Abstract

Quantitative in-line X-ray phase-contrast imaging methods seek to reconstruct separate images that depict an object’s projected absorption and refractive properties. An understanding of the statistical properties of the reconstructed images can facilitate the identification of optimal imaging parameters for specific diagnostic tasks. However, the statistical properties of quantitative X-ray phase-contrast imaging remain largely unexplored. In this work, we derive analytic expressions that describe the second-order statistics of the reconstructed absorption and phase images. Concepts from statistical decision theory are applied to demonstrate how the statistical properties of images corresponding to distinct imaging geometries can influence signal detectability.

© 2009 Optical Society of America

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    [Crossref] [PubMed]
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    [Crossref]
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    [Crossref]
  37. M. Eckstein, J. Bartroff, C. Abbey, J. Whiting, and F. Bochud, “Automated computer evaluation and optimization of image compression of x-ray coronary angiograms for signal known exactly detection tasks,” Opt. Express 11(5), 460–475 (2003). URL http://www.opticsexpress.org/abstract.cfm?URI=oe-11-5-460.
    [Crossref]
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    [Crossref]

2008 (2)

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(10), 4556–66 (2008).
[Crossref]

T. E. Gureyev, Y. I. Nesterets, A. W. Stevenson, P. R. Miller, A. Pogany, and S. W. Wilkins, “Some simple rules for contrast, signal-to-noise and resolution in in-line x-ray phase-contrast imaging,” Opt. Express 16(5), 3223–3241 (2008). URL http://www.opticsexpress.org/abstract.cfm?URI=oe-16-5-3223.
[Crossref]

2007 (3)

C. K. Abbey and M. P. Eckstein, “Classification images for simple detection and discrimination tasks in correlated noise,” J. Opt. Soc. Am. A 24, B110–B124 (2007).
[Crossref]

D. Shi and M. A. Anastasio, “Intensity diffraction tomography with fixed detector plane,” Opt. Eng. 46, 107,003 (2007).
[Crossref]

C.-Y. Chou, Y. Huang, D. Shi, and M. A. Anastasio, “Image reconstruction in quantitative X-ray phase-contrast imaging employing multiple measurements,” Opt. Express 15(16), 10,002–10,025 (2007). URL http://www.opticsexpress.org/abstract.cfm?URI=oe-15-16-10002.

2005 (1)

X. Wu, H. Liu, and A. Yan, “Optimization of X-ray phase-contrast imaging based on in-line holography,” Nucl. Instrum. Meth. B 234, 563–572 (2005).
[Crossref]

2004 (5)

T. E. Gureyev, D. M. Paganin, A. W. Stevenson, S. Mayo, and S. Wilkins, “Generalized eikonal of partially coherent beams and its use in quantitative imaging,” Phys. Rev. Lett. 93(6), 068,103 (2004).

D. M. Paganin, T. E. Gureyev, K. M. Pavlov, R. A. Lewis, and M. Kitchen, “Quantitative phase retrieval using coherent imaging systems with linear transfer functions,” Opt. Commun. 234, 87–105 (2004).
[Crossref]

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

R. A. Lewis, “Medical phase contrast x-ray imaging: current status and future prospects,” Phys. Med. Biol. 49(16), 3573–3583 (2004). URL http://stacks.iop.org/0031-9155/49/3573.
[Crossref]

B. D. Arhatari, A. P. Mancuso, A. G. Peele, and K. A. Nugent, “Phase contrast radiography: Image modelling and optimization,” Rev. Sci. Instrum. 75, 5271–5276 (2004).
[Crossref]

2003 (5)

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]

E. F. Donnelly, R. R. Price, and D. R. Pickens, “Characterization of the phase-contrast radiography edge-enhancement effect in a cabinet x-ray system,” Med. Phys. 30, 2292–2296 (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]

M. Eckstein, J. Bartroff, C. Abbey, J. Whiting, and F. Bochud, “Automated computer evaluation and optimization of image compression of x-ray coronary angiograms for signal known exactly detection tasks,” Opt. Express 11(5), 460–475 (2003). URL http://www.opticsexpress.org/abstract.cfm?URI=oe-11-5-460.
[Crossref]

A. H. Baydush, D. M. Catarious, C. K. Abbey, and C. E. Floyd, “Computer aided detection of masses in mammography using subregion Hotelling observers,” Med. Phys. 30(7), 1781–1787 (2003). URL http://link.aip.org/link/?MPH/30/1781/1.
[Crossref]

2001 (2)

A. R. Pineda and H. H. Barrett, “What does DQE say about lesion detectability in digital radiography?” in Proc. of SPIE, vol.  4320, pp. 561–569 (2001).
[Crossref]

T. E. Gureyev, S. Mayo, S. W. Wilkins, D. Paganin, and A. W. Stevenson, “Quantitative In-Line Phase-Contrast Imaging with Multienergy X Rays,” Phys. Rev. Lett. 86, 5827–5830 (2001).
[Crossref] [PubMed]

2000 (1)

A. Barty, K. Nugent, A. Roberts, and D. Paganin, “Quantitative phase tomography,” Opt. Commun. 175(4), 329–336 (2000).
[Crossref]

1999 (1)

P. Cloetens, W. Ludwig, J. Baruchel, D. Dyck, J. Landuyt, J. P. Guigay, and M. Schlenker, “Holotomography: Quantitative phase tomography with micrometer resolution using hard synchrotron radiation x rays,” Appl. Phys. Lett. 75, 29,132 (1999).
[Crossref]

1998 (2)

F. Arfelli, M. Assante, V. Bonvicini, A. Bravin, G. Cantatore, E. Castelli, L. D. Palma, M. D. Michiel, R. Longo, A. Olivo, S. Pani, D. Pontoni, P. Poropat, M. Prest, A. Rashevsky, G. Tromba, A. Vacchi, E. Vallazza, and F. Zanconati, “Low-dose phase contrast x-ray medical imaging,” Phys. Med. Biol. 43(10), 2845–2852 (1998). URL http://stacks.iop.org/0031-9155/43/2845.
[Crossref]

M. A. Anastasio, M. Kupinski, and X. Pan, “Noise properties of reconstructed images in ultrasound diffraction tomography,” IEEE Trans. Nucl. Sci. 45, 2216–2223 (1998).
[Crossref]

1997 (1)

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

1996 (3)

T. Davis, D. Gao, T. E. Gureyev, A. Stevenson, and S. Wilkins, “Phase-contrast imaging of weakly absorbing materials using hard X-rays,” Nature (London) 373, 335–338 (1996).

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

T. E. Gureyev and K. A. Nugent, “Phase retrieval with the transport-of-intensity equation. II. Orthogonal series solution for nonuniform illumination,” J. Opt. Soc. Am. A 13(8), 1670–1682 (1996). URL http://josaa.osa.org/abstract.cfm?URI=josaa-13-8-1670.
[Crossref]

1993 (1)

H. H. Barrett, J. Yao, J. P. Rolland, and K. J. Myers, “Model observers for assessment of image quality,” Proc. Natl. Acad. Sci. 90, 9758–9765 (1993).
[Crossref] [PubMed]

1990 (1)

1986 (1)

W. E. Smith and H. H. Barrett, “Hotelling trace criterion as a figure of merit for the optimization of imaging systems,” J. Opt. Soc. Am. A 3(5), 717–725 (1986). URL http://josaa.osa.org/abstract.cfm?URI=josaa-3-5-717.
[Crossref]

1985 (2)

R. F. Wagner and D. G. Brown, “Unified SNR analysis of medical imaging systems,” Phys. Med. Biol. 30(6), 489–518 (1985). URL http://stacks.iop.org/0031-9155/30/489.
[Crossref]

K. J. Myers, H. H. Barrett, M. C. Borgstrom, D. D. Patton, and G. W. Seeley, “Effect of noise correlation on detectability of disk signals in medical imaging,” J. Opt. Soc. Am. A 2, 1752–1759 (1985).
[Crossref] [PubMed]

1977 (1)

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

1970 (1)

Abbey, C.

M. Eckstein, J. Bartroff, C. Abbey, J. Whiting, and F. Bochud, “Automated computer evaluation and optimization of image compression of x-ray coronary angiograms for signal known exactly detection tasks,” Opt. Express 11(5), 460–475 (2003). URL http://www.opticsexpress.org/abstract.cfm?URI=oe-11-5-460.
[Crossref]

Abbey, C. K.

C. K. Abbey and M. P. Eckstein, “Classification images for simple detection and discrimination tasks in correlated noise,” J. Opt. Soc. Am. A 24, B110–B124 (2007).
[Crossref]

A. H. Baydush, D. M. Catarious, C. K. Abbey, and C. E. Floyd, “Computer aided detection of masses in mammography using subregion Hotelling observers,” Med. Phys. 30(7), 1781–1787 (2003). URL http://link.aip.org/link/?MPH/30/1781/1.
[Crossref]

Anastasio, M. A.

D. Shi and M. A. Anastasio, “Intensity diffraction tomography with fixed detector plane,” Opt. Eng. 46, 107,003 (2007).
[Crossref]

C.-Y. Chou, Y. Huang, D. Shi, and M. A. Anastasio, “Image reconstruction in quantitative X-ray phase-contrast imaging employing multiple measurements,” Opt. Express 15(16), 10,002–10,025 (2007). URL http://www.opticsexpress.org/abstract.cfm?URI=oe-15-16-10002.

M. A. Anastasio, M. Kupinski, and X. Pan, “Noise properties of reconstructed images in ultrasound diffraction tomography,” IEEE Trans. Nucl. Sci. 45, 2216–2223 (1998).
[Crossref]

Arfelli, F.

F. Arfelli, M. Assante, V. Bonvicini, A. Bravin, G. Cantatore, E. Castelli, L. D. Palma, M. D. Michiel, R. Longo, A. Olivo, S. Pani, D. Pontoni, P. Poropat, M. Prest, A. Rashevsky, G. Tromba, A. Vacchi, E. Vallazza, and F. Zanconati, “Low-dose phase contrast x-ray medical imaging,” Phys. Med. Biol. 43(10), 2845–2852 (1998). URL http://stacks.iop.org/0031-9155/43/2845.
[Crossref]

Arhatari, B. D.

B. D. Arhatari, A. P. Mancuso, A. G. Peele, and K. A. Nugent, “Phase contrast radiography: Image modelling and optimization,” Rev. Sci. Instrum. 75, 5271–5276 (2004).
[Crossref]

Arsenault, H.

Assante, M.

F. Arfelli, M. Assante, V. Bonvicini, A. Bravin, G. Cantatore, E. Castelli, L. D. Palma, M. D. Michiel, R. Longo, A. Olivo, S. Pani, D. Pontoni, P. Poropat, M. Prest, A. Rashevsky, G. Tromba, A. Vacchi, E. Vallazza, and F. Zanconati, “Low-dose phase contrast x-ray medical imaging,” Phys. Med. Biol. 43(10), 2845–2852 (1998). URL http://stacks.iop.org/0031-9155/43/2845.
[Crossref]

Barnea, Z.

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

Barrett, H.

H. Barrett and K. Myers, Foundations of Image Science (Wiley Series in Pure and Applied Optics, 2004).

Barrett, H. H.

A. R. Pineda and H. H. Barrett, “What does DQE say about lesion detectability in digital radiography?” in Proc. of SPIE, vol.  4320, pp. 561–569 (2001).
[Crossref]

H. H. Barrett, J. Yao, J. P. Rolland, and K. J. Myers, “Model observers for assessment of image quality,” Proc. Natl. Acad. Sci. 90, 9758–9765 (1993).
[Crossref] [PubMed]

H. H. Barrett, “Objective assessment of image quality: effects of quantum noise and object variability,” J. Opt. Soc. Am. A 7, 1266–1278 (1990).
[Crossref] [PubMed]

W. E. Smith and H. H. Barrett, “Hotelling trace criterion as a figure of merit for the optimization of imaging systems,” J. Opt. Soc. Am. A 3(5), 717–725 (1986). URL http://josaa.osa.org/abstract.cfm?URI=josaa-3-5-717.
[Crossref]

K. J. Myers, H. H. Barrett, M. C. Borgstrom, D. D. Patton, and G. W. Seeley, “Effect of noise correlation on detectability of disk signals in medical imaging,” J. Opt. Soc. Am. A 2, 1752–1759 (1985).
[Crossref] [PubMed]

Bartroff, J.

M. Eckstein, J. Bartroff, C. Abbey, J. Whiting, and F. Bochud, “Automated computer evaluation and optimization of image compression of x-ray coronary angiograms for signal known exactly detection tasks,” Opt. Express 11(5), 460–475 (2003). URL http://www.opticsexpress.org/abstract.cfm?URI=oe-11-5-460.
[Crossref]

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]

A. Barty, K. Nugent, A. Roberts, and D. Paganin, “Quantitative phase tomography,” Opt. Commun. 175(4), 329–336 (2000).
[Crossref]

Baruchel, J.

P. Cloetens, W. Ludwig, J. Baruchel, D. Dyck, J. Landuyt, J. P. Guigay, and M. Schlenker, “Holotomography: Quantitative phase tomography with micrometer resolution using hard synchrotron radiation x rays,” Appl. Phys. Lett. 75, 29,132 (1999).
[Crossref]

Baydush, A. H.

A. H. Baydush, D. M. Catarious, C. K. Abbey, and C. E. Floyd, “Computer aided detection of masses in mammography using subregion Hotelling observers,” Med. Phys. 30(7), 1781–1787 (2003). URL http://link.aip.org/link/?MPH/30/1781/1.
[Crossref]

Bochud, F.

M. Eckstein, J. Bartroff, C. Abbey, J. Whiting, and F. Bochud, “Automated computer evaluation and optimization of image compression of x-ray coronary angiograms for signal known exactly detection tasks,” Opt. Express 11(5), 460–475 (2003). URL http://www.opticsexpress.org/abstract.cfm?URI=oe-11-5-460.
[Crossref]

Bonvicini, V.

F. Arfelli, M. Assante, V. Bonvicini, A. Bravin, G. Cantatore, E. Castelli, L. D. Palma, M. D. Michiel, R. Longo, A. Olivo, S. Pani, D. Pontoni, P. Poropat, M. Prest, A. Rashevsky, G. Tromba, A. Vacchi, E. Vallazza, and F. Zanconati, “Low-dose phase contrast x-ray medical imaging,” Phys. Med. Biol. 43(10), 2845–2852 (1998). URL http://stacks.iop.org/0031-9155/43/2845.
[Crossref]

Borgstrom, M. C.

Bravin, A.

F. Arfelli, M. Assante, V. Bonvicini, A. Bravin, G. Cantatore, E. Castelli, L. D. Palma, M. D. Michiel, R. Longo, A. Olivo, S. Pani, D. Pontoni, P. Poropat, M. Prest, A. Rashevsky, G. Tromba, A. Vacchi, E. Vallazza, and F. Zanconati, “Low-dose phase contrast x-ray medical imaging,” Phys. Med. Biol. 43(10), 2845–2852 (1998). URL http://stacks.iop.org/0031-9155/43/2845.
[Crossref]

Brown, D. G.

R. F. Wagner and D. G. Brown, “Unified SNR analysis of medical imaging systems,” Phys. Med. Biol. 30(6), 489–518 (1985). URL http://stacks.iop.org/0031-9155/30/489.
[Crossref]

Cantatore, G.

F. Arfelli, M. Assante, V. Bonvicini, A. Bravin, G. Cantatore, E. Castelli, L. D. Palma, M. D. Michiel, R. Longo, A. Olivo, S. Pani, D. Pontoni, P. Poropat, M. Prest, A. Rashevsky, G. Tromba, A. Vacchi, E. Vallazza, and F. Zanconati, “Low-dose phase contrast x-ray medical imaging,” Phys. Med. Biol. 43(10), 2845–2852 (1998). URL http://stacks.iop.org/0031-9155/43/2845.
[Crossref]

Castelli, E.

F. Arfelli, M. Assante, V. Bonvicini, A. Bravin, G. Cantatore, E. Castelli, L. D. Palma, M. D. Michiel, R. Longo, A. Olivo, S. Pani, D. Pontoni, P. Poropat, M. Prest, A. Rashevsky, G. Tromba, A. Vacchi, E. Vallazza, and F. Zanconati, “Low-dose phase contrast x-ray medical imaging,” Phys. Med. Biol. 43(10), 2845–2852 (1998). URL http://stacks.iop.org/0031-9155/43/2845.
[Crossref]

Catarious, D. M.

A. H. Baydush, D. M. Catarious, C. K. Abbey, and C. E. Floyd, “Computer aided detection of masses in mammography using subregion Hotelling observers,” Med. Phys. 30(7), 1781–1787 (2003). URL http://link.aip.org/link/?MPH/30/1781/1.
[Crossref]

Chou, C.-Y.

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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(10), 4556–66 (2008).
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E. F. Donnelly, R. R. Price, and D. R. Pickens, “Characterization of the phase-contrast radiography edge-enhancement effect in a cabinet x-ray system,” Med. Phys. 30, 2292–2296 (2003).
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P. Cloetens, W. Ludwig, J. Baruchel, D. Dyck, J. Landuyt, J. P. Guigay, and M. Schlenker, “Holotomography: Quantitative phase tomography with micrometer resolution using hard synchrotron radiation x rays,” Appl. Phys. Lett. 75, 29,132 (1999).
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T. E. Gureyev, Y. I. Nesterets, A. W. Stevenson, P. R. Miller, A. Pogany, and S. W. Wilkins, “Some simple rules for contrast, signal-to-noise and resolution in in-line x-ray phase-contrast imaging,” Opt. Express 16(5), 3223–3241 (2008). URL http://www.opticsexpress.org/abstract.cfm?URI=oe-16-5-3223.
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D. M. Paganin, T. E. Gureyev, K. M. Pavlov, R. A. Lewis, and M. Kitchen, “Quantitative phase retrieval using coherent imaging systems with linear transfer functions,” Opt. Commun. 234, 87–105 (2004).
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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(10), 4556–66 (2008).
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D. M. Paganin, T. E. Gureyev, K. M. Pavlov, R. A. Lewis, and M. Kitchen, “Quantitative phase retrieval using coherent imaging systems with linear transfer functions,” Opt. Commun. 234, 87–105 (2004).
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Ludwig, W.

P. Cloetens, W. Ludwig, J. Baruchel, D. Dyck, J. Landuyt, J. P. Guigay, and M. Schlenker, “Holotomography: Quantitative phase tomography with micrometer resolution using hard synchrotron radiation x rays,” Appl. Phys. Lett. 75, 29,132 (1999).
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B. D. Arhatari, A. P. Mancuso, A. G. Peele, and K. A. Nugent, “Phase contrast radiography: Image modelling and optimization,” Rev. Sci. Instrum. 75, 5271–5276 (2004).
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Mayo, S.

T. E. Gureyev, D. M. Paganin, A. W. Stevenson, S. Mayo, and S. Wilkins, “Generalized eikonal of partially coherent beams and its use in quantitative imaging,” Phys. Rev. Lett. 93(6), 068,103 (2004).

T. E. Gureyev, S. Mayo, S. W. Wilkins, D. Paganin, and A. W. Stevenson, “Quantitative In-Line Phase-Contrast Imaging with Multienergy X Rays,” Phys. Rev. Lett. 86, 5827–5830 (2001).
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Miller, P. R.

T. E. Gureyev, Y. I. Nesterets, A. W. Stevenson, P. R. Miller, A. Pogany, and S. W. Wilkins, “Some simple rules for contrast, signal-to-noise and resolution in in-line x-ray phase-contrast imaging,” Opt. Express 16(5), 3223–3241 (2008). URL http://www.opticsexpress.org/abstract.cfm?URI=oe-16-5-3223.
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Nesterets, Y. I.

T. E. Gureyev, Y. I. Nesterets, A. W. Stevenson, P. R. Miller, A. Pogany, and S. W. Wilkins, “Some simple rules for contrast, signal-to-noise and resolution in in-line x-ray phase-contrast imaging,” Opt. Express 16(5), 3223–3241 (2008). URL http://www.opticsexpress.org/abstract.cfm?URI=oe-16-5-3223.
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A. Barty, K. Nugent, A. Roberts, and D. Paganin, “Quantitative phase tomography,” Opt. Commun. 175(4), 329–336 (2000).
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B. D. Arhatari, A. P. Mancuso, A. G. Peele, and K. A. Nugent, “Phase contrast radiography: Image modelling and optimization,” Rev. Sci. Instrum. 75, 5271–5276 (2004).
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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).
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T. E. Gureyev and K. A. Nugent, “Phase retrieval with the transport-of-intensity equation. II. Orthogonal series solution for nonuniform illumination,” J. Opt. Soc. Am. A 13(8), 1670–1682 (1996). URL http://josaa.osa.org/abstract.cfm?URI=josaa-13-8-1670.
[Crossref]

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

Olivo, A.

F. Arfelli, M. Assante, V. Bonvicini, A. Bravin, G. Cantatore, E. Castelli, L. D. Palma, M. D. Michiel, R. Longo, A. Olivo, S. Pani, D. Pontoni, P. Poropat, M. Prest, A. Rashevsky, G. Tromba, A. Vacchi, E. Vallazza, and F. Zanconati, “Low-dose phase contrast x-ray medical imaging,” Phys. Med. Biol. 43(10), 2845–2852 (1998). URL http://stacks.iop.org/0031-9155/43/2845.
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Paganin, D.

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]

T. E. Gureyev, S. Mayo, S. W. Wilkins, D. Paganin, and A. W. Stevenson, “Quantitative In-Line Phase-Contrast Imaging with Multienergy X Rays,” Phys. Rev. Lett. 86, 5827–5830 (2001).
[Crossref] [PubMed]

A. Barty, K. Nugent, A. Roberts, and D. Paganin, “Quantitative phase tomography,” Opt. Commun. 175(4), 329–336 (2000).
[Crossref]

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

Paganin, D. M.

D. M. Paganin, T. E. Gureyev, K. M. Pavlov, R. A. Lewis, and M. Kitchen, “Quantitative phase retrieval using coherent imaging systems with linear transfer functions,” Opt. Commun. 234, 87–105 (2004).
[Crossref]

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

T. E. Gureyev, D. M. Paganin, A. W. Stevenson, S. Mayo, and S. Wilkins, “Generalized eikonal of partially coherent beams and its use in quantitative imaging,” Phys. Rev. Lett. 93(6), 068,103 (2004).

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Palma, L. D.

F. Arfelli, M. Assante, V. Bonvicini, A. Bravin, G. Cantatore, E. Castelli, L. D. Palma, M. D. Michiel, R. Longo, A. Olivo, S. Pani, D. Pontoni, P. Poropat, M. Prest, A. Rashevsky, G. Tromba, A. Vacchi, E. Vallazza, and F. Zanconati, “Low-dose phase contrast x-ray medical imaging,” Phys. Med. Biol. 43(10), 2845–2852 (1998). URL http://stacks.iop.org/0031-9155/43/2845.
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Pan, X.

M. A. Anastasio, M. Kupinski, and X. Pan, “Noise properties of reconstructed images in ultrasound diffraction tomography,” IEEE Trans. Nucl. Sci. 45, 2216–2223 (1998).
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Pani, S.

F. Arfelli, M. Assante, V. Bonvicini, A. Bravin, G. Cantatore, E. Castelli, L. D. Palma, M. D. Michiel, R. Longo, A. Olivo, S. Pani, D. Pontoni, P. Poropat, M. Prest, A. Rashevsky, G. Tromba, A. Vacchi, E. Vallazza, and F. Zanconati, “Low-dose phase contrast x-ray medical imaging,” Phys. Med. Biol. 43(10), 2845–2852 (1998). URL http://stacks.iop.org/0031-9155/43/2845.
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Pavlov, K. M.

D. M. Paganin, T. E. Gureyev, K. M. Pavlov, R. A. Lewis, and M. Kitchen, “Quantitative phase retrieval using coherent imaging systems with linear transfer functions,” Opt. Commun. 234, 87–105 (2004).
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Peele, A. G.

B. D. Arhatari, A. P. Mancuso, A. G. Peele, and K. A. Nugent, “Phase contrast radiography: Image modelling and optimization,” Rev. Sci. Instrum. 75, 5271–5276 (2004).
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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(10), 4556–66 (2008).
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Pickens, D. R.

E. F. Donnelly, R. R. Price, and D. R. Pickens, “Characterization of the phase-contrast radiography edge-enhancement effect in a cabinet x-ray system,” Med. Phys. 30, 2292–2296 (2003).
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T. E. Gureyev, Y. I. Nesterets, A. W. Stevenson, P. R. Miller, A. Pogany, and S. W. Wilkins, “Some simple rules for contrast, signal-to-noise and resolution in in-line x-ray phase-contrast imaging,” Opt. Express 16(5), 3223–3241 (2008). URL http://www.opticsexpress.org/abstract.cfm?URI=oe-16-5-3223.
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T. E. Gureyev, A. Pogany, D. M. Paganin, and S. W. Wilkins, “Linear algorithms for phase retrieval in the Fresnel region,” Opt. Commun. 231, 53–70 (2004).
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A. Pogany, D. Gao, and S. W. Wilkins, “Contrast and resolution in imaging with a microfocus x-ray source,” Rev. Sci. Instrum. 68, 2774–2782 (1997).
[Crossref]

Y. Nesterets, S. Wilkins, T. Gureyev, A. Pogany, and A. Stevenson, “On the optimization of experimental parameters for x-ray in-line phase-contrast imaging,” Rev. Sci. Instrum.76(9) (2005).

Pontoni, D.

F. Arfelli, M. Assante, V. Bonvicini, A. Bravin, G. Cantatore, E. Castelli, L. D. Palma, M. D. Michiel, R. Longo, A. Olivo, S. Pani, D. Pontoni, P. Poropat, M. Prest, A. Rashevsky, G. Tromba, A. Vacchi, E. Vallazza, and F. Zanconati, “Low-dose phase contrast x-ray medical imaging,” Phys. Med. Biol. 43(10), 2845–2852 (1998). URL http://stacks.iop.org/0031-9155/43/2845.
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F. Arfelli, M. Assante, V. Bonvicini, A. Bravin, G. Cantatore, E. Castelli, L. D. Palma, M. D. Michiel, R. Longo, A. Olivo, S. Pani, D. Pontoni, P. Poropat, M. Prest, A. Rashevsky, G. Tromba, A. Vacchi, E. Vallazza, and F. Zanconati, “Low-dose phase contrast x-ray medical imaging,” Phys. Med. Biol. 43(10), 2845–2852 (1998). URL http://stacks.iop.org/0031-9155/43/2845.
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Prest, M.

F. Arfelli, M. Assante, V. Bonvicini, A. Bravin, G. Cantatore, E. Castelli, L. D. Palma, M. D. Michiel, R. Longo, A. Olivo, S. Pani, D. Pontoni, P. Poropat, M. Prest, A. Rashevsky, G. Tromba, A. Vacchi, E. Vallazza, and F. Zanconati, “Low-dose phase contrast x-ray medical imaging,” Phys. Med. Biol. 43(10), 2845–2852 (1998). URL http://stacks.iop.org/0031-9155/43/2845.
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Price, R. R.

E. F. Donnelly, R. R. Price, and D. R. Pickens, “Characterization of the phase-contrast radiography edge-enhancement effect in a cabinet x-ray system,” Med. Phys. 30, 2292–2296 (2003).
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F. Arfelli, M. Assante, V. Bonvicini, A. Bravin, G. Cantatore, E. Castelli, L. D. Palma, M. D. Michiel, R. Longo, A. Olivo, S. Pani, D. Pontoni, P. Poropat, M. Prest, A. Rashevsky, G. Tromba, A. Vacchi, E. Vallazza, and F. Zanconati, “Low-dose phase contrast x-ray medical imaging,” Phys. Med. Biol. 43(10), 2845–2852 (1998). URL http://stacks.iop.org/0031-9155/43/2845.
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A. Barty, K. Nugent, A. Roberts, and D. Paganin, “Quantitative phase tomography,” Opt. Commun. 175(4), 329–336 (2000).
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T. Davis, D. Gao, T. E. Gureyev, A. Stevenson, and S. Wilkins, “Phase-contrast imaging of weakly absorbing materials using hard X-rays,” Nature (London) 373, 335–338 (1996).

Y. Nesterets, S. Wilkins, T. Gureyev, A. Pogany, and A. Stevenson, “On the optimization of experimental parameters for x-ray in-line phase-contrast imaging,” Rev. Sci. Instrum.76(9) (2005).

Stevenson, A. W.

T. E. Gureyev, Y. I. Nesterets, A. W. Stevenson, P. R. Miller, A. Pogany, and S. W. Wilkins, “Some simple rules for contrast, signal-to-noise and resolution in in-line x-ray phase-contrast imaging,” Opt. Express 16(5), 3223–3241 (2008). URL http://www.opticsexpress.org/abstract.cfm?URI=oe-16-5-3223.
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T. E. Gureyev, D. M. Paganin, A. W. Stevenson, S. Mayo, and S. Wilkins, “Generalized eikonal of partially coherent beams and its use in quantitative imaging,” Phys. Rev. Lett. 93(6), 068,103 (2004).

T. E. Gureyev, S. Mayo, S. W. Wilkins, D. Paganin, and A. W. Stevenson, “Quantitative In-Line Phase-Contrast Imaging with Multienergy X Rays,” Phys. Rev. Lett. 86, 5827–5830 (2001).
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Tromba, G.

F. Arfelli, M. Assante, V. Bonvicini, A. Bravin, G. Cantatore, E. Castelli, L. D. Palma, M. D. Michiel, R. Longo, A. Olivo, S. Pani, D. Pontoni, P. Poropat, M. Prest, A. Rashevsky, G. Tromba, A. Vacchi, E. Vallazza, and F. Zanconati, “Low-dose phase contrast x-ray medical imaging,” Phys. Med. Biol. 43(10), 2845–2852 (1998). URL http://stacks.iop.org/0031-9155/43/2845.
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F. Arfelli, M. Assante, V. Bonvicini, A. Bravin, G. Cantatore, E. Castelli, L. D. Palma, M. D. Michiel, R. Longo, A. Olivo, S. Pani, D. Pontoni, P. Poropat, M. Prest, A. Rashevsky, G. Tromba, A. Vacchi, E. Vallazza, and F. Zanconati, “Low-dose phase contrast x-ray medical imaging,” Phys. Med. Biol. 43(10), 2845–2852 (1998). URL http://stacks.iop.org/0031-9155/43/2845.
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Vallazza, E.

F. Arfelli, M. Assante, V. Bonvicini, A. Bravin, G. Cantatore, E. Castelli, L. D. Palma, M. D. Michiel, R. Longo, A. Olivo, S. Pani, D. Pontoni, P. Poropat, M. Prest, A. Rashevsky, G. Tromba, A. Vacchi, E. Vallazza, and F. Zanconati, “Low-dose phase contrast x-ray medical imaging,” Phys. Med. Biol. 43(10), 2845–2852 (1998). URL http://stacks.iop.org/0031-9155/43/2845.
[Crossref]

Wagner, R. F.

R. F. Wagner and D. G. Brown, “Unified SNR analysis of medical imaging systems,” Phys. Med. Biol. 30(6), 489–518 (1985). URL http://stacks.iop.org/0031-9155/30/489.
[Crossref]

Whiting, J.

M. Eckstein, J. Bartroff, C. Abbey, J. Whiting, and F. Bochud, “Automated computer evaluation and optimization of image compression of x-ray coronary angiograms for signal known exactly detection tasks,” Opt. Express 11(5), 460–475 (2003). URL http://www.opticsexpress.org/abstract.cfm?URI=oe-11-5-460.
[Crossref]

Wilkins, S.

T. E. Gureyev, D. M. Paganin, A. W. Stevenson, S. Mayo, and S. Wilkins, “Generalized eikonal of partially coherent beams and its use in quantitative imaging,” Phys. Rev. Lett. 93(6), 068,103 (2004).

T. Davis, D. Gao, T. E. Gureyev, A. Stevenson, and S. Wilkins, “Phase-contrast imaging of weakly absorbing materials using hard X-rays,” Nature (London) 373, 335–338 (1996).

Y. Nesterets, S. Wilkins, T. Gureyev, A. Pogany, and A. Stevenson, “On the optimization of experimental parameters for x-ray in-line phase-contrast imaging,” Rev. Sci. Instrum.76(9) (2005).

Wilkins, S. W.

T. E. Gureyev, Y. I. Nesterets, A. W. Stevenson, P. R. Miller, A. Pogany, and S. W. Wilkins, “Some simple rules for contrast, signal-to-noise and resolution in in-line x-ray phase-contrast imaging,” Opt. Express 16(5), 3223–3241 (2008). URL http://www.opticsexpress.org/abstract.cfm?URI=oe-16-5-3223.
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T. E. Gureyev, A. Pogany, D. M. Paganin, and S. W. Wilkins, “Linear algorithms for phase retrieval in the Fresnel region,” Opt. Commun. 231, 53–70 (2004).
[Crossref]

T. E. Gureyev, S. Mayo, S. W. Wilkins, D. Paganin, and A. W. Stevenson, “Quantitative In-Line Phase-Contrast Imaging with Multienergy X Rays,” Phys. Rev. Lett. 86, 5827–5830 (2001).
[Crossref] [PubMed]

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

Wu, X.

X. Wu, H. Liu, and A. Yan, “Optimization of X-ray phase-contrast imaging based on in-line holography,” Nucl. Instrum. Meth. B 234, 563–572 (2005).
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X. Wu and H. Liu, “Clinical implementation of X-ray phase-contrast imaging: Theoretical foundations and design considerations,” Med. Phys. 30, 2169–2179 (2003).
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Yan, A.

X. Wu, H. Liu, and A. Yan, “Optimization of X-ray phase-contrast imaging based on in-line holography,” Nucl. Instrum. Meth. B 234, 563–572 (2005).
[Crossref]

Yao, J.

H. H. Barrett, J. Yao, J. P. Rolland, and K. J. Myers, “Model observers for assessment of image quality,” Proc. Natl. Acad. Sci. 90, 9758–9765 (1993).
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Zanconati, F.

F. Arfelli, M. Assante, V. Bonvicini, A. Bravin, G. Cantatore, E. Castelli, L. D. Palma, M. D. Michiel, R. Longo, A. Olivo, S. Pani, D. Pontoni, P. Poropat, M. Prest, A. Rashevsky, G. Tromba, A. Vacchi, E. Vallazza, and F. Zanconati, “Low-dose phase contrast x-ray medical imaging,” Phys. Med. Biol. 43(10), 2845–2852 (1998). URL http://stacks.iop.org/0031-9155/43/2845.
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Appl. Phys. Lett. (1)

P. Cloetens, W. Ludwig, J. Baruchel, D. Dyck, J. Landuyt, J. P. Guigay, and M. Schlenker, “Holotomography: Quantitative phase tomography with micrometer resolution using hard synchrotron radiation x rays,” Appl. Phys. Lett. 75, 29,132 (1999).
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IEEE Trans. Nucl. Sci. (1)

M. A. Anastasio, M. Kupinski, and X. Pan, “Noise properties of reconstructed images in ultrasound diffraction tomography,” IEEE Trans. Nucl. Sci. 45, 2216–2223 (1998).
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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).
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H. H. Barrett, “Objective assessment of image quality: effects of quantum noise and object variability,” J. Opt. Soc. Am. A 7, 1266–1278 (1990).
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K. J. Myers, H. H. Barrett, M. C. Borgstrom, D. D. Patton, and G. W. Seeley, “Effect of noise correlation on detectability of disk signals in medical imaging,” J. Opt. Soc. Am. A 2, 1752–1759 (1985).
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T. E. Gureyev and K. A. Nugent, “Phase retrieval with the transport-of-intensity equation. II. Orthogonal series solution for nonuniform illumination,” J. Opt. Soc. Am. A 13(8), 1670–1682 (1996). URL http://josaa.osa.org/abstract.cfm?URI=josaa-13-8-1670.
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Med. Phys. (4)

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(10), 4556–66 (2008).
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E. F. Donnelly, R. R. Price, and D. R. Pickens, “Characterization of the phase-contrast radiography edge-enhancement effect in a cabinet x-ray system,” Med. Phys. 30, 2292–2296 (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).
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A. H. Baydush, D. M. Catarious, C. K. Abbey, and C. E. Floyd, “Computer aided detection of masses in mammography using subregion Hotelling observers,” Med. Phys. 30(7), 1781–1787 (2003). URL http://link.aip.org/link/?MPH/30/1781/1.
[Crossref]

Nature (London) (1)

T. Davis, D. Gao, T. E. Gureyev, A. Stevenson, and S. Wilkins, “Phase-contrast imaging of weakly absorbing materials using hard X-rays,” Nature (London) 373, 335–338 (1996).

Nucl. Instrum. Meth. B (1)

X. Wu, H. Liu, and A. Yan, “Optimization of X-ray phase-contrast imaging based on in-line holography,” Nucl. Instrum. Meth. B 234, 563–572 (2005).
[Crossref]

Opt. Commun. (3)

D. M. Paganin, T. E. Gureyev, K. M. Pavlov, R. A. Lewis, and M. Kitchen, “Quantitative phase retrieval using coherent imaging systems with linear transfer functions,” Opt. Commun. 234, 87–105 (2004).
[Crossref]

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

A. Barty, K. Nugent, A. Roberts, and D. Paganin, “Quantitative phase tomography,” Opt. Commun. 175(4), 329–336 (2000).
[Crossref]

Opt. Eng. (1)

D. Shi and M. A. Anastasio, “Intensity diffraction tomography with fixed detector plane,” Opt. Eng. 46, 107,003 (2007).
[Crossref]

Opt. Express (3)

M. Eckstein, J. Bartroff, C. Abbey, J. Whiting, and F. Bochud, “Automated computer evaluation and optimization of image compression of x-ray coronary angiograms for signal known exactly detection tasks,” Opt. Express 11(5), 460–475 (2003). URL http://www.opticsexpress.org/abstract.cfm?URI=oe-11-5-460.
[Crossref]

T. E. Gureyev, Y. I. Nesterets, A. W. Stevenson, P. R. Miller, A. Pogany, and S. W. Wilkins, “Some simple rules for contrast, signal-to-noise and resolution in in-line x-ray phase-contrast imaging,” Opt. Express 16(5), 3223–3241 (2008). URL http://www.opticsexpress.org/abstract.cfm?URI=oe-16-5-3223.
[Crossref]

C.-Y. Chou, Y. Huang, D. Shi, and M. A. Anastasio, “Image reconstruction in quantitative X-ray phase-contrast imaging employing multiple measurements,” Opt. Express 15(16), 10,002–10,025 (2007). URL http://www.opticsexpress.org/abstract.cfm?URI=oe-15-16-10002.

Optik (1)

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

Phys. Med. Biol. (3)

R. A. Lewis, “Medical phase contrast x-ray imaging: current status and future prospects,” Phys. Med. Biol. 49(16), 3573–3583 (2004). URL http://stacks.iop.org/0031-9155/49/3573.
[Crossref]

F. Arfelli, M. Assante, V. Bonvicini, A. Bravin, G. Cantatore, E. Castelli, L. D. Palma, M. D. Michiel, R. Longo, A. Olivo, S. Pani, D. Pontoni, P. Poropat, M. Prest, A. Rashevsky, G. Tromba, A. Vacchi, E. Vallazza, and F. Zanconati, “Low-dose phase contrast x-ray medical imaging,” Phys. Med. Biol. 43(10), 2845–2852 (1998). URL http://stacks.iop.org/0031-9155/43/2845.
[Crossref]

R. F. Wagner and D. G. Brown, “Unified SNR analysis of medical imaging systems,” Phys. Med. Biol. 30(6), 489–518 (1985). URL http://stacks.iop.org/0031-9155/30/489.
[Crossref]

Phys. Rev. Lett. (3)

T. E. Gureyev, D. M. Paganin, A. W. Stevenson, S. Mayo, and S. Wilkins, “Generalized eikonal of partially coherent beams and its use in quantitative imaging,” Phys. Rev. Lett. 93(6), 068,103 (2004).

T. E. Gureyev, S. Mayo, S. W. Wilkins, D. Paganin, and A. W. Stevenson, “Quantitative In-Line Phase-Contrast Imaging with Multienergy X Rays,” Phys. Rev. Lett. 86, 5827–5830 (2001).
[Crossref] [PubMed]

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

Proc. Natl. Acad. Sci. (1)

H. H. Barrett, J. Yao, J. P. Rolland, and K. J. Myers, “Model observers for assessment of image quality,” Proc. Natl. Acad. Sci. 90, 9758–9765 (1993).
[Crossref] [PubMed]

Rev. Sci. Instrum. (2)

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

B. D. Arhatari, A. P. Mancuso, A. G. Peele, and K. A. Nugent, “Phase contrast radiography: Image modelling and optimization,” Rev. Sci. Instrum. 75, 5271–5276 (2004).
[Crossref]

Other (6)

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

H. Barrett and K. Myers, Foundations of Image Science (Wiley Series in Pure and Applied Optics, 2004).

Y. Nesterets, S. Wilkins, T. Gureyev, A. Pogany, and A. Stevenson, “On the optimization of experimental parameters for x-ray in-line phase-contrast imaging,” Rev. Sci. Instrum.76(9) (2005).

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

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

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Figures (7)

Fig. 1.
Fig. 1.

A schematic of a generic X-ray phase-contrast imaging system.

Fig. 2.
Fig. 2.

Estimates of ϕm,n [r, s] reconstructed from noisy data for imaging geometries with detector spacings Δz=(a) 28, (b) 69, (c) 106, and (d) 167. The corresponding noisy estimates of Am,n [r, s] are contained in subfigures (e)–(h).

Fig. 3.
Fig. 3.

Images of theoretical and empirical estimates of Cov{ϕ 1,2[r, s],ϕ 1,2[0,0]} are displayed in subfigures (a) and (b), respectively. The corresponding values of Cov{A 1,2[r, s],A 1,2[0,0]} are displayed in subfigures (c) and (d).

Fig. 4.
Fig. 4.

Autocovariance profiles corresponding to the images in Fig. 3. Subfigure (a) contains the theoretically and empirically determined autocovariance profiles Cov{ϕ 1,2[0, s],ϕ 1,2[0,0]}, which are depicted by solid and dashed curves, respectively. The corresponding autocovariance profiles Cov{A 1,2[0, s],A 1,2[0,0]} are shown in subfigure (b).

Fig. 5.
Fig. 5.

The autocovariance profiles (a) Cov{ϕm,n [0, s],ϕm,n [0,0]} and (b) Cov{Am,n [0, s],Am,n [0,0)} corresponding to different detector spacings Δz. The results for detector pairs (1,2), (1,3), (1,4), (1,5) and (1,6) are denoted by solid, dashed, dashed-dotted, dotted and thin solid curves, respectively. The variance of the Gaussian noise was σ 2=10%.

Fig. 6.
Fig. 6.

ROC curves for (a) phase images ϕm,n [r, s] and (b) absorption images Am,n [r, s], where the variance of the Gaussian noise was σ 2=1%. The solid, dashed, dashed-dotted, dotted and thin solid curves denote detector pairs (1,2), (1,3), (1,4), (1,5) and (1,6), respectively.

Fig. 7.
Fig. 7.

ROC for phase images ϕm,n [r, s] in which the variance of the Gaussian noise was σ 2=10%. The solid, dashed, dashed-dotted, dotted and thin solid curves denote detector pairs (1,2), (1,3), (1,4), (1,5) and (1,6), respectively.

Equations (51)

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Ut(x,y)=Ui(x,y)exp[A(x,y)+jϕ(x,y)],
ϕ(x,y)=kLdzδ(r),
A(x,y)=kLdzβ(r),
Im(x,y)=Ut(x,y)*2hm(x,y)2,
Km(x,y)Im(x,y)Ii1,
ϕ˜(u,v)= d x d y exp (j2π(ux+vy))ϕ(x,y),
A˜(u,v)= d x d y exp (j2π(ux+vy))A(x,y),
ϕ˜m,n(u,v)=cos(πλznf2)K˜m(u,v)cos(πλzmf2)K˜n(u,v)2sin[πλf2(znzm)],
A˜m,n(u,v)=sin(πλznf2)K˜m(u,v)sin(πλzmf2)K˜n(u,v)2sin[πλf2(znzm)],
ϕ˜m,n(u,v)=K˜m(u,v)K˜n(u,v)2πλf2(znzm),
A˜m,n(u,v)=znK˜m(u,v)zmK˜n(u,v)2(znzm).
Cov{ϕ˜m,n(u,v),ϕ˜m,n(u,v)}=Cov{K˜m(u,v),K˜m(u,v)}+Cov{K˜n(u,v),K˜n(u,v)}4π2λ2(znzm)2f2f'2,
Cov {K˜m(u,v),K˜m(u,v)}
= d x d y d x d y exp [j2π[(uu)x+(vv)y]]Cov{Km(x,y),Km(x,y)}.
Cov{Km(x,y),Km(x,y)}=Var {Km(x,y)} δ (xx)δ(yy)δm,m,
Cov {ϕ˜m,n(u,v),ϕ˜m,n(u,v)}
=14π2λ2(znzm)2f2f2
× d x d y exp [j2π[(uu)x+(vv)y]] [Var{Km(x,y)}+Var{Kn(x,y)}] .
Cov{A˜m,n(u,v),A˜m,n(u',v')}=14(znzm)2
×[zn2Cov{K˜m(u,v),K˜m(u',v')}+zm2Cov{K˜n(u,v),K˜n(u',v')}].
Cov{A˜m,n(u,v),A˜m,n(u,v)}
=14(znzm)2dxdyexp[j2π[(uu)x+(vv)y]][zn2Var{Km(x,y)}+zm2Var{Kn(x,y)}].
Cov{ϕm,n(x,y),ϕm,n(x,y)}= d u d v exp [j2π(ux+vy)]
×dudvexp[j2π(ux+vy)]Cov{ϕ˜m,n(u,v),ϕ˜m,n(u,v)},
Cov{Am,n(x,y),Am,n(x,y)}= d u d v exp [j2π(ux+vy)]
×dudvexp[j2π(ux+vy)]Cov{A˜m,n(u,v),A˜m,n(u,v)}.
Im[r,s]=Im(x,y)x=rΔx,y=sΔy,
Im[r,s]=Im0[r,s]+nm[r,s],
Cov{nm[r,s],nm[r,s]}=σ2[r,s;zm]δr,rδs,sδm,m,
Cov{Km[r,s],Km[r,s]}=σ2[r,s;zm]Ii2δr,rδs,sδm,m,
I˜m[p,q]=I˜m0[p,q]+n˜m[p,q],
I˜m[p,q]=Σr=0N1Σs=0N1Im[r,s]exp[j2πN(pr+qs)],
I˜m0[p,q]=Σr=0N1Σs=0N1Im0[r,s]exp[j2πN(pr+qs)],
n˜m[p,q]=Σr=0N1Σs=0N1nm[r,s]exp[j2πN(pr+qs)],
K˜m(u,v)u=pΔu,v=qΔvL2N2K˜m[p,q],
Cov{K˜m(u,v),K˜m(u,v)}u=pΔu,v=qΔvu=pΔu,v=qΔv L4N4Cov{K˜m[p,q],K˜m[p,q]}
=L4N4Σr=0N1Σs=0N1exp[j2πN(r(pp)+s(qq))]Var{Km[r,s]}
= L4N4Ii2 Σr=0N1 Σs=0N1 exp [j2πN(r(pp)+s(qq))] σ2 [r,s,zm] .
Cov{ϕ˜m,n(u,v),ϕ˜m,n(u,v)}u=pΔu,v=qΔvu=pΔu,v=qΔv L4N4Cov{ϕ˜m,n[p,q],ϕ˜m,n[p,q]}
=L4N4Cov{K˜m[p,q],K˜m[p,q]}+Cov{K˜n[p,q],K˜n[p,q]}4π2λ2(znzm)2fd2fd2 ,
Cov {ϕm,n(x,y),ϕm,n(x,y)}x=rΔx,y=sΔyx'=rΔx,y=sΔy Cov {ϕm,n[r,s],ϕm,n[r,s]}
=1N4Σp=0N1Σq=0N1exp[j2πN(pr+qs)]
×Σp'=0N1Σq'=0N1exp[j2πN(pr+qs)]Cov{ϕ˜m,n[p,q],ϕ˜m,n[p,q]}.
Cov {A˜m,n(u,v),A˜m,n(u,v)}u=pΔu,v=qΔvu=pΔu,v=qΔv L4N4Cov {A˜m,n[p,q],A˜m,n[p,q]}
=L4N414(znzm)2[zn2Cov{K˜m[p,q],K˜m[p,q]}+zm2Cov{K˜n[p,q],K˜n[p,q]}],
Cov {Am,n(x,y),Am,n(x,y)}x=rΔx,y=sΔyx=rΔx,y=sΔy Cov {Am,n[r,s],Am,n[r,s]}
=1N4Σp=0N1Σq=0N1exp[j2πN(pr+qs)]
×Σp=0N1Σq=0N1exp[j2πN(pr+qs)]Cov{A˜m,n[p,q],A˜m,n[p,q]}.
T(g)=wg,
wKg1 Δ g̅ .
1(znzm)2

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