Abstract

An iterative phase retrieval algorithm was previously investigated for in-line x-ray phase imaging. Through detailed theoretical analysis and computer simulations, we now discuss the limitations, robustness, and efficiency of the algorithm. The iterative algorithm was proved robust against imaging noise but sensitive to the variations of several system parameters. It is also efficient in terms of calculation time. It was shown that the algorithm can be applied to phase retrieval based on one phase-contrast image and one attenuation image, or two phase-contrast images; in both cases, the two images can be obtained either by one detector in two exposures, or by two detectors in only one exposure as in the dual-detector scheme.

© 2008 Optical Society of America

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    [CrossRef] [PubMed]

2007 (3)

2006 (3)

T. E. Gureyev, Y. L. Nesterets, D. M. Paganin, A. Pogany, and S. W. Wilkins, “Linear algorithms for phase retrieval in the Fresnel region. 2. Partially coherent illumination,” Opt. Commun. 259, 569-580 (2006).
[CrossRef]

P. Cloetens, R. Mache, M. Schlenker, and S. Lerbs-Mache, “Quantitative phase tomography of Arabidopsis seeds reveals intercellular void network,” Proc. Natl. Acad. Sci. USA 103, 14,626-14,630 (2006).
[CrossRef]

A. Olivo and R. Speller, “Experimental validation of a simple model capable of predicting the phase contrast imaging capabilities of any x-ray imaging system,” Phys. Med. Biol. 51, 3015-3030 (2006).
[CrossRef] [PubMed]

2005 (8)

E. F. Donnelly, R. R. Price, and D. R. Pickens, “Experimental validation of the Wigner distributions theory of phase-contrast imaging,” Med. Phys. 32, 928-931 (2005).
[CrossRef] [PubMed]

R. Toth, J. C. Kieffer, S. Fourmaux, T. Ozaki, and A. Krol, “In-line phase-contrast imaging with a laser-based hard x-ray source,” Rev. Sci. Instrum. 76, 083701 (2005).
[CrossRef]

T. Q. Xiao, A. Bergamaschi, D. Dreossi, R. Longo, A. Olivo, S. Pani, L. Rigon, T. Rokvic, C. Venanzi, and E. Castelli, “Effect of spatial coherence on application of in-line phase contrast imaging to synchrotron radiation mammography,” Nucl. Instrum. Methods Phys. Res. A 548, 155-162 (2005).
[CrossRef]

S. Zabler, P. Cloetens, J.-P. Guigay, J. Baruchel, and M. Schlenker, “Optimization of phase contrast imaging using hard x rays,” Rev. Sci. Instrum. 76, 073,705 (2005).
[CrossRef]

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

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

X. Wu and H. Liu, “Phase-space formulation for phase-contrast x-ray imaging,” Appl. Opt. 44, 5847-5854 (2005).
[CrossRef] [PubMed]

B. D. Arhatari, K. A. Nugent, A. G. Peele, and J. Thornton, “Phase contrast radiography. II. Imaging of complex objects,” Rev. Sci. Instrum. 76, 113, 704 (2005).
[CrossRef]

2004 (5)

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

R. A. Lewis, “Medical phase contrast x-ray imaging: current status and future prospects,” Phys. Med. Biol. 49, 3573-3583 (2004).
[CrossRef] [PubMed]

X. Wu and H. Liu, “A dual detector approach for x-ray attenuation and phase imaging,” J. X-Ray Sci. Technol. 12, 35-42(2004).

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

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

2003 (5)

E. F. Donnelly, R. R. Price, and D. R. Pickens, “Dual focal-spot imaging for phase extraction in phase-contrast radiography,” Med. Phys. 30, 2292-2296 (2003).
[CrossRef] [PubMed]

E. F. Donnelly, R. R. Price, and D. R. Pickens, “Quantification of the effect of system and object parameters on edge enhancement in phase-contrast radiography,” Med. Phys. 30, 2888-2896 (2003).
[CrossRef] [PubMed]

M. T. Freedman, S.-C. B. Lo, C. Honda, E. Makariou, G. Sisney, E. Pien, H. Ohara, A. Ishisaka, and F. Shimada, “Phase contrast digital mammography using molybdenum x-ray: clinical implications in detectability improvement,” Proc. SPIE 5030, 533-540 (2003).
[CrossRef]

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

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]

2002 (2)

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

H. Ohara, C. Honda, A. Ishisaka, and F. Shimada, “Image quality in digital phase contrast imaging using a tungsten anode x-ray tube with small focal-spot size,” Proc. SPIE 4682, 713-723 (2002).
[CrossRef]

2000 (1)

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

P. Cloetens, W. Ludwig, J. Baruchel, D. V. Dyck, J. V. Landuyt, J. P. Guigay, and M. Schlenker, “Holotomography: quantitative phase tomography with micrometer resolution using hard synchrotron radiation x rays,” Appl. Phys. Lett. 75, 2912-2914 (1999).
[CrossRef]

1998 (1)

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, 2845-2852 (1998).
[CrossRef] [PubMed]

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)

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

S. W. Wilkins, T. E. Gureyev, D. Gao, A. Pogany, and A. W. Stevenson, “Phase-contrast imaging using polychromatic hard X-rays,” Nature 384, 335-338 (1996).
[CrossRef]

1995 (2)

T. J. Davis, T. E. Gureyev, D. Gao, A. W. Stevenson, and S. W. Wilkins, “X-ray image contrast from a simple phase object,” Phys. Rev. Lett. 74, 3173-3176 (1995).
[CrossRef] [PubMed]

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

Arfelli, F.

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

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, 2845-2852 (1998).
[CrossRef] [PubMed]

Arhatari, B. D.

B. D. Arhatari, K. A. Nugent, A. G. Peele, and J. Thornton, “Phase contrast radiography. II. Imaging of complex objects,” Rev. Sci. Instrum. 76, 113, 704 (2005).
[CrossRef]

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, 2845-2852 (1998).
[CrossRef] [PubMed]

Baruchel, J.

S. Zabler, P. Cloetens, J.-P. Guigay, J. Baruchel, and M. Schlenker, “Optimization of phase contrast imaging using hard x rays,” Rev. Sci. Instrum. 76, 073,705 (2005).
[CrossRef]

P. Cloetens, W. Ludwig, J. Baruchel, D. V. Dyck, J. V. Landuyt, J. P. Guigay, and M. Schlenker, “Holotomography: quantitative phase tomography with micrometer resolution using hard synchrotron radiation x rays,” Appl. Phys. Lett. 75, 2912-2914 (1999).
[CrossRef]

Bergamaschi, A.

T. Q. Xiao, A. Bergamaschi, D. Dreossi, R. Longo, A. Olivo, S. Pani, L. Rigon, T. Rokvic, C. Venanzi, and E. Castelli, “Effect of spatial coherence on application of in-line phase contrast imaging to synchrotron radiation mammography,” Nucl. Instrum. Methods Phys. Res. A 548, 155-162 (2005).
[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]

Boistel, R.

Bonvicini, V.

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

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, 2845-2852 (1998).
[CrossRef] [PubMed]

Bravin, A.

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

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, 2845-2852 (1998).
[CrossRef] [PubMed]

Cantatore, G.

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

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, 2845-2852 (1998).
[CrossRef] [PubMed]

Castelli, E.

T. Q. Xiao, A. Bergamaschi, D. Dreossi, R. Longo, A. Olivo, S. Pani, L. Rigon, T. Rokvic, C. Venanzi, and E. Castelli, “Effect of spatial coherence on application of in-line phase contrast imaging to synchrotron radiation mammography,” Nucl. Instrum. Methods Phys. Res. A 548, 155-162 (2005).
[CrossRef]

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

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, 2845-2852 (1998).
[CrossRef] [PubMed]

Cloetens, P.

J. P. Guigay, M. Langer, R. Boistel, and P. Cloetens, “Mixed transfer function and transport of intensity approach for phase retrieval in the Fresnel region.” Opt. Lett. 32, 1617-1619 (2007).
[CrossRef] [PubMed]

P. Cloetens, R. Mache, M. Schlenker, and S. Lerbs-Mache, “Quantitative phase tomography of Arabidopsis seeds reveals intercellular void network,” Proc. Natl. Acad. Sci. USA 103, 14,626-14,630 (2006).
[CrossRef]

S. Zabler, P. Cloetens, J.-P. Guigay, J. Baruchel, and M. Schlenker, “Optimization of phase contrast imaging using hard x rays,” Rev. Sci. Instrum. 76, 073,705 (2005).
[CrossRef]

P. Cloetens, W. Ludwig, J. Baruchel, D. V. Dyck, J. V. Landuyt, J. P. Guigay, and M. Schlenker, “Holotomography: quantitative phase tomography with micrometer resolution using hard synchrotron radiation x rays,” Appl. Phys. Lett. 75, 2912-2914 (1999).
[CrossRef]

Cookson, D. F.

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

Dalla Palma, L.

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

Davis, T. J.

T. J. Davis, T. E. Gureyev, D. Gao, A. W. Stevenson, and S. W. Wilkins, “X-ray image contrast from a simple phase object,” Phys. Rev. Lett. 74, 3173-3176 (1995).
[CrossRef] [PubMed]

Deans, A. E.

X. Wu, A. E. Deans, and H. Liu, Biomedical Photonics Handbook (CRC Press, 2003), Chap. 26.

Di Michiel, M.

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

Donnelly, E. F.

E. F. Donnelly, R. R. Price, and D. R. Pickens, “Experimental validation of the Wigner distributions theory of phase-contrast imaging,” Med. Phys. 32, 928-931 (2005).
[CrossRef] [PubMed]

E. F. Donnelly, R. R. Price, and D. R. Pickens, “Dual focal-spot imaging for phase extraction in phase-contrast radiography,” Med. Phys. 30, 2292-2296 (2003).
[CrossRef] [PubMed]

E. F. Donnelly, R. R. Price, and D. R. Pickens, “Quantification of the effect of system and object parameters on edge enhancement in phase-contrast radiography,” Med. Phys. 30, 2888-2896 (2003).
[CrossRef] [PubMed]

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

Dreossi, D.

T. Q. Xiao, A. Bergamaschi, D. Dreossi, R. Longo, A. Olivo, S. Pani, L. Rigon, T. Rokvic, C. Venanzi, and E. Castelli, “Effect of spatial coherence on application of in-line phase contrast imaging to synchrotron radiation mammography,” Nucl. Instrum. Methods Phys. Res. A 548, 155-162 (2005).
[CrossRef]

Dyck, D. V.

P. Cloetens, W. Ludwig, J. Baruchel, D. V. Dyck, J. V. Landuyt, J. P. Guigay, and M. Schlenker, “Holotomography: quantitative phase tomography with micrometer resolution using hard synchrotron radiation x rays,” Appl. Phys. Lett. 75, 2912-2914 (1999).
[CrossRef]

Fabrizioli, M.

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

Fourmaux, S.

R. Toth, J. C. Kieffer, S. Fourmaux, T. Ozaki, and A. Krol, “In-line phase-contrast imaging with a laser-based hard x-ray source,” Rev. Sci. Instrum. 76, 083701 (2005).
[CrossRef]

Freedman, M. T.

M. T. Freedman, S.-C. B. Lo, C. Honda, E. Makariou, G. Sisney, E. Pien, H. Ohara, A. Ishisaka, and F. Shimada, “Phase contrast digital mammography using molybdenum x-ray: clinical implications in detectability improvement,” Proc. SPIE 5030, 533-540 (2003).
[CrossRef]

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]

S. W. Wilkins, T. E. Gureyev, D. Gao, A. Pogany, and A. W. Stevenson, “Phase-contrast imaging using polychromatic hard X-rays,” Nature 384, 335-338 (1996).
[CrossRef]

T. J. Davis, T. E. Gureyev, D. Gao, A. W. Stevenson, and S. W. Wilkins, “X-ray image contrast from a simple phase object,” Phys. Rev. Lett. 74, 3173-3176 (1995).
[CrossRef] [PubMed]

Guigay, J. P.

J. P. Guigay, M. Langer, R. Boistel, and P. Cloetens, “Mixed transfer function and transport of intensity approach for phase retrieval in the Fresnel region.” Opt. Lett. 32, 1617-1619 (2007).
[CrossRef] [PubMed]

P. Cloetens, W. Ludwig, J. Baruchel, D. V. Dyck, J. V. Landuyt, J. P. Guigay, and M. Schlenker, “Holotomography: quantitative phase tomography with micrometer resolution using hard synchrotron radiation x rays,” Appl. Phys. Lett. 75, 2912-2914 (1999).
[CrossRef]

Guigay, J.-P.

S. Zabler, P. Cloetens, J.-P. Guigay, J. Baruchel, and M. Schlenker, “Optimization of phase contrast imaging using hard x rays,” Rev. Sci. Instrum. 76, 073,705 (2005).
[CrossRef]

Gureyev, T. E.

T. E. Gureyev, Y. L. Nesterets, D. M. Paganin, A. Pogany, and S. W. Wilkins, “Linear algorithms for phase retrieval in the Fresnel region. 2. Partially coherent illumination,” Opt. Commun. 259, 569-580 (2006).
[CrossRef]

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

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

S. W. Wilkins, T. E. Gureyev, D. Gao, A. Pogany, and A. W. Stevenson, “Phase-contrast imaging using polychromatic hard X-rays,” Nature 384, 335-338 (1996).
[CrossRef]

T. J. Davis, T. E. Gureyev, D. Gao, A. W. Stevenson, and S. W. Wilkins, “X-ray image contrast from a simple phase object,” Phys. Rev. Lett. 74, 3173-3176 (1995).
[CrossRef] [PubMed]

Honda, C.

M. T. Freedman, S.-C. B. Lo, C. Honda, E. Makariou, G. Sisney, E. Pien, H. Ohara, A. Ishisaka, and F. Shimada, “Phase contrast digital mammography using molybdenum x-ray: clinical implications in detectability improvement,” Proc. SPIE 5030, 533-540 (2003).
[CrossRef]

H. Ohara, C. Honda, A. Ishisaka, and F. Shimada, “Image quality in digital phase contrast imaging using a tungsten anode x-ray tube with small focal-spot size,” Proc. SPIE 4682, 713-723 (2002).
[CrossRef]

Ishisaka, A.

M. T. Freedman, S.-C. B. Lo, C. Honda, E. Makariou, G. Sisney, E. Pien, H. Ohara, A. Ishisaka, and F. Shimada, “Phase contrast digital mammography using molybdenum x-ray: clinical implications in detectability improvement,” Proc. SPIE 5030, 533-540 (2003).
[CrossRef]

H. Ohara, C. Honda, A. Ishisaka, and F. Shimada, “Image quality in digital phase contrast imaging using a tungsten anode x-ray tube with small focal-spot size,” Proc. SPIE 4682, 713-723 (2002).
[CrossRef]

Kieffer, J. C.

R. Toth, J. C. Kieffer, S. Fourmaux, T. Ozaki, and A. Krol, “In-line phase-contrast imaging with a laser-based hard x-ray source,” Rev. Sci. Instrum. 76, 083701 (2005).
[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]

Krol, A.

R. Toth, J. C. Kieffer, S. Fourmaux, T. Ozaki, and A. Krol, “In-line phase-contrast imaging with a laser-based hard x-ray source,” Rev. Sci. Instrum. 76, 083701 (2005).
[CrossRef]

Landuyt, J. V.

P. Cloetens, W. Ludwig, J. Baruchel, D. V. Dyck, J. V. Landuyt, J. P. Guigay, and M. Schlenker, “Holotomography: quantitative phase tomography with micrometer resolution using hard synchrotron radiation x rays,” Appl. Phys. Lett. 75, 2912-2914 (1999).
[CrossRef]

Langer, M.

Lerbs-Mache, S.

P. Cloetens, R. Mache, M. Schlenker, and S. Lerbs-Mache, “Quantitative phase tomography of Arabidopsis seeds reveals intercellular void network,” Proc. Natl. Acad. Sci. USA 103, 14,626-14,630 (2006).
[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).
[CrossRef] [PubMed]

Liu, H.

F. B. Meng, A. Yan, G. Zhou, X. Wu, and H. Liu, “Development of a dual-detector X-ray imaging system for phase retrieval study,” Nucl. Instrum. Methods Phys. Res. B 254, 300-306(2007).
[CrossRef]

F. B. Meng, H. Liu, and X. Wu, “An iterative phase retrieval algorithm for in-line x-ray phase imaging,” Opt. Express 15, 8383-8390 (2007).
[CrossRef] [PubMed]

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

X. Wu and H. Liu, “Phase-space formulation for phase-contrast x-ray imaging,” Appl. Opt. 44, 5847-5854 (2005).
[CrossRef] [PubMed]

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

X. Wu and H. Liu, “A dual detector approach for x-ray attenuation and phase imaging,” J. X-Ray Sci. Technol. 12, 35-42(2004).

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

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

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

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]

X. Wu, A. E. Deans, and H. Liu, Biomedical Photonics Handbook (CRC Press, 2003), Chap. 26.

Lo, S.-C. B.

M. T. Freedman, S.-C. B. Lo, C. Honda, E. Makariou, G. Sisney, E. Pien, H. Ohara, A. Ishisaka, and F. Shimada, “Phase contrast digital mammography using molybdenum x-ray: clinical implications in detectability improvement,” Proc. SPIE 5030, 533-540 (2003).
[CrossRef]

Longo, R.

T. Q. Xiao, A. Bergamaschi, D. Dreossi, R. Longo, A. Olivo, S. Pani, L. Rigon, T. Rokvic, C. Venanzi, and E. Castelli, “Effect of spatial coherence on application of in-line phase contrast imaging to synchrotron radiation mammography,” Nucl. Instrum. Methods Phys. Res. A 548, 155-162 (2005).
[CrossRef]

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

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, 2845-2852 (1998).
[CrossRef] [PubMed]

Ludwig, W.

P. Cloetens, W. Ludwig, J. Baruchel, D. V. Dyck, J. V. Landuyt, J. P. Guigay, and M. Schlenker, “Holotomography: quantitative phase tomography with micrometer resolution using hard synchrotron radiation x rays,” Appl. Phys. Lett. 75, 2912-2914 (1999).
[CrossRef]

Mache, R.

P. Cloetens, R. Mache, M. Schlenker, and S. Lerbs-Mache, “Quantitative phase tomography of Arabidopsis seeds reveals intercellular void network,” Proc. Natl. Acad. Sci. USA 103, 14,626-14,630 (2006).
[CrossRef]

Makariou, E.

M. T. Freedman, S.-C. B. Lo, C. Honda, E. Makariou, G. Sisney, E. Pien, H. Ohara, A. Ishisaka, and F. Shimada, “Phase contrast digital mammography using molybdenum x-ray: clinical implications in detectability improvement,” Proc. SPIE 5030, 533-540 (2003).
[CrossRef]

Mayo, S. C.

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

Meng, F. B.

F. B. Meng, A. Yan, G. Zhou, X. Wu, and H. Liu, “Development of a dual-detector X-ray imaging system for phase retrieval study,” Nucl. Instrum. Methods Phys. Res. B 254, 300-306(2007).
[CrossRef]

F. B. Meng, H. Liu, and X. Wu, “An iterative phase retrieval algorithm for in-line x-ray phase imaging,” Opt. Express 15, 8383-8390 (2007).
[CrossRef] [PubMed]

Menk, R. H.

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

Michiel, M. 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, 2845-2852 (1998).
[CrossRef] [PubMed]

Nesterets, Y. L.

T. E. Gureyev, Y. L. Nesterets, D. M. Paganin, A. Pogany, and S. W. Wilkins, “Linear algorithms for phase retrieval in the Fresnel region. 2. Partially coherent illumination,” Opt. Commun. 259, 569-580 (2006).
[CrossRef]

Nugent, K. A.

B. D. Arhatari, K. A. Nugent, A. G. Peele, and J. Thornton, “Phase contrast radiography. II. Imaging of complex objects,” Rev. Sci. Instrum. 76, 113, 704 (2005).
[CrossRef]

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

Ohara, H.

M. T. Freedman, S.-C. B. Lo, C. Honda, E. Makariou, G. Sisney, E. Pien, H. Ohara, A. Ishisaka, and F. Shimada, “Phase contrast digital mammography using molybdenum x-ray: clinical implications in detectability improvement,” Proc. SPIE 5030, 533-540 (2003).
[CrossRef]

H. Ohara, C. Honda, A. Ishisaka, and F. Shimada, “Image quality in digital phase contrast imaging using a tungsten anode x-ray tube with small focal-spot size,” Proc. SPIE 4682, 713-723 (2002).
[CrossRef]

Olivo, A.

A. Olivo and R. Speller, “Experimental validation of a simple model capable of predicting the phase contrast imaging capabilities of any x-ray imaging system,” Phys. Med. Biol. 51, 3015-3030 (2006).
[CrossRef] [PubMed]

T. Q. Xiao, A. Bergamaschi, D. Dreossi, R. Longo, A. Olivo, S. Pani, L. Rigon, T. Rokvic, C. Venanzi, and E. Castelli, “Effect of spatial coherence on application of in-line phase contrast imaging to synchrotron radiation mammography,” Nucl. Instrum. Methods Phys. Res. A 548, 155-162 (2005).
[CrossRef]

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

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, 2845-2852 (1998).
[CrossRef] [PubMed]

Ozaki, T.

R. Toth, J. C. Kieffer, S. Fourmaux, T. Ozaki, and A. Krol, “In-line phase-contrast imaging with a laser-based hard x-ray source,” Rev. Sci. Instrum. 76, 083701 (2005).
[CrossRef]

Paganin, D. M.

T. E. Gureyev, Y. L. Nesterets, D. M. Paganin, A. Pogany, and S. W. Wilkins, “Linear algorithms for phase retrieval in the Fresnel region. 2. Partially coherent illumination,” Opt. Commun. 259, 569-580 (2006).
[CrossRef]

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

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, 2845-2852 (1998).
[CrossRef] [PubMed]

Pani, S.

T. Q. Xiao, A. Bergamaschi, D. Dreossi, R. Longo, A. Olivo, S. Pani, L. Rigon, T. Rokvic, C. Venanzi, and E. Castelli, “Effect of spatial coherence on application of in-line phase contrast imaging to synchrotron radiation mammography,” Nucl. Instrum. Methods Phys. Res. A 548, 155-162 (2005).
[CrossRef]

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

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, 2845-2852 (1998).
[CrossRef] [PubMed]

Peele, A. G.

B. D. Arhatari, K. A. Nugent, A. G. Peele, and J. Thornton, “Phase contrast radiography. II. Imaging of complex objects,” Rev. Sci. Instrum. 76, 113, 704 (2005).
[CrossRef]

Pickens, D. R.

E. F. Donnelly, R. R. Price, and D. R. Pickens, “Experimental validation of the Wigner distributions theory of phase-contrast imaging,” Med. Phys. 32, 928-931 (2005).
[CrossRef] [PubMed]

E. F. Donnelly, R. R. Price, and D. R. Pickens, “Quantification of the effect of system and object parameters on edge enhancement in phase-contrast radiography,” Med. Phys. 30, 2888-2896 (2003).
[CrossRef] [PubMed]

E. F. Donnelly, R. R. Price, and D. R. Pickens, “Dual focal-spot imaging for phase extraction in phase-contrast radiography,” Med. Phys. 30, 2292-2296 (2003).
[CrossRef] [PubMed]

Pien, E.

M. T. Freedman, S.-C. B. Lo, C. Honda, E. Makariou, G. Sisney, E. Pien, H. Ohara, A. Ishisaka, and F. Shimada, “Phase contrast digital mammography using molybdenum x-ray: clinical implications in detectability improvement,” Proc. SPIE 5030, 533-540 (2003).
[CrossRef]

Pogany, A.

T. E. Gureyev, Y. L. Nesterets, D. M. Paganin, A. Pogany, and S. W. Wilkins, “Linear algorithms for phase retrieval in the Fresnel region. 2. Partially coherent illumination,” Opt. Commun. 259, 569-580 (2006).
[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]

S. W. Wilkins, T. E. Gureyev, D. Gao, A. Pogany, and A. W. Stevenson, “Phase-contrast imaging using polychromatic hard X-rays,” Nature 384, 335-338 (1996).
[CrossRef]

Pontoni, D.

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

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, 2845-2852 (1998).
[CrossRef] [PubMed]

Poropat, P.

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

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, 2845-2852 (1998).
[CrossRef] [PubMed]

Prest, M.

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

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, 2845-2852 (1998).
[CrossRef] [PubMed]

Price, R. R.

E. F. Donnelly, R. R. Price, and D. R. Pickens, “Experimental validation of the Wigner distributions theory of phase-contrast imaging,” Med. Phys. 32, 928-931 (2005).
[CrossRef] [PubMed]

E. F. Donnelly, R. R. Price, and D. R. Pickens, “Dual focal-spot imaging for phase extraction in phase-contrast radiography,” Med. Phys. 30, 2292-2296 (2003).
[CrossRef] [PubMed]

E. F. Donnelly, R. R. Price, and D. R. Pickens, “Quantification of the effect of system and object parameters on edge enhancement in phase-contrast radiography,” Med. Phys. 30, 2888-2896 (2003).
[CrossRef] [PubMed]

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

Rashevsky, A.

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

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, 2845-2852 (1998).
[CrossRef] [PubMed]

Ratti, M.

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

Rigon, L.

T. Q. Xiao, A. Bergamaschi, D. Dreossi, R. Longo, A. Olivo, S. Pani, L. Rigon, T. Rokvic, C. Venanzi, and E. Castelli, “Effect of spatial coherence on application of in-line phase contrast imaging to synchrotron radiation mammography,” Nucl. Instrum. Methods Phys. Res. A 548, 155-162 (2005).
[CrossRef]

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

Rokvic, T.

T. Q. Xiao, A. Bergamaschi, D. Dreossi, R. Longo, A. Olivo, S. Pani, L. Rigon, T. Rokvic, C. Venanzi, and E. Castelli, “Effect of spatial coherence on application of in-line phase contrast imaging to synchrotron radiation mammography,” Nucl. Instrum. Methods Phys. Res. A 548, 155-162 (2005).
[CrossRef]

Schlenker, M.

P. Cloetens, R. Mache, M. Schlenker, and S. Lerbs-Mache, “Quantitative phase tomography of Arabidopsis seeds reveals intercellular void network,” Proc. Natl. Acad. Sci. USA 103, 14,626-14,630 (2006).
[CrossRef]

S. Zabler, P. Cloetens, J.-P. Guigay, J. Baruchel, and M. Schlenker, “Optimization of phase contrast imaging using hard x rays,” Rev. Sci. Instrum. 76, 073,705 (2005).
[CrossRef]

P. Cloetens, W. Ludwig, J. Baruchel, D. V. Dyck, J. V. Landuyt, J. P. Guigay, and M. Schlenker, “Holotomography: quantitative phase tomography with micrometer resolution using hard synchrotron radiation x rays,” Appl. Phys. Lett. 75, 2912-2914 (1999).
[CrossRef]

Shimada, F.

M. T. Freedman, S.-C. B. Lo, C. Honda, E. Makariou, G. Sisney, E. Pien, H. Ohara, A. Ishisaka, and F. Shimada, “Phase contrast digital mammography using molybdenum x-ray: clinical implications in detectability improvement,” Proc. SPIE 5030, 533-540 (2003).
[CrossRef]

H. Ohara, C. Honda, A. Ishisaka, and F. Shimada, “Image quality in digital phase contrast imaging using a tungsten anode x-ray tube with small focal-spot size,” Proc. SPIE 4682, 713-723 (2002).
[CrossRef]

Sisney, G.

M. T. Freedman, S.-C. B. Lo, C. Honda, E. Makariou, G. Sisney, E. Pien, H. Ohara, A. Ishisaka, and F. Shimada, “Phase contrast digital mammography using molybdenum x-ray: clinical implications in detectability improvement,” Proc. SPIE 5030, 533-540 (2003).
[CrossRef]

Snigirev, A.

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

Snigireva, I.

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

Speller, R.

A. Olivo and R. Speller, “Experimental validation of a simple model capable of predicting the phase contrast imaging capabilities of any x-ray imaging system,” Phys. Med. Biol. 51, 3015-3030 (2006).
[CrossRef] [PubMed]

Stevenson, A. W.

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

S. W. Wilkins, T. E. Gureyev, D. Gao, A. Pogany, and A. W. Stevenson, “Phase-contrast imaging using polychromatic hard X-rays,” Nature 384, 335-338 (1996).
[CrossRef]

T. J. Davis, T. E. Gureyev, D. Gao, A. W. Stevenson, and S. W. Wilkins, “X-ray image contrast from a simple phase object,” Phys. Rev. Lett. 74, 3173-3176 (1995).
[CrossRef] [PubMed]

Thornton, J.

B. D. Arhatari, K. A. Nugent, A. G. Peele, and J. Thornton, “Phase contrast radiography. II. Imaging of complex objects,” Rev. Sci. Instrum. 76, 113, 704 (2005).
[CrossRef]

Toth, R.

R. Toth, J. C. Kieffer, S. Fourmaux, T. Ozaki, and A. Krol, “In-line phase-contrast imaging with a laser-based hard x-ray source,” Rev. Sci. Instrum. 76, 083701 (2005).
[CrossRef]

Tromba, G.

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

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, 2845-2852 (1998).
[CrossRef] [PubMed]

Vacchi, A.

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

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, 2845-2852 (1998).
[CrossRef] [PubMed]

Vallazza, E.

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

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, 2845-2852 (1998).
[CrossRef] [PubMed]

Venanzi, C.

T. Q. Xiao, A. Bergamaschi, D. Dreossi, R. Longo, A. Olivo, S. Pani, L. Rigon, T. Rokvic, C. Venanzi, and E. Castelli, “Effect of spatial coherence on application of in-line phase contrast imaging to synchrotron radiation mammography,” Nucl. Instrum. Methods Phys. Res. A 548, 155-162 (2005).
[CrossRef]

Wilkins, S. W.

T. E. Gureyev, Y. L. Nesterets, D. M. Paganin, A. Pogany, and S. W. Wilkins, “Linear algorithms for phase retrieval in the Fresnel region. 2. Partially coherent illumination,” Opt. Commun. 259, 569-580 (2006).
[CrossRef]

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

S. W. Wilkins, T. E. Gureyev, D. Gao, A. Pogany, and A. W. Stevenson, “Phase-contrast imaging using polychromatic hard X-rays,” Nature 384, 335-338 (1996).
[CrossRef]

T. J. Davis, T. E. Gureyev, D. Gao, A. W. Stevenson, and S. W. Wilkins, “X-ray image contrast from a simple phase object,” Phys. Rev. Lett. 74, 3173-3176 (1995).
[CrossRef] [PubMed]

Wu, X.

F. B. Meng, A. Yan, G. Zhou, X. Wu, and H. Liu, “Development of a dual-detector X-ray imaging system for phase retrieval study,” Nucl. Instrum. Methods Phys. Res. B 254, 300-306(2007).
[CrossRef]

F. B. Meng, H. Liu, and X. Wu, “An iterative phase retrieval algorithm for in-line x-ray phase imaging,” Opt. Express 15, 8383-8390 (2007).
[CrossRef] [PubMed]

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

X. Wu and H. Liu, “Phase-space formulation for phase-contrast x-ray imaging,” Appl. Opt. 44, 5847-5854 (2005).
[CrossRef] [PubMed]

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

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

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

X. Wu and H. Liu, “A dual detector approach for x-ray attenuation and phase imaging,” J. X-Ray Sci. Technol. 12, 35-42(2004).

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]

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

X. Wu, A. E. Deans, and H. Liu, Biomedical Photonics Handbook (CRC Press, 2003), Chap. 26.

Xiao, T. Q.

T. Q. Xiao, A. Bergamaschi, D. Dreossi, R. Longo, A. Olivo, S. Pani, L. Rigon, T. Rokvic, C. Venanzi, and E. Castelli, “Effect of spatial coherence on application of in-line phase contrast imaging to synchrotron radiation mammography,” Nucl. Instrum. Methods Phys. Res. A 548, 155-162 (2005).
[CrossRef]

Yan, A.

F. B. Meng, A. Yan, G. Zhou, X. Wu, and H. Liu, “Development of a dual-detector X-ray imaging system for phase retrieval study,” Nucl. Instrum. Methods Phys. Res. B 254, 300-306(2007).
[CrossRef]

Yan, A. M.

Zabler, S.

S. Zabler, P. Cloetens, J.-P. Guigay, J. Baruchel, and M. Schlenker, “Optimization of phase contrast imaging using hard x rays,” Rev. Sci. Instrum. 76, 073,705 (2005).
[CrossRef]

Zanconati, F.

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

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, 2845-2852 (1998).
[CrossRef] [PubMed]

Zhou, G.

F. B. Meng, A. Yan, G. Zhou, X. Wu, and H. Liu, “Development of a dual-detector X-ray imaging system for phase retrieval study,” Nucl. Instrum. Methods Phys. Res. B 254, 300-306(2007).
[CrossRef]

Appl. Opt. (1)

Appl. Phys. Lett. (1)

P. Cloetens, W. Ludwig, J. Baruchel, D. V. Dyck, J. V. Landuyt, J. P. Guigay, and M. Schlenker, “Holotomography: quantitative phase tomography with micrometer resolution using hard synchrotron radiation x rays,” Appl. Phys. Lett. 75, 2912-2914 (1999).
[CrossRef]

J. X-Ray Sci. Technol. (2)

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

X. Wu and H. Liu, “A dual detector approach for x-ray attenuation and phase imaging,” J. X-Ray Sci. Technol. 12, 35-42(2004).

Med. Phys. (7)

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

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

E. F. Donnelly, R. R. Price, and D. R. Pickens, “Experimental validation of the Wigner distributions theory of phase-contrast imaging,” Med. Phys. 32, 928-931 (2005).
[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]

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

E. F. Donnelly, R. R. Price, and D. R. Pickens, “Dual focal-spot imaging for phase extraction in phase-contrast radiography,” Med. Phys. 30, 2292-2296 (2003).
[CrossRef] [PubMed]

E. F. Donnelly, R. R. Price, and D. R. Pickens, “Quantification of the effect of system and object parameters on edge enhancement in phase-contrast radiography,” Med. Phys. 30, 2888-2896 (2003).
[CrossRef] [PubMed]

Nature (1)

S. W. Wilkins, T. E. Gureyev, D. Gao, A. Pogany, and A. W. Stevenson, “Phase-contrast imaging using polychromatic hard X-rays,” Nature 384, 335-338 (1996).
[CrossRef]

Nucl. Instrum. Methods Phys. Res. A (1)

T. Q. Xiao, A. Bergamaschi, D. Dreossi, R. Longo, A. Olivo, S. Pani, L. Rigon, T. Rokvic, C. Venanzi, and E. Castelli, “Effect of spatial coherence on application of in-line phase contrast imaging to synchrotron radiation mammography,” Nucl. Instrum. Methods Phys. Res. A 548, 155-162 (2005).
[CrossRef]

Nucl. Instrum. Methods Phys. Res. B (1)

F. B. Meng, A. Yan, G. Zhou, X. Wu, and H. Liu, “Development of a dual-detector X-ray imaging system for phase retrieval study,” Nucl. Instrum. Methods Phys. Res. B 254, 300-306(2007).
[CrossRef]

Opt. Commun. (1)

T. E. Gureyev, Y. L. Nesterets, D. M. Paganin, A. Pogany, and S. W. Wilkins, “Linear algorithms for phase retrieval in the Fresnel region. 2. Partially coherent illumination,” Opt. Commun. 259, 569-580 (2006).
[CrossRef]

Opt. Express (2)

Opt. Lett. (2)

Phys. Med. Biol. (4)

A. Olivo and R. Speller, “Experimental validation of a simple model capable of predicting the phase contrast imaging capabilities of any x-ray imaging system,” Phys. Med. Biol. 51, 3015-3030 (2006).
[CrossRef] [PubMed]

R. A. Lewis, “Medical phase contrast x-ray imaging: current status and future prospects,” Phys. Med. Biol. 49, 3573-3583 (2004).
[CrossRef] [PubMed]

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, 2845-2852 (1998).
[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]

Phys. Rev. Lett. (3)

T. J. Davis, T. E. Gureyev, D. Gao, A. W. Stevenson, and S. W. Wilkins, “X-ray image contrast from a simple phase object,” Phys. Rev. Lett. 74, 3173-3176 (1995).
[CrossRef] [PubMed]

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

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

Proc. Natl. Acad. Sci. USA (1)

P. Cloetens, R. Mache, M. Schlenker, and S. Lerbs-Mache, “Quantitative phase tomography of Arabidopsis seeds reveals intercellular void network,” Proc. Natl. Acad. Sci. USA 103, 14,626-14,630 (2006).
[CrossRef]

Proc. SPIE (2)

H. Ohara, C. Honda, A. Ishisaka, and F. Shimada, “Image quality in digital phase contrast imaging using a tungsten anode x-ray tube with small focal-spot size,” Proc. SPIE 4682, 713-723 (2002).
[CrossRef]

M. T. Freedman, S.-C. B. Lo, C. Honda, E. Makariou, G. Sisney, E. Pien, H. Ohara, A. Ishisaka, and F. Shimada, “Phase contrast digital mammography using molybdenum x-ray: clinical implications in detectability improvement,” Proc. SPIE 5030, 533-540 (2003).
[CrossRef]

Radiology (1)

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

Rev. Sci. Instrum. (5)

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

B. D. Arhatari, K. A. Nugent, A. G. Peele, and J. Thornton, “Phase contrast radiography. II. Imaging of complex objects,” Rev. Sci. Instrum. 76, 113, 704 (2005).
[CrossRef]

S. Zabler, P. Cloetens, J.-P. Guigay, J. Baruchel, and M. Schlenker, “Optimization of phase contrast imaging using hard x rays,” Rev. Sci. Instrum. 76, 073,705 (2005).
[CrossRef]

R. Toth, J. C. Kieffer, S. Fourmaux, T. Ozaki, and A. Krol, “In-line phase-contrast imaging with a laser-based hard x-ray source,” Rev. Sci. Instrum. 76, 083701 (2005).
[CrossRef]

Other (1)

X. Wu, A. E. Deans, and H. Liu, Biomedical Photonics Handbook (CRC Press, 2003), Chap. 26.

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Figures (6)

Fig. 1
Fig. 1

Schematic experimental configuration of the in-line x-ray phase imaging. The distance between the x-ray source and the object plane is R 1 , and that between the object and the image planes is R 2 . The geometric magnification can then be calculated as M = ( R 1 + R 2 ) / R 1 .

Fig. 2
Fig. 2

Attenuation and phase maps used in the computer simulations. The attenuation map has a mean value of 0.96 and a standard deviation (SD) of 0.04; the phase map has a mean value of 10.87 rad and a SD of 10.52 rad .

Fig. 3
Fig. 3

Simulated attenuation-based image I 1 and phase-contrast image I 2 for the virtual object in Fig. 2 under ideal experimental conditions, i.e., without any noises or errors.

Fig. 4
Fig. 4

Retrieved phase map from Fig. 3 using iterative algorithm Eq. (3). The good agreements between this figure and Fig. 2 shows the effectiveness of the algorithm.

Fig. 5
Fig. 5

Retrieved phase maps with consideration of some experimental factors.

Fig. 6
Fig. 6

Efficiency tests of the iterative algorithm. (a) Total time consumption versus iteration count and (b) per iteration and noniteration time consumption for images with different size.

Tables (1)

Tables Icon

Table 1 Effects of the Errors in Several Experimental Factors on the Iterative Phase Retrieval

Equations (22)

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

F ^ [ I bp ( ρ ) ] = I 0 · OTF ( u M ) { cos ( α u 2 ) F ^ [ A 2 ( ρ ) ] + 2 sin ( α u 2 ) F ^ [ A 2 ( ρ ) ϕ ( ρ ) ] + i λ R 2 2 M sin ( α u 2 ) u · F ^ [ A 2 ( ρ ) ] + i λ R 2 M cos ( α u 2 ) u · F ^ [ ϕ ( ρ ) A 2 ( ρ ) ] } ,
OTF ( u M ) = OTF G . U . ( u M ) · OTF det ( u M ) ,
F ^ [ A 2 ] = I ˜ 1 , F ^ [ A 2 ϕ n + 1 ] = I ˜ 2 cos ( α u 2 ) F ^ [ A 2 ] T 4 ( A 2 , ϕ n , R 2 ) 2 sin ( α u 2 ) ,
1 r exp ( i k r ) 1 z exp ( i k z i k ρ 2 2 z ) ,
ρ 4 4 λ z 3 .
exp ( i π ( ρ ρ ) 2 / λ z ) ,
| δ ϕ | ( 2 π | ρ ρ | · | d ρ | / λ z ) .
| ρ ρ | λ z / 2 | d ρ | .
| ρ ρ | λ z 2 σ .
T ( ρ ) = exp ( i ϕ ( ρ ) μ ( ρ ) / 2 ) = A ( ρ ) e i ϕ ( ρ ) ,
ϕ ( ρ ) = 2 π λ δ ( r ) d z = λ r e ρ e , p ( ρ ) ,
μ ( ρ ) = 4 π λ β ( r ) d z .
E t ( ρ ) = E i ( ρ ) · T ( ρ ) ,
exp ( i ϕ ( ρ ) i ϕ ( ρ λ R 2 u M ) ) 1 + i ϕ ( ρ ) i ϕ ( ρ λ R 2 u M ) ,
A ( ρ ± λ R 2 u M ) A ( ρ ) ± λ R 2 u M A ( ρ ) .
λ R 2 u max 2 M 1.
| ϕ ( ρ ) | M λ R 2 u max .
PV = a ln X + b .
[ A 2 ( ρ ; R 2 ) , ϕ ( ρ ; R 2 ) ] = IterAlgo1 ( I 2 , I 2 ) , F ^ [ A ( ρ ) exp ( i ϕ ( ρ ) ) ] = F ^ [ A ( ρ ; R 2 ) exp ( i ϕ ( ρ ; R 2 ) ) ] F ^ [ exp ( i π ρ 2 ) / λ R 2 ] .
A total 2 ( ρ ) = A 2 ( ρ ) · A D 1 2 ( ρ ) , ϕ total ( ρ ) = ϕ ( ρ ) + ϕ D 1 ( ρ ) .
[ A total 2 ( ρ ) , ϕ total ( ρ ) ] = IterAlog1 ( I 1 · A D 1 2 , I 2 ) , A 2 ( ρ ) = A total 2 ( ρ ) / A D 1 2 ( ρ ) , ϕ ( ρ ) = ϕ total ( ρ ) ϕ D 1 ( ρ ) ,
[ A total 2 ( ρ ) , ϕ total ( ρ ) ] = IterAlgo1 ( I 2 · A D 1 2 , I 2 ) , A 2 ( ρ ; R 2 ) = A total 2 ( ρ ) / A D 1 2 ( ρ ) , ϕ ( ρ ; R 2 ) = ϕ total ( ρ ) ϕ D 1 ( ρ ) , F ^ [ A ( ρ ) exp ( i ϕ ( ρ ) ) ] = F ^ [ A ( ρ ; R 2 ) exp ( i ϕ ( ρ ; R 2 ) ) ] F ^ [ exp ( i π ρ 2 ) / λ R 2 ] .

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