt. Soc. Am. A 12, 1942 - 1946 (1995).
[CrossRef]

Hubbell, J. H.

J. H. Hubbell, W. I. Veigele, E. A. Briggs,  et al., "Atomic form factors, incohoerent scattering functions, and photon scattering cross sections," J. Phys. Chem. Ref. Data 4, 471-538 (1975).
[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.

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 and H. Liu, "Phase-space evolution of x-ray coherence in phase-sensitive imaging," Appl. Opt. 47, E44 - E52 (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]

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]

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. Tech. 12, 35-42 (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. Tech.  11, 33 - 42 (2003).

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]

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]

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. Commun. 259, 569 - 580 (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 - 2965 (1996).
[CrossRef]

T. Gureyev and K. 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]

T. Gureyev, A. Roberts, and K. Nugent, "Partially coherent fields, the transport-of-intensity equation, and phase uniqueness," J. Opt. Soc. Am. A 12, 1942 - 1946 (1995).
[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]

Oxley, M.

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

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 - 2965 (1996).
[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. Commun. 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]

Roberts, A.

Roddier, C.

Roddier, F.

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]

Saxton, W. O.

R. W. Gerchberg and W. O. Saxton, "A practical algorithm for the determination of the phase from image and diffraction plane pictures," Optik 35, 237 - 246 (1972).

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]

Seldin, J.

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]

Teague, M.

Tucker, D. M.

X. Wu, E. L. Gingold, G. T. Barnes and D. M. Tucker, "Normalized average glandular dose in Molybdenum target-Rhodium filter and Rhodium target-Rhodium filter mammography," Radiology 193, 83 - 89 (1994).
[PubMed]

X. Wu, G. T. Barnes and D. M. Tucker, "Spectral dependence of glandular tissue dose in screen-film mammography," Radiology 179, 143 - 148 (1991).
[PubMed]

Veigele, W. I.

J. H. Hubbell, W. I. Veigele, E. A. Briggs,  et al., "Atomic form factors, incohoerent scattering functions, and photon scattering cross sections," J. Phys. Chem. Ref. Data 4, 471-538 (1975).
[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. Commun. 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 and A. Yan, "Phase Retrieval From One Single Phase Contrast X-Ray Image," Opt. Express  17, 11187 - 11196 (2009).
[CrossRef]

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]

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

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] [PubMed]

X. Wu and H. Liu, "A dual detector approach for X-ray attenuation and phase imaging," J. X-ray Sci. Tech. 12, 35-42 (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. Tech.  11, 33 - 42 (2003).

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, E. L. Gingold, G. T. Barnes and D. M. Tucker, "Normalized average glandular dose in Molybdenum target-Rhodium filter and Rhodium target-Rhodium filter mammography," Radiology 193, 83 - 89 (1994).
[PubMed]

X. Wu, G. T. Barnes and D. M. Tucker, "Spectral dependence of glandular tissue dose in screen-film mammography," Radiology 179, 143 - 148 (1991).
[PubMed]

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]

Appl. Opt. (3)

Eur. J. Radiology (1)

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]

IEEE Trans. Biomed. Eng. (1)

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]

J. Microsc. (1)

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. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (4)

J. Phys. Chem. Ref. Data (1)

J. H. Hubbell, W. I. Veigele, E. A. Briggs,  et al., "Atomic form factors, incohoerent scattering functions, and photon scattering cross sections," J. Phys. Chem. Ref. Data 4, 471-538 (1975).
[CrossRef]

J. X-ray Sci. Tech (1)

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

J. X-ray Sci. Tech. (1)

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

Med. Phys. (3)

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

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]

Nature (1)

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. (2)

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. Commun. 259, 569 - 580 (2006).
[CrossRef]

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

Opt. Express (4)

Opt. Lett. (3)

Optik (1)

R. W. Gerchberg and W. O. Saxton, "A practical algorithm for the determination of the phase from image and diffraction plane pictures," Optik 35, 237 - 246 (1972).

Phy. Rev. Lett. (2)

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 - 2965 (1996).
[CrossRef]

Physica D (1)

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

PNAS (1)

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]

Radiology (3)

X. Wu, G. T. Barnes and D. M. Tucker, "Spectral dependence of glandular tissue dose in screen-film mammography," Radiology 179, 143 - 148 (1991).
[PubMed]

X. Wu, E. L. Gingold, G. T. Barnes and D. M. Tucker, "Normalized average glandular dose in Molybdenum target-Rhodium filter and Rhodium target-Rhodium filter mammography," Radiology 193, 83 - 89 (1994).
[PubMed]

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

Rev. Sci. Instrum. (2)

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]

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]

Other (4)

N. Dyson, X-Rays in Atomic and Nuclear Physics (Longman Scientific and Technical, Essex, UK, 1973).

X. Wu, A. Dean, and H. Liu, Biomedical Photonics Handbook, (CRC Press, Tampa, Fla., 2003) Chap. 26, pp. 26-1 - 26-34.

L. Rudin, "Images, numerical analysis of singularities and shock filters," Report #TR:5250:87, Caltech, C,S, Dept. (1987).

A. Tychonoff and V. Arsenin, Solution of Ill-posed Problems (Winston & Sons, Washington, 1977).

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Figures (9)

Fig. 1.
Fig. 1.

Flow chart of APBA

Fig. 2.
Fig. 2.

Example “Lena” images used in measuring the closeness between images. (a) ϕ 1, (b) ϕ 2, (c) ϕ 3

Fig. 3.
Fig. 3.

Image manifest of (a) A 2 pe,coh, (b) A 2 KN and (c) A 2 0 when x-ray energy equals 35.5 keV.

Fig. 4.
Fig. 4.

Profiles of A 2 pe,coh, the solid lines, and A 2 KN, the dotted lines, when x-ray energy equals (a) 18.5 keV, (b) 35.5 keV and (c) 59.5 keV.

Fig. 5.
Fig. 5.

Image representation of the inputs generated from the simulation model and Fresnel propagation. (a) the phase map ϕ; (b) the attenuation map A 2 0; and (c) the normalized Fresnel propagated phase contrast image I with object to detector distance R 2 = 26 in (0.66 m).

Fig. 6.
Fig. 6.

Comparison of the performance of the GS algorithm and APBA. (a) plot of the accuracy measures with respect to iteration steps. The plot with solid line represents the APBA. The one with dashed line represents the GS algorithm; (b) recovered phase map with the GS algorithm after 100 iterations; (c) recovered phase map with APBA after 100 iterations.

Fig. 7.
Fig. 7.

Comparison of APBA and the TIE algorithm with pure data. (a) plot of the accuracy measure with respect to iteration steps. The plot with solid line represents the APBA. The one with dashed line represents the TIE algorithm. It needs 1110 steps for the TV measure, 0.00215226, of APBA to achieve to the TV measure, 0.00215269, of TIE; (b) recovered phase using the TIE algorithm; (c) recovered phase using APBA after 1500 iteration steps.

Fig. 8.
Fig. 8.

Comparison of the TIE algorithm and APBA with noise added. (a) True phase map ϕ; (b) attenuation map A 2 0; (c) the normalized phase contrast image I; (d) recovered phase map with the TIE algorithm, no Tikhonov regularization is used; (e) recovered phase map with the TIE algorithm with Tikhonov regularization; (f) recovered phase map with APBA after 10 iteration steps. In the simulation, the acquired data is assumed to have a level of δb = 0.03% detector noise and one pixel misalignment between A 2 0 and I horizontally.

Fig. 9.
Fig. 9.

Profiles, along a line passing through the microcalcifications, of the recovered phase using APBA, the solid line, and using the TIE algorithm with Tikhonov regularization, the dashed line, in Case 3. The dash-dotted line is the true phase.

Tables (1)

Tables Icon

Table 1. TV comparison of the TIE algorithm and APBA. In the table, κ is the Tikhonov regularization parameter, Δ represents the sampling step-size in FT-space.

Equations (19)

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

ϕ ( r ) = ( h c E ) r e ρ e ( r , z ) dz = ( hc E ) r e ρ e , p ( r ) ,
̂ ( I ) ( u M ; R 1 + R 2 ) = I in { cos ( π λ R 2 M u 2 ) · ̂ ( 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 ) ] · ̂ ( A 0 2 ϕ )
cos ( π λ R 2 M u 2 ) · λ R 2 2 π M · ̂ ( · ( A 0 2 ϕ ) )
λ R 2 4 π M sin ( π λ R 2 M u 2 ) · ̂ ( 2 A 0 2 ) } ,
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 ) } .
A KN ( r ) = exp [ σ KN 2 ρ e , p ( r ) ] , ϕ ( r ) = λ r e ρ e , p ,
σ KN ( E photon ) = 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 ) = 𝔇 ( I ) = ̂ 1 ( ̂ ( I ) 1 + 4 π 2 k ˜ u 2 ) , ϕ ( r ) = ( λ r e σ KN ) ln ( A KN 2 ( r ) ) ,
k ˜ = λ R 2 2 π M · λ r e σ KN ,
A 0 ( r ) = A KN ( r ) · A pe , coh ( r ) ,
A 0 = A KN δ A , δ A = A KN ( 1 A pe , coh ) ,
δ A = A KN · ( 1 P ) , P = A 0 A KN .
𝔉 𝔯 ( T ) ( r ) = 1 λ R 2 2 exp [ i π M λ R 2 ( r M ξ ) 2 ] T ( ξ ) d ξ .
std ( g , f ) : = [ Ω ( g ( r ) f ( r ) μ ) 2 d r ] 1 2 V ( Ω )
TV ( g , f ) : = Ω ( g f ) d r V ( Ω ) = Ω ( ( g f ) x ) 2 + ( ( g f ) y ) 2 d r V ( Ω ) ,
ϕ ( r ) = 2 π M λ R 2 2 { · [ [ 2 ( I A 0 2 ) ] A 0 2 ] } ,
x κ = arg min x X A x y Y 2 + κ x X 2 .

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