Abstract

Various X-ray phase-contrast imaging techniques have been developed and applied over the last twenty years in different domains, such as material sciences, biology and medicine. However, no comprehensive inter-comparison exists in the literature. We present here a theoretical study that compares three among the most used techniques: propagation-based imaging (PBI), analyzer-based imaging (ABI) and grating interferometry (GI). These techniques are evaluated in terms of signal-to-noise ratio, figure of merit and spatial resolution. Both area and edge signals are considered. Dependences upon the object properties (absorption, phase shift) and the experimental acquisition parameters (energy, system point-spread function etc.) are derived and discussed. The results obtained from this analysis can be used as the reference for determining the most suitable technique for a given application.

© 2012 OSA

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2011

E. Castelli, M. Tonutti, F. Arfelli, R. Longo, E. Quaia, L. Rigon, D. Sanabor, F. Zanconati, D. Dreossi, A. Abrami, E. Quai, P. Bregant, K. Casarin, V. Chenda, R. H. Menk, T. Rokvic, A. Vascotto, G. Tromba, and M. A. Cova, “Mammography with synchrotron radiation: first clinical experience with phase-detection technique,” Radiology 259(3), 684–694 (2011).
[CrossRef] [PubMed]

P. C. Diemoz, P. Coan, I. Zanette, A. Bravin, S. Lang, C. Glaser, and T. Weitkamp, “A simplified approach for computed tomography with an X-ray grating interferometer,” Opt. Express 19(3), 1691–1698 (2011).
[CrossRef] [PubMed]

A. Olivo, K. Ignatyev, P. R. T. Munro, and R. D. Speller, “Noninterferometric phase-contrast images obtained with incoherent x-ray sources,” Appl. Opt. 50(12), 1765–1769 (2011).
[CrossRef] [PubMed]

2010

P. Coan, F. Bamberg, P. C. Diemoz, A. Bravin, K. Timpert, E. Mützel, J. G. Raya, S. Adam-Neumair, M. F. Reiser, and C. Glaser, “Characterization of osteoarthritic and normal human patella cartilage by computed tomography X-ray phase-contrast imaging: a feasibility study,” Invest. Radiol. 45(7), 437–444 (2010).
[PubMed]

G. Schulz, T. Weitkamp, I. Zanette, F. Pfeiffer, F. Beckmann, C. David, S. Rutishauser, E. Reznikova, and B. Mueller, “High-resolution tomographic imaging of a human cerebellum: comparison of absorption and grating-based phase contrast,” J. R. Soc. Interface 7(53), 1665–1676 (2010).
[CrossRef] [PubMed]

2008

2006

F. Pfeiffer, T. Weitkamp, O. Bunk, and C. David, “Phase retrieval and differential phase-contrast imaging with low-brilliance X-ray sources,” Nat. Phys. 2(4), 258–261 (2006).
[CrossRef]

Y. I. Nesterets, P. Coan, T. E. Gureyev, A. Bravin, P. Cloetens, and S. W. Wilkins, “On qualitative and quantitative analysis in analyser-based imaging,” Acta Crystallogr. A 62(4), 296–308 (2006).
[CrossRef] [PubMed]

2005

E. Pagot, S. Fiedler, P. Cloetens, A. Bravin, P. Coan, K. Fezzaa, J. Baruchel, J. Härtwig, K. von Smitten, M. Leidenius, M. L. Karjalainen-Lindsberg, and J. Keyriläinen, “Quantitative comparison between two phase contrast techniques: diffraction enhanced imaging and phase propagation imaging,” Phys. Med. Biol. 50(4), 709–724 (2005).
[CrossRef] [PubMed]

P. Coan, E. Pagot, S. Fiedler, P. Cloetens, J. Baruchel, and A. Bravin, “Phase-contrast X-ray imaging combining free space propagation and Bragg diffraction,” J. Synchrotron Radiat. 12(2), 241–245 (2005).
[CrossRef] [PubMed]

T. Weitkamp, A. Diaz, C. David, F. Pfeiffer, M. Stampanoni, P. Cloetens, and E. Ziegler, “X-ray phase imaging with a grating interferometer,” Opt. Express 13(16), 6296–6304 (2005).
[CrossRef] [PubMed]

2004

K. M. Pavlov, T. E. Gureyev, D. Paganin, Y. I. Nesterets, M. J. Morgan, and R. A. Lewis, “Linear systems with slowly varying transfer functions and their application to x-ray phase-contrast imaging,” J. Phys. D Appl. Phys. 37(19), 2746–2750 (2004).
[CrossRef]

R. Tanuma and M. Ohsawa, “Submicron-resolved X-ray topography using asymmetric-reflection magnifiers,” Spectrochim. Acta, Part B 59(10-11), 1549–1555 (2004).
[CrossRef]

2003

A. Bravin, “Exploiting the X-ray refraction contrast with an analyser: the state of the art,” J. Phys. D Appl. Phys. 36(10A), A24–A29 (2003).
[CrossRef]

2002

C. David, B. Nöhammer, H. Solak, and E. Ziegler, “Differential X-ray phase contrast imaging using a shearing interferometer,” Appl. Phys. Lett. 81(17), 3287–3289 (2002).
[CrossRef]

2000

J. M. Boone, K. K. Lindfors, V. N. Cooper, and J. A. Seibert, “Scatter/primary in mammography: comprehensive results,” Med. Phys. 27(10), 2408–2416 (2000).
[CrossRef] [PubMed]

1998

F. Arfelli, V. Bonvicini, A. Bravin, G. Cantatore, E. Castelli, L. Dalla Palma, M. Di Michiel, R. Longo, A. Olivo, S. Pani, D. Pontoni, P. Poropat, M. Prest, A. Rashevsky, G. Tromba, and A. Vacchi, “Mammography of a phantom and breast tissue with synchrotron radiation and a linear-array silicon detector,” Radiology 208(3), 709–715 (1998).
[PubMed]

1997

T. E. Gureyev and S. W. Wilkins, “Regimes of X-ray phase-contrast imaging with perfect crystals,” Nuovo Cimento D 19(2-4), 545–552 (1997).
[CrossRef]

D. Chapman, W. Thomlinson, R. E. Johnston, D. Washburn, E. Pisano, N. Gmur, Z. Zhong, R. Menk, F. Arfelli, and D. Sayers, “Diffraction enhanced x-ray imaging,” Phys. Med. Biol. 42(11), 2015–2025 (1997).
[CrossRef] [PubMed]

J. Persliden and G. A. Carlsson, “Scatter rejection by air gaps in diagnostic radiology. Calculations using a Monte Carlo collision density method and consideration of molecular interference in coherent scattering,” Phys. Med. Biol. 42(1), 155–175 (1997).
[CrossRef] [PubMed]

M. Sanchez del Rio, C. Ferrero, and V. Mocella, “Computer simulations of bent perfect crystal diffraction profiles,” Proc. SPIE 3152, 312–323 (1997), http://www.esrf.eu/UsersAndScience/Experiments/TBS/ SciSoft/xop2.3 .
[CrossRef]

1996

A. Momose, T. Takeda, Y. Itai, and K. Hirano, “Phase-contrast X-ray computed tomography for observing biological soft tissues,” Nat. Med. 2(4), 473–475 (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(6607), 335–338 (1996).
[CrossRef]

1995

A. Snigirev, I. Snigireva, V. Kohn, S. Kuznetsov, and I. Schelokov, “On the possibility of x-ray phase contrast microimaging by coherent high-energy synchrotron radiation,” Rev. Sci. Instrum. 66(12), 5486–5492 (1995).
[CrossRef]

1982

1980

E. Förster, K. Goetz, and P. Zaumseil, “Double crystal diffractometry for the characterization of targets for laser fusion experiments,” Krist. Tech. 15(8), 937–945 (1980).
[CrossRef]

1965

U. Bonse and M. Hart, “An X-ray interferometer,” Appl. Phys. Lett. 6(8), 155–156 (1965).
[CrossRef]

Abrami, A.

E. Castelli, M. Tonutti, F. Arfelli, R. Longo, E. Quaia, L. Rigon, D. Sanabor, F. Zanconati, D. Dreossi, A. Abrami, E. Quai, P. Bregant, K. Casarin, V. Chenda, R. H. Menk, T. Rokvic, A. Vascotto, G. Tromba, and M. A. Cova, “Mammography with synchrotron radiation: first clinical experience with phase-detection technique,” Radiology 259(3), 684–694 (2011).
[CrossRef] [PubMed]

Adam-Neumair, S.

P. Coan, F. Bamberg, P. C. Diemoz, A. Bravin, K. Timpert, E. Mützel, J. G. Raya, S. Adam-Neumair, M. F. Reiser, and C. Glaser, “Characterization of osteoarthritic and normal human patella cartilage by computed tomography X-ray phase-contrast imaging: a feasibility study,” Invest. Radiol. 45(7), 437–444 (2010).
[PubMed]

Arfelli, F.

E. Castelli, M. Tonutti, F. Arfelli, R. Longo, E. Quaia, L. Rigon, D. Sanabor, F. Zanconati, D. Dreossi, A. Abrami, E. Quai, P. Bregant, K. Casarin, V. Chenda, R. H. Menk, T. Rokvic, A. Vascotto, G. Tromba, and M. A. Cova, “Mammography with synchrotron radiation: first clinical experience with phase-detection technique,” Radiology 259(3), 684–694 (2011).
[CrossRef] [PubMed]

F. Arfelli, V. Bonvicini, A. Bravin, G. Cantatore, E. Castelli, L. Dalla Palma, M. Di Michiel, R. Longo, A. Olivo, S. Pani, D. Pontoni, P. Poropat, M. Prest, A. Rashevsky, G. Tromba, and A. Vacchi, “Mammography of a phantom and breast tissue with synchrotron radiation and a linear-array silicon detector,” Radiology 208(3), 709–715 (1998).
[PubMed]

D. Chapman, W. Thomlinson, R. E. Johnston, D. Washburn, E. Pisano, N. Gmur, Z. Zhong, R. Menk, F. Arfelli, and D. Sayers, “Diffraction enhanced x-ray imaging,” Phys. Med. Biol. 42(11), 2015–2025 (1997).
[CrossRef] [PubMed]

Bamberg, F.

P. Coan, F. Bamberg, P. C. Diemoz, A. Bravin, K. Timpert, E. Mützel, J. G. Raya, S. Adam-Neumair, M. F. Reiser, and C. Glaser, “Characterization of osteoarthritic and normal human patella cartilage by computed tomography X-ray phase-contrast imaging: a feasibility study,” Invest. Radiol. 45(7), 437–444 (2010).
[PubMed]

Baruchel, J.

E. Pagot, S. Fiedler, P. Cloetens, A. Bravin, P. Coan, K. Fezzaa, J. Baruchel, J. Härtwig, K. von Smitten, M. Leidenius, M. L. Karjalainen-Lindsberg, and J. Keyriläinen, “Quantitative comparison between two phase contrast techniques: diffraction enhanced imaging and phase propagation imaging,” Phys. Med. Biol. 50(4), 709–724 (2005).
[CrossRef] [PubMed]

P. Coan, E. Pagot, S. Fiedler, P. Cloetens, J. Baruchel, and A. Bravin, “Phase-contrast X-ray imaging combining free space propagation and Bragg diffraction,” J. Synchrotron Radiat. 12(2), 241–245 (2005).
[CrossRef] [PubMed]

Beckmann, F.

G. Schulz, T. Weitkamp, I. Zanette, F. Pfeiffer, F. Beckmann, C. David, S. Rutishauser, E. Reznikova, and B. Mueller, “High-resolution tomographic imaging of a human cerebellum: comparison of absorption and grating-based phase contrast,” J. R. Soc. Interface 7(53), 1665–1676 (2010).
[CrossRef] [PubMed]

Bonse, U.

U. Bonse and M. Hart, “An X-ray interferometer,” Appl. Phys. Lett. 6(8), 155–156 (1965).
[CrossRef]

Bonvicini, V.

F. Arfelli, V. Bonvicini, A. Bravin, G. Cantatore, E. Castelli, L. Dalla Palma, M. Di Michiel, R. Longo, A. Olivo, S. Pani, D. Pontoni, P. Poropat, M. Prest, A. Rashevsky, G. Tromba, and A. Vacchi, “Mammography of a phantom and breast tissue with synchrotron radiation and a linear-array silicon detector,” Radiology 208(3), 709–715 (1998).
[PubMed]

Boone, J. M.

J. M. Boone, K. K. Lindfors, V. N. Cooper, and J. A. Seibert, “Scatter/primary in mammography: comprehensive results,” Med. Phys. 27(10), 2408–2416 (2000).
[CrossRef] [PubMed]

Bravin, A.

P. C. Diemoz, P. Coan, I. Zanette, A. Bravin, S. Lang, C. Glaser, and T. Weitkamp, “A simplified approach for computed tomography with an X-ray grating interferometer,” Opt. Express 19(3), 1691–1698 (2011).
[CrossRef] [PubMed]

P. Coan, F. Bamberg, P. C. Diemoz, A. Bravin, K. Timpert, E. Mützel, J. G. Raya, S. Adam-Neumair, M. F. Reiser, and C. Glaser, “Characterization of osteoarthritic and normal human patella cartilage by computed tomography X-ray phase-contrast imaging: a feasibility study,” Invest. Radiol. 45(7), 437–444 (2010).
[PubMed]

Y. I. Nesterets, P. Coan, T. E. Gureyev, A. Bravin, P. Cloetens, and S. W. Wilkins, “On qualitative and quantitative analysis in analyser-based imaging,” Acta Crystallogr. A 62(4), 296–308 (2006).
[CrossRef] [PubMed]

E. Pagot, S. Fiedler, P. Cloetens, A. Bravin, P. Coan, K. Fezzaa, J. Baruchel, J. Härtwig, K. von Smitten, M. Leidenius, M. L. Karjalainen-Lindsberg, and J. Keyriläinen, “Quantitative comparison between two phase contrast techniques: diffraction enhanced imaging and phase propagation imaging,” Phys. Med. Biol. 50(4), 709–724 (2005).
[CrossRef] [PubMed]

P. Coan, E. Pagot, S. Fiedler, P. Cloetens, J. Baruchel, and A. Bravin, “Phase-contrast X-ray imaging combining free space propagation and Bragg diffraction,” J. Synchrotron Radiat. 12(2), 241–245 (2005).
[CrossRef] [PubMed]

A. Bravin, “Exploiting the X-ray refraction contrast with an analyser: the state of the art,” J. Phys. D Appl. Phys. 36(10A), A24–A29 (2003).
[CrossRef]

F. Arfelli, V. Bonvicini, A. Bravin, G. Cantatore, E. Castelli, L. Dalla Palma, M. Di Michiel, R. Longo, A. Olivo, S. Pani, D. Pontoni, P. Poropat, M. Prest, A. Rashevsky, G. Tromba, and A. Vacchi, “Mammography of a phantom and breast tissue with synchrotron radiation and a linear-array silicon detector,” Radiology 208(3), 709–715 (1998).
[PubMed]

Bregant, P.

E. Castelli, M. Tonutti, F. Arfelli, R. Longo, E. Quaia, L. Rigon, D. Sanabor, F. Zanconati, D. Dreossi, A. Abrami, E. Quai, P. Bregant, K. Casarin, V. Chenda, R. H. Menk, T. Rokvic, A. Vascotto, G. Tromba, and M. A. Cova, “Mammography with synchrotron radiation: first clinical experience with phase-detection technique,” Radiology 259(3), 684–694 (2011).
[CrossRef] [PubMed]

Bunk, O.

F. Pfeiffer, T. Weitkamp, O. Bunk, and C. David, “Phase retrieval and differential phase-contrast imaging with low-brilliance X-ray sources,” Nat. Phys. 2(4), 258–261 (2006).
[CrossRef]

Cantatore, G.

F. Arfelli, V. Bonvicini, A. Bravin, G. Cantatore, E. Castelli, L. Dalla Palma, M. Di Michiel, R. Longo, A. Olivo, S. Pani, D. Pontoni, P. Poropat, M. Prest, A. Rashevsky, G. Tromba, and A. Vacchi, “Mammography of a phantom and breast tissue with synchrotron radiation and a linear-array silicon detector,” Radiology 208(3), 709–715 (1998).
[PubMed]

Carlsson, G. A.

J. Persliden and G. A. Carlsson, “Scatter rejection by air gaps in diagnostic radiology. Calculations using a Monte Carlo collision density method and consideration of molecular interference in coherent scattering,” Phys. Med. Biol. 42(1), 155–175 (1997).
[CrossRef] [PubMed]

Casarin, K.

E. Castelli, M. Tonutti, F. Arfelli, R. Longo, E. Quaia, L. Rigon, D. Sanabor, F. Zanconati, D. Dreossi, A. Abrami, E. Quai, P. Bregant, K. Casarin, V. Chenda, R. H. Menk, T. Rokvic, A. Vascotto, G. Tromba, and M. A. Cova, “Mammography with synchrotron radiation: first clinical experience with phase-detection technique,” Radiology 259(3), 684–694 (2011).
[CrossRef] [PubMed]

Castelli, E.

E. Castelli, M. Tonutti, F. Arfelli, R. Longo, E. Quaia, L. Rigon, D. Sanabor, F. Zanconati, D. Dreossi, A. Abrami, E. Quai, P. Bregant, K. Casarin, V. Chenda, R. H. Menk, T. Rokvic, A. Vascotto, G. Tromba, and M. A. Cova, “Mammography with synchrotron radiation: first clinical experience with phase-detection technique,” Radiology 259(3), 684–694 (2011).
[CrossRef] [PubMed]

F. Arfelli, V. Bonvicini, A. Bravin, G. Cantatore, E. Castelli, L. Dalla Palma, M. Di Michiel, R. Longo, A. Olivo, S. Pani, D. Pontoni, P. Poropat, M. Prest, A. Rashevsky, G. Tromba, and A. Vacchi, “Mammography of a phantom and breast tissue with synchrotron radiation and a linear-array silicon detector,” Radiology 208(3), 709–715 (1998).
[PubMed]

Chapman, D.

D. Chapman, W. Thomlinson, R. E. Johnston, D. Washburn, E. Pisano, N. Gmur, Z. Zhong, R. Menk, F. Arfelli, and D. Sayers, “Diffraction enhanced x-ray imaging,” Phys. Med. Biol. 42(11), 2015–2025 (1997).
[CrossRef] [PubMed]

Chenda, V.

E. Castelli, M. Tonutti, F. Arfelli, R. Longo, E. Quaia, L. Rigon, D. Sanabor, F. Zanconati, D. Dreossi, A. Abrami, E. Quai, P. Bregant, K. Casarin, V. Chenda, R. H. Menk, T. Rokvic, A. Vascotto, G. Tromba, and M. A. Cova, “Mammography with synchrotron radiation: first clinical experience with phase-detection technique,” Radiology 259(3), 684–694 (2011).
[CrossRef] [PubMed]

Cloetens, P.

Y. I. Nesterets, P. Coan, T. E. Gureyev, A. Bravin, P. Cloetens, and S. W. Wilkins, “On qualitative and quantitative analysis in analyser-based imaging,” Acta Crystallogr. A 62(4), 296–308 (2006).
[CrossRef] [PubMed]

T. Weitkamp, A. Diaz, C. David, F. Pfeiffer, M. Stampanoni, P. Cloetens, and E. Ziegler, “X-ray phase imaging with a grating interferometer,” Opt. Express 13(16), 6296–6304 (2005).
[CrossRef] [PubMed]

E. Pagot, S. Fiedler, P. Cloetens, A. Bravin, P. Coan, K. Fezzaa, J. Baruchel, J. Härtwig, K. von Smitten, M. Leidenius, M. L. Karjalainen-Lindsberg, and J. Keyriläinen, “Quantitative comparison between two phase contrast techniques: diffraction enhanced imaging and phase propagation imaging,” Phys. Med. Biol. 50(4), 709–724 (2005).
[CrossRef] [PubMed]

P. Coan, E. Pagot, S. Fiedler, P. Cloetens, J. Baruchel, and A. Bravin, “Phase-contrast X-ray imaging combining free space propagation and Bragg diffraction,” J. Synchrotron Radiat. 12(2), 241–245 (2005).
[CrossRef] [PubMed]

Coan, P.

P. C. Diemoz, P. Coan, I. Zanette, A. Bravin, S. Lang, C. Glaser, and T. Weitkamp, “A simplified approach for computed tomography with an X-ray grating interferometer,” Opt. Express 19(3), 1691–1698 (2011).
[CrossRef] [PubMed]

P. Coan, F. Bamberg, P. C. Diemoz, A. Bravin, K. Timpert, E. Mützel, J. G. Raya, S. Adam-Neumair, M. F. Reiser, and C. Glaser, “Characterization of osteoarthritic and normal human patella cartilage by computed tomography X-ray phase-contrast imaging: a feasibility study,” Invest. Radiol. 45(7), 437–444 (2010).
[PubMed]

Y. I. Nesterets, P. Coan, T. E. Gureyev, A. Bravin, P. Cloetens, and S. W. Wilkins, “On qualitative and quantitative analysis in analyser-based imaging,” Acta Crystallogr. A 62(4), 296–308 (2006).
[CrossRef] [PubMed]

E. Pagot, S. Fiedler, P. Cloetens, A. Bravin, P. Coan, K. Fezzaa, J. Baruchel, J. Härtwig, K. von Smitten, M. Leidenius, M. L. Karjalainen-Lindsberg, and J. Keyriläinen, “Quantitative comparison between two phase contrast techniques: diffraction enhanced imaging and phase propagation imaging,” Phys. Med. Biol. 50(4), 709–724 (2005).
[CrossRef] [PubMed]

P. Coan, E. Pagot, S. Fiedler, P. Cloetens, J. Baruchel, and A. Bravin, “Phase-contrast X-ray imaging combining free space propagation and Bragg diffraction,” J. Synchrotron Radiat. 12(2), 241–245 (2005).
[CrossRef] [PubMed]

Cooper, V. N.

J. M. Boone, K. K. Lindfors, V. N. Cooper, and J. A. Seibert, “Scatter/primary in mammography: comprehensive results,” Med. Phys. 27(10), 2408–2416 (2000).
[CrossRef] [PubMed]

Cova, M. A.

E. Castelli, M. Tonutti, F. Arfelli, R. Longo, E. Quaia, L. Rigon, D. Sanabor, F. Zanconati, D. Dreossi, A. Abrami, E. Quai, P. Bregant, K. Casarin, V. Chenda, R. H. Menk, T. Rokvic, A. Vascotto, G. Tromba, and M. A. Cova, “Mammography with synchrotron radiation: first clinical experience with phase-detection technique,” Radiology 259(3), 684–694 (2011).
[CrossRef] [PubMed]

Dalla Palma, L.

F. Arfelli, V. Bonvicini, A. Bravin, G. Cantatore, E. Castelli, L. Dalla Palma, M. Di Michiel, R. Longo, A. Olivo, S. Pani, D. Pontoni, P. Poropat, M. Prest, A. Rashevsky, G. Tromba, and A. Vacchi, “Mammography of a phantom and breast tissue with synchrotron radiation and a linear-array silicon detector,” Radiology 208(3), 709–715 (1998).
[PubMed]

David, C.

G. Schulz, T. Weitkamp, I. Zanette, F. Pfeiffer, F. Beckmann, C. David, S. Rutishauser, E. Reznikova, and B. Mueller, “High-resolution tomographic imaging of a human cerebellum: comparison of absorption and grating-based phase contrast,” J. R. Soc. Interface 7(53), 1665–1676 (2010).
[CrossRef] [PubMed]

F. Pfeiffer, T. Weitkamp, O. Bunk, and C. David, “Phase retrieval and differential phase-contrast imaging with low-brilliance X-ray sources,” Nat. Phys. 2(4), 258–261 (2006).
[CrossRef]

T. Weitkamp, A. Diaz, C. David, F. Pfeiffer, M. Stampanoni, P. Cloetens, and E. Ziegler, “X-ray phase imaging with a grating interferometer,” Opt. Express 13(16), 6296–6304 (2005).
[CrossRef] [PubMed]

C. David, B. Nöhammer, H. Solak, and E. Ziegler, “Differential X-ray phase contrast imaging using a shearing interferometer,” Appl. Phys. Lett. 81(17), 3287–3289 (2002).
[CrossRef]

Di Michiel, M.

F. Arfelli, V. Bonvicini, A. Bravin, G. Cantatore, E. Castelli, L. Dalla Palma, M. Di Michiel, R. Longo, A. Olivo, S. Pani, D. Pontoni, P. Poropat, M. Prest, A. Rashevsky, G. Tromba, and A. Vacchi, “Mammography of a phantom and breast tissue with synchrotron radiation and a linear-array silicon detector,” Radiology 208(3), 709–715 (1998).
[PubMed]

Diaz, A.

Diemoz, P. C.

P. C. Diemoz, P. Coan, I. Zanette, A. Bravin, S. Lang, C. Glaser, and T. Weitkamp, “A simplified approach for computed tomography with an X-ray grating interferometer,” Opt. Express 19(3), 1691–1698 (2011).
[CrossRef] [PubMed]

P. Coan, F. Bamberg, P. C. Diemoz, A. Bravin, K. Timpert, E. Mützel, J. G. Raya, S. Adam-Neumair, M. F. Reiser, and C. Glaser, “Characterization of osteoarthritic and normal human patella cartilage by computed tomography X-ray phase-contrast imaging: a feasibility study,” Invest. Radiol. 45(7), 437–444 (2010).
[PubMed]

Dreossi, D.

E. Castelli, M. Tonutti, F. Arfelli, R. Longo, E. Quaia, L. Rigon, D. Sanabor, F. Zanconati, D. Dreossi, A. Abrami, E. Quai, P. Bregant, K. Casarin, V. Chenda, R. H. Menk, T. Rokvic, A. Vascotto, G. Tromba, and M. A. Cova, “Mammography with synchrotron radiation: first clinical experience with phase-detection technique,” Radiology 259(3), 684–694 (2011).
[CrossRef] [PubMed]

Ferrero, C.

M. Sanchez del Rio, C. Ferrero, and V. Mocella, “Computer simulations of bent perfect crystal diffraction profiles,” Proc. SPIE 3152, 312–323 (1997), http://www.esrf.eu/UsersAndScience/Experiments/TBS/ SciSoft/xop2.3 .
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Fezzaa, K.

E. Pagot, S. Fiedler, P. Cloetens, A. Bravin, P. Coan, K. Fezzaa, J. Baruchel, J. Härtwig, K. von Smitten, M. Leidenius, M. L. Karjalainen-Lindsberg, and J. Keyriläinen, “Quantitative comparison between two phase contrast techniques: diffraction enhanced imaging and phase propagation imaging,” Phys. Med. Biol. 50(4), 709–724 (2005).
[CrossRef] [PubMed]

Fiedler, S.

E. Pagot, S. Fiedler, P. Cloetens, A. Bravin, P. Coan, K. Fezzaa, J. Baruchel, J. Härtwig, K. von Smitten, M. Leidenius, M. L. Karjalainen-Lindsberg, and J. Keyriläinen, “Quantitative comparison between two phase contrast techniques: diffraction enhanced imaging and phase propagation imaging,” Phys. Med. Biol. 50(4), 709–724 (2005).
[CrossRef] [PubMed]

P. Coan, E. Pagot, S. Fiedler, P. Cloetens, J. Baruchel, and A. Bravin, “Phase-contrast X-ray imaging combining free space propagation and Bragg diffraction,” J. Synchrotron Radiat. 12(2), 241–245 (2005).
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Förster, E.

E. Förster, K. Goetz, and P. Zaumseil, “Double crystal diffractometry for the characterization of targets for laser fusion experiments,” Krist. Tech. 15(8), 937–945 (1980).
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Gao, D.

S. W. Wilkins, T. E. Gureyev, D. Gao, A. Pogany, and A. W. Stevenson, “Phase-contrast imaging using polychromatic hard X-rays,” Nature 384(6607), 335–338 (1996).
[CrossRef]

Glaser, C.

P. C. Diemoz, P. Coan, I. Zanette, A. Bravin, S. Lang, C. Glaser, and T. Weitkamp, “A simplified approach for computed tomography with an X-ray grating interferometer,” Opt. Express 19(3), 1691–1698 (2011).
[CrossRef] [PubMed]

P. Coan, F. Bamberg, P. C. Diemoz, A. Bravin, K. Timpert, E. Mützel, J. G. Raya, S. Adam-Neumair, M. F. Reiser, and C. Glaser, “Characterization of osteoarthritic and normal human patella cartilage by computed tomography X-ray phase-contrast imaging: a feasibility study,” Invest. Radiol. 45(7), 437–444 (2010).
[PubMed]

Gmur, N.

D. Chapman, W. Thomlinson, R. E. Johnston, D. Washburn, E. Pisano, N. Gmur, Z. Zhong, R. Menk, F. Arfelli, and D. Sayers, “Diffraction enhanced x-ray imaging,” Phys. Med. Biol. 42(11), 2015–2025 (1997).
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Goetz, K.

E. Förster, K. Goetz, and P. Zaumseil, “Double crystal diffractometry for the characterization of targets for laser fusion experiments,” Krist. Tech. 15(8), 937–945 (1980).
[CrossRef]

Gureyev, T. E.

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).
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Y. I. Nesterets, P. Coan, T. E. Gureyev, A. Bravin, P. Cloetens, and S. W. Wilkins, “On qualitative and quantitative analysis in analyser-based imaging,” Acta Crystallogr. A 62(4), 296–308 (2006).
[CrossRef] [PubMed]

K. M. Pavlov, T. E. Gureyev, D. Paganin, Y. I. Nesterets, M. J. Morgan, and R. A. Lewis, “Linear systems with slowly varying transfer functions and their application to x-ray phase-contrast imaging,” J. Phys. D Appl. Phys. 37(19), 2746–2750 (2004).
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T. E. Gureyev and S. W. Wilkins, “Regimes of X-ray phase-contrast imaging with perfect crystals,” Nuovo Cimento D 19(2-4), 545–552 (1997).
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S. W. Wilkins, T. E. Gureyev, D. Gao, A. Pogany, and A. W. Stevenson, “Phase-contrast imaging using polychromatic hard X-rays,” Nature 384(6607), 335–338 (1996).
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Hart, M.

U. Bonse and M. Hart, “An X-ray interferometer,” Appl. Phys. Lett. 6(8), 155–156 (1965).
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Härtwig, J.

E. Pagot, S. Fiedler, P. Cloetens, A. Bravin, P. Coan, K. Fezzaa, J. Baruchel, J. Härtwig, K. von Smitten, M. Leidenius, M. L. Karjalainen-Lindsberg, and J. Keyriläinen, “Quantitative comparison between two phase contrast techniques: diffraction enhanced imaging and phase propagation imaging,” Phys. Med. Biol. 50(4), 709–724 (2005).
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Hirano, K.

A. Momose, T. Takeda, Y. Itai, and K. Hirano, “Phase-contrast X-ray computed tomography for observing biological soft tissues,” Nat. Med. 2(4), 473–475 (1996).
[CrossRef] [PubMed]

Ignatyev, K.

Itai, Y.

A. Momose, T. Takeda, Y. Itai, and K. Hirano, “Phase-contrast X-ray computed tomography for observing biological soft tissues,” Nat. Med. 2(4), 473–475 (1996).
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Johnston, R. E.

D. Chapman, W. Thomlinson, R. E. Johnston, D. Washburn, E. Pisano, N. Gmur, Z. Zhong, R. Menk, F. Arfelli, and D. Sayers, “Diffraction enhanced x-ray imaging,” Phys. Med. Biol. 42(11), 2015–2025 (1997).
[CrossRef] [PubMed]

Karjalainen-Lindsberg, M. L.

E. Pagot, S. Fiedler, P. Cloetens, A. Bravin, P. Coan, K. Fezzaa, J. Baruchel, J. Härtwig, K. von Smitten, M. Leidenius, M. L. Karjalainen-Lindsberg, and J. Keyriläinen, “Quantitative comparison between two phase contrast techniques: diffraction enhanced imaging and phase propagation imaging,” Phys. Med. Biol. 50(4), 709–724 (2005).
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Keyriläinen, J.

E. Pagot, S. Fiedler, P. Cloetens, A. Bravin, P. Coan, K. Fezzaa, J. Baruchel, J. Härtwig, K. von Smitten, M. Leidenius, M. L. Karjalainen-Lindsberg, and J. Keyriläinen, “Quantitative comparison between two phase contrast techniques: diffraction enhanced imaging and phase propagation imaging,” Phys. Med. Biol. 50(4), 709–724 (2005).
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Kohn, V.

A. Snigirev, I. Snigireva, V. Kohn, S. Kuznetsov, and I. Schelokov, “On the possibility of x-ray phase contrast microimaging by coherent high-energy synchrotron radiation,” Rev. Sci. Instrum. 66(12), 5486–5492 (1995).
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Kuznetsov, S.

A. Snigirev, I. Snigireva, V. Kohn, S. Kuznetsov, and I. Schelokov, “On the possibility of x-ray phase contrast microimaging by coherent high-energy synchrotron radiation,” Rev. Sci. Instrum. 66(12), 5486–5492 (1995).
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Lang, S.

Leidenius, M.

E. Pagot, S. Fiedler, P. Cloetens, A. Bravin, P. Coan, K. Fezzaa, J. Baruchel, J. Härtwig, K. von Smitten, M. Leidenius, M. L. Karjalainen-Lindsberg, and J. Keyriläinen, “Quantitative comparison between two phase contrast techniques: diffraction enhanced imaging and phase propagation imaging,” Phys. Med. Biol. 50(4), 709–724 (2005).
[CrossRef] [PubMed]

Lewis, R. A.

K. M. Pavlov, T. E. Gureyev, D. Paganin, Y. I. Nesterets, M. J. Morgan, and R. A. Lewis, “Linear systems with slowly varying transfer functions and their application to x-ray phase-contrast imaging,” J. Phys. D Appl. Phys. 37(19), 2746–2750 (2004).
[CrossRef]

Lindfors, K. K.

J. M. Boone, K. K. Lindfors, V. N. Cooper, and J. A. Seibert, “Scatter/primary in mammography: comprehensive results,” Med. Phys. 27(10), 2408–2416 (2000).
[CrossRef] [PubMed]

Longo, R.

E. Castelli, M. Tonutti, F. Arfelli, R. Longo, E. Quaia, L. Rigon, D. Sanabor, F. Zanconati, D. Dreossi, A. Abrami, E. Quai, P. Bregant, K. Casarin, V. Chenda, R. H. Menk, T. Rokvic, A. Vascotto, G. Tromba, and M. A. Cova, “Mammography with synchrotron radiation: first clinical experience with phase-detection technique,” Radiology 259(3), 684–694 (2011).
[CrossRef] [PubMed]

F. Arfelli, V. Bonvicini, A. Bravin, G. Cantatore, E. Castelli, L. Dalla Palma, M. Di Michiel, R. Longo, A. Olivo, S. Pani, D. Pontoni, P. Poropat, M. Prest, A. Rashevsky, G. Tromba, and A. Vacchi, “Mammography of a phantom and breast tissue with synchrotron radiation and a linear-array silicon detector,” Radiology 208(3), 709–715 (1998).
[PubMed]

Menk, R.

D. Chapman, W. Thomlinson, R. E. Johnston, D. Washburn, E. Pisano, N. Gmur, Z. Zhong, R. Menk, F. Arfelli, and D. Sayers, “Diffraction enhanced x-ray imaging,” Phys. Med. Biol. 42(11), 2015–2025 (1997).
[CrossRef] [PubMed]

Menk, R. H.

E. Castelli, M. Tonutti, F. Arfelli, R. Longo, E. Quaia, L. Rigon, D. Sanabor, F. Zanconati, D. Dreossi, A. Abrami, E. Quai, P. Bregant, K. Casarin, V. Chenda, R. H. Menk, T. Rokvic, A. Vascotto, G. Tromba, and M. A. Cova, “Mammography with synchrotron radiation: first clinical experience with phase-detection technique,” Radiology 259(3), 684–694 (2011).
[CrossRef] [PubMed]

Miller, P. R.

Mocella, V.

M. Sanchez del Rio, C. Ferrero, and V. Mocella, “Computer simulations of bent perfect crystal diffraction profiles,” Proc. SPIE 3152, 312–323 (1997), http://www.esrf.eu/UsersAndScience/Experiments/TBS/ SciSoft/xop2.3 .
[CrossRef]

Momose, A.

A. Momose, T. Takeda, Y. Itai, and K. Hirano, “Phase-contrast X-ray computed tomography for observing biological soft tissues,” Nat. Med. 2(4), 473–475 (1996).
[CrossRef] [PubMed]

Morgan, M. J.

K. M. Pavlov, T. E. Gureyev, D. Paganin, Y. I. Nesterets, M. J. Morgan, and R. A. Lewis, “Linear systems with slowly varying transfer functions and their application to x-ray phase-contrast imaging,” J. Phys. D Appl. Phys. 37(19), 2746–2750 (2004).
[CrossRef]

Mueller, B.

G. Schulz, T. Weitkamp, I. Zanette, F. Pfeiffer, F. Beckmann, C. David, S. Rutishauser, E. Reznikova, and B. Mueller, “High-resolution tomographic imaging of a human cerebellum: comparison of absorption and grating-based phase contrast,” J. R. Soc. Interface 7(53), 1665–1676 (2010).
[CrossRef] [PubMed]

Munro, P. R. T.

Mützel, E.

P. Coan, F. Bamberg, P. C. Diemoz, A. Bravin, K. Timpert, E. Mützel, J. G. Raya, S. Adam-Neumair, M. F. Reiser, and C. Glaser, “Characterization of osteoarthritic and normal human patella cartilage by computed tomography X-ray phase-contrast imaging: a feasibility study,” Invest. Radiol. 45(7), 437–444 (2010).
[PubMed]

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).
[CrossRef] [PubMed]

Y. I. Nesterets, P. Coan, T. E. Gureyev, A. Bravin, P. Cloetens, and S. W. Wilkins, “On qualitative and quantitative analysis in analyser-based imaging,” Acta Crystallogr. A 62(4), 296–308 (2006).
[CrossRef] [PubMed]

K. M. Pavlov, T. E. Gureyev, D. Paganin, Y. I. Nesterets, M. J. Morgan, and R. A. Lewis, “Linear systems with slowly varying transfer functions and their application to x-ray phase-contrast imaging,” J. Phys. D Appl. Phys. 37(19), 2746–2750 (2004).
[CrossRef]

Nöhammer, B.

C. David, B. Nöhammer, H. Solak, and E. Ziegler, “Differential X-ray phase contrast imaging using a shearing interferometer,” Appl. Phys. Lett. 81(17), 3287–3289 (2002).
[CrossRef]

Ohsawa, M.

R. Tanuma and M. Ohsawa, “Submicron-resolved X-ray topography using asymmetric-reflection magnifiers,” Spectrochim. Acta, Part B 59(10-11), 1549–1555 (2004).
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Olivo, A.

A. Olivo, K. Ignatyev, P. R. T. Munro, and R. D. Speller, “Noninterferometric phase-contrast images obtained with incoherent x-ray sources,” Appl. Opt. 50(12), 1765–1769 (2011).
[CrossRef] [PubMed]

F. Arfelli, V. Bonvicini, A. Bravin, G. Cantatore, E. Castelli, L. Dalla Palma, M. Di Michiel, R. Longo, A. Olivo, S. Pani, D. Pontoni, P. Poropat, M. Prest, A. Rashevsky, G. Tromba, and A. Vacchi, “Mammography of a phantom and breast tissue with synchrotron radiation and a linear-array silicon detector,” Radiology 208(3), 709–715 (1998).
[PubMed]

Paganin, D.

K. M. Pavlov, T. E. Gureyev, D. Paganin, Y. I. Nesterets, M. J. Morgan, and R. A. Lewis, “Linear systems with slowly varying transfer functions and their application to x-ray phase-contrast imaging,” J. Phys. D Appl. Phys. 37(19), 2746–2750 (2004).
[CrossRef]

Pagot, E.

P. Coan, E. Pagot, S. Fiedler, P. Cloetens, J. Baruchel, and A. Bravin, “Phase-contrast X-ray imaging combining free space propagation and Bragg diffraction,” J. Synchrotron Radiat. 12(2), 241–245 (2005).
[CrossRef] [PubMed]

E. Pagot, S. Fiedler, P. Cloetens, A. Bravin, P. Coan, K. Fezzaa, J. Baruchel, J. Härtwig, K. von Smitten, M. Leidenius, M. L. Karjalainen-Lindsberg, and J. Keyriläinen, “Quantitative comparison between two phase contrast techniques: diffraction enhanced imaging and phase propagation imaging,” Phys. Med. Biol. 50(4), 709–724 (2005).
[CrossRef] [PubMed]

Pani, S.

F. Arfelli, V. Bonvicini, A. Bravin, G. Cantatore, E. Castelli, L. Dalla Palma, M. Di Michiel, R. Longo, A. Olivo, S. Pani, D. Pontoni, P. Poropat, M. Prest, A. Rashevsky, G. Tromba, and A. Vacchi, “Mammography of a phantom and breast tissue with synchrotron radiation and a linear-array silicon detector,” Radiology 208(3), 709–715 (1998).
[PubMed]

Pavlov, K. M.

K. M. Pavlov, T. E. Gureyev, D. Paganin, Y. I. Nesterets, M. J. Morgan, and R. A. Lewis, “Linear systems with slowly varying transfer functions and their application to x-ray phase-contrast imaging,” J. Phys. D Appl. Phys. 37(19), 2746–2750 (2004).
[CrossRef]

Persliden, J.

J. Persliden and G. A. Carlsson, “Scatter rejection by air gaps in diagnostic radiology. Calculations using a Monte Carlo collision density method and consideration of molecular interference in coherent scattering,” Phys. Med. Biol. 42(1), 155–175 (1997).
[CrossRef] [PubMed]

Pfeiffer, F.

G. Schulz, T. Weitkamp, I. Zanette, F. Pfeiffer, F. Beckmann, C. David, S. Rutishauser, E. Reznikova, and B. Mueller, “High-resolution tomographic imaging of a human cerebellum: comparison of absorption and grating-based phase contrast,” J. R. Soc. Interface 7(53), 1665–1676 (2010).
[CrossRef] [PubMed]

F. Pfeiffer, T. Weitkamp, O. Bunk, and C. David, “Phase retrieval and differential phase-contrast imaging with low-brilliance X-ray sources,” Nat. Phys. 2(4), 258–261 (2006).
[CrossRef]

T. Weitkamp, A. Diaz, C. David, F. Pfeiffer, M. Stampanoni, P. Cloetens, and E. Ziegler, “X-ray phase imaging with a grating interferometer,” Opt. Express 13(16), 6296–6304 (2005).
[CrossRef] [PubMed]

Pisano, E.

D. Chapman, W. Thomlinson, R. E. Johnston, D. Washburn, E. Pisano, N. Gmur, Z. Zhong, R. Menk, F. Arfelli, and D. Sayers, “Diffraction enhanced x-ray imaging,” Phys. Med. Biol. 42(11), 2015–2025 (1997).
[CrossRef] [PubMed]

Pogany, A.

Pontoni, D.

F. Arfelli, V. Bonvicini, A. Bravin, G. Cantatore, E. Castelli, L. Dalla Palma, M. Di Michiel, R. Longo, A. Olivo, S. Pani, D. Pontoni, P. Poropat, M. Prest, A. Rashevsky, G. Tromba, and A. Vacchi, “Mammography of a phantom and breast tissue with synchrotron radiation and a linear-array silicon detector,” Radiology 208(3), 709–715 (1998).
[PubMed]

Poropat, P.

F. Arfelli, V. Bonvicini, A. Bravin, G. Cantatore, E. Castelli, L. Dalla Palma, M. Di Michiel, R. Longo, A. Olivo, S. Pani, D. Pontoni, P. Poropat, M. Prest, A. Rashevsky, G. Tromba, and A. Vacchi, “Mammography of a phantom and breast tissue with synchrotron radiation and a linear-array silicon detector,” Radiology 208(3), 709–715 (1998).
[PubMed]

Prest, M.

F. Arfelli, V. Bonvicini, A. Bravin, G. Cantatore, E. Castelli, L. Dalla Palma, M. Di Michiel, R. Longo, A. Olivo, S. Pani, D. Pontoni, P. Poropat, M. Prest, A. Rashevsky, G. Tromba, and A. Vacchi, “Mammography of a phantom and breast tissue with synchrotron radiation and a linear-array silicon detector,” Radiology 208(3), 709–715 (1998).
[PubMed]

Quai, E.

E. Castelli, M. Tonutti, F. Arfelli, R. Longo, E. Quaia, L. Rigon, D. Sanabor, F. Zanconati, D. Dreossi, A. Abrami, E. Quai, P. Bregant, K. Casarin, V. Chenda, R. H. Menk, T. Rokvic, A. Vascotto, G. Tromba, and M. A. Cova, “Mammography with synchrotron radiation: first clinical experience with phase-detection technique,” Radiology 259(3), 684–694 (2011).
[CrossRef] [PubMed]

Quaia, E.

E. Castelli, M. Tonutti, F. Arfelli, R. Longo, E. Quaia, L. Rigon, D. Sanabor, F. Zanconati, D. Dreossi, A. Abrami, E. Quai, P. Bregant, K. Casarin, V. Chenda, R. H. Menk, T. Rokvic, A. Vascotto, G. Tromba, and M. A. Cova, “Mammography with synchrotron radiation: first clinical experience with phase-detection technique,” Radiology 259(3), 684–694 (2011).
[CrossRef] [PubMed]

Rashevsky, A.

F. Arfelli, V. Bonvicini, A. Bravin, G. Cantatore, E. Castelli, L. Dalla Palma, M. Di Michiel, R. Longo, A. Olivo, S. Pani, D. Pontoni, P. Poropat, M. Prest, A. Rashevsky, G. Tromba, and A. Vacchi, “Mammography of a phantom and breast tissue with synchrotron radiation and a linear-array silicon detector,” Radiology 208(3), 709–715 (1998).
[PubMed]

Raya, J. G.

P. Coan, F. Bamberg, P. C. Diemoz, A. Bravin, K. Timpert, E. Mützel, J. G. Raya, S. Adam-Neumair, M. F. Reiser, and C. Glaser, “Characterization of osteoarthritic and normal human patella cartilage by computed tomography X-ray phase-contrast imaging: a feasibility study,” Invest. Radiol. 45(7), 437–444 (2010).
[PubMed]

Reiser, M. F.

P. Coan, F. Bamberg, P. C. Diemoz, A. Bravin, K. Timpert, E. Mützel, J. G. Raya, S. Adam-Neumair, M. F. Reiser, and C. Glaser, “Characterization of osteoarthritic and normal human patella cartilage by computed tomography X-ray phase-contrast imaging: a feasibility study,” Invest. Radiol. 45(7), 437–444 (2010).
[PubMed]

Reznikova, E.

G. Schulz, T. Weitkamp, I. Zanette, F. Pfeiffer, F. Beckmann, C. David, S. Rutishauser, E. Reznikova, and B. Mueller, “High-resolution tomographic imaging of a human cerebellum: comparison of absorption and grating-based phase contrast,” J. R. Soc. Interface 7(53), 1665–1676 (2010).
[CrossRef] [PubMed]

Rigon, L.

E. Castelli, M. Tonutti, F. Arfelli, R. Longo, E. Quaia, L. Rigon, D. Sanabor, F. Zanconati, D. Dreossi, A. Abrami, E. Quai, P. Bregant, K. Casarin, V. Chenda, R. H. Menk, T. Rokvic, A. Vascotto, G. Tromba, and M. A. Cova, “Mammography with synchrotron radiation: first clinical experience with phase-detection technique,” Radiology 259(3), 684–694 (2011).
[CrossRef] [PubMed]

Rokvic, T.

E. Castelli, M. Tonutti, F. Arfelli, R. Longo, E. Quaia, L. Rigon, D. Sanabor, F. Zanconati, D. Dreossi, A. Abrami, E. Quai, P. Bregant, K. Casarin, V. Chenda, R. H. Menk, T. Rokvic, A. Vascotto, G. Tromba, and M. A. Cova, “Mammography with synchrotron radiation: first clinical experience with phase-detection technique,” Radiology 259(3), 684–694 (2011).
[CrossRef] [PubMed]

Rutishauser, S.

G. Schulz, T. Weitkamp, I. Zanette, F. Pfeiffer, F. Beckmann, C. David, S. Rutishauser, E. Reznikova, and B. Mueller, “High-resolution tomographic imaging of a human cerebellum: comparison of absorption and grating-based phase contrast,” J. R. Soc. Interface 7(53), 1665–1676 (2010).
[CrossRef] [PubMed]

Sanabor, D.

E. Castelli, M. Tonutti, F. Arfelli, R. Longo, E. Quaia, L. Rigon, D. Sanabor, F. Zanconati, D. Dreossi, A. Abrami, E. Quai, P. Bregant, K. Casarin, V. Chenda, R. H. Menk, T. Rokvic, A. Vascotto, G. Tromba, and M. A. Cova, “Mammography with synchrotron radiation: first clinical experience with phase-detection technique,” Radiology 259(3), 684–694 (2011).
[CrossRef] [PubMed]

Sanchez del Rio, M.

M. Sanchez del Rio, C. Ferrero, and V. Mocella, “Computer simulations of bent perfect crystal diffraction profiles,” Proc. SPIE 3152, 312–323 (1997), http://www.esrf.eu/UsersAndScience/Experiments/TBS/ SciSoft/xop2.3 .
[CrossRef]

Sayers, D.

D. Chapman, W. Thomlinson, R. E. Johnston, D. Washburn, E. Pisano, N. Gmur, Z. Zhong, R. Menk, F. Arfelli, and D. Sayers, “Diffraction enhanced x-ray imaging,” Phys. Med. Biol. 42(11), 2015–2025 (1997).
[CrossRef] [PubMed]

Schelokov, I.

A. Snigirev, I. Snigireva, V. Kohn, S. Kuznetsov, and I. Schelokov, “On the possibility of x-ray phase contrast microimaging by coherent high-energy synchrotron radiation,” Rev. Sci. Instrum. 66(12), 5486–5492 (1995).
[CrossRef]

Schulz, G.

G. Schulz, T. Weitkamp, I. Zanette, F. Pfeiffer, F. Beckmann, C. David, S. Rutishauser, E. Reznikova, and B. Mueller, “High-resolution tomographic imaging of a human cerebellum: comparison of absorption and grating-based phase contrast,” J. R. Soc. Interface 7(53), 1665–1676 (2010).
[CrossRef] [PubMed]

Seibert, J. A.

J. M. Boone, K. K. Lindfors, V. N. Cooper, and J. A. Seibert, “Scatter/primary in mammography: comprehensive results,” Med. Phys. 27(10), 2408–2416 (2000).
[CrossRef] [PubMed]

Snigirev, A.

A. Snigirev, I. Snigireva, V. Kohn, S. Kuznetsov, and I. Schelokov, “On the possibility of x-ray phase contrast microimaging by coherent high-energy synchrotron radiation,” Rev. Sci. Instrum. 66(12), 5486–5492 (1995).
[CrossRef]

Snigireva, I.

A. Snigirev, I. Snigireva, V. Kohn, S. Kuznetsov, and I. Schelokov, “On the possibility of x-ray phase contrast microimaging by coherent high-energy synchrotron radiation,” Rev. Sci. Instrum. 66(12), 5486–5492 (1995).
[CrossRef]

Solak, H.

C. David, B. Nöhammer, H. Solak, and E. Ziegler, “Differential X-ray phase contrast imaging using a shearing interferometer,” Appl. Phys. Lett. 81(17), 3287–3289 (2002).
[CrossRef]

Speller, R. D.

Stampanoni, M.

Stevenson, A. W.

Takeda, T.

A. Momose, T. Takeda, Y. Itai, and K. Hirano, “Phase-contrast X-ray computed tomography for observing biological soft tissues,” Nat. Med. 2(4), 473–475 (1996).
[CrossRef] [PubMed]

Tanuma, R.

R. Tanuma and M. Ohsawa, “Submicron-resolved X-ray topography using asymmetric-reflection magnifiers,” Spectrochim. Acta, Part B 59(10-11), 1549–1555 (2004).
[CrossRef]

Teague, M. R.

Thomlinson, W.

D. Chapman, W. Thomlinson, R. E. Johnston, D. Washburn, E. Pisano, N. Gmur, Z. Zhong, R. Menk, F. Arfelli, and D. Sayers, “Diffraction enhanced x-ray imaging,” Phys. Med. Biol. 42(11), 2015–2025 (1997).
[CrossRef] [PubMed]

Timpert, K.

P. Coan, F. Bamberg, P. C. Diemoz, A. Bravin, K. Timpert, E. Mützel, J. G. Raya, S. Adam-Neumair, M. F. Reiser, and C. Glaser, “Characterization of osteoarthritic and normal human patella cartilage by computed tomography X-ray phase-contrast imaging: a feasibility study,” Invest. Radiol. 45(7), 437–444 (2010).
[PubMed]

Tonutti, M.

E. Castelli, M. Tonutti, F. Arfelli, R. Longo, E. Quaia, L. Rigon, D. Sanabor, F. Zanconati, D. Dreossi, A. Abrami, E. Quai, P. Bregant, K. Casarin, V. Chenda, R. H. Menk, T. Rokvic, A. Vascotto, G. Tromba, and M. A. Cova, “Mammography with synchrotron radiation: first clinical experience with phase-detection technique,” Radiology 259(3), 684–694 (2011).
[CrossRef] [PubMed]

Tromba, G.

E. Castelli, M. Tonutti, F. Arfelli, R. Longo, E. Quaia, L. Rigon, D. Sanabor, F. Zanconati, D. Dreossi, A. Abrami, E. Quai, P. Bregant, K. Casarin, V. Chenda, R. H. Menk, T. Rokvic, A. Vascotto, G. Tromba, and M. A. Cova, “Mammography with synchrotron radiation: first clinical experience with phase-detection technique,” Radiology 259(3), 684–694 (2011).
[CrossRef] [PubMed]

F. Arfelli, V. Bonvicini, A. Bravin, G. Cantatore, E. Castelli, L. Dalla Palma, M. Di Michiel, R. Longo, A. Olivo, S. Pani, D. Pontoni, P. Poropat, M. Prest, A. Rashevsky, G. Tromba, and A. Vacchi, “Mammography of a phantom and breast tissue with synchrotron radiation and a linear-array silicon detector,” Radiology 208(3), 709–715 (1998).
[PubMed]

Vacchi, A.

F. Arfelli, V. Bonvicini, A. Bravin, G. Cantatore, E. Castelli, L. Dalla Palma, M. Di Michiel, R. Longo, A. Olivo, S. Pani, D. Pontoni, P. Poropat, M. Prest, A. Rashevsky, G. Tromba, and A. Vacchi, “Mammography of a phantom and breast tissue with synchrotron radiation and a linear-array silicon detector,” Radiology 208(3), 709–715 (1998).
[PubMed]

Vascotto, A.

E. Castelli, M. Tonutti, F. Arfelli, R. Longo, E. Quaia, L. Rigon, D. Sanabor, F. Zanconati, D. Dreossi, A. Abrami, E. Quai, P. Bregant, K. Casarin, V. Chenda, R. H. Menk, T. Rokvic, A. Vascotto, G. Tromba, and M. A. Cova, “Mammography with synchrotron radiation: first clinical experience with phase-detection technique,” Radiology 259(3), 684–694 (2011).
[CrossRef] [PubMed]

von Smitten, K.

E. Pagot, S. Fiedler, P. Cloetens, A. Bravin, P. Coan, K. Fezzaa, J. Baruchel, J. Härtwig, K. von Smitten, M. Leidenius, M. L. Karjalainen-Lindsberg, and J. Keyriläinen, “Quantitative comparison between two phase contrast techniques: diffraction enhanced imaging and phase propagation imaging,” Phys. Med. Biol. 50(4), 709–724 (2005).
[CrossRef] [PubMed]

Washburn, D.

D. Chapman, W. Thomlinson, R. E. Johnston, D. Washburn, E. Pisano, N. Gmur, Z. Zhong, R. Menk, F. Arfelli, and D. Sayers, “Diffraction enhanced x-ray imaging,” Phys. Med. Biol. 42(11), 2015–2025 (1997).
[CrossRef] [PubMed]

Weitkamp, T.

P. C. Diemoz, P. Coan, I. Zanette, A. Bravin, S. Lang, C. Glaser, and T. Weitkamp, “A simplified approach for computed tomography with an X-ray grating interferometer,” Opt. Express 19(3), 1691–1698 (2011).
[CrossRef] [PubMed]

G. Schulz, T. Weitkamp, I. Zanette, F. Pfeiffer, F. Beckmann, C. David, S. Rutishauser, E. Reznikova, and B. Mueller, “High-resolution tomographic imaging of a human cerebellum: comparison of absorption and grating-based phase contrast,” J. R. Soc. Interface 7(53), 1665–1676 (2010).
[CrossRef] [PubMed]

F. Pfeiffer, T. Weitkamp, O. Bunk, and C. David, “Phase retrieval and differential phase-contrast imaging with low-brilliance X-ray sources,” Nat. Phys. 2(4), 258–261 (2006).
[CrossRef]

T. Weitkamp, A. Diaz, C. David, F. Pfeiffer, M. Stampanoni, P. Cloetens, and E. Ziegler, “X-ray phase imaging with a grating interferometer,” Opt. Express 13(16), 6296–6304 (2005).
[CrossRef] [PubMed]

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).
[CrossRef] [PubMed]

Y. I. Nesterets, P. Coan, T. E. Gureyev, A. Bravin, P. Cloetens, and S. W. Wilkins, “On qualitative and quantitative analysis in analyser-based imaging,” Acta Crystallogr. A 62(4), 296–308 (2006).
[CrossRef] [PubMed]

T. E. Gureyev and S. W. Wilkins, “Regimes of X-ray phase-contrast imaging with perfect crystals,” Nuovo Cimento D 19(2-4), 545–552 (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(6607), 335–338 (1996).
[CrossRef]

Zanconati, F.

E. Castelli, M. Tonutti, F. Arfelli, R. Longo, E. Quaia, L. Rigon, D. Sanabor, F. Zanconati, D. Dreossi, A. Abrami, E. Quai, P. Bregant, K. Casarin, V. Chenda, R. H. Menk, T. Rokvic, A. Vascotto, G. Tromba, and M. A. Cova, “Mammography with synchrotron radiation: first clinical experience with phase-detection technique,” Radiology 259(3), 684–694 (2011).
[CrossRef] [PubMed]

Zanette, I.

P. C. Diemoz, P. Coan, I. Zanette, A. Bravin, S. Lang, C. Glaser, and T. Weitkamp, “A simplified approach for computed tomography with an X-ray grating interferometer,” Opt. Express 19(3), 1691–1698 (2011).
[CrossRef] [PubMed]

G. Schulz, T. Weitkamp, I. Zanette, F. Pfeiffer, F. Beckmann, C. David, S. Rutishauser, E. Reznikova, and B. Mueller, “High-resolution tomographic imaging of a human cerebellum: comparison of absorption and grating-based phase contrast,” J. R. Soc. Interface 7(53), 1665–1676 (2010).
[CrossRef] [PubMed]

Zaumseil, P.

E. Förster, K. Goetz, and P. Zaumseil, “Double crystal diffractometry for the characterization of targets for laser fusion experiments,” Krist. Tech. 15(8), 937–945 (1980).
[CrossRef]

Zhong, Z.

D. Chapman, W. Thomlinson, R. E. Johnston, D. Washburn, E. Pisano, N. Gmur, Z. Zhong, R. Menk, F. Arfelli, and D. Sayers, “Diffraction enhanced x-ray imaging,” Phys. Med. Biol. 42(11), 2015–2025 (1997).
[CrossRef] [PubMed]

Ziegler, E.

T. Weitkamp, A. Diaz, C. David, F. Pfeiffer, M. Stampanoni, P. Cloetens, and E. Ziegler, “X-ray phase imaging with a grating interferometer,” Opt. Express 13(16), 6296–6304 (2005).
[CrossRef] [PubMed]

C. David, B. Nöhammer, H. Solak, and E. Ziegler, “Differential X-ray phase contrast imaging using a shearing interferometer,” Appl. Phys. Lett. 81(17), 3287–3289 (2002).
[CrossRef]

Acta Crystallogr. A

Y. I. Nesterets, P. Coan, T. E. Gureyev, A. Bravin, P. Cloetens, and S. W. Wilkins, “On qualitative and quantitative analysis in analyser-based imaging,” Acta Crystallogr. A 62(4), 296–308 (2006).
[CrossRef] [PubMed]

Appl. Opt.

Appl. Phys. Lett.

U. Bonse and M. Hart, “An X-ray interferometer,” Appl. Phys. Lett. 6(8), 155–156 (1965).
[CrossRef]

C. David, B. Nöhammer, H. Solak, and E. Ziegler, “Differential X-ray phase contrast imaging using a shearing interferometer,” Appl. Phys. Lett. 81(17), 3287–3289 (2002).
[CrossRef]

Invest. Radiol.

P. Coan, F. Bamberg, P. C. Diemoz, A. Bravin, K. Timpert, E. Mützel, J. G. Raya, S. Adam-Neumair, M. F. Reiser, and C. Glaser, “Characterization of osteoarthritic and normal human patella cartilage by computed tomography X-ray phase-contrast imaging: a feasibility study,” Invest. Radiol. 45(7), 437–444 (2010).
[PubMed]

J. Opt. Soc. Am.

J. Phys. D Appl. Phys.

A. Bravin, “Exploiting the X-ray refraction contrast with an analyser: the state of the art,” J. Phys. D Appl. Phys. 36(10A), A24–A29 (2003).
[CrossRef]

K. M. Pavlov, T. E. Gureyev, D. Paganin, Y. I. Nesterets, M. J. Morgan, and R. A. Lewis, “Linear systems with slowly varying transfer functions and their application to x-ray phase-contrast imaging,” J. Phys. D Appl. Phys. 37(19), 2746–2750 (2004).
[CrossRef]

J. R. Soc. Interface

G. Schulz, T. Weitkamp, I. Zanette, F. Pfeiffer, F. Beckmann, C. David, S. Rutishauser, E. Reznikova, and B. Mueller, “High-resolution tomographic imaging of a human cerebellum: comparison of absorption and grating-based phase contrast,” J. R. Soc. Interface 7(53), 1665–1676 (2010).
[CrossRef] [PubMed]

J. Synchrotron Radiat.

P. Coan, E. Pagot, S. Fiedler, P. Cloetens, J. Baruchel, and A. Bravin, “Phase-contrast X-ray imaging combining free space propagation and Bragg diffraction,” J. Synchrotron Radiat. 12(2), 241–245 (2005).
[CrossRef] [PubMed]

Krist. Tech.

E. Förster, K. Goetz, and P. Zaumseil, “Double crystal diffractometry for the characterization of targets for laser fusion experiments,” Krist. Tech. 15(8), 937–945 (1980).
[CrossRef]

Med. Phys.

J. M. Boone, K. K. Lindfors, V. N. Cooper, and J. A. Seibert, “Scatter/primary in mammography: comprehensive results,” Med. Phys. 27(10), 2408–2416 (2000).
[CrossRef] [PubMed]

Nat. Med.

A. Momose, T. Takeda, Y. Itai, and K. Hirano, “Phase-contrast X-ray computed tomography for observing biological soft tissues,” Nat. Med. 2(4), 473–475 (1996).
[CrossRef] [PubMed]

Nat. Phys.

F. Pfeiffer, T. Weitkamp, O. Bunk, and C. David, “Phase retrieval and differential phase-contrast imaging with low-brilliance X-ray sources,” Nat. Phys. 2(4), 258–261 (2006).
[CrossRef]

Nature

S. W. Wilkins, T. E. Gureyev, D. Gao, A. Pogany, and A. W. Stevenson, “Phase-contrast imaging using polychromatic hard X-rays,” Nature 384(6607), 335–338 (1996).
[CrossRef]

Nuovo Cimento D

T. E. Gureyev and S. W. Wilkins, “Regimes of X-ray phase-contrast imaging with perfect crystals,” Nuovo Cimento D 19(2-4), 545–552 (1997).
[CrossRef]

Opt. Express

Phys. Med. Biol.

J. Persliden and G. A. Carlsson, “Scatter rejection by air gaps in diagnostic radiology. Calculations using a Monte Carlo collision density method and consideration of molecular interference in coherent scattering,” Phys. Med. Biol. 42(1), 155–175 (1997).
[CrossRef] [PubMed]

E. Pagot, S. Fiedler, P. Cloetens, A. Bravin, P. Coan, K. Fezzaa, J. Baruchel, J. Härtwig, K. von Smitten, M. Leidenius, M. L. Karjalainen-Lindsberg, and J. Keyriläinen, “Quantitative comparison between two phase contrast techniques: diffraction enhanced imaging and phase propagation imaging,” Phys. Med. Biol. 50(4), 709–724 (2005).
[CrossRef] [PubMed]

D. Chapman, W. Thomlinson, R. E. Johnston, D. Washburn, E. Pisano, N. Gmur, Z. Zhong, R. Menk, F. Arfelli, and D. Sayers, “Diffraction enhanced x-ray imaging,” Phys. Med. Biol. 42(11), 2015–2025 (1997).
[CrossRef] [PubMed]

Proc. SPIE

M. Sanchez del Rio, C. Ferrero, and V. Mocella, “Computer simulations of bent perfect crystal diffraction profiles,” Proc. SPIE 3152, 312–323 (1997), http://www.esrf.eu/UsersAndScience/Experiments/TBS/ SciSoft/xop2.3 .
[CrossRef]

Radiology

F. Arfelli, V. Bonvicini, A. Bravin, G. Cantatore, E. Castelli, L. Dalla Palma, M. Di Michiel, R. Longo, A. Olivo, S. Pani, D. Pontoni, P. Poropat, M. Prest, A. Rashevsky, G. Tromba, and A. Vacchi, “Mammography of a phantom and breast tissue with synchrotron radiation and a linear-array silicon detector,” Radiology 208(3), 709–715 (1998).
[PubMed]

E. Castelli, M. Tonutti, F. Arfelli, R. Longo, E. Quaia, L. Rigon, D. Sanabor, F. Zanconati, D. Dreossi, A. Abrami, E. Quai, P. Bregant, K. Casarin, V. Chenda, R. H. Menk, T. Rokvic, A. Vascotto, G. Tromba, and M. A. Cova, “Mammography with synchrotron radiation: first clinical experience with phase-detection technique,” Radiology 259(3), 684–694 (2011).
[CrossRef] [PubMed]

Rev. Sci. Instrum.

A. Snigirev, I. Snigireva, V. Kohn, S. Kuznetsov, and I. Schelokov, “On the possibility of x-ray phase contrast microimaging by coherent high-energy synchrotron radiation,” Rev. Sci. Instrum. 66(12), 5486–5492 (1995).
[CrossRef]

Spectrochim. Acta, Part B

R. Tanuma and M. Ohsawa, “Submicron-resolved X-ray topography using asymmetric-reflection magnifiers,” Spectrochim. Acta, Part B 59(10-11), 1549–1555 (2004).
[CrossRef]

Other

A. L. Evans, The Evaluation of Medical Images (Adam Hilger Ltd, Bristol, 1981).

M. Born and E. Wolf, Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light (Cambridge University Press, Cambridge, 1999).

S. Webb, The Physics of Medical Imaging (Institute of Physics Publishing, Bristol, 1988).

T. Matsushita and H. Hashizume, “X-Ray monochromators,” in Handbook on Synchrotron Radiation, E. Koch, ed. (North Holland Publishing Company, New York, 1983), pp. 261–314.

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Figures (3)

Fig. 1
Fig. 1

(a) Profile of a pure phase object. (b) Typical corresponding PBI signal that can be calculated from Eq. (1). (c) Typical ABI / GI signal that can be calculated from Eqs. (4 and 6). Values have been normalized by the average background intensity Iback. Poissonian noise has been added to the theoretical profiles. Maximum (Imax) and minimum (Imin) values for the intensity at the edge, as well as the integral of the edge signal, are also drawn in the figure.

Fig. 2
Fig. 2

(a) Profile of the theoretical RC, in the case of monochromator and analyzer in Si(111) Bragg reflection geometry, (b) profile of its first derivative, and (c) ratio between the first derivative and the square root of the RC. All quantities have been calculated with XOP [30] for an X-ray energy of 26 keV.

Fig. 3
Fig. 3

(a) Intensity profile in GI as a function of the gratings relative displacement, for a visibility of 30%, (b) Normalized FoM values calculated by using Eq. (24) in the range [–π/2, 3/2π] for the quantity ψ + 2πyG/p2. The different positions on the fringe profile are indicated.

Tables (1)

Tables Icon

Table 1 Peak edge FoM and integral edge FoM calculated in PBI, ABI and GI techniques.

Equations (36)

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

I PBI = M 2 I 0 T( 1 λd 2π 2 ϕ )
I ABI = M 2 M asym I 0 TR( θ an +Δ θ y )
I ABI M 2 M asym I 0 T[ R( θ an )+R'( θ an )Δ θ y ]
I GI M 2 I 0 T GR T[ G( y G )+G'( y G )Δ θ y ]
P sys = ( 2π σ sys,x σ sys,y ) 1 exp[ x 2 / 2 σ sys,y 2 y 2 / 2 σ sys,y 2 ]
σ sysPBI,x,y 2 = ( M1 ) 2 M 2 σ src,x,y 2 + M 2 σ det,x,y 2
σ sysABI,x 2 = ( M1 ) 2 M 2 σ src,x 2 + M 2 σ det,x 2
σ sysABI,y 2 = ( M1 ) 2 M 2 σ src,y 2 + M 2 M asym 2 σ det,y 2 + M 2 M asym 2 σ an,y 2
SN R area A ( I obj I back ) st d 2 ( I obj )+st d 2 ( I back ) A ( I obj I back ) I obj + I back
FoM SNR D
D= K dose I 0
Fo M area = A ( I obj I back ) K dose I 0 ( I obj + I back )
SN R edgepeak A ( I max I min ) 2st d 2 ( I back ) A ( I max I min ) 2 I back
Fo M edgepeak = A ( I max I min ) 2 K dose I 0 I back
SN R int.edge a +a dy x 0 x 1 dx | I( x,y ) I back | 2 a +a dy x 0 x 1 dx I back
Fo M int.edge = a +a dy x 0 x 1 dx | I( x,y ) I back | 2 K dose I 0 a +a dy x 0 x 1 dx I back
Fo M PBI,abs = A [ exp( dz μ obj )exp( dz μ back ) ] K dose ( exp( dz μ obj )+exp( dz μ back ) ) A ( dz μ back dz μ obj ) K dose ( 2 dz μ back dz μ obj )
Fo M ABI,abs = R( θ an ) Fo M PBI,abs
Fo M GI,abs = T GR G( y G ) Fo M PBI,abs
Fo M PBI,peak λd A 2π 2 K dose ( max 2 ϕ min 2 ϕ )
Fo M ABI,peak A 2 K dose R'( θ an ) R( θ an ) Δ θ y
Fo M GI,peak A T GR G'( y G ;Δ θ y =0 ) 2 K dose G( y G ;Δ θ y =0 ) Δ θ y = 2 π A T GR K dose V d Talbot p 2 cos( ψ+ 2π p 2 y G ) 1+sin( ψ+ 2π p 2 y G ) Δ θ y
Fo M ABI,peak Fo M GI,peak = R'( θ an ) R( θ an ) G( y G ;Δ θ y =0 ) G'( y G ;Δ θ y =0 ) T GR
Fo M int.edge = L x a +a dy| I( y ) I back | 4 K dose a I 0 I back
ϕ=( H P obj )( y ) ϕ step
I( y ) I back I 0 λd 2π 2 ϕ( y ) P sys,PBI ( x,y )
2 ϕ P sys,PBI = ϕ step δ( y ) y ( P obj P sys )= ϕ step yexp[ y 2 / 2 σ PBI 2 ] ( σ PBI 3 2π ) 1
Fo M PBI,int.edge = dλ L x 2π σ PBI 2π K dose | ϕ step | 1 a [ 1exp( a 2 2 σ PBI 2 ) ]
Fo M PBI,int.edge 3.88 10 2 dλ L x σ PBI σ PBI K dose | ϕ step |
I( x,y ) I back = I 0 R'( θ an ) λ 2π ϕ( y ) y P sys,ABI ( x,y )
ϕ y P sys,ABI = ϕ step δ( y )( P obj P sys )= ϕ step ( 2π σ ABI ) 1 exp[ y 2 / 2 σ ABI 2 ]
Fo M ABI,int.edge = λ L x R'( θ an )| ϕ step | 4π K dose R( θ an ) ( 1 a a +a ( 2π σ ABI ) 1 exp[ y 2 / 2 σ ABI 2 ] )
Fo M ABI,int.edge 5.37 10 2 λ L x σ ABI K dose R'( θ an ) R( θ an ) | ϕ step |
Fo M GI,int.edge 5.37 10 2 λ L x T GR σ GI K dose G'( y G ) G( y G ) | ϕ step |
Fo M ABI,int.edge Fo M GI,int.edge = σ GI σ ABI R'( θ an ) R( θ an ) G( y G ) T GR G'( y G )
Fo M ABI,int.edge Fo M PBI,int.edge 1.38 R'( θ an ) R( θ an ) 1 d σ PBI σ PBI σ ABI

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