Abstract

Phase-space formulation based on the Wigner distribution has been presented for analyzing phase-contrast image formation. Based on the statistical nature and affine canonical covariance of Wigner distributions in the phase space, we show that the partial coherence effects of incident x-ray wave field on image intensity are simply accounted for by a multiplication factor, which is the reduced complex degree of coherence of the incident x-ray wave field. We show especially that with the undulator sources one cannot obtain the phase-contrast intensity by summing over the contributions from all electron positions, since the van Cittert–Zernike theorem fails in general for undulators. We derive a comprehensive formula that quantifies the effects of partial spatial coherence, polychromatic spectrum, body attenuation, imaging-detector resolution, and radiation dose on phase-contrast visibility in clinical imaging. The results of our computer modeling and simulations show how the formula can provide design guidelines and optimal parameters for clinical x-ray phase-contrast imaging systems.

© 2005 Optical Society of America

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References

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  1. A. Snigirev, I. Snigireva, V. Kohn, S. Kuznetsov, I. Schelokov, “On the possibilities of x-ray phase contrast microimaging by coherent high-energy synchrotron radiation,” Rev. Sci. Instrum. 66, 5486–5492 (1995).
    [CrossRef]
  2. F. Arfelli, V. Bonvicini, A. Bravin, G. Cantatore, E. Castelli, L. Dalla Palma, M. Di Michiel, M. Fabrizioli, R. Longo, R. H. Menk, A. Olivo, S. Pani, D. Pontoni, P. Poropat, M. Prest, A. Rashevsky, M. Ratti, L. Rigon, G. Tromba, A. Vacchi, E. Vallazza, F. Zanconati, “Mammography with synchrotron radiation: phase-detection techniques,” Radiology 215, 286–293 (2000).
    [CrossRef]
  3. X. Wu, A. Dean, H. Liu, “X-ray diagnostic techniques,” in Biomedical Photonics Handbook, T. VoDinh, ed. (CRC, Tampa, Fla., 2003), Chap. 26, pp. 26-1–26-34.
  4. S. Wilkins, T. Gureyev, D. Gao, A. Pogany, A. Stevenson, “Phase contrast imaging using polychromatic hard x-ray,” Nature 384, 335–338 (1996).
    [CrossRef]
  5. A. Pogany, D. Gao, S. W. Wilkins, “Contrast and resolution in imaging with a microfocus x-ray source,” Rev. Sci. Instrum. 68, 2774–2782 (1997).
    [CrossRef]
  6. C. J. Kotre, I. P. Birch, “Phase contrast enhancement of x-ray mammography: a design study,” Phys. Med. Biol. 44, 2853–2866 (1999).
    [CrossRef] [PubMed]
  7. E. Donnelly, R. Price, “Effect of kVp on edge-enhancement index in phase-contrast radiography,” Med. Phys. 29, 999–1002 (2002).
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef]
  22. X. Wu, H. Liu, “An experimental method of determining relative phase-contrast factor for x-ray imaging systems,” Med. Phys. 31, 997–1002 (2004).
    [CrossRef]
  23. F. Ouandji, E. Potter, W. R. Chen, Y. Li, D. Tang, H. Liu, “Characterization of a CCD-based digital x-ray imaging system for small animal studies: properties of spatial resolution,” Appl. Opt. 41, 2420–2427 (2002).
    [CrossRef] [PubMed]
  24. X. Wu, G. Barnes, D. Tucker, “Spectral dependence of glandular tissue dose in screen-film mammography,” Radiology 179, 143–148 (1991).
    [PubMed]
  25. X. Wu, E. Gingold, G. Barnes, D. Tucker, “Normalized average glandular dose in molybdenum target–rhodium filter and rhodium target–rhodium filter mammography,” Radiology 193, 83–89 (1994).
    [PubMed]

2004 (3)

X. Wu, H. Liu, “A new theory of phase-contrast x-ray imaging based on Wigner distributions,” Med. Phys. 31, 2378–2384 (2004).
[CrossRef] [PubMed]

T. Gureyev, A. Pogany, D. Paganin, S. Wilkins, “Linear algorithms for phase retrieval in the Fresnel region,” Opt. Commun. 231, 53–70 (2004).
[CrossRef]

X. Wu, H. Liu, “An experimental method of determining relative phase-contrast factor for x-ray imaging systems,” Med. Phys. 31, 997–1002 (2004).
[CrossRef]

2003 (2)

X. Wu, H. Liu, “A general formalism for x-ray phase contrast imaging,” J. X-Ray Sci. Technol. 11, 33–42 (2003).

X. Wu, H. Liu, “Clinical implementation of phase contrast x-ray imaging: theoretical foundation and design considerations,” Med. Phys. 30, 2169–2179 (2003).
[CrossRef] [PubMed]

2002 (2)

2000 (1)

F. Arfelli, V. Bonvicini, A. Bravin, G. Cantatore, E. Castelli, L. Dalla Palma, M. Di Michiel, M. Fabrizioli, R. Longo, R. H. Menk, A. Olivo, S. Pani, D. Pontoni, P. Poropat, M. Prest, A. Rashevsky, M. Ratti, L. Rigon, G. Tromba, A. Vacchi, E. Vallazza, F. Zanconati, “Mammography with synchrotron radiation: phase-detection techniques,” Radiology 215, 286–293 (2000).
[CrossRef]

1999 (3)

C. J. Kotre, I. P. Birch, “Phase contrast enhancement of x-ray mammography: a design study,” Phys. Med. Biol. 44, 2853–2866 (1999).
[CrossRef] [PubMed]

A. Wax, S. Bali, G. A. Alphonse, J. E. Thomas, “Characterizing the coherence of broadband sources using optical phase space contours,” J. Biomed. Opt. 4, 482–489 (1999).
[CrossRef] [PubMed]

K. F. Lee, F. Reil, S. Bali, A. Wax, J. E. Thomas, “Heterodyne measurement of Wigner distributions for classical optical fields,” Opt. Lett. 24, 1370–1372 (1999).
[CrossRef]

1998 (1)

1997 (1)

A. Pogany, D. Gao, S. W. Wilkins, “Contrast and resolution in imaging with a microfocus x-ray source,” Rev. Sci. Instrum. 68, 2774–2782 (1997).
[CrossRef]

1996 (1)

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

1995 (2)

A. Snigirev, I. Snigireva, V. Kohn, S. Kuznetsov, I. Schelokov, “On the possibilities of x-ray phase contrast microimaging by coherent high-energy synchrotron radiation,” Rev. Sci. Instrum. 66, 5486–5492 (1995).
[CrossRef]

R. Coisson, “Spatial coherence of synchrotron radiation,” Appl. Opt. 34, 904–908 (1995).
[CrossRef] [PubMed]

1994 (1)

X. Wu, E. Gingold, G. Barnes, D. Tucker, “Normalized average glandular dose in molybdenum target–rhodium filter and rhodium target–rhodium filter mammography,” Radiology 193, 83–89 (1994).
[PubMed]

1991 (1)

X. Wu, G. Barnes, D. Tucker, “Spectral dependence of glandular tissue dose in screen-film mammography,” Radiology 179, 143–148 (1991).
[PubMed]

1986 (1)

R. G. Littlejohn, “The semiclassical evolution of wave packets,” Phys. Rep. 138, 193–291 (1986).
[CrossRef]

1983 (1)

1977 (1)

J. Guigay, “Fourier transform analysis of Fresnel diffraction and in-line holograms,” Optik (Stuttgart) 49, 121–125 (1977).

Alphonse, G. A.

A. Wax, S. Bali, G. A. Alphonse, J. E. Thomas, “Characterizing the coherence of broadband sources using optical phase space contours,” J. Biomed. Opt. 4, 482–489 (1999).
[CrossRef] [PubMed]

Arfelli, F.

F. Arfelli, V. Bonvicini, A. Bravin, G. Cantatore, E. Castelli, L. Dalla Palma, M. Di Michiel, M. Fabrizioli, R. Longo, R. H. Menk, A. Olivo, S. Pani, D. Pontoni, P. Poropat, M. Prest, A. Rashevsky, M. Ratti, L. Rigon, G. Tromba, A. Vacchi, E. Vallazza, F. Zanconati, “Mammography with synchrotron radiation: phase-detection techniques,” Radiology 215, 286–293 (2000).
[CrossRef]

Bali, S.

A. Wax, S. Bali, G. A. Alphonse, J. E. Thomas, “Characterizing the coherence of broadband sources using optical phase space contours,” J. Biomed. Opt. 4, 482–489 (1999).
[CrossRef] [PubMed]

K. F. Lee, F. Reil, S. Bali, A. Wax, J. E. Thomas, “Heterodyne measurement of Wigner distributions for classical optical fields,” Opt. Lett. 24, 1370–1372 (1999).
[CrossRef]

Barnes, G.

X. Wu, E. Gingold, G. Barnes, D. Tucker, “Normalized average glandular dose in molybdenum target–rhodium filter and rhodium target–rhodium filter mammography,” Radiology 193, 83–89 (1994).
[PubMed]

X. Wu, G. Barnes, D. Tucker, “Spectral dependence of glandular tissue dose in screen-film mammography,” Radiology 179, 143–148 (1991).
[PubMed]

Birch, I. P.

C. J. Kotre, I. P. Birch, “Phase contrast enhancement of x-ray mammography: a design study,” Phys. Med. Biol. 44, 2853–2866 (1999).
[CrossRef] [PubMed]

Bonvicini, V.

F. Arfelli, V. Bonvicini, A. Bravin, G. Cantatore, E. Castelli, L. Dalla Palma, M. Di Michiel, M. Fabrizioli, R. Longo, R. H. Menk, A. Olivo, S. Pani, D. Pontoni, P. Poropat, M. Prest, A. Rashevsky, M. Ratti, L. Rigon, G. Tromba, A. Vacchi, E. Vallazza, F. Zanconati, “Mammography with synchrotron radiation: phase-detection techniques,” Radiology 215, 286–293 (2000).
[CrossRef]

Born, M.

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

Bravin, A.

F. Arfelli, V. Bonvicini, A. Bravin, G. Cantatore, E. Castelli, L. Dalla Palma, M. Di Michiel, M. Fabrizioli, R. Longo, R. H. Menk, A. Olivo, S. Pani, D. Pontoni, P. Poropat, M. Prest, A. Rashevsky, M. Ratti, L. Rigon, G. Tromba, A. Vacchi, E. Vallazza, F. Zanconati, “Mammography with synchrotron radiation: phase-detection techniques,” Radiology 215, 286–293 (2000).
[CrossRef]

Cantatore, G.

F. Arfelli, V. Bonvicini, A. Bravin, G. Cantatore, E. Castelli, L. Dalla Palma, M. Di Michiel, M. Fabrizioli, R. Longo, R. H. Menk, A. Olivo, S. Pani, D. Pontoni, P. Poropat, M. Prest, A. Rashevsky, M. Ratti, L. Rigon, G. Tromba, A. Vacchi, E. Vallazza, F. Zanconati, “Mammography with synchrotron radiation: phase-detection techniques,” Radiology 215, 286–293 (2000).
[CrossRef]

Castelli, E.

F. Arfelli, V. Bonvicini, A. Bravin, G. Cantatore, E. Castelli, L. Dalla Palma, M. Di Michiel, M. Fabrizioli, R. Longo, R. H. Menk, A. Olivo, S. Pani, D. Pontoni, P. Poropat, M. Prest, A. Rashevsky, M. Ratti, L. Rigon, G. Tromba, A. Vacchi, E. Vallazza, F. Zanconati, “Mammography with synchrotron radiation: phase-detection techniques,” Radiology 215, 286–293 (2000).
[CrossRef]

Chen, W. R.

Coisson, R.

Dalla Palma, L.

F. Arfelli, V. Bonvicini, A. Bravin, G. Cantatore, E. Castelli, L. Dalla Palma, M. Di Michiel, M. Fabrizioli, R. Longo, R. H. Menk, A. Olivo, S. Pani, D. Pontoni, P. Poropat, M. Prest, A. Rashevsky, M. Ratti, L. Rigon, G. Tromba, A. Vacchi, E. Vallazza, F. Zanconati, “Mammography with synchrotron radiation: phase-detection techniques,” Radiology 215, 286–293 (2000).
[CrossRef]

Dean, A.

X. Wu, A. Dean, H. Liu, “X-ray diagnostic techniques,” in Biomedical Photonics Handbook, T. VoDinh, ed. (CRC, Tampa, Fla., 2003), Chap. 26, pp. 26-1–26-34.

Di Michiel, M.

F. Arfelli, V. Bonvicini, A. Bravin, G. Cantatore, E. Castelli, L. Dalla Palma, M. Di Michiel, M. Fabrizioli, R. Longo, R. H. Menk, A. Olivo, S. Pani, D. Pontoni, P. Poropat, M. Prest, A. Rashevsky, M. Ratti, L. Rigon, G. Tromba, A. Vacchi, E. Vallazza, F. Zanconati, “Mammography with synchrotron radiation: phase-detection techniques,” Radiology 215, 286–293 (2000).
[CrossRef]

Donnelly, E.

E. Donnelly, R. Price, “Effect of kVp on edge-enhancement index in phase-contrast radiography,” Med. Phys. 29, 999–1002 (2002).
[CrossRef] [PubMed]

Fabrizioli, M.

F. Arfelli, V. Bonvicini, A. Bravin, G. Cantatore, E. Castelli, L. Dalla Palma, M. Di Michiel, M. Fabrizioli, R. Longo, R. H. Menk, A. Olivo, S. Pani, D. Pontoni, P. Poropat, M. Prest, A. Rashevsky, M. Ratti, L. Rigon, G. Tromba, A. Vacchi, E. Vallazza, F. Zanconati, “Mammography with synchrotron radiation: phase-detection techniques,” Radiology 215, 286–293 (2000).
[CrossRef]

Gao, D.

A. Pogany, D. Gao, S. W. Wilkins, “Contrast and resolution in imaging with a microfocus x-ray source,” Rev. Sci. Instrum. 68, 2774–2782 (1997).
[CrossRef]

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

Gingold, E.

X. Wu, E. Gingold, G. Barnes, D. Tucker, “Normalized average glandular dose in molybdenum target–rhodium filter and rhodium target–rhodium filter mammography,” Radiology 193, 83–89 (1994).
[PubMed]

Guigay, J.

J. Guigay, “Fourier transform analysis of Fresnel diffraction and in-line holograms,” Optik (Stuttgart) 49, 121–125 (1977).

Gureyev, T.

T. Gureyev, A. Pogany, D. Paganin, S. Wilkins, “Linear algorithms for phase retrieval in the Fresnel region,” Opt. Commun. 231, 53–70 (2004).
[CrossRef]

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

Kohn, V.

A. Snigirev, I. Snigireva, V. Kohn, S. Kuznetsov, I. Schelokov, “On the possibilities of x-ray phase contrast microimaging by coherent high-energy synchrotron radiation,” Rev. Sci. Instrum. 66, 5486–5492 (1995).
[CrossRef]

Kotre, C. J.

C. J. Kotre, I. P. Birch, “Phase contrast enhancement of x-ray mammography: a design study,” Phys. Med. Biol. 44, 2853–2866 (1999).
[CrossRef] [PubMed]

Kuznetsov, S.

A. Snigirev, I. Snigireva, V. Kohn, S. Kuznetsov, I. Schelokov, “On the possibilities of x-ray phase contrast microimaging by coherent high-energy synchrotron radiation,” Rev. Sci. Instrum. 66, 5486–5492 (1995).
[CrossRef]

Lee, K. F.

Li, Y.

Littlejohn, R. G.

R. G. Littlejohn, “The semiclassical evolution of wave packets,” Phys. Rep. 138, 193–291 (1986).
[CrossRef]

Liu, H.

X. Wu, H. Liu, “An experimental method of determining relative phase-contrast factor for x-ray imaging systems,” Med. Phys. 31, 997–1002 (2004).
[CrossRef]

X. Wu, H. Liu, “A new theory of phase-contrast x-ray imaging based on Wigner distributions,” Med. Phys. 31, 2378–2384 (2004).
[CrossRef] [PubMed]

X. Wu, H. Liu, “Clinical implementation of phase contrast x-ray imaging: theoretical foundation and design considerations,” Med. Phys. 30, 2169–2179 (2003).
[CrossRef] [PubMed]

X. Wu, H. Liu, “A general formalism for x-ray phase contrast imaging,” J. X-Ray Sci. Technol. 11, 33–42 (2003).

F. Ouandji, E. Potter, W. R. Chen, Y. Li, D. Tang, H. Liu, “Characterization of a CCD-based digital x-ray imaging system for small animal studies: properties of spatial resolution,” Appl. Opt. 41, 2420–2427 (2002).
[CrossRef] [PubMed]

X. Wu, A. Dean, H. Liu, “X-ray diagnostic techniques,” in Biomedical Photonics Handbook, T. VoDinh, ed. (CRC, Tampa, Fla., 2003), Chap. 26, pp. 26-1–26-34.

Longo, R.

F. Arfelli, V. Bonvicini, A. Bravin, G. Cantatore, E. Castelli, L. Dalla Palma, M. Di Michiel, M. Fabrizioli, R. Longo, R. H. Menk, A. Olivo, S. Pani, D. Pontoni, P. Poropat, M. Prest, A. Rashevsky, M. Ratti, L. Rigon, G. Tromba, A. Vacchi, E. Vallazza, F. Zanconati, “Mammography with synchrotron radiation: phase-detection techniques,” Radiology 215, 286–293 (2000).
[CrossRef]

Mandel, L.

L. Mandel, E. Wolf, Optical Coherence and Quantum Optics (Cambridge University Press, Cambridge, 1995).
[CrossRef]

Menk, R. H.

F. Arfelli, V. Bonvicini, A. Bravin, G. Cantatore, E. Castelli, L. Dalla Palma, M. Di Michiel, M. Fabrizioli, R. Longo, R. H. Menk, A. Olivo, S. Pani, D. Pontoni, P. Poropat, M. Prest, A. Rashevsky, M. Ratti, L. Rigon, G. Tromba, A. Vacchi, E. Vallazza, F. Zanconati, “Mammography with synchrotron radiation: phase-detection techniques,” Radiology 215, 286–293 (2000).
[CrossRef]

Olivo, A.

F. Arfelli, V. Bonvicini, A. Bravin, G. Cantatore, E. Castelli, L. Dalla Palma, M. Di Michiel, M. Fabrizioli, R. Longo, R. H. Menk, A. Olivo, S. Pani, D. Pontoni, P. Poropat, M. Prest, A. Rashevsky, M. Ratti, L. Rigon, G. Tromba, A. Vacchi, E. Vallazza, F. Zanconati, “Mammography with synchrotron radiation: phase-detection techniques,” Radiology 215, 286–293 (2000).
[CrossRef]

Ouandji, F.

Paganin, D.

T. Gureyev, A. Pogany, D. Paganin, S. Wilkins, “Linear algorithms for phase retrieval in the Fresnel region,” Opt. Commun. 231, 53–70 (2004).
[CrossRef]

Pani, S.

F. Arfelli, V. Bonvicini, A. Bravin, G. Cantatore, E. Castelli, L. Dalla Palma, M. Di Michiel, M. Fabrizioli, R. Longo, R. H. Menk, A. Olivo, S. Pani, D. Pontoni, P. Poropat, M. Prest, A. Rashevsky, M. Ratti, L. Rigon, G. Tromba, A. Vacchi, E. Vallazza, F. Zanconati, “Mammography with synchrotron radiation: phase-detection techniques,” Radiology 215, 286–293 (2000).
[CrossRef]

Pogany, A.

T. Gureyev, A. Pogany, D. Paganin, S. Wilkins, “Linear algorithms for phase retrieval in the Fresnel region,” Opt. Commun. 231, 53–70 (2004).
[CrossRef]

A. Pogany, D. Gao, S. W. Wilkins, “Contrast and resolution in imaging with a microfocus x-ray source,” Rev. Sci. Instrum. 68, 2774–2782 (1997).
[CrossRef]

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

Pontoni, D.

F. Arfelli, V. Bonvicini, A. Bravin, G. Cantatore, E. Castelli, L. Dalla Palma, M. Di Michiel, M. Fabrizioli, R. Longo, R. H. Menk, A. Olivo, S. Pani, D. Pontoni, P. Poropat, M. Prest, A. Rashevsky, M. Ratti, L. Rigon, G. Tromba, A. Vacchi, E. Vallazza, F. Zanconati, “Mammography with synchrotron radiation: phase-detection techniques,” Radiology 215, 286–293 (2000).
[CrossRef]

Poropat, P.

F. Arfelli, V. Bonvicini, A. Bravin, G. Cantatore, E. Castelli, L. Dalla Palma, M. Di Michiel, M. Fabrizioli, R. Longo, R. H. Menk, A. Olivo, S. Pani, D. Pontoni, P. Poropat, M. Prest, A. Rashevsky, M. Ratti, L. Rigon, G. Tromba, A. Vacchi, E. Vallazza, F. Zanconati, “Mammography with synchrotron radiation: phase-detection techniques,” Radiology 215, 286–293 (2000).
[CrossRef]

Potter, E.

Prest, M.

F. Arfelli, V. Bonvicini, A. Bravin, G. Cantatore, E. Castelli, L. Dalla Palma, M. Di Michiel, M. Fabrizioli, R. Longo, R. H. Menk, A. Olivo, S. Pani, D. Pontoni, P. Poropat, M. Prest, A. Rashevsky, M. Ratti, L. Rigon, G. Tromba, A. Vacchi, E. Vallazza, F. Zanconati, “Mammography with synchrotron radiation: phase-detection techniques,” Radiology 215, 286–293 (2000).
[CrossRef]

Price, R.

E. Donnelly, R. Price, “Effect of kVp on edge-enhancement index in phase-contrast radiography,” Med. Phys. 29, 999–1002 (2002).
[CrossRef] [PubMed]

Rashevsky, A.

F. Arfelli, V. Bonvicini, A. Bravin, G. Cantatore, E. Castelli, L. Dalla Palma, M. Di Michiel, M. Fabrizioli, R. Longo, R. H. Menk, A. Olivo, S. Pani, D. Pontoni, P. Poropat, M. Prest, A. Rashevsky, M. Ratti, L. Rigon, G. Tromba, A. Vacchi, E. Vallazza, F. Zanconati, “Mammography with synchrotron radiation: phase-detection techniques,” Radiology 215, 286–293 (2000).
[CrossRef]

Ratti, M.

F. Arfelli, V. Bonvicini, A. Bravin, G. Cantatore, E. Castelli, L. Dalla Palma, M. Di Michiel, M. Fabrizioli, R. Longo, R. H. Menk, A. Olivo, S. Pani, D. Pontoni, P. Poropat, M. Prest, A. Rashevsky, M. Ratti, L. Rigon, G. Tromba, A. Vacchi, E. Vallazza, F. Zanconati, “Mammography with synchrotron radiation: phase-detection techniques,” Radiology 215, 286–293 (2000).
[CrossRef]

Reil, F.

Rigon, L.

F. Arfelli, V. Bonvicini, A. Bravin, G. Cantatore, E. Castelli, L. Dalla Palma, M. Di Michiel, M. Fabrizioli, R. Longo, R. H. Menk, A. Olivo, S. Pani, D. Pontoni, P. Poropat, M. Prest, A. Rashevsky, M. Ratti, L. Rigon, G. Tromba, A. Vacchi, E. Vallazza, F. Zanconati, “Mammography with synchrotron radiation: phase-detection techniques,” Radiology 215, 286–293 (2000).
[CrossRef]

Schelokov, I.

A. Snigirev, I. Snigireva, V. Kohn, S. Kuznetsov, I. Schelokov, “On the possibilities of x-ray phase contrast microimaging by coherent high-energy synchrotron radiation,” Rev. Sci. Instrum. 66, 5486–5492 (1995).
[CrossRef]

Snigirev, A.

A. Snigirev, I. Snigireva, V. Kohn, S. Kuznetsov, I. Schelokov, “On the possibilities of x-ray phase contrast microimaging by coherent high-energy synchrotron radiation,” Rev. Sci. Instrum. 66, 5486–5492 (1995).
[CrossRef]

Snigireva, I.

A. Snigirev, I. Snigireva, V. Kohn, S. Kuznetsov, I. Schelokov, “On the possibilities of x-ray phase contrast microimaging by coherent high-energy synchrotron radiation,” Rev. Sci. Instrum. 66, 5486–5492 (1995).
[CrossRef]

Stevenson, A.

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

Tang, D.

Teague, M.

Thomas, J. E.

Tromba, G.

F. Arfelli, V. Bonvicini, A. Bravin, G. Cantatore, E. Castelli, L. Dalla Palma, M. Di Michiel, M. Fabrizioli, R. Longo, R. H. Menk, A. Olivo, S. Pani, D. Pontoni, P. Poropat, M. Prest, A. Rashevsky, M. Ratti, L. Rigon, G. Tromba, A. Vacchi, E. Vallazza, F. Zanconati, “Mammography with synchrotron radiation: phase-detection techniques,” Radiology 215, 286–293 (2000).
[CrossRef]

Tucker, D.

X. Wu, E. Gingold, G. Barnes, D. Tucker, “Normalized average glandular dose in molybdenum target–rhodium filter and rhodium target–rhodium filter mammography,” Radiology 193, 83–89 (1994).
[PubMed]

X. Wu, G. Barnes, D. Tucker, “Spectral dependence of glandular tissue dose in screen-film mammography,” Radiology 179, 143–148 (1991).
[PubMed]

Vacchi, A.

F. Arfelli, V. Bonvicini, A. Bravin, G. Cantatore, E. Castelli, L. Dalla Palma, M. Di Michiel, M. Fabrizioli, R. Longo, R. H. Menk, A. Olivo, S. Pani, D. Pontoni, P. Poropat, M. Prest, A. Rashevsky, M. Ratti, L. Rigon, G. Tromba, A. Vacchi, E. Vallazza, F. Zanconati, “Mammography with synchrotron radiation: phase-detection techniques,” Radiology 215, 286–293 (2000).
[CrossRef]

Vallazza, E.

F. Arfelli, V. Bonvicini, A. Bravin, G. Cantatore, E. Castelli, L. Dalla Palma, M. Di Michiel, M. Fabrizioli, R. Longo, R. H. Menk, A. Olivo, S. Pani, D. Pontoni, P. Poropat, M. Prest, A. Rashevsky, M. Ratti, L. Rigon, G. Tromba, A. Vacchi, E. Vallazza, F. Zanconati, “Mammography with synchrotron radiation: phase-detection techniques,” Radiology 215, 286–293 (2000).
[CrossRef]

Wax, A.

Wiedemann, H.

H. Wiedemann, Synchrotron Radiation (Springer-Verlag, Berlin, 2003).
[CrossRef]

Wilkins, S.

T. Gureyev, A. Pogany, D. Paganin, S. Wilkins, “Linear algorithms for phase retrieval in the Fresnel region,” Opt. Commun. 231, 53–70 (2004).
[CrossRef]

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

Wilkins, S. W.

A. Pogany, D. Gao, S. W. Wilkins, “Contrast and resolution in imaging with a microfocus x-ray source,” Rev. Sci. Instrum. 68, 2774–2782 (1997).
[CrossRef]

Wolf, E.

L. Mandel, E. Wolf, Optical Coherence and Quantum Optics (Cambridge University Press, Cambridge, 1995).
[CrossRef]

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

Wu, X.

X. Wu, H. Liu, “An experimental method of determining relative phase-contrast factor for x-ray imaging systems,” Med. Phys. 31, 997–1002 (2004).
[CrossRef]

X. Wu, H. Liu, “A new theory of phase-contrast x-ray imaging based on Wigner distributions,” Med. Phys. 31, 2378–2384 (2004).
[CrossRef] [PubMed]

X. Wu, H. Liu, “Clinical implementation of phase contrast x-ray imaging: theoretical foundation and design considerations,” Med. Phys. 30, 2169–2179 (2003).
[CrossRef] [PubMed]

X. Wu, H. Liu, “A general formalism for x-ray phase contrast imaging,” J. X-Ray Sci. Technol. 11, 33–42 (2003).

X. Wu, E. Gingold, G. Barnes, D. Tucker, “Normalized average glandular dose in molybdenum target–rhodium filter and rhodium target–rhodium filter mammography,” Radiology 193, 83–89 (1994).
[PubMed]

X. Wu, G. Barnes, D. Tucker, “Spectral dependence of glandular tissue dose in screen-film mammography,” Radiology 179, 143–148 (1991).
[PubMed]

X. Wu, A. Dean, H. Liu, “X-ray diagnostic techniques,” in Biomedical Photonics Handbook, T. VoDinh, ed. (CRC, Tampa, Fla., 2003), Chap. 26, pp. 26-1–26-34.

Zanconati, F.

F. Arfelli, V. Bonvicini, A. Bravin, G. Cantatore, E. Castelli, L. Dalla Palma, M. Di Michiel, M. Fabrizioli, R. Longo, R. H. Menk, A. Olivo, S. Pani, D. Pontoni, P. Poropat, M. Prest, A. Rashevsky, M. Ratti, L. Rigon, G. Tromba, A. Vacchi, E. Vallazza, F. Zanconati, “Mammography with synchrotron radiation: phase-detection techniques,” Radiology 215, 286–293 (2000).
[CrossRef]

Appl. Opt. (2)

J. Biomed. Opt. (1)

A. Wax, S. Bali, G. A. Alphonse, J. E. Thomas, “Characterizing the coherence of broadband sources using optical phase space contours,” J. Biomed. Opt. 4, 482–489 (1999).
[CrossRef] [PubMed]

J. Opt. Soc. Am. (1)

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

J. X-Ray Sci. Technol. (1)

X. Wu, H. Liu, “A general formalism for x-ray phase contrast imaging,” J. X-Ray Sci. Technol. 11, 33–42 (2003).

Med. Phys. (4)

X. Wu, H. Liu, “Clinical implementation of phase contrast x-ray imaging: theoretical foundation and design considerations,” Med. Phys. 30, 2169–2179 (2003).
[CrossRef] [PubMed]

X. Wu, H. Liu, “A new theory of phase-contrast x-ray imaging based on Wigner distributions,” Med. Phys. 31, 2378–2384 (2004).
[CrossRef] [PubMed]

E. Donnelly, R. Price, “Effect of kVp on edge-enhancement index in phase-contrast radiography,” Med. Phys. 29, 999–1002 (2002).
[CrossRef] [PubMed]

X. Wu, H. Liu, “An experimental method of determining relative phase-contrast factor for x-ray imaging systems,” Med. Phys. 31, 997–1002 (2004).
[CrossRef]

Nature (1)

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

Opt. Commun. (1)

T. Gureyev, A. Pogany, D. Paganin, S. Wilkins, “Linear algorithms for phase retrieval in the Fresnel region,” Opt. Commun. 231, 53–70 (2004).
[CrossRef]

Opt. Lett. (1)

Optik (Stuttgart) (1)

J. Guigay, “Fourier transform analysis of Fresnel diffraction and in-line holograms,” Optik (Stuttgart) 49, 121–125 (1977).

Phys. Med. Biol. (1)

C. J. Kotre, I. P. Birch, “Phase contrast enhancement of x-ray mammography: a design study,” Phys. Med. Biol. 44, 2853–2866 (1999).
[CrossRef] [PubMed]

Phys. Rep. (1)

R. G. Littlejohn, “The semiclassical evolution of wave packets,” Phys. Rep. 138, 193–291 (1986).
[CrossRef]

Radiology (3)

X. Wu, G. Barnes, D. Tucker, “Spectral dependence of glandular tissue dose in screen-film mammography,” Radiology 179, 143–148 (1991).
[PubMed]

X. Wu, E. Gingold, G. Barnes, D. Tucker, “Normalized average glandular dose in molybdenum target–rhodium filter and rhodium target–rhodium filter mammography,” Radiology 193, 83–89 (1994).
[PubMed]

F. Arfelli, V. Bonvicini, A. Bravin, G. Cantatore, E. Castelli, L. Dalla Palma, M. Di Michiel, M. Fabrizioli, R. Longo, R. H. Menk, A. Olivo, S. Pani, D. Pontoni, P. Poropat, M. Prest, A. Rashevsky, M. Ratti, L. Rigon, G. Tromba, A. Vacchi, E. Vallazza, F. Zanconati, “Mammography with synchrotron radiation: phase-detection techniques,” Radiology 215, 286–293 (2000).
[CrossRef]

Rev. Sci. Instrum. (2)

A. Pogany, D. Gao, S. W. Wilkins, “Contrast and resolution in imaging with a microfocus x-ray source,” Rev. Sci. Instrum. 68, 2774–2782 (1997).
[CrossRef]

A. Snigirev, I. Snigireva, V. Kohn, S. Kuznetsov, I. Schelokov, “On the possibilities of x-ray phase contrast microimaging by coherent high-energy synchrotron radiation,” Rev. Sci. Instrum. 66, 5486–5492 (1995).
[CrossRef]

Other (4)

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

X. Wu, A. Dean, H. Liu, “X-ray diagnostic techniques,” in Biomedical Photonics Handbook, T. VoDinh, ed. (CRC, Tampa, Fla., 2003), Chap. 26, pp. 26-1–26-34.

H. Wiedemann, Synchrotron Radiation (Springer-Verlag, Berlin, 2003).
[CrossRef]

L. Mandel, E. Wolf, Optical Coherence and Quantum Optics (Cambridge University Press, Cambridge, 1995).
[CrossRef]

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Figures (4)

Fig. 1
Fig. 1

Schematic of the undulator.

Fig. 2
Fig. 2

Average glandular doses of the breast as a function of R1 for constant pixel noise in the images.

Fig. 3
Fig. 3

|RPF(u)| for a phase-contrast mammography system with focal spot sizes of 0.01, 0.03, and 0.05 mm and with a detector pitch of 0.025 mm.

Fig. 4
Fig. 4

|RPF(u)| for a phase-contrast mammography system with focal spot sizes of 0.03 and 0.05 and detector pitches of 0.025 and 0.04 mm.

Equations (29)

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n = 1 - δ + i β .
ϕ = - 2 π λ δ ( s ) d s ,             μ p = 4 π λ β ( s ) d s ,
W ( r , u ; z ) = J ( r + q / 2 , r - q / 2 ; z ) × exp ( - i 2 π q · u ) d q ,
W z + λ u · r W = 0.
W ( r , u ; z ) = W [ r - λ ( z - R 1 ) u , u ; R 1 ] .
I ( r ; R 1 + R 2 ) = W ( r , u ; R 1 + R 2 ) d u = W ( r - λ R 2 u , u ; R 1 ) d u .
T ( r ) = A ( r ) e i ϕ ( r ) = exp [ i ϕ ( r ) - μ p ( r ) 2 ] ,
I ˜ ( u ; R 1 + R 2 ) = J i n ( s + λ R 2 u / 2 , s - λ R 2 u / 2 ) × T ( s + λ R 2 u / 2 ) T * ( s - λ R 2 u / 2 ) × exp ( i 2 π s · u ) d s ,
μ i n ( s 1 , s 2 ) = J i n ( s 1 , s 2 ) I i n ( s 1 ) I i n ( s 2 ) .
I ˜ ( u ; R 1 + R 2 ) = I i n ( s + λ R 2 u / 2 ) I i n ( s - λ R 2 u / 2 ) × μ i n ( s + λ R 2 u / 2 , s - λ R 2 u / 2 ) × T ( s + λ R 2 u / 2 ) × T * ( s - λ R 2 u / 2 ) exp ( i 2 π s · u ) d s .
I ˜ ( u ; R 1 + R 2 ) = I i n μ i n ( s + λ R 2 u / 2 , s - λ R 2 u / 2 ) × T ( s + λ R 2 u / 2 ) T * ( s - λ R 2 u / 2 ) × exp ( i 2 π s · u ) d s .
μ i n ( s 1 , s 2 ) = exp ( i s 1 2 - s 2 2 λ R 1 ) μ ˜ i n ( s 1 - s 2 ) ,
μ ˜ i n ( λ R 2 u M ) = I source ( ξ ) exp [ i 2 π ξ ( M - 1 ) u M ] d ξ I source ( ξ ) d ξ ,
I ˜ ( u M ; R 1 + R 2 ) = I i n u ˜ i n ( λ R 2 u M ) T ( s + λ R 2 u 2 M ) × T * ( s - λ R 2 u 2 M ) exp ( i 2 π s · u ) d s .
μ ˜ i n ( λ R 2 u M ) = O T F G . U . ( u M ) .
1 / γ m N u ,
exp [ i ( s 1 2 - s 2 2 ) / λ R 1 ] μ ˜ i n ( s 1 - s 2 ) .
μ ˜ i n ( s 1 - s 2 ) = exp { - 1 2 [ ( s 1 x - s 2 x ) 2 σ μ x 2 + ( s 1 y - s 2 y ) 2 σ μ y 2 ] } ,
σ μ x = ( λ R 1 2 π ) 1 ( σ e x 2 + 1 / 4 u 1 x 2 ) - 1 4 ( 2 π σ e x / λ ) 2 + 4 u 1 x 2 ,
I ˜ ( u M ; R 1 + R 2 ) = I i n u ˜ i n ( λ R 2 u M ) × A ( s + λ R 2 u 2 M ) A ( s - λ R 2 u 2 M ) × exp [ i ϕ ( r + λ R 2 u 2 M ) ] - i ϕ ( r - λ R 2 u 2 M ) × exp ( i 2 π s · u ) d s .
| ϕ ( r + λ R 2 u 2 M ) - ϕ ( r + λ R 2 u 2 M ) | 1.
exp [ i ϕ ( r + λ R 2 u 2 M ) - i ϕ ( r - λ R 2 u 2 M ) ] 1 + i [ ϕ ( r + λ R 2 u 2 M ) - ϕ ( r + λ R 2 u 2 M ) ] .
A ( r + λ R 2 u 2 M ) A ( r ) + λ R 2 2 M u · A ( r ) .
I ˜ ( u M ; R 1 + R 2 ) = I i n u ˜ i n ( λ R 2 u M ) OTF det ( u M ) × { cos ( π λ R 2 u 2 M ) [ FT ( A 2 ) - i λ R 2 M u · FT ( ϕ A 2 ) ] + 2 sin ( π λ R 2 u 2 M ) [ FT ( A 2 ϕ ) + i λ R 2 4 M u · FT ( A 2 ) ] } ,
I ˜ ( r ; R 1 + R 2 ) = I i n M 2 FT - 1 [ μ ˜ i n ( λ R 2 u M ) OTF det ( u M ) ] { A 2 ( r M ) - λ R 2 2 π M · [ A 2 ϕ ( r M ) ] } ,
I ˜ ( u M ; R 1 + R 2 ) = I i n [ OTF G . U . ( u M ) OTF det ( u M ) FT ( A o T 2 ) - 2 r e RPF ( u ) FT ( A o T 2 ρ e ) ] ,
RPF ( u ) = c 2 h 2 OTF G . U . ( u M ) OTF det ( u M ) × [ π R 2 u 2 M E 2 S E x i t ( E ) d E ] ,
| OTF G . U . ( u M ) | = 2 J 1 [ π f ( M - 1 ) u / M ] π f ( M - 1 ) u / M ,
| OTF det ( u M ) | = sin c ( p a x u x M ) sin c ( p a y u y M ) ,

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