Abstract

We propose an algorithm for tomographic reconstruction of the refractive index map of an object translated across a fan-shaped X-ray beam. We adopt a forward image model valid under the non-paraxial condition, and use a unique mapping of the acquired projection images to reduce the computational cost. Even though the imaging setup affords only a limited angular coverage, our algorithm provides accurate refractive index values by employing the positivity and piecewise-smoothness constraints.

© 2013 OSA

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  1. A. Stanton, “Wilhelm Conrad Röntgen on a new kind of rays: translation of a paper read before the Würzburg Physical and Medical Society, 1895,” Nature53, 274–276 (1896).
  2. C. A. Helms, Fundamentals of Skeletal Radiology (Saunders, 2005).
  3. E. D. Pisano, M. J. Yaffe, and C. M. Kuzmiak, Digital Mammography (Lippincott Williams & Wilkins, 2004).
  4. H. Vogel and D. Haller, “Luggage and shipped goods,” Eur. J. Radiol.63(2), 242–253 (2007).
    [CrossRef] [PubMed]
  5. J. H. Hubbell, Wm. J. Veigele, E. A. Briggs, R. T. Brown, D. T. Cromer, and R. J. Howerton, “Atomic form factors, incoherent scattering functions, and photon scattering cross sections,” J. Phys. Chem. Ref. Data4(3), 471–538 (1975).
    [CrossRef]
  6. D. M. Paganin, Coherent X-ray Optics (Oxford University, 2006).
  7. A. Momose, T. Takeda, and Y. Itai, “Blood vessels: depiction at phase-contrast X-ray imaging without contrast agents in the mouse and rat-feasibility study,” Radiology217(2), 593–596 (2000).
    [PubMed]
  8. S. W. Wilkins, T. E. Gureyev, D. Gao, A. Pogany, and A. Stevenson, “Phase-contrast imaging using polychromatic hard X-rays,” Nature384(6607), 335–338 (1996).
    [CrossRef]
  9. U. Bonse and F. Beckmann, “Multiple-beam X-ray interferometry for phase-contrast microtomography,” J. Synchrotron Radiat.8(1), 1–5 (2001).
    [CrossRef] [PubMed]
  10. M. Ando, A. Maksimenko, H. Sugiyama, W. Pattanasiriwisawa, K. Hyodo, and C. Uyama, “Simple X-ray dark-and bright-field imaging using achromatic Laue optics,” Jpn. J. Appl. Phys.41(Part 2, No. 9A/B), L1016–L1018 (2002).
    [CrossRef]
  11. Y. S. Kashyap, P. S. Yadav, T. Roy, P. S. Sarkar, M. Shukla, and A. Sinha, “Laboratory-based X-ray phase-contrast imaging technique for material and medical science applications,” Appl. Radiat. Isot.66(8), 1083–1090 (2008).
    [CrossRef] [PubMed]
  12. Z. Zaprazny, D. Korytar, V. Ac, P. Konopka, and J. Bielecki, “Phase contrast imaging of lightweight objects using microfocus X-ray source and high resolution CCD camera,” J. Instrum.7(03), C03005 (2012).
    [CrossRef]
  13. T. Weitkamp, C. David, O. Bunk, J. Bruder, P. Cloetens, and F. Pfeiffer, “X-ray phase radiography and tomography of soft tissue using grating interferometry,” Eur. J. Radiol.68(3Suppl), S13–S17 (2008).
    [CrossRef] [PubMed]
  14. F. Pfeiffer, M. Bech, O. Bunk, P. Kraft, E. F. Eikenberry, Ch. Brönnimann, C. Grünzweig, and C. David, “Hard-X-ray dark-field imaging using a grating interferometer,” Nat. Mater.7(2), 134–137 (2008).
    [CrossRef] [PubMed]
  15. A. Olivo and R. Speller, “A coded-aperture technique allowing x-ray phase contrast imaging with laboratory sources,” Appl. Phys. Lett.91(7), 074106 (2007).
    [CrossRef]
  16. H. Wen, E. E. Bennett, M. M. Hegedus, and S. C. Carroll, “Spatial harmonic imaging of X-ray scattering--initial results,” IEEE Trans. Med. Imaging27(8), 997–1002 (2008).
    [CrossRef] [PubMed]
  17. A. Bravin, P. Coan, and P. Suortti, “X-ray phase-contrast imaging: from pre-clinical applications towards clinics,” Phys. Med. Biol.58(1), R1–R35 (2013).
    [CrossRef] [PubMed]
  18. A. C. Kak and M. Slaney, Principles of Computerized Tomographic Imaging (Society for Industrial and Applied Mathematics, 1988).
  19. A. Momose, W. Yashiro, Y. Takeda, Y. Suzuki, and T. Hattori, “Phase tomography by X-ray Talbot interferometry for biological imaging,” Jpn. J. Appl. Phys.45(6A), 5254–5262 (2006).
    [CrossRef]
  20. F. Pfeiffer, C. Kottler, O. Bunk, and C. David, “Hard X-ray phase tomography with low-brilliance sources,” Phys. Rev. Lett.98(10), 108105 (2007).
    [CrossRef] [PubMed]
  21. S. Mayo, T. Davis, T. Gureyev, P. Miller, D. Paganin, A. Pogany, A. Stevenson, and S. Wilkins, “X-ray phase-contrast microscopy and microtomography,” Opt. Express11(19), 2289–2302 (2003).
    [CrossRef] [PubMed]
  22. G. Donges and R. Dietrich, “Baggage inspection system,” U. S. patent 4,759,047 (1988).
  23. R. Dietrich, “Baggage inspection system,” U. S. patent 4,783,794 (1988).
  24. J. M. Park, E. A. Franken, M. Garg, L. L. Fajardo, and L. T. Niklason, “Breast tomosynthesis: present considerations and future applications,” Radiographics27(Suppl 1), S231–S240 (2007).
    [CrossRef] [PubMed]
  25. D. Nahamoo, S. Pan, and A. C. Kak, “Synthetic aperture diffraction tomography and its interpolation-free computer implementation,” IEEE Trans. Sonics Ultrason.31(4), 218–229 (1984).
    [CrossRef]
  26. T. Yasuda, T. Yasui, T. Araki, and E. Abraham, “Real-time two-dimensional terahertz tomography of moving objects,” Opt. Commun.267(1), 128–136 (2006).
    [CrossRef]
  27. N. Lue, W. Choi, G. Popescu, K. Badizadegan, R. R. Dasari, and M. S. Feld, “Synthetic aperture tomographic phase microscopy for 3D imaging of live cells in translational motion,” Opt. Express16(20), 16240–16246 (2008).
    [CrossRef] [PubMed]
  28. Y. Sung, C. J. R. Sheppard, G. Barbastathis, M. Ando, and R. Gupta, “Full-wave approach for X-ray phase imaging,” Opt. Express21(15), 17547–17557 (2013).
    [CrossRef] [PubMed]
  29. A. Burvall, U. Lundström, P. A. C. Takman, D. H. Larsson, and H. M. Hertz, “Phase retrieval in X-ray phase-contrast imaging suitable for tomography,” Opt. Express19(11), 10359–10376 (2011).
    [CrossRef] [PubMed]
  30. M. R. Teague, “Deterministic phase retrieval: a Green’s function solution,” J. Opt. Soc. Am.73(11), 1434–1441 (1983).
    [CrossRef]
  31. N. Streibl, “Phase imaging by the transport equation of intensity,” Opt. Commun.49(1), 6–10 (1984).
    [CrossRef]
  32. B. Henke, E. Gullikson, and J. C. Davis, “X-ray Interactions: Photoabsorption, Scattering, Transmission, and Reflection at E = 50-30,000 eV, Z = 1-92,” At. Data Nucl. Data Tables54(2), 181–342 (1993).
    [CrossRef]
  33. Y. Sung and G. Barbastathis, “Rytov approximation for x-ray phase imaging,” Opt. Express21(3), 2674–2682 (2013).
    [CrossRef] [PubMed]
  34. E. Wolf, “Three-dimensional structure determination of semi-transparent objects from holographic data,” Opt. Commun.1(4), 153–156 (1969).
    [CrossRef]
  35. Y. Sung, W. Choi, C. Fang-Yen, K. Badizadegan, R. R. Dasari, and M. S. Feld, “Optical diffraction tomography for high resolution live cell imaging,” Opt. Express17(1), 266–277 (2009).
    [CrossRef] [PubMed]
  36. P. Charbonnier, L. Blanc-Feraud, G. Aubert, and M. Barlaud, “Deterministic edge-preserving regularization in computed imaging,” IEEE Trans. Image Process.6(2), 298–311 (1997).
    [CrossRef] [PubMed]
  37. Y. Sung and R. R. Dasari, “Deterministic regularization of three-dimensional optical diffraction tomography,” J. Opt. Soc. Am. A28(8), 1554–1561 (2011).
    [CrossRef] [PubMed]
  38. M. Bertero and P. Boccacci, Introduction to Inverse Problems in Imaging (Taylor & Francis, 1998).
  39. L. Ritschl, F. Bergner, and M. Kachelrieß, “A new approach to limited angle tomography using the compressed sensing framework,” Proc. SPIE7622, 76222H, 76222H-9 (2010).
    [CrossRef]

2013

2012

Z. Zaprazny, D. Korytar, V. Ac, P. Konopka, and J. Bielecki, “Phase contrast imaging of lightweight objects using microfocus X-ray source and high resolution CCD camera,” J. Instrum.7(03), C03005 (2012).
[CrossRef]

2011

2010

L. Ritschl, F. Bergner, and M. Kachelrieß, “A new approach to limited angle tomography using the compressed sensing framework,” Proc. SPIE7622, 76222H, 76222H-9 (2010).
[CrossRef]

2009

2008

N. Lue, W. Choi, G. Popescu, K. Badizadegan, R. R. Dasari, and M. S. Feld, “Synthetic aperture tomographic phase microscopy for 3D imaging of live cells in translational motion,” Opt. Express16(20), 16240–16246 (2008).
[CrossRef] [PubMed]

H. Wen, E. E. Bennett, M. M. Hegedus, and S. C. Carroll, “Spatial harmonic imaging of X-ray scattering--initial results,” IEEE Trans. Med. Imaging27(8), 997–1002 (2008).
[CrossRef] [PubMed]

T. Weitkamp, C. David, O. Bunk, J. Bruder, P. Cloetens, and F. Pfeiffer, “X-ray phase radiography and tomography of soft tissue using grating interferometry,” Eur. J. Radiol.68(3Suppl), S13–S17 (2008).
[CrossRef] [PubMed]

F. Pfeiffer, M. Bech, O. Bunk, P. Kraft, E. F. Eikenberry, Ch. Brönnimann, C. Grünzweig, and C. David, “Hard-X-ray dark-field imaging using a grating interferometer,” Nat. Mater.7(2), 134–137 (2008).
[CrossRef] [PubMed]

Y. S. Kashyap, P. S. Yadav, T. Roy, P. S. Sarkar, M. Shukla, and A. Sinha, “Laboratory-based X-ray phase-contrast imaging technique for material and medical science applications,” Appl. Radiat. Isot.66(8), 1083–1090 (2008).
[CrossRef] [PubMed]

2007

H. Vogel and D. Haller, “Luggage and shipped goods,” Eur. J. Radiol.63(2), 242–253 (2007).
[CrossRef] [PubMed]

A. Olivo and R. Speller, “A coded-aperture technique allowing x-ray phase contrast imaging with laboratory sources,” Appl. Phys. Lett.91(7), 074106 (2007).
[CrossRef]

F. Pfeiffer, C. Kottler, O. Bunk, and C. David, “Hard X-ray phase tomography with low-brilliance sources,” Phys. Rev. Lett.98(10), 108105 (2007).
[CrossRef] [PubMed]

J. M. Park, E. A. Franken, M. Garg, L. L. Fajardo, and L. T. Niklason, “Breast tomosynthesis: present considerations and future applications,” Radiographics27(Suppl 1), S231–S240 (2007).
[CrossRef] [PubMed]

2006

A. Momose, W. Yashiro, Y. Takeda, Y. Suzuki, and T. Hattori, “Phase tomography by X-ray Talbot interferometry for biological imaging,” Jpn. J. Appl. Phys.45(6A), 5254–5262 (2006).
[CrossRef]

T. Yasuda, T. Yasui, T. Araki, and E. Abraham, “Real-time two-dimensional terahertz tomography of moving objects,” Opt. Commun.267(1), 128–136 (2006).
[CrossRef]

2003

2002

M. Ando, A. Maksimenko, H. Sugiyama, W. Pattanasiriwisawa, K. Hyodo, and C. Uyama, “Simple X-ray dark-and bright-field imaging using achromatic Laue optics,” Jpn. J. Appl. Phys.41(Part 2, No. 9A/B), L1016–L1018 (2002).
[CrossRef]

2001

U. Bonse and F. Beckmann, “Multiple-beam X-ray interferometry for phase-contrast microtomography,” J. Synchrotron Radiat.8(1), 1–5 (2001).
[CrossRef] [PubMed]

2000

A. Momose, T. Takeda, and Y. Itai, “Blood vessels: depiction at phase-contrast X-ray imaging without contrast agents in the mouse and rat-feasibility study,” Radiology217(2), 593–596 (2000).
[PubMed]

1997

P. Charbonnier, L. Blanc-Feraud, G. Aubert, and M. Barlaud, “Deterministic edge-preserving regularization in computed imaging,” IEEE Trans. Image Process.6(2), 298–311 (1997).
[CrossRef] [PubMed]

1996

S. W. Wilkins, T. E. Gureyev, D. Gao, A. Pogany, and A. Stevenson, “Phase-contrast imaging using polychromatic hard X-rays,” Nature384(6607), 335–338 (1996).
[CrossRef]

1993

B. Henke, E. Gullikson, and J. C. Davis, “X-ray Interactions: Photoabsorption, Scattering, Transmission, and Reflection at E = 50-30,000 eV, Z = 1-92,” At. Data Nucl. Data Tables54(2), 181–342 (1993).
[CrossRef]

1984

N. Streibl, “Phase imaging by the transport equation of intensity,” Opt. Commun.49(1), 6–10 (1984).
[CrossRef]

D. Nahamoo, S. Pan, and A. C. Kak, “Synthetic aperture diffraction tomography and its interpolation-free computer implementation,” IEEE Trans. Sonics Ultrason.31(4), 218–229 (1984).
[CrossRef]

1983

1975

J. H. Hubbell, Wm. J. Veigele, E. A. Briggs, R. T. Brown, D. T. Cromer, and R. J. Howerton, “Atomic form factors, incoherent scattering functions, and photon scattering cross sections,” J. Phys. Chem. Ref. Data4(3), 471–538 (1975).
[CrossRef]

1969

E. Wolf, “Three-dimensional structure determination of semi-transparent objects from holographic data,” Opt. Commun.1(4), 153–156 (1969).
[CrossRef]

1896

A. Stanton, “Wilhelm Conrad Röntgen on a new kind of rays: translation of a paper read before the Würzburg Physical and Medical Society, 1895,” Nature53, 274–276 (1896).

Abraham, E.

T. Yasuda, T. Yasui, T. Araki, and E. Abraham, “Real-time two-dimensional terahertz tomography of moving objects,” Opt. Commun.267(1), 128–136 (2006).
[CrossRef]

Ac, V.

Z. Zaprazny, D. Korytar, V. Ac, P. Konopka, and J. Bielecki, “Phase contrast imaging of lightweight objects using microfocus X-ray source and high resolution CCD camera,” J. Instrum.7(03), C03005 (2012).
[CrossRef]

Ando, M.

Y. Sung, C. J. R. Sheppard, G. Barbastathis, M. Ando, and R. Gupta, “Full-wave approach for X-ray phase imaging,” Opt. Express21(15), 17547–17557 (2013).
[CrossRef] [PubMed]

M. Ando, A. Maksimenko, H. Sugiyama, W. Pattanasiriwisawa, K. Hyodo, and C. Uyama, “Simple X-ray dark-and bright-field imaging using achromatic Laue optics,” Jpn. J. Appl. Phys.41(Part 2, No. 9A/B), L1016–L1018 (2002).
[CrossRef]

Araki, T.

T. Yasuda, T. Yasui, T. Araki, and E. Abraham, “Real-time two-dimensional terahertz tomography of moving objects,” Opt. Commun.267(1), 128–136 (2006).
[CrossRef]

Aubert, G.

P. Charbonnier, L. Blanc-Feraud, G. Aubert, and M. Barlaud, “Deterministic edge-preserving regularization in computed imaging,” IEEE Trans. Image Process.6(2), 298–311 (1997).
[CrossRef] [PubMed]

Badizadegan, K.

Barbastathis, G.

Barlaud, M.

P. Charbonnier, L. Blanc-Feraud, G. Aubert, and M. Barlaud, “Deterministic edge-preserving regularization in computed imaging,” IEEE Trans. Image Process.6(2), 298–311 (1997).
[CrossRef] [PubMed]

Bech, M.

F. Pfeiffer, M. Bech, O. Bunk, P. Kraft, E. F. Eikenberry, Ch. Brönnimann, C. Grünzweig, and C. David, “Hard-X-ray dark-field imaging using a grating interferometer,” Nat. Mater.7(2), 134–137 (2008).
[CrossRef] [PubMed]

Beckmann, F.

U. Bonse and F. Beckmann, “Multiple-beam X-ray interferometry for phase-contrast microtomography,” J. Synchrotron Radiat.8(1), 1–5 (2001).
[CrossRef] [PubMed]

Bennett, E. E.

H. Wen, E. E. Bennett, M. M. Hegedus, and S. C. Carroll, “Spatial harmonic imaging of X-ray scattering--initial results,” IEEE Trans. Med. Imaging27(8), 997–1002 (2008).
[CrossRef] [PubMed]

Bergner, F.

L. Ritschl, F. Bergner, and M. Kachelrieß, “A new approach to limited angle tomography using the compressed sensing framework,” Proc. SPIE7622, 76222H, 76222H-9 (2010).
[CrossRef]

Bielecki, J.

Z. Zaprazny, D. Korytar, V. Ac, P. Konopka, and J. Bielecki, “Phase contrast imaging of lightweight objects using microfocus X-ray source and high resolution CCD camera,” J. Instrum.7(03), C03005 (2012).
[CrossRef]

Blanc-Feraud, L.

P. Charbonnier, L. Blanc-Feraud, G. Aubert, and M. Barlaud, “Deterministic edge-preserving regularization in computed imaging,” IEEE Trans. Image Process.6(2), 298–311 (1997).
[CrossRef] [PubMed]

Bonse, U.

U. Bonse and F. Beckmann, “Multiple-beam X-ray interferometry for phase-contrast microtomography,” J. Synchrotron Radiat.8(1), 1–5 (2001).
[CrossRef] [PubMed]

Bravin, A.

A. Bravin, P. Coan, and P. Suortti, “X-ray phase-contrast imaging: from pre-clinical applications towards clinics,” Phys. Med. Biol.58(1), R1–R35 (2013).
[CrossRef] [PubMed]

Briggs, E. A.

J. H. Hubbell, Wm. J. Veigele, E. A. Briggs, R. T. Brown, D. T. Cromer, and R. J. Howerton, “Atomic form factors, incoherent scattering functions, and photon scattering cross sections,” J. Phys. Chem. Ref. Data4(3), 471–538 (1975).
[CrossRef]

Brönnimann, Ch.

F. Pfeiffer, M. Bech, O. Bunk, P. Kraft, E. F. Eikenberry, Ch. Brönnimann, C. Grünzweig, and C. David, “Hard-X-ray dark-field imaging using a grating interferometer,” Nat. Mater.7(2), 134–137 (2008).
[CrossRef] [PubMed]

Brown, R. T.

J. H. Hubbell, Wm. J. Veigele, E. A. Briggs, R. T. Brown, D. T. Cromer, and R. J. Howerton, “Atomic form factors, incoherent scattering functions, and photon scattering cross sections,” J. Phys. Chem. Ref. Data4(3), 471–538 (1975).
[CrossRef]

Bruder, J.

T. Weitkamp, C. David, O. Bunk, J. Bruder, P. Cloetens, and F. Pfeiffer, “X-ray phase radiography and tomography of soft tissue using grating interferometry,” Eur. J. Radiol.68(3Suppl), S13–S17 (2008).
[CrossRef] [PubMed]

Bunk, O.

F. Pfeiffer, M. Bech, O. Bunk, P. Kraft, E. F. Eikenberry, Ch. Brönnimann, C. Grünzweig, and C. David, “Hard-X-ray dark-field imaging using a grating interferometer,” Nat. Mater.7(2), 134–137 (2008).
[CrossRef] [PubMed]

T. Weitkamp, C. David, O. Bunk, J. Bruder, P. Cloetens, and F. Pfeiffer, “X-ray phase radiography and tomography of soft tissue using grating interferometry,” Eur. J. Radiol.68(3Suppl), S13–S17 (2008).
[CrossRef] [PubMed]

F. Pfeiffer, C. Kottler, O. Bunk, and C. David, “Hard X-ray phase tomography with low-brilliance sources,” Phys. Rev. Lett.98(10), 108105 (2007).
[CrossRef] [PubMed]

Burvall, A.

Carroll, S. C.

H. Wen, E. E. Bennett, M. M. Hegedus, and S. C. Carroll, “Spatial harmonic imaging of X-ray scattering--initial results,” IEEE Trans. Med. Imaging27(8), 997–1002 (2008).
[CrossRef] [PubMed]

Charbonnier, P.

P. Charbonnier, L. Blanc-Feraud, G. Aubert, and M. Barlaud, “Deterministic edge-preserving regularization in computed imaging,” IEEE Trans. Image Process.6(2), 298–311 (1997).
[CrossRef] [PubMed]

Choi, W.

Cloetens, P.

T. Weitkamp, C. David, O. Bunk, J. Bruder, P. Cloetens, and F. Pfeiffer, “X-ray phase radiography and tomography of soft tissue using grating interferometry,” Eur. J. Radiol.68(3Suppl), S13–S17 (2008).
[CrossRef] [PubMed]

Coan, P.

A. Bravin, P. Coan, and P. Suortti, “X-ray phase-contrast imaging: from pre-clinical applications towards clinics,” Phys. Med. Biol.58(1), R1–R35 (2013).
[CrossRef] [PubMed]

Cromer, D. T.

J. H. Hubbell, Wm. J. Veigele, E. A. Briggs, R. T. Brown, D. T. Cromer, and R. J. Howerton, “Atomic form factors, incoherent scattering functions, and photon scattering cross sections,” J. Phys. Chem. Ref. Data4(3), 471–538 (1975).
[CrossRef]

Dasari, R. R.

David, C.

T. Weitkamp, C. David, O. Bunk, J. Bruder, P. Cloetens, and F. Pfeiffer, “X-ray phase radiography and tomography of soft tissue using grating interferometry,” Eur. J. Radiol.68(3Suppl), S13–S17 (2008).
[CrossRef] [PubMed]

F. Pfeiffer, M. Bech, O. Bunk, P. Kraft, E. F. Eikenberry, Ch. Brönnimann, C. Grünzweig, and C. David, “Hard-X-ray dark-field imaging using a grating interferometer,” Nat. Mater.7(2), 134–137 (2008).
[CrossRef] [PubMed]

F. Pfeiffer, C. Kottler, O. Bunk, and C. David, “Hard X-ray phase tomography with low-brilliance sources,” Phys. Rev. Lett.98(10), 108105 (2007).
[CrossRef] [PubMed]

Davis, J. C.

B. Henke, E. Gullikson, and J. C. Davis, “X-ray Interactions: Photoabsorption, Scattering, Transmission, and Reflection at E = 50-30,000 eV, Z = 1-92,” At. Data Nucl. Data Tables54(2), 181–342 (1993).
[CrossRef]

Davis, T.

Eikenberry, E. F.

F. Pfeiffer, M. Bech, O. Bunk, P. Kraft, E. F. Eikenberry, Ch. Brönnimann, C. Grünzweig, and C. David, “Hard-X-ray dark-field imaging using a grating interferometer,” Nat. Mater.7(2), 134–137 (2008).
[CrossRef] [PubMed]

Fajardo, L. L.

J. M. Park, E. A. Franken, M. Garg, L. L. Fajardo, and L. T. Niklason, “Breast tomosynthesis: present considerations and future applications,” Radiographics27(Suppl 1), S231–S240 (2007).
[CrossRef] [PubMed]

Fang-Yen, C.

Feld, M. S.

Franken, E. A.

J. M. Park, E. A. Franken, M. Garg, L. L. Fajardo, and L. T. Niklason, “Breast tomosynthesis: present considerations and future applications,” Radiographics27(Suppl 1), S231–S240 (2007).
[CrossRef] [PubMed]

Gao, D.

S. W. Wilkins, T. E. Gureyev, D. Gao, A. Pogany, and A. Stevenson, “Phase-contrast imaging using polychromatic hard X-rays,” Nature384(6607), 335–338 (1996).
[CrossRef]

Garg, M.

J. M. Park, E. A. Franken, M. Garg, L. L. Fajardo, and L. T. Niklason, “Breast tomosynthesis: present considerations and future applications,” Radiographics27(Suppl 1), S231–S240 (2007).
[CrossRef] [PubMed]

Grünzweig, C.

F. Pfeiffer, M. Bech, O. Bunk, P. Kraft, E. F. Eikenberry, Ch. Brönnimann, C. Grünzweig, and C. David, “Hard-X-ray dark-field imaging using a grating interferometer,” Nat. Mater.7(2), 134–137 (2008).
[CrossRef] [PubMed]

Gullikson, E.

B. Henke, E. Gullikson, and J. C. Davis, “X-ray Interactions: Photoabsorption, Scattering, Transmission, and Reflection at E = 50-30,000 eV, Z = 1-92,” At. Data Nucl. Data Tables54(2), 181–342 (1993).
[CrossRef]

Gupta, R.

Gureyev, T.

Gureyev, T. E.

S. W. Wilkins, T. E. Gureyev, D. Gao, A. Pogany, and A. Stevenson, “Phase-contrast imaging using polychromatic hard X-rays,” Nature384(6607), 335–338 (1996).
[CrossRef]

Haller, D.

H. Vogel and D. Haller, “Luggage and shipped goods,” Eur. J. Radiol.63(2), 242–253 (2007).
[CrossRef] [PubMed]

Hattori, T.

A. Momose, W. Yashiro, Y. Takeda, Y. Suzuki, and T. Hattori, “Phase tomography by X-ray Talbot interferometry for biological imaging,” Jpn. J. Appl. Phys.45(6A), 5254–5262 (2006).
[CrossRef]

Hegedus, M. M.

H. Wen, E. E. Bennett, M. M. Hegedus, and S. C. Carroll, “Spatial harmonic imaging of X-ray scattering--initial results,” IEEE Trans. Med. Imaging27(8), 997–1002 (2008).
[CrossRef] [PubMed]

Henke, B.

B. Henke, E. Gullikson, and J. C. Davis, “X-ray Interactions: Photoabsorption, Scattering, Transmission, and Reflection at E = 50-30,000 eV, Z = 1-92,” At. Data Nucl. Data Tables54(2), 181–342 (1993).
[CrossRef]

Hertz, H. M.

Howerton, R. J.

J. H. Hubbell, Wm. J. Veigele, E. A. Briggs, R. T. Brown, D. T. Cromer, and R. J. Howerton, “Atomic form factors, incoherent scattering functions, and photon scattering cross sections,” J. Phys. Chem. Ref. Data4(3), 471–538 (1975).
[CrossRef]

Hubbell, J. H.

J. H. Hubbell, Wm. J. Veigele, E. A. Briggs, R. T. Brown, D. T. Cromer, and R. J. Howerton, “Atomic form factors, incoherent scattering functions, and photon scattering cross sections,” J. Phys. Chem. Ref. Data4(3), 471–538 (1975).
[CrossRef]

Hyodo, K.

M. Ando, A. Maksimenko, H. Sugiyama, W. Pattanasiriwisawa, K. Hyodo, and C. Uyama, “Simple X-ray dark-and bright-field imaging using achromatic Laue optics,” Jpn. J. Appl. Phys.41(Part 2, No. 9A/B), L1016–L1018 (2002).
[CrossRef]

Itai, Y.

A. Momose, T. Takeda, and Y. Itai, “Blood vessels: depiction at phase-contrast X-ray imaging without contrast agents in the mouse and rat-feasibility study,” Radiology217(2), 593–596 (2000).
[PubMed]

Kachelrieß, M.

L. Ritschl, F. Bergner, and M. Kachelrieß, “A new approach to limited angle tomography using the compressed sensing framework,” Proc. SPIE7622, 76222H, 76222H-9 (2010).
[CrossRef]

Kak, A. C.

D. Nahamoo, S. Pan, and A. C. Kak, “Synthetic aperture diffraction tomography and its interpolation-free computer implementation,” IEEE Trans. Sonics Ultrason.31(4), 218–229 (1984).
[CrossRef]

Kashyap, Y. S.

Y. S. Kashyap, P. S. Yadav, T. Roy, P. S. Sarkar, M. Shukla, and A. Sinha, “Laboratory-based X-ray phase-contrast imaging technique for material and medical science applications,” Appl. Radiat. Isot.66(8), 1083–1090 (2008).
[CrossRef] [PubMed]

Konopka, P.

Z. Zaprazny, D. Korytar, V. Ac, P. Konopka, and J. Bielecki, “Phase contrast imaging of lightweight objects using microfocus X-ray source and high resolution CCD camera,” J. Instrum.7(03), C03005 (2012).
[CrossRef]

Korytar, D.

Z. Zaprazny, D. Korytar, V. Ac, P. Konopka, and J. Bielecki, “Phase contrast imaging of lightweight objects using microfocus X-ray source and high resolution CCD camera,” J. Instrum.7(03), C03005 (2012).
[CrossRef]

Kottler, C.

F. Pfeiffer, C. Kottler, O. Bunk, and C. David, “Hard X-ray phase tomography with low-brilliance sources,” Phys. Rev. Lett.98(10), 108105 (2007).
[CrossRef] [PubMed]

Kraft, P.

F. Pfeiffer, M. Bech, O. Bunk, P. Kraft, E. F. Eikenberry, Ch. Brönnimann, C. Grünzweig, and C. David, “Hard-X-ray dark-field imaging using a grating interferometer,” Nat. Mater.7(2), 134–137 (2008).
[CrossRef] [PubMed]

Larsson, D. H.

Lue, N.

Lundström, U.

Maksimenko, A.

M. Ando, A. Maksimenko, H. Sugiyama, W. Pattanasiriwisawa, K. Hyodo, and C. Uyama, “Simple X-ray dark-and bright-field imaging using achromatic Laue optics,” Jpn. J. Appl. Phys.41(Part 2, No. 9A/B), L1016–L1018 (2002).
[CrossRef]

Mayo, S.

Miller, P.

Momose, A.

A. Momose, W. Yashiro, Y. Takeda, Y. Suzuki, and T. Hattori, “Phase tomography by X-ray Talbot interferometry for biological imaging,” Jpn. J. Appl. Phys.45(6A), 5254–5262 (2006).
[CrossRef]

A. Momose, T. Takeda, and Y. Itai, “Blood vessels: depiction at phase-contrast X-ray imaging without contrast agents in the mouse and rat-feasibility study,” Radiology217(2), 593–596 (2000).
[PubMed]

Nahamoo, D.

D. Nahamoo, S. Pan, and A. C. Kak, “Synthetic aperture diffraction tomography and its interpolation-free computer implementation,” IEEE Trans. Sonics Ultrason.31(4), 218–229 (1984).
[CrossRef]

Niklason, L. T.

J. M. Park, E. A. Franken, M. Garg, L. L. Fajardo, and L. T. Niklason, “Breast tomosynthesis: present considerations and future applications,” Radiographics27(Suppl 1), S231–S240 (2007).
[CrossRef] [PubMed]

Olivo, A.

A. Olivo and R. Speller, “A coded-aperture technique allowing x-ray phase contrast imaging with laboratory sources,” Appl. Phys. Lett.91(7), 074106 (2007).
[CrossRef]

Paganin, D.

Pan, S.

D. Nahamoo, S. Pan, and A. C. Kak, “Synthetic aperture diffraction tomography and its interpolation-free computer implementation,” IEEE Trans. Sonics Ultrason.31(4), 218–229 (1984).
[CrossRef]

Park, J. M.

J. M. Park, E. A. Franken, M. Garg, L. L. Fajardo, and L. T. Niklason, “Breast tomosynthesis: present considerations and future applications,” Radiographics27(Suppl 1), S231–S240 (2007).
[CrossRef] [PubMed]

Pattanasiriwisawa, W.

M. Ando, A. Maksimenko, H. Sugiyama, W. Pattanasiriwisawa, K. Hyodo, and C. Uyama, “Simple X-ray dark-and bright-field imaging using achromatic Laue optics,” Jpn. J. Appl. Phys.41(Part 2, No. 9A/B), L1016–L1018 (2002).
[CrossRef]

Pfeiffer, F.

T. Weitkamp, C. David, O. Bunk, J. Bruder, P. Cloetens, and F. Pfeiffer, “X-ray phase radiography and tomography of soft tissue using grating interferometry,” Eur. J. Radiol.68(3Suppl), S13–S17 (2008).
[CrossRef] [PubMed]

F. Pfeiffer, M. Bech, O. Bunk, P. Kraft, E. F. Eikenberry, Ch. Brönnimann, C. Grünzweig, and C. David, “Hard-X-ray dark-field imaging using a grating interferometer,” Nat. Mater.7(2), 134–137 (2008).
[CrossRef] [PubMed]

F. Pfeiffer, C. Kottler, O. Bunk, and C. David, “Hard X-ray phase tomography with low-brilliance sources,” Phys. Rev. Lett.98(10), 108105 (2007).
[CrossRef] [PubMed]

Pogany, A.

S. Mayo, T. Davis, T. Gureyev, P. Miller, D. Paganin, A. Pogany, A. Stevenson, and S. Wilkins, “X-ray phase-contrast microscopy and microtomography,” Opt. Express11(19), 2289–2302 (2003).
[CrossRef] [PubMed]

S. W. Wilkins, T. E. Gureyev, D. Gao, A. Pogany, and A. Stevenson, “Phase-contrast imaging using polychromatic hard X-rays,” Nature384(6607), 335–338 (1996).
[CrossRef]

Popescu, G.

Ritschl, L.

L. Ritschl, F. Bergner, and M. Kachelrieß, “A new approach to limited angle tomography using the compressed sensing framework,” Proc. SPIE7622, 76222H, 76222H-9 (2010).
[CrossRef]

Roy, T.

Y. S. Kashyap, P. S. Yadav, T. Roy, P. S. Sarkar, M. Shukla, and A. Sinha, “Laboratory-based X-ray phase-contrast imaging technique for material and medical science applications,” Appl. Radiat. Isot.66(8), 1083–1090 (2008).
[CrossRef] [PubMed]

Sarkar, P. S.

Y. S. Kashyap, P. S. Yadav, T. Roy, P. S. Sarkar, M. Shukla, and A. Sinha, “Laboratory-based X-ray phase-contrast imaging technique for material and medical science applications,” Appl. Radiat. Isot.66(8), 1083–1090 (2008).
[CrossRef] [PubMed]

Sheppard, C. J. R.

Shukla, M.

Y. S. Kashyap, P. S. Yadav, T. Roy, P. S. Sarkar, M. Shukla, and A. Sinha, “Laboratory-based X-ray phase-contrast imaging technique for material and medical science applications,” Appl. Radiat. Isot.66(8), 1083–1090 (2008).
[CrossRef] [PubMed]

Sinha, A.

Y. S. Kashyap, P. S. Yadav, T. Roy, P. S. Sarkar, M. Shukla, and A. Sinha, “Laboratory-based X-ray phase-contrast imaging technique for material and medical science applications,” Appl. Radiat. Isot.66(8), 1083–1090 (2008).
[CrossRef] [PubMed]

Speller, R.

A. Olivo and R. Speller, “A coded-aperture technique allowing x-ray phase contrast imaging with laboratory sources,” Appl. Phys. Lett.91(7), 074106 (2007).
[CrossRef]

Stanton, A.

A. Stanton, “Wilhelm Conrad Röntgen on a new kind of rays: translation of a paper read before the Würzburg Physical and Medical Society, 1895,” Nature53, 274–276 (1896).

Stevenson, A.

S. Mayo, T. Davis, T. Gureyev, P. Miller, D. Paganin, A. Pogany, A. Stevenson, and S. Wilkins, “X-ray phase-contrast microscopy and microtomography,” Opt. Express11(19), 2289–2302 (2003).
[CrossRef] [PubMed]

S. W. Wilkins, T. E. Gureyev, D. Gao, A. Pogany, and A. Stevenson, “Phase-contrast imaging using polychromatic hard X-rays,” Nature384(6607), 335–338 (1996).
[CrossRef]

Streibl, N.

N. Streibl, “Phase imaging by the transport equation of intensity,” Opt. Commun.49(1), 6–10 (1984).
[CrossRef]

Sugiyama, H.

M. Ando, A. Maksimenko, H. Sugiyama, W. Pattanasiriwisawa, K. Hyodo, and C. Uyama, “Simple X-ray dark-and bright-field imaging using achromatic Laue optics,” Jpn. J. Appl. Phys.41(Part 2, No. 9A/B), L1016–L1018 (2002).
[CrossRef]

Sung, Y.

Suortti, P.

A. Bravin, P. Coan, and P. Suortti, “X-ray phase-contrast imaging: from pre-clinical applications towards clinics,” Phys. Med. Biol.58(1), R1–R35 (2013).
[CrossRef] [PubMed]

Suzuki, Y.

A. Momose, W. Yashiro, Y. Takeda, Y. Suzuki, and T. Hattori, “Phase tomography by X-ray Talbot interferometry for biological imaging,” Jpn. J. Appl. Phys.45(6A), 5254–5262 (2006).
[CrossRef]

Takeda, T.

A. Momose, T. Takeda, and Y. Itai, “Blood vessels: depiction at phase-contrast X-ray imaging without contrast agents in the mouse and rat-feasibility study,” Radiology217(2), 593–596 (2000).
[PubMed]

Takeda, Y.

A. Momose, W. Yashiro, Y. Takeda, Y. Suzuki, and T. Hattori, “Phase tomography by X-ray Talbot interferometry for biological imaging,” Jpn. J. Appl. Phys.45(6A), 5254–5262 (2006).
[CrossRef]

Takman, P. A. C.

Teague, M. R.

Uyama, C.

M. Ando, A. Maksimenko, H. Sugiyama, W. Pattanasiriwisawa, K. Hyodo, and C. Uyama, “Simple X-ray dark-and bright-field imaging using achromatic Laue optics,” Jpn. J. Appl. Phys.41(Part 2, No. 9A/B), L1016–L1018 (2002).
[CrossRef]

Veigele, Wm. J.

J. H. Hubbell, Wm. J. Veigele, E. A. Briggs, R. T. Brown, D. T. Cromer, and R. J. Howerton, “Atomic form factors, incoherent scattering functions, and photon scattering cross sections,” J. Phys. Chem. Ref. Data4(3), 471–538 (1975).
[CrossRef]

Vogel, H.

H. Vogel and D. Haller, “Luggage and shipped goods,” Eur. J. Radiol.63(2), 242–253 (2007).
[CrossRef] [PubMed]

Weitkamp, T.

T. Weitkamp, C. David, O. Bunk, J. Bruder, P. Cloetens, and F. Pfeiffer, “X-ray phase radiography and tomography of soft tissue using grating interferometry,” Eur. J. Radiol.68(3Suppl), S13–S17 (2008).
[CrossRef] [PubMed]

Wen, H.

H. Wen, E. E. Bennett, M. M. Hegedus, and S. C. Carroll, “Spatial harmonic imaging of X-ray scattering--initial results,” IEEE Trans. Med. Imaging27(8), 997–1002 (2008).
[CrossRef] [PubMed]

Wilkins, S.

Wilkins, S. W.

S. W. Wilkins, T. E. Gureyev, D. Gao, A. Pogany, and A. Stevenson, “Phase-contrast imaging using polychromatic hard X-rays,” Nature384(6607), 335–338 (1996).
[CrossRef]

Wolf, E.

E. Wolf, “Three-dimensional structure determination of semi-transparent objects from holographic data,” Opt. Commun.1(4), 153–156 (1969).
[CrossRef]

Yadav, P. S.

Y. S. Kashyap, P. S. Yadav, T. Roy, P. S. Sarkar, M. Shukla, and A. Sinha, “Laboratory-based X-ray phase-contrast imaging technique for material and medical science applications,” Appl. Radiat. Isot.66(8), 1083–1090 (2008).
[CrossRef] [PubMed]

Yashiro, W.

A. Momose, W. Yashiro, Y. Takeda, Y. Suzuki, and T. Hattori, “Phase tomography by X-ray Talbot interferometry for biological imaging,” Jpn. J. Appl. Phys.45(6A), 5254–5262 (2006).
[CrossRef]

Yasuda, T.

T. Yasuda, T. Yasui, T. Araki, and E. Abraham, “Real-time two-dimensional terahertz tomography of moving objects,” Opt. Commun.267(1), 128–136 (2006).
[CrossRef]

Yasui, T.

T. Yasuda, T. Yasui, T. Araki, and E. Abraham, “Real-time two-dimensional terahertz tomography of moving objects,” Opt. Commun.267(1), 128–136 (2006).
[CrossRef]

Zaprazny, Z.

Z. Zaprazny, D. Korytar, V. Ac, P. Konopka, and J. Bielecki, “Phase contrast imaging of lightweight objects using microfocus X-ray source and high resolution CCD camera,” J. Instrum.7(03), C03005 (2012).
[CrossRef]

Appl. Phys. Lett.

A. Olivo and R. Speller, “A coded-aperture technique allowing x-ray phase contrast imaging with laboratory sources,” Appl. Phys. Lett.91(7), 074106 (2007).
[CrossRef]

Appl. Radiat. Isot.

Y. S. Kashyap, P. S. Yadav, T. Roy, P. S. Sarkar, M. Shukla, and A. Sinha, “Laboratory-based X-ray phase-contrast imaging technique for material and medical science applications,” Appl. Radiat. Isot.66(8), 1083–1090 (2008).
[CrossRef] [PubMed]

At. Data Nucl. Data Tables

B. Henke, E. Gullikson, and J. C. Davis, “X-ray Interactions: Photoabsorption, Scattering, Transmission, and Reflection at E = 50-30,000 eV, Z = 1-92,” At. Data Nucl. Data Tables54(2), 181–342 (1993).
[CrossRef]

Eur. J. Radiol.

T. Weitkamp, C. David, O. Bunk, J. Bruder, P. Cloetens, and F. Pfeiffer, “X-ray phase radiography and tomography of soft tissue using grating interferometry,” Eur. J. Radiol.68(3Suppl), S13–S17 (2008).
[CrossRef] [PubMed]

H. Vogel and D. Haller, “Luggage and shipped goods,” Eur. J. Radiol.63(2), 242–253 (2007).
[CrossRef] [PubMed]

IEEE Trans. Image Process.

P. Charbonnier, L. Blanc-Feraud, G. Aubert, and M. Barlaud, “Deterministic edge-preserving regularization in computed imaging,” IEEE Trans. Image Process.6(2), 298–311 (1997).
[CrossRef] [PubMed]

IEEE Trans. Med. Imaging

H. Wen, E. E. Bennett, M. M. Hegedus, and S. C. Carroll, “Spatial harmonic imaging of X-ray scattering--initial results,” IEEE Trans. Med. Imaging27(8), 997–1002 (2008).
[CrossRef] [PubMed]

IEEE Trans. Sonics Ultrason.

D. Nahamoo, S. Pan, and A. C. Kak, “Synthetic aperture diffraction tomography and its interpolation-free computer implementation,” IEEE Trans. Sonics Ultrason.31(4), 218–229 (1984).
[CrossRef]

J. Instrum.

Z. Zaprazny, D. Korytar, V. Ac, P. Konopka, and J. Bielecki, “Phase contrast imaging of lightweight objects using microfocus X-ray source and high resolution CCD camera,” J. Instrum.7(03), C03005 (2012).
[CrossRef]

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

J. Phys. Chem. Ref. Data

J. H. Hubbell, Wm. J. Veigele, E. A. Briggs, R. T. Brown, D. T. Cromer, and R. J. Howerton, “Atomic form factors, incoherent scattering functions, and photon scattering cross sections,” J. Phys. Chem. Ref. Data4(3), 471–538 (1975).
[CrossRef]

J. Synchrotron Radiat.

U. Bonse and F. Beckmann, “Multiple-beam X-ray interferometry for phase-contrast microtomography,” J. Synchrotron Radiat.8(1), 1–5 (2001).
[CrossRef] [PubMed]

Jpn. J. Appl. Phys.

M. Ando, A. Maksimenko, H. Sugiyama, W. Pattanasiriwisawa, K. Hyodo, and C. Uyama, “Simple X-ray dark-and bright-field imaging using achromatic Laue optics,” Jpn. J. Appl. Phys.41(Part 2, No. 9A/B), L1016–L1018 (2002).
[CrossRef]

A. Momose, W. Yashiro, Y. Takeda, Y. Suzuki, and T. Hattori, “Phase tomography by X-ray Talbot interferometry for biological imaging,” Jpn. J. Appl. Phys.45(6A), 5254–5262 (2006).
[CrossRef]

Nat. Mater.

F. Pfeiffer, M. Bech, O. Bunk, P. Kraft, E. F. Eikenberry, Ch. Brönnimann, C. Grünzweig, and C. David, “Hard-X-ray dark-field imaging using a grating interferometer,” Nat. Mater.7(2), 134–137 (2008).
[CrossRef] [PubMed]

Nature

S. W. Wilkins, T. E. Gureyev, D. Gao, A. Pogany, and A. Stevenson, “Phase-contrast imaging using polychromatic hard X-rays,” Nature384(6607), 335–338 (1996).
[CrossRef]

A. Stanton, “Wilhelm Conrad Röntgen on a new kind of rays: translation of a paper read before the Würzburg Physical and Medical Society, 1895,” Nature53, 274–276 (1896).

Opt. Commun.

T. Yasuda, T. Yasui, T. Araki, and E. Abraham, “Real-time two-dimensional terahertz tomography of moving objects,” Opt. Commun.267(1), 128–136 (2006).
[CrossRef]

N. Streibl, “Phase imaging by the transport equation of intensity,” Opt. Commun.49(1), 6–10 (1984).
[CrossRef]

E. Wolf, “Three-dimensional structure determination of semi-transparent objects from holographic data,” Opt. Commun.1(4), 153–156 (1969).
[CrossRef]

Opt. Express

Phys. Med. Biol.

A. Bravin, P. Coan, and P. Suortti, “X-ray phase-contrast imaging: from pre-clinical applications towards clinics,” Phys. Med. Biol.58(1), R1–R35 (2013).
[CrossRef] [PubMed]

Phys. Rev. Lett.

F. Pfeiffer, C. Kottler, O. Bunk, and C. David, “Hard X-ray phase tomography with low-brilliance sources,” Phys. Rev. Lett.98(10), 108105 (2007).
[CrossRef] [PubMed]

Proc. SPIE

L. Ritschl, F. Bergner, and M. Kachelrieß, “A new approach to limited angle tomography using the compressed sensing framework,” Proc. SPIE7622, 76222H, 76222H-9 (2010).
[CrossRef]

Radiographics

J. M. Park, E. A. Franken, M. Garg, L. L. Fajardo, and L. T. Niklason, “Breast tomosynthesis: present considerations and future applications,” Radiographics27(Suppl 1), S231–S240 (2007).
[CrossRef] [PubMed]

Radiology

A. Momose, T. Takeda, and Y. Itai, “Blood vessels: depiction at phase-contrast X-ray imaging without contrast agents in the mouse and rat-feasibility study,” Radiology217(2), 593–596 (2000).
[PubMed]

Other

C. A. Helms, Fundamentals of Skeletal Radiology (Saunders, 2005).

E. D. Pisano, M. J. Yaffe, and C. M. Kuzmiak, Digital Mammography (Lippincott Williams & Wilkins, 2004).

D. M. Paganin, Coherent X-ray Optics (Oxford University, 2006).

A. C. Kak and M. Slaney, Principles of Computerized Tomographic Imaging (Society for Industrial and Applied Mathematics, 1988).

M. Bertero and P. Boccacci, Introduction to Inverse Problems in Imaging (Taylor & Francis, 1998).

G. Donges and R. Dietrich, “Baggage inspection system,” U. S. patent 4,759,047 (1988).

R. Dietrich, “Baggage inspection system,” U. S. patent 4,783,794 (1988).

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Figures (5)

Fig. 1
Fig. 1

Schematic diagram of X-ray phase tomography set-up for a moving sample.

Fig. 2
Fig. 2

(a) Example of a numerically generated phase profile for a cylinder (1mm dia.) with negligible absorption (δ = 2.56 × 10−7, β = 0.00), and located at a = 0.289 m. In this simulation, the source energy is 30 keV, detector pitch is 5 μm, and R1 = R2 = 0.5 m. (b) 2-D mapping of the data acquired for different object locations along the x-axis.

Fig. 3
Fig. 3

(a) Graphical representation of how to perform the mapping encoded by Eq. (4) on the data shown in Fig. 2(b). (b) The result of applying the mapping Eq. (4) to the diffracted fields measured for the cylinder considered in this study. The unwrapped phase is plotted.

Fig. 4
Fig. 4

(a) Spatial frequency spectrum of the cylinder obtained with the Fourier mapping. The amplitude of the spectrum in the logarithmic scale of base 10 is plotted. The maximum angle of incidence corresponds to 60°. (b) Refractive index map of the cylinder reconstructed from Fig. 4(a). (c) Refractive index histogram of the map in Fig. 4(b). (d) Spatial frequency spectrum of the cylinder obtained with the iterative reconstruction algorithm proposed in this study. The amplitude of the spectrum in the logarithmic scale of base 10 is plotted. (e) Refractive index map of the cylinder reconstructed from Fig. 4(d). (f) Refractive index histogram of the map in Fig. 4(e).

Fig. 5
Fig. 5

(a) Original refractive index map of the phantom considered in this study. The cylinder of diameter 200 mm is filled with water; it contains four cylinders (diameter 40 mm) of different materials (polypropylene, Mylar, Teflon, and PMMA). The refractive index values of the materials at the energy used in this study (30 keV) are listed in Table 1. (b) Reconstructed map of the refractive index after 1000 iterations. (c) Reconstruction using the Fourier mapping and direct inversion (without regularization). (d) Mean refractive index value within each cylinder region plotted at each iteration step. The value was normalized by the true value for each material.

Tables (1)

Tables Icon

Table 1 List of materials and refractive index values (at 30 keV) for the phantom considered in this study.

Equations (10)

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

n=1δ+iβ,
u( x;a )=( A s /r ) e ikr exp[ u ¯ (s) ( x;a ) ],
u ¯ (s) ( x;a )= ( i4πw ) 1 f ˜ ( U,W )exp{ i2π[ U( xa )+W R 2 ] }dU .
U s ( s;α )= u ¯ (s) ( α;α/M s ).
U s ( s;α )= ( i4πw ) 1 f ˜ ( U,W )exp{ i2π[ U( s+αα' )+W R 2 ] }dU ,
f ˜ ( U,W )=i4π( W+ m 1 /λ )exp{ i2π[ U( αα' )+W R 2 ] } U ˜ s ( U;α ),
A n f= U s ( s; α n ), n = 1, 2, , N,
Φ μ ( f; g 1 , g 2 ,, g N )= 1 2 n=1 N A n f U s ( s; α n ) 2 +μJ( f ),
f ( k+1 ) = f ( k ) +τ n ( A n g n A n A n f ( k ) ) τμJ( f ( k ) ),
n ( k+1 ) n ( k ) 2 <ε,

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