Abstract

We derive a Fourier formulation of coded-aperture x-ray phase-contrast imaging, based on the wave theory of optics in the Fresnel approximation. We use this model to develop a flexible, efficient, and general simulation algorithm that can be easily adapted to other implementations of x-ray phase contrast imaging. Likewise, the algorithm enables a simple extension to 2D aperture designs, different acquisition schemes, etc. Problems related to numerical implementation of the algorithm are analyzed in detail, and simple rules are derived that enable us to avoid or at least mitigate them. Finally, comparisons with experimental data and data obtained with a different simulation algorithm are presented to validate the model and demonstrate its advantages in practical implementations. This also enabled us to demonstrate an increase in computational speed of more than one order of magnitude over a previous algorithm.

© 2013 Optical Society of America

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  1. A. Olivo, “Recent patents in x-ray phase contrast imaging,” Recent Pat. Biomed. Eng. 3, 95–106 (2010).
    [CrossRef]
  2. A. Bravin, P. Coan, and P. Suortti, “X-ray phase-contrast imaging: from pre-clinical applications towards clinics,” Phys. Med. Biol. 58, R1–R35 (2013).
    [CrossRef]
  3. U. Bonse and M. Hart, “An x-ray interferometer,” Appl. Phys. Lett. 6, 155–156 (1965).
    [CrossRef]
  4. A. Momose, T. Takeda, Y. Itai, and K. Hirano, “Phase-contrast x-ray computed tomography for observing biological soft tissues,” Nat. Med. 2, 473–475 (1996).
    [CrossRef]
  5. A. Snigirev, I. Snigireva, V. Kohn, S. Kuznetsov, and 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]
  6. S. W. Wilkins, T. E. Gureyev, D. Gao, A. Pogany, and A. W. Stevenson, “Phase-contrast imaging using polychromatic hard x-rays,” Nature 384, 335–338 (1996).
    [CrossRef]
  7. V. N. Ingal and E. A. Beliaevskaya, “X-ray plane-wave topography observation of the phase contrast from a non-crystalline object,” J. Phys. D 28, 2314–2317 (1995).
    [CrossRef]
  8. D. Chapman, W. Thomlinson, R. E. Johnston, D. Washburn, E. Pisano, N. Gmur, Z. Zhong, R. Menk, F. Arfelli, and D. Sayers, “Diffraction enhanced x-ray imaging,” Phys. Med. Biol. 42, 2015–2025 (1997).
    [CrossRef]
  9. C. David, B. Nohammer, H. H. Solak, and E. Ziegler, “Differential x-ray phase contrast imaging using a shearing interferometer,” Appl. Phys. Lett. 81, 3287–3289 (2002).
    [CrossRef]
  10. A. Momose, S. Kawamoto, I. Koyama, Y. Hamaishi, K. Takai, and Y. Suzuki, “Demonstration of x-ray Talbot interferometry,” Jpn. J. Appl. Phys. 42, L866–L868 (2003).
    [CrossRef]
  11. F. Pfeiffer, T. Weitkamp, O. Bunk, and C. David, “Phase retrieval and differential phase-contrast imaging with low-brilliance x-ray sources,” Nat. Phys. 2, 258–261 (2006).
    [CrossRef]
  12. A. Olivo and R. Speller, “A coded-aperture approach allowing x-ray phase contrast imaging with conventional sources,” Appl. Phys. Lett. 91, 074106 (2007).
    [CrossRef]
  13. A. Olivo, F. Arfelli, G. Cantatore, R. Longo, R. H. Menk, S. Pani, M. Prest, P. Poropat, L. Rigon, G. Tromba, E. Vallazza, and E. Castelli, “An innovative digital imaging set-up allowing a low-dose approach to phase contrast applications in the medical field,” Med. Phys. 28, 1610–1619 (2001).
    [CrossRef]
  14. P. R. T. Munro, K. Ignatyev, R. D. Speller, and A. Olivo, “Phase and absorption retrieval using incoherent x-ray sources,” Proc. Natl. Am. Sci. 109, 13922–13927 (2012).
    [CrossRef]
  15. A. Olivo, K. Ignatyev, P. R. T. Munro, and R. D. Speller, “Noninterferometric phase-contrast images obtained with incoherent x-ray sources,” Appl. Opt. 50, 1765–1769 (2011).
    [CrossRef]
  16. K. Ignatyev, P. R. T. Munro, D. Chana, R. D. Speller, and A. Olivo, “Coded apertures allow high-energy x-ray phase contrast imaging with laboratory sources,” J. Appl. Phys. 110, 014906 (2011).
    [CrossRef]
  17. P. R. T. Munro, K. Ignatyev, R. D. Speller, and A. Olivo, “The relationship between wave and geometrical optics model of coded aperture type x-ray phase contrast imaging systems,” Opt. Express 18, 4103–4117 (2010).
    [CrossRef]
  18. J. M. Cowley and A. F. Moodie, “The scattering of electrons by atoms and crystals. I. A new theoretical approach,” Acta Crystallogr. 10, 609–619 (1957).
    [CrossRef]
  19. P. R. T. Munro, L. Rigon, K. Ignatyev, F. C. Lopez, D. Dreossi, R. D. Speller, and A. Olivo, “A quantitative, non-interferometric x-ray phase contrast imaging technique,” Opt. Express 21, 647–661 (2013).
    [CrossRef]
  20. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, 1996).
  21. D. M. Paganin, Coherent X-Ray Optics (Oxford University, 2006).
  22. B. Henke, E. Gullikson, and J. Davis, “X-ray interactions: photoabsorption, scattering, transmission and reflection at E = 50–30,000  eV, Z = 1–92,” At. Data Nucl. Data Tables 54, 181–342 (1993).
    [CrossRef]

2013 (2)

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

P. R. T. Munro, L. Rigon, K. Ignatyev, F. C. Lopez, D. Dreossi, R. D. Speller, and A. Olivo, “A quantitative, non-interferometric x-ray phase contrast imaging technique,” Opt. Express 21, 647–661 (2013).
[CrossRef]

2012 (1)

P. R. T. Munro, K. Ignatyev, R. D. Speller, and A. Olivo, “Phase and absorption retrieval using incoherent x-ray sources,” Proc. Natl. Am. Sci. 109, 13922–13927 (2012).
[CrossRef]

2011 (2)

K. Ignatyev, P. R. T. Munro, D. Chana, R. D. Speller, and A. Olivo, “Coded apertures allow high-energy x-ray phase contrast imaging with laboratory sources,” J. Appl. Phys. 110, 014906 (2011).
[CrossRef]

A. Olivo, K. Ignatyev, P. R. T. Munro, and R. D. Speller, “Noninterferometric phase-contrast images obtained with incoherent x-ray sources,” Appl. Opt. 50, 1765–1769 (2011).
[CrossRef]

2010 (2)

2007 (1)

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

2006 (1)

F. Pfeiffer, T. Weitkamp, O. Bunk, and C. David, “Phase retrieval and differential phase-contrast imaging with low-brilliance x-ray sources,” Nat. Phys. 2, 258–261 (2006).
[CrossRef]

2003 (1)

A. Momose, S. Kawamoto, I. Koyama, Y. Hamaishi, K. Takai, and Y. Suzuki, “Demonstration of x-ray Talbot interferometry,” Jpn. J. Appl. Phys. 42, L866–L868 (2003).
[CrossRef]

2002 (1)

C. David, B. Nohammer, H. H. Solak, and E. Ziegler, “Differential x-ray phase contrast imaging using a shearing interferometer,” Appl. Phys. Lett. 81, 3287–3289 (2002).
[CrossRef]

2001 (1)

A. Olivo, F. Arfelli, G. Cantatore, R. Longo, R. H. Menk, S. Pani, M. Prest, P. Poropat, L. Rigon, G. Tromba, E. Vallazza, and E. Castelli, “An innovative digital imaging set-up allowing a low-dose approach to phase contrast applications in the medical field,” Med. Phys. 28, 1610–1619 (2001).
[CrossRef]

1997 (1)

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

1996 (2)

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

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

1995 (2)

A. Snigirev, I. Snigireva, V. Kohn, S. Kuznetsov, and 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 and E. A. Beliaevskaya, “X-ray plane-wave topography observation of the phase contrast from a non-crystalline object,” J. Phys. D 28, 2314–2317 (1995).
[CrossRef]

1993 (1)

B. Henke, E. Gullikson, and J. Davis, “X-ray interactions: photoabsorption, scattering, transmission and reflection at E = 50–30,000  eV, Z = 1–92,” At. Data Nucl. Data Tables 54, 181–342 (1993).
[CrossRef]

1965 (1)

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

1957 (1)

J. M. Cowley and A. F. Moodie, “The scattering of electrons by atoms and crystals. I. A new theoretical approach,” Acta Crystallogr. 10, 609–619 (1957).
[CrossRef]

Arfelli, F.

A. Olivo, F. Arfelli, G. Cantatore, R. Longo, R. H. Menk, S. Pani, M. Prest, P. Poropat, L. Rigon, G. Tromba, E. Vallazza, and E. Castelli, “An innovative digital imaging set-up allowing a low-dose approach to phase contrast applications in the medical field,” Med. Phys. 28, 1610–1619 (2001).
[CrossRef]

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

Beliaevskaya, E. A.

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

Bonse, U.

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

Bravin, A.

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

Bunk, O.

F. Pfeiffer, T. Weitkamp, O. Bunk, and C. David, “Phase retrieval and differential phase-contrast imaging with low-brilliance x-ray sources,” Nat. Phys. 2, 258–261 (2006).
[CrossRef]

Cantatore, G.

A. Olivo, F. Arfelli, G. Cantatore, R. Longo, R. H. Menk, S. Pani, M. Prest, P. Poropat, L. Rigon, G. Tromba, E. Vallazza, and E. Castelli, “An innovative digital imaging set-up allowing a low-dose approach to phase contrast applications in the medical field,” Med. Phys. 28, 1610–1619 (2001).
[CrossRef]

Castelli, E.

A. Olivo, F. Arfelli, G. Cantatore, R. Longo, R. H. Menk, S. Pani, M. Prest, P. Poropat, L. Rigon, G. Tromba, E. Vallazza, and E. Castelli, “An innovative digital imaging set-up allowing a low-dose approach to phase contrast applications in the medical field,” Med. Phys. 28, 1610–1619 (2001).
[CrossRef]

Chana, D.

K. Ignatyev, P. R. T. Munro, D. Chana, R. D. Speller, and A. Olivo, “Coded apertures allow high-energy x-ray phase contrast imaging with laboratory sources,” J. Appl. Phys. 110, 014906 (2011).
[CrossRef]

Chapman, D.

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

Coan, P.

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

Cowley, J. M.

J. M. Cowley and A. F. Moodie, “The scattering of electrons by atoms and crystals. I. A new theoretical approach,” Acta Crystallogr. 10, 609–619 (1957).
[CrossRef]

David, C.

F. Pfeiffer, T. Weitkamp, O. Bunk, and C. David, “Phase retrieval and differential phase-contrast imaging with low-brilliance x-ray sources,” Nat. Phys. 2, 258–261 (2006).
[CrossRef]

C. David, B. Nohammer, H. H. Solak, and E. Ziegler, “Differential x-ray phase contrast imaging using a shearing interferometer,” Appl. Phys. Lett. 81, 3287–3289 (2002).
[CrossRef]

Davis, J.

B. Henke, E. Gullikson, and J. Davis, “X-ray interactions: photoabsorption, scattering, transmission and reflection at E = 50–30,000  eV, Z = 1–92,” At. Data Nucl. Data Tables 54, 181–342 (1993).
[CrossRef]

Dreossi, D.

Gao, D.

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

Gmur, N.

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

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, 1996).

Gullikson, E.

B. Henke, E. Gullikson, and J. Davis, “X-ray interactions: photoabsorption, scattering, transmission and reflection at E = 50–30,000  eV, Z = 1–92,” At. Data Nucl. Data Tables 54, 181–342 (1993).
[CrossRef]

Gureyev, T. E.

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

Hamaishi, Y.

A. Momose, S. Kawamoto, I. Koyama, Y. Hamaishi, K. Takai, and Y. Suzuki, “Demonstration of x-ray Talbot interferometry,” Jpn. J. Appl. Phys. 42, L866–L868 (2003).
[CrossRef]

Hart, M.

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

Henke, B.

B. Henke, E. Gullikson, and J. Davis, “X-ray interactions: photoabsorption, scattering, transmission and reflection at E = 50–30,000  eV, Z = 1–92,” At. Data Nucl. Data Tables 54, 181–342 (1993).
[CrossRef]

Hirano, K.

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

Ignatyev, K.

Ingal, V. N.

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

Itai, Y.

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

Johnston, R. E.

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

Kawamoto, S.

A. Momose, S. Kawamoto, I. Koyama, Y. Hamaishi, K. Takai, and Y. Suzuki, “Demonstration of x-ray Talbot interferometry,” Jpn. J. Appl. Phys. 42, L866–L868 (2003).
[CrossRef]

Kohn, V.

A. Snigirev, I. Snigireva, V. Kohn, S. Kuznetsov, and 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, S. Kawamoto, I. Koyama, Y. Hamaishi, K. Takai, and Y. Suzuki, “Demonstration of x-ray Talbot interferometry,” Jpn. J. Appl. Phys. 42, L866–L868 (2003).
[CrossRef]

Kuznetsov, S.

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

Longo, R.

A. Olivo, F. Arfelli, G. Cantatore, R. Longo, R. H. Menk, S. Pani, M. Prest, P. Poropat, L. Rigon, G. Tromba, E. Vallazza, and E. Castelli, “An innovative digital imaging set-up allowing a low-dose approach to phase contrast applications in the medical field,” Med. Phys. 28, 1610–1619 (2001).
[CrossRef]

Lopez, F. C.

Menk, R.

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

Menk, R. H.

A. Olivo, F. Arfelli, G. Cantatore, R. Longo, R. H. Menk, S. Pani, M. Prest, P. Poropat, L. Rigon, G. Tromba, E. Vallazza, and E. Castelli, “An innovative digital imaging set-up allowing a low-dose approach to phase contrast applications in the medical field,” Med. Phys. 28, 1610–1619 (2001).
[CrossRef]

Momose, A.

A. Momose, S. Kawamoto, I. Koyama, Y. Hamaishi, K. Takai, and Y. Suzuki, “Demonstration of x-ray Talbot interferometry,” Jpn. J. Appl. Phys. 42, L866–L868 (2003).
[CrossRef]

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

Moodie, A. F.

J. M. Cowley and A. F. Moodie, “The scattering of electrons by atoms and crystals. I. A new theoretical approach,” Acta Crystallogr. 10, 609–619 (1957).
[CrossRef]

Munro, P. R. T.

Nohammer, B.

C. David, B. Nohammer, H. H. Solak, and E. Ziegler, “Differential x-ray phase contrast imaging using a shearing interferometer,” Appl. Phys. Lett. 81, 3287–3289 (2002).
[CrossRef]

Olivo, A.

P. R. T. Munro, L. Rigon, K. Ignatyev, F. C. Lopez, D. Dreossi, R. D. Speller, and A. Olivo, “A quantitative, non-interferometric x-ray phase contrast imaging technique,” Opt. Express 21, 647–661 (2013).
[CrossRef]

P. R. T. Munro, K. Ignatyev, R. D. Speller, and A. Olivo, “Phase and absorption retrieval using incoherent x-ray sources,” Proc. Natl. Am. Sci. 109, 13922–13927 (2012).
[CrossRef]

K. Ignatyev, P. R. T. Munro, D. Chana, R. D. Speller, and A. Olivo, “Coded apertures allow high-energy x-ray phase contrast imaging with laboratory sources,” J. Appl. Phys. 110, 014906 (2011).
[CrossRef]

A. Olivo, K. Ignatyev, P. R. T. Munro, and R. D. Speller, “Noninterferometric phase-contrast images obtained with incoherent x-ray sources,” Appl. Opt. 50, 1765–1769 (2011).
[CrossRef]

P. R. T. Munro, K. Ignatyev, R. D. Speller, and A. Olivo, “The relationship between wave and geometrical optics model of coded aperture type x-ray phase contrast imaging systems,” Opt. Express 18, 4103–4117 (2010).
[CrossRef]

A. Olivo, “Recent patents in x-ray phase contrast imaging,” Recent Pat. Biomed. Eng. 3, 95–106 (2010).
[CrossRef]

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

A. Olivo, F. Arfelli, G. Cantatore, R. Longo, R. H. Menk, S. Pani, M. Prest, P. Poropat, L. Rigon, G. Tromba, E. Vallazza, and E. Castelli, “An innovative digital imaging set-up allowing a low-dose approach to phase contrast applications in the medical field,” Med. Phys. 28, 1610–1619 (2001).
[CrossRef]

Paganin, D. M.

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

Pani, S.

A. Olivo, F. Arfelli, G. Cantatore, R. Longo, R. H. Menk, S. Pani, M. Prest, P. Poropat, L. Rigon, G. Tromba, E. Vallazza, and E. Castelli, “An innovative digital imaging set-up allowing a low-dose approach to phase contrast applications in the medical field,” Med. Phys. 28, 1610–1619 (2001).
[CrossRef]

Pfeiffer, F.

F. Pfeiffer, T. Weitkamp, O. Bunk, and C. David, “Phase retrieval and differential phase-contrast imaging with low-brilliance x-ray sources,” Nat. Phys. 2, 258–261 (2006).
[CrossRef]

Pisano, E.

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

Pogany, A.

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

Poropat, P.

A. Olivo, F. Arfelli, G. Cantatore, R. Longo, R. H. Menk, S. Pani, M. Prest, P. Poropat, L. Rigon, G. Tromba, E. Vallazza, and E. Castelli, “An innovative digital imaging set-up allowing a low-dose approach to phase contrast applications in the medical field,” Med. Phys. 28, 1610–1619 (2001).
[CrossRef]

Prest, M.

A. Olivo, F. Arfelli, G. Cantatore, R. Longo, R. H. Menk, S. Pani, M. Prest, P. Poropat, L. Rigon, G. Tromba, E. Vallazza, and E. Castelli, “An innovative digital imaging set-up allowing a low-dose approach to phase contrast applications in the medical field,” Med. Phys. 28, 1610–1619 (2001).
[CrossRef]

Rigon, L.

P. R. T. Munro, L. Rigon, K. Ignatyev, F. C. Lopez, D. Dreossi, R. D. Speller, and A. Olivo, “A quantitative, non-interferometric x-ray phase contrast imaging technique,” Opt. Express 21, 647–661 (2013).
[CrossRef]

A. Olivo, F. Arfelli, G. Cantatore, R. Longo, R. H. Menk, S. Pani, M. Prest, P. Poropat, L. Rigon, G. Tromba, E. Vallazza, and E. Castelli, “An innovative digital imaging set-up allowing a low-dose approach to phase contrast applications in the medical field,” Med. Phys. 28, 1610–1619 (2001).
[CrossRef]

Sayers, D.

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

Schelokov, I.

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

Snigirev, A.

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

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

Solak, H. H.

C. David, B. Nohammer, H. H. Solak, and E. Ziegler, “Differential x-ray phase contrast imaging using a shearing interferometer,” Appl. Phys. Lett. 81, 3287–3289 (2002).
[CrossRef]

Speller, R.

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

Speller, R. D.

Stevenson, A. W.

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

Suortti, P.

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

Suzuki, Y.

A. Momose, S. Kawamoto, I. Koyama, Y. Hamaishi, K. Takai, and Y. Suzuki, “Demonstration of x-ray Talbot interferometry,” Jpn. J. Appl. Phys. 42, L866–L868 (2003).
[CrossRef]

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D. Chapman, W. Thomlinson, R. E. Johnston, D. Washburn, E. Pisano, N. Gmur, Z. Zhong, R. Menk, F. Arfelli, and D. Sayers, “Diffraction enhanced x-ray imaging,” Phys. Med. Biol. 42, 2015–2025 (1997).
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Figures (8)

Fig. 1.
Fig. 1.

Schematic diagram of CA XPCi system.

Fig. 2.
Fig. 2.

Modulus of H^z,eff(ξ): the blue curve is calculated numerically, while the green curve is calculated analytically with the approximation discussed in the text. Parameters used in the simulation: z=2m, L=2mm, and λ=0.31Å (E=40keV).

Fig. 3.
Fig. 3.

Simulations of CA XPCi profiles of a polypropylene wire, obtained with different numbers of sampling points: δxblue=3.4nm is derived in accordance with Eq. (26), δxgreen=15δxblue, δxred=30δxblue. Only the first simulation (blue) provides correct results. Parameters used in the simulation: monochromatic Gaussian distributed source (FWHM=60μm, E=30keV); z1=1.6m, z2=0.4m; sample mask with 12 μm apertures and a period of 80 μm; detector mask with 20 μm apertures and a period of 100 μm; 50% illuminated fraction; pixel size=100μm; wire diameter=140μm; number of dithering steps (number of subpixel sample displacements)=10 (each step=8μm).

Fig. 4.
Fig. 4.

Simulations of CA XPCi profiles of a polypropylene wire, obtained using different dimensions for the sampled space. The blue curve is calculated considering a sampled space L=LD+RM1+RSr, which leads to correct results; the green curve is calculated with a sampled space L=LD, which causes errors in the simulated profile. Parameters used in the simulation: monochromatic Gaussian distributed source (FWHM=1μm, E=30keV); z1=100m, z2=0.1m; sample mask with 20 μm apertures and a period of 120 μm; detector mask with 20 μm apertures and a period of 120 μm; 50% illuminated fraction; pixel size=120μm; wire diameter=160μm; number of dithering steps=10 (each step=12μm).

Fig. 5.
Fig. 5.

Comparison between simulated results obtained with the proposed algorithm (blue curves) and the one described in [17] (green curves). (a) Polypropylene wire: monochromatic Gaussian distributed source (FWHM=60μm, E=30keV); z1=1.6m, z2=0.4m; sample mask with 20 μm apertures and 80 μm period; detector mask with 50 μm apertures and 100 μm period; 50% illuminated fraction; pixel size=100μm; wire diameter=260μm; number of dithering steps=40 (each step=2μm). (b) Aluminum wire: monochromatic Gaussian distributed source (FWHM=1μm, E=20keV); z1=0.1m, z2=1m; sample mask with 3.4 μm apertures and 13.6 μm period; detector mask with 75 μm apertures and 150 μm period; 50% illuminated fraction; pixel size=150μm; wire diameter=14μm; dithering steps=40 (each step=0.34μm).

Fig. 6.
Fig. 6.

Ratio between the computational times for the algorithm described in [17] and the one presented here, as a function of the sample dimension. Parameters used in the simulation: monochromatic Gaussian distributed source (FWHM=60μm, E=30keV); z1=1.6m, z2=0.4m; sample mask with 40 μm apertures and 80 μm period; detector mask with 50 μm apertures and 100 μm period; 50% illuminated fraction; pixel size=100μm; polypropylene wire sample; dithering steps=5 (each step=16μm).

Fig. 7.
Fig. 7.

Image of different wires acquired with synchrotron radiation using the CA XPCi method.

Fig. 8.
Fig. 8.

Comparison between experimental data (intensity profiles along the black vertical line in Fig. 7) and simulation results.

Tables (1)

Tables Icon

Table 1. Properties of Wires Highlighted in Fig. 7

Equations (27)

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EB(x,y)=EA(x,y)exp(ik|r|)iλ|r|cosαdxdy
EB(x,y)=EA(x,y)*HΔz(x,y),
Hz(x,y)=exp(ikz)iλzexp(ikx2+y22z).
Eout(x,y)=Tobj(x,y)Ein(x,y),
Tobj(x,y)=exp[iϕ(x,y)M(x,y)],
ϕ(x,y)=kLδ(x,y,z)dz;M(x,y)=kLβ(x,y,z)dz
EM2(x)=A0[H1(x)TM1(x)Tobj(x)]*H2(x),
Hz(xxs)=iλzexp(ikz)Hz(x)Hz(xs)exp(ikxxs/z),
EM2(x,xs)=C(x)EM2(x+z2z1xs),
C(x)=exp[ik2z1(1z2z1)xs2ikz1xxs].
ID(x)=S(xs)|EM2(x,xs)TM2(x)|2dxs,
ID(x)=[Sr(x)*Ip(x)]|TM2(x)|2,
In=xnxn+PID(x)dx,
f(x)*g(x)=F1{F[f(x)]F[g(x)]},
Hz(x)exp(i2πx22zλ),H^z(ξ)exp(iπzλξ2);
Hz,eff(x)exp(i2πx22zλ)rect(xL)
H^z,eff(ξ)exp(iπzλξ2)*sin(πLξ)πξ=exp[iπzλ(ξη)2]sin(πLη)πηdη.
|H^z,eff(ξ)||exp(i2πzλξη)sin(πLη)πηdη|=rect(zλξL).
WH=2L/(zλ).
TM(x)=rect(xA),T^M(ξ)=sin(πAξ)πξ;
WM=2×103πA;
Tobj(x)exp(x22σo2)*rect(xO),T^obj(ξ)exp(2π2σo2ξ2)sin(πOξ)πξ;
Wobj=min{2×103πO,[6ln10πσo2]1/2}.
Sr(x)exp(x22σr2),S^r(ξ)exp(2π2σr2ξ2);
WSr=[6ln10πσr2]1/2.
max(WH1+2WM1,WH2,WSr)<1/Δx,WIp=2min(WH1+2WM1,WH2)<1/Δx,min(WSr,WIp)+2WM2<1/Δx.
fl*¯gl=FFT1{FFT[fl]FFT[gl]},

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