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).

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).

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).

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]

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]

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]

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).

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).
[CrossRef] [PubMed]

A. Barty, K. A. Nugent, D. Paganin, and A. Roberts, “Quantitative optical phase microscopy,” Opt. Lett. 23, 817–819 (1998).
[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]

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).
[CrossRef] [PubMed]

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]

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

Paterson, D.

Peele, A. G.

Pfeiffer, F.

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]

Schlenker, M.

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]

Scholten, R. E.

Slaney, M.

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

Snigirev, A.

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).

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).
[CrossRef]

Wu, X.

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]

Zabler, S.

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).
[CrossRef] [PubMed]

Adv. Phys. (1)

K. A. Nugent, “Coherent methods in the X-ray sciences,” Adv. Phys. 59, 1–99 (2010).
[CrossRef]

App. Phys. Lett. (1)

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]

Appl. Opt. (1)

Appl. Phys. Lett. (1)

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

J. Appl. Phys. (1)

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]

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]

J. Opt. (Paris) (1)

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

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (3)

J. Phys. D: Appl. Phys. (2)

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

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)

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

Opt. Express (5)

Opt. Lett. (1)

Optik (1)

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

Phys. Med. Biol. (2)

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

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]

Phys. Today (1)

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

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).

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).

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

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

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