Abstract

In-line phase-contrast X-ray imaging provides images where both absorption and refraction contribute. For quantitative analysis of these images, the phase needs to be retrieved numerically. There are many phase-retrieval methods available. Those suitable for phase-contrast tomography, i.e., non-iterative phase-retrieval methods that use only one image at each projection angle, all follow the same pattern though derived in different ways. We outline this pattern and use it to compare the methods to each other, considering only phase-retrieval performance and not the additional effects of tomographic reconstruction. We also outline derivations, approximations and assumptions, and show which methods are similar or identical and how they relate to each other. A simple scheme for choosing reconstruction method is presented, and numerical phase-retrieval performed for all methods.

© 2011 OSA

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2010 (2)

2009 (2)

T. E. Gureyev, S. C. Mayo, D. E. Myers, Ya. Nesterets, D. M. Paganin, A. Pogany, A. W. Stevenson, and S. W. Wilkins, “Refracting Röntgen’s rays: propagation-based X-ray phase contrast for biomedical imaging,” J. Appl. Phys. 105, 102005 (2009).
[CrossRef]

See, e.g., P. SuetensFundamentals of Medical Imaging (Cambridge Univ Press, 2009).
[CrossRef]

2008 (1)

2007 (3)

T. Tuohimaa, M. Otendal, and H. M. Hertz, “Phase-contrast X-ray imaging with a liquid-metal-jet-anode micro-focus source,” App. Phys. Lett. 91, 074104 (2007).
[CrossRef]

G. R. Myers, S. C. Mayo, T. E. Gureyev, D. M. Paganin, and S. W. Wilkins, “Polychromatic cone-beam phase-contrast tomography,” Phys. Rev. A 76, 045804 (2007).
[CrossRef]

K. A. Nugent, “X-ray noninterferometric phase imaging: a unified picture,” J. Opt. Soc. Am. A 24, 536–546 (2007).
[CrossRef]

2006 (3)

A. Groso, R. Abela, and M. Stampanoni, “Implementation of a fast method for high resolution phase contrast tomography,” Opt. Express 14, 8103–8110 (2006).
[CrossRef] [PubMed]

D. M. Paganin, Coherent X-Ray Optics (Oxford Science Publications, 2006).
[CrossRef]

F. Pfeiffer, T. Weitkamp, O. Bunk, and C. David, “Phase retrieval and differential phase-contrast imaging with low-brilliance X-ray sources,” Nat. Phys. 2, 258–261 (2006).
[CrossRef]

2005 (3)

A. Momose, “Recent advances in X-ray phase imaging,” Jpn. J. Appl. Phys. 44, 6355–6367 (2005).
[CrossRef]

S. Zabler, P. Cloetens, J.-P. Guigay, and J. Baruchel, “Optimization of phase contrast imaging using hard x-rays,” Rev. Sci. Intstrum. 76, 073705 (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]

2004 (4)

2002 (1)

D. Paganin, S. C. Mayo, T. E. Gureyev, P. R. Wilkins, and S. W. 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)

A. C. Kak and M. Slaney, Principles of Computerized Tomographic Imaging (Siam, 2001).
[CrossRef]

2000 (1)

R. Fitzgerald, “Phase-sensitive X-ray imaging,” Phys. Today 53(7), 23–26 (2000).
[CrossRef]

1999 (1)

A. V. Bronnikov, “Reconstruction formulas for phase-contrast imaging,” Opt. Commun. 171, 239–244 (1999).
[CrossRef]

1998 (1)

1997 (1)

D. Chapman, W. Thomlinson, R. E. Johnston, D. Washburn, E. Pisano, N. Gmür, Z. Zhong, R. Menk, F. Arfelli, and D. Sayers, “Diffraction enhanced X-ray imaging,” Phys. Med. Biol. 42, 2015–2025 (1997).
[CrossRef] [PubMed]

1996 (4)

S. W. Wilkins, T. E. Gureyev, D. Gao, A. Pogany, and A. W. Stevenson, “Phase-constrast imaging using poly-chromatic hard X-rays,” Nature 384, 335–338 (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]

P. Cloetens, R. Barrett, J. Baruchel, J. P. Guigay, and M. Schlenker, “Phase objects in synchrotron radiation hard X-ray imaging,” J. Phys. D: Appl. Phys. 29, 133–146 (1996).
[CrossRef]

J. W. Goodman, Introduction to Fourier Optics , 2nd ed. (McGraw-Hill, 1996).

1995 (4)

J. M. Cowley, Diffraction Physics (Elsevier, 1995).

T. J. Davis, D. Gao, T. E. Gureyev, A. W. Stevenson, and S. W. Wilkins, “Phase-contrast imaging of weakly absorbing materials using hard X-rays,” Nature 373, 595–598 (1995).
[CrossRef]

V. N. Ingal and E. A. Beliaevskaya, “X-ray plane-wave topographyobservation of the phase contrast from a non-crystalline object,” J. Phys. D: Appl. Phys. 28, 2314–2317 (1995).
[CrossRef]

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

1987 (1)

1986 (1)

1983 (1)

1982 (1)

R. Grella, “Fresnel propagation and diffraction and paraxial wave equation,” J. Opt. (Paris) 13, 367–364 (1982).

1977 (1)

J. P. Guigay, “Fourier transform analysis of Fresnel diffraction patterns and in-line holograms,” Optik 49, 121–125 (1977).

1965 (1)

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

Abela, R.

Arfelli, F.

D. Chapman, W. Thomlinson, R. E. Johnston, D. Washburn, E. Pisano, N. Gmür, Z. Zhong, R. Menk, F. Arfelli, and D. Sayers, “Diffraction enhanced X-ray imaging,” Phys. Med. Biol. 42, 2015–2025 (1997).
[CrossRef] [PubMed]

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]

Barrett, R.

P. Cloetens, R. Barrett, J. Baruchel, J. P. Guigay, and M. Schlenker, “Phase objects in synchrotron radiation hard X-ray imaging,” J. Phys. D: Appl. Phys. 29, 133–146 (1996).
[CrossRef]

Barty, A.

Baruchel, J.

S. Zabler, P. Cloetens, J.-P. Guigay, and J. Baruchel, “Optimization of phase contrast imaging using hard x-rays,” Rev. Sci. Intstrum. 76, 073705 (2005).
[CrossRef]

P. Cloetens, R. Barrett, J. Baruchel, J. P. Guigay, and M. Schlenker, “Phase objects in synchrotron radiation hard X-ray imaging,” J. Phys. D: Appl. Phys. 29, 133–146 (1996).
[CrossRef]

Bastiaans, M. J.

Beliaevskaya, E. A.

V. N. Ingal and E. A. Beliaevskaya, “X-ray plane-wave topographyobservation of the phase contrast from a non-crystalline object,” J. Phys. D: Appl. Phys. 28, 2314–2317 (1995).
[CrossRef]

Beltran, M. A.

Bonse, U.

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

Bronnikov, A. V.

A. V. Bronnikov, “Reconstruction formulas for phase-contrast imaging,” Opt. Commun. 171, 239–244 (1999).
[CrossRef]

Bunk, O.

F. Pfeiffer, T. Weitkamp, O. Bunk, and C. David, “Phase retrieval and differential phase-contrast imaging with low-brilliance X-ray sources,” Nat. Phys. 2, 258–261 (2006).
[CrossRef]

Chapman, D.

D. Chapman, W. Thomlinson, R. E. Johnston, D. Washburn, E. Pisano, N. Gmür, Z. Zhong, R. Menk, F. Arfelli, and D. Sayers, “Diffraction enhanced X-ray imaging,” Phys. Med. Biol. 42, 2015–2025 (1997).
[CrossRef] [PubMed]

Cloetens, P.

S. Zabler, P. Cloetens, J.-P. Guigay, and J. Baruchel, “Optimization of phase contrast imaging using hard x-rays,” Rev. Sci. Intstrum. 76, 073705 (2005).
[CrossRef]

P. Cloetens, R. Barrett, J. Baruchel, J. P. Guigay, and M. Schlenker, “Phase objects in synchrotron radiation hard X-ray imaging,” J. Phys. D: Appl. Phys. 29, 133–146 (1996).
[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 (Elsevier, 1995).

David, C.

F. Pfeiffer, T. Weitkamp, O. Bunk, and C. David, “Phase retrieval and differential phase-contrast imaging with low-brilliance X-ray sources,” Nat. Phys. 2, 258–261 (2006).
[CrossRef]

Davis, T. J.

T. E. Gureyev, T. J. Davis, A. Pogany, S. C. Mayo, and S. W. Wilkins, “Optical phase retrieval by use of first Born- and Rytov-type approximations,” Appl. Opt. 43, 2418–2430 (2004).
[CrossRef] [PubMed]

T. J. Davis, D. Gao, T. E. Gureyev, A. W. Stevenson, and S. W. Wilkins, “Phase-contrast imaging of weakly absorbing materials using hard X-rays,” Nature 373, 595–598 (1995).
[CrossRef]

Dhal, B. B.

Fitzgerald, R.

R. Fitzgerald, “Phase-sensitive X-ray imaging,” Phys. Today 53(7), 23–26 (2000).
[CrossRef]

Gao, D.

S. W. Wilkins, T. E. Gureyev, D. Gao, A. Pogany, and A. W. Stevenson, “Phase-constrast imaging using poly-chromatic hard X-rays,” Nature 384, 335–338 (1996).
[CrossRef]

T. J. Davis, D. Gao, T. E. Gureyev, A. W. Stevenson, and S. W. Wilkins, “Phase-contrast imaging of weakly absorbing materials using hard X-rays,” Nature 373, 595–598 (1995).
[CrossRef]

Gmür, N.

D. Chapman, W. Thomlinson, R. E. Johnston, D. Washburn, E. Pisano, N. Gmür, Z. Zhong, R. Menk, F. Arfelli, and D. Sayers, “Diffraction enhanced X-ray imaging,” Phys. Med. Biol. 42, 2015–2025 (1997).
[CrossRef] [PubMed]

Gonsalves, R. A.

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics , 2nd ed. (McGraw-Hill, 1996).

Grella, R.

R. Grella, “Fresnel propagation and diffraction and paraxial wave equation,” J. Opt. (Paris) 13, 367–364 (1982).

Groso, A.

Guigay, J. P.

P. Cloetens, R. Barrett, J. Baruchel, J. P. Guigay, and M. Schlenker, “Phase objects in synchrotron radiation hard X-ray imaging,” J. Phys. D: Appl. Phys. 29, 133–146 (1996).
[CrossRef]

J. P. Guigay, “Fourier transform analysis of Fresnel diffraction patterns and in-line holograms,” Optik 49, 121–125 (1977).

Guigay, J.-P.

S. Zabler, P. Cloetens, J.-P. Guigay, and J. Baruchel, “Optimization of phase contrast imaging using hard x-rays,” Rev. Sci. Intstrum. 76, 073705 (2005).
[CrossRef]

Gureyev, T. E.

T. E. Gureyev, S. C. Mayo, D. E. Myers, Ya. Nesterets, D. M. Paganin, A. Pogany, A. W. Stevenson, and S. W. Wilkins, “Refracting Röntgen’s rays: propagation-based X-ray phase contrast for biomedical imaging,” J. Appl. Phys. 105, 102005 (2009).
[CrossRef]

T. E. Gureyev, Y. I. Nesterets, A. W. Stevenson, P. R. Miller, A. Pogany, and S. W. Stevenson, “Some simple rules for contrast, signal-to-noise and resolution in in-line phase-contrast imaging,” Opt. Express 16, 3223–3241 (2008).
[CrossRef] [PubMed]

G. R. Myers, S. C. Mayo, T. E. Gureyev, D. M. Paganin, and S. W. Wilkins, “Polychromatic cone-beam phase-contrast tomography,” Phys. Rev. A 76, 045804 (2007).
[CrossRef]

T. E. Gureyev, T. J. Davis, A. Pogany, S. C. Mayo, and S. W. Wilkins, “Optical phase retrieval by use of first Born- and Rytov-type approximations,” Appl. Opt. 43, 2418–2430 (2004).
[CrossRef] [PubMed]

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

S. W. Wilkins, T. E. Gureyev, D. Gao, A. Pogany, and A. W. Stevenson, “Phase-constrast imaging using poly-chromatic hard X-rays,” Nature 384, 335–338 (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]

T. J. Davis, D. Gao, T. E. Gureyev, A. W. Stevenson, and S. W. Wilkins, “Phase-contrast imaging of weakly absorbing materials using hard X-rays,” Nature 373, 595–598 (1995).
[CrossRef]

Hart, M.

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

Hayes, J. P.

Hertz, H. M.

T. Tuohimaa, M. Otendal, and H. M. Hertz, “Phase-contrast X-ray imaging with a liquid-metal-jet-anode micro-focus source,” App. Phys. Lett. 91, 074104 (2007).
[CrossRef]

Ingal, V. N.

V. N. Ingal and E. A. Beliaevskaya, “X-ray plane-wave topographyobservation of the phase contrast from a non-crystalline object,” J. Phys. D: Appl. Phys. 28, 2314–2317 (1995).
[CrossRef]

Johnston, R. E.

D. Chapman, W. Thomlinson, R. E. Johnston, D. Washburn, E. Pisano, N. Gmür, Z. Zhong, R. Menk, F. Arfelli, and D. Sayers, “Diffraction enhanced X-ray imaging,” Phys. Med. Biol. 42, 2015–2025 (1997).
[CrossRef] [PubMed]

Kak, A. C.

A. C. Kak and M. Slaney, Principles of Computerized Tomographic Imaging (Siam, 2001).
[CrossRef]

Kitchen, M. J.

Kohn, V.

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

Lewis, R. A.

R. A. Lewis, “Medical phase contrast X-ray imaging: current status and future prospects,” Phys. Med. Biol. 49, 3573–3583 (2004).
[CrossRef] [PubMed]

Liu, H.

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]

Mancuso, A. P.

Mayo, S. C.

T. E. Gureyev, S. C. Mayo, D. E. Myers, Ya. Nesterets, D. M. Paganin, A. Pogany, A. W. Stevenson, and S. W. Wilkins, “Refracting Röntgen’s rays: propagation-based X-ray phase contrast for biomedical imaging,” J. Appl. Phys. 105, 102005 (2009).
[CrossRef]

G. R. Myers, S. C. Mayo, T. E. Gureyev, D. M. Paganin, and S. W. Wilkins, “Polychromatic cone-beam phase-contrast tomography,” Phys. Rev. A 76, 045804 (2007).
[CrossRef]

T. E. Gureyev, T. J. Davis, A. Pogany, S. C. Mayo, and S. W. Wilkins, “Optical phase retrieval by use of first Born- and Rytov-type approximations,” Appl. Opt. 43, 2418–2430 (2004).
[CrossRef] [PubMed]

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

Menk, R.

D. Chapman, W. Thomlinson, R. E. Johnston, D. Washburn, E. Pisano, N. Gmür, Z. Zhong, R. Menk, F. Arfelli, and D. Sayers, “Diffraction enhanced X-ray imaging,” Phys. Med. Biol. 42, 2015–2025 (1997).
[CrossRef] [PubMed]

Miller, P. R.

Momose, A.

A. Momose, “Recent advances in X-ray phase imaging,” Jpn. J. Appl. Phys. 44, 6355–6367 (2005).
[CrossRef]

Myers, D. E.

T. E. Gureyev, S. C. Mayo, D. E. Myers, Ya. Nesterets, D. M. Paganin, A. Pogany, A. W. Stevenson, and S. W. Wilkins, “Refracting Röntgen’s rays: propagation-based X-ray phase contrast for biomedical imaging,” J. Appl. Phys. 105, 102005 (2009).
[CrossRef]

Myers, G. R.

G. R. Myers, S. C. Mayo, T. E. Gureyev, D. M. Paganin, and S. W. Wilkins, “Polychromatic cone-beam phase-contrast tomography,” Phys. Rev. A 76, 045804 (2007).
[CrossRef]

Nesterets, Y. I.

Nesterets, Ya.

T. E. Gureyev, S. C. Mayo, D. E. Myers, Ya. Nesterets, D. M. Paganin, A. Pogany, A. W. Stevenson, and S. W. Wilkins, “Refracting Röntgen’s rays: propagation-based X-ray phase contrast for biomedical imaging,” J. Appl. Phys. 105, 102005 (2009).
[CrossRef]

Nugent, K. A.

Otendal, M.

T. Tuohimaa, M. Otendal, and H. M. Hertz, “Phase-contrast X-ray imaging with a liquid-metal-jet-anode micro-focus source,” App. Phys. Lett. 91, 074104 (2007).
[CrossRef]

Paganin, D.

D. Paganin, S. C. Mayo, T. E. Gureyev, P. R. Wilkins, and S. W. Wilkins, “Simultaneous phase and amplitude extraction from a single defocused image of a homogeneous object,” J. Microsc. 206, 33–40 (2002).
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A. Barty, K. A. Nugent, D. Paganin, and A. Roberts, “Quantitative optical phase microscopy,” Opt. Lett. 23, 817–819 (1998).
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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.

M. A. Beltran, D. M. Paganin, K. Uesugi, and M. J. Kitchen, “2D and 3D X-ray phase retrieval of multi-material objects using a single defocus distance,” Opt. Express 18, 6423–6436 (2010).
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T. E. Gureyev, S. C. Mayo, D. E. Myers, Ya. Nesterets, D. M. Paganin, A. Pogany, A. W. Stevenson, and S. W. Wilkins, “Refracting Röntgen’s rays: propagation-based X-ray phase contrast for biomedical imaging,” J. Appl. Phys. 105, 102005 (2009).
[CrossRef]

G. R. Myers, S. C. Mayo, T. E. Gureyev, D. M. Paganin, and S. W. Wilkins, “Polychromatic cone-beam phase-contrast tomography,” Phys. Rev. A 76, 045804 (2007).
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D. M. Paganin, Coherent X-Ray Optics (Oxford Science Publications, 2006).
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F. Pfeiffer, T. Weitkamp, O. Bunk, and C. David, “Phase retrieval and differential phase-contrast imaging with low-brilliance X-ray sources,” Nat. Phys. 2, 258–261 (2006).
[CrossRef]

Pisano, E.

D. Chapman, W. Thomlinson, R. E. Johnston, D. Washburn, E. Pisano, N. Gmür, Z. Zhong, R. Menk, F. Arfelli, and D. Sayers, “Diffraction enhanced X-ray imaging,” Phys. Med. Biol. 42, 2015–2025 (1997).
[CrossRef] [PubMed]

Pogany, A.

T. E. Gureyev, S. C. Mayo, D. E. Myers, Ya. Nesterets, D. M. Paganin, A. Pogany, A. W. Stevenson, and S. W. Wilkins, “Refracting Röntgen’s rays: propagation-based X-ray phase contrast for biomedical imaging,” J. Appl. Phys. 105, 102005 (2009).
[CrossRef]

T. E. Gureyev, Y. I. Nesterets, A. W. Stevenson, P. R. Miller, A. Pogany, and S. W. Stevenson, “Some simple rules for contrast, signal-to-noise and resolution in in-line phase-contrast imaging,” Opt. Express 16, 3223–3241 (2008).
[CrossRef] [PubMed]

T. E. Gureyev, T. J. Davis, A. Pogany, S. C. Mayo, and S. W. Wilkins, “Optical phase retrieval by use of first Born- and Rytov-type approximations,” Appl. Opt. 43, 2418–2430 (2004).
[CrossRef] [PubMed]

S. W. Wilkins, T. E. Gureyev, D. Gao, A. Pogany, and A. W. Stevenson, “Phase-constrast imaging using poly-chromatic hard X-rays,” Nature 384, 335–338 (1996).
[CrossRef]

Roberts, A.

Sayers, D.

D. Chapman, W. Thomlinson, R. E. Johnston, D. Washburn, E. Pisano, N. Gmür, Z. Zhong, R. Menk, F. Arfelli, and D. Sayers, “Diffraction enhanced X-ray imaging,” Phys. Med. Biol. 42, 2015–2025 (1997).
[CrossRef] [PubMed]

Schelokov, I.

A. Snigirev, I. Snigireva, V. Kohn, S. Kuznetsov, and I. Schelokov, “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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P. Cloetens, R. Barrett, J. Baruchel, J. P. Guigay, and M. Schlenker, “Phase objects in synchrotron radiation hard X-ray imaging,” J. Phys. D: Appl. Phys. 29, 133–146 (1996).
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A. C. Kak and M. Slaney, Principles of Computerized Tomographic Imaging (Siam, 2001).
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A. Snigirev, I. Snigireva, V. Kohn, S. Kuznetsov, and I. Schelokov, “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. Schelokov, “On the possibilities of X-ray phase contrast microimaging by coherent high-energy synchrotron radiation,” Rev. Sci. Instrum. 66, 5486–5492 (1995).
[CrossRef]

Stampanoni, M.

Stevenson, A. W.

T. E. Gureyev, S. C. Mayo, D. E. Myers, Ya. Nesterets, D. M. Paganin, A. Pogany, A. W. Stevenson, and S. W. Wilkins, “Refracting Röntgen’s rays: propagation-based X-ray phase contrast for biomedical imaging,” J. Appl. Phys. 105, 102005 (2009).
[CrossRef]

T. E. Gureyev, Y. I. Nesterets, A. W. Stevenson, P. R. Miller, A. Pogany, and S. W. Stevenson, “Some simple rules for contrast, signal-to-noise and resolution in in-line phase-contrast imaging,” Opt. Express 16, 3223–3241 (2008).
[CrossRef] [PubMed]

S. W. Wilkins, T. E. Gureyev, D. Gao, A. Pogany, and A. W. Stevenson, “Phase-constrast imaging using poly-chromatic hard X-rays,” Nature 384, 335–338 (1996).
[CrossRef]

T. J. Davis, D. Gao, T. E. Gureyev, A. W. Stevenson, and S. W. Wilkins, “Phase-contrast imaging of weakly absorbing materials using hard X-rays,” Nature 373, 595–598 (1995).
[CrossRef]

Stevenson, S. W.

Suetens, P.

See, e.g., P. SuetensFundamentals of Medical Imaging (Cambridge Univ Press, 2009).
[CrossRef]

Teague, M. R.

Thomlinson, W.

D. Chapman, W. Thomlinson, R. E. Johnston, D. Washburn, E. Pisano, N. Gmür, Z. Zhong, R. Menk, F. Arfelli, and D. Sayers, “Diffraction enhanced X-ray imaging,” Phys. Med. Biol. 42, 2015–2025 (1997).
[CrossRef] [PubMed]

Tran, C. Q.

Tuohimaa, T.

T. Tuohimaa, M. Otendal, and H. M. Hertz, “Phase-contrast X-ray imaging with a liquid-metal-jet-anode micro-focus source,” App. Phys. Lett. 91, 074104 (2007).
[CrossRef]

Turner, L. D.

Uesugi, K.

Washburn, D.

D. Chapman, W. Thomlinson, R. E. Johnston, D. Washburn, E. Pisano, N. Gmür, Z. Zhong, R. Menk, F. Arfelli, and D. Sayers, “Diffraction enhanced X-ray imaging,” Phys. Med. Biol. 42, 2015–2025 (1997).
[CrossRef] [PubMed]

Weitkamp, T.

F. Pfeiffer, T. Weitkamp, O. Bunk, and C. David, “Phase retrieval and differential phase-contrast imaging with low-brilliance X-ray sources,” Nat. Phys. 2, 258–261 (2006).
[CrossRef]

Wilkins, P. R.

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

Wilkins, S. W.

T. E. Gureyev, S. C. Mayo, D. E. Myers, Ya. Nesterets, D. M. Paganin, A. Pogany, A. W. Stevenson, and S. W. Wilkins, “Refracting Röntgen’s rays: propagation-based X-ray phase contrast for biomedical imaging,” J. Appl. Phys. 105, 102005 (2009).
[CrossRef]

G. R. Myers, S. C. Mayo, T. E. Gureyev, D. M. Paganin, and S. W. Wilkins, “Polychromatic cone-beam phase-contrast tomography,” Phys. Rev. A 76, 045804 (2007).
[CrossRef]

T. E. Gureyev, T. J. Davis, A. Pogany, S. C. Mayo, and S. W. Wilkins, “Optical phase retrieval by use of first Born- and Rytov-type approximations,” Appl. Opt. 43, 2418–2430 (2004).
[CrossRef] [PubMed]

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

S. W. Wilkins, T. E. Gureyev, D. Gao, A. Pogany, and A. W. Stevenson, “Phase-constrast imaging using poly-chromatic hard X-rays,” Nature 384, 335–338 (1996).
[CrossRef]

T. J. Davis, D. Gao, T. E. Gureyev, A. W. Stevenson, and S. W. Wilkins, “Phase-contrast imaging of weakly absorbing materials using hard X-rays,” Nature 373, 595–598 (1995).
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[CrossRef] [PubMed]

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S. Zabler, P. Cloetens, J.-P. Guigay, and J. Baruchel, “Optimization of phase contrast imaging using hard x-rays,” Rev. Sci. Intstrum. 76, 073705 (2005).
[CrossRef]

Zhong, Z.

D. Chapman, W. Thomlinson, R. E. Johnston, D. Washburn, E. Pisano, N. Gmür, Z. Zhong, R. Menk, F. Arfelli, and D. Sayers, “Diffraction enhanced X-ray imaging,” Phys. Med. Biol. 42, 2015–2025 (1997).
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[CrossRef]

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[CrossRef]

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[CrossRef]

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[CrossRef]

J. Microsc. (1)

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

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P. Cloetens, R. Barrett, J. Baruchel, J. P. Guigay, and M. Schlenker, “Phase objects in synchrotron radiation hard X-ray imaging,” J. Phys. D: Appl. Phys. 29, 133–146 (1996).
[CrossRef]

V. N. Ingal and E. A. Beliaevskaya, “X-ray plane-wave topographyobservation of the phase contrast from a non-crystalline object,” J. Phys. D: Appl. Phys. 28, 2314–2317 (1995).
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Med. Phys. (1)

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]

Nat. Phys. (1)

F. Pfeiffer, T. Weitkamp, O. Bunk, and C. David, “Phase retrieval and differential phase-contrast imaging with low-brilliance X-ray sources,” Nat. Phys. 2, 258–261 (2006).
[CrossRef]

Nature (2)

S. W. Wilkins, T. E. Gureyev, D. Gao, A. Pogany, and A. W. Stevenson, “Phase-constrast imaging using poly-chromatic hard X-rays,” Nature 384, 335–338 (1996).
[CrossRef]

T. J. Davis, D. Gao, T. E. Gureyev, A. W. Stevenson, and S. W. Wilkins, “Phase-contrast imaging of weakly absorbing materials using hard X-rays,” Nature 373, 595–598 (1995).
[CrossRef]

Opt. Commun. (1)

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Optik (1)

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R. A. Lewis, “Medical phase contrast X-ray imaging: current status and future prospects,” Phys. Med. Biol. 49, 3573–3583 (2004).
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D. Chapman, W. Thomlinson, R. E. Johnston, D. Washburn, E. Pisano, N. Gmür, Z. Zhong, R. Menk, F. Arfelli, and D. Sayers, “Diffraction enhanced X-ray imaging,” Phys. Med. Biol. 42, 2015–2025 (1997).
[CrossRef] [PubMed]

Phys. Rev. A (1)

G. R. Myers, S. C. Mayo, T. E. Gureyev, D. M. Paganin, and S. W. Wilkins, “Polychromatic cone-beam phase-contrast tomography,” Phys. Rev. A 76, 045804 (2007).
[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]

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Rev. Sci. Instrum. (1)

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

Rev. Sci. Intstrum. (1)

S. Zabler, P. Cloetens, J.-P. Guigay, and J. Baruchel, “Optimization of phase contrast imaging using hard x-rays,” Rev. Sci. Intstrum. 76, 073705 (2005).
[CrossRef]

Other (6)

A. C. Kak and M. Slaney, Principles of Computerized Tomographic Imaging (Siam, 2001).
[CrossRef]

J. W. Goodman, Introduction to Fourier Optics , 2nd ed. (McGraw-Hill, 1996).

U. Lundström, P. A. C. Takman, L. Scott, H. Brismar, and H. M. Hertz, “Low-dose high-resolution laboratory phase-contrast X-ray imaging,” manuscript in preparation.

D. M. Paganin, Coherent X-Ray Optics (Oxford Science Publications, 2006).
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J. M. Cowley, Diffraction Physics (Elsevier, 1995).

See, e.g., P. SuetensFundamentals of Medical Imaging (Cambridge Univ Press, 2009).
[CrossRef]

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Figures (9)

Fig. 1
Fig. 1

Illustration of in-line phase contrast. Extending the propagation distance after passage through the object lets intensity differences due to refraction develop.

Fig. 2
Fig. 2

(a) Phase contribution of an object consisting of polystyrene cylinders and spheres in air. (b) Simulated phase-contrast image of the object. (c) Phase retrieved from (b), using the single-material phase-retrieval method. Color scale is linear from minimum (black) to maximum (white).

Fig. 3
Fig. 3

The process of phase retrieval followed by tomographic reconstruction.

Fig. 4
Fig. 4

The process of phase retrieval and tomographic reconstruction performed together.

Fig. 5
Fig. 5

Procedure for choosing your phase-retrieval method.

Fig. 6
Fig. 6

Retrieved phase of the object in Fig. 2(a) from simulated noise-free phase-contrast images, for R 1 = 0.6 m, R 2 = 2.4 m, photon energy 15 keV, and simulated detector pixel size 9 μm. The phase is retrieved using (a) the Bronnikov method, (b) the modified Bronnikov method for α = 1.0 · 10−3, (c) the phase-attenuation duality, (d) the single-material method, (e) the Fourier-Born method using γ = 5.0 · 10−4, and (f) the Fourier-Rytov method using γ = 5.0 · 10−4. All color scales are linear ranging from −6 to 13 radians for (a), −2 to 7 radians for (b) and (d)–(f), and −3 to 10 radians for (c). Part (g) shows line profiles, taken along the white line in figures (a)–(f), for all six methods.

Fig. 7
Fig. 7

Retrieved phase of the object in Fig. 2(a) from simulated images (a) with a pixel SNR of 4, under the same conditions as Fig. 6. The phase is retrieved using (b) the Bronnikov method, (c) the modified Bronnikov method for α = 1.0 · 10−3, (d) the single-material method, (e) the Fourier-Born method using γ = 5.0·10−4, and (f) the Fourier-Rytov method using γ = 5.0 · 10−4. For the Fourier methods a Tikhonov’s regularization term η = 10−6 was used. All color scales are linear ranging from 0.3 to 2 in normalized pixel intensity for (a), −14 to 22 radians in (b), and −7 to 11 radians in (c)–(f). Part (g) shows line profiles, taken along the white line in figures (b)–(f), for all five methods.

Fig. 8
Fig. 8

Some special cases of phase retrieval. (a) Retrieved phase of the object in Fig. 2 from a simulated noise-free phase-contrast image at 100 keV, using the phase-attenuation duality method. (b) Simulated noise-free phase-contrast image of an object consisting of a 400 μm square rod of PMMA containing 100 μm diameter spheres of water (upper), teflon (middle), and air (lower). (c) Phase retrieved from the image in (b), assuming the encasing material is PMMA and the material of interest is teflon, using the two-material method. (d) Simulated noise-free phase-contrast image of a 20 μm cylinder at R 1 = 6m and R 2 = 24m. (e) Phase retrieved from (d) using the Fourier-Rytov method (γ = 5.0 · 10−4) with regularization term η = 10−2. (f) Same as (e), except the single-material method is used for phase retrieval.

Fig. 9
Fig. 9

Phase retrieval on experimental data, in this case blood vessels in a rat kidney using CO2 as contrast medium. (a) Phase-contrast image of the blood vessels, taken at source-to-object distance R 1 = 0.6 m and object-to-detector distance R 2 = 2.4 m at a photon energy centered at around 15 keV, using a detector of pixel size 9 μm. (b) Phase retrieved using the Bronnikov method. (c) Phase retrieved using the modified Bronnikov method for α = 2.1 · 10−3. (d) Phase retrieved using the single-material method. (e) Phase retrieved using the Fourier method in the Born approximation, for γ = 1.0 · 10−3 and regularization parameter η = 1 · 10−6. (f) Same as (e), except in the Rytov approximation. All color scales are linear ranging from 0.5 to 1.3 in normalized pixel intensity for (a), −70 to 70 radians in (b), and −7 to 3 radians in (c)–(f).

Tables (3)

Tables Icon

Table 1 Methods of Phase Retrieval Suitable for In-Line Phase-Contrast Tomography, and Their Properties 1

Tables Icon

Table 2 Approximations Made in the Derivation of the Methods of Table 1 1

Tables Icon

Table 3 Least Mean Square Error of Normalized Retrieved Phase Images for Different Noise Levels 1

Equations (7)

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

φ ( r ) = f ( 1 { H p [ g ( I ) ] } )
I ( r ) = I in exp [ d z μ ( r , z ) ]
φ ( r ) = 2 π λ d z δ ( r , z )
f ^ ( w )  = d 2 r f ( r ) exp ( 2 π r w )
f ( r ) = d 2 u f ^ ( w ) exp ( 2 π w r ) .
H wien ( w ) = h p * ( w ) h p ( w ) h p * ( w )  + η
I R 1 ( M r , R 2 ) = 1 M 2 I ( r , R 2 M ) ,

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