Abstract

X-ray phase-contrast tomography (PCT) methods seek to quantitatively reconstruct separate images that depict an object’s absorption and refractive contrasts. Most PCT reconstruction algorithms generally operate by explicitly or implicitly performing the decoupling of the projected absorption and phase properties at each tomographic view angle by use of a phase-retrieval formula. However, the presence of zero-frequency singularity in the Fourier-based phase retrieval formulas will lead to a strong noise amplification in the projection estimate and the subsequent refractive image obtained using conventional algorithms like filtered backprojection (FBP). Tomographic reconstruction by use of statistical methods can account for the noise model and a priori information, and thereby can produce images with better quality over conventional filtered backprojection algorithms. In this work, we demonstrate an iterative image reconstruction method that exploits the second-order statistical properties of the projection data can mitigate noise amplification in PCT. The autocovariance function of the reconstructed refractive images was empirically computed and shows smaller and shorter noise correlation compared to those obtained using the FBP and unweighted penalized least-squares methods. Concepts from statistical decision theory are applied to demonstrate that the statistical properties of images produced by our method can improve signal detectability.

© 2011 OSA

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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
  26. D. Snyder and M. Miller, “The use of sieves to stabilize images produced with the EM algorithm for emission tomography,” IEEE Trans. Nucl. Sci. 32, 3864–3872 (1985).
    [CrossRef]
  27. J. Fessler and S. Booth, “Conjugate-gradient preconditioning methods for shift-variant PET image reconstruction,” IEEE Trans. Image Process. 8, 688–699 (1999).
    [CrossRef]
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    [CrossRef]
  31. E. Mumcuoglu, R. Leahy, S. R. Cherry, and Z. Zhou, “Fast gradient-based methods for Bayesian reconstruction of transmission and emission PET images,” IEEE Trans. Med. Imag. 13, 687–701 (1994).
    [CrossRef]

2010 (1)

C.-Y. Chou and M. A. Anastasio, “Noise texture and signal detectability in propagation-based x-ray phase-contrast tomography,” Med. Phys. 37, 270 (2010).
[CrossRef] [PubMed]

2009 (2)

C.-Y. Chou and M. A. Anastasio, “Statistical properties of X-ray phase-contrast tomography,” in Annual International Conference of the IEEE Engineering in Medicine and Biology Society, 2009. EMBC 2009, pp. 6648 –6650 (2009).
[CrossRef] [PubMed]

C.-Y. Chou and M. A. Anastasio, “Influence of imaging geometry on noise texture in quantitative in-line X-ray phase-contrast imaging,” Opt. Express 17, 14,466–14,480 (2009).
[CrossRef]

2008 (1)

M. Langer, P. Cloetens, J. P. Guigay, and F. Peyrin, “Quantitative comparison of direct phase retrieval algorithms in in-line phase tomography,” Med. Phys. 35, 4556–66 (2008).
[CrossRef] [PubMed]

2006 (2)

2004 (2)

R. A. Lewis, “Medical phase contrast x-ray imaging: current status and future prospects,” Phys. Med. Biol. 49, 3573–3583 (2004). URL http://stacks.iop.org/0031-9155/49/3573 .
[CrossRef] [PubMed]

S. Fiedler, A. Bravin, J. Keyrilainen, M. Fernandaz, P. Suortti, W. Thomlinson, M. Tenhenun, P. Virkkunen, and M. Karjalainen-Lindsberg, “Imaging lobular breast carcinoma: comparison of synchrotron radiation CT-DEI technique with clinical CT, mammography and histology,” Phys. Med. Biol. 49, 1–15 (2004).
[CrossRef]

2003 (4)

S. Mayo, T. Davis, T. Gureyev, P. Miller, D. Paganin, A. Pogany, A. Stevenson, and S. Wilkins, “X-ray phase-contrast microscopy and microtomography,” Opt. Express 11, 2289–2302 (2003).
[CrossRef] [PubMed]

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

D. Paganin, A. Barty, P. J. McMahon, and K. A. Nugent, “Quantitative phase-amplitude microscopy III. The effects of noise,” J. Microsc. 214, 51–61 (2003).
[CrossRef]

P. McMahon, A. Peele, D. Paterson, K. A. Nugent, A. Snigirev, T. Weitkamp, and C. Rau, “X-ray tomographic imaging of the complex refractive index,” Appl. Phys. Lett. 83, 1480–1482 (2003).
[CrossRef]

2002 (2)

P. Cloetens, W. Ludwig, E. Boller, L. Helfen, L. Salvo, R. Mache, and M. Schlenker, “Quantitative phase-contrast tomography using coherent synchrotron radiation,” in Developments in X-Ray Tomography III, U. Bonse, ed., Proc. SPIE 4503, 82–91 (2002).

A. V. Bronnikov, “Theory of quantitative phase-contrast computed tomography,” J. Opt. Soc. Am. A 19, 472–480 (2002).
[CrossRef]

2001 (1)

P. Cloetens, W. Ludwig, E. Boller, L. Helfen, L. Salvo, R. Mache, and M. Schlenker, “Quantitative phase contrast tomography using coherent synchrotron radiation,” in Developments in X-Ray Tomography III, U. Bonse, ed., Proc. SPIE 4503, 82–91 (2001).

2000 (1)

F. Arfelli, V. Bonvicini, A. Bravin, G. Cantatore, E. Castelli, L. D. Palma, M. D. 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, and F. Zanconati, “Mammography with synchrotron radiation: phase-detection techniques,” Radiol. 215, 286–293 (2000).

1999 (2)

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

J. Fessler and S. Booth, “Conjugate-gradient preconditioning methods for shift-variant PET image reconstruction,” IEEE Trans. Image Process. 8, 688–699 (1999).
[CrossRef]

1997 (1)

P. Cloetens, M. Pateyron-Salome, J. Y. Buffiere, G. Peix, J. Baruchel, F. Peyrin, and M. Schlenker, “Observation of microstructure and damage in materials by phase sensitive radiography and tomography,” J. Appl. Phys. 81, 5878–5886 (1997).
[CrossRef]

1996 (1)

K. A. Nugent, T. E. Gureyev, D. F. Cookson, D. M. Paganin, and Z. Barnea, “Quantitative phase imaging using hard x-rays,” Phys. Rev. Lett. 77, 2961–2964 (1996).
[CrossRef] [PubMed]

1994 (1)

E. Mumcuoglu, R. Leahy, S. R. Cherry, and Z. Zhou, “Fast gradient-based methods for Bayesian reconstruction of transmission and emission PET images,” IEEE Trans. Med. Imag. 13, 687–701 (1994).
[CrossRef]

1992 (1)

R. Lewitt, “Alternatives to voxels for image representation in iterative reconstruction algorithms,” Phys. Med. Biol. 37, 705–716 (1992).
[CrossRef] [PubMed]

1985 (1)

D. Snyder and M. Miller, “The use of sieves to stabilize images produced with the EM algorithm for emission tomography,” IEEE Trans. Nucl. Sci. 32, 3864–3872 (1985).
[CrossRef]

1977 (1)

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

Anastasio, M. A.

C.-Y. Chou and M. A. Anastasio, “Noise texture and signal detectability in propagation-based x-ray phase-contrast tomography,” Med. Phys. 37, 270 (2010).
[CrossRef] [PubMed]

C.-Y. Chou and M. A. Anastasio, “Influence of imaging geometry on noise texture in quantitative in-line X-ray phase-contrast imaging,” Opt. Express 17, 14,466–14,480 (2009).
[CrossRef]

C.-Y. Chou and M. A. Anastasio, “Statistical properties of X-ray phase-contrast tomography,” in Annual International Conference of the IEEE Engineering in Medicine and Biology Society, 2009. EMBC 2009, pp. 6648 –6650 (2009).
[CrossRef] [PubMed]

C.-Y. Chou and M. A. Anastasio, “Influence of imaging geometry on noise texture in x-ray in-line phase-contrast imaging,” in Medical Imaging 2008: Physics of Medical Imaging, J. Hsieh and E. Samei, eds., Proc. SPIE6913, 69131Z (2008). URL http://link.aip.org/link/?PSI/6913/69131Z/1 .

Arfelli, F.

F. Arfelli, V. Bonvicini, A. Bravin, G. Cantatore, E. Castelli, L. D. Palma, M. D. 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, and F. Zanconati, “Mammography with synchrotron radiation: phase-detection techniques,” Radiol. 215, 286–293 (2000).

Barnea, Z.

K. A. Nugent, T. E. Gureyev, D. F. Cookson, D. M. Paganin, and Z. Barnea, “Quantitative phase imaging using hard x-rays,” Phys. Rev. Lett. 77, 2961–2964 (1996).
[CrossRef] [PubMed]

Barrettt, H. H.

H. H. Barrettt and K. J. Myers, Foundations of Image Science, Wiley Series in Pure and Applied Optics (John Wiley & Sons, Inc., Hoboken, New Jersey, 2004).

Barty, A.

D. Paganin, A. Barty, P. J. McMahon, and K. A. Nugent, “Quantitative phase-amplitude microscopy III. The effects of noise,” J. Microsc. 214, 51–61 (2003).
[CrossRef]

Baruchel, J.

P. Cloetens, M. Pateyron-Salome, J. Y. Buffiere, G. Peix, J. Baruchel, F. Peyrin, and M. Schlenker, “Observation of microstructure and damage in materials by phase sensitive radiography and tomography,” J. Appl. Phys. 81, 5878–5886 (1997).
[CrossRef]

Bertero, M.

M. Bertero, Introduction to inverse problems in imaging (Taylor & Francis, 1998).
[CrossRef]

Birch, I. P.

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

Boller, E.

P. Cloetens, W. Ludwig, E. Boller, L. Helfen, L. Salvo, R. Mache, and M. Schlenker, “Quantitative phase-contrast tomography using coherent synchrotron radiation,” in Developments in X-Ray Tomography III, U. Bonse, ed., Proc. SPIE 4503, 82–91 (2002).

P. Cloetens, W. Ludwig, E. Boller, L. Helfen, L. Salvo, R. Mache, and M. Schlenker, “Quantitative phase contrast tomography using coherent synchrotron radiation,” in Developments in X-Ray Tomography III, U. Bonse, ed., Proc. SPIE 4503, 82–91 (2001).

Bonvicini, V.

F. Arfelli, V. Bonvicini, A. Bravin, G. Cantatore, E. Castelli, L. D. Palma, M. D. 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, and F. Zanconati, “Mammography with synchrotron radiation: phase-detection techniques,” Radiol. 215, 286–293 (2000).

Booth, S.

J. Fessler and S. Booth, “Conjugate-gradient preconditioning methods for shift-variant PET image reconstruction,” IEEE Trans. Image Process. 8, 688–699 (1999).
[CrossRef]

Bravin, A.

S. Fiedler, A. Bravin, J. Keyrilainen, M. Fernandaz, P. Suortti, W. Thomlinson, M. Tenhenun, P. Virkkunen, and M. Karjalainen-Lindsberg, “Imaging lobular breast carcinoma: comparison of synchrotron radiation CT-DEI technique with clinical CT, mammography and histology,” Phys. Med. Biol. 49, 1–15 (2004).
[CrossRef]

F. Arfelli, V. Bonvicini, A. Bravin, G. Cantatore, E. Castelli, L. D. Palma, M. D. 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, and F. Zanconati, “Mammography with synchrotron radiation: phase-detection techniques,” Radiol. 215, 286–293 (2000).

Bronnikov, A. V.

Buffiere, J. Y.

P. Cloetens, M. Pateyron-Salome, J. Y. Buffiere, G. Peix, J. Baruchel, F. Peyrin, and M. Schlenker, “Observation of microstructure and damage in materials by phase sensitive radiography and tomography,” J. Appl. Phys. 81, 5878–5886 (1997).
[CrossRef]

Cantatore, G.

F. Arfelli, V. Bonvicini, A. Bravin, G. Cantatore, E. Castelli, L. D. Palma, M. D. 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, and F. Zanconati, “Mammography with synchrotron radiation: phase-detection techniques,” Radiol. 215, 286–293 (2000).

Castelli, E.

F. Arfelli, V. Bonvicini, A. Bravin, G. Cantatore, E. Castelli, L. D. Palma, M. D. 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, and F. Zanconati, “Mammography with synchrotron radiation: phase-detection techniques,” Radiol. 215, 286–293 (2000).

Cherry, S. R.

E. Mumcuoglu, R. Leahy, S. R. Cherry, and Z. Zhou, “Fast gradient-based methods for Bayesian reconstruction of transmission and emission PET images,” IEEE Trans. Med. Imag. 13, 687–701 (1994).
[CrossRef]

Chou, C.-Y.

C.-Y. Chou and M. A. Anastasio, “Noise texture and signal detectability in propagation-based x-ray phase-contrast tomography,” Med. Phys. 37, 270 (2010).
[CrossRef] [PubMed]

C.-Y. Chou and M. A. Anastasio, “Statistical properties of X-ray phase-contrast tomography,” in Annual International Conference of the IEEE Engineering in Medicine and Biology Society, 2009. EMBC 2009, pp. 6648 –6650 (2009).
[CrossRef] [PubMed]

C.-Y. Chou and M. A. Anastasio, “Influence of imaging geometry on noise texture in quantitative in-line X-ray phase-contrast imaging,” Opt. Express 17, 14,466–14,480 (2009).
[CrossRef]

C.-Y. Chou and M. A. Anastasio, “Influence of imaging geometry on noise texture in x-ray in-line phase-contrast imaging,” in Medical Imaging 2008: Physics of Medical Imaging, J. Hsieh and E. Samei, eds., Proc. SPIE6913, 69131Z (2008). URL http://link.aip.org/link/?PSI/6913/69131Z/1 .

Cloetens, P.

M. Langer, P. Cloetens, J. P. Guigay, and F. Peyrin, “Quantitative comparison of direct phase retrieval algorithms in in-line phase tomography,” Med. Phys. 35, 4556–66 (2008).
[CrossRef] [PubMed]

P. Cloetens, W. Ludwig, E. Boller, L. Helfen, L. Salvo, R. Mache, and M. Schlenker, “Quantitative phase-contrast tomography using coherent synchrotron radiation,” in Developments in X-Ray Tomography III, U. Bonse, ed., Proc. SPIE 4503, 82–91 (2002).

P. Cloetens, W. Ludwig, E. Boller, L. Helfen, L. Salvo, R. Mache, and M. Schlenker, “Quantitative phase contrast tomography using coherent synchrotron radiation,” in Developments in X-Ray Tomography III, U. Bonse, ed., Proc. SPIE 4503, 82–91 (2001).

P. Cloetens, M. Pateyron-Salome, J. Y. Buffiere, G. Peix, J. Baruchel, F. Peyrin, and M. Schlenker, “Observation of microstructure and damage in materials by phase sensitive radiography and tomography,” J. Appl. Phys. 81, 5878–5886 (1997).
[CrossRef]

P. Cloetens, “Contribution to Phase Contrast Imaging, Reconstruction and Tomography with Hard Synchrotron Radiation: Principles, Implementation and Applications,” Ph.D. thesis, Vrije Universiteit Brussel (1999).

Cookson, D. F.

K. A. Nugent, T. E. Gureyev, D. F. Cookson, D. M. Paganin, and Z. Barnea, “Quantitative phase imaging using hard x-rays,” Phys. Rev. Lett. 77, 2961–2964 (1996).
[CrossRef] [PubMed]

Davis, T.

Defrise, M.

M. Defrise and G. T. Gullberg, “Image reconstruction,” Phys. Med. Biol. 51, R139 (2006). URL http://stacks.iop.org/0031-9155/51/i=13/a=R09 .
[CrossRef] [PubMed]

Dougherty, G. R.

W. D. Stanley, G. R. Dougherty, and R. Dougherty, Digital Signal Processing, 2nd ed. (Reston Publishing Company, Inc., Reston, VA, 1984).

Dougherty, R.

W. D. Stanley, G. R. Dougherty, and R. Dougherty, Digital Signal Processing, 2nd ed. (Reston Publishing Company, Inc., Reston, VA, 1984).

Fabrizioli, M.

F. Arfelli, V. Bonvicini, A. Bravin, G. Cantatore, E. Castelli, L. D. Palma, M. D. 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, and F. Zanconati, “Mammography with synchrotron radiation: phase-detection techniques,” Radiol. 215, 286–293 (2000).

Fernandaz, M.

S. Fiedler, A. Bravin, J. Keyrilainen, M. Fernandaz, P. Suortti, W. Thomlinson, M. Tenhenun, P. Virkkunen, and M. Karjalainen-Lindsberg, “Imaging lobular breast carcinoma: comparison of synchrotron radiation CT-DEI technique with clinical CT, mammography and histology,” Phys. Med. Biol. 49, 1–15 (2004).
[CrossRef]

Fessler, J.

J. Fessler and S. Booth, “Conjugate-gradient preconditioning methods for shift-variant PET image reconstruction,” IEEE Trans. Image Process. 8, 688–699 (1999).
[CrossRef]

Fiedler, S.

S. Fiedler, A. Bravin, J. Keyrilainen, M. Fernandaz, P. Suortti, W. Thomlinson, M. Tenhenun, P. Virkkunen, and M. Karjalainen-Lindsberg, “Imaging lobular breast carcinoma: comparison of synchrotron radiation CT-DEI technique with clinical CT, mammography and histology,” Phys. Med. Biol. 49, 1–15 (2004).
[CrossRef]

Guigay, J. P.

M. Langer, P. Cloetens, J. P. Guigay, and F. Peyrin, “Quantitative comparison of direct phase retrieval algorithms in in-line phase tomography,” Med. Phys. 35, 4556–66 (2008).
[CrossRef] [PubMed]

Guigay, J.-P.

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

Gullberg, G. T.

M. Defrise and G. T. Gullberg, “Image reconstruction,” Phys. Med. Biol. 51, R139 (2006). URL http://stacks.iop.org/0031-9155/51/i=13/a=R09 .
[CrossRef] [PubMed]

Gureyev, T.

Gureyev, T. E.

Helfen, L.

P. Cloetens, W. Ludwig, E. Boller, L. Helfen, L. Salvo, R. Mache, and M. Schlenker, “Quantitative phase-contrast tomography using coherent synchrotron radiation,” in Developments in X-Ray Tomography III, U. Bonse, ed., Proc. SPIE 4503, 82–91 (2002).

P. Cloetens, W. Ludwig, E. Boller, L. Helfen, L. Salvo, R. Mache, and M. Schlenker, “Quantitative phase contrast tomography using coherent synchrotron radiation,” in Developments in X-Ray Tomography III, U. Bonse, ed., Proc. SPIE 4503, 82–91 (2001).

Karjalainen-Lindsberg, M.

S. Fiedler, A. Bravin, J. Keyrilainen, M. Fernandaz, P. Suortti, W. Thomlinson, M. Tenhenun, P. Virkkunen, and M. Karjalainen-Lindsberg, “Imaging lobular breast carcinoma: comparison of synchrotron radiation CT-DEI technique with clinical CT, mammography and histology,” Phys. Med. Biol. 49, 1–15 (2004).
[CrossRef]

Keyrilainen, J.

S. Fiedler, A. Bravin, J. Keyrilainen, M. Fernandaz, P. Suortti, W. Thomlinson, M. Tenhenun, P. Virkkunen, and M. Karjalainen-Lindsberg, “Imaging lobular breast carcinoma: comparison of synchrotron radiation CT-DEI technique with clinical CT, mammography and histology,” Phys. Med. Biol. 49, 1–15 (2004).
[CrossRef]

Kotre, C. J.

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

Langer, M.

M. Langer, P. Cloetens, J. P. Guigay, and F. Peyrin, “Quantitative comparison of direct phase retrieval algorithms in in-line phase tomography,” Med. Phys. 35, 4556–66 (2008).
[CrossRef] [PubMed]

Leahy, R.

E. Mumcuoglu, R. Leahy, S. R. Cherry, and Z. Zhou, “Fast gradient-based methods for Bayesian reconstruction of transmission and emission PET images,” IEEE Trans. Med. Imag. 13, 687–701 (1994).
[CrossRef]

Lewis, R. A.

R. A. Lewis, “Medical phase contrast x-ray imaging: current status and future prospects,” Phys. Med. Biol. 49, 3573–3583 (2004). URL http://stacks.iop.org/0031-9155/49/3573 .
[CrossRef] [PubMed]

Lewitt, R.

R. Lewitt, “Alternatives to voxels for image representation in iterative reconstruction algorithms,” Phys. Med. Biol. 37, 705–716 (1992).
[CrossRef] [PubMed]

Liu, H.

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

Longo, R.

F. Arfelli, V. Bonvicini, A. Bravin, G. Cantatore, E. Castelli, L. D. Palma, M. D. 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, and F. Zanconati, “Mammography with synchrotron radiation: phase-detection techniques,” Radiol. 215, 286–293 (2000).

Ludwig, W.

P. Cloetens, W. Ludwig, E. Boller, L. Helfen, L. Salvo, R. Mache, and M. Schlenker, “Quantitative phase-contrast tomography using coherent synchrotron radiation,” in Developments in X-Ray Tomography III, U. Bonse, ed., Proc. SPIE 4503, 82–91 (2002).

P. Cloetens, W. Ludwig, E. Boller, L. Helfen, L. Salvo, R. Mache, and M. Schlenker, “Quantitative phase contrast tomography using coherent synchrotron radiation,” in Developments in X-Ray Tomography III, U. Bonse, ed., Proc. SPIE 4503, 82–91 (2001).

Mache, R.

P. Cloetens, W. Ludwig, E. Boller, L. Helfen, L. Salvo, R. Mache, and M. Schlenker, “Quantitative phase-contrast tomography using coherent synchrotron radiation,” in Developments in X-Ray Tomography III, U. Bonse, ed., Proc. SPIE 4503, 82–91 (2002).

P. Cloetens, W. Ludwig, E. Boller, L. Helfen, L. Salvo, R. Mache, and M. Schlenker, “Quantitative phase contrast tomography using coherent synchrotron radiation,” in Developments in X-Ray Tomography III, U. Bonse, ed., Proc. SPIE 4503, 82–91 (2001).

Mayo, S.

McMahon, P.

P. McMahon, A. Peele, D. Paterson, K. A. Nugent, A. Snigirev, T. Weitkamp, and C. Rau, “X-ray tomographic imaging of the complex refractive index,” Appl. Phys. Lett. 83, 1480–1482 (2003).
[CrossRef]

McMahon, P. J.

D. Paganin, A. Barty, P. J. McMahon, and K. A. Nugent, “Quantitative phase-amplitude microscopy III. The effects of noise,” J. Microsc. 214, 51–61 (2003).
[CrossRef]

Menk, R. H.

F. Arfelli, V. Bonvicini, A. Bravin, G. Cantatore, E. Castelli, L. D. Palma, M. D. 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, and F. Zanconati, “Mammography with synchrotron radiation: phase-detection techniques,” Radiol. 215, 286–293 (2000).

Michiel, M. D.

F. Arfelli, V. Bonvicini, A. Bravin, G. Cantatore, E. Castelli, L. D. Palma, M. D. 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, and F. Zanconati, “Mammography with synchrotron radiation: phase-detection techniques,” Radiol. 215, 286–293 (2000).

Miller, M.

D. Snyder and M. Miller, “The use of sieves to stabilize images produced with the EM algorithm for emission tomography,” IEEE Trans. Nucl. Sci. 32, 3864–3872 (1985).
[CrossRef]

Miller, P.

Mumcuoglu, E.

E. Mumcuoglu, R. Leahy, S. R. Cherry, and Z. Zhou, “Fast gradient-based methods for Bayesian reconstruction of transmission and emission PET images,” IEEE Trans. Med. Imag. 13, 687–701 (1994).
[CrossRef]

Myers, K. J.

H. H. Barrettt and K. J. Myers, Foundations of Image Science, Wiley Series in Pure and Applied Optics (John Wiley & Sons, Inc., Hoboken, New Jersey, 2004).

Nesterets, Y. I.

Nugent, K. A.

D. Paganin, A. Barty, P. J. McMahon, and K. A. Nugent, “Quantitative phase-amplitude microscopy III. The effects of noise,” J. Microsc. 214, 51–61 (2003).
[CrossRef]

P. McMahon, A. Peele, D. Paterson, K. A. Nugent, A. Snigirev, T. Weitkamp, and C. Rau, “X-ray tomographic imaging of the complex refractive index,” Appl. Phys. Lett. 83, 1480–1482 (2003).
[CrossRef]

K. A. Nugent, T. E. Gureyev, D. F. Cookson, D. M. Paganin, and Z. Barnea, “Quantitative phase imaging using hard x-rays,” Phys. Rev. Lett. 77, 2961–2964 (1996).
[CrossRef] [PubMed]

Olivo, A.

F. Arfelli, V. Bonvicini, A. Bravin, G. Cantatore, E. Castelli, L. D. Palma, M. D. 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, and F. Zanconati, “Mammography with synchrotron radiation: phase-detection techniques,” Radiol. 215, 286–293 (2000).

Paganin, D.

S. Mayo, T. Davis, T. Gureyev, P. Miller, D. Paganin, A. Pogany, A. Stevenson, and S. Wilkins, “X-ray phase-contrast microscopy and microtomography,” Opt. Express 11, 2289–2302 (2003).
[CrossRef] [PubMed]

D. Paganin, A. Barty, P. J. McMahon, and K. A. Nugent, “Quantitative phase-amplitude microscopy III. The effects of noise,” J. Microsc. 214, 51–61 (2003).
[CrossRef]

Paganin, D. M.

Palma, L. D.

F. Arfelli, V. Bonvicini, A. Bravin, G. Cantatore, E. Castelli, L. D. Palma, M. D. 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, and F. Zanconati, “Mammography with synchrotron radiation: phase-detection techniques,” Radiol. 215, 286–293 (2000).

Pani, S.

F. Arfelli, V. Bonvicini, A. Bravin, G. Cantatore, E. Castelli, L. D. Palma, M. D. 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, and F. Zanconati, “Mammography with synchrotron radiation: phase-detection techniques,” Radiol. 215, 286–293 (2000).

Papoulis, A.

A. Papoulis and S. U. Pillai, Probability, Random Variables, and Stochastic Processes (McGraw Hill, New York, 2002).

Paterson, D.

P. McMahon, A. Peele, D. Paterson, K. A. Nugent, A. Snigirev, T. Weitkamp, and C. Rau, “X-ray tomographic imaging of the complex refractive index,” Appl. Phys. Lett. 83, 1480–1482 (2003).
[CrossRef]

Pateyron-Salome, M.

P. Cloetens, M. Pateyron-Salome, J. Y. Buffiere, G. Peix, J. Baruchel, F. Peyrin, and M. Schlenker, “Observation of microstructure and damage in materials by phase sensitive radiography and tomography,” J. Appl. Phys. 81, 5878–5886 (1997).
[CrossRef]

Peele, A.

P. McMahon, A. Peele, D. Paterson, K. A. Nugent, A. Snigirev, T. Weitkamp, and C. Rau, “X-ray tomographic imaging of the complex refractive index,” Appl. Phys. Lett. 83, 1480–1482 (2003).
[CrossRef]

Peix, G.

P. Cloetens, M. Pateyron-Salome, J. Y. Buffiere, G. Peix, J. Baruchel, F. Peyrin, and M. Schlenker, “Observation of microstructure and damage in materials by phase sensitive radiography and tomography,” J. Appl. Phys. 81, 5878–5886 (1997).
[CrossRef]

Peyrin, F.

M. Langer, P. Cloetens, J. P. Guigay, and F. Peyrin, “Quantitative comparison of direct phase retrieval algorithms in in-line phase tomography,” Med. Phys. 35, 4556–66 (2008).
[CrossRef] [PubMed]

P. Cloetens, M. Pateyron-Salome, J. Y. Buffiere, G. Peix, J. Baruchel, F. Peyrin, and M. Schlenker, “Observation of microstructure and damage in materials by phase sensitive radiography and tomography,” J. Appl. Phys. 81, 5878–5886 (1997).
[CrossRef]

Pillai, S. U.

A. Papoulis and S. U. Pillai, Probability, Random Variables, and Stochastic Processes (McGraw Hill, New York, 2002).

Pogany, A.

Pontoni, D.

F. Arfelli, V. Bonvicini, A. Bravin, G. Cantatore, E. Castelli, L. D. Palma, M. D. 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, and F. Zanconati, “Mammography with synchrotron radiation: phase-detection techniques,” Radiol. 215, 286–293 (2000).

Poropat, P.

F. Arfelli, V. Bonvicini, A. Bravin, G. Cantatore, E. Castelli, L. D. Palma, M. D. 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, and F. Zanconati, “Mammography with synchrotron radiation: phase-detection techniques,” Radiol. 215, 286–293 (2000).

Prest, M.

F. Arfelli, V. Bonvicini, A. Bravin, G. Cantatore, E. Castelli, L. D. Palma, M. D. 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, and F. Zanconati, “Mammography with synchrotron radiation: phase-detection techniques,” Radiol. 215, 286–293 (2000).

Rashevsky, A.

F. Arfelli, V. Bonvicini, A. Bravin, G. Cantatore, E. Castelli, L. D. Palma, M. D. 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, and F. Zanconati, “Mammography with synchrotron radiation: phase-detection techniques,” Radiol. 215, 286–293 (2000).

Ratti, M.

F. Arfelli, V. Bonvicini, A. Bravin, G. Cantatore, E. Castelli, L. D. Palma, M. D. 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, and F. Zanconati, “Mammography with synchrotron radiation: phase-detection techniques,” Radiol. 215, 286–293 (2000).

Rau, C.

P. McMahon, A. Peele, D. Paterson, K. A. Nugent, A. Snigirev, T. Weitkamp, and C. Rau, “X-ray tomographic imaging of the complex refractive index,” Appl. Phys. Lett. 83, 1480–1482 (2003).
[CrossRef]

Rigon, L.

F. Arfelli, V. Bonvicini, A. Bravin, G. Cantatore, E. Castelli, L. D. Palma, M. D. 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, and F. Zanconati, “Mammography with synchrotron radiation: phase-detection techniques,” Radiol. 215, 286–293 (2000).

Salvo, L.

P. Cloetens, W. Ludwig, E. Boller, L. Helfen, L. Salvo, R. Mache, and M. Schlenker, “Quantitative phase-contrast tomography using coherent synchrotron radiation,” in Developments in X-Ray Tomography III, U. Bonse, ed., Proc. SPIE 4503, 82–91 (2002).

P. Cloetens, W. Ludwig, E. Boller, L. Helfen, L. Salvo, R. Mache, and M. Schlenker, “Quantitative phase contrast tomography using coherent synchrotron radiation,” in Developments in X-Ray Tomography III, U. Bonse, ed., Proc. SPIE 4503, 82–91 (2001).

Schlenker, M.

P. Cloetens, W. Ludwig, E. Boller, L. Helfen, L. Salvo, R. Mache, and M. Schlenker, “Quantitative phase-contrast tomography using coherent synchrotron radiation,” in Developments in X-Ray Tomography III, U. Bonse, ed., Proc. SPIE 4503, 82–91 (2002).

P. Cloetens, W. Ludwig, E. Boller, L. Helfen, L. Salvo, R. Mache, and M. Schlenker, “Quantitative phase contrast tomography using coherent synchrotron radiation,” in Developments in X-Ray Tomography III, U. Bonse, ed., Proc. SPIE 4503, 82–91 (2001).

P. Cloetens, M. Pateyron-Salome, J. Y. Buffiere, G. Peix, J. Baruchel, F. Peyrin, and M. Schlenker, “Observation of microstructure and damage in materials by phase sensitive radiography and tomography,” J. Appl. Phys. 81, 5878–5886 (1997).
[CrossRef]

Snigirev, A.

P. McMahon, A. Peele, D. Paterson, K. A. Nugent, A. Snigirev, T. Weitkamp, and C. Rau, “X-ray tomographic imaging of the complex refractive index,” Appl. Phys. Lett. 83, 1480–1482 (2003).
[CrossRef]

Snyder, D.

D. Snyder and M. Miller, “The use of sieves to stabilize images produced with the EM algorithm for emission tomography,” IEEE Trans. Nucl. Sci. 32, 3864–3872 (1985).
[CrossRef]

Stanley, W. D.

W. D. Stanley, G. R. Dougherty, and R. Dougherty, Digital Signal Processing, 2nd ed. (Reston Publishing Company, Inc., Reston, VA, 1984).

Stevenson, A.

Suortti, P.

S. Fiedler, A. Bravin, J. Keyrilainen, M. Fernandaz, P. Suortti, W. Thomlinson, M. Tenhenun, P. Virkkunen, and M. Karjalainen-Lindsberg, “Imaging lobular breast carcinoma: comparison of synchrotron radiation CT-DEI technique with clinical CT, mammography and histology,” Phys. Med. Biol. 49, 1–15 (2004).
[CrossRef]

Tenhenun, M.

S. Fiedler, A. Bravin, J. Keyrilainen, M. Fernandaz, P. Suortti, W. Thomlinson, M. Tenhenun, P. Virkkunen, and M. Karjalainen-Lindsberg, “Imaging lobular breast carcinoma: comparison of synchrotron radiation CT-DEI technique with clinical CT, mammography and histology,” Phys. Med. Biol. 49, 1–15 (2004).
[CrossRef]

Thomlinson, W.

S. Fiedler, A. Bravin, J. Keyrilainen, M. Fernandaz, P. Suortti, W. Thomlinson, M. Tenhenun, P. Virkkunen, and M. Karjalainen-Lindsberg, “Imaging lobular breast carcinoma: comparison of synchrotron radiation CT-DEI technique with clinical CT, mammography and histology,” Phys. Med. Biol. 49, 1–15 (2004).
[CrossRef]

Tromba, G.

F. Arfelli, V. Bonvicini, A. Bravin, G. Cantatore, E. Castelli, L. D. Palma, M. D. 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, and F. Zanconati, “Mammography with synchrotron radiation: phase-detection techniques,” Radiol. 215, 286–293 (2000).

Vacchi, A.

F. Arfelli, V. Bonvicini, A. Bravin, G. Cantatore, E. Castelli, L. D. Palma, M. D. 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, and F. Zanconati, “Mammography with synchrotron radiation: phase-detection techniques,” Radiol. 215, 286–293 (2000).

Vallazza, E.

F. Arfelli, V. Bonvicini, A. Bravin, G. Cantatore, E. Castelli, L. D. Palma, M. D. 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, and F. Zanconati, “Mammography with synchrotron radiation: phase-detection techniques,” Radiol. 215, 286–293 (2000).

Virkkunen, P.

S. Fiedler, A. Bravin, J. Keyrilainen, M. Fernandaz, P. Suortti, W. Thomlinson, M. Tenhenun, P. Virkkunen, and M. Karjalainen-Lindsberg, “Imaging lobular breast carcinoma: comparison of synchrotron radiation CT-DEI technique with clinical CT, mammography and histology,” Phys. Med. Biol. 49, 1–15 (2004).
[CrossRef]

Weitkamp, T.

P. McMahon, A. Peele, D. Paterson, K. A. Nugent, A. Snigirev, T. Weitkamp, and C. Rau, “X-ray tomographic imaging of the complex refractive index,” Appl. Phys. Lett. 83, 1480–1482 (2003).
[CrossRef]

Wilkins, S.

Wilkins, S. W.

Wu, X.

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

Zanconati, F.

F. Arfelli, V. Bonvicini, A. Bravin, G. Cantatore, E. Castelli, L. D. Palma, M. D. 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, and F. Zanconati, “Mammography with synchrotron radiation: phase-detection techniques,” Radiol. 215, 286–293 (2000).

Zhou, Z.

E. Mumcuoglu, R. Leahy, S. R. Cherry, and Z. Zhou, “Fast gradient-based methods for Bayesian reconstruction of transmission and emission PET images,” IEEE Trans. Med. Imag. 13, 687–701 (1994).
[CrossRef]

Annual International Conference of the IEEE Engineering in Medicine and Biology Society, 2009. EMBC 2009 (1)

C.-Y. Chou and M. A. Anastasio, “Statistical properties of X-ray phase-contrast tomography,” in Annual International Conference of the IEEE Engineering in Medicine and Biology Society, 2009. EMBC 2009, pp. 6648 –6650 (2009).
[CrossRef] [PubMed]

Appl. Phys. Lett. (1)

P. McMahon, A. Peele, D. Paterson, K. A. Nugent, A. Snigirev, T. Weitkamp, and C. Rau, “X-ray tomographic imaging of the complex refractive index,” Appl. Phys. Lett. 83, 1480–1482 (2003).
[CrossRef]

IEEE Trans. Image Process. (1)

J. Fessler and S. Booth, “Conjugate-gradient preconditioning methods for shift-variant PET image reconstruction,” IEEE Trans. Image Process. 8, 688–699 (1999).
[CrossRef]

IEEE Trans. Med. Imag. (1)

E. Mumcuoglu, R. Leahy, S. R. Cherry, and Z. Zhou, “Fast gradient-based methods for Bayesian reconstruction of transmission and emission PET images,” IEEE Trans. Med. Imag. 13, 687–701 (1994).
[CrossRef]

IEEE Trans. Nucl. Sci. (1)

D. Snyder and M. Miller, “The use of sieves to stabilize images produced with the EM algorithm for emission tomography,” IEEE Trans. Nucl. Sci. 32, 3864–3872 (1985).
[CrossRef]

J. Appl. Phys. (1)

P. Cloetens, M. Pateyron-Salome, J. Y. Buffiere, G. Peix, J. Baruchel, F. Peyrin, and M. Schlenker, “Observation of microstructure and damage in materials by phase sensitive radiography and tomography,” J. Appl. Phys. 81, 5878–5886 (1997).
[CrossRef]

J. Microsc. (1)

D. Paganin, A. Barty, P. J. McMahon, and K. A. Nugent, “Quantitative phase-amplitude microscopy III. The effects of noise,” J. Microsc. 214, 51–61 (2003).
[CrossRef]

J. Opt. Soc. Am. A (2)

Med. Phys. (3)

C.-Y. Chou and M. A. Anastasio, “Noise texture and signal detectability in propagation-based x-ray phase-contrast tomography,” Med. Phys. 37, 270 (2010).
[CrossRef] [PubMed]

M. Langer, P. Cloetens, J. P. Guigay, and F. Peyrin, “Quantitative comparison of direct phase retrieval algorithms in in-line phase tomography,” Med. Phys. 35, 4556–66 (2008).
[CrossRef] [PubMed]

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

Opt. Express (2)

C.-Y. Chou and M. A. Anastasio, “Influence of imaging geometry on noise texture in quantitative in-line X-ray phase-contrast imaging,” Opt. Express 17, 14,466–14,480 (2009).
[CrossRef]

S. Mayo, T. Davis, T. Gureyev, P. Miller, D. Paganin, A. Pogany, A. Stevenson, and S. Wilkins, “X-ray phase-contrast microscopy and microtomography,” Opt. Express 11, 2289–2302 (2003).
[CrossRef] [PubMed]

Optik (1)

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

Phys. Med. Biol. (5)

R. Lewitt, “Alternatives to voxels for image representation in iterative reconstruction algorithms,” Phys. Med. Biol. 37, 705–716 (1992).
[CrossRef] [PubMed]

M. Defrise and G. T. Gullberg, “Image reconstruction,” Phys. Med. Biol. 51, R139 (2006). URL http://stacks.iop.org/0031-9155/51/i=13/a=R09 .
[CrossRef] [PubMed]

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

R. A. Lewis, “Medical phase contrast x-ray imaging: current status and future prospects,” Phys. Med. Biol. 49, 3573–3583 (2004). URL http://stacks.iop.org/0031-9155/49/3573 .
[CrossRef] [PubMed]

S. Fiedler, A. Bravin, J. Keyrilainen, M. Fernandaz, P. Suortti, W. Thomlinson, M. Tenhenun, P. Virkkunen, and M. Karjalainen-Lindsberg, “Imaging lobular breast carcinoma: comparison of synchrotron radiation CT-DEI technique with clinical CT, mammography and histology,” Phys. Med. Biol. 49, 1–15 (2004).
[CrossRef]

Phys. Rev. Lett. (1)

K. A. Nugent, T. E. Gureyev, D. F. Cookson, D. M. Paganin, and Z. Barnea, “Quantitative phase imaging using hard x-rays,” Phys. Rev. Lett. 77, 2961–2964 (1996).
[CrossRef] [PubMed]

Proc. SPIE (2)

P. Cloetens, W. Ludwig, E. Boller, L. Helfen, L. Salvo, R. Mache, and M. Schlenker, “Quantitative phase contrast tomography using coherent synchrotron radiation,” in Developments in X-Ray Tomography III, U. Bonse, ed., Proc. SPIE 4503, 82–91 (2001).

P. Cloetens, W. Ludwig, E. Boller, L. Helfen, L. Salvo, R. Mache, and M. Schlenker, “Quantitative phase-contrast tomography using coherent synchrotron radiation,” in Developments in X-Ray Tomography III, U. Bonse, ed., Proc. SPIE 4503, 82–91 (2002).

Radiol. (1)

F. Arfelli, V. Bonvicini, A. Bravin, G. Cantatore, E. Castelli, L. D. Palma, M. D. 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, and F. Zanconati, “Mammography with synchrotron radiation: phase-detection techniques,” Radiol. 215, 286–293 (2000).

Other (7)

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

A. Papoulis and S. U. Pillai, Probability, Random Variables, and Stochastic Processes (McGraw Hill, New York, 2002).

H. H. Barrettt and K. J. Myers, Foundations of Image Science, Wiley Series in Pure and Applied Optics (John Wiley & Sons, Inc., Hoboken, New Jersey, 2004).

M. Bertero, Introduction to inverse problems in imaging (Taylor & Francis, 1998).
[CrossRef]

P. Cloetens, “Contribution to Phase Contrast Imaging, Reconstruction and Tomography with Hard Synchrotron Radiation: Principles, Implementation and Applications,” Ph.D. thesis, Vrije Universiteit Brussel (1999).

W. D. Stanley, G. R. Dougherty, and R. Dougherty, Digital Signal Processing, 2nd ed. (Reston Publishing Company, Inc., Reston, VA, 1984).

C.-Y. Chou and M. A. Anastasio, “Influence of imaging geometry on noise texture in x-ray in-line phase-contrast imaging,” in Medical Imaging 2008: Physics of Medical Imaging, J. Hsieh and E. Samei, eds., Proc. SPIE6913, 69131Z (2008). URL http://link.aip.org/link/?PSI/6913/69131Z/1 .

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Figures (9)

Fig. 1
Fig. 1

A schematic of the imaging geometry of propagation-based X-ray phase-contrast tomography.

Fig. 2
Fig. 2

Normalized eigenspectra of the weighted (solid curve) and unweighted (dashed curve) Hessian matrices.

Fig. 3
Fig. 3

The images reconstructed by use of the (a) FBP and PWLS algorithms employing (b) 10, (c) 50, (d) 90 iterations. The bottom row contains images reconstructed using the PLS method employing (e) 3, (f) 7, (g) 11, (h) 15 iterations. The standard deviation of the additive Gaussian noise and regularization parameter were set as σ = 1% and η = 0.0434 for the normalized matrices RWR and RR, respectively.

Fig. 4
Fig. 4

The cost function and normalized mean square error at each iteration for the PWLS method are contained in subfigures (a) and (b). The corresponding cost function and normalized mean square error for the PLS method are contained in subfigures (c) and (d), respectively. The additive Gaussian noise and regularization parameter were set as σ = 1% and η = 0.0434 for the normalized matrices RWR and RR, respectively.

Fig. 5
Fig. 5

Top row shows the autocovariance functions Cov{a1,2[0, s,t],a1,2[0, 0,0]} reconstructed by use of the (a) FBP and PWLS methods after (b) 10, (c) 50, and (d) 90 iterations. The Gaussian noise and regularization parameter were set as σ = 1% and η = 0.0434 for the normalized matrices RWR and RR, respectively.

Fig. 6
Fig. 6

Autocovariance profiles Cov{a1,2[0, s,0], a1,2[0, 0,0]} corresponding to refractive images produced by use of the PWLS, PLS and FBP algorithms are denoted by solid, solid with circle marker, and dashed curves, respectively. Solid curves denote the PWLS results estimated after (a) 10, (b) 50, and (c) 90 iterations, while the curve with circle marker for each subfigure denotes the PLS result in the 3-rd iteration. The standard deviation of the Gaussian noise and the regularization parameter were set as σ = 10% and η = 1.4942 for the normalized matrices RWR and RR, respectively.

Fig. 7
Fig. 7

Autocovariance profiles Cov{a1,2[0, s,0], a1,2[0, 0,0]} corresponding to refractive images generated by use of the (a) FBP (dashed curve) algorithm and PLS (solid curve with circle marker) algorithm at the 3-rd iteration, and by use of the (b) PWLS algorithm after 10 iterations (dash-dotted curve) and after 90 iterations (solid curve), respectively. The standard deviation of the Gaussian noise was σ = 10% and the regularization parameter was η = 1.4942 for the normalized matrices RWR and RR, respectively.

Fig. 8
Fig. 8

The numerical phantoms corresponding to (a) signal present and (b) signal absent cases.

Fig. 9
Fig. 9

The ROC curves for refractive images generated by use of the PWLS (solid curve), PLS (solid curve with circle marker) and FBP (dashed curve) algorithms when the standard deviation of the additive Gaussian noise and the regularization parameter were set as σ = 10% and η = 1.4942 for the normalized matrices RWR and RR, respectively.

Equations (28)

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n ( r ) = 1 a ( r ) + j β ( r ) ,
U t ( x , y r ; z r = 0 , θ ) = U i ( x , y r ; θ ) exp [ A ( x , y r ; θ ) + j ϕ ( x , y r ; θ ) ] ,
A ( x , y r ) = ( 2 π / λ ) L β ( r ) d z r
ϕ ( x , y r ) = ( 2 π / λ ) L a ( r ) d z r ,
I m ( x , y r ; θ ) = | U t ( x , y r ; z r = 0 , θ ) * * h z m ( x , y r ) | 2 ,
K m ( x , y r ; θ ) = I m ( x , y r ; θ ) I i ( x , y r ; θ ) 1 ,
I m [ r , s r ; k ] = I ( x , y r ; z r = z m , θ ) | x = r Δ d , y r = s r Δ d , θ = k Δ θ ,
K ˜ m [ p , q r ; k ] = r = 0 N 1 s r = 0 N 1 K m [ r , s r ; k ] exp [ j 2 π N ( p r + q r s r ) ]
K m [ r , s r ; k ] = I m [ r , s r ; k ] / I i 1 ,
K ˜ m ( u , v r ; θ ) | u = p L , v r = q r L , θ = k Δ θ Δ d 2 K ˜ m [ p , q r ; k ] .
A ˜ m , n ( u = p L , v r = q r L ; θ = k Δ θ ) = Δ d 2 D m , n ( p L , q r L ) [ sin ( π λ z n f d 2 ) K ˜ m [ p , q r ; k ] sin ( π λ z m f d 2 ) K ˜ n [ p , q r ; k ] ]
ϕ ˜ m , n ( u = p L , v r = q r L ; θ = k Δ θ ) = Δ d 2 D m , n ( p L , q r L ) [ cos ( π λ z n f d 2 ) K ˜ m [ p , q r ; k ] cos ( π λ z m f d 2 ) K ˜ n [ p , q r ; k ] ] ,
P [ r , s r ; k ] 1 L 2 p = N 2 N 2 1 q r = N 2 N 2 1 P ˜ ( u = p L , v r = q r L ; θ = k Δ θ ) exp [ j 2 π N ( p r + q r s r ) ] .
f a = n = 0 N 2 1 α n ψ n ,
α n = V d 3 r ψ n ( r ) f ( r ) ,
g = R α ,
Φ ( α ) = 1 2 ( g R α ) W ( g R α ) + η U ( α ) ,
U ( α ) = 1 2 n = 0 N 2 1 k 𝒩 n ( α n α k ) 2 ,
α ^ = arg min α Φ ( α ) .
I m [ r , s r ] = I m 0 [ r , s r ] + n m [ r , s r ] ,
Cov { n m [ r , s r ] , n m [ r , s r ] } = Var { n m [ r , s r ] } δ r r δ s s r δ m m ,
Cov { ϕ m , n ( x , y r ) , ϕ m , n ( x , y r ) } | x = r Δ d , y r = s r Δ d x = r Δ d , y r = s r Δ d Cov { ϕ m , n [ r , s r ] , ϕ m , n [ r , s r ] } = 1 N 4 p = 0 N 1 q r = 0 N 1 exp [ j 2 π N ( p r + q r s r ) ] × p = 0 N 1 q r = 0 N 1 exp [ j 2 π N ( p r + q r s r ) ] Cov { ϕ ˜ m , n [ p , q r ] , ϕ ˜ m , n [ p , q r ] } ,
Cov { A m , n ( x , y r ) , A m , n ( x , y r ) } | x = r Δ d , y r = s r Δ d x = r Δ d , y r = s r Δ d Cov { A m , n [ r , s r ] , A m , n [ r , s r ] } = 1 N 4 p = 0 N 1 q r = 0 N 1 exp [ j 2 π N ( p r + q r s r ) ] × p = 0 N 1 q r = 0 N 1 exp [ j 2 π N ( p r + q r s r ) ] Cov { A ˜ m , n [ p , q r ] , A ˜ m , n [ p , q r ] } .
K g = ( K 1 0 0 0 K 2 0 0 0 K N v )
K g 1 = ( K 1 0 0 0 K 2 0 0 0 K N v ) 1 = ( K 1 1 0 0 0 K 2 1 0 0 0 K N v 1 ) .
H W = R W R ,
H = R R .
Cov { α ^ n , α ^ n } = 1 J 1 [ i = 1 J α ^ n i α ^ n i 1 J i = 1 J α ^ n i i = 1 J α ^ n i ] ,

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