Abstract

Phase contrast in the object plane of a phase object is retrieved from intensity contrast at a {\sl single} object-detector distance. Expanding intensity contrast and phase shift in the detector plane in powers of object-detector distance, phase retrieval is performed beyond the solution to the linearized transport-of-intensity equation. The expansion coefficients are determined by the entire paraxial wave equation. The Laplacian of the phase shift in the object plane thus is written as a local expression linear in the intensity contrast and nonlinear in the phase shift in the object plane. A perturbative approach to this expression is proposed and tested with simulated phantom data.

© 2010 Optical Society of America

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    [CrossRef]
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  3. J.-P. Guigay, "The ambiguity function in diffraction and isoplanatic imaging by partially coherent beams," Opt. Commun. 26, 136 (1978).
    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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  10. C. David, B. Nöhammer, H. H. Solak, and E. Ziegler, "Differential x-ray phase contrast imaging using a shearing interferometer," Appl. Phys. Lett. 81, 3287 (2002).
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    [CrossRef]
  16. S. W. Wilkins, T. E. Gureyev, D. Gao, A. Pogany, and A. W. Stevenson, "Phase-contrast imaging using polychromatic hard X-rays," Nature 384, 335 (1996).
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    [CrossRef]
  29. M. Boone, Y. De Witte, M. Dierick, J. Van den Bulcke, J. Vlassenbroeck, and L. Van Hoorebeke, "Practical use of the modified Bronnikov algorithm in micro-CT," Nucl. Instrum. Methods Phys. Res. B 267, 1182 (2009).
    [CrossRef]
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2009 (1)

M. Boone, Y. De Witte, M. Dierick, J. Van den Bulcke, J. Vlassenbroeck, and L. Van Hoorebeke, "Practical use of the modified Bronnikov algorithm in micro-CT," Nucl. Instrum. Methods Phys. Res. B 267, 1182 (2009).
[CrossRef]

2008 (1)

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

2006 (4)

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

T. E. Gureyev, Ya. I. Nesterets, D. M. Paganin, A. Pogany, and S. W. Wilkins, "Linear algorithms for phase retrieval in the Fresnel region. 2. Partially coherent illumination," Opt. Commun. 259, 569 (2006).
[CrossRef]

T. E. Gureyev, D. M. Paganin, G. R. Myers, Ya. I. Nesterets, and S. W. Wilkins, "Phase-and-amplitude computer tomography," Appl. Phys. Lett. 89, 034102 (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 (2006).
[CrossRef]

2004 (3)

L. D. Turner, B. Dhal, J. Hayes, A. Mancuso, K. Nugent, D. Paterson, R. Scholten, C. Tran, and A. Peele, "X-ray phase imaging: Demonstration of extended conditions for homogeneous objects," Opt. Express 12, 2960 (2004).
[CrossRef]

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

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

2002 (2)

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

C. David, B. Nöhammer, H. H. Solak, and E. Ziegler, "Differential x-ray phase contrast imaging using a shearing interferometer," Appl. Phys. Lett. 81, 3287 (2002).
[CrossRef]

1998 (1)

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

1996 (3)

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

V. A. Bushuev, V. N. Ingal, and E. A. Belyaevskaya, "Dynamical Theory of Images Generated by Noncrystalline Objects for the Method of Phase-Dispersive Introscopy," Kristallografiya 41, 808 (1996).

M. Op de Beeck, D. Van Dyck, and W. Coene, "Wave function reconstruction in HRTEM: the parabola method," Ultramicroscopy 64, 167 (1996) and refs. therein.
[CrossRef]

1995 (5)

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

A. Snigirev, I. Snigireva, V. Kohn, S. Kuznetsov, and I. Schelokov, "On the possibilities of xray phase contrast microimaging by coherent high energy synchrotron radiation," Rev. Sci. Instrum. 66(12), 5486 (1995).
[CrossRef]

T. E. Gureyev, A. Roberts, and K. A. Nugent, "Phase retrieval with the transport-of-intensity equation: matrix solution with use of Zernike polynomials," J. Opt. Soc. Am. A 12, 1942 (1995).
[CrossRef]

A. Momose, "Demonstration of phase-contrast X-ray computed tomography using an X-ray interferometer," Nucl. Instrum. Methods Phys. Res. A 352, 622 (1995).
[CrossRef]

F. Beckmann, U. Bonse, F. Busch, O. Günnewig, and T. Biermann, "A novel system for X-ray phase-contrast microtomography," HASYLAB Annual Report II, 691 (1995).

1980 (1)

E. Förster, K. Goetz, and P. Zaumseil, "Double crystal diffractometry for the characterization of targets for laser fusion experiments," Krist. Tech. 1, 937 (1980).
[CrossRef]

1978 (1)

J.-P. Guigay, "The ambiguity function in diffraction and isoplanatic imaging by partially coherent beams," Opt. Commun. 26, 136 (1978).
[CrossRef]

1977 (1)

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

1974 (1)

A. Papoulis, "Ambiguity function in Fourier optics," J. Opt. Soc. Am. 64, 779 (1974).
[CrossRef]

1965 (1)

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

1942 (1)

F. Zernike, "Phase-contrast, a new method for microscopic observation of transparent objects. Part I," Physica 9, 686 (1942).
[CrossRef]

Abela, R.

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

Beckmann, F.

F. Beckmann, U. Bonse, F. Busch, O. Günnewig, and T. Biermann, "A novel system for X-ray phase-contrast microtomography," HASYLAB Annual Report II, 691 (1995).

Belyaevskaya, E. A.

V. A. Bushuev, V. N. Ingal, and E. A. Belyaevskaya, "Dynamical Theory of Images Generated by Noncrystalline Objects for the Method of Phase-Dispersive Introscopy," Kristallografiya 41, 808 (1996).

Biermann, T.

F. Beckmann, U. Bonse, F. Busch, O. Günnewig, and T. Biermann, "A novel system for X-ray phase-contrast microtomography," HASYLAB Annual Report II, 691 (1995).

Bonse, U.

F. Beckmann, U. Bonse, F. Busch, O. Günnewig, and T. Biermann, "A novel system for X-ray phase-contrast microtomography," HASYLAB Annual Report II, 691 (1995).

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

Boone, M.

M. Boone, Y. De Witte, M. Dierick, J. Van den Bulcke, J. Vlassenbroeck, and L. Van Hoorebeke, "Practical use of the modified Bronnikov algorithm in micro-CT," Nucl. Instrum. Methods Phys. Res. B 267, 1182 (2009).
[CrossRef]

Bronnikov, A. V.

A. V. Bronnikov, "Theory of quantitative phase-contrast computed tomography," J. Opt. Soc. Am. A 19, 472 (2002).
[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 (2006).
[CrossRef]

Busch, F.

F. Beckmann, U. Bonse, F. Busch, O. Günnewig, and T. Biermann, "A novel system for X-ray phase-contrast microtomography," HASYLAB Annual Report II, 691 (1995).

Bushuev, V. A.

V. A. Bushuev, V. N. Ingal, and E. A. Belyaevskaya, "Dynamical Theory of Images Generated by Noncrystalline Objects for the Method of Phase-Dispersive Introscopy," Kristallografiya 41, 808 (1996).

Cloetens, P.

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

Coene, W.

M. Op de Beeck, D. Van Dyck, and W. Coene, "Wave function reconstruction in HRTEM: the parabola method," Ultramicroscopy 64, 167 (1996) and refs. therein.
[CrossRef]

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 (2006).
[CrossRef]

C. David, B. Nöhammer, H. H. Solak, and E. Ziegler, "Differential x-ray phase contrast imaging using a shearing interferometer," Appl. Phys. Lett. 81, 3287 (2002).
[CrossRef]

Davis, T. J.

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

De Witte, Y.

M. Boone, Y. De Witte, M. Dierick, J. Van den Bulcke, J. Vlassenbroeck, and L. Van Hoorebeke, "Practical use of the modified Bronnikov algorithm in micro-CT," Nucl. Instrum. Methods Phys. Res. B 267, 1182 (2009).
[CrossRef]

Dhal, B.

L. D. Turner, B. Dhal, J. Hayes, A. Mancuso, K. Nugent, D. Paterson, R. Scholten, C. Tran, and A. Peele, "X-ray phase imaging: Demonstration of extended conditions for homogeneous objects," Opt. Express 12, 2960 (2004).
[CrossRef]

Dierick, M.

M. Boone, Y. De Witte, M. Dierick, J. Van den Bulcke, J. Vlassenbroeck, and L. Van Hoorebeke, "Practical use of the modified Bronnikov algorithm in micro-CT," Nucl. Instrum. Methods Phys. Res. B 267, 1182 (2009).
[CrossRef]

Förster, E.

E. Förster, K. Goetz, and P. Zaumseil, "Double crystal diffractometry for the characterization of targets for laser fusion experiments," Krist. Tech. 1, 937 (1980).
[CrossRef]

Gao, D.

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

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

Goetz, K.

E. Förster, K. Goetz, and P. Zaumseil, "Double crystal diffractometry for the characterization of targets for laser fusion experiments," Krist. Tech. 1, 937 (1980).
[CrossRef]

Groso, A.

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

Guigay, J.-P.

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

J.-P. Guigay, "The ambiguity function in diffraction and isoplanatic imaging by partially coherent beams," Opt. Commun. 26, 136 (1978).
[CrossRef]

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

Günnewig, O.

F. Beckmann, U. Bonse, F. Busch, O. Günnewig, and T. Biermann, "A novel system for X-ray phase-contrast microtomography," HASYLAB Annual Report II, 691 (1995).

Gureyev, T. E.

T. E. Gureyev, D. M. Paganin, G. R. Myers, Ya. I. Nesterets, and S. W. Wilkins, "Phase-and-amplitude computer tomography," Appl. Phys. Lett. 89, 034102 (2006).
[CrossRef]

T. E. Gureyev, Ya. I. Nesterets, D. M. Paganin, A. Pogany, and S. W. Wilkins, "Linear algorithms for phase retrieval in the Fresnel region. 2. Partially coherent illumination," Opt. Commun. 259, 569 (2006).
[CrossRef]

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

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

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

T. E. Gureyev, A. Roberts, and K. A. Nugent, "Phase retrieval with the transport-of-intensity equation: matrix solution with use of Zernike polynomials," J. Opt. Soc. Am. A 12, 1942 (1995).
[CrossRef]

Hart, M.

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

Hayes, J.

L. D. Turner, B. Dhal, J. Hayes, A. Mancuso, K. Nugent, D. Paterson, R. Scholten, C. Tran, and A. Peele, "X-ray phase imaging: Demonstration of extended conditions for homogeneous objects," Opt. Express 12, 2960 (2004).
[CrossRef]

Ingal, V. N.

V. A. Bushuev, V. N. Ingal, and E. A. Belyaevskaya, "Dynamical Theory of Images Generated by Noncrystalline Objects for the Method of Phase-Dispersive Introscopy," Kristallografiya 41, 808 (1996).

Kohn, V.

A. Snigirev, I. Snigireva, V. Kohn, S. Kuznetsov, and I. Schelokov, "On the possibilities of xray phase contrast microimaging by coherent high energy synchrotron radiation," Rev. Sci. Instrum. 66(12), 5486 (1995).
[CrossRef]

Kuznetsov, S.

A. Snigirev, I. Snigireva, V. Kohn, S. Kuznetsov, and I. Schelokov, "On the possibilities of xray phase contrast microimaging by coherent high energy synchrotron radiation," Rev. Sci. Instrum. 66(12), 5486 (1995).
[CrossRef]

Langer, M.

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

Liu, H.

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

Mancuso, A.

L. D. Turner, B. Dhal, J. Hayes, A. Mancuso, K. Nugent, D. Paterson, R. Scholten, C. Tran, and A. Peele, "X-ray phase imaging: Demonstration of extended conditions for homogeneous objects," Opt. Express 12, 2960 (2004).
[CrossRef]

Momose, A.

A. Momose, "Demonstration of phase-contrast X-ray computed tomography using an X-ray interferometer," Nucl. Instrum. Methods Phys. Res. A 352, 622 (1995).
[CrossRef]

Myers, G. R.

T. E. Gureyev, D. M. Paganin, G. R. Myers, Ya. I. Nesterets, and S. W. Wilkins, "Phase-and-amplitude computer tomography," Appl. Phys. Lett. 89, 034102 (2006).
[CrossRef]

Nesterets, Ya. I.

T. E. Gureyev, D. M. Paganin, G. R. Myers, Ya. I. Nesterets, and S. W. Wilkins, "Phase-and-amplitude computer tomography," Appl. Phys. Lett. 89, 034102 (2006).
[CrossRef]

T. E. Gureyev, Ya. I. Nesterets, D. M. Paganin, A. Pogany, and S. W. Wilkins, "Linear algorithms for phase retrieval in the Fresnel region. 2. Partially coherent illumination," Opt. Commun. 259, 569 (2006).
[CrossRef]

Nöhammer, B.

C. David, B. Nöhammer, H. H. Solak, and E. Ziegler, "Differential x-ray phase contrast imaging using a shearing interferometer," Appl. Phys. Lett. 81, 3287 (2002).
[CrossRef]

Nugent, K.

L. D. Turner, B. Dhal, J. Hayes, A. Mancuso, K. Nugent, D. Paterson, R. Scholten, C. Tran, and A. Peele, "X-ray phase imaging: Demonstration of extended conditions for homogeneous objects," Opt. Express 12, 2960 (2004).
[CrossRef]

Nugent, K. A.

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

T. E. Gureyev, A. Roberts, and K. A. Nugent, "Phase retrieval with the transport-of-intensity equation: matrix solution with use of Zernike polynomials," J. Opt. Soc. Am. A 12, 1942 (1995).
[CrossRef]

Op de Beeck, M.

M. Op de Beeck, D. Van Dyck, and W. Coene, "Wave function reconstruction in HRTEM: the parabola method," Ultramicroscopy 64, 167 (1996) and refs. therein.
[CrossRef]

Paganin, D.

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

Paganin, D. M.

T. E. Gureyev, D. M. Paganin, G. R. Myers, Ya. I. Nesterets, and S. W. Wilkins, "Phase-and-amplitude computer tomography," Appl. Phys. Lett. 89, 034102 (2006).
[CrossRef]

T. E. Gureyev, Ya. I. Nesterets, D. M. Paganin, A. Pogany, and S. W. Wilkins, "Linear algorithms for phase retrieval in the Fresnel region. 2. Partially coherent illumination," Opt. Commun. 259, 569 (2006).
[CrossRef]

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

Papoulis, A.

A. Papoulis, "Ambiguity function in Fourier optics," J. Opt. Soc. Am. 64, 779 (1974).
[CrossRef]

Paterson, D.

L. D. Turner, B. Dhal, J. Hayes, A. Mancuso, K. Nugent, D. Paterson, R. Scholten, C. Tran, and A. Peele, "X-ray phase imaging: Demonstration of extended conditions for homogeneous objects," Opt. Express 12, 2960 (2004).
[CrossRef]

Peele, A.

L. D. Turner, B. Dhal, J. Hayes, A. Mancuso, K. Nugent, D. Paterson, R. Scholten, C. Tran, and A. Peele, "X-ray phase imaging: Demonstration of extended conditions for homogeneous objects," Opt. Express 12, 2960 (2004).
[CrossRef]

Peyrin, F.

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

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 (2006).
[CrossRef]

Pogany, A.

T. E. Gureyev, Ya. I. Nesterets, D. M. Paganin, A. Pogany, and S. W. Wilkins, "Linear algorithms for phase retrieval in the Fresnel region. 2. Partially coherent illumination," Opt. Commun. 259, 569 (2006).
[CrossRef]

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

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

Roberts, A.

T. E. Gureyev, A. Roberts, and K. A. Nugent, "Phase retrieval with the transport-of-intensity equation: matrix solution with use of Zernike polynomials," J. Opt. Soc. Am. A 12, 1942 (1995).
[CrossRef]

Schelokov, I.

A. Snigirev, I. Snigireva, V. Kohn, S. Kuznetsov, and I. Schelokov, "On the possibilities of xray phase contrast microimaging by coherent high energy synchrotron radiation," Rev. Sci. Instrum. 66(12), 5486 (1995).
[CrossRef]

Scholten, R.

L. D. Turner, B. Dhal, J. Hayes, A. Mancuso, K. Nugent, D. Paterson, R. Scholten, C. Tran, and A. Peele, "X-ray phase imaging: Demonstration of extended conditions for homogeneous objects," Opt. Express 12, 2960 (2004).
[CrossRef]

Snigirev, A.

A. Snigirev, I. Snigireva, V. Kohn, S. Kuznetsov, and I. Schelokov, "On the possibilities of xray phase contrast microimaging by coherent high energy synchrotron radiation," Rev. Sci. Instrum. 66(12), 5486 (1995).
[CrossRef]

Snigireva, I.

A. Snigirev, I. Snigireva, V. Kohn, S. Kuznetsov, and I. Schelokov, "On the possibilities of xray phase contrast microimaging by coherent high energy synchrotron radiation," Rev. Sci. Instrum. 66(12), 5486 (1995).
[CrossRef]

Solak, H. H.

C. David, B. Nöhammer, H. H. Solak, and E. Ziegler, "Differential x-ray phase contrast imaging using a shearing interferometer," Appl. Phys. Lett. 81, 3287 (2002).
[CrossRef]

Stampanoni, M.

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

Stevenson, A. W.

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

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

Tran, C.

L. D. Turner, B. Dhal, J. Hayes, A. Mancuso, K. Nugent, D. Paterson, R. Scholten, C. Tran, and A. Peele, "X-ray phase imaging: Demonstration of extended conditions for homogeneous objects," Opt. Express 12, 2960 (2004).
[CrossRef]

Turner, L. D.

L. D. Turner, B. Dhal, J. Hayes, A. Mancuso, K. Nugent, D. Paterson, R. Scholten, C. Tran, and A. Peele, "X-ray phase imaging: Demonstration of extended conditions for homogeneous objects," Opt. Express 12, 2960 (2004).
[CrossRef]

Van den Bulcke, J.

M. Boone, Y. De Witte, M. Dierick, J. Van den Bulcke, J. Vlassenbroeck, and L. Van Hoorebeke, "Practical use of the modified Bronnikov algorithm in micro-CT," Nucl. Instrum. Methods Phys. Res. B 267, 1182 (2009).
[CrossRef]

Van Dyck, D.

M. Op de Beeck, D. Van Dyck, and W. Coene, "Wave function reconstruction in HRTEM: the parabola method," Ultramicroscopy 64, 167 (1996) and refs. therein.
[CrossRef]

Van Hoorebeke, L.

M. Boone, Y. De Witte, M. Dierick, J. Van den Bulcke, J. Vlassenbroeck, and L. Van Hoorebeke, "Practical use of the modified Bronnikov algorithm in micro-CT," Nucl. Instrum. Methods Phys. Res. B 267, 1182 (2009).
[CrossRef]

Vlassenbroeck, J.

M. Boone, Y. De Witte, M. Dierick, J. Van den Bulcke, J. Vlassenbroeck, and L. Van Hoorebeke, "Practical use of the modified Bronnikov algorithm in micro-CT," Nucl. Instrum. Methods Phys. Res. B 267, 1182 (2009).
[CrossRef]

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 (2006).
[CrossRef]

Wilkins, S. W.

T. E. Gureyev, D. M. Paganin, G. R. Myers, Ya. I. Nesterets, and S. W. Wilkins, "Phase-and-amplitude computer tomography," Appl. Phys. Lett. 89, 034102 (2006).
[CrossRef]

T. E. Gureyev, Ya. I. Nesterets, D. M. Paganin, A. Pogany, and S. W. Wilkins, "Linear algorithms for phase retrieval in the Fresnel region. 2. Partially coherent illumination," Opt. Commun. 259, 569 (2006).
[CrossRef]

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

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

Wilkins, W.

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

Wu, X.

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

Zaumseil, P.

E. Förster, K. Goetz, and P. Zaumseil, "Double crystal diffractometry for the characterization of targets for laser fusion experiments," Krist. Tech. 1, 937 (1980).
[CrossRef]

Zernike, F.

F. Zernike, "Phase-contrast, a new method for microscopic observation of transparent objects. Part I," Physica 9, 686 (1942).
[CrossRef]

Ziegler, E.

C. David, B. Nöhammer, H. H. Solak, and E. Ziegler, "Differential x-ray phase contrast imaging using a shearing interferometer," Appl. Phys. Lett. 81, 3287 (2002).
[CrossRef]

Appl. Phys. Lett. (3)

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

C. David, B. Nöhammer, H. H. Solak, and E. Ziegler, "Differential x-ray phase contrast imaging using a shearing interferometer," Appl. Phys. Lett. 81, 3287 (2002).
[CrossRef]

T. E. Gureyev, D. M. Paganin, G. R. Myers, Ya. I. Nesterets, and S. W. Wilkins, "Phase-and-amplitude computer tomography," Appl. Phys. Lett. 89, 034102 (2006).
[CrossRef]

HASYLAB Annual Report (1)

F. Beckmann, U. Bonse, F. Busch, O. Günnewig, and T. Biermann, "A novel system for X-ray phase-contrast microtomography," HASYLAB Annual Report II, 691 (1995).

J. Opt. Soc. Am. (1)

A. Papoulis, "Ambiguity function in Fourier optics," J. Opt. Soc. Am. 64, 779 (1974).
[CrossRef]

J. Opt. Soc. Am. A (2)

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

T. E. Gureyev, A. Roberts, and K. A. Nugent, "Phase retrieval with the transport-of-intensity equation: matrix solution with use of Zernike polynomials," J. Opt. Soc. Am. A 12, 1942 (1995).
[CrossRef]

Krist. Tech. (1)

E. Förster, K. Goetz, and P. Zaumseil, "Double crystal diffractometry for the characterization of targets for laser fusion experiments," Krist. Tech. 1, 937 (1980).
[CrossRef]

Kristallografiya (1)

V. A. Bushuev, V. N. Ingal, and E. A. Belyaevskaya, "Dynamical Theory of Images Generated by Noncrystalline Objects for the Method of Phase-Dispersive Introscopy," Kristallografiya 41, 808 (1996).

Med. Phys. (2)

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

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

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 (2006).
[CrossRef]

Nature (2)

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

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

Nucl. Instrum. Methods Phys. Res. A (1)

A. Momose, "Demonstration of phase-contrast X-ray computed tomography using an X-ray interferometer," Nucl. Instrum. Methods Phys. Res. A 352, 622 (1995).
[CrossRef]

Nucl. Instrum. Methods Phys. Res. B (1)

M. Boone, Y. De Witte, M. Dierick, J. Van den Bulcke, J. Vlassenbroeck, and L. Van Hoorebeke, "Practical use of the modified Bronnikov algorithm in micro-CT," Nucl. Instrum. Methods Phys. Res. B 267, 1182 (2009).
[CrossRef]

Opt. Commun. (3)

T. E. Gureyev, Ya. I. Nesterets, D. M. Paganin, A. Pogany, and S. W. Wilkins, "Linear algorithms for phase retrieval in the Fresnel region. 2. Partially coherent illumination," Opt. Commun. 259, 569 (2006).
[CrossRef]

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

J.-P. Guigay, "The ambiguity function in diffraction and isoplanatic imaging by partially coherent beams," Opt. Commun. 26, 136 (1978).
[CrossRef]

Opt. Express (2)

L. D. Turner, B. Dhal, J. Hayes, A. Mancuso, K. Nugent, D. Paterson, R. Scholten, C. Tran, and A. Peele, "X-ray phase imaging: Demonstration of extended conditions for homogeneous objects," Opt. Express 12, 2960 (2004).
[CrossRef]

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

Optik (1)

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

Phys. Rev. Lett. (1)

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

Physica (1)

F. Zernike, "Phase-contrast, a new method for microscopic observation of transparent objects. Part I," Physica 9, 686 (1942).
[CrossRef]

Rev. Sci. Instrum. (1)

A. Snigirev, I. Snigireva, V. Kohn, S. Kuznetsov, and I. Schelokov, "On the possibilities of xray phase contrast microimaging by coherent high energy synchrotron radiation," Rev. Sci. Instrum. 66(12), 5486 (1995).
[CrossRef]

Ultramicroscopy (1)

M. Op de Beeck, D. Van Dyck, and W. Coene, "Wave function reconstruction in HRTEM: the parabola method," Ultramicroscopy 64, 167 (1996) and refs. therein.
[CrossRef]

Other (5)

P. Cloetens, Contribution to Phase Contrast Imaging, Reconstruction and Tomography with Hard Synchrotron Radiation, PhD thesis at Vrije Universiteit Brussel (1999) and references therein.

J.-P. Guigay, R. H. Wade, and C. Delpha, "Optical diffraction of Lorentz microscope images," Proceedings of the 25th meeting of the Electron Microscopy and Analysis Group, W. C. Nixon ed. (The Institute of Physics, London, 1971), pp. 238-239.

M. Langer, Phase Retrieval in the Fresnel Region for Hard X-ray Tomography, PhD thesis at L’Insitut National des Sciences Appliquees de Lyon (2008) and references therein.

D. Paganin, Coherent X-Ray Optics, Oxford University Press (2006).

M. Born, and E. Wolf, Principles of Optics, 6th ed., Pergamon Press, Oxford, New York (1980).

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