Abstract

This paper presents theoretical and numerical studies of diffraction tomography using hard x rays, from the viewpoint of imaging and reconstruction methods for cell imaging. The proposed system employs a single-perfect-crystal analyzer in symmetric Laue-case transmission geometry to efficiently detect the higher spatial frequency components of an object’s refractive-index distribution, and to effectively suppress interference between the unperturbated wave field and the wave field diffracted by the object. This system features acquisition of a single projection by a single exposure using a simple geometry and aggressive use of diffracted x rays. We present the physical description of the imaging method using the Fourier diffraction theorem derived from the Born approximation. First, we demonstrate that the reconstruction leads to the phase-retrieval problem. We then describe a reconstruction algorithm based on the classical Gerchberg–Saxton–Fienup algorithm. Finally, we show the efficacy of this system by computer simulation. Our simulation demonstrates that the imaging system delineates microstructure 3.5μm in diameter in a phase object 400μm in diameter.

© 2005 Optical Society of America

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  1. R. Y. Tsien, “Imaging imaging’s future,” Nat. Rev. Mol. Cell Biol. 4, SS16–21 (2004).
  2. R. E. Jacobs, C. Papan, S. Ruffines, J. M. Tyszka, S. E. Fraser, “MRI: volumetric imaging for vital imaging and atlas construction,” Nat. Rev. Mol. Cell Biol. 4, SS10–16 (2004).
  3. U. Bonse, M. Hart, “An x-ray interferometer,” Appl. Phys. Lett. 6, 155–156 (1965).
    [CrossRef]
  4. M. Ando, S. Hosoya, “An attempt at x-ray phase-contrast microscopy,” in Proceedings 6th International Conference of X-ray Optics and Microanalysis, G. Shinoda, K. Kohra, and T. Ichinokawa, eds. (University of Tokyo Press, 1972), pp. 63–68.
  5. T. J. Davis, T. E. Gureyev, D. Gao, A. W. Stevenson, S. W. Wilkins, “X-ray image contrast from a simple phase object,” Phys. Rev. Lett. 74, 3173–3176 (1995).
    [CrossRef] [PubMed]
  6. P. Cloetens, R. Barret, J. Baruchel, J.-P. Guigay, M. Schlenker, “Phase objects in synchrotron radiation hard x-ray imaging,” J. Phys. D 29, 133–146 (1996).
    [CrossRef]
  7. C. Raven, A. Snigirev, I. Snigireva, P. Spanne, A. Souvorov, V. Kohn, “Phase-contrast microtomography with coherent high-energy synchrotron x-rays,” Appl. Phys. Lett. 69, 1826–1828 (1996).
    [CrossRef]
  8. A. Momose, T. Takeda, Y. Itai, K. Hirano, “Phase-contrast x-ray computed tomography for observing biological soft tissues,” Nat. Med. 2, 473–475 (1996).
    [CrossRef] [PubMed]
  9. F. A. Dilmanian, Z. Zhong, B. Ren, X. Y. Wu, L. D. Chapman, I. Orion, W. C. Tomlinson, “Computed tomography of x-ray index of refraction using the diffraction enhanced imaging method,” Phys. Med. Biol. 45, 933–946 (2000).
    [CrossRef] [PubMed]
  10. M. N. Wernick, O. Wirjadi, D. Chapman, Z. Zhong, N. P. Galatsanos, Y. Yang, J. G. Brankov, O. Oltulu, M. A. Anastasio, C. Muehleman, “Multiple- image radiography,” Phys. Med. Biol. 48, 3875–3895 (2003).
    [CrossRef]
  11. T. J. Davis, D. Gao, T. E. Gureyev, A. W. Stevenson, S. W. Wilkins, “Phase-contrast imaging of weakly absorbing materials using hard x-rays,” Nature 373, 595–598 (1995).
    [CrossRef]
  12. 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]
  13. V. N. Ingal, E. A. Beliaevskaya, “X-ray plane-wave topography observation of the phase-contrast from noncrystalline object,” J. Phys. D 28, 2314–2317 (1995).
    [CrossRef]
  14. V. A. Somenkov, A. K. Tkalich, S. S. Shilstein, “Refraction contrast in X-ray introscopy,” Sov. Phys. Tech. Phys. 61, 197–201 (1991).
  15. V. V. Protopopov, “On the possibility of X-ray refractive radiography using multilayer mirrors with resonant absorption,” Opt. Commun. 174, 13–18 (2000).
    [CrossRef]
  16. V. V. Protopopov, J. Sobota, “X-ray dark-field refraction-contrast imaging of micro-objects,” Opt. Commun. 213, 267–279 (2002).
    [CrossRef]
  17. M. Ando, A. Maksimenko, H. Sugiyama, W. Pattanasiriwisawa, K. Hyodo, C. Uyama, “Simple x-ray dark- and bright-field imaging using achromatic Laue optics,” Jpn. J. Appl. Phys., Part 1 41, L1016–L1018 (2002).
    [CrossRef]
  18. E. Wolf, “Three dimensional structure determination of semi transparent objects from holographic data,” Opt. Commun. 1, 153–156 (1969).
    [CrossRef]
  19. R. K. Mueller, M. Kaveh, R. D. Inverson, “A new approach to acoustic tomography using diffraction techniques,” in Acoustical Imaging, A. F. Metherell, ed. (Plenum, 1980).
    [CrossRef]
  20. A. J. Devaney, “A filtered backpropagation algorithm for diffraction tomography,” Ultrason. Imaging 4, 336–350 (1982).
    [CrossRef] [PubMed]
  21. J. R. Fienup, “Phase retrieval algorithms: a comparison,” Appl. Opt. 21, 2758–2769 (1982).
    [CrossRef] [PubMed]
  22. R. W. Gerchberg, “Super-resolution through error energy reduction,” Opt. Acta 21, 709–720 (1974).
    [CrossRef]
  23. J. Cheng, S. Han, “Diffraction tomography reconstruction algorithms for quantitative imaging of phase objects,” J. Opt. Soc. Am. A 18, 1460–1464 (2001).
    [CrossRef]
  24. A. C. Kak, M. Slaney, Principles of Computerized Tomographic Imaging (Society for Industrial and Applied Mathematics, 2001), Vol. 33.
    [CrossRef]
  25. A. Authier, Dynamical Theory of X-ray Diffraction (Oxford U. Press, 2001).
  26. A. Makasimenko, H. Sugiyama, K. Hirano, T. Yuasa, M. Ando, “Dark-field imaging using an asymmetric Bragg case transmission analyzer,” Meas. Sci. Technol. 15, 1251–1254 (2004).
    [CrossRef]
  27. A. Momose, I. Koyama, Y. Hamaishi, H. Yoshikawa, T. Takeda, J. Wu, Y. Itai, K. Takai, K. Uesugi, Y. Suzuki, “Phase-contrast microtomography using an x-ray interferometer having a 40-μm analyzer,” J. Phys. IV 104, 599–602 (2003).
  28. M. M. Bronstein, A. M. Bronstein, M. Zibulevsky, H. Azhari, “Reconstruction in Diffraction Ultrasound Tomography Using Nonuniform FFT,” IEEE Trans. Med. Imaging 21, 1395–1401 (2002).
    [CrossRef]
  29. R. D. Spal, “Submicrometer resolution hard x-ray holography with asymmetric Bragg diffraction microscopy,” Phys. Rev. Lett. 86, 3044–3047 (2001).
    [CrossRef] [PubMed]

2004 (3)

R. Y. Tsien, “Imaging imaging’s future,” Nat. Rev. Mol. Cell Biol. 4, SS16–21 (2004).

R. E. Jacobs, C. Papan, S. Ruffines, J. M. Tyszka, S. E. Fraser, “MRI: volumetric imaging for vital imaging and atlas construction,” Nat. Rev. Mol. Cell Biol. 4, SS10–16 (2004).

A. Makasimenko, H. Sugiyama, K. Hirano, T. Yuasa, M. Ando, “Dark-field imaging using an asymmetric Bragg case transmission analyzer,” Meas. Sci. Technol. 15, 1251–1254 (2004).
[CrossRef]

2003 (2)

A. Momose, I. Koyama, Y. Hamaishi, H. Yoshikawa, T. Takeda, J. Wu, Y. Itai, K. Takai, K. Uesugi, Y. Suzuki, “Phase-contrast microtomography using an x-ray interferometer having a 40-μm analyzer,” J. Phys. IV 104, 599–602 (2003).

M. N. Wernick, O. Wirjadi, D. Chapman, Z. Zhong, N. P. Galatsanos, Y. Yang, J. G. Brankov, O. Oltulu, M. A. Anastasio, C. Muehleman, “Multiple- image radiography,” Phys. Med. Biol. 48, 3875–3895 (2003).
[CrossRef]

2002 (3)

V. V. Protopopov, J. Sobota, “X-ray dark-field refraction-contrast imaging of micro-objects,” Opt. Commun. 213, 267–279 (2002).
[CrossRef]

M. Ando, A. Maksimenko, H. Sugiyama, W. Pattanasiriwisawa, K. Hyodo, C. Uyama, “Simple x-ray dark- and bright-field imaging using achromatic Laue optics,” Jpn. J. Appl. Phys., Part 1 41, L1016–L1018 (2002).
[CrossRef]

M. M. Bronstein, A. M. Bronstein, M. Zibulevsky, H. Azhari, “Reconstruction in Diffraction Ultrasound Tomography Using Nonuniform FFT,” IEEE Trans. Med. Imaging 21, 1395–1401 (2002).
[CrossRef]

2001 (2)

R. D. Spal, “Submicrometer resolution hard x-ray holography with asymmetric Bragg diffraction microscopy,” Phys. Rev. Lett. 86, 3044–3047 (2001).
[CrossRef] [PubMed]

J. Cheng, S. Han, “Diffraction tomography reconstruction algorithms for quantitative imaging of phase objects,” J. Opt. Soc. Am. A 18, 1460–1464 (2001).
[CrossRef]

2000 (2)

V. V. Protopopov, “On the possibility of X-ray refractive radiography using multilayer mirrors with resonant absorption,” Opt. Commun. 174, 13–18 (2000).
[CrossRef]

F. A. Dilmanian, Z. Zhong, B. Ren, X. Y. Wu, L. D. Chapman, I. Orion, W. C. Tomlinson, “Computed tomography of x-ray index of refraction using the diffraction enhanced imaging method,” Phys. Med. Biol. 45, 933–946 (2000).
[CrossRef] [PubMed]

1996 (3)

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

C. Raven, A. Snigirev, I. Snigireva, P. Spanne, A. Souvorov, V. Kohn, “Phase-contrast microtomography with coherent high-energy synchrotron x-rays,” Appl. Phys. Lett. 69, 1826–1828 (1996).
[CrossRef]

A. Momose, T. Takeda, Y. Itai, K. Hirano, “Phase-contrast x-ray computed tomography for observing biological soft tissues,” Nat. Med. 2, 473–475 (1996).
[CrossRef] [PubMed]

1995 (4)

T. J. Davis, D. Gao, T. E. Gureyev, A. W. Stevenson, S. W. Wilkins, “Phase-contrast imaging of weakly absorbing materials using hard x-rays,” Nature 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]

V. N. Ingal, E. A. Beliaevskaya, “X-ray plane-wave topography observation of the phase-contrast from noncrystalline object,” J. Phys. D 28, 2314–2317 (1995).
[CrossRef]

T. J. Davis, T. E. Gureyev, D. Gao, A. W. Stevenson, S. W. Wilkins, “X-ray image contrast from a simple phase object,” Phys. Rev. Lett. 74, 3173–3176 (1995).
[CrossRef] [PubMed]

1991 (1)

V. A. Somenkov, A. K. Tkalich, S. S. Shilstein, “Refraction contrast in X-ray introscopy,” Sov. Phys. Tech. Phys. 61, 197–201 (1991).

1982 (2)

A. J. Devaney, “A filtered backpropagation algorithm for diffraction tomography,” Ultrason. Imaging 4, 336–350 (1982).
[CrossRef] [PubMed]

J. R. Fienup, “Phase retrieval algorithms: a comparison,” Appl. Opt. 21, 2758–2769 (1982).
[CrossRef] [PubMed]

1974 (1)

R. W. Gerchberg, “Super-resolution through error energy reduction,” Opt. Acta 21, 709–720 (1974).
[CrossRef]

1969 (1)

E. Wolf, “Three dimensional structure determination of semi transparent objects from holographic data,” Opt. Commun. 1, 153–156 (1969).
[CrossRef]

1965 (1)

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

Anastasio, M. A.

M. N. Wernick, O. Wirjadi, D. Chapman, Z. Zhong, N. P. Galatsanos, Y. Yang, J. G. Brankov, O. Oltulu, M. A. Anastasio, C. Muehleman, “Multiple- image radiography,” Phys. Med. Biol. 48, 3875–3895 (2003).
[CrossRef]

Ando, M.

A. Makasimenko, H. Sugiyama, K. Hirano, T. Yuasa, M. Ando, “Dark-field imaging using an asymmetric Bragg case transmission analyzer,” Meas. Sci. Technol. 15, 1251–1254 (2004).
[CrossRef]

M. Ando, A. Maksimenko, H. Sugiyama, W. Pattanasiriwisawa, K. Hyodo, C. Uyama, “Simple x-ray dark- and bright-field imaging using achromatic Laue optics,” Jpn. J. Appl. Phys., Part 1 41, L1016–L1018 (2002).
[CrossRef]

M. Ando, S. Hosoya, “An attempt at x-ray phase-contrast microscopy,” in Proceedings 6th International Conference of X-ray Optics and Microanalysis, G. Shinoda, K. Kohra, and T. Ichinokawa, eds. (University of Tokyo Press, 1972), pp. 63–68.

Authier, A.

A. Authier, Dynamical Theory of X-ray Diffraction (Oxford U. Press, 2001).

Azhari, H.

M. M. Bronstein, A. M. Bronstein, M. Zibulevsky, H. Azhari, “Reconstruction in Diffraction Ultrasound Tomography Using Nonuniform FFT,” IEEE Trans. Med. Imaging 21, 1395–1401 (2002).
[CrossRef]

Barret, R.

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

Baruchel, J.

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

Beliaevskaya, E. A.

V. N. Ingal, E. A. Beliaevskaya, “X-ray plane-wave topography observation of the phase-contrast from noncrystalline object,” J. Phys. D 28, 2314–2317 (1995).
[CrossRef]

Bonse, U.

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

Brankov, J. G.

M. N. Wernick, O. Wirjadi, D. Chapman, Z. Zhong, N. P. Galatsanos, Y. Yang, J. G. Brankov, O. Oltulu, M. A. Anastasio, C. Muehleman, “Multiple- image radiography,” Phys. Med. Biol. 48, 3875–3895 (2003).
[CrossRef]

Bronstein, A. M.

M. M. Bronstein, A. M. Bronstein, M. Zibulevsky, H. Azhari, “Reconstruction in Diffraction Ultrasound Tomography Using Nonuniform FFT,” IEEE Trans. Med. Imaging 21, 1395–1401 (2002).
[CrossRef]

Bronstein, M. M.

M. M. Bronstein, A. M. Bronstein, M. Zibulevsky, H. Azhari, “Reconstruction in Diffraction Ultrasound Tomography Using Nonuniform FFT,” IEEE Trans. Med. Imaging 21, 1395–1401 (2002).
[CrossRef]

Chapman, D.

M. N. Wernick, O. Wirjadi, D. Chapman, Z. Zhong, N. P. Galatsanos, Y. Yang, J. G. Brankov, O. Oltulu, M. A. Anastasio, C. Muehleman, “Multiple- image radiography,” Phys. Med. Biol. 48, 3875–3895 (2003).
[CrossRef]

Chapman, L. D.

F. A. Dilmanian, Z. Zhong, B. Ren, X. Y. Wu, L. D. Chapman, I. Orion, W. C. Tomlinson, “Computed tomography of x-ray index of refraction using the diffraction enhanced imaging method,” Phys. Med. Biol. 45, 933–946 (2000).
[CrossRef] [PubMed]

Cheng, J.

Cloetens, P.

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

Davis, T. J.

T. J. Davis, T. E. Gureyev, D. Gao, A. W. Stevenson, S. W. Wilkins, “X-ray image contrast from a simple phase object,” Phys. Rev. Lett. 74, 3173–3176 (1995).
[CrossRef] [PubMed]

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

Devaney, A. J.

A. J. Devaney, “A filtered backpropagation algorithm for diffraction tomography,” Ultrason. Imaging 4, 336–350 (1982).
[CrossRef] [PubMed]

Dilmanian, F. A.

F. A. Dilmanian, Z. Zhong, B. Ren, X. Y. Wu, L. D. Chapman, I. Orion, W. C. Tomlinson, “Computed tomography of x-ray index of refraction using the diffraction enhanced imaging method,” Phys. Med. Biol. 45, 933–946 (2000).
[CrossRef] [PubMed]

Fienup, J. R.

Fraser, S. E.

R. E. Jacobs, C. Papan, S. Ruffines, J. M. Tyszka, S. E. Fraser, “MRI: volumetric imaging for vital imaging and atlas construction,” Nat. Rev. Mol. Cell Biol. 4, SS10–16 (2004).

Galatsanos, N. P.

M. N. Wernick, O. Wirjadi, D. Chapman, Z. Zhong, N. P. Galatsanos, Y. Yang, J. G. Brankov, O. Oltulu, M. A. Anastasio, C. Muehleman, “Multiple- image radiography,” Phys. Med. Biol. 48, 3875–3895 (2003).
[CrossRef]

Gao, D.

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

T. J. Davis, T. E. Gureyev, D. Gao, A. W. Stevenson, S. W. Wilkins, “X-ray image contrast from a simple phase object,” Phys. Rev. Lett. 74, 3173–3176 (1995).
[CrossRef] [PubMed]

Gerchberg, R. W.

R. W. Gerchberg, “Super-resolution through error energy reduction,” Opt. Acta 21, 709–720 (1974).
[CrossRef]

Guigay, J.-P.

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

Gureyev, T. E.

T. J. Davis, T. E. Gureyev, D. Gao, A. W. Stevenson, S. W. Wilkins, “X-ray image contrast from a simple phase object,” Phys. Rev. Lett. 74, 3173–3176 (1995).
[CrossRef] [PubMed]

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

Hamaishi, Y.

A. Momose, I. Koyama, Y. Hamaishi, H. Yoshikawa, T. Takeda, J. Wu, Y. Itai, K. Takai, K. Uesugi, Y. Suzuki, “Phase-contrast microtomography using an x-ray interferometer having a 40-μm analyzer,” J. Phys. IV 104, 599–602 (2003).

Han, S.

Hart, M.

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

Hirano, K.

A. Makasimenko, H. Sugiyama, K. Hirano, T. Yuasa, M. Ando, “Dark-field imaging using an asymmetric Bragg case transmission analyzer,” Meas. Sci. Technol. 15, 1251–1254 (2004).
[CrossRef]

A. Momose, T. Takeda, Y. Itai, K. Hirano, “Phase-contrast x-ray computed tomography for observing biological soft tissues,” Nat. Med. 2, 473–475 (1996).
[CrossRef] [PubMed]

Hosoya, S.

M. Ando, S. Hosoya, “An attempt at x-ray phase-contrast microscopy,” in Proceedings 6th International Conference of X-ray Optics and Microanalysis, G. Shinoda, K. Kohra, and T. Ichinokawa, eds. (University of Tokyo Press, 1972), pp. 63–68.

Hyodo, K.

M. Ando, A. Maksimenko, H. Sugiyama, W. Pattanasiriwisawa, K. Hyodo, C. Uyama, “Simple x-ray dark- and bright-field imaging using achromatic Laue optics,” Jpn. J. Appl. Phys., Part 1 41, L1016–L1018 (2002).
[CrossRef]

Ingal, V. N.

V. N. Ingal, E. A. Beliaevskaya, “X-ray plane-wave topography observation of the phase-contrast from noncrystalline object,” J. Phys. D 28, 2314–2317 (1995).
[CrossRef]

Inverson, R. D.

R. K. Mueller, M. Kaveh, R. D. Inverson, “A new approach to acoustic tomography using diffraction techniques,” in Acoustical Imaging, A. F. Metherell, ed. (Plenum, 1980).
[CrossRef]

Itai, Y.

A. Momose, I. Koyama, Y. Hamaishi, H. Yoshikawa, T. Takeda, J. Wu, Y. Itai, K. Takai, K. Uesugi, Y. Suzuki, “Phase-contrast microtomography using an x-ray interferometer having a 40-μm analyzer,” J. Phys. IV 104, 599–602 (2003).

A. Momose, T. Takeda, Y. Itai, K. Hirano, “Phase-contrast x-ray computed tomography for observing biological soft tissues,” Nat. Med. 2, 473–475 (1996).
[CrossRef] [PubMed]

Jacobs, R. E.

R. E. Jacobs, C. Papan, S. Ruffines, J. M. Tyszka, S. E. Fraser, “MRI: volumetric imaging for vital imaging and atlas construction,” Nat. Rev. Mol. Cell Biol. 4, SS10–16 (2004).

Kak, A. C.

A. C. Kak, M. Slaney, Principles of Computerized Tomographic Imaging (Society for Industrial and Applied Mathematics, 2001), Vol. 33.
[CrossRef]

Kaveh, M.

R. K. Mueller, M. Kaveh, R. D. Inverson, “A new approach to acoustic tomography using diffraction techniques,” in Acoustical Imaging, A. F. Metherell, ed. (Plenum, 1980).
[CrossRef]

Kohn, V.

C. Raven, A. Snigirev, I. Snigireva, P. Spanne, A. Souvorov, V. Kohn, “Phase-contrast microtomography with coherent high-energy synchrotron x-rays,” Appl. Phys. Lett. 69, 1826–1828 (1996).
[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]

Koyama, I.

A. Momose, I. Koyama, Y. Hamaishi, H. Yoshikawa, T. Takeda, J. Wu, Y. Itai, K. Takai, K. Uesugi, Y. Suzuki, “Phase-contrast microtomography using an x-ray interferometer having a 40-μm analyzer,” J. Phys. IV 104, 599–602 (2003).

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]

Makasimenko, A.

A. Makasimenko, H. Sugiyama, K. Hirano, T. Yuasa, M. Ando, “Dark-field imaging using an asymmetric Bragg case transmission analyzer,” Meas. Sci. Technol. 15, 1251–1254 (2004).
[CrossRef]

Maksimenko, A.

M. Ando, A. Maksimenko, H. Sugiyama, W. Pattanasiriwisawa, K. Hyodo, C. Uyama, “Simple x-ray dark- and bright-field imaging using achromatic Laue optics,” Jpn. J. Appl. Phys., Part 1 41, L1016–L1018 (2002).
[CrossRef]

Momose, A.

A. Momose, I. Koyama, Y. Hamaishi, H. Yoshikawa, T. Takeda, J. Wu, Y. Itai, K. Takai, K. Uesugi, Y. Suzuki, “Phase-contrast microtomography using an x-ray interferometer having a 40-μm analyzer,” J. Phys. IV 104, 599–602 (2003).

A. Momose, T. Takeda, Y. Itai, K. Hirano, “Phase-contrast x-ray computed tomography for observing biological soft tissues,” Nat. Med. 2, 473–475 (1996).
[CrossRef] [PubMed]

Muehleman, C.

M. N. Wernick, O. Wirjadi, D. Chapman, Z. Zhong, N. P. Galatsanos, Y. Yang, J. G. Brankov, O. Oltulu, M. A. Anastasio, C. Muehleman, “Multiple- image radiography,” Phys. Med. Biol. 48, 3875–3895 (2003).
[CrossRef]

Mueller, R. K.

R. K. Mueller, M. Kaveh, R. D. Inverson, “A new approach to acoustic tomography using diffraction techniques,” in Acoustical Imaging, A. F. Metherell, ed. (Plenum, 1980).
[CrossRef]

Oltulu, O.

M. N. Wernick, O. Wirjadi, D. Chapman, Z. Zhong, N. P. Galatsanos, Y. Yang, J. G. Brankov, O. Oltulu, M. A. Anastasio, C. Muehleman, “Multiple- image radiography,” Phys. Med. Biol. 48, 3875–3895 (2003).
[CrossRef]

Orion, I.

F. A. Dilmanian, Z. Zhong, B. Ren, X. Y. Wu, L. D. Chapman, I. Orion, W. C. Tomlinson, “Computed tomography of x-ray index of refraction using the diffraction enhanced imaging method,” Phys. Med. Biol. 45, 933–946 (2000).
[CrossRef] [PubMed]

Papan, C.

R. E. Jacobs, C. Papan, S. Ruffines, J. M. Tyszka, S. E. Fraser, “MRI: volumetric imaging for vital imaging and atlas construction,” Nat. Rev. Mol. Cell Biol. 4, SS10–16 (2004).

Pattanasiriwisawa, W.

M. Ando, A. Maksimenko, H. Sugiyama, W. Pattanasiriwisawa, K. Hyodo, C. Uyama, “Simple x-ray dark- and bright-field imaging using achromatic Laue optics,” Jpn. J. Appl. Phys., Part 1 41, L1016–L1018 (2002).
[CrossRef]

Protopopov, V. V.

V. V. Protopopov, J. Sobota, “X-ray dark-field refraction-contrast imaging of micro-objects,” Opt. Commun. 213, 267–279 (2002).
[CrossRef]

V. V. Protopopov, “On the possibility of X-ray refractive radiography using multilayer mirrors with resonant absorption,” Opt. Commun. 174, 13–18 (2000).
[CrossRef]

Raven, C.

C. Raven, A. Snigirev, I. Snigireva, P. Spanne, A. Souvorov, V. Kohn, “Phase-contrast microtomography with coherent high-energy synchrotron x-rays,” Appl. Phys. Lett. 69, 1826–1828 (1996).
[CrossRef]

Ren, B.

F. A. Dilmanian, Z. Zhong, B. Ren, X. Y. Wu, L. D. Chapman, I. Orion, W. C. Tomlinson, “Computed tomography of x-ray index of refraction using the diffraction enhanced imaging method,” Phys. Med. Biol. 45, 933–946 (2000).
[CrossRef] [PubMed]

Ruffines, S.

R. E. Jacobs, C. Papan, S. Ruffines, J. M. Tyszka, S. E. Fraser, “MRI: volumetric imaging for vital imaging and atlas construction,” Nat. Rev. Mol. Cell Biol. 4, SS10–16 (2004).

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

Shilstein, S. S.

V. A. Somenkov, A. K. Tkalich, S. S. Shilstein, “Refraction contrast in X-ray introscopy,” Sov. Phys. Tech. Phys. 61, 197–201 (1991).

Slaney, M.

A. C. Kak, M. Slaney, Principles of Computerized Tomographic Imaging (Society for Industrial and Applied Mathematics, 2001), Vol. 33.
[CrossRef]

Snigirev, A.

C. Raven, A. Snigirev, I. Snigireva, P. Spanne, A. Souvorov, V. Kohn, “Phase-contrast microtomography with coherent high-energy synchrotron x-rays,” Appl. Phys. Lett. 69, 1826–1828 (1996).
[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]

Snigireva, I.

C. Raven, A. Snigirev, I. Snigireva, P. Spanne, A. Souvorov, V. Kohn, “Phase-contrast microtomography with coherent high-energy synchrotron x-rays,” Appl. Phys. Lett. 69, 1826–1828 (1996).
[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]

Sobota, J.

V. V. Protopopov, J. Sobota, “X-ray dark-field refraction-contrast imaging of micro-objects,” Opt. Commun. 213, 267–279 (2002).
[CrossRef]

Somenkov, V. A.

V. A. Somenkov, A. K. Tkalich, S. S. Shilstein, “Refraction contrast in X-ray introscopy,” Sov. Phys. Tech. Phys. 61, 197–201 (1991).

Souvorov, A.

C. Raven, A. Snigirev, I. Snigireva, P. Spanne, A. Souvorov, V. Kohn, “Phase-contrast microtomography with coherent high-energy synchrotron x-rays,” Appl. Phys. Lett. 69, 1826–1828 (1996).
[CrossRef]

Spal, R. D.

R. D. Spal, “Submicrometer resolution hard x-ray holography with asymmetric Bragg diffraction microscopy,” Phys. Rev. Lett. 86, 3044–3047 (2001).
[CrossRef] [PubMed]

Spanne, P.

C. Raven, A. Snigirev, I. Snigireva, P. Spanne, A. Souvorov, V. Kohn, “Phase-contrast microtomography with coherent high-energy synchrotron x-rays,” Appl. Phys. Lett. 69, 1826–1828 (1996).
[CrossRef]

Stevenson, A. W.

T. J. Davis, T. E. Gureyev, D. Gao, A. W. Stevenson, S. W. Wilkins, “X-ray image contrast from a simple phase object,” Phys. Rev. Lett. 74, 3173–3176 (1995).
[CrossRef] [PubMed]

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

Sugiyama, H.

A. Makasimenko, H. Sugiyama, K. Hirano, T. Yuasa, M. Ando, “Dark-field imaging using an asymmetric Bragg case transmission analyzer,” Meas. Sci. Technol. 15, 1251–1254 (2004).
[CrossRef]

M. Ando, A. Maksimenko, H. Sugiyama, W. Pattanasiriwisawa, K. Hyodo, C. Uyama, “Simple x-ray dark- and bright-field imaging using achromatic Laue optics,” Jpn. J. Appl. Phys., Part 1 41, L1016–L1018 (2002).
[CrossRef]

Suzuki, Y.

A. Momose, I. Koyama, Y. Hamaishi, H. Yoshikawa, T. Takeda, J. Wu, Y. Itai, K. Takai, K. Uesugi, Y. Suzuki, “Phase-contrast microtomography using an x-ray interferometer having a 40-μm analyzer,” J. Phys. IV 104, 599–602 (2003).

Takai, K.

A. Momose, I. Koyama, Y. Hamaishi, H. Yoshikawa, T. Takeda, J. Wu, Y. Itai, K. Takai, K. Uesugi, Y. Suzuki, “Phase-contrast microtomography using an x-ray interferometer having a 40-μm analyzer,” J. Phys. IV 104, 599–602 (2003).

Takeda, T.

A. Momose, I. Koyama, Y. Hamaishi, H. Yoshikawa, T. Takeda, J. Wu, Y. Itai, K. Takai, K. Uesugi, Y. Suzuki, “Phase-contrast microtomography using an x-ray interferometer having a 40-μm analyzer,” J. Phys. IV 104, 599–602 (2003).

A. Momose, T. Takeda, Y. Itai, K. Hirano, “Phase-contrast x-ray computed tomography for observing biological soft tissues,” Nat. Med. 2, 473–475 (1996).
[CrossRef] [PubMed]

Tkalich, A. K.

V. A. Somenkov, A. K. Tkalich, S. S. Shilstein, “Refraction contrast in X-ray introscopy,” Sov. Phys. Tech. Phys. 61, 197–201 (1991).

Tomlinson, W. C.

F. A. Dilmanian, Z. Zhong, B. Ren, X. Y. Wu, L. D. Chapman, I. Orion, W. C. Tomlinson, “Computed tomography of x-ray index of refraction using the diffraction enhanced imaging method,” Phys. Med. Biol. 45, 933–946 (2000).
[CrossRef] [PubMed]

Tsien, R. Y.

R. Y. Tsien, “Imaging imaging’s future,” Nat. Rev. Mol. Cell Biol. 4, SS16–21 (2004).

Tyszka, J. M.

R. E. Jacobs, C. Papan, S. Ruffines, J. M. Tyszka, S. E. Fraser, “MRI: volumetric imaging for vital imaging and atlas construction,” Nat. Rev. Mol. Cell Biol. 4, SS10–16 (2004).

Uesugi, K.

A. Momose, I. Koyama, Y. Hamaishi, H. Yoshikawa, T. Takeda, J. Wu, Y. Itai, K. Takai, K. Uesugi, Y. Suzuki, “Phase-contrast microtomography using an x-ray interferometer having a 40-μm analyzer,” J. Phys. IV 104, 599–602 (2003).

Uyama, C.

M. Ando, A. Maksimenko, H. Sugiyama, W. Pattanasiriwisawa, K. Hyodo, C. Uyama, “Simple x-ray dark- and bright-field imaging using achromatic Laue optics,” Jpn. J. Appl. Phys., Part 1 41, L1016–L1018 (2002).
[CrossRef]

Wernick, M. N.

M. N. Wernick, O. Wirjadi, D. Chapman, Z. Zhong, N. P. Galatsanos, Y. Yang, J. G. Brankov, O. Oltulu, M. A. Anastasio, C. Muehleman, “Multiple- image radiography,” Phys. Med. Biol. 48, 3875–3895 (2003).
[CrossRef]

Wilkins, S. W.

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

T. J. Davis, T. E. Gureyev, D. Gao, A. W. Stevenson, S. W. Wilkins, “X-ray image contrast from a simple phase object,” Phys. Rev. Lett. 74, 3173–3176 (1995).
[CrossRef] [PubMed]

Wirjadi, O.

M. N. Wernick, O. Wirjadi, D. Chapman, Z. Zhong, N. P. Galatsanos, Y. Yang, J. G. Brankov, O. Oltulu, M. A. Anastasio, C. Muehleman, “Multiple- image radiography,” Phys. Med. Biol. 48, 3875–3895 (2003).
[CrossRef]

Wolf, E.

E. Wolf, “Three dimensional structure determination of semi transparent objects from holographic data,” Opt. Commun. 1, 153–156 (1969).
[CrossRef]

Wu, J.

A. Momose, I. Koyama, Y. Hamaishi, H. Yoshikawa, T. Takeda, J. Wu, Y. Itai, K. Takai, K. Uesugi, Y. Suzuki, “Phase-contrast microtomography using an x-ray interferometer having a 40-μm analyzer,” J. Phys. IV 104, 599–602 (2003).

Wu, X. Y.

F. A. Dilmanian, Z. Zhong, B. Ren, X. Y. Wu, L. D. Chapman, I. Orion, W. C. Tomlinson, “Computed tomography of x-ray index of refraction using the diffraction enhanced imaging method,” Phys. Med. Biol. 45, 933–946 (2000).
[CrossRef] [PubMed]

Yang, Y.

M. N. Wernick, O. Wirjadi, D. Chapman, Z. Zhong, N. P. Galatsanos, Y. Yang, J. G. Brankov, O. Oltulu, M. A. Anastasio, C. Muehleman, “Multiple- image radiography,” Phys. Med. Biol. 48, 3875–3895 (2003).
[CrossRef]

Yoshikawa, H.

A. Momose, I. Koyama, Y. Hamaishi, H. Yoshikawa, T. Takeda, J. Wu, Y. Itai, K. Takai, K. Uesugi, Y. Suzuki, “Phase-contrast microtomography using an x-ray interferometer having a 40-μm analyzer,” J. Phys. IV 104, 599–602 (2003).

Yuasa, T.

A. Makasimenko, H. Sugiyama, K. Hirano, T. Yuasa, M. Ando, “Dark-field imaging using an asymmetric Bragg case transmission analyzer,” Meas. Sci. Technol. 15, 1251–1254 (2004).
[CrossRef]

Zhong, Z.

M. N. Wernick, O. Wirjadi, D. Chapman, Z. Zhong, N. P. Galatsanos, Y. Yang, J. G. Brankov, O. Oltulu, M. A. Anastasio, C. Muehleman, “Multiple- image radiography,” Phys. Med. Biol. 48, 3875–3895 (2003).
[CrossRef]

F. A. Dilmanian, Z. Zhong, B. Ren, X. Y. Wu, L. D. Chapman, I. Orion, W. C. Tomlinson, “Computed tomography of x-ray index of refraction using the diffraction enhanced imaging method,” Phys. Med. Biol. 45, 933–946 (2000).
[CrossRef] [PubMed]

Zibulevsky, M.

M. M. Bronstein, A. M. Bronstein, M. Zibulevsky, H. Azhari, “Reconstruction in Diffraction Ultrasound Tomography Using Nonuniform FFT,” IEEE Trans. Med. Imaging 21, 1395–1401 (2002).
[CrossRef]

Appl. Opt. (1)

Appl. Phys. Lett. (2)

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

C. Raven, A. Snigirev, I. Snigireva, P. Spanne, A. Souvorov, V. Kohn, “Phase-contrast microtomography with coherent high-energy synchrotron x-rays,” Appl. Phys. Lett. 69, 1826–1828 (1996).
[CrossRef]

IEEE Trans. Med. Imaging (1)

M. M. Bronstein, A. M. Bronstein, M. Zibulevsky, H. Azhari, “Reconstruction in Diffraction Ultrasound Tomography Using Nonuniform FFT,” IEEE Trans. Med. Imaging 21, 1395–1401 (2002).
[CrossRef]

J. Opt. Soc. Am. A (1)

J. Phys. D (2)

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

V. N. Ingal, E. A. Beliaevskaya, “X-ray plane-wave topography observation of the phase-contrast from noncrystalline object,” J. Phys. D 28, 2314–2317 (1995).
[CrossRef]

J. Phys. IV (1)

A. Momose, I. Koyama, Y. Hamaishi, H. Yoshikawa, T. Takeda, J. Wu, Y. Itai, K. Takai, K. Uesugi, Y. Suzuki, “Phase-contrast microtomography using an x-ray interferometer having a 40-μm analyzer,” J. Phys. IV 104, 599–602 (2003).

Jpn. J. Appl. Phys., Part 1 (1)

M. Ando, A. Maksimenko, H. Sugiyama, W. Pattanasiriwisawa, K. Hyodo, C. Uyama, “Simple x-ray dark- and bright-field imaging using achromatic Laue optics,” Jpn. J. Appl. Phys., Part 1 41, L1016–L1018 (2002).
[CrossRef]

Meas. Sci. Technol. (1)

A. Makasimenko, H. Sugiyama, K. Hirano, T. Yuasa, M. Ando, “Dark-field imaging using an asymmetric Bragg case transmission analyzer,” Meas. Sci. Technol. 15, 1251–1254 (2004).
[CrossRef]

Nat. Med. (1)

A. Momose, T. Takeda, Y. Itai, K. Hirano, “Phase-contrast x-ray computed tomography for observing biological soft tissues,” Nat. Med. 2, 473–475 (1996).
[CrossRef] [PubMed]

Nat. Rev. Mol. Cell Biol. (2)

R. Y. Tsien, “Imaging imaging’s future,” Nat. Rev. Mol. Cell Biol. 4, SS16–21 (2004).

R. E. Jacobs, C. Papan, S. Ruffines, J. M. Tyszka, S. E. Fraser, “MRI: volumetric imaging for vital imaging and atlas construction,” Nat. Rev. Mol. Cell Biol. 4, SS10–16 (2004).

Nature (1)

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

Opt. Acta (1)

R. W. Gerchberg, “Super-resolution through error energy reduction,” Opt. Acta 21, 709–720 (1974).
[CrossRef]

Opt. Commun. (3)

E. Wolf, “Three dimensional structure determination of semi transparent objects from holographic data,” Opt. Commun. 1, 153–156 (1969).
[CrossRef]

V. V. Protopopov, “On the possibility of X-ray refractive radiography using multilayer mirrors with resonant absorption,” Opt. Commun. 174, 13–18 (2000).
[CrossRef]

V. V. Protopopov, J. Sobota, “X-ray dark-field refraction-contrast imaging of micro-objects,” Opt. Commun. 213, 267–279 (2002).
[CrossRef]

Phys. Med. Biol. (2)

F. A. Dilmanian, Z. Zhong, B. Ren, X. Y. Wu, L. D. Chapman, I. Orion, W. C. Tomlinson, “Computed tomography of x-ray index of refraction using the diffraction enhanced imaging method,” Phys. Med. Biol. 45, 933–946 (2000).
[CrossRef] [PubMed]

M. N. Wernick, O. Wirjadi, D. Chapman, Z. Zhong, N. P. Galatsanos, Y. Yang, J. G. Brankov, O. Oltulu, M. A. Anastasio, C. Muehleman, “Multiple- image radiography,” Phys. Med. Biol. 48, 3875–3895 (2003).
[CrossRef]

Phys. Rev. Lett. (2)

T. J. Davis, T. E. Gureyev, D. Gao, A. W. Stevenson, S. W. Wilkins, “X-ray image contrast from a simple phase object,” Phys. Rev. Lett. 74, 3173–3176 (1995).
[CrossRef] [PubMed]

R. D. Spal, “Submicrometer resolution hard x-ray holography with asymmetric Bragg diffraction microscopy,” Phys. Rev. Lett. 86, 3044–3047 (2001).
[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]

Sov. Phys. Tech. Phys. (1)

V. A. Somenkov, A. K. Tkalich, S. S. Shilstein, “Refraction contrast in X-ray introscopy,” Sov. Phys. Tech. Phys. 61, 197–201 (1991).

Ultrason. Imaging (1)

A. J. Devaney, “A filtered backpropagation algorithm for diffraction tomography,” Ultrason. Imaging 4, 336–350 (1982).
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Other (4)

A. C. Kak, M. Slaney, Principles of Computerized Tomographic Imaging (Society for Industrial and Applied Mathematics, 2001), Vol. 33.
[CrossRef]

A. Authier, Dynamical Theory of X-ray Diffraction (Oxford U. Press, 2001).

R. K. Mueller, M. Kaveh, R. D. Inverson, “A new approach to acoustic tomography using diffraction techniques,” in Acoustical Imaging, A. F. Metherell, ed. (Plenum, 1980).
[CrossRef]

M. Ando, S. Hosoya, “An attempt at x-ray phase-contrast microscopy,” in Proceedings 6th International Conference of X-ray Optics and Microanalysis, G. Shinoda, K. Kohra, and T. Ichinokawa, eds. (University of Tokyo Press, 1972), pp. 63–68.

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Figures (6)

Fig. 1
Fig. 1

(a) Schematic of the imaging system. o ( x , z ) is a 2D object function related to refractive-index variations inside the object. The symmetric Laue case transmission analyzer without absorption is set between z = l 0 and l 1 in order to select only the wave diffracted by the object from the unperturbated wave, so that the x z plane is perpendicular to the crystal surface, and that the negative z axis and the normal vector of the crystal surface are equal to the Bragg angle of the analyzer θ B . A linear array of square-law detectors is set at z = l 1 so that the z axis is perpendicular to the detector surface. A phase object located near the origin and irradiated with a plane wave propagating in the direction toward the positive z axis from a source diffracts the incident wave field, where k is a wavenumber vector of the incident plane wave. The diffracted x-ray wave impinges on the symmetric Laue case transmission analyzer to undergo amplitude and phase modulation. The x ray refracted through the analyzer reaches the linear array of square-law detectors. (b) The Fourier diffraction theorem relates the 1D Fourier transform of a diffracted projection on line z = l 1 to the 2D Fourier transform of the object function o ( x , z ) along a semicircular arc. When an object function o ( x , z ) is illuminated with a plane wave as shown in (a), the Fourier transform of the forwardscattered field measured on the line z = l 1 gives the values related to the 2D transform O ( ξ , ζ ) of the object function along a semicircular arc in the frequency domain.

Fig. 2
Fig. 2

(a) Rotation of the x z coordinate system by the incident angle ϕ, defined as the angle between the positive z axis and the incident propagating vector k. (b) Rotation of the ξ ζ coordinate system by the incident angle ϕ. When an object function o ( x , z ) is illuminated with a plane wave having the propagating vector k ϕ , as shown in (a), the Fourier transform of the forwardscattered field measured on the line z = l 1 gives the information related to the 2D transform O ( ξ , ζ ) of the object function along a semicircular arc in the frequency domain, as shown here, by using the Fourier diffraction theorem. If the object is illuminated by plane waves from many directions over 180   deg , that is, while changing the incident angle ϕ, the resulting circular arcs fill up the frequency domain. The tomographic image related to the object function may then be recovered, because the information over the entire frequency domain was collected.

Fig. 3
Fig. 3

(a) Block diagram of the iterative reconstruction algorithm based on the Gerchberg–Saxton–Fienup algorithm. (b) Block diagram of the detailed update procedure shown as the hatched box in (a).

Fig. 4
Fig. 4

Amplitude properties X ( W ) of the Laue analyzer for incident energy of 35 keV as a function of W when H Λ = 0.5 and 1.0, where H and Λ are the crystal thickness and Λ = λ cos θ B P χ G , respectively. Parameter values are in Section 4. The black and gray curves correspond to H Λ = 0.5 and 1.0, respectively.

Fig. 5
Fig. 5

Point E corresponds to the spatial frequency of the object at which the Laue analyzer transmits the x ray diffracted by the object beam most efficiently. Thus, the diffracted beam having propagation vector S E carries the information at point E.

Fig. 6
Fig. 6

(a) Original numerical phantom as described in the text. (b) Reconstructed image using the proposed method. (c) Image filtered by the Laue high-pass filter.

Equations (43)

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u ( r ) = g ( r r ) o ( r ) u 0 ( r ) d r ,
g ( r r ) = i 4 H 0 ( k 0 r r ) ,
o ( r ) = k 0 2 ( n 2 ( r ) 1 ) .
o ( r ) = 2 k 0 2 δ ( r ) ,
U ( ξ , z = l 0 ) = i exp ( i k 0 2 ξ 2 l 0 ) 2 k 0 2 ξ 2 O ( ξ , k 0 2 ξ 2 k 0 ) .
U ( ξ , z l 0 ) = U ( ξ , z = l 0 ) exp [ i k 0 2 ξ 2 ( z l 0 ) ] .
u ( x , z l 0 ) = ( 1 2 π ) U ( ξ , z = l 0 ) exp [ i ξ x + i k 0 2 ξ 2 ( z l 0 ) ] d ξ .
V ( k 0 sin θ , z = l 1 ) = 1 2 π F ( θ ) U ( k 0 sin θ , z = l 0 ) exp [ i k 0 cos θ ( l 1 l 0 ) ] ,
X ( θ ) = W ( θ ) 2 + cos 2 ( π H W ( θ ) 2 + 1 Λ ) W ( θ ) 2 + 1 ,
ϴ ( θ ) = arctan [ W ( θ ) W ( θ ) 2 + 1 tan ( π H W ( θ ) 2 + 1 Λ ) ] ,
V ( k 0 sin θ , z = l 1 ) = i exp ( i k 0 cos θ l 1 ) 2 k 0 cos θ F ( θ ) O ( k 0 sin θ , k 0 cos θ k 0 ) ,
W ( θ ) = 2 Λ θ sin θ B λ = k 0 θ Λ sin θ B π k 0 sin θ Λ sin θ B π = ξ Λ sin θ B π .
V ( ξ , z = l 1 ) = i exp ( i k 0 2 ξ 2 l 1 ) 2 k 0 2 ξ 2 F ( ξ ) O ( ξ , k 0 2 ξ 2 k 0 ) i exp ( i k 0 l 1 ) 2 k 0 F ( ξ ) O ( ξ , 0 ) ,
V ( ξ , z = l 1 ) = i k 0 exp ( i k 0 l 1 ) F ( ξ ) D ( ξ , 0 ) ,
V ( ξ ϕ ) V ( ξ ϕ , z = l 1 ) = i exp ( i k 0 l 1 ) 2 k 0 F ( ξ ϕ ) O ( ξ ϕ , ζ ϕ = 0 ) = i k 0 exp ( i k 0 l 1 ) F ( ξ ϕ ) D ( ξ ϕ , ζ ϕ = 0 ) .
D * ( ρ ) = D ( ρ ) ,
O * ( ρ ) = O ( ρ ) ,
X ( ρ ) = X ( ρ )
ϴ ( ρ ) = ϴ ( ρ ) .
F * ( ρ ) = X ( ρ ) exp [ i ϴ ( ρ ) ] = X ( ρ ) exp [ i ϴ ( ρ ) ] = F ( ρ ) .
P ( j ) = { P 0 Δ ϕ ( j ) ( N 2 Δ ρ ) , P 0 Δ ϕ ( j ) [ ( N 2 + 1 ) Δ ρ ] , , P 0 Δ ϕ ( j ) [ ( N 2 1 ) Δ ρ ] ; P 1 Δ ϕ ( j ) ( N 2 Δ ρ ) , P 1 Δ ϕ ( j ) ( ( N 2 + 1 ) Δ ρ ) , , P 1 Δ ϕ ( j ) [ ( N 2 1 ) Δ ρ ] ; ; P k Δ ϕ ( j ) ( N 2 Δ ρ ) , P k Δ ϕ ( j ) [ ( N 2 + 1 ) Δ ρ ] , , P k Δ ϕ ( j ) [ ( N 2 1 ) Δ ρ ] ; ; P ( M 1 ) Δ ϕ ( j ) ( N 2 Δ ρ ) , P ( M 1 ) Δ ϕ ( j ) [ ( N 2 + 1 ) Δ ρ ] , , P ( M 1 ) Δ ϕ ( j ) [ ( N 2 1 ) Δ ρ ] } T ,
v = { v 0 Δ ϕ ( N 2 Δ ρ ) , v 0 Δ ϕ [ ( N 2 + 1 ) Δ ρ ] , , v 0 Δ ϕ [ ( N 2 1 ) Δ ρ ] ; v 1 Δ ϕ ( N 2 Δ ρ ) , v 1 Δ ϕ [ ( N 2 + 1 ) Δ ρ ] , , v 1 Δ ϕ [ ( N 2 1 ) Δ ρ ] ; ; v k Δ ϕ ( N 2 Δ ρ ) , v k Δ ϕ [ ( N 2 + 1 ) Δ ρ ] , , v k Δ ϕ [ ( N 2 1 ) Δ ρ ] ; ; v ( M 1 ) Δ ϕ ( N 2 Δ ρ ) , v ( M 1 ) Δ ϕ [ ( N 2 + 1 ) Δ ρ ] , , v ( M 1 ) Δ ϕ [ ( N 2 1 ) Δ ρ ] } T .
E O 2 ( j ) = r q ( j + 1 ) ( r ) q ( j ) ( r ) 2 ,
E F 2 ( j ) = 1 L 2 ρ Q ( j ) ( ρ ) Q ( j ) ( ρ ) 2 .
E F 2 ( j ) = 1 L 2 ρ Q ( j ) ( ρ ) Q ( j ) ( ρ ) 2 = r Q ( j ) ( r ) q ( j ) ( r ) 2 .
q ( j + 1 ) ( r ) q ( j ) ( r ) q ( j ) ( r ) q ( j ) ( r ) .
E O 2 ( j ) E F 2 ( j ) .
E O 2 ( j ) = r q ( j + 1 ) ( r ) q ( j ) ( r ) 2 = 1 L 2 ρ Q ( j + 1 ) ( ρ ) Q ( j ) ( ρ ) 2 .
Q ( j + 1 ) ( ρ ) Q ( j + 1 ) ( ρ ) Q ( j + 1 ) ( ρ ) Q ( j ) ( ρ ) .
E F 2 ( j + 1 ) E O 2 ( j ) .
E F 2 ( j + 1 ) E O 2 ( j ) E F 2 ( j ) .
Q ( j ) = { Q ( j ) ( L 2 Δ ρ , L 2 Δ ρ ) , Q ( j ) [ ( L 2 + 1 ) Δ ρ , L 2 Δ ρ ] , , Q ( j ) [ ( L 2 1 ) Δ ρ , L 2 Δ ρ ] ; Q ( j ) [ L 2 Δ ρ , ( L 2 + 1 ) Δ ρ ] , Q ( j ) [ ( L 2 + 1 ) Δ ρ , ( L 2 + 1 ) Δ ρ ] , , Q ( j ) [ ( L 2 1 ) Δ ρ , ( L 2 + 1 ) Δ ρ ] ; ; Q ( j ) [ L 2 Δ ρ , ( L 2 + t ) Δ ρ ] , Q ( j ) [ ( L 2 + 1 ) Δ ρ , ( L 2 + t ) Δ ρ ] , , Q ( j ) [ ( L 2 1 ) Δ ρ , ( L 2 + t ) Δ ρ ] ; ; Q ( j ) [ L 2 Δ ρ , ( L 2 1 ) Δ ρ ] , Q ( j ) [ ( L 2 + 1 ) Δ ρ , ( L 2 1 ) Δ ρ ] , , Q ( j ) [ ( L 2 1 ) Δ ρ , ( L 2 1 ) Δ ρ ] } T .
x L 2 = T G x M N ,
y M N = T R y L 2 ,
T G T R = I M N ,
T R T G = I L 2 ,
E F 2 ( j ) = 1 L 2 ( Q ( j ) Q ( j ) ) T ( Q ( j ) Q ( j ) ) = 1 L 2 Q ( j ) Q ( j ) L 2 .
p ( j ) = F 1 D 1 T G Q ( j ) ,
Q ( j ) = T R F 1 D p ( j ) .
E F 2 ( j ) = 1 L 2 T R F 1 D ( p ( j ) p ( j ) ) L 2 = 1 L 2 F 1 D ( p ( j ) p ( j ) ) M N = M N L 2 p ( j ) p ( j ) M N ,
E F 2 ( j ) = M N L 2 ( p ( j ) p ( j ) ) T ( p ( j ) p ( j ) ) = M N L 2 k = 0 M 1 m = N 2 N 2 1 p k ( j ) ( m ) p k ( j ) ( m ) 2 .
p k ( j ) ( m ) = v k ( m ) p k ( j ) ( m ) p k ( j ) ( m ) ,
E F 2 ( j ) = M N L 2 k = 0 M 1 m = N 2 N 2 1 p k ( j ) ( m ) v k ( m ) p k ( j ) ( m ) p k ( j ) ( m ) 2 = M N L 2 k = 0 M 1 m = N 2 N 2 1 p k ( j ) ( m ) v k ( m ) 2 .

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