Abstract

Phase contrast X-ray imaging is increasingly popular in the past decade. In order to acquire phase contrast X-ray images, different types of imaging mechanisms have been proposed. Among them, in-line phase contrast X-ray imaging shows the highest potential because of its simplicity. In the study of in-line phase contrast imaging, based on different physical assumptions, many non-iterative phase retrieval methods, such as Bronnikov method, modified Bronnikov method, phase-attenuation duality (PAD) method, single-material method, and two-material method have been proposed. The main step of the non-iterative methods is a filtering process, thus different methods involve different filter design. In this paper we showed that every filter applied in the methods listed above is indeed the minimizer of a L2-norm regularization problem. In addition, two methods were proposed to overcome the over smoothing problem owing to the nature of L2-norm regularization.

© 2015 Optical Society of America

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References

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    [Crossref]
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    [Crossref]
  34. M. D. Wong, X. Wu, and H. Liu, “Image quality and dose efficiency of high energy phase sensitive x-ray imaging: Phantom studies,” J. Xray Sci. Technol. 22, 321–334 (2014).
    [PubMed]
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    [Crossref]
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    [Crossref]
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    [Crossref]
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    [Crossref]
  40. L. A. Feldkemp, L. C. Davis, and J. W. Kress, “Practical cone beam algorithm,” J. Opt. Soc. Am. 1, 612–619 (1984).
    [Crossref]
  41. T. E. Gureyev, T. J. Davis, A. Pogany, S. C. Mayo, and S. W. Wilkins, “Optical phase retrieval by use of first Born- and Rytov-type approximation,” Appl. Opt. 43, 2418–2430 (2004).
    [Crossref] [PubMed]
  42. A. V. Bronnikov, “Reconstruction formulas for phase-contrast tomography,” Opt. Commum. 171, 239–244 (1999).
    [Crossref]
  43. X. Wu and H. Liu, “An experimental method of determining relative phase-contrast factor for x-ray imaging system,” Med. Phys. 31, 997–1002 (2004).
    [Crossref]
  44. X. Wu, H. Liu, and A. Yan, “System design consideration for a x-ray phase-contrast imaging system based on in-line holography,” Proc. of SPIE, Vol. 5630, (2005).
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    [Crossref]
  46. M. Langer, P. Cloetens, and F. Peyrin, “Regularization of phase retrieval with phase-attenuation duality prior for 3-D holotomography,” IEEE Trans. Image Process. 19(9), 2428–2436 (2010).
    [Crossref] [PubMed]

2014 (3)

D. Wu, A. Yan, Y. Li, W. R. Chen, X. Wu, and H. Liu, “Phantom study based on a high-energy in-line phase contrast tomosynthesis prototype,” Proc. SPIE 8944, 89440 (2014).
[Crossref]

M. D. Wong, X. Wu, and H. Liu, “Image quality and dose efficiency of high energy phase sensitive x-ray imaging: Phantom studies,” J. Xray Sci. Technol. 22, 321–334 (2014).
[PubMed]

A. Pan, L. Xu, J. C. Petrucelli, R. Gupta, B. Singh, and G. Barbastahis, “Contrast enhancement in X-ray phase contrast tomography,” Opt. Express 22, 18021–18026 (2014).
[Crossref]

2013 (1)

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

2012 (4)

S. C. Mayo, A. W. Stevenson, and S. W. Wilkins, “In-line phase-contrast X-ray imaging and tomography for Material Science,” Material 5, 937–965 (2012).
[Crossref]

A. Olivo, P. C. Diemoz, and A. Bravin, “Amplification of the phase contrast signal at very high x-ray energies,” Opt. Lett. 37, 915–917 (2012).
[Crossref] [PubMed]

X. Guo, X. Liu, M. Gu, C. Ni, S. Huang, and B. Liu, “Polychromatic x-ray in-line phase-contrast tomography for soft tissue,” Europhysics Lett. 98, 14001 (2012).
[Crossref]

X. Wu, A. Yan, and H. Liu, “X-ray phase-shifted-based method of volumetric breast density measurement,” Med. Phys. 39, pp. 4239–4244 (2012).
[Crossref] [PubMed]

2011 (2)

2010 (3)

M. A. Beltran, D. M. Paganin, K. Uesugi, and M. J. Kitchen, “2D and 3D X-ray phase retrieval of multi-material objects using a single defocus distance,” Opt. Express 18, 6423–6436 (2010).
[Crossref] [PubMed]

M. Wong, X. Wu, and H. Liu, “Image quality comparison of high energy phase contrast x-ray images with low energy conventional images: phantom studies,” Proc. SPIE 7563, 756305 (2010).
[Crossref]

M. Langer, P. Cloetens, and F. Peyrin, “Regularization of phase retrieval with phase-attenuation duality prior for 3-D holotomography,” IEEE Trans. Image Process. 19(9), 2428–2436 (2010).
[Crossref] [PubMed]

2009 (2)

P. Rodríguez and B. Wohlberg, “Efficient minimization method for a generalized total variation functional,” Tran. IEEE Imag. Process. 18, 322–332 (2009).
[Crossref]

X. Wu and A. Yan, “Phase retrieval from one single phase contrast X-ray image,” Opt. Express 17(13), 11187–96 (2009).
[Crossref] [PubMed]

2008 (2)

S. Ahn and R. M. Leahy, “Analysis of resolution and noise properties of nonquadratically regularized image reconstruction methods for PET,” Trans. IEEE Med. Imag. 27, pp. 413–424 (2008).
[Crossref]

Y. I. Nesterests and S. W. Wilkins, “Phase-contrast imaging using a scanning-doublegrating configuration,” Opt. Express 16, 5849–5867 (2008).
[Crossref]

2007 (1)

2006 (2)

A. Groso, R. Abela, and M. Stampanoi, “Implementation of a fast method for high resolution phase contrast tomography,” Opt. Express 14, 8103–8110 (2006).
[Crossref] [PubMed]

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

2005 (2)

2004 (3)

A. Chambolle, “An algorithm for total variation minimization and application,” J. Math. Imag. Vis. 20, 89–97 (2004).
[Crossref]

T. E. Gureyev, T. J. Davis, A. Pogany, S. C. Mayo, and S. W. Wilkins, “Optical phase retrieval by use of first Born- and Rytov-type approximation,” Appl. Opt. 43, 2418–2430 (2004).
[Crossref] [PubMed]

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

2003 (3)

A. Bravin, “Exploiting he X-ray refraction contrast with an analyzer: the state of the art,” J. Phys. D: Appl. Phys. 36, A24–A29 (2003).
[Crossref]

A. Momose, “Phase-sensitive imaging and phase tomography using x-ray interferometer,” Opt. Express 11, 2303–2314 (2003).
[Crossref] [PubMed]

A. Momose, “Demonstration of x-ray Talbot interferometry,” Japan J. Appl. Phys. 44, 6355–6367 (2003).
[Crossref]

2002 (2)

D. Paganin, S.C. Mayo, T. E. Gureyev, P. R. Wilkins, and S. W. Wilkins, “Simultaneous phase and amplitude extraction from a single defocused image of a homogeneous object,” J. Microsc. 206, 33–40 (2002).
[Crossref] [PubMed]

Z. Chen and S. Haykin, “On different facets of regularization theory,” Neural Comp. 14, 2791–2846 (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 application in the medical field,” Med. Phys. 28, 1610–1619 (2001).
[Crossref] [PubMed]

1999 (1)

A. V. Bronnikov, “Reconstruction formulas for phase-contrast tomography,” Opt. Commum. 171, 239–244 (1999).
[Crossref]

1998 (1)

M. J. Black, G. Sapiro, D. H. Marimont, and D. Heeger, “Robust anisotropic diffusion,” Trans. IEEE Imag. Process. 7, 421–432 (1998).
[Crossref]

1997 (3)

P. Charbonnier, L. Blanc-Féraud, G. Aubert, and M. Barlaud, “Deterministic edge-preserving regularization in computed image,” Trans. IEEE Imag. Process. 6, 298–311 (1997).
[Crossref]

A. Chambolle and P.-L. Lions, “Image recovery via total variation minimization and related problems,” Numerische Mathematik 76, 167–188 (1997).
[Crossref]

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

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]

Y. You, W. Xu, A. Tannenbaum, and M. Kaveh, “Behavioral analysis of anisotropic diffusion in image processing,” IEEE Trans. Image Process. 5, 1539–1592 (1996).
[Crossref] [PubMed]

1995 (1)

A. Momose, “Demonstration of phase-contrast x-ray computed-tomography using an x-ray interferometer,” Nucl. Instrum. Methods A 352, 622–628 (1995).
[Crossref]

1992 (1)

L. I. Rudin, S. Osher, and E. Fatemi, “Nonlinear total variation based noise removal algorithms,” Physica D: Nonlinear Phenomena 60259–268 (1992).
[Crossref]

1984 (1)

L. A. Feldkemp, L. C. Davis, and J. W. Kress, “Practical cone beam algorithm,” J. Opt. Soc. Am. 1, 612–619 (1984).
[Crossref]

1965 (1)

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

Abela, R.

Ahn, S.

S. Ahn and R. M. Leahy, “Analysis of resolution and noise properties of nonquadratically regularized image reconstruction methods for PET,” Trans. IEEE Med. Imag. 27, pp. 413–424 (2008).
[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 application in the medical field,” Med. Phys. 28, 1610–1619 (2001).
[Crossref] [PubMed]

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

Aubert, G.

P. Charbonnier, L. Blanc-Féraud, G. Aubert, and M. Barlaud, “Deterministic edge-preserving regularization in computed image,” Trans. IEEE Imag. Process. 6, 298–311 (1997).
[Crossref]

Ay, M. R.

A. Mehranian, M. R. Ay, A. Rahmim, and H. Zaidi, “Metal artifact Reduction in CT-based attenuation correction of PET using Sobolev sinogram restoration,” in Proceedings of IEEE Conference on Nuclear Science Symposium (IEEE, 2011), pp. 2936–2942.

Barbastahis, G.

A. Pan, L. Xu, J. C. Petrucelli, R. Gupta, B. Singh, and G. Barbastahis, “Contrast enhancement in X-ray phase contrast tomography,” Opt. Express 22, 18021–18026 (2014).
[Crossref]

Barlaud, M.

P. Charbonnier, L. Blanc-Féraud, G. Aubert, and M. Barlaud, “Deterministic edge-preserving regularization in computed image,” Trans. IEEE Imag. Process. 6, 298–311 (1997).
[Crossref]

Beltran, M. A.

Black, M. J.

M. J. Black, G. Sapiro, D. H. Marimont, and D. Heeger, “Robust anisotropic diffusion,” Trans. IEEE Imag. Process. 7, 421–432 (1998).
[Crossref]

Blanc-Féraud, L.

P. Charbonnier, L. Blanc-Féraud, G. Aubert, and M. Barlaud, “Deterministic edge-preserving regularization in computed image,” Trans. IEEE Imag. Process. 6, 298–311 (1997).
[Crossref]

Bonse, U.

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

Bougleux, S.

G. Peyré, S. Bougleux, and L. D. Cohen, “Non-local regularization of inverse problems,” Inverse Problem and Imaging 5, 511–530 (2011).
[Crossref]

Bravin, A.

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

A. Olivo, P. C. Diemoz, and A. Bravin, “Amplification of the phase contrast signal at very high x-ray energies,” Opt. Lett. 37, 915–917 (2012).
[Crossref] [PubMed]

A. Bravin, “Exploiting he X-ray refraction contrast with an analyzer: the state of the art,” J. Phys. D: Appl. Phys. 36, A24–A29 (2003).
[Crossref]

Bronnikov, A. V.

A. V. Bronnikov, “Reconstruction formulas for phase-contrast tomography,” Opt. Commum. 171, 239–244 (1999).
[Crossref]

Bunk, O.

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

Burvall, A.

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 application 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 application in the medical field,” Med. Phys. 28, 1610–1619 (2001).
[Crossref] [PubMed]

Chambolle, A.

A. Chambolle, “An algorithm for total variation minimization and application,” J. Math. Imag. Vis. 20, 89–97 (2004).
[Crossref]

A. Chambolle and P.-L. Lions, “Image recovery via total variation minimization and related problems,” Numerische Mathematik 76, 167–188 (1997).
[Crossref]

Chapman, D.

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

Charbonnier, P.

P. Charbonnier, L. Blanc-Féraud, G. Aubert, and M. Barlaud, “Deterministic edge-preserving regularization in computed image,” Trans. IEEE Imag. Process. 6, 298–311 (1997).
[Crossref]

Chen, W. R.

D. Wu, A. Yan, Y. Li, W. R. Chen, X. Wu, and H. Liu, “Phantom study based on a high-energy in-line phase contrast tomosynthesis prototype,” Proc. SPIE 8944, 89440 (2014).
[Crossref]

Chen, Z.

Z. Chen and S. Haykin, “On different facets of regularization theory,” Neural Comp. 14, 2791–2846 (2002).
[Crossref]

Cloetens, P.

M. Langer, P. Cloetens, and F. Peyrin, “Regularization of phase retrieval with phase-attenuation duality prior for 3-D holotomography,” IEEE Trans. Image Process. 19(9), 2428–2436 (2010).
[Crossref] [PubMed]

Coan, P.

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

Cohen, L. D.

G. Peyré, S. Bougleux, and L. D. Cohen, “Non-local regularization of inverse problems,” Inverse Problem and Imaging 5, 511–530 (2011).
[Crossref]

David, C.

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

Davis, L. C.

L. A. Feldkemp, L. C. Davis, and J. W. Kress, “Practical cone beam algorithm,” J. Opt. Soc. Am. 1, 612–619 (1984).
[Crossref]

Davis, T. J.

Diemoz, P. C.

Fatemi, E.

L. I. Rudin, S. Osher, and E. Fatemi, “Nonlinear total variation based noise removal algorithms,” Physica D: Nonlinear Phenomena 60259–268 (1992).
[Crossref]

Feldkemp, L. A.

L. A. Feldkemp, L. C. Davis, and J. W. Kress, “Practical cone beam algorithm,” J. Opt. Soc. Am. 1, 612–619 (1984).
[Crossref]

Fessler, J. A.

J. A. Fessler and W. L. Rogers, “Resolution properties of regularized image reconstruction methods,” Tech. Rep. 297, Comm. and Sign. Proc. Lab., Dept. of EECS, Univ. of Michigan, Ann Arbor, MI, (1995).

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]

Gmür, N.

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D. Chapman, W. Thomlinson, R. E. Johnston, D. Washburn, E. Pisano, N. Gmür, 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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Liu, H.

D. Wu, A. Yan, Y. Li, W. R. Chen, X. Wu, and H. Liu, “Phantom study based on a high-energy in-line phase contrast tomosynthesis prototype,” Proc. SPIE 8944, 89440 (2014).
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X. Wu and H. Liu, “X-Ray cone-beam phase tomography formulas based on phase-attenuation duality,” Opt. Express 13(16), 6000–6014 (2005).
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X. Wu and H. Liu, “An experimental method of determining relative phase-contrast factor for x-ray imaging system,” Med. Phys. 31, 997–1002 (2004).
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X. Wu, H. Liu, and A. Yan, “System design consideration for a x-ray phase-contrast imaging system based on in-line holography,” Proc. of SPIE, Vol. 5630, (2005).

X. Wu and H. Liu, “X-ray phase-contrast imaging: phase reconstruction,” in Proceedings of IEEE Conference on Engineering in Medicine and Biology (IEEE, 2005), pp. 1786–1789.

Liu, X.

X. Guo, X. Liu, M. Gu, C. Ni, S. Huang, and B. Liu, “Polychromatic x-ray in-line phase-contrast tomography for soft tissue,” Europhysics Lett. 98, 14001 (2012).
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S. C. Mayo, A. W. Stevenson, and S. W. Wilkins, “In-line phase-contrast X-ray imaging and tomography for Material Science,” Material 5, 937–965 (2012).
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D. Paganin, S.C. Mayo, T. E. Gureyev, P. R. Wilkins, and S. W. Wilkins, “Simultaneous phase and amplitude extraction from a single defocused image of a homogeneous object,” J. Microsc. 206, 33–40 (2002).
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Menk, R.

D. Chapman, W. Thomlinson, R. E. Johnston, D. Washburn, E. Pisano, N. Gmür, 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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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 application in the medical field,” Med. Phys. 28, 1610–1619 (2001).
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Paganin, D. M.

Pan, A.

A. Pan, L. Xu, J. C. Petrucelli, R. Gupta, B. Singh, and G. Barbastahis, “Contrast enhancement in X-ray phase contrast tomography,” Opt. Express 22, 18021–18026 (2014).
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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 application in the medical field,” Med. Phys. 28, 1610–1619 (2001).
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Petrucelli, J. C.

A. Pan, L. Xu, J. C. Petrucelli, R. Gupta, B. Singh, and G. Barbastahis, “Contrast enhancement in X-ray phase contrast tomography,” Opt. Express 22, 18021–18026 (2014).
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G. Peyré, S. Bougleux, and L. D. Cohen, “Non-local regularization of inverse problems,” Inverse Problem and Imaging 5, 511–530 (2011).
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M. Langer, P. Cloetens, and F. Peyrin, “Regularization of phase retrieval with phase-attenuation duality prior for 3-D holotomography,” IEEE Trans. Image Process. 19(9), 2428–2436 (2010).
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D. Chapman, W. Thomlinson, R. E. Johnston, D. Washburn, E. Pisano, N. Gmür, 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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T. E. Gureyev, T. J. Davis, A. Pogany, S. C. Mayo, and S. W. Wilkins, “Optical phase retrieval by use of first Born- and Rytov-type approximation,” Appl. Opt. 43, 2418–2430 (2004).
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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).
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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 application in the medical field,” Med. Phys. 28, 1610–1619 (2001).
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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 application in the medical field,” Med. Phys. 28, 1610–1619 (2001).
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Rahmim, A.

A. Mehranian, M. R. Ay, A. Rahmim, and H. Zaidi, “Metal artifact Reduction in CT-based attenuation correction of PET using Sobolev sinogram restoration,” in Proceedings of IEEE Conference on Nuclear Science Symposium (IEEE, 2011), pp. 2936–2942.

Rigon, L.

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 application in the medical field,” Med. Phys. 28, 1610–1619 (2001).
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L. I. Rudin, S. Osher, and E. Fatemi, “Nonlinear total variation based noise removal algorithms,” Physica D: Nonlinear Phenomena 60259–268 (1992).
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M. J. Black, G. Sapiro, D. H. Marimont, and D. Heeger, “Robust anisotropic diffusion,” Trans. IEEE Imag. Process. 7, 421–432 (1998).
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Sayers, D.

D. Chapman, W. Thomlinson, R. E. Johnston, D. Washburn, E. Pisano, N. Gmür, 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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A. Pan, L. Xu, J. C. Petrucelli, R. Gupta, B. Singh, and G. Barbastahis, “Contrast enhancement in X-ray phase contrast tomography,” Opt. Express 22, 18021–18026 (2014).
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Stevenson, A. W.

S. C. Mayo, A. W. Stevenson, and S. W. Wilkins, “In-line phase-contrast X-ray imaging and tomography for Material Science,” Material 5, 937–965 (2012).
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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).
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Thomlinson, W.

D. Chapman, W. Thomlinson, R. E. Johnston, D. Washburn, E. Pisano, N. Gmür, 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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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 application in the medical field,” Med. Phys. 28, 1610–1619 (2001).
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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 application in the medical field,” Med. Phys. 28, 1610–1619 (2001).
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Washburn, D.

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

Weitkamp, T.

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

Wilkins, P. R.

D. Paganin, S.C. Mayo, T. E. Gureyev, P. R. Wilkins, and S. W. Wilkins, “Simultaneous phase and amplitude extraction from a single defocused image of a homogeneous object,” J. Microsc. 206, 33–40 (2002).
[Crossref] [PubMed]

Wilkins, S. W.

S. C. Mayo, A. W. Stevenson, and S. W. Wilkins, “In-line phase-contrast X-ray imaging and tomography for Material Science,” Material 5, 937–965 (2012).
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Y. I. Nesterests and S. W. Wilkins, “Phase-contrast imaging using a scanning-doublegrating configuration,” Opt. Express 16, 5849–5867 (2008).
[Crossref]

T. E. Gureyev, T. J. Davis, A. Pogany, S. C. Mayo, and S. W. Wilkins, “Optical phase retrieval by use of first Born- and Rytov-type approximation,” Appl. Opt. 43, 2418–2430 (2004).
[Crossref] [PubMed]

D. Paganin, S.C. Mayo, T. E. Gureyev, P. R. Wilkins, and S. W. Wilkins, “Simultaneous phase and amplitude extraction from a single defocused image of a homogeneous object,” J. Microsc. 206, 33–40 (2002).
[Crossref] [PubMed]

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]

Wohlberg, B.

P. Rodríguez and B. Wohlberg, “Efficient minimization method for a generalized total variation functional,” Tran. IEEE Imag. Process. 18, 322–332 (2009).
[Crossref]

Wong, M.

M. Wong, X. Wu, and H. Liu, “Image quality comparison of high energy phase contrast x-ray images with low energy conventional images: phantom studies,” Proc. SPIE 7563, 756305 (2010).
[Crossref]

Wong, M. D.

M. D. Wong, X. Wu, and H. Liu, “Image quality and dose efficiency of high energy phase sensitive x-ray imaging: Phantom studies,” J. Xray Sci. Technol. 22, 321–334 (2014).
[PubMed]

Wu, D.

D. Wu, A. Yan, Y. Li, W. R. Chen, X. Wu, and H. Liu, “Phantom study based on a high-energy in-line phase contrast tomosynthesis prototype,” Proc. SPIE 8944, 89440 (2014).
[Crossref]

Wu, X.

M. D. Wong, X. Wu, and H. Liu, “Image quality and dose efficiency of high energy phase sensitive x-ray imaging: Phantom studies,” J. Xray Sci. Technol. 22, 321–334 (2014).
[PubMed]

D. Wu, A. Yan, Y. Li, W. R. Chen, X. Wu, and H. Liu, “Phantom study based on a high-energy in-line phase contrast tomosynthesis prototype,” Proc. SPIE 8944, 89440 (2014).
[Crossref]

X. Wu, A. Yan, and H. Liu, “X-ray phase-shifted-based method of volumetric breast density measurement,” Med. Phys. 39, pp. 4239–4244 (2012).
[Crossref] [PubMed]

M. Wong, X. Wu, and H. Liu, “Image quality comparison of high energy phase contrast x-ray images with low energy conventional images: phantom studies,” Proc. SPIE 7563, 756305 (2010).
[Crossref]

X. Wu and A. Yan, “Phase retrieval from one single phase contrast X-ray image,” Opt. Express 17(13), 11187–96 (2009).
[Crossref] [PubMed]

X. Wu, H. Liu, and A. Yan, “X-ray phase attenuation duality and phase retrieval,” Opt. Lett. 30(4), 379–381 (2005).
[Crossref] [PubMed]

X. Wu and H. Liu, “X-Ray cone-beam phase tomography formulas based on phase-attenuation duality,” Opt. Express 13(16), 6000–6014 (2005).
[Crossref] [PubMed]

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

X. Wu, H. Liu, and A. Yan, “System design consideration for a x-ray phase-contrast imaging system based on in-line holography,” Proc. of SPIE, Vol. 5630, (2005).

X. Wu and H. Liu, “X-ray phase-contrast imaging: phase reconstruction,” in Proceedings of IEEE Conference on Engineering in Medicine and Biology (IEEE, 2005), pp. 1786–1789.

Xu, L.

A. Pan, L. Xu, J. C. Petrucelli, R. Gupta, B. Singh, and G. Barbastahis, “Contrast enhancement in X-ray phase contrast tomography,” Opt. Express 22, 18021–18026 (2014).
[Crossref]

Xu, W.

Y. You, W. Xu, A. Tannenbaum, and M. Kaveh, “Behavioral analysis of anisotropic diffusion in image processing,” IEEE Trans. Image Process. 5, 1539–1592 (1996).
[Crossref] [PubMed]

Yan, A.

D. Wu, A. Yan, Y. Li, W. R. Chen, X. Wu, and H. Liu, “Phantom study based on a high-energy in-line phase contrast tomosynthesis prototype,” Proc. SPIE 8944, 89440 (2014).
[Crossref]

X. Wu, A. Yan, and H. Liu, “X-ray phase-shifted-based method of volumetric breast density measurement,” Med. Phys. 39, pp. 4239–4244 (2012).
[Crossref] [PubMed]

X. Wu and A. Yan, “Phase retrieval from one single phase contrast X-ray image,” Opt. Express 17(13), 11187–96 (2009).
[Crossref] [PubMed]

X. Wu, H. Liu, and A. Yan, “X-ray phase attenuation duality and phase retrieval,” Opt. Lett. 30(4), 379–381 (2005).
[Crossref] [PubMed]

X. Wu, H. Liu, and A. Yan, “System design consideration for a x-ray phase-contrast imaging system based on in-line holography,” Proc. of SPIE, Vol. 5630, (2005).

You, Y.

Y. You, W. Xu, A. Tannenbaum, and M. Kaveh, “Behavioral analysis of anisotropic diffusion in image processing,” IEEE Trans. Image Process. 5, 1539–1592 (1996).
[Crossref] [PubMed]

Zaidi, H.

A. Mehranian, M. R. Ay, A. Rahmim, and H. Zaidi, “Metal artifact Reduction in CT-based attenuation correction of PET using Sobolev sinogram restoration,” in Proceedings of IEEE Conference on Nuclear Science Symposium (IEEE, 2011), pp. 2936–2942.

Zhong, Z.

D. Chapman, W. Thomlinson, R. E. Johnston, D. Washburn, E. Pisano, N. Gmür, 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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Appl. Opt. (1)

Appl. Phys. Lett. (1)

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

Europhysics Lett. (1)

X. Guo, X. Liu, M. Gu, C. Ni, S. Huang, and B. Liu, “Polychromatic x-ray in-line phase-contrast tomography for soft tissue,” Europhysics Lett. 98, 14001 (2012).
[Crossref]

IEEE Trans. Image Process. (2)

Y. You, W. Xu, A. Tannenbaum, and M. Kaveh, “Behavioral analysis of anisotropic diffusion in image processing,” IEEE Trans. Image Process. 5, 1539–1592 (1996).
[Crossref] [PubMed]

M. Langer, P. Cloetens, and F. Peyrin, “Regularization of phase retrieval with phase-attenuation duality prior for 3-D holotomography,” IEEE Trans. Image Process. 19(9), 2428–2436 (2010).
[Crossref] [PubMed]

Inverse Problem and Imaging (1)

G. Peyré, S. Bougleux, and L. D. Cohen, “Non-local regularization of inverse problems,” Inverse Problem and Imaging 5, 511–530 (2011).
[Crossref]

J. Math. Imag. Vis. (1)

A. Chambolle, “An algorithm for total variation minimization and application,” J. Math. Imag. Vis. 20, 89–97 (2004).
[Crossref]

J. Microsc. (1)

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Figures (4)

Fig. 1
Fig. 1 Illustration of the experimental phantom.
Fig. 2
Fig. 2 The images of phase contrast CT of the phantom were illustrated. From left to right, the images are reconstructed from the phase images computed using Wu’s method, the proposed filter (p = 1.9), and gTV (p = 1.9), respectively. The display window were all [0.11 0.45]×1023cm−3.
Fig. 3
Fig. 3 CT slices reconstructed from computed results were shown. From left to right, the CT images were reconstructed from the projection data computed using Wu’s method, the proposed filter (p = 1.9) and gTV method (p = 1.9), respectively.
Fig. 4
Fig. 4 Details of CT images reconstructed from phase image and computed results were shown. The tomography reconstructed from original projection data was illustrated in (a). Two chosen areas, indicated by red and yellow retangles in (a) were were enlarged. Images (b)–(d) corresponds to the red rectangular area and (e)–(g) corresponds to the yellow rectangular area. From (b)–(d) and (e)–(g), the images were from the CT slice reconstructed from projection data computed using Wu’s method, the proposed filter (p=1.9), and gTV (p=1.9), respectively. Compared with (b), both (c) and (d) kept more details. In comparison with (e), edges in both (f) and (g) were sharper. Yellow arrows indicated the perceptual differences.

Tables (1)

Tables Icon

Table 1 Quantitative measurements of the reconstructed images of phase contrast.

Equations (30)

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A ( x , y ) = exp [ ( 4 π / λ ) β ( x , y , z ) d z ]
ϕ ( x , y ) = ( 2 π / λ ) δ ( x , y , z ) d z
arg min f 1 2 y T f 2 + τ J ( f ) ,
arg min f 1 2 y f 2 + τ P f 2 ,
v p = ( i j | v ( i , j ) | p ) 1 / p
J ( f ) = f ( x ) 2 d x ,
min f 1 2 y f 2 + τ f ( x ) 2 d x ,
y f τ Δ f = 0
( f ) ( ω ) = ( y ) ( ω ) 1 + τ ω 2
ϕ ( r ) = λ r e ρ e , p ,
A ( r ) = exp ( α KN ρ e , p ( r ) ) ,
σ KN = 1 + η η 2 [ 2 ( 1 + η ) 1 + 2 η 1 η ln ( 1 + 2 η ) ] + 1 2 η ln ( 1 + 2 η ) 1 + 3 η ( 1 + 2 η ) 2
( I ) ( A ) [ cos ( π λ R 2 u 2 M ) + ( 2 λ r e σ KN + π λ R 2 u 2 M ) sin ( π λ R 2 u 2 M ) ] ,
( I ) = ( A ) [ 1 + ( 2 λ r e σ KN ) ( π λ R 2 u M ) ]
ϕ ( r ) = γ ln ( 1 { ( I ) 1 + 2 γ ( π λ R 2 u 2 M ) } ) ,
( A ) = ( I ) 1 + τ ( M , λ ) u 2
τ ( M , λ ) = 2 π λ 2 r e σ KN R ( M 1 ) M 2
min A 1 2 I A 2 + τ A ( x ) 2 d x
( f ) ( ω ) = ( y ) ( ω ) 1 + ( 2 τ / p ) ω p
min A 1 2 I A 2 + τ T V A ( x ) p d x
τ T V = 2 τ p × 2 L d
δ = ρ e r e 2 c 2 2 π E 2
y f κ ( Δ ) p / 2 f = 0
1 2 π λ R 2 u 2 + α
1 α ( 1 1 + 2 π λ R 2 u 2 / α )
min f = 1 2 I f 2 + 2 π λ R 2 α f ( x ) 2 d x
1 2 π R 2 u 2
[ 1 + 4 π 2 R 2 ( δ 1 δ 2 μ 1 μ 2 ) u 2 ] 1
min A = 1 2 I A 2 + 4 π 2 R 2 ( δ 1 δ 2 μ 1 μ 2 ) A ( x ) 2 d x
min A = 1 2 I A 2 + ( 4 π 2 R 2 δ 1 μ 1 ) A ( x ) 2 d x

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