Abstract

We report on what are believed to be the first full-scale images obtained with the coded aperture concept, which uses conventional x-ray sources without the need to collimate/aperture their output. We discuss the differences in the underpinning physical principles with respect to other methods, and explain why these might lead to a more efficient use of the source. In particular, we discuss how the evaluation of the first imaging system provided promising indications on the method’s potential to detect details invisible to conventional absorption methods, use an increased average x-ray energy, and reduce exposure times—all important aspects with regards to real-world implementations.

© 2011 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. T. J. Davis, D. Gao, T. E. Gureyev, A. W. Stevenson, and S. W. Wilkins, “Phase-contrast imaging of weakly absorbing materials using hard x-rays,” Nature 373, 595–598 (1995).
    [CrossRef]
  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, 5486–5892 (1995).
    [CrossRef]
  3. R. Lewis, “Medical phase-contrast x-ray imaging: current status and future prospects,” Phys. Med. Biol. 49, 3573–3583(2004).
    [CrossRef] [PubMed]
  4. F. Pfeiffer, T. Weitkamp, O. Bunk, and C. David, “Phase retrieval and differential phase-contrast imaging with low-brilliance x-ray sources,” Nature Phys. 2, 258–261 (2006).
    [CrossRef]
  5. 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]
  6. A. Olivo, D. Chana, and R. Speller, “A preliminary investigation of the potential of phase contrast x-ray imaging in the field of homeland security,” J. Phys. D 41, 225503 (2008).
    [CrossRef]
  7. A. Olivo and R. Speller, “Modelling of a novel x-ray phase contrast imaging technique based on coded apertures,” Phys. Med. Biol. 52, 6555–6573 (2007).
    [CrossRef] [PubMed]
  8. M. Born and E. Wolf, Principles of Optics, 6th ed.(Pergamon, 1980).
  9. 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] [PubMed]
  10. J. F. Clauser, “Ultrahigh resolution interferometric x-ray imaging,” U.S. patent 5812629 (22 September 1998).
  11. A. Momose, A. W. Yashiro, T. Takeda, Y. Suzuki, and T. Hattori, “Phase tomography by x-ray Talbot interferometry for biological imaging,” Jpn. J. Appl. Phys. 45, 5254–5262(2006).
    [CrossRef]
  12. A. Olivo and R. Speller, “Experimental validation of a simple model capable of predicting the phase contrast imaging capabilities of an x-ray imaging system,” Phys. Med. Biol. 51, 3015–3030 (2006).
    [CrossRef] [PubMed]
  13. A. Peterzol, A. Olivo, L. Rigon, S. Pani, and D. Dreossi, “The effects of the imaging system on the validity limits of the ray-optical approach to phase contrast imaging,” Med. Phys. 32, 3617–3627 (2005).
    [CrossRef]
  14. 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]
  15. D. Chapman, W. Thomlinson, R. E. Johnson, 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] [PubMed]
  16. D. J. Vine, D. M. Paganin, K. M. Pavlov, J. Kräusslich, O. Wehrhan, I. Uschmann, and E. Förster, “Analyzer-based phase contrast imaging and phase retrieval using a rotating anode x-ray source,” Appl. Phys. Lett. 91, 254110 (2007).
    [CrossRef]
  17. 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] [PubMed]
  18. A. Olivo and R. Speller, “Image formation principles in coded-aperture based x-ray phase contrast imaging,” Phys. Med. Biol. 53, 6461–6474 (2008).
    [CrossRef] [PubMed]
  19. P. R. T. Munro, K. Ignatyev, R. D. Speller, and A. Olivo, “The relationship between wave and geometrical optics models of coded aperture type x-ray phase contrast imaging systems,” Opt. Express 18, 4103–4117 (2010).
    [CrossRef] [PubMed]
  20. A. Olivo, K. Ignatyev, P. R. T. Munro, and R. D. Speller, “Design and realization of a coded-aperture based x-ray phase contrast imaging for homeland security applications,” Nucl. Instrum. Methods Phys. Res. A 610, 604–614 (2009).
    [CrossRef]
  21. J. C. Buckland-Wright, “A new high-definition microfocal x-ray unit,” Br. J. Radiol. 62, 201–208 (1989).
    [CrossRef] [PubMed]
  22. F. Pfeiffer, M. Bech, O. Bunk, P. Kraft, E. F. Eikenberry, C. Brönnimann, C. Grünzweig, and C. David, “Hard-x-ray dark-field imaging using a grating interferometer,” Nat. Mater. 7, 134–137 (2008).
    [CrossRef] [PubMed]
  23. A. Momose, W. Yashiro, H. Kuwabara, and K. Kawabata, “Grating-based x-ray phase imaging using multiline x-ray source,” Jpn. J. Appl. Phys. 48, 076512 (2009).
    [CrossRef]
  24. X. Wu and H. Liu, “Clarification of aspects in in-line phase-sensitive x-ray imaging,” Med. Phys. 34, 737–743 (2007).
    [CrossRef] [PubMed]
  25. A. Olivo, S. E. Bohndiek, J. A. Griffiths, A. Konstantinidis, and R. D. Speller, “A non-free-space propagation x-ray phase contrast imaging method sensitive to phase effects in two directions simultaneously,” Appl. Phys. Lett. 94, 044108 (2009).
    [CrossRef]

2010 (1)

2009 (3)

A. Olivo, S. E. Bohndiek, J. A. Griffiths, A. Konstantinidis, and R. D. Speller, “A non-free-space propagation x-ray phase contrast imaging method sensitive to phase effects in two directions simultaneously,” Appl. Phys. Lett. 94, 044108 (2009).
[CrossRef]

A. Momose, W. Yashiro, H. Kuwabara, and K. Kawabata, “Grating-based x-ray phase imaging using multiline x-ray source,” Jpn. J. Appl. Phys. 48, 076512 (2009).
[CrossRef]

A. Olivo, K. Ignatyev, P. R. T. Munro, and R. D. Speller, “Design and realization of a coded-aperture based x-ray phase contrast imaging for homeland security applications,” Nucl. Instrum. Methods Phys. Res. A 610, 604–614 (2009).
[CrossRef]

2008 (3)

A. Olivo, D. Chana, and R. Speller, “A preliminary investigation of the potential of phase contrast x-ray imaging in the field of homeland security,” J. Phys. D 41, 225503 (2008).
[CrossRef]

A. Olivo and R. Speller, “Image formation principles in coded-aperture based x-ray phase contrast imaging,” Phys. Med. Biol. 53, 6461–6474 (2008).
[CrossRef] [PubMed]

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

2007 (4)

X. Wu and H. Liu, “Clarification of aspects in in-line phase-sensitive x-ray imaging,” Med. Phys. 34, 737–743 (2007).
[CrossRef] [PubMed]

A. Olivo and R. Speller, “Modelling of a novel x-ray phase contrast imaging technique based on coded apertures,” Phys. Med. Biol. 52, 6555–6573 (2007).
[CrossRef] [PubMed]

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]

D. J. Vine, D. M. Paganin, K. M. Pavlov, J. Kräusslich, O. Wehrhan, I. Uschmann, and E. Förster, “Analyzer-based phase contrast imaging and phase retrieval using a rotating anode x-ray source,” Appl. Phys. Lett. 91, 254110 (2007).
[CrossRef]

2006 (3)

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

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

A. Olivo and R. Speller, “Experimental validation of a simple model capable of predicting the phase contrast imaging capabilities of an x-ray imaging system,” Phys. Med. Biol. 51, 3015–3030 (2006).
[CrossRef] [PubMed]

2005 (1)

A. Peterzol, A. Olivo, L. Rigon, S. Pani, and D. Dreossi, “The effects of the imaging system on the validity limits of the ray-optical approach to phase contrast imaging,” Med. Phys. 32, 3617–3627 (2005).
[CrossRef]

2004 (1)

R. Lewis, “Medical phase-contrast x-ray imaging: current status and future prospects,” Phys. Med. Biol. 49, 3573–3583(2004).
[CrossRef] [PubMed]

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] [PubMed]

1997 (1)

D. Chapman, W. Thomlinson, R. E. Johnson, 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] [PubMed]

1996 (1)

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] [PubMed]

1995 (3)

T. J. Davis, D. Gao, T. E. Gureyev, A. W. Stevenson, and 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, and I. Schelokov, “On the possibilities of x-ray phase contrast microimaging by coherent high-energy synchrotron radiation,” Rev. Sci. Instrum. 66, 5486–5892 (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]

1989 (1)

J. C. Buckland-Wright, “A new high-definition microfocal x-ray unit,” Br. J. Radiol. 62, 201–208 (1989).
[CrossRef] [PubMed]

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] [PubMed]

D. Chapman, W. Thomlinson, R. E. Johnson, 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] [PubMed]

Bech, M.

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

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]

Bohndiek, S. E.

A. Olivo, S. E. Bohndiek, J. A. Griffiths, A. Konstantinidis, and R. D. Speller, “A non-free-space propagation x-ray phase contrast imaging method sensitive to phase effects in two directions simultaneously,” Appl. Phys. Lett. 94, 044108 (2009).
[CrossRef]

Born, M.

M. Born and E. Wolf, Principles of Optics, 6th ed.(Pergamon, 1980).

Brönnimann, C.

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

Buckland-Wright, J. C.

J. C. Buckland-Wright, “A new high-definition microfocal x-ray unit,” Br. J. Radiol. 62, 201–208 (1989).
[CrossRef] [PubMed]

Bunk, O.

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

F. Pfeiffer, T. Weitkamp, O. Bunk, and C. David, “Phase retrieval and differential phase-contrast imaging with low-brilliance x-ray sources,” Nature 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] [PubMed]

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] [PubMed]

Chana, D.

A. Olivo, D. Chana, and R. Speller, “A preliminary investigation of the potential of phase contrast x-ray imaging in the field of homeland security,” J. Phys. D 41, 225503 (2008).
[CrossRef]

Chapman, D.

D. Chapman, W. Thomlinson, R. E. Johnson, 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] [PubMed]

Clauser, J. F.

J. F. Clauser, “Ultrahigh resolution interferometric x-ray imaging,” U.S. patent 5812629 (22 September 1998).

David, C.

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

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

Davis, T. J.

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

Dreossi, D.

A. Peterzol, A. Olivo, L. Rigon, S. Pani, and D. Dreossi, “The effects of the imaging system on the validity limits of the ray-optical approach to phase contrast imaging,” Med. Phys. 32, 3617–3627 (2005).
[CrossRef]

Eikenberry, E. F.

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

Förster, E.

D. J. Vine, D. M. Paganin, K. M. Pavlov, J. Kräusslich, O. Wehrhan, I. Uschmann, and E. Förster, “Analyzer-based phase contrast imaging and phase retrieval using a rotating anode x-ray source,” Appl. Phys. Lett. 91, 254110 (2007).
[CrossRef]

Gao, D.

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

Gmur, N.

D. Chapman, W. Thomlinson, R. E. Johnson, 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] [PubMed]

Griffiths, J. A.

A. Olivo, S. E. Bohndiek, J. A. Griffiths, A. Konstantinidis, and R. D. Speller, “A non-free-space propagation x-ray phase contrast imaging method sensitive to phase effects in two directions simultaneously,” Appl. Phys. Lett. 94, 044108 (2009).
[CrossRef]

Grünzweig, C.

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

Gureyev, T. E.

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

Hattori, T.

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

Ignatyev, K.

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

A. Olivo, K. Ignatyev, P. R. T. Munro, and R. D. Speller, “Design and realization of a coded-aperture based x-ray phase contrast imaging for homeland security applications,” Nucl. Instrum. Methods Phys. Res. A 610, 604–614 (2009).
[CrossRef]

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] [PubMed]

Johnson, R. E.

D. Chapman, W. Thomlinson, R. E. Johnson, 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] [PubMed]

Kawabata, K.

A. Momose, W. Yashiro, H. Kuwabara, and K. Kawabata, “Grating-based x-ray phase imaging using multiline x-ray source,” Jpn. J. Appl. Phys. 48, 076512 (2009).
[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, 5486–5892 (1995).
[CrossRef]

Konstantinidis, A.

A. Olivo, S. E. Bohndiek, J. A. Griffiths, A. Konstantinidis, and R. D. Speller, “A non-free-space propagation x-ray phase contrast imaging method sensitive to phase effects in two directions simultaneously,” Appl. Phys. Lett. 94, 044108 (2009).
[CrossRef]

Kraft, P.

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

Kräusslich, J.

D. J. Vine, D. M. Paganin, K. M. Pavlov, J. Kräusslich, O. Wehrhan, I. Uschmann, and E. Förster, “Analyzer-based phase contrast imaging and phase retrieval using a rotating anode x-ray source,” Appl. Phys. Lett. 91, 254110 (2007).
[CrossRef]

Kuwabara, H.

A. Momose, W. Yashiro, H. Kuwabara, and K. Kawabata, “Grating-based x-ray phase imaging using multiline x-ray source,” Jpn. J. Appl. Phys. 48, 076512 (2009).
[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, 5486–5892 (1995).
[CrossRef]

Lewis, R.

R. Lewis, “Medical phase-contrast x-ray imaging: current status and future prospects,” Phys. Med. Biol. 49, 3573–3583(2004).
[CrossRef] [PubMed]

Liu, H.

X. Wu and H. Liu, “Clarification of aspects in in-line phase-sensitive x-ray imaging,” Med. Phys. 34, 737–743 (2007).
[CrossRef] [PubMed]

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] [PubMed]

Menk, R.

D. Chapman, W. Thomlinson, R. E. Johnson, 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] [PubMed]

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] [PubMed]

Momose, A.

A. Momose, W. Yashiro, H. Kuwabara, and K. Kawabata, “Grating-based x-ray phase imaging using multiline x-ray source,” Jpn. J. Appl. Phys. 48, 076512 (2009).
[CrossRef]

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

Munro, P. R. T.

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

A. Olivo, K. Ignatyev, P. R. T. Munro, and R. D. Speller, “Design and realization of a coded-aperture based x-ray phase contrast imaging for homeland security applications,” Nucl. Instrum. Methods Phys. Res. A 610, 604–614 (2009).
[CrossRef]

Olivo, A.

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

A. Olivo, K. Ignatyev, P. R. T. Munro, and R. D. Speller, “Design and realization of a coded-aperture based x-ray phase contrast imaging for homeland security applications,” Nucl. Instrum. Methods Phys. Res. A 610, 604–614 (2009).
[CrossRef]

A. Olivo, S. E. Bohndiek, J. A. Griffiths, A. Konstantinidis, and R. D. Speller, “A non-free-space propagation x-ray phase contrast imaging method sensitive to phase effects in two directions simultaneously,” Appl. Phys. Lett. 94, 044108 (2009).
[CrossRef]

A. Olivo, D. Chana, and R. Speller, “A preliminary investigation of the potential of phase contrast x-ray imaging in the field of homeland security,” J. Phys. D 41, 225503 (2008).
[CrossRef]

A. Olivo and R. Speller, “Image formation principles in coded-aperture based x-ray phase contrast imaging,” Phys. Med. Biol. 53, 6461–6474 (2008).
[CrossRef] [PubMed]

A. Olivo and R. Speller, “Modelling of a novel x-ray phase contrast imaging technique based on coded apertures,” Phys. Med. Biol. 52, 6555–6573 (2007).
[CrossRef] [PubMed]

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 and R. Speller, “Experimental validation of a simple model capable of predicting the phase contrast imaging capabilities of an x-ray imaging system,” Phys. Med. Biol. 51, 3015–3030 (2006).
[CrossRef] [PubMed]

A. Peterzol, A. Olivo, L. Rigon, S. Pani, and D. Dreossi, “The effects of the imaging system on the validity limits of the ray-optical approach to phase contrast imaging,” Med. Phys. 32, 3617–3627 (2005).
[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] [PubMed]

Paganin, D. M.

D. J. Vine, D. M. Paganin, K. M. Pavlov, J. Kräusslich, O. Wehrhan, I. Uschmann, and E. Förster, “Analyzer-based phase contrast imaging and phase retrieval using a rotating anode x-ray source,” Appl. Phys. Lett. 91, 254110 (2007).
[CrossRef]

Pani, S.

A. Peterzol, A. Olivo, L. Rigon, S. Pani, and D. Dreossi, “The effects of the imaging system on the validity limits of the ray-optical approach to phase contrast imaging,” Med. Phys. 32, 3617–3627 (2005).
[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] [PubMed]

Pavlov, K. M.

D. J. Vine, D. M. Paganin, K. M. Pavlov, J. Kräusslich, O. Wehrhan, I. Uschmann, and E. Förster, “Analyzer-based phase contrast imaging and phase retrieval using a rotating anode x-ray source,” Appl. Phys. Lett. 91, 254110 (2007).
[CrossRef]

Peterzol, A.

A. Peterzol, A. Olivo, L. Rigon, S. Pani, and D. Dreossi, “The effects of the imaging system on the validity limits of the ray-optical approach to phase contrast imaging,” Med. Phys. 32, 3617–3627 (2005).
[CrossRef]

Pfeiffer, F.

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

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

Pisano, E.

D. Chapman, W. Thomlinson, R. E. Johnson, 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] [PubMed]

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] [PubMed]

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] [PubMed]

Rigon, L.

A. Peterzol, A. Olivo, L. Rigon, S. Pani, and D. Dreossi, “The effects of the imaging system on the validity limits of the ray-optical approach to phase contrast imaging,” Med. Phys. 32, 3617–3627 (2005).
[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] [PubMed]

Sayers, D.

D. Chapman, W. Thomlinson, R. E. Johnson, 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] [PubMed]

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, 5486–5892 (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, 5486–5892 (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, 5486–5892 (1995).
[CrossRef]

Speller, R.

A. Olivo, D. Chana, and R. Speller, “A preliminary investigation of the potential of phase contrast x-ray imaging in the field of homeland security,” J. Phys. D 41, 225503 (2008).
[CrossRef]

A. Olivo and R. Speller, “Image formation principles in coded-aperture based x-ray phase contrast imaging,” Phys. Med. Biol. 53, 6461–6474 (2008).
[CrossRef] [PubMed]

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 and R. Speller, “Modelling of a novel x-ray phase contrast imaging technique based on coded apertures,” Phys. Med. Biol. 52, 6555–6573 (2007).
[CrossRef] [PubMed]

A. Olivo and R. Speller, “Experimental validation of a simple model capable of predicting the phase contrast imaging capabilities of an x-ray imaging system,” Phys. Med. Biol. 51, 3015–3030 (2006).
[CrossRef] [PubMed]

Speller, R. D.

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

A. Olivo, S. E. Bohndiek, J. A. Griffiths, A. Konstantinidis, and R. D. Speller, “A non-free-space propagation x-ray phase contrast imaging method sensitive to phase effects in two directions simultaneously,” Appl. Phys. Lett. 94, 044108 (2009).
[CrossRef]

A. Olivo, K. Ignatyev, P. R. T. Munro, and R. D. Speller, “Design and realization of a coded-aperture based x-ray phase contrast imaging for homeland security applications,” Nucl. Instrum. Methods Phys. Res. A 610, 604–614 (2009).
[CrossRef]

Stevenson, A. W.

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

Suzuki, Y.

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

Takeda, T.

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

Thomlinson, W.

D. Chapman, W. Thomlinson, R. E. Johnson, 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] [PubMed]

Tromba, 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] [PubMed]

Uschmann, I.

D. J. Vine, D. M. Paganin, K. M. Pavlov, J. Kräusslich, O. Wehrhan, I. Uschmann, and E. Förster, “Analyzer-based phase contrast imaging and phase retrieval using a rotating anode x-ray source,” Appl. Phys. Lett. 91, 254110 (2007).
[CrossRef]

Vallazza, 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] [PubMed]

Vine, D. J.

D. J. Vine, D. M. Paganin, K. M. Pavlov, J. Kräusslich, O. Wehrhan, I. Uschmann, and E. Förster, “Analyzer-based phase contrast imaging and phase retrieval using a rotating anode x-ray source,” Appl. Phys. Lett. 91, 254110 (2007).
[CrossRef]

Washburn, D.

D. Chapman, W. Thomlinson, R. E. Johnson, 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] [PubMed]

Wehrhan, O.

D. J. Vine, D. M. Paganin, K. M. Pavlov, J. Kräusslich, O. Wehrhan, I. Uschmann, and E. Förster, “Analyzer-based phase contrast imaging and phase retrieval using a rotating anode x-ray source,” Appl. Phys. Lett. 91, 254110 (2007).
[CrossRef]

Weitkamp, T.

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

Wilkins, S. W.

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

Wolf, E.

M. Born and E. Wolf, Principles of Optics, 6th ed.(Pergamon, 1980).

Wu, X.

X. Wu and H. Liu, “Clarification of aspects in in-line phase-sensitive x-ray imaging,” Med. Phys. 34, 737–743 (2007).
[CrossRef] [PubMed]

Yashiro, A. W.

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

Yashiro, W.

A. Momose, W. Yashiro, H. Kuwabara, and K. Kawabata, “Grating-based x-ray phase imaging using multiline x-ray source,” Jpn. J. Appl. Phys. 48, 076512 (2009).
[CrossRef]

Zhong, Z.

D. Chapman, W. Thomlinson, R. E. Johnson, 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] [PubMed]

Appl. Phys. Lett. (3)

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]

D. J. Vine, D. M. Paganin, K. M. Pavlov, J. Kräusslich, O. Wehrhan, I. Uschmann, and E. Förster, “Analyzer-based phase contrast imaging and phase retrieval using a rotating anode x-ray source,” Appl. Phys. Lett. 91, 254110 (2007).
[CrossRef]

A. Olivo, S. E. Bohndiek, J. A. Griffiths, A. Konstantinidis, and R. D. Speller, “A non-free-space propagation x-ray phase contrast imaging method sensitive to phase effects in two directions simultaneously,” Appl. Phys. Lett. 94, 044108 (2009).
[CrossRef]

Br. J. Radiol. (1)

J. C. Buckland-Wright, “A new high-definition microfocal x-ray unit,” Br. J. Radiol. 62, 201–208 (1989).
[CrossRef] [PubMed]

J. Phys. D (2)

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]

A. Olivo, D. Chana, and R. Speller, “A preliminary investigation of the potential of phase contrast x-ray imaging in the field of homeland security,” J. Phys. D 41, 225503 (2008).
[CrossRef]

Jpn. J. Appl. Phys. (2)

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

A. Momose, W. Yashiro, H. Kuwabara, and K. Kawabata, “Grating-based x-ray phase imaging using multiline x-ray source,” Jpn. J. Appl. Phys. 48, 076512 (2009).
[CrossRef]

Med. Phys. (3)

X. Wu and H. Liu, “Clarification of aspects in in-line phase-sensitive x-ray imaging,” Med. Phys. 34, 737–743 (2007).
[CrossRef] [PubMed]

A. Peterzol, A. Olivo, L. Rigon, S. Pani, and D. Dreossi, “The effects of the imaging system on the validity limits of the ray-optical approach to phase contrast imaging,” Med. Phys. 32, 3617–3627 (2005).
[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] [PubMed]

Nat. Mater. (1)

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

Nat. Med. (1)

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] [PubMed]

Nature (1)

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

Nature Phys. (1)

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

Nucl. Instrum. Methods Phys. Res. A (1)

A. Olivo, K. Ignatyev, P. R. T. Munro, and R. D. Speller, “Design and realization of a coded-aperture based x-ray phase contrast imaging for homeland security applications,” Nucl. Instrum. Methods Phys. Res. A 610, 604–614 (2009).
[CrossRef]

Opt. Express (1)

Phys. Med. Biol. (5)

R. Lewis, “Medical phase-contrast x-ray imaging: current status and future prospects,” Phys. Med. Biol. 49, 3573–3583(2004).
[CrossRef] [PubMed]

A. Olivo and R. Speller, “Modelling of a novel x-ray phase contrast imaging technique based on coded apertures,” Phys. Med. Biol. 52, 6555–6573 (2007).
[CrossRef] [PubMed]

A. Olivo and R. Speller, “Experimental validation of a simple model capable of predicting the phase contrast imaging capabilities of an x-ray imaging system,” Phys. Med. Biol. 51, 3015–3030 (2006).
[CrossRef] [PubMed]

A. Olivo and R. Speller, “Image formation principles in coded-aperture based x-ray phase contrast imaging,” Phys. Med. Biol. 53, 6461–6474 (2008).
[CrossRef] [PubMed]

D. Chapman, W. Thomlinson, R. E. Johnson, 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] [PubMed]

Rev. Sci. Instrum. (1)

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, 5486–5892 (1995).
[CrossRef]

Other (2)

M. Born and E. Wolf, Principles of Optics, 6th ed.(Pergamon, 1980).

J. F. Clauser, “Ultrahigh resolution interferometric x-ray imaging,” U.S. patent 5812629 (22 September 1998).

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (6)

Fig. 1
Fig. 1

Schematic of the edge illumination working principle. (a) and (b) show the image formation principle for a single pixel and a single collimated beam, aligned with the edge of the pixel. Photons deviated due to refraction in the sample can change their status from undetected to detected (a) or vice versa (b), creating the positive and negative peaks of differential XPCi.

Fig. 2
Fig. 2

Schematic of the coded aperture setup: two sets of coded apertures (SA, sample apertures, and DA, detector apertures) allow repeating the situation depicted in Fig. 1 for all rows (or columns) of an area detector. The sample is placed immediately downstream of the sample apertures, almost in contact with them. P is the detector pixel size and Δ P the fraction of the pixel directly illuminated by radiation: this can be changed simply by displacing SA with respect to DA. A smaller Δ P results in increased sensitivity but also in increased exposure time.

Fig. 3
Fig. 3

Scanning electron microscope image of a small area on the top of the detector mask (apertures in dark gray). The pitch has been slightly reduced with respect to the desired one ( 100 mm ) to allow for a small distance between mask and detector, hence facilitating mask positioning and alignment.

Fig. 4
Fig. 4

Coded aperture XPCi versus conventional absorption imaging. (a) and (b) show the coded aperture XPCi and the conventional absorption image, respectively, of the same sample, obtained at the same photon statistics, i.e., at the same level of background noise.

Fig. 5
Fig. 5

Undithered coded aperture XPCi image. Images presented in Fig. 2 were “dithered”, i.e., combinations of a series of images taken at various subpixel displacements. Here an image taken without the dithering procedure shows, despite a coarser overall resolution, that all details are still detected. This allows further reducing the exposure time.

Fig. 6
Fig. 6

High-energy coded aperture XPCi image. This was taken at 100 instead of 40 kVp : the increased average beam energy causes the expected reduction in the contrast, but despite this all details are still detected.

Equations (1)

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

α = λ 2 π | x , y Φ | = | x , y [ object δ ( x , y , z ) d z ] | ,

Metrics