Abstract

The first Born and Rytov approximations of scattering theory are introduced in their less familiar near-field versions. Two algorithms for phase retrieval based on these approximations are then described. It is shown theoretically and by numerical simulations that, despite the differences in their formulation, the two algorithms deliver fairly similar results when used for optical phase retrieval in the near and intermediate fields. The algorithms are applied to derive explicit solutions to four phase-retrieval problems of practical relevance to quantitative phase-contrast imaging and tomography. An example of successful phase reconstruction by use of the Born-type algorithm with an experimental x-ray image is presented.

© 2004 Optical Society of America

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    [CrossRef] [PubMed]
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2002

D. R. Luke, J. V. Burke, R. G. Lyon, “Optical wavefront reconstruction: theory and numerical methods,” SIAM (Soc. Ind. Appl. Math.) Rev. 44, 169–224 (2002).

J. Miao, T. Ishikawa, B. Johnson, E. H. Anderson, B. Lai, K. O. Hodgson, “High resolution 3D x-ray diffraction microscopy,” Phys. Rev. Lett. 89, 088303 (2002).
[CrossRef] [PubMed]

J. C. Spence, U. Weierstall, M. Howells, “Phase recovery and lensless imaging by iterative methods in optical, x-ray and electron diffraction,” Philos. Trans. R. Soc. London Ser. A 360, 875–895 (2002).
[CrossRef]

S. C. Mayo, P. R. Miller, S. W. Wilkins, T. J. Davis, D. Gao, T. E. Gureyev, D. Paganin, D. J. Parry, A. Pogany, A. W. Stevenson, “Quantitative x-ray projection microscopy: phase-contrast and multi-spectral imaging,” J. Microsc. 207, 79–96 (2002).
[CrossRef] [PubMed]

J. Miao, T. Ohsuna, O. Terasaki, K. O. Hodgson, M. A. O’Keefe, “Atomic resolution three-dimensional electron diffraction microscopy,” Phys. Rev. Lett. 89, 155502 (2002).
[CrossRef] [PubMed]

D. Paganin, S. C. Mayo, T. E. Gureyev, P. R. Miller, 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

T. E. Gureyev, S. Mayo, S. W. Wilkins, D. Paganin, A. W. Stevenson, “Quantitative in-line phase-contrast imaging with multi-energy x-rays,” Phys. Rev. Lett. 86, 5827–5830 (2001).
[CrossRef] [PubMed]

I. K. Robinson, I. A. Vartanyants, G. J. Williams, M. A. Pfeifer, J. A. Pitney, “Reconstruction of the shapes of gold nanocrystals using coherent x-ray diffraction,” Phys. Rev. Lett. 87, 195505 (2001).
[CrossRef] [PubMed]

K. A. Nugent, D. Paganin, T. E. Gureyev, “A phase odyssey,” Phys. Today 54, 27–32 (2001).
[CrossRef]

J. C. H. Spence, M. Howells, L. D. Marks, J. Miao, “Lensless imaging: a workshop on ‘new approaches to the phase problem for non-periodic objects,’” Ultramicroscopy 90, 1–6 (2001).
[CrossRef]

2000

S. Bajt, A. Barty, K. A. Nugent, M. McCartney, M. Wall, D. Paganin, “Quantitative phase-sensitive imaging in a transmission electron microscope,” Ultramicroscopy 83, 67–73 (2000).
[CrossRef] [PubMed]

D. T. Miller, “Retinal imaging and vision at the frontiers of adaptive optics,” Phys. Today 53, 31–36 (2000).
[CrossRef]

1999

J. Miao, P. Charalambous, J. Kirz, D. Sayre, “Extending the methodology of x-ray crystallography to allow imaging of micrometer-sized non-crystalline specimens,” Nature 400, 342–344 (1999).
[CrossRef]

T. E. Gureyev, C. Raven, A. Snigirev, I. Snigireva, S. W. Wilkins, “Hard x-ray quantitative non-interferometric phase-contrast microscopy,” J. Physics D 32, 563–567 (1999).
[CrossRef]

P. Cloetens, W. Ludwig, J. Baruchel, D. Van Dyck, J. Van Landuyt, J. P. Guigay, M. Schlenker, “Holotomography: quantitative phase tomography with micrometer resolution using hard synchrotron radiation x rays,” Appl. Phys. Lett. 75, 2912–2914 (1999).
[CrossRef]

1998

1997

A. Pogany, D. Gao, S. W. Wilkins, “Contrast and resolution in imaging with a microfocus x-ray source,” Rev. Sci. Instrum. 68, 2774–2782 (1997).
[CrossRef]

1996

K. A. Nugent, T. E. Gureyev, D. F. Cookson, D. Paganin, Z. Barnea, “Quantitative phase imaging using hard x rays,” Phys. Rev. Lett. 77, 2961–2964 (1996).
[CrossRef] [PubMed]

M. Op de Beeck, D. Van Dyck, W. Coene, “Wave function reconstruction in HRTEM: the parabola method,” Ultramicroscopy 64, 167–183 (1996).
[CrossRef]

1995

T. C. Wedberg, J. J. Stamnes, “Comparison of phase-retrieval methods for optical diffraction tomography,” Pure Appl. Opt. 4, 39–54 (1995).
[CrossRef]

1994

M. H. Maleki, A. J. Devaney, “Noniterative reconstruction of complex-valued objects from two intensity measurements,” Opt. Eng. 33, 3243–3253 (1994).
[CrossRef]

T. J. Davis, “A unified treatment of small-angle x-ray scattering, x-ray refraction and absorption using the Rytov approximation,” Acta Crystallogr. Sect. A 50, 686–690 (1994).
[CrossRef]

1993

1990

1987

D. Van Dyck, W. Coene, “A new procedure for wave function restoration in high resolution electron microscopy,” Optik (Stuttgart) 77, 125–128 (1987).

1983

1982

1977

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

1972

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

1942

F. Zernike, “Phase contrast, a new method for the microscopic observation of transparent objects,” Physica 9, 686–698 (1942).
[CrossRef]

Anderson, E. H.

J. Miao, T. Ishikawa, B. Johnson, E. H. Anderson, B. Lai, K. O. Hodgson, “High resolution 3D x-ray diffraction microscopy,” Phys. Rev. Lett. 89, 088303 (2002).
[CrossRef] [PubMed]

Arsenin, V.

A. N. Tichonov, V. Arsenin, Solutions of Ill-Posed Problems (Wiley, New York, 1977).

Bajt, S.

S. Bajt, A. Barty, K. A. Nugent, M. McCartney, M. Wall, D. Paganin, “Quantitative phase-sensitive imaging in a transmission electron microscope,” Ultramicroscopy 83, 67–73 (2000).
[CrossRef] [PubMed]

Barnea, Z.

K. A. Nugent, T. E. Gureyev, D. F. Cookson, D. Paganin, Z. Barnea, “Quantitative phase imaging using hard x rays,” Phys. Rev. Lett. 77, 2961–2964 (1996).
[CrossRef] [PubMed]

Barty, A.

S. Bajt, A. Barty, K. A. Nugent, M. McCartney, M. Wall, D. Paganin, “Quantitative phase-sensitive imaging in a transmission electron microscope,” Ultramicroscopy 83, 67–73 (2000).
[CrossRef] [PubMed]

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

Baruchel, J.

P. Cloetens, W. Ludwig, J. Baruchel, D. Van Dyck, J. Van Landuyt, J. P. Guigay, M. Schlenker, “Holotomography: quantitative phase tomography with micrometer resolution using hard synchrotron radiation x rays,” Appl. Phys. Lett. 75, 2912–2914 (1999).
[CrossRef]

Bertero, M.

M. Bertero, P. Boccacci, Introduction to Inverse Problems in Imaging (Institute of Physics, Bristol, UK, 1998).
[CrossRef]

Boccacci, P.

M. Bertero, P. Boccacci, Introduction to Inverse Problems in Imaging (Institute of Physics, Bristol, UK, 1998).
[CrossRef]

Born, M.

M. Born, E. Wolf, Principles of Optics, 7th ed. (Cambridge U. Press, Cambridge, UK, 1999).

Burke, J. V.

D. R. Luke, J. V. Burke, R. G. Lyon, “Optical wavefront reconstruction: theory and numerical methods,” SIAM (Soc. Ind. Appl. Math.) Rev. 44, 169–224 (2002).

Charalambous, P.

J. Miao, P. Charalambous, J. Kirz, D. Sayre, “Extending the methodology of x-ray crystallography to allow imaging of micrometer-sized non-crystalline specimens,” Nature 400, 342–344 (1999).
[CrossRef]

Cloetens, P.

P. Cloetens, W. Ludwig, J. Baruchel, D. Van Dyck, J. Van Landuyt, J. P. Guigay, M. Schlenker, “Holotomography: quantitative phase tomography with micrometer resolution using hard synchrotron radiation x rays,” Appl. Phys. Lett. 75, 2912–2914 (1999).
[CrossRef]

Coene, W.

M. Op de Beeck, D. Van Dyck, W. Coene, “Wave function reconstruction in HRTEM: the parabola method,” Ultramicroscopy 64, 167–183 (1996).
[CrossRef]

D. Van Dyck, W. Coene, “A new procedure for wave function restoration in high resolution electron microscopy,” Optik (Stuttgart) 77, 125–128 (1987).

Cookson, D. F.

K. A. Nugent, T. E. Gureyev, D. F. Cookson, D. Paganin, 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, 3rd ed. (Elsevier, Amsterdam, 1995).

Davis, T. J.

S. C. Mayo, P. R. Miller, S. W. Wilkins, T. J. Davis, D. Gao, T. E. Gureyev, D. Paganin, D. J. Parry, A. Pogany, A. W. Stevenson, “Quantitative x-ray projection microscopy: phase-contrast and multi-spectral imaging,” J. Microsc. 207, 79–96 (2002).
[CrossRef] [PubMed]

T. J. Davis, “A unified treatment of small-angle x-ray scattering, x-ray refraction and absorption using the Rytov approximation,” Acta Crystallogr. Sect. A 50, 686–690 (1994).
[CrossRef]

Devaney, A. J.

M. H. Maleki, A. J. Devaney, “Noniterative reconstruction of complex-valued objects from two intensity measurements,” Opt. Eng. 33, 3243–3253 (1994).
[CrossRef]

M. H. Maleki, A. J. Devaney, “Phase-retrieval and intensity-only reconstruction algorithms for optical diffraction tomography,” J. Opt. Soc. Am. A 10, 1086–1092 (1993).
[CrossRef]

Fienup, J. R.

Gao, D.

S. C. Mayo, P. R. Miller, S. W. Wilkins, T. J. Davis, D. Gao, T. E. Gureyev, D. Paganin, D. J. Parry, A. Pogany, A. W. Stevenson, “Quantitative x-ray projection microscopy: phase-contrast and multi-spectral imaging,” J. Microsc. 207, 79–96 (2002).
[CrossRef] [PubMed]

A. Pogany, D. Gao, S. W. Wilkins, “Contrast and resolution in imaging with a microfocus x-ray source,” Rev. Sci. Instrum. 68, 2774–2782 (1997).
[CrossRef]

Gerchberg, R. W.

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

Guigay, J. P.

P. Cloetens, W. Ludwig, J. Baruchel, D. Van Dyck, J. Van Landuyt, J. P. Guigay, M. Schlenker, “Holotomography: quantitative phase tomography with micrometer resolution using hard synchrotron radiation x rays,” Appl. Phys. Lett. 75, 2912–2914 (1999).
[CrossRef]

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

Gureyev, T. E.

S. C. Mayo, P. R. Miller, S. W. Wilkins, T. J. Davis, D. Gao, T. E. Gureyev, D. Paganin, D. J. Parry, A. Pogany, A. W. Stevenson, “Quantitative x-ray projection microscopy: phase-contrast and multi-spectral imaging,” J. Microsc. 207, 79–96 (2002).
[CrossRef] [PubMed]

D. Paganin, S. C. Mayo, T. E. Gureyev, P. R. Miller, 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, D. Paganin, T. E. Gureyev, “A phase odyssey,” Phys. Today 54, 27–32 (2001).
[CrossRef]

T. E. Gureyev, S. Mayo, S. W. Wilkins, D. Paganin, A. W. Stevenson, “Quantitative in-line phase-contrast imaging with multi-energy x-rays,” Phys. Rev. Lett. 86, 5827–5830 (2001).
[CrossRef] [PubMed]

T. E. Gureyev, C. Raven, A. Snigirev, I. Snigireva, S. W. Wilkins, “Hard x-ray quantitative non-interferometric phase-contrast microscopy,” J. Physics D 32, 563–567 (1999).
[CrossRef]

K. A. Nugent, T. E. Gureyev, D. F. Cookson, D. Paganin, Z. Barnea, “Quantitative phase imaging using hard x rays,” Phys. Rev. Lett. 77, 2961–2964 (1996).
[CrossRef] [PubMed]

Hodgson, K. O.

J. Miao, T. Ohsuna, O. Terasaki, K. O. Hodgson, M. A. O’Keefe, “Atomic resolution three-dimensional electron diffraction microscopy,” Phys. Rev. Lett. 89, 155502 (2002).
[CrossRef] [PubMed]

J. Miao, T. Ishikawa, B. Johnson, E. H. Anderson, B. Lai, K. O. Hodgson, “High resolution 3D x-ray diffraction microscopy,” Phys. Rev. Lett. 89, 088303 (2002).
[CrossRef] [PubMed]

Howells, M.

J. C. Spence, U. Weierstall, M. Howells, “Phase recovery and lensless imaging by iterative methods in optical, x-ray and electron diffraction,” Philos. Trans. R. Soc. London Ser. A 360, 875–895 (2002).
[CrossRef]

J. C. H. Spence, M. Howells, L. D. Marks, J. Miao, “Lensless imaging: a workshop on ‘new approaches to the phase problem for non-periodic objects,’” Ultramicroscopy 90, 1–6 (2001).
[CrossRef]

Ishikawa, T.

J. Miao, T. Ishikawa, B. Johnson, E. H. Anderson, B. Lai, K. O. Hodgson, “High resolution 3D x-ray diffraction microscopy,” Phys. Rev. Lett. 89, 088303 (2002).
[CrossRef] [PubMed]

Johnson, B.

J. Miao, T. Ishikawa, B. Johnson, E. H. Anderson, B. Lai, K. O. Hodgson, “High resolution 3D x-ray diffraction microscopy,” Phys. Rev. Lett. 89, 088303 (2002).
[CrossRef] [PubMed]

Kirz, J.

J. Miao, P. Charalambous, J. Kirz, D. Sayre, “Extending the methodology of x-ray crystallography to allow imaging of micrometer-sized non-crystalline specimens,” Nature 400, 342–344 (1999).
[CrossRef]

Lai, B.

J. Miao, T. Ishikawa, B. Johnson, E. H. Anderson, B. Lai, K. O. Hodgson, “High resolution 3D x-ray diffraction microscopy,” Phys. Rev. Lett. 89, 088303 (2002).
[CrossRef] [PubMed]

Ludwig, W.

P. Cloetens, W. Ludwig, J. Baruchel, D. Van Dyck, J. Van Landuyt, J. P. Guigay, M. Schlenker, “Holotomography: quantitative phase tomography with micrometer resolution using hard synchrotron radiation x rays,” Appl. Phys. Lett. 75, 2912–2914 (1999).
[CrossRef]

Luke, D. R.

D. R. Luke, J. V. Burke, R. G. Lyon, “Optical wavefront reconstruction: theory and numerical methods,” SIAM (Soc. Ind. Appl. Math.) Rev. 44, 169–224 (2002).

Lyon, R. G.

D. R. Luke, J. V. Burke, R. G. Lyon, “Optical wavefront reconstruction: theory and numerical methods,” SIAM (Soc. Ind. Appl. Math.) Rev. 44, 169–224 (2002).

Maleki, M. H.

M. H. Maleki, A. J. Devaney, “Noniterative reconstruction of complex-valued objects from two intensity measurements,” Opt. Eng. 33, 3243–3253 (1994).
[CrossRef]

M. H. Maleki, A. J. Devaney, “Phase-retrieval and intensity-only reconstruction algorithms for optical diffraction tomography,” J. Opt. Soc. Am. A 10, 1086–1092 (1993).
[CrossRef]

Marks, L. D.

J. C. H. Spence, M. Howells, L. D. Marks, J. Miao, “Lensless imaging: a workshop on ‘new approaches to the phase problem for non-periodic objects,’” Ultramicroscopy 90, 1–6 (2001).
[CrossRef]

Mayo, S.

T. E. Gureyev, S. Mayo, S. W. Wilkins, D. Paganin, A. W. Stevenson, “Quantitative in-line phase-contrast imaging with multi-energy x-rays,” Phys. Rev. Lett. 86, 5827–5830 (2001).
[CrossRef] [PubMed]

Mayo, S. C.

S. C. Mayo, P. R. Miller, S. W. Wilkins, T. J. Davis, D. Gao, T. E. Gureyev, D. Paganin, D. J. Parry, A. Pogany, A. W. Stevenson, “Quantitative x-ray projection microscopy: phase-contrast and multi-spectral imaging,” J. Microsc. 207, 79–96 (2002).
[CrossRef] [PubMed]

D. Paganin, S. C. Mayo, T. E. Gureyev, P. R. Miller, 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]

McCartney, M.

S. Bajt, A. Barty, K. A. Nugent, M. McCartney, M. Wall, D. Paganin, “Quantitative phase-sensitive imaging in a transmission electron microscope,” Ultramicroscopy 83, 67–73 (2000).
[CrossRef] [PubMed]

Miao, J.

J. Miao, T. Ohsuna, O. Terasaki, K. O. Hodgson, M. A. O’Keefe, “Atomic resolution three-dimensional electron diffraction microscopy,” Phys. Rev. Lett. 89, 155502 (2002).
[CrossRef] [PubMed]

J. Miao, T. Ishikawa, B. Johnson, E. H. Anderson, B. Lai, K. O. Hodgson, “High resolution 3D x-ray diffraction microscopy,” Phys. Rev. Lett. 89, 088303 (2002).
[CrossRef] [PubMed]

J. C. H. Spence, M. Howells, L. D. Marks, J. Miao, “Lensless imaging: a workshop on ‘new approaches to the phase problem for non-periodic objects,’” Ultramicroscopy 90, 1–6 (2001).
[CrossRef]

J. Miao, P. Charalambous, J. Kirz, D. Sayre, “Extending the methodology of x-ray crystallography to allow imaging of micrometer-sized non-crystalline specimens,” Nature 400, 342–344 (1999).
[CrossRef]

Millane, R. P.

Miller, D. T.

D. T. Miller, “Retinal imaging and vision at the frontiers of adaptive optics,” Phys. Today 53, 31–36 (2000).
[CrossRef]

Miller, P. R.

S. C. Mayo, P. R. Miller, S. W. Wilkins, T. J. Davis, D. Gao, T. E. Gureyev, D. Paganin, D. J. Parry, A. Pogany, A. W. Stevenson, “Quantitative x-ray projection microscopy: phase-contrast and multi-spectral imaging,” J. Microsc. 207, 79–96 (2002).
[CrossRef] [PubMed]

D. Paganin, S. C. Mayo, T. E. Gureyev, P. R. Miller, 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]

Nieto-Vesperinas, M.

M. Nieto-Vesperinas, Scattering and Diffraction in Physical Optics (Wiley, New York, 1991).

Nugent, K. A.

K. A. Nugent, D. Paganin, T. E. Gureyev, “A phase odyssey,” Phys. Today 54, 27–32 (2001).
[CrossRef]

S. Bajt, A. Barty, K. A. Nugent, M. McCartney, M. Wall, D. Paganin, “Quantitative phase-sensitive imaging in a transmission electron microscope,” Ultramicroscopy 83, 67–73 (2000).
[CrossRef] [PubMed]

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

K. A. Nugent, T. E. Gureyev, D. F. Cookson, D. Paganin, Z. Barnea, “Quantitative phase imaging using hard x rays,” Phys. Rev. Lett. 77, 2961–2964 (1996).
[CrossRef] [PubMed]

O’Keefe, M. A.

J. Miao, T. Ohsuna, O. Terasaki, K. O. Hodgson, M. A. O’Keefe, “Atomic resolution three-dimensional electron diffraction microscopy,” Phys. Rev. Lett. 89, 155502 (2002).
[CrossRef] [PubMed]

Ohsuna, T.

J. Miao, T. Ohsuna, O. Terasaki, K. O. Hodgson, M. A. O’Keefe, “Atomic resolution three-dimensional electron diffraction microscopy,” Phys. Rev. Lett. 89, 155502 (2002).
[CrossRef] [PubMed]

Op de Beeck, M.

M. Op de Beeck, D. Van Dyck, W. Coene, “Wave function reconstruction in HRTEM: the parabola method,” Ultramicroscopy 64, 167–183 (1996).
[CrossRef]

Paganin, D.

S. C. Mayo, P. R. Miller, S. W. Wilkins, T. J. Davis, D. Gao, T. E. Gureyev, D. Paganin, D. J. Parry, A. Pogany, A. W. Stevenson, “Quantitative x-ray projection microscopy: phase-contrast and multi-spectral imaging,” J. Microsc. 207, 79–96 (2002).
[CrossRef] [PubMed]

D. Paganin, S. C. Mayo, T. E. Gureyev, P. R. Miller, 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, D. Paganin, T. E. Gureyev, “A phase odyssey,” Phys. Today 54, 27–32 (2001).
[CrossRef]

T. E. Gureyev, S. Mayo, S. W. Wilkins, D. Paganin, A. W. Stevenson, “Quantitative in-line phase-contrast imaging with multi-energy x-rays,” Phys. Rev. Lett. 86, 5827–5830 (2001).
[CrossRef] [PubMed]

S. Bajt, A. Barty, K. A. Nugent, M. McCartney, M. Wall, D. Paganin, “Quantitative phase-sensitive imaging in a transmission electron microscope,” Ultramicroscopy 83, 67–73 (2000).
[CrossRef] [PubMed]

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

K. A. Nugent, T. E. Gureyev, D. F. Cookson, D. Paganin, Z. Barnea, “Quantitative phase imaging using hard x rays,” Phys. Rev. Lett. 77, 2961–2964 (1996).
[CrossRef] [PubMed]

Parry, D. J.

S. C. Mayo, P. R. Miller, S. W. Wilkins, T. J. Davis, D. Gao, T. E. Gureyev, D. Paganin, D. J. Parry, A. Pogany, A. W. Stevenson, “Quantitative x-ray projection microscopy: phase-contrast and multi-spectral imaging,” J. Microsc. 207, 79–96 (2002).
[CrossRef] [PubMed]

Pfeifer, M. A.

I. K. Robinson, I. A. Vartanyants, G. J. Williams, M. A. Pfeifer, J. A. Pitney, “Reconstruction of the shapes of gold nanocrystals using coherent x-ray diffraction,” Phys. Rev. Lett. 87, 195505 (2001).
[CrossRef] [PubMed]

Pitney, J. A.

I. K. Robinson, I. A. Vartanyants, G. J. Williams, M. A. Pfeifer, J. A. Pitney, “Reconstruction of the shapes of gold nanocrystals using coherent x-ray diffraction,” Phys. Rev. Lett. 87, 195505 (2001).
[CrossRef] [PubMed]

Pogany, A.

S. C. Mayo, P. R. Miller, S. W. Wilkins, T. J. Davis, D. Gao, T. E. Gureyev, D. Paganin, D. J. Parry, A. Pogany, A. W. Stevenson, “Quantitative x-ray projection microscopy: phase-contrast and multi-spectral imaging,” J. Microsc. 207, 79–96 (2002).
[CrossRef] [PubMed]

A. Pogany, D. Gao, S. W. Wilkins, “Contrast and resolution in imaging with a microfocus x-ray source,” Rev. Sci. Instrum. 68, 2774–2782 (1997).
[CrossRef]

Raven, C.

T. E. Gureyev, C. Raven, A. Snigirev, I. Snigireva, S. W. Wilkins, “Hard x-ray quantitative non-interferometric phase-contrast microscopy,” J. Physics D 32, 563–567 (1999).
[CrossRef]

Roberts, A.

Robinson, I. K.

I. K. Robinson, I. A. Vartanyants, G. J. Williams, M. A. Pfeifer, J. A. Pitney, “Reconstruction of the shapes of gold nanocrystals using coherent x-ray diffraction,” Phys. Rev. Lett. 87, 195505 (2001).
[CrossRef] [PubMed]

Roddier, F.

Saleh, B. E. A.

B. E. A. Saleh, M. C. Teich, Fundamentals of Photonics (Wiley, New York, 1991).
[CrossRef]

Saxton, W. O.

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

Sayre, D.

J. Miao, P. Charalambous, J. Kirz, D. Sayre, “Extending the methodology of x-ray crystallography to allow imaging of micrometer-sized non-crystalline specimens,” Nature 400, 342–344 (1999).
[CrossRef]

Schlenker, M.

P. Cloetens, W. Ludwig, J. Baruchel, D. Van Dyck, J. Van Landuyt, J. P. Guigay, M. Schlenker, “Holotomography: quantitative phase tomography with micrometer resolution using hard synchrotron radiation x rays,” Appl. Phys. Lett. 75, 2912–2914 (1999).
[CrossRef]

Snigirev, A.

T. E. Gureyev, C. Raven, A. Snigirev, I. Snigireva, S. W. Wilkins, “Hard x-ray quantitative non-interferometric phase-contrast microscopy,” J. Physics D 32, 563–567 (1999).
[CrossRef]

Snigireva, I.

T. E. Gureyev, C. Raven, A. Snigirev, I. Snigireva, S. W. Wilkins, “Hard x-ray quantitative non-interferometric phase-contrast microscopy,” J. Physics D 32, 563–567 (1999).
[CrossRef]

Spence, J. C.

J. C. Spence, U. Weierstall, M. Howells, “Phase recovery and lensless imaging by iterative methods in optical, x-ray and electron diffraction,” Philos. Trans. R. Soc. London Ser. A 360, 875–895 (2002).
[CrossRef]

Spence, J. C. H.

J. C. H. Spence, M. Howells, L. D. Marks, J. Miao, “Lensless imaging: a workshop on ‘new approaches to the phase problem for non-periodic objects,’” Ultramicroscopy 90, 1–6 (2001).
[CrossRef]

Stamnes, J. J.

T. C. Wedberg, J. J. Stamnes, “Comparison of phase-retrieval methods for optical diffraction tomography,” Pure Appl. Opt. 4, 39–54 (1995).
[CrossRef]

Stevenson, A. W.

S. C. Mayo, P. R. Miller, S. W. Wilkins, T. J. Davis, D. Gao, T. E. Gureyev, D. Paganin, D. J. Parry, A. Pogany, A. W. Stevenson, “Quantitative x-ray projection microscopy: phase-contrast and multi-spectral imaging,” J. Microsc. 207, 79–96 (2002).
[CrossRef] [PubMed]

T. E. Gureyev, S. Mayo, S. W. Wilkins, D. Paganin, A. W. Stevenson, “Quantitative in-line phase-contrast imaging with multi-energy x-rays,” Phys. Rev. Lett. 86, 5827–5830 (2001).
[CrossRef] [PubMed]

Tatarskii, V. I.

V. I. Tatarskii, Wave Propagation in a Turbulent Medium (Dover, New York, 1967).

Teague, M. R.

Teich, M. C.

B. E. A. Saleh, M. C. Teich, Fundamentals of Photonics (Wiley, New York, 1991).
[CrossRef]

Terasaki, O.

J. Miao, T. Ohsuna, O. Terasaki, K. O. Hodgson, M. A. O’Keefe, “Atomic resolution three-dimensional electron diffraction microscopy,” Phys. Rev. Lett. 89, 155502 (2002).
[CrossRef] [PubMed]

Tichonov, A. N.

A. N. Tichonov, V. Arsenin, Solutions of Ill-Posed Problems (Wiley, New York, 1977).

Van Dyck, D.

P. Cloetens, W. Ludwig, J. Baruchel, D. Van Dyck, J. Van Landuyt, J. P. Guigay, M. Schlenker, “Holotomography: quantitative phase tomography with micrometer resolution using hard synchrotron radiation x rays,” Appl. Phys. Lett. 75, 2912–2914 (1999).
[CrossRef]

M. Op de Beeck, D. Van Dyck, W. Coene, “Wave function reconstruction in HRTEM: the parabola method,” Ultramicroscopy 64, 167–183 (1996).
[CrossRef]

D. Van Dyck, W. Coene, “A new procedure for wave function restoration in high resolution electron microscopy,” Optik (Stuttgart) 77, 125–128 (1987).

Van Landuyt, J.

P. Cloetens, W. Ludwig, J. Baruchel, D. Van Dyck, J. Van Landuyt, J. P. Guigay, M. Schlenker, “Holotomography: quantitative phase tomography with micrometer resolution using hard synchrotron radiation x rays,” Appl. Phys. Lett. 75, 2912–2914 (1999).
[CrossRef]

Vartanyants, I. A.

I. K. Robinson, I. A. Vartanyants, G. J. Williams, M. A. Pfeifer, J. A. Pitney, “Reconstruction of the shapes of gold nanocrystals using coherent x-ray diffraction,” Phys. Rev. Lett. 87, 195505 (2001).
[CrossRef] [PubMed]

Wall, M.

S. Bajt, A. Barty, K. A. Nugent, M. McCartney, M. Wall, D. Paganin, “Quantitative phase-sensitive imaging in a transmission electron microscope,” Ultramicroscopy 83, 67–73 (2000).
[CrossRef] [PubMed]

Wedberg, T. C.

T. C. Wedberg, J. J. Stamnes, “Comparison of phase-retrieval methods for optical diffraction tomography,” Pure Appl. Opt. 4, 39–54 (1995).
[CrossRef]

Weierstall, U.

J. C. Spence, U. Weierstall, M. Howells, “Phase recovery and lensless imaging by iterative methods in optical, x-ray and electron diffraction,” Philos. Trans. R. Soc. London Ser. A 360, 875–895 (2002).
[CrossRef]

Wilkins, S. W.

D. Paganin, S. C. Mayo, T. E. Gureyev, P. R. Miller, 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. C. Mayo, P. R. Miller, S. W. Wilkins, T. J. Davis, D. Gao, T. E. Gureyev, D. Paganin, D. J. Parry, A. Pogany, A. W. Stevenson, “Quantitative x-ray projection microscopy: phase-contrast and multi-spectral imaging,” J. Microsc. 207, 79–96 (2002).
[CrossRef] [PubMed]

T. E. Gureyev, S. Mayo, S. W. Wilkins, D. Paganin, A. W. Stevenson, “Quantitative in-line phase-contrast imaging with multi-energy x-rays,” Phys. Rev. Lett. 86, 5827–5830 (2001).
[CrossRef] [PubMed]

T. E. Gureyev, C. Raven, A. Snigirev, I. Snigireva, S. W. Wilkins, “Hard x-ray quantitative non-interferometric phase-contrast microscopy,” J. Physics D 32, 563–567 (1999).
[CrossRef]

A. Pogany, D. Gao, S. W. Wilkins, “Contrast and resolution in imaging with a microfocus x-ray source,” Rev. Sci. Instrum. 68, 2774–2782 (1997).
[CrossRef]

Williams, G. J.

I. K. Robinson, I. A. Vartanyants, G. J. Williams, M. A. Pfeifer, J. A. Pitney, “Reconstruction of the shapes of gold nanocrystals using coherent x-ray diffraction,” Phys. Rev. Lett. 87, 195505 (2001).
[CrossRef] [PubMed]

Wolf, E.

M. Born, E. Wolf, Principles of Optics, 7th ed. (Cambridge U. Press, Cambridge, UK, 1999).

Zernike, F.

F. Zernike, “Phase contrast, a new method for the microscopic observation of transparent objects,” Physica 9, 686–698 (1942).
[CrossRef]

Acta Crystallogr. Sect. A

T. J. Davis, “A unified treatment of small-angle x-ray scattering, x-ray refraction and absorption using the Rytov approximation,” Acta Crystallogr. Sect. A 50, 686–690 (1994).
[CrossRef]

Appl. Opt.

Appl. Phys. Lett.

P. Cloetens, W. Ludwig, J. Baruchel, D. Van Dyck, J. Van Landuyt, J. P. Guigay, M. Schlenker, “Holotomography: quantitative phase tomography with micrometer resolution using hard synchrotron radiation x rays,” Appl. Phys. Lett. 75, 2912–2914 (1999).
[CrossRef]

J. Microsc.

S. C. Mayo, P. R. Miller, S. W. Wilkins, T. J. Davis, D. Gao, T. E. Gureyev, D. Paganin, D. J. Parry, A. Pogany, A. W. Stevenson, “Quantitative x-ray projection microscopy: phase-contrast and multi-spectral imaging,” J. Microsc. 207, 79–96 (2002).
[CrossRef] [PubMed]

D. Paganin, S. C. Mayo, T. E. Gureyev, P. R. Miller, 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. Soc. Am.

J. Opt. Soc. Am. A

J. Physics D

T. E. Gureyev, C. Raven, A. Snigirev, I. Snigireva, S. W. Wilkins, “Hard x-ray quantitative non-interferometric phase-contrast microscopy,” J. Physics D 32, 563–567 (1999).
[CrossRef]

Nature

J. Miao, P. Charalambous, J. Kirz, D. Sayre, “Extending the methodology of x-ray crystallography to allow imaging of micrometer-sized non-crystalline specimens,” Nature 400, 342–344 (1999).
[CrossRef]

Opt. Eng.

M. H. Maleki, A. J. Devaney, “Noniterative reconstruction of complex-valued objects from two intensity measurements,” Opt. Eng. 33, 3243–3253 (1994).
[CrossRef]

Opt. Lett.

Optik (Stuttgart)

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

D. Van Dyck, W. Coene, “A new procedure for wave function restoration in high resolution electron microscopy,” Optik (Stuttgart) 77, 125–128 (1987).

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

Philos. Trans. R. Soc. London Ser. A

J. C. Spence, U. Weierstall, M. Howells, “Phase recovery and lensless imaging by iterative methods in optical, x-ray and electron diffraction,” Philos. Trans. R. Soc. London Ser. A 360, 875–895 (2002).
[CrossRef]

Phys. Rev. Lett.

I. K. Robinson, I. A. Vartanyants, G. J. Williams, M. A. Pfeifer, J. A. Pitney, “Reconstruction of the shapes of gold nanocrystals using coherent x-ray diffraction,” Phys. Rev. Lett. 87, 195505 (2001).
[CrossRef] [PubMed]

J. Miao, T. Ohsuna, O. Terasaki, K. O. Hodgson, M. A. O’Keefe, “Atomic resolution three-dimensional electron diffraction microscopy,” Phys. Rev. Lett. 89, 155502 (2002).
[CrossRef] [PubMed]

T. E. Gureyev, S. Mayo, S. W. Wilkins, D. Paganin, A. W. Stevenson, “Quantitative in-line phase-contrast imaging with multi-energy x-rays,” Phys. Rev. Lett. 86, 5827–5830 (2001).
[CrossRef] [PubMed]

K. A. Nugent, T. E. Gureyev, D. F. Cookson, D. Paganin, Z. Barnea, “Quantitative phase imaging using hard x rays,” Phys. Rev. Lett. 77, 2961–2964 (1996).
[CrossRef] [PubMed]

J. Miao, T. Ishikawa, B. Johnson, E. H. Anderson, B. Lai, K. O. Hodgson, “High resolution 3D x-ray diffraction microscopy,” Phys. Rev. Lett. 89, 088303 (2002).
[CrossRef] [PubMed]

Phys. Today

K. A. Nugent, D. Paganin, T. E. Gureyev, “A phase odyssey,” Phys. Today 54, 27–32 (2001).
[CrossRef]

D. T. Miller, “Retinal imaging and vision at the frontiers of adaptive optics,” Phys. Today 53, 31–36 (2000).
[CrossRef]

Physica

F. Zernike, “Phase contrast, a new method for the microscopic observation of transparent objects,” Physica 9, 686–698 (1942).
[CrossRef]

Pure Appl. Opt.

T. C. Wedberg, J. J. Stamnes, “Comparison of phase-retrieval methods for optical diffraction tomography,” Pure Appl. Opt. 4, 39–54 (1995).
[CrossRef]

Rev. Sci. Instrum.

A. Pogany, D. Gao, S. W. Wilkins, “Contrast and resolution in imaging with a microfocus x-ray source,” Rev. Sci. Instrum. 68, 2774–2782 (1997).
[CrossRef]

SIAM (Soc. Ind. Appl. Math.) Rev.

D. R. Luke, J. V. Burke, R. G. Lyon, “Optical wavefront reconstruction: theory and numerical methods,” SIAM (Soc. Ind. Appl. Math.) Rev. 44, 169–224 (2002).

Ultramicroscopy

S. Bajt, A. Barty, K. A. Nugent, M. McCartney, M. Wall, D. Paganin, “Quantitative phase-sensitive imaging in a transmission electron microscope,” Ultramicroscopy 83, 67–73 (2000).
[CrossRef] [PubMed]

J. C. H. Spence, M. Howells, L. D. Marks, J. Miao, “Lensless imaging: a workshop on ‘new approaches to the phase problem for non-periodic objects,’” Ultramicroscopy 90, 1–6 (2001).
[CrossRef]

M. Op de Beeck, D. Van Dyck, W. Coene, “Wave function reconstruction in HRTEM: the parabola method,” Ultramicroscopy 64, 167–183 (1996).
[CrossRef]

Other

M. Nieto-Vesperinas, Scattering and Diffraction in Physical Optics (Wiley, New York, 1991).

V. I. Tatarskii, Wave Propagation in a Turbulent Medium (Dover, New York, 1967).

A. N. Tichonov, V. Arsenin, Solutions of Ill-Posed Problems (Wiley, New York, 1977).

M. Bertero, P. Boccacci, Introduction to Inverse Problems in Imaging (Institute of Physics, Bristol, UK, 1998).
[CrossRef]

B. E. A. Saleh, M. C. Teich, Fundamentals of Photonics (Wiley, New York, 1991).
[CrossRef]

M. Born, E. Wolf, Principles of Optics, 7th ed. (Cambridge U. Press, Cambridge, UK, 1999).

J. M. Cowley, Diffraction Physics, 3rd ed. (Elsevier, Amsterdam, 1995).

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Figures (12)

Fig. 1
Fig. 1

Imaging geometry of the phase-retrieval methods.

Fig. 2
Fig. 2

Simulated phase distribution in the object plane used in the tests of problem 1.

Fig. 3
Fig. 3

Image at the propagation distance z = 1 m corresponding to the initial phase from Fig. 2 and uniform initial intensity.

Fig. 4
Fig. 4

Phase with the maximum original variation of 0.01 rad retrieved by the first Born method from the image data from Fig. 3.

Fig. 5
Fig. 5

Phase with the maximum original variation of 1 rad retrieved by the first Born method from the image data from Fig. 3.

Fig. 6
Fig. 6

Simulated phase distribution in the object plane used in the tests of problems 2–4.

Fig. 7
Fig. 7

Image at the propagation distance z = 1 m corresponding to problem 2.

Fig. 8
Fig. 8

Object-plane phase distribution corresponding to the maximum variation of 0.1 rad in the original phase data and reconstructed by use of the Rytov algorithm with α = 0.001 from the image data from Fig. 7.

Fig. 9
Fig. 9

Object-plane intensity distribution corresponding to the maximum object-plane absorption of 2% reconstructed by use of the Born solution for problem 3.

Fig. 10
Fig. 10

Phase reconstructed from the image data corresponding to problem 2 with 1% noise by use of the Born method.

Fig. 11
Fig. 11

(a) X-ray in-line image of a cluster of 1-μm latex spheres and (b) cross section through the region of (a) marked by the white rectangle.

Fig. 12
Fig. 12

(a) Object-plane phase distribution reconstructed by use of the Born method from the image data of Fig. 11(a) and (b) cross section through the region of (a) marked by the white rectangle.

Tables (6)

Tables Icon

Table 1 Relative Error E in the Reconstructed Phase by Use of Eq. (28) as a Function of the Phase Magnitude and Regularization Parametera

Tables Icon

Table 2 Relative Error E in the Reconstructed Phase by Use of the Regularized Version of Eq. (29) as a Function of the Phase Magnitude and Regularization Parametera

Tables Icon

Table 3 Relative Error E in the Reconstructed Phase by Use of the Regularized Version of Eq. (30) as a Function of the Phase Magnitude and Regularization Parametera

Tables Icon

Table 4 Relative Error E in the Reconstructed Phase by Use of the Regularized Version of Eq. (31) as a Function of the Phase Magnitude and Regularization Parametera

Tables Icon

Table 5 Relative Error E in the Reconstructed Phase by Use of Eq. (2) as a Function of Defocus Distance

Tables Icon

Table 6 Relative Error E in the Reconstructed Phase by Use of the Regularized Version of Eq. (27) as a Function of Noise Level

Equations (35)

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

2U+k2n2U=0,
2U+k¯2U=-2k¯2δn/nU,
Ux, y, z=Ain expik¯z+ik¯-kH×1+ψzx, y, |ψzx, y|  1,
ψzx, y= aξ, ηexpixξ+yη+z+Hζ-k¯dξdη,
aξ, η=ik¯28π2ζ  2δn/n¯exp-ixξ+yη+z+Hζ-k¯dxdydz.
|ζ-k¯H|  1,
aξ, ηik8π2   2δnx, y, zdz×exp-ixξ+yηdxdy.
Ux, y, 0Ain expik¯-kH×1+ik  δnx, y, zdz.
 ψ0x, ydxdy=a0, 0=ik8π2  δnx, y, zdxdydz=0.
Ux, y, zAin expik¯z+ik¯-kHexp ψzx, y,
2ik¯zψz+2ψz+|ψz|2+2k¯2δn/n¯=0.
2ik¯zψz+2ψz=-2k¯2δn/n¯.
Ux, y, 0Ain expik¯-kH×expik  δnx, y, zdz.
Ux, y, z=expikzUzx, y,
2ikzUzx, y=-2Uzx, y,
Uzx, y=iλz-1  expiπ/λzx-x2+y-y2U0x, ydxdy.
Uzx, y=A1+ψzx, y,
Izx, yIout1+2 Re ψzx, y,
Re ψzx, y=1/λz  sinπ/λzx-x2+y-y2Re ψ0x, y+cosπ/λzx-x2+y-y2Im ψ0x, ydxdy.
 expi2πxξ+yηexpiπ/λzx2+y2dxdy=iλz expiπλzξ2+η2,
Iz/Iout-1/2ξ, η=cosπλzξ2+η2 ×Re ψ0ξ, η+sinπλzξ2+η2Im ψ0ξ, η,
Uzx, y=A exp ψzx, y,
Izx, y=Iout exp2 Re ψzx, y.
2ikzψzx, y=-2ψzx, y-|ψzx, y|2.
|ψzx, y|2  |2ψzx, y|,
lnIz/Iout/2ξ, η=cosπλzξ2+η2 ×Re ψ0ξ, η+sinπλzξ2+η2Im ψ0ξ, η,
Fˆzξ, η=Iz/Iout-1/2ξ, η
Fˆzξ, η=lnIz/Iout/2ξ, η
Im ψ0ξ, η=Fˆzξ, η/sinzρ2,
Im ψ0ξ, η=Fˆzξ, ηsinzρ2/sin2zρ2+α,
Re ψ0ξ, η=Fˆzξ, η/1+γ2 coszρ2-arctan γ.
Re ψ0ξ, ηIm ψ0ξ, η=1sinz2-z1ρ2×sinz2ρ2-sinz1ρ2-cosz2ρ2cosz1ρ2×Fˆz1ξ, ηFˆz2ξ, η,
Re ψ0ξ, η; λ1Im ψ0ξ, η; λ1=1det M ×σ sinzσρ12-sinzρ12-σ3 coszσρ12coszρ12 ×Fˆzξ, η; λ1Fˆzξ, η; λ2,
Ux, y, 0=Uinx, y, 0Qav1+ψ0x, y.
E=100%i,j|ϕijrec-ϕijtrue|2/i,j|ϕijtrue|21/2,

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