Abstract

Phase-contrast x-ray computed tomography (CT) is an emerging imaging technique that can be implemented at third-generation synchrotron radiation sources or by using a microfocus x-ray source. Promising results have recently been obtained in materials science and medicine. At the same time, the lack of a mathematical theory comparable with that of conventional CT limits the progress in this field. Such a theory is now suggested, establishing a fundamental relation between the three-dimensional Radon transform of the object function and the two-dimensional Radon transform of the phase-contrast projection. A reconstruction algorithm is derived in the form of a filtered backprojection. The filter function is given in the space and spatial-frequency domains. The theory suggested enables one to quantitatively determine the refractive index of a weakly absorbing medium from x-ray intensity data measured in the near-field region. The results of computer simulations are discussed.

© 2002 Optical Society of America

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  1. D. Gabor, “A new microscopic principle,” Nature (London) 161, 777–778 (1948).
    [CrossRef]
  2. A. Snigirev, I. Snigireva, V. Kohn, S. Kuznetsov, 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]
  3. P. Cloetens, M. Pateyron-Salome, J. Y. Buffiere, G. Peix, J. Baruchel, F. Peyrin, M. Schlenker, “Observation of microstructure and damage in materials by phase sensitive radiography and tomography,” J. Appl. Phys. 81, 5878–5886 (1997).
    [CrossRef]
  4. T. Davis, D. Gao, T. Gureyev, A. Stevenson, S. Wilkins, “Phase contrast imaging of weakly absorbing materials using hard x-rays,” Nature (London) 373, 595–598 (1995).
    [CrossRef]
  5. S. W. Wilkins, T. E. Gureyev, D. Gao, A. Pogany, A. W. Stevenson, “Phase-contrast imaging using polychromatic hard x-rays,” Nature (London) 384, 335–338 (1996).
    [CrossRef]
  6. D. Chapman, W. Thomlinson, R. E. Johnston, D. Washburn, E. Pisano, N. Gmur, Z. Zhong, R. Menk, F. Ardeli, D. Sayers, “Diffraction enhanced x-ray imaging,” Phys. Med. Biol. 42, 2015–2025 (1997).
    [CrossRef] [PubMed]
  7. D. Chapman, W. Thomlinson, Z. Zhong, R. E. Johnston, E. Pisano, D. Washburn, D. Sayers, C. Serge, “Diffraction enhanced imaging applied to materials science and medicine,” Synchr. Radiat. News, 11, 4–11 (1998).
    [CrossRef]
  8. D. Chapman, E. Pisano, W. Thomlinson, Z. Zhong, R. E. Johnston, D. Washburn, D. Sayers, K. Malinowska, “Medical applications of diffraction enhanced imaging,” Breast Dis. 10, 197–201 (1998).
  9. P. Spanne, C. Raven, I. Snigireva, A. Snigirev, “In-line holography and phase-contrast microtomography with high energy x-rays,” Phys. Med. Biol. 44, 741–750 (1999).
    [CrossRef] [PubMed]
  10. P. Cloetens, W. Ludwig, J. Baruchel, D. Van Dyke, 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]
  11. T. E. Gureyev, C. Raven, A. Snigirev, I. Snigireva, S. W. Wilkins, “Hard x-ray quantitative non-interferometric phase-contrast microscopy,” J. Phys. D 32, 563–567 (1999).
    [CrossRef]
  12. T. E. Gureyev, K. A. Nugent, “Phase retrieval with the transport-of-intensity equation. II. Orthogonal series solution for nonuniform illumination,” J. Opt. Soc. Am. A 13, 1670–1682 (1996).
    [CrossRef]
  13. 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]
  14. T. E. Gureyev, S. W. Wilkins, “On x-ray phase imaging with a point source,” J. Opt. Soc. Am. A 15, 579–585 (1998).
    [CrossRef]
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    [CrossRef]
  16. J. R. Fienup, “Phase retrieval algorithms: a comparison,” Appl. Opt. 21, 2758–2769 (1982).
    [CrossRef] [PubMed]
  17. A. V. Bronnikov, B. O. Maier, N. G. Preobrazhenskii, “Discrete Wigner transform in the phase problem of optics,” Opt. Spectrosc. 70, 512–516 (1991).
  18. A. V. Bronnikov, “Reconstruction formulas in phase-contrast tomography,” Opt. Commun. 171, 239–244 (1999).
    [CrossRef]
  19. J. M. Cowley, Diffraction Physics (North-Holland, Amsterdam, 1981).
  20. A. C. Kak, M. Slaney, Principles of Computerized Tomographic Imaging (Institute of Electrical and Electronics Engineers, New York1988).
  21. A. V. Bronnikov, “Cone-beam reconstruction by backprojection and filtering,” J. Opt. Soc. Am. A 17, 1993–2000 (2000).
    [CrossRef]

2000 (2)

A. Barty, K. A. Nugent, A. Roberts, D. Paganin, “Quantitative phase tomography,” Opt. Commun. 175, 329–336 (2000).
[CrossRef]

A. V. Bronnikov, “Cone-beam reconstruction by backprojection and filtering,” J. Opt. Soc. Am. A 17, 1993–2000 (2000).
[CrossRef]

1999 (4)

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

P. Spanne, C. Raven, I. Snigireva, A. Snigirev, “In-line holography and phase-contrast microtomography with high energy x-rays,” Phys. Med. Biol. 44, 741–750 (1999).
[CrossRef] [PubMed]

P. Cloetens, W. Ludwig, J. Baruchel, D. Van Dyke, 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]

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

1998 (3)

T. E. Gureyev, S. W. Wilkins, “On x-ray phase imaging with a point source,” J. Opt. Soc. Am. A 15, 579–585 (1998).
[CrossRef]

D. Chapman, W. Thomlinson, Z. Zhong, R. E. Johnston, E. Pisano, D. Washburn, D. Sayers, C. Serge, “Diffraction enhanced imaging applied to materials science and medicine,” Synchr. Radiat. News, 11, 4–11 (1998).
[CrossRef]

D. Chapman, E. Pisano, W. Thomlinson, Z. Zhong, R. E. Johnston, D. Washburn, D. Sayers, K. Malinowska, “Medical applications of diffraction enhanced imaging,” Breast Dis. 10, 197–201 (1998).

1997 (2)

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

P. Cloetens, M. Pateyron-Salome, J. Y. Buffiere, G. Peix, J. Baruchel, F. Peyrin, M. Schlenker, “Observation of microstructure and damage in materials by phase sensitive radiography and tomography,” J. Appl. Phys. 81, 5878–5886 (1997).
[CrossRef]

1996 (3)

S. W. Wilkins, T. E. Gureyev, D. Gao, A. Pogany, A. W. Stevenson, “Phase-contrast imaging using polychromatic hard x-rays,” Nature (London) 384, 335–338 (1996).
[CrossRef]

T. E. Gureyev, K. A. Nugent, “Phase retrieval with the transport-of-intensity equation. II. Orthogonal series solution for nonuniform illumination,” J. Opt. Soc. Am. A 13, 1670–1682 (1996).
[CrossRef]

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]

1995 (2)

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

A. Snigirev, I. Snigireva, V. Kohn, S. Kuznetsov, 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]

1991 (1)

A. V. Bronnikov, B. O. Maier, N. G. Preobrazhenskii, “Discrete Wigner transform in the phase problem of optics,” Opt. Spectrosc. 70, 512–516 (1991).

1982 (1)

1948 (1)

D. Gabor, “A new microscopic principle,” Nature (London) 161, 777–778 (1948).
[CrossRef]

Ardeli, F.

D. Chapman, W. Thomlinson, R. E. Johnston, D. Washburn, E. Pisano, N. Gmur, Z. Zhong, R. Menk, F. Ardeli, 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, Z. Barnea, “Quantitative phase imaging using hard x rays,” Phys. Rev. Lett. 77, 2961–2964 (1996).
[CrossRef] [PubMed]

Barty, A.

A. Barty, K. A. Nugent, A. Roberts, D. Paganin, “Quantitative phase tomography,” Opt. Commun. 175, 329–336 (2000).
[CrossRef]

Baruchel, J.

P. Cloetens, W. Ludwig, J. Baruchel, D. Van Dyke, 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]

P. Cloetens, M. Pateyron-Salome, J. Y. Buffiere, G. Peix, J. Baruchel, F. Peyrin, M. Schlenker, “Observation of microstructure and damage in materials by phase sensitive radiography and tomography,” J. Appl. Phys. 81, 5878–5886 (1997).
[CrossRef]

Bronnikov, A. V.

A. V. Bronnikov, “Cone-beam reconstruction by backprojection and filtering,” J. Opt. Soc. Am. A 17, 1993–2000 (2000).
[CrossRef]

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

A. V. Bronnikov, B. O. Maier, N. G. Preobrazhenskii, “Discrete Wigner transform in the phase problem of optics,” Opt. Spectrosc. 70, 512–516 (1991).

Buffiere, J. Y.

P. Cloetens, M. Pateyron-Salome, J. Y. Buffiere, G. Peix, J. Baruchel, F. Peyrin, M. Schlenker, “Observation of microstructure and damage in materials by phase sensitive radiography and tomography,” J. Appl. Phys. 81, 5878–5886 (1997).
[CrossRef]

Chapman, D.

D. Chapman, W. Thomlinson, Z. Zhong, R. E. Johnston, E. Pisano, D. Washburn, D. Sayers, C. Serge, “Diffraction enhanced imaging applied to materials science and medicine,” Synchr. Radiat. News, 11, 4–11 (1998).
[CrossRef]

D. Chapman, E. Pisano, W. Thomlinson, Z. Zhong, R. E. Johnston, D. Washburn, D. Sayers, K. Malinowska, “Medical applications of diffraction enhanced imaging,” Breast Dis. 10, 197–201 (1998).

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

Cloetens, P.

P. Cloetens, W. Ludwig, J. Baruchel, D. Van Dyke, 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]

P. Cloetens, M. Pateyron-Salome, J. Y. Buffiere, G. Peix, J. Baruchel, F. Peyrin, M. Schlenker, “Observation of microstructure and damage in materials by phase sensitive radiography and tomography,” J. Appl. Phys. 81, 5878–5886 (1997).
[CrossRef]

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 (North-Holland, Amsterdam, 1981).

Davis, T.

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

Fienup, J. R.

Gabor, D.

D. Gabor, “A new microscopic principle,” Nature (London) 161, 777–778 (1948).
[CrossRef]

Gao, D.

S. W. Wilkins, T. E. Gureyev, D. Gao, A. Pogany, A. W. Stevenson, “Phase-contrast imaging using polychromatic hard x-rays,” Nature (London) 384, 335–338 (1996).
[CrossRef]

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

Gmur, N.

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

Guigay, J. P.

P. Cloetens, W. Ludwig, J. Baruchel, D. Van Dyke, 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]

Gureyev, T.

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

Gureyev, T. E.

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

T. E. Gureyev, S. W. Wilkins, “On x-ray phase imaging with a point source,” J. Opt. Soc. Am. A 15, 579–585 (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]

T. E. Gureyev, K. A. Nugent, “Phase retrieval with the transport-of-intensity equation. II. Orthogonal series solution for nonuniform illumination,” J. Opt. Soc. Am. A 13, 1670–1682 (1996).
[CrossRef]

S. W. Wilkins, T. E. Gureyev, D. Gao, A. Pogany, A. W. Stevenson, “Phase-contrast imaging using polychromatic hard x-rays,” Nature (London) 384, 335–338 (1996).
[CrossRef]

Johnston, R. E.

D. Chapman, W. Thomlinson, Z. Zhong, R. E. Johnston, E. Pisano, D. Washburn, D. Sayers, C. Serge, “Diffraction enhanced imaging applied to materials science and medicine,” Synchr. Radiat. News, 11, 4–11 (1998).
[CrossRef]

D. Chapman, E. Pisano, W. Thomlinson, Z. Zhong, R. E. Johnston, D. Washburn, D. Sayers, K. Malinowska, “Medical applications of diffraction enhanced imaging,” Breast Dis. 10, 197–201 (1998).

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

Kak, A. C.

A. C. Kak, M. Slaney, Principles of Computerized Tomographic Imaging (Institute of Electrical and Electronics Engineers, New York1988).

Kohn, V.

A. Snigirev, I. Snigireva, V. Kohn, S. Kuznetsov, 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, 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]

Ludwig, W.

P. Cloetens, W. Ludwig, J. Baruchel, D. Van Dyke, 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]

Maier, B. O.

A. V. Bronnikov, B. O. Maier, N. G. Preobrazhenskii, “Discrete Wigner transform in the phase problem of optics,” Opt. Spectrosc. 70, 512–516 (1991).

Malinowska, K.

D. Chapman, E. Pisano, W. Thomlinson, Z. Zhong, R. E. Johnston, D. Washburn, D. Sayers, K. Malinowska, “Medical applications of diffraction enhanced imaging,” Breast Dis. 10, 197–201 (1998).

Menk, R.

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

Nugent, K. A.

A. Barty, K. A. Nugent, A. Roberts, D. Paganin, “Quantitative phase tomography,” Opt. Commun. 175, 329–336 (2000).
[CrossRef]

T. E. Gureyev, K. A. Nugent, “Phase retrieval with the transport-of-intensity equation. II. Orthogonal series solution for nonuniform illumination,” J. Opt. Soc. Am. A 13, 1670–1682 (1996).
[CrossRef]

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]

Paganin, D.

A. Barty, K. A. Nugent, A. Roberts, D. Paganin, “Quantitative phase tomography,” Opt. Commun. 175, 329–336 (2000).
[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]

Pateyron-Salome, M.

P. Cloetens, M. Pateyron-Salome, J. Y. Buffiere, G. Peix, J. Baruchel, F. Peyrin, M. Schlenker, “Observation of microstructure and damage in materials by phase sensitive radiography and tomography,” J. Appl. Phys. 81, 5878–5886 (1997).
[CrossRef]

Peix, G.

P. Cloetens, M. Pateyron-Salome, J. Y. Buffiere, G. Peix, J. Baruchel, F. Peyrin, M. Schlenker, “Observation of microstructure and damage in materials by phase sensitive radiography and tomography,” J. Appl. Phys. 81, 5878–5886 (1997).
[CrossRef]

Peyrin, F.

P. Cloetens, M. Pateyron-Salome, J. Y. Buffiere, G. Peix, J. Baruchel, F. Peyrin, M. Schlenker, “Observation of microstructure and damage in materials by phase sensitive radiography and tomography,” J. Appl. Phys. 81, 5878–5886 (1997).
[CrossRef]

Pisano, E.

D. Chapman, W. Thomlinson, Z. Zhong, R. E. Johnston, E. Pisano, D. Washburn, D. Sayers, C. Serge, “Diffraction enhanced imaging applied to materials science and medicine,” Synchr. Radiat. News, 11, 4–11 (1998).
[CrossRef]

D. Chapman, E. Pisano, W. Thomlinson, Z. Zhong, R. E. Johnston, D. Washburn, D. Sayers, K. Malinowska, “Medical applications of diffraction enhanced imaging,” Breast Dis. 10, 197–201 (1998).

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

Pogany, A.

S. W. Wilkins, T. E. Gureyev, D. Gao, A. Pogany, A. W. Stevenson, “Phase-contrast imaging using polychromatic hard x-rays,” Nature (London) 384, 335–338 (1996).
[CrossRef]

Preobrazhenskii, N. G.

A. V. Bronnikov, B. O. Maier, N. G. Preobrazhenskii, “Discrete Wigner transform in the phase problem of optics,” Opt. Spectrosc. 70, 512–516 (1991).

Raven, C.

P. Spanne, C. Raven, I. Snigireva, A. Snigirev, “In-line holography and phase-contrast microtomography with high energy x-rays,” Phys. Med. Biol. 44, 741–750 (1999).
[CrossRef] [PubMed]

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

Roberts, A.

A. Barty, K. A. Nugent, A. Roberts, D. Paganin, “Quantitative phase tomography,” Opt. Commun. 175, 329–336 (2000).
[CrossRef]

Sayers, D.

D. Chapman, E. Pisano, W. Thomlinson, Z. Zhong, R. E. Johnston, D. Washburn, D. Sayers, K. Malinowska, “Medical applications of diffraction enhanced imaging,” Breast Dis. 10, 197–201 (1998).

D. Chapman, W. Thomlinson, Z. Zhong, R. E. Johnston, E. Pisano, D. Washburn, D. Sayers, C. Serge, “Diffraction enhanced imaging applied to materials science and medicine,” Synchr. Radiat. News, 11, 4–11 (1998).
[CrossRef]

D. Chapman, W. Thomlinson, R. E. Johnston, D. Washburn, E. Pisano, N. Gmur, Z. Zhong, R. Menk, F. Ardeli, 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, 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, W. Ludwig, J. Baruchel, D. Van Dyke, 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]

P. Cloetens, M. Pateyron-Salome, J. Y. Buffiere, G. Peix, J. Baruchel, F. Peyrin, M. Schlenker, “Observation of microstructure and damage in materials by phase sensitive radiography and tomography,” J. Appl. Phys. 81, 5878–5886 (1997).
[CrossRef]

Serge, C.

D. Chapman, W. Thomlinson, Z. Zhong, R. E. Johnston, E. Pisano, D. Washburn, D. Sayers, C. Serge, “Diffraction enhanced imaging applied to materials science and medicine,” Synchr. Radiat. News, 11, 4–11 (1998).
[CrossRef]

Slaney, M.

A. C. Kak, M. Slaney, Principles of Computerized Tomographic Imaging (Institute of Electrical and Electronics Engineers, New York1988).

Snigirev, A.

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

P. Spanne, C. Raven, I. Snigireva, A. Snigirev, “In-line holography and phase-contrast microtomography with high energy x-rays,” Phys. Med. Biol. 44, 741–750 (1999).
[CrossRef] [PubMed]

A. Snigirev, I. Snigireva, V. Kohn, S. Kuznetsov, 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.

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

P. Spanne, C. Raven, I. Snigireva, A. Snigirev, “In-line holography and phase-contrast microtomography with high energy x-rays,” Phys. Med. Biol. 44, 741–750 (1999).
[CrossRef] [PubMed]

A. Snigirev, I. Snigireva, V. Kohn, S. Kuznetsov, 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]

Spanne, P.

P. Spanne, C. Raven, I. Snigireva, A. Snigirev, “In-line holography and phase-contrast microtomography with high energy x-rays,” Phys. Med. Biol. 44, 741–750 (1999).
[CrossRef] [PubMed]

Stevenson, A.

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

Stevenson, A. W.

S. W. Wilkins, T. E. Gureyev, D. Gao, A. Pogany, A. W. Stevenson, “Phase-contrast imaging using polychromatic hard x-rays,” Nature (London) 384, 335–338 (1996).
[CrossRef]

Thomlinson, W.

D. Chapman, W. Thomlinson, Z. Zhong, R. E. Johnston, E. Pisano, D. Washburn, D. Sayers, C. Serge, “Diffraction enhanced imaging applied to materials science and medicine,” Synchr. Radiat. News, 11, 4–11 (1998).
[CrossRef]

D. Chapman, E. Pisano, W. Thomlinson, Z. Zhong, R. E. Johnston, D. Washburn, D. Sayers, K. Malinowska, “Medical applications of diffraction enhanced imaging,” Breast Dis. 10, 197–201 (1998).

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

Van Dyke, D.

P. Cloetens, W. Ludwig, J. Baruchel, D. Van Dyke, 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]

Van Landuyt, J.

P. Cloetens, W. Ludwig, J. Baruchel, D. Van Dyke, 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]

Washburn, D.

D. Chapman, E. Pisano, W. Thomlinson, Z. Zhong, R. E. Johnston, D. Washburn, D. Sayers, K. Malinowska, “Medical applications of diffraction enhanced imaging,” Breast Dis. 10, 197–201 (1998).

D. Chapman, W. Thomlinson, Z. Zhong, R. E. Johnston, E. Pisano, D. Washburn, D. Sayers, C. Serge, “Diffraction enhanced imaging applied to materials science and medicine,” Synchr. Radiat. News, 11, 4–11 (1998).
[CrossRef]

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

Wilkins, S.

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

Wilkins, S. W.

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

T. E. Gureyev, S. W. Wilkins, “On x-ray phase imaging with a point source,” J. Opt. Soc. Am. A 15, 579–585 (1998).
[CrossRef]

S. W. Wilkins, T. E. Gureyev, D. Gao, A. Pogany, A. W. Stevenson, “Phase-contrast imaging using polychromatic hard x-rays,” Nature (London) 384, 335–338 (1996).
[CrossRef]

Zhong, Z.

D. Chapman, W. Thomlinson, Z. Zhong, R. E. Johnston, E. Pisano, D. Washburn, D. Sayers, C. Serge, “Diffraction enhanced imaging applied to materials science and medicine,” Synchr. Radiat. News, 11, 4–11 (1998).
[CrossRef]

D. Chapman, E. Pisano, W. Thomlinson, Z. Zhong, R. E. Johnston, D. Washburn, D. Sayers, K. Malinowska, “Medical applications of diffraction enhanced imaging,” Breast Dis. 10, 197–201 (1998).

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

Appl. Opt. (1)

Appl. Phys. Lett. (1)

P. Cloetens, W. Ludwig, J. Baruchel, D. Van Dyke, 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]

Breast Dis. (1)

D. Chapman, E. Pisano, W. Thomlinson, Z. Zhong, R. E. Johnston, D. Washburn, D. Sayers, K. Malinowska, “Medical applications of diffraction enhanced imaging,” Breast Dis. 10, 197–201 (1998).

J. Appl. Phys. (1)

P. Cloetens, M. Pateyron-Salome, J. Y. Buffiere, G. Peix, J. Baruchel, F. Peyrin, M. Schlenker, “Observation of microstructure and damage in materials by phase sensitive radiography and tomography,” J. Appl. Phys. 81, 5878–5886 (1997).
[CrossRef]

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

J. Phys. D (1)

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

Nature (London) (3)

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

S. W. Wilkins, T. E. Gureyev, D. Gao, A. Pogany, A. W. Stevenson, “Phase-contrast imaging using polychromatic hard x-rays,” Nature (London) 384, 335–338 (1996).
[CrossRef]

D. Gabor, “A new microscopic principle,” Nature (London) 161, 777–778 (1948).
[CrossRef]

Opt. Commun. (2)

A. Barty, K. A. Nugent, A. Roberts, D. Paganin, “Quantitative phase tomography,” Opt. Commun. 175, 329–336 (2000).
[CrossRef]

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

Opt. Spectrosc. (1)

A. V. Bronnikov, B. O. Maier, N. G. Preobrazhenskii, “Discrete Wigner transform in the phase problem of optics,” Opt. Spectrosc. 70, 512–516 (1991).

Phys. Med. Biol. (2)

P. Spanne, C. Raven, I. Snigireva, A. Snigirev, “In-line holography and phase-contrast microtomography with high energy x-rays,” Phys. Med. Biol. 44, 741–750 (1999).
[CrossRef] [PubMed]

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

Phys. Rev. Lett. (1)

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]

Rev. Sci. Instrum. (1)

A. Snigirev, I. Snigireva, V. Kohn, S. Kuznetsov, 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]

Synchr. Radiat. News (1)

D. Chapman, W. Thomlinson, Z. Zhong, R. E. Johnston, E. Pisano, D. Washburn, D. Sayers, C. Serge, “Diffraction enhanced imaging applied to materials science and medicine,” Synchr. Radiat. News, 11, 4–11 (1998).
[CrossRef]

Other (2)

J. M. Cowley, Diffraction Physics (North-Holland, Amsterdam, 1981).

A. C. Kak, M. Slaney, Principles of Computerized Tomographic Imaging (Institute of Electrical and Electronics Engineers, New York1988).

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Figures (11)

Fig. 1
Fig. 1

Principle of coherent imaging. The absorption-contrast image is observed in the contact print plane. The phase-contrast images are observed in the near field of the Fresnel diffraction region.

Fig. 2
Fig. 2

Coordinate system for the object and the plane of observation.

Fig. 3
Fig. 3

Normalized discrete transfer function Q(ξ, η) of the reconstruction filter.

Fig. 4
Fig. 4

Computer phantom of the object function. Twelve representative transversal cross sections are shown. The color map is [-10.5×10-7, white; 0, black].

Fig. 5
Fig. 5

Intensity data I0z(x, y) for z=0, 0.6, 1.25, 2.5, 5, 10, 20, and 40 cm. The object is invisible in the contact print plane (z=0), but it becomes visible in the near-field region.

Fig. 6
Fig. 6

Object function reconstructed for d=2.5 cm.

Fig. 7
Fig. 7

Profiles across the object: (a) the phantom (the dashed line) and the object function reconstructed with d=2.5 cm and (b) the object functions reconstructed with d=1.25, 5, 10, and 20 cm.

Fig. 8
Fig. 8

Intensity data at d=2.5 cm and θ=0: (a) noiseless, (b) 1% noise, (c) 2% noise, (d) 5% noise.

Fig. 9
Fig. 9

Central transversal cross section of the object function reconstructed from (a) noiseless data, (b) data with 1% noise, (c) data with 2% noise, and (d) data with 5% noise.

Fig. 10
Fig. 10

Mixed phase and amplitude object: (a) sampled function μθ/2 and (b) sampled phase function ϕ0.

Fig. 11
Fig. 11

Mixed phase and amplitude object: (a) intensity in the contact print plane, (b) intensity at d=2.5 cm, (c) central cross section of the object reconstructed without compensation for the absorption, (d) central cross section of the object reconstructed by the suggested algorithm.

Equations (36)

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Uθ(x, y)=Tθ(x, y)Ui,
Tθ(x, y)=Mθ(x, y)exp[iϕθ(x, y)],
μθ(x, y)=2μ(x1, x2, y)×δ(x-x1 cos θ-x2 sin θ)dx1dx2
ϕθ(x, y)=2πλ2f(x1, x2, y)×δ(x-x1 cos θ-x2 sin θ)dx1dx2.
Iθz(x, y)=|hz ** Uθ|2,
hz(x, y)=exp(ikz)iλzexpiπλz(x2+y2)
Hz(ξ, η)=exp(ikz)exp[-iπλz(ξ2+η2)],
H˜z(ξ, η)=exp(ikz)[1-iπλz(ξ2+η2)],
Iθz(x, y)1+iλz4π2Uθ2,
2Uθ=(aθ+ibθ)Uθ,
aθ=-122μθ+14μθx2+14μθy2-ϕθx2-ϕθy2,
bθ=2ϕθ-μθxϕθx-μθyϕθy.
Iθz1+iλz4π(aθ+ibθ)Uθ2=Iθ01+λz4π(iaθ-bθ)2=Iθ01-λz2πbθ+λ2z216π2(aθ2+bθ2)Iθ01-λz2π2ϕθ+λz2πμθxϕθy+μθyϕθx,
Iθd(x, y)=Iθ01-λd2π2ϕθ(x, y).
gˆ(s, ω)=2g(x, y)δ(s-x sin ω-y cos ω)dxdy
fˆ(s, θ, ω)=3f(x1, x2, x3) δ(s-(x1 cos θ+x2 sin θ)×sin ω-x3 cos ω)dx1dx2dx3
gθ(x, y)=Iθd(x, y)/Iθ0-1,
2s2fˆ(s, θ, ω)=-1dgˆθ(s, ω).
ϕˆθ(s, ω)=ϕθ(x, x3)δ(s-x sin ω-x3 cos ω)dxdx3=2πλf(x1, x2, x3)×δ(x-x1 cos θ-x2 sin θ)dx1dx2×δ(s-x sin ω-x3 cos ω)dxdx3=2πλf(x1, x2, x3)dx1dx2dx3×δ(x-x1 cos θ-x2 sin θ)×δ(s-x sin ω-x3 cos ω)dx=2πλf(x1, x2, x3)dx1dx2dx3×δ(s-(x1 cos θ+x2 sin θ)×sin ω-x3 cos ω),
fˆ(s, θ, ω)=λ2πϕˆθ(s, ω).
2s2fˆ(s, θ, ω)=λ2π2s2ϕˆθ(s, ω).
2s2ϕˆθ(s, ω)=22ϕθ(x, y)×δ(s-x sin ω-y cos ω)dxdy,
2s2fˆ(s, θ, ω)=λ2π22ϕθ(x, y)×δ(s-x sin ω-y cos ω)dxdy.
2ϕθ(x, y)=-2πλdgθ(x, y)
f(x1, x2, x3)=-14π20πsin ω dω0πdθ 2s2fˆ(s, θ, ω),
s=(x1 cos θ+x2 sin θ)sin ω+x3 cos ω.
f(x1, x2, x3)=14π2d0πsin ω dω0πdθ gˆθ(s, ω),
f(x1, x2, x3)=14π2d0πdθgθ(x, y)dxdy×0πδ((x1 cos θ+x2 sin θ-x)sin ω+(x3-y)cos ω)sin ω dω.
[(x1 cos θ+x2 sin θ-x)2+(x3-y)2]1/2.
γ=tan-1x1 cos θ+x2 sin θ-xx3-y,
cos γ=x3-y[(x1 cos θ+x2 sin θ-x)2+(x3-y)2]1/2,
sin γ=x1 cos θ+x2 sin θ-x[(x1 cos θ+x2 sin θ-x)2+(x3-y)2]1/2.
0πδ((x1 cos θ+x2 sin θ-x)sin ω+(x3-y)cos ω)sin ω dω=1[(x1 cos θ+x2 sin θ-x)2+(x3-y)2]1/20πδ(sin γ sin ω+cos γ cos ω)sin ω dω=1[(x1 cos θ+x2 sin θ-x)2+(x3-y)2]1/20πδ(cos(ω-γ))sin ω dω=1|sin(ω-γ)|[(x1 cos θ+x2 sin θ-x)2+(x3-y)2]1/20πδ(ω-ω)sin ω dω=|x3-y|(x1 cos θ+x2 sin θ-x)2+(x3-y)2,
q(x, y)=|y|x2+y2,
f(x1, x2, x3)=14π2d0πdθq ** gθ,
Q(ξ, η)=|y|x2+y2exp[-i2π(xξ+yη)]dxdy=12 exp(-y|ξ|)exp(-i2πyη)dy=|ξ|ξ2+η2,

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