Abstract

We present a theoretical and experimental comparison of three X-ray phase-contrast techniques: propagation-based imaging, analyzer-based imaging and grating interferometry. The signal-to-noise ratio and the figure of merit are quantitatively compared for the three techniques on the same phantoms and using the same X-ray source and detector. Principal dependencies of the signal upon the numerous acquisition parameters, the spatial resolution and X-ray energy are discussed in detail. The sensitivity of each technique, in terms of the smallest detectable phase shift, is also evaluated.

© 2012 OSA

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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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2012 (1)

2011 (4)

2010 (1)

V. Revol, C. Kottler, R. Kaufmann, U. Straumann, and C. Urban, “Noise analysis of grating-based x-ray differential phase contrast imaging,” Rev. Sci. Instrum. 81(7), 073709 (2010).
[CrossRef] [PubMed]

2008 (2)

2006 (4)

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]

P. Coan, A. Peterzol, S. Fiedler, C. Ponchut, J. C. Labiche, and A. Bravin, “Evaluation of imaging performance of a taper optics CCD ҅FReLoN’ camera designed for medical imaging,” J. Synchrotron Radiat. 13(3), 260–270 (2006).
[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]

D. Shimao, H. Sugiyama, T. Kunisada, and M. Ando, “Articular cartilage depicted at optimized angular position of Laue angular analyzer by X-ray dark-field imaging,” Appl. Radiat. Isot. 64(8), 868–874 (2006).
[CrossRef] [PubMed]

2005 (3)

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]

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 (1)

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]

2002 (1)

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]

1997 (3)

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

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

M. Sanchez del Rio, C. Ferrero, and V. Mocella, “Computer simulations of bent perfect crystal diffraction profiles,” Proc. SPIE 3151, 312–323 (1997), http://www.esrf.eu/UsersAndScience/Experiments/TBS/ SciSoft/xop2.3 .
[CrossRef]

1996 (2)

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 (2)

T. Davis, D. Gao, T. Gureyev, A. Stevenson, and S. Wilkins, “Phase-contrast imaging of weakly absorbing materials using hard X-rays,” Nature 373(6515), 595–598 (1995).
[CrossRef]

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]

1993 (1)

B. L. Henke, E. M. Gullikson, and J. C. Davis, “X-ray interactions: photoabsorption, scattering, transmission, and reflection at E=50-30000 eV, Z=1-92,” At. Data Nucl. Data Tables 54(2), 181–342 (1993).
[CrossRef]

1982 (1)

Allison, B. J.

M. J. Kitchen, D. M. Paganin, K. Uesugi, B. J. Allison, R. A. Lewis, S. B. Hooper, and K. M. Pavlov, “Phase contrast image segmentation using a Laue analyser crystal,” Phys. Med. Biol. 56(3), 515–534 (2011).
[CrossRef] [PubMed]

Ando, M.

D. Shimao, H. Sugiyama, T. Kunisada, and M. Ando, “Articular cartilage depicted at optimized angular position of Laue angular analyzer by X-ray dark-field imaging,” Appl. Radiat. Isot. 64(8), 868–874 (2006).
[CrossRef] [PubMed]

Arfelli, F.

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

Bravin, A.

P. C. Diemoz, A. Bravin, and P. Coan, “Theoretical comparison of three X-ray phase-contrast imaging techniques: propagation-based imaging, analyzer-based imaging and grating interferometry,” Opt. Express 20(3), 2789–2805 (2012).
[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]

P. Coan, A. Peterzol, S. Fiedler, C. Ponchut, J. C. Labiche, and A. Bravin, “Evaluation of imaging performance of a taper optics CCD ҅FReLoN’ camera designed for medical imaging,” J. Synchrotron Radiat. 13(3), 260–270 (2006).
[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]

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]

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]

Chapman, D.

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

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]

Coan, P.

P. C. Diemoz, A. Bravin, and P. Coan, “Theoretical comparison of three X-ray phase-contrast imaging techniques: propagation-based imaging, analyzer-based imaging and grating interferometry,” Opt. Express 20(3), 2789–2805 (2012).
[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]

P. Coan, A. Peterzol, S. Fiedler, C. Ponchut, J. C. Labiche, and A. Bravin, “Evaluation of imaging performance of a taper optics CCD ҅FReLoN’ camera designed for medical imaging,” J. Synchrotron Radiat. 13(3), 260–270 (2006).
[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]

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]

David, C.

P. Modregger, B. R. Pinzer, T. Thüring, S. Rutishauser, C. David, and M. Stampanoni, “Sensitivity of X-ray grating interferometry,” Opt. Express 19(19), 18324–18338 (2011).
[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]

Davis, J. C.

B. L. Henke, E. M. Gullikson, and J. C. Davis, “X-ray interactions: photoabsorption, scattering, transmission, and reflection at E=50-30000 eV, Z=1-92,” At. Data Nucl. Data Tables 54(2), 181–342 (1993).
[CrossRef]

Davis, T.

T. Davis, D. Gao, T. Gureyev, A. Stevenson, and S. Wilkins, “Phase-contrast imaging of weakly absorbing materials using hard X-rays,” Nature 373(6515), 595–598 (1995).
[CrossRef]

Diaz, A.

Diemoz, P. C.

Ferrero, C.

M. Sanchez del Rio, C. Ferrero, and V. Mocella, “Computer simulations of bent perfect crystal diffraction profiles,” Proc. SPIE 3151, 312–323 (1997), http://www.esrf.eu/UsersAndScience/Experiments/TBS/ SciSoft/xop2.3 .
[CrossRef]

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.

P. Coan, A. Peterzol, S. Fiedler, C. Ponchut, J. C. Labiche, and A. Bravin, “Evaluation of imaging performance of a taper optics CCD ҅FReLoN’ camera designed for medical imaging,” J. Synchrotron Radiat. 13(3), 260–270 (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]

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]

T. Davis, D. Gao, T. Gureyev, A. Stevenson, and S. Wilkins, “Phase-contrast imaging of weakly absorbing materials using hard X-rays,” Nature 373(6515), 595–598 (1995).
[CrossRef]

Glaser, C.

Gmür, N.

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

Gullikson, E. M.

B. L. Henke, E. M. Gullikson, and J. C. Davis, “X-ray interactions: photoabsorption, scattering, transmission, and reflection at E=50-30000 eV, Z=1-92,” At. Data Nucl. Data Tables 54(2), 181–342 (1993).
[CrossRef]

Gureyev, T.

T. Davis, D. Gao, T. Gureyev, A. Stevenson, and S. Wilkins, “Phase-contrast imaging of weakly absorbing materials using hard X-rays,” Nature 373(6515), 595–598 (1995).
[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).
[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]

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

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]

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).
[CrossRef] [PubMed]

Henke, B. L.

B. L. Henke, E. M. Gullikson, and J. C. Davis, “X-ray interactions: photoabsorption, scattering, transmission, and reflection at E=50-30000 eV, Z=1-92,” At. Data Nucl. Data Tables 54(2), 181–342 (1993).
[CrossRef]

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]

Hooper, S. B.

M. J. Kitchen, D. M. Paganin, K. Uesugi, B. J. Allison, R. A. Lewis, S. B. Hooper, and K. M. Pavlov, “Phase contrast image segmentation using a Laue analyser crystal,” Phys. Med. Biol. 56(3), 515–534 (2011).
[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).
[CrossRef] [PubMed]

Johnston, R. E.

D. Chapman, W. Thomlinson, R. E. Johnston, D. Washburn, E. Pisano, N. Gmür, 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).
[CrossRef] [PubMed]

Kaufmann, R.

V. Revol, C. Kottler, R. Kaufmann, U. Straumann, and C. Urban, “Noise analysis of grating-based x-ray differential phase contrast imaging,” Rev. Sci. Instrum. 81(7), 073709 (2010).
[CrossRef] [PubMed]

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).
[CrossRef] [PubMed]

Kitchen, M. J.

M. J. Kitchen, D. M. Paganin, K. Uesugi, B. J. Allison, R. A. Lewis, S. B. Hooper, and K. M. Pavlov, “Phase contrast image segmentation using a Laue analyser crystal,” Phys. Med. Biol. 56(3), 515–534 (2011).
[CrossRef] [PubMed]

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).
[CrossRef]

Kottler, C.

V. Revol, C. Kottler, R. Kaufmann, U. Straumann, and C. Urban, “Noise analysis of grating-based x-ray differential phase contrast imaging,” Rev. Sci. Instrum. 81(7), 073709 (2010).
[CrossRef] [PubMed]

Kunisada, T.

D. Shimao, H. Sugiyama, T. Kunisada, and M. Ando, “Articular cartilage depicted at optimized angular position of Laue angular analyzer by X-ray dark-field imaging,” Appl. Radiat. Isot. 64(8), 868–874 (2006).
[CrossRef] [PubMed]

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).
[CrossRef]

Labiche, J. C.

P. Coan, A. Peterzol, S. Fiedler, C. Ponchut, J. C. Labiche, and A. Bravin, “Evaluation of imaging performance of a taper optics CCD ҅FReLoN’ camera designed for medical imaging,” J. Synchrotron Radiat. 13(3), 260–270 (2006).
[CrossRef] [PubMed]

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.

M. J. Kitchen, D. M. Paganin, K. Uesugi, B. J. Allison, R. A. Lewis, S. B. Hooper, and K. M. Pavlov, “Phase contrast image segmentation using a Laue analyser crystal,” Phys. Med. Biol. 56(3), 515–534 (2011).
[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]

Menk, R.

D. Chapman, W. Thomlinson, R. E. Johnston, D. Washburn, E. Pisano, N. Gmür, Z. Zhong, R. Menk, F. Arfelli, and D. Sayers, “Diffraction enhanced x-ray imaging,” Phys. Med. Biol. 42(11), 2015–2025 (1997).
[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 3151, 312–323 (1997), http://www.esrf.eu/UsersAndScience/Experiments/TBS/ SciSoft/xop2.3 .
[CrossRef]

Modregger, P.

Momose, A.

W. Yashiro, Y. Takeda, and A. Momose, “Efficiency of capturing a phase image using cone-beam x-ray Talbot interferometry,” J. Opt. Soc. Am. A 25(8), 2025–2039 (2008).
[CrossRef] [PubMed]

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]

Munro, P. R. T.

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]

Olivo, A.

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]

Paganin, D. M.

M. J. Kitchen, D. M. Paganin, K. Uesugi, B. J. Allison, R. A. Lewis, S. B. Hooper, and K. M. Pavlov, “Phase contrast image segmentation using a Laue analyser crystal,” Phys. Med. Biol. 56(3), 515–534 (2011).
[CrossRef] [PubMed]

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]

Pavlov, K. M.

M. J. Kitchen, D. M. Paganin, K. Uesugi, B. J. Allison, R. A. Lewis, S. B. Hooper, and K. M. Pavlov, “Phase contrast image segmentation using a Laue analyser crystal,” Phys. Med. Biol. 56(3), 515–534 (2011).
[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]

Peterzol, A.

P. Coan, A. Peterzol, S. Fiedler, C. Ponchut, J. C. Labiche, and A. Bravin, “Evaluation of imaging performance of a taper optics CCD ҅FReLoN’ camera designed for medical imaging,” J. Synchrotron Radiat. 13(3), 260–270 (2006).
[CrossRef] [PubMed]

Pfeiffer, F.

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]

Pinzer, B. R.

Pisano, E.

D. Chapman, W. Thomlinson, R. E. Johnston, D. Washburn, E. Pisano, N. Gmür, 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.

Ponchut, C.

P. Coan, A. Peterzol, S. Fiedler, C. Ponchut, J. C. Labiche, and A. Bravin, “Evaluation of imaging performance of a taper optics CCD ҅FReLoN’ camera designed for medical imaging,” J. Synchrotron Radiat. 13(3), 260–270 (2006).
[CrossRef] [PubMed]

Revol, V.

V. Revol, C. Kottler, R. Kaufmann, U. Straumann, and C. Urban, “Noise analysis of grating-based x-ray differential phase contrast imaging,” Rev. Sci. Instrum. 81(7), 073709 (2010).
[CrossRef] [PubMed]

Rutishauser, S.

Sanchez del Rio, M.

M. Sanchez del Rio, C. Ferrero, and V. Mocella, “Computer simulations of bent perfect crystal diffraction profiles,” Proc. SPIE 3151, 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. Gmür, 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]

Shimao, D.

D. Shimao, H. Sugiyama, T. Kunisada, and M. Ando, “Articular cartilage depicted at optimized angular position of Laue angular analyzer by X-ray dark-field imaging,” Appl. Radiat. Isot. 64(8), 868–874 (2006).
[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.

T. Davis, D. Gao, T. Gureyev, A. Stevenson, and S. Wilkins, “Phase-contrast imaging of weakly absorbing materials using hard X-rays,” Nature 373(6515), 595–598 (1995).
[CrossRef]

Stevenson, A. W.

Straumann, U.

V. Revol, C. Kottler, R. Kaufmann, U. Straumann, and C. Urban, “Noise analysis of grating-based x-ray differential phase contrast imaging,” Rev. Sci. Instrum. 81(7), 073709 (2010).
[CrossRef] [PubMed]

Sugiyama, H.

D. Shimao, H. Sugiyama, T. Kunisada, and M. Ando, “Articular cartilage depicted at optimized angular position of Laue angular analyzer by X-ray dark-field imaging,” Appl. Radiat. Isot. 64(8), 868–874 (2006).
[CrossRef] [PubMed]

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]

Takeda, Y.

Teague, M. R.

Thomlinson, W.

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

Thüring, T.

Uesugi, K.

M. J. Kitchen, D. M. Paganin, K. Uesugi, B. J. Allison, R. A. Lewis, S. B. Hooper, and K. M. Pavlov, “Phase contrast image segmentation using a Laue analyser crystal,” Phys. Med. Biol. 56(3), 515–534 (2011).
[CrossRef] [PubMed]

Urban, C.

V. Revol, C. Kottler, R. Kaufmann, U. Straumann, and C. Urban, “Noise analysis of grating-based x-ray differential phase contrast imaging,” Rev. Sci. Instrum. 81(7), 073709 (2010).
[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. Gmür, 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.

Wilkins, S.

T. Davis, D. Gao, T. Gureyev, A. Stevenson, and S. Wilkins, “Phase-contrast imaging of weakly absorbing materials using hard X-rays,” Nature 373(6515), 595–598 (1995).
[CrossRef]

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 Cim. 19D, 545–552 (1997).

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]

Yashiro, W.

Zanette, I.

Zhong, Z.

D. Chapman, W. Thomlinson, R. E. Johnston, D. Washburn, E. Pisano, N. Gmür, 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 (1)

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. (1)

Appl. Phys. Lett. (1)

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]

Appl. Radiat. Isot. (1)

D. Shimao, H. Sugiyama, T. Kunisada, and M. Ando, “Articular cartilage depicted at optimized angular position of Laue angular analyzer by X-ray dark-field imaging,” Appl. Radiat. Isot. 64(8), 868–874 (2006).
[CrossRef] [PubMed]

At. Data Nucl. Data Tables (1)

B. L. Henke, E. M. Gullikson, and J. C. Davis, “X-ray interactions: photoabsorption, scattering, transmission, and reflection at E=50-30000 eV, Z=1-92,” At. Data Nucl. Data Tables 54(2), 181–342 (1993).
[CrossRef]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (1)

J. Phys. D Appl. Phys. (1)

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. Synchrotron Radiat. (2)

P. Coan, A. Peterzol, S. Fiedler, C. Ponchut, J. C. Labiche, and A. Bravin, “Evaluation of imaging performance of a taper optics CCD ҅FReLoN’ camera designed for medical imaging,” J. Synchrotron Radiat. 13(3), 260–270 (2006).
[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]

Nat. Med. (1)

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. (1)

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 (2)

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]

T. Davis, D. Gao, T. Gureyev, A. Stevenson, and S. Wilkins, “Phase-contrast imaging of weakly absorbing materials using hard X-rays,” Nature 373(6515), 595–598 (1995).
[CrossRef]

Nuovo Cim. (1)

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

Opt. Express (5)

Phys. Med. Biol. (3)

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]

M. J. Kitchen, D. M. Paganin, K. Uesugi, B. J. Allison, R. A. Lewis, S. B. Hooper, and K. M. Pavlov, “Phase contrast image segmentation using a Laue analyser crystal,” Phys. Med. Biol. 56(3), 515–534 (2011).
[CrossRef] [PubMed]

D. Chapman, W. Thomlinson, R. E. Johnston, D. Washburn, E. Pisano, N. Gmür, 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 (1)

M. Sanchez del Rio, C. Ferrero, and V. Mocella, “Computer simulations of bent perfect crystal diffraction profiles,” Proc. SPIE 3151, 312–323 (1997), http://www.esrf.eu/UsersAndScience/Experiments/TBS/ SciSoft/xop2.3 .
[CrossRef]

Rev. Sci. Instrum. (2)

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]

V. Revol, C. Kottler, R. Kaufmann, U. Straumann, and C. Urban, “Noise analysis of grating-based x-ray differential phase contrast imaging,” Rev. Sci. Instrum. 81(7), 073709 (2010).
[CrossRef] [PubMed]

Other (3)

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.

E. Pagot, “Quantitative comparison between two phase contrast techniques for mammography,” PhD Thesis, Université Joseph Fourier, Grenoble (2004).

M. Bech, “X-ray imaging with a grating interferometer,” PhD Thesis, University of Copenhagen (2009).

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Figures (9)

Fig. 1
Fig. 1

(a) Theoretical ABI RC in the case of monochromator and analyzer in Si(333) Bragg reflection geometry, and for an X-ray energy of 26 keV. (b) First derivative of the RC, (c) ratio between the first derivative of the RC and its square root, (d) second derivative of the RC. The RC values have been calculated with XOP [17].

Fig. 2
Fig. 2

(a) Plot of the function G in the GI technique, for a second grating period p2 = 2 µm, a Talbot distance dTalbot = 125 mm, a visibility V = 0.34 and a shift of the sinusoid ψ = 0. (b) First derivative of G, (c) ratio between the first derivative of G and its square root, (d) second derivative of G.

Fig. 3
Fig. 3

PB images acquired at 3 different sample-to-detector distances: expanded views in small regions of interest in the 350 µm diameter nylon wire and in the groove in the Lucite phantom.

Fig. 4
Fig. 4

Calculated values of the figure of merit (FoM) of the PBI edge signal for three different objects: (a) the 200 μm diameter nylon wire, (b) the 350 µm diameter nylon wire, (c) the groove in the Lucite phantom. Values at both the top and bottom edges are reported.

Fig. 5
Fig. 5

AB images acquired at 5 different orientations of the analyzer crystal: expanded views in small regions of interest in the 350 µm diameter nylon wire and in the groove in the Lucite phantom.

Fig. 6
Fig. 6

Calculated values of the figure of merit (FoM) of the ABI edge signal for three different objects: (a) the 200 μm diameter nylon wire, (b) the 350 µm diameter nylon wire, (c) the groove in Lucite. Values at both the top and bottom edges are reported. The zero on the x axis corresponds to the Bragg condition (top) for the analyzer crystal.

Fig. 7
Fig. 7

GI images acquired at 9 different relative positions of the two gratings along one fringe period: expanded view in small regions of interest in the 350 µm diameter nylon wire and in the groove in the Lucite phantom.

Fig. 8
Fig. 8

Calculated values of the figure of merit (FoM) of the GI edge signal for three different objects: (a) the 200 μm diameter nylon wire, (b) the 350 µm diameter nylon wire, (c) the groove in the Lucite phantom. Values at both the top and bottom edges are reported.

Fig. 9
Fig. 9

Refraction angle maps calculated from AB images (with “Gaussian Curve Fitting” algorithm) and from GI images (phase-stepping method). a) 350 μm diameter nylon wire in ABI and b) in GI, c) vertical profiles across the wire, d) groove in Lucite in ABI and e) in GI, f) vertical profiles across the groove.

Tables (2)

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Table 1 Measured values of the skin dose in air for the images acquisition

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Table 2 Peak edge FoMs calculated for the refraction signal

Equations (27)

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I PBI = I 0 T( 1 λd 2π 2 ϕ )
I ABI = I 0 TR( θ an +Δ θ y )
I GI = I 0 T GR T[ 1+Vsin( ψ+ 2π p 2 y G + 2π S Δ θ y ) ]= I 0 T GR TG( y G ;Δ θ y )
V= 8 π 2 e 2 π 2 ( σ proj / p 2 ) 2 n=1 1 ( 2n1 ) 2 e 2 π 2 ( σ proj / p 2 ) 2 ( 4 n 2 4n ) 8 π 2 e 2 π 2 ( σ proj / p 2 ) 2
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,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+Vsin( ψ+ 2π p 2 y G ) Δ θ y
f= 1+Vsin( ψ+ 2π p 2 y G ) | cos( ψ+ 2π p 2 y G ) |
sin( ψ+ 2π p 2 y G )= 1 V ( 1 1 V 2 )
f min = V 2 1 V 2 4 V 2 1+ 1 V 2
Fo M GI,peak,max =2π A T GR K dose d Talbot p 2 V 2 1+ 1 V 2 1 V 2 4 Δ θ y
Fo M ABI,peak,top A 2 2 K dose R ( θ an ) R( θ an ) ( Δ θ y ) 2
Fo M GI,peak,topandbottom A T GR 2 2 K dose G ( y G ;Δ θ y =0 ) G( y G ;Δ θ y =0 ) ( Δ θ y ) 2
Fo M PBI,int.edge =3.88 10 2 dλ L x σ PBI σ PBI K dose ϕ step
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
| SN R PBI,int.edge |= I 0 K dose | Fo M PBI,int.edge |=3.88 10 2 dλ L x I 0 σ PBI σ PBI | ϕ step | K SNR
| ϕ step | ϕ min,PBI 25.76 K SNR σ PBI σ PBI dλ L x I 0
| ϕ step | ϕ min,ABI =18.62 K SNR R( θ an ) σ ABI λ| R ( θ an ) | L x I 0
| ϕ step | ϕ min,GI =18.62 K SNR G( y G ) σ GI λ| G ( y G ) | L x T GR I 0
ϕ min,PBI ϕ min,ABI =1.38 σ PBI σ PBI d σ ABI | R ( θ an ) | R( θ an ) = Fo M ABI,int.edge Fo M PBI,int.edge
ϕ min,GI ϕ min,ABI = σ GI T GR σ ABI G( y G ) | G ( y G ) | | R ( θ an ) | R( θ an ) = Fo M ABI,int.edge Fo M GI,int.edge
ϕ min,GI ϕ min,ABI = p 2 2 2 π T GR d Talbot 1 V 2 4 V 2 1+ 1 V 2 | R ( θ an,max ) | R( θ an,max )

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