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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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(10), 4556-66 (2008).
[CrossRef]

2007 (2)

2005 (2)

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]

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).

2004 (4)

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), 068103 (2004).

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 (3)

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]

E. F. Donnelly, R. R. Price, and D. R. Pickens, "Characterization of the phase-contrast radiography edgeenhancement effect in a cabinet x-ray system," Med. Phys. 30, 2292-2296 (2003).
[CrossRef] [PubMed]

2001 (1)

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 (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).
[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 (2)

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]

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)

1985 (1)

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. K.

Anastasio, M. A.

D. Shi and M. A. Anastasio, "Intensity diffraction tomography with fixed detector plane," Opt. Eng.  46, 107003 (2007).
[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]

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.

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.

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]

Borgstrom, M. C.

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).
[CrossRef]

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]

Cookson, D.

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]

Davis, T.

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).

Donnelly, E. F.

E. F. Donnelly, R. R. Price, and D. R. Pickens, "Characterization of the phase-contrast radiography edgeenhancement effect in a cabinet x-ray system," Med. Phys. 30, 2292-2296 (2003).
[CrossRef] [PubMed]

Dyck, D.

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]

Eckstein, M. P.

Gao, D.

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]

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).

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(10), 4556-66 (2008).
[CrossRef]

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]

Guigay, J.-P.

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

Gureyev, T.

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).

Gureyev, T. E.

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]

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, 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), 068103 (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]

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]

Kitchen, 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]

Kupinski, M.

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]

Landuyt, 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]

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(10), 4556-66 (2008).
[CrossRef]

Lewis, R. A.

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]

Liu, H.

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]

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]

Lowenthal, S.

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).
[CrossRef]

Mancuso, A. P.

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]

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), 068103 (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]

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]

Myers, K. J.

Nesterets, Y.

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).

Nugent, K.

A. Barty, K. Nugent, A. Roberts, and D. Paganin, "Quantitative phase tomography," Opt. Commun. 175(4), 329-336 (2000).
[CrossRef]

Nugent, K. A.

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]

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]

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.

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), 068103 (2004).

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).
[CrossRef]

Patton, D. D.

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).
[CrossRef]

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).
[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(10), 4556-66 (2008).
[CrossRef]

Pickens, D. R.

E. F. Donnelly, R. R. Price, and D. R. Pickens, "Characterization of the phase-contrast radiography edgeenhancement effect in a cabinet x-ray system," Med. Phys. 30, 2292-2296 (2003).
[CrossRef] [PubMed]

Pogany, A.

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).

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. 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]

Price, R. R.

E. F. Donnelly, R. R. Price, and D. R. Pickens, "Characterization of the phase-contrast radiography edgeenhancement effect in a cabinet x-ray system," Med. Phys. 30, 2292-2296 (2003).
[CrossRef] [PubMed]

Roberts, A.

A. Barty, K. Nugent, A. Roberts, and D. Paganin, "Quantitative phase tomography," Opt. Commun. 175(4), 329-336 (2000).
[CrossRef]

Rolland, J. P.

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]

Schlenker, M.

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]

Seeley, G. W.

Shi, D.

D. Shi and M. A. Anastasio, "Intensity diffraction tomography with fixed detector plane," Opt. Eng.  46, 107003 (2007).
[CrossRef]

Stevenson, A.

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).

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).

Stevenson, A. W.

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), 068103 (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]

Wilkins, S.

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).

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), 068103 (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).

Wilkins, S. W.

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]

Wilkins, S.W.

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]

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).
[CrossRef]

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]

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).
[CrossRef] [PubMed]

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).
[CrossRef]

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).
[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. (1)

J. Opt. Soc. Am. A (3)

Med. Phys. (3)

E. F. Donnelly, R. R. Price, and D. R. Pickens, "Characterization of the phase-contrast radiography edgeenhancement 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. 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]

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, 107003 (2007).
[CrossRef]

Optik (1)

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

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), 068103 (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. (3)

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).

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 (15)

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]

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

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).

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]

C.-Y. Chou, Y. Huang, D. Shi, and M. A. Anastasio, "Image reconstruction in quantitative X-ray phasecontrast 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.

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]

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).

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]

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]

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]

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]

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

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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)

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

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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