Abstract

In this paper we present a single-image phase retrieval algorithm for multi-material samples, developed for the edge illumination (EI) X-ray phase contrast imaging method. The theoretical derivation is provided, along with any assumptions made. The algorithm is evaluated quantitatively using both simulated and experimental results from a computed tomography (CT) scan using the EI laboratory implementation. Qualitative CT results are provided for a biological sample containing both bone and soft-tissue. Using a single EI image per projection and knowledge of the complex refractive index, the algorithm can accurately retrieve the interface between a given pair of materials. A composite CT slice can be created by splicing together multiple CT reconstructions, each retrieved for a different pair of materials.

Published by The Optical Society under the terms of the Creative Commons Attribution 4.0 License. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.

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    [Crossref]
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2017 (1)

M. Topperwien, M. Krenkel, D. Vincenz, F. Stober, A. M. Oelschlegel, J. Goldschmidt, and T. Salditt, “Three-dimensional mouse brain cytoarchitecture revealed by laboratory-based x-ray phase-contrast tomography,” Sci. Rep. 7, 42847 (2017).
[Crossref] [PubMed]

2016 (1)

D. Macindoe, M. J. Kitchen, S. C. Irvine, A. Fouras, and K. S. Morgan, “Requirements for dynamical differential phase contrast x-ray imaging with a laboratory source,” Phys. Med. Biol. 61, 8720–8735 (2016).
[Crossref] [PubMed]

2015 (2)

M. Endrizzi, F. A. Vittoria, G. Kallon, D. Basta, P. C. Diemoz, A. Vincenzi, P. Delogu, R. Bellazzini, and A. Olivo, “Achromatic approach to phase-based multi-modal imaging with conventional x-ray sources,” Opt. Exp. 23, 16473–16480 (2015).
[Crossref]

P. C. Diemoz, F. A. Vittoria, C. K. Hagen, M. Endrizzi, P. Coan, E. Brun, U. H. Wagner, C. Rau, I. K. Robinson, A. Bravin, and A. Olivo, “Single-image phase retrieval using an edge illumination x-ray phase-contrast imaging setup,” J. Synchrotron Radiat. 22, 1072–1077 (2015).
[Crossref] [PubMed]

2014 (7)

C. K. Hagen, P. C. Diemoz, M. Endrizzi, and A. Olivo, “The effect of the spatial sampling rate on quantitative phase information extracted from planar and tomographic edge illumination x-ray phase contrast images,” J. Phys. D. Appl. Phys. 47, 455401 (2014).
[Crossref]

M. Endrizzi, P. C. Diemoz, C. K. Hagen, F. A. Vittoria, P. R. T. Munro, L. Rigon, D. Dreossi, F. Arfelli, F. C. M. Lopez, R. Longo, M. Marenzana, P. Delogu, A. Vincenzi, L. D. Ruvo, G. Spandre, A. Brez, R. Bellazzini, and A. Olivo, “Edge-illumination X-ray phase contrast imaging: matching the imaging method to the detector technology,” J. Instrum. 9, C11004 (2014).
[Crossref]

A. Horng, E. Brun, A. Mittone, S. Gasilov, L. Weber, T. Geith, S. Adam-neumair, S. D. Auweter, A. Bravin, M. F. Reiser, and P. Coan, “Cartilage and soft tissue imaging using x-rays,” Invest. Radiol. 49, 627–634 (2014).
[Crossref] [PubMed]

P. Diemoz, F. Vittoria, and A. Olivo, “Spatial resolution of edge illumination x-ray phase-contrast imaging,” Opt. Express 22, 15514–29 (2014).
[Crossref] [PubMed]

C. K. Hagen, P. R. T. Munro, M. Endrizzi, P. C. Diemoz, and A. Olivo, “Low-dose phase contrast tomography with conventional x-ray sources,” Med. Phys. 41, 070701 (2014).
[Crossref] [PubMed]

P. C. Diemoz and A. Olivo, “On the origin of contrast in edge illumination x-ray phase-contrast imaging,” Opt. Exp. 22, 28199–28214 (2014).
[Crossref]

S. W. Wilkins, Y. I. Nesterets, T. E. Gureyev, S. C. Mayo, A. Pogany, and A. W. Stevenson, “On the evolution and relative merits of hard X-ray phase-contrast imaging methods,” Phil. Trans. R. Soc. 372, 20130021 (2014).
[Crossref]

2013 (5)

A. Bravin, P. Coan, and P. Suortti, “X-ray phase-contrast imaging: from pre-clinical applications towards clinics,” Phys. Med. Biol. 58, R1–R35 (2013).
[Crossref]

F. A. Vittoria, P. C. Diemoz, M. Endrizzi, L. Rigon, F. C. Lopez, D. Dreossi, P. R. T. Munro, and A. Olivo, “Strategies for efficient and fast wave optics simulation of coded-aperture and other x-ray phase-contrast imaging methods,” Appl. Opt. 52, 6940–6947 (2013).
[Crossref] [PubMed]

R. Bellazzini, G. Spandre, A. Brez, M. Minuti, M. Pinchera, and P. Mozzo, “Chromatic X-ray imaging with a fine pitch CdTe sensor coupled to a large area photon counting pixel ASIC,” J. Instrum. 8, C02028 (2013).
[Crossref]

P. Munro and A. Olivo, “X-ray phase-contrast imaging with polychromatic sources and the concept of effective energy,” Phys. Rev. A 87, 053838 (2013).
[Crossref]

P. C. Diemoz, M. Endrizzi, C. E. Zapata, Z. D. Pesic, C. Rau, A. Bravin, I. K. Robinson, and A. Olivo, “X-ray phase-contrast imaging with nanoradian angular resolution,” Phys. Rev. Lett. 110, 138105 (2013).
[Crossref] [PubMed]

2012 (2)

P. R. T. Munro, K. Ignatyev, R. D. Speller, and A. Olivo, “Phase and absorption retrieval using incoherent x-ray sources,” Proc. Natl. Acad. Sci. 109, 13922–13927 (2012).
[Crossref] [PubMed]

S. C. Mayo, A. W. Stevenson, and S. W. Wilkins, “In-line phase-contrast x-ray imaging and tomography for materials science,” Materials 5, 937–965 (2012).
[Crossref]

2011 (3)

K. S. Morgan, D. M. Paganin, and K. K.W. Siu, “Quantitative single-exposure x-ray phase contrast imaging using a single attenuation grid,” Opt. Exp. 19, 19781–19789 (2011).
[Crossref]

M. A. Beltran, D. M. Paganin, K. K. W. Siu, A. Fouras, S. B. Hooper, D. H. Reser, and M. J. Kitchen, “Interface-specific x-ray phase retrieval tomography of complex biological organs,” Phys. Med. Biol. 56, 7353–7369 (2011).
[Crossref] [PubMed]

K. Ignatyev, P. R. T. Munro, R. D. Speller, and A. Olivo, “Effects of signal diffusion on x-ray phase contrast images,” Rev. Sci. Instrum. 82, 073702 (2011).
[Crossref] [PubMed]

2010 (2)

P. R. T. Munro, K. Ignatyev, R. D. Speller, and A. Olivo, “Design of a novel phase contrast x-ray imaging system for mammography,” Phys. Med. Biol. 55, 4169–4185 (2010).
[Crossref] [PubMed]

M. A. Beltran, D. M. Paganin, K. Uesugi, and M. J. Kitchen, “2D and 3D X-ray phase retrieval of multi-material objects using a single defocus distance,” Opt. Express 18, 6423–6436 (2010).
[Crossref] [PubMed]

2008 (2)

2007 (3)

A. Olivo and R. Speller, “A coded-aperture technique allowing x-ray phase contrast imaging with conventional sources,” Appl. Phys. Lett. 91, 074106 (2007).
[Crossref]

A. Olivo and R. Speller, “Modelling of a novel x-ray phase contrast imaging technique based on coded apertures,” Phys. Med. Biol. 52, 6555–6573 (2007).
[Crossref] [PubMed]

F. Pfeiffer, C. Kottler, O. Bunk, and C. David, “Hard x-ray phase tomography with low-brilliance sources,” Phys. Rev. Lett. 98, 108105 (2007).
[Crossref] [PubMed]

2006 (1)

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

2003 (1)

A. Momose, S. Kawamoto, I. Koyama, Y. Hamaishi, K. Takai, and Y. Suzuki, “Demonstration of x-ray talbot interferometry,” Jpn. J. Appl. Phys. 42, L866–L868 (2003).
[Crossref]

2002 (1)

D. Paganin, S. C. Mayo, T. E. Gureyev, P. R. Miller, and S. W. Wilkins, “Simultaneous phase and amplitude extraction from a single defocused image of a homogeneous object,” J. Microsc. 206, 33–40 (2002).
[Crossref] [PubMed]

2001 (1)

A. Olivo, F. Arfelli, G. Cantatore, R. Longo, R. H. Menk, S. Pani, M. Prest, P. Poropat, L. Rigon, G. Tromba, E. Vallazza, and E. Castelli, “An innovative digital imaging set-up allowing a low-dose approach to phase contrast applications in the medical field,” Med. Phys. 28, 1610–1619 (2001).
[Crossref] [PubMed]

1996 (1)

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

1995 (1)

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

1983 (1)

1979 (1)

N. Otsu, “A threshold selection method from gray-level histograms,” IEEE Trans. Syst. Man. Cybern. 9, 62–66 (1979).
[Crossref]

Adam-neumair, S.

A. Horng, E. Brun, A. Mittone, S. Gasilov, L. Weber, T. Geith, S. Adam-neumair, S. D. Auweter, A. Bravin, M. F. Reiser, and P. Coan, “Cartilage and soft tissue imaging using x-rays,” Invest. Radiol. 49, 627–634 (2014).
[Crossref] [PubMed]

Arfelli, F.

M. Endrizzi, P. C. Diemoz, C. K. Hagen, F. A. Vittoria, P. R. T. Munro, L. Rigon, D. Dreossi, F. Arfelli, F. C. M. Lopez, R. Longo, M. Marenzana, P. Delogu, A. Vincenzi, L. D. Ruvo, G. Spandre, A. Brez, R. Bellazzini, and A. Olivo, “Edge-illumination X-ray phase contrast imaging: matching the imaging method to the detector technology,” J. Instrum. 9, C11004 (2014).
[Crossref]

A. Olivo, F. Arfelli, G. Cantatore, R. Longo, R. H. Menk, S. Pani, M. Prest, P. Poropat, L. Rigon, G. Tromba, E. Vallazza, and E. Castelli, “An innovative digital imaging set-up allowing a low-dose approach to phase contrast applications in the medical field,” Med. Phys. 28, 1610–1619 (2001).
[Crossref] [PubMed]

Auweter, S. D.

A. Horng, E. Brun, A. Mittone, S. Gasilov, L. Weber, T. Geith, S. Adam-neumair, S. D. Auweter, A. Bravin, M. F. Reiser, and P. Coan, “Cartilage and soft tissue imaging using x-rays,” Invest. Radiol. 49, 627–634 (2014).
[Crossref] [PubMed]

Basta, D.

M. Endrizzi, F. A. Vittoria, G. Kallon, D. Basta, P. C. Diemoz, A. Vincenzi, P. Delogu, R. Bellazzini, and A. Olivo, “Achromatic approach to phase-based multi-modal imaging with conventional x-ray sources,” Opt. Exp. 23, 16473–16480 (2015).
[Crossref]

Bellazzini, R.

M. Endrizzi, F. A. Vittoria, G. Kallon, D. Basta, P. C. Diemoz, A. Vincenzi, P. Delogu, R. Bellazzini, and A. Olivo, “Achromatic approach to phase-based multi-modal imaging with conventional x-ray sources,” Opt. Exp. 23, 16473–16480 (2015).
[Crossref]

M. Endrizzi, P. C. Diemoz, C. K. Hagen, F. A. Vittoria, P. R. T. Munro, L. Rigon, D. Dreossi, F. Arfelli, F. C. M. Lopez, R. Longo, M. Marenzana, P. Delogu, A. Vincenzi, L. D. Ruvo, G. Spandre, A. Brez, R. Bellazzini, and A. Olivo, “Edge-illumination X-ray phase contrast imaging: matching the imaging method to the detector technology,” J. Instrum. 9, C11004 (2014).
[Crossref]

R. Bellazzini, G. Spandre, A. Brez, M. Minuti, M. Pinchera, and P. Mozzo, “Chromatic X-ray imaging with a fine pitch CdTe sensor coupled to a large area photon counting pixel ASIC,” J. Instrum. 8, C02028 (2013).
[Crossref]

P. C. Diemoz, C. K. Hagen, M. Endrizzi, M. Minuti, R. Bellazzini, L. Urbani, P. De Coppi, and A. Olivo, “Single-shot X-ray phase-contrast computed tomography with non-microfocal laboratory sources,” Phys. Rev. Appl. (2017, in press).

Beltran, M. A.

M. A. Beltran, D. M. Paganin, K. K. W. Siu, A. Fouras, S. B. Hooper, D. H. Reser, and M. J. Kitchen, “Interface-specific x-ray phase retrieval tomography of complex biological organs,” Phys. Med. Biol. 56, 7353–7369 (2011).
[Crossref] [PubMed]

M. A. Beltran, D. M. Paganin, K. Uesugi, and M. J. Kitchen, “2D and 3D X-ray phase retrieval of multi-material objects using a single defocus distance,” Opt. Express 18, 6423–6436 (2010).
[Crossref] [PubMed]

Bravin, A.

P. C. Diemoz, F. A. Vittoria, C. K. Hagen, M. Endrizzi, P. Coan, E. Brun, U. H. Wagner, C. Rau, I. K. Robinson, A. Bravin, and A. Olivo, “Single-image phase retrieval using an edge illumination x-ray phase-contrast imaging setup,” J. Synchrotron Radiat. 22, 1072–1077 (2015).
[Crossref] [PubMed]

A. Horng, E. Brun, A. Mittone, S. Gasilov, L. Weber, T. Geith, S. Adam-neumair, S. D. Auweter, A. Bravin, M. F. Reiser, and P. Coan, “Cartilage and soft tissue imaging using x-rays,” Invest. Radiol. 49, 627–634 (2014).
[Crossref] [PubMed]

P. C. Diemoz, M. Endrizzi, C. E. Zapata, Z. D. Pesic, C. Rau, A. Bravin, I. K. Robinson, and A. Olivo, “X-ray phase-contrast imaging with nanoradian angular resolution,” Phys. Rev. Lett. 110, 138105 (2013).
[Crossref] [PubMed]

A. Bravin, P. Coan, and P. Suortti, “X-ray phase-contrast imaging: from pre-clinical applications towards clinics,” Phys. Med. Biol. 58, R1–R35 (2013).
[Crossref]

Brez, A.

M. Endrizzi, P. C. Diemoz, C. K. Hagen, F. A. Vittoria, P. R. T. Munro, L. Rigon, D. Dreossi, F. Arfelli, F. C. M. Lopez, R. Longo, M. Marenzana, P. Delogu, A. Vincenzi, L. D. Ruvo, G. Spandre, A. Brez, R. Bellazzini, and A. Olivo, “Edge-illumination X-ray phase contrast imaging: matching the imaging method to the detector technology,” J. Instrum. 9, C11004 (2014).
[Crossref]

R. Bellazzini, G. Spandre, A. Brez, M. Minuti, M. Pinchera, and P. Mozzo, “Chromatic X-ray imaging with a fine pitch CdTe sensor coupled to a large area photon counting pixel ASIC,” J. Instrum. 8, C02028 (2013).
[Crossref]

Brun, E.

P. C. Diemoz, F. A. Vittoria, C. K. Hagen, M. Endrizzi, P. Coan, E. Brun, U. H. Wagner, C. Rau, I. K. Robinson, A. Bravin, and A. Olivo, “Single-image phase retrieval using an edge illumination x-ray phase-contrast imaging setup,” J. Synchrotron Radiat. 22, 1072–1077 (2015).
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A. Horng, E. Brun, A. Mittone, S. Gasilov, L. Weber, T. Geith, S. Adam-neumair, S. D. Auweter, A. Bravin, M. F. Reiser, and P. Coan, “Cartilage and soft tissue imaging using x-rays,” Invest. Radiol. 49, 627–634 (2014).
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Bunk, O.

F. Pfeiffer, C. Kottler, O. Bunk, and C. David, “Hard x-ray phase tomography with low-brilliance sources,” Phys. Rev. Lett. 98, 108105 (2007).
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F. Pfeiffer, T. Weitkamp, O. Bunk, and C. David, “Phase retrieval and differential phase-contrast imaging with low-brilliance x-ray sources,” Nature Phys. 2, 258–261 (2006).
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Cantatore, G.

A. Olivo, F. Arfelli, G. Cantatore, R. Longo, R. H. Menk, S. Pani, M. Prest, P. Poropat, L. Rigon, G. Tromba, E. Vallazza, and E. Castelli, “An innovative digital imaging set-up allowing a low-dose approach to phase contrast applications in the medical field,” Med. Phys. 28, 1610–1619 (2001).
[Crossref] [PubMed]

Castelli, E.

A. Olivo, F. Arfelli, G. Cantatore, R. Longo, R. H. Menk, S. Pani, M. Prest, P. Poropat, L. Rigon, G. Tromba, E. Vallazza, and E. Castelli, “An innovative digital imaging set-up allowing a low-dose approach to phase contrast applications in the medical field,” Med. Phys. 28, 1610–1619 (2001).
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Coan, P.

P. C. Diemoz, F. A. Vittoria, C. K. Hagen, M. Endrizzi, P. Coan, E. Brun, U. H. Wagner, C. Rau, I. K. Robinson, A. Bravin, and A. Olivo, “Single-image phase retrieval using an edge illumination x-ray phase-contrast imaging setup,” J. Synchrotron Radiat. 22, 1072–1077 (2015).
[Crossref] [PubMed]

A. Horng, E. Brun, A. Mittone, S. Gasilov, L. Weber, T. Geith, S. Adam-neumair, S. D. Auweter, A. Bravin, M. F. Reiser, and P. Coan, “Cartilage and soft tissue imaging using x-rays,” Invest. Radiol. 49, 627–634 (2014).
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A. Bravin, P. Coan, and P. Suortti, “X-ray phase-contrast imaging: from pre-clinical applications towards clinics,” Phys. Med. Biol. 58, R1–R35 (2013).
[Crossref]

Coppi, P. De

P. C. Diemoz, C. K. Hagen, M. Endrizzi, M. Minuti, R. Bellazzini, L. Urbani, P. De Coppi, and A. Olivo, “Single-shot X-ray phase-contrast computed tomography with non-microfocal laboratory sources,” Phys. Rev. Appl. (2017, in press).

David, C.

F. Pfeiffer, C. Kottler, O. Bunk, and C. David, “Hard x-ray phase tomography with low-brilliance sources,” Phys. Rev. Lett. 98, 108105 (2007).
[Crossref] [PubMed]

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

Delogu, P.

M. Endrizzi, F. A. Vittoria, G. Kallon, D. Basta, P. C. Diemoz, A. Vincenzi, P. Delogu, R. Bellazzini, and A. Olivo, “Achromatic approach to phase-based multi-modal imaging with conventional x-ray sources,” Opt. Exp. 23, 16473–16480 (2015).
[Crossref]

M. Endrizzi, P. C. Diemoz, C. K. Hagen, F. A. Vittoria, P. R. T. Munro, L. Rigon, D. Dreossi, F. Arfelli, F. C. M. Lopez, R. Longo, M. Marenzana, P. Delogu, A. Vincenzi, L. D. Ruvo, G. Spandre, A. Brez, R. Bellazzini, and A. Olivo, “Edge-illumination X-ray phase contrast imaging: matching the imaging method to the detector technology,” J. Instrum. 9, C11004 (2014).
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Diemoz, P.

Diemoz, P. C.

P. C. Diemoz, F. A. Vittoria, C. K. Hagen, M. Endrizzi, P. Coan, E. Brun, U. H. Wagner, C. Rau, I. K. Robinson, A. Bravin, and A. Olivo, “Single-image phase retrieval using an edge illumination x-ray phase-contrast imaging setup,” J. Synchrotron Radiat. 22, 1072–1077 (2015).
[Crossref] [PubMed]

M. Endrizzi, F. A. Vittoria, G. Kallon, D. Basta, P. C. Diemoz, A. Vincenzi, P. Delogu, R. Bellazzini, and A. Olivo, “Achromatic approach to phase-based multi-modal imaging with conventional x-ray sources,” Opt. Exp. 23, 16473–16480 (2015).
[Crossref]

C. K. Hagen, P. R. T. Munro, M. Endrizzi, P. C. Diemoz, and A. Olivo, “Low-dose phase contrast tomography with conventional x-ray sources,” Med. Phys. 41, 070701 (2014).
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P. C. Diemoz and A. Olivo, “On the origin of contrast in edge illumination x-ray phase-contrast imaging,” Opt. Exp. 22, 28199–28214 (2014).
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M. Endrizzi, P. C. Diemoz, C. K. Hagen, F. A. Vittoria, P. R. T. Munro, L. Rigon, D. Dreossi, F. Arfelli, F. C. M. Lopez, R. Longo, M. Marenzana, P. Delogu, A. Vincenzi, L. D. Ruvo, G. Spandre, A. Brez, R. Bellazzini, and A. Olivo, “Edge-illumination X-ray phase contrast imaging: matching the imaging method to the detector technology,” J. Instrum. 9, C11004 (2014).
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C. K. Hagen, P. C. Diemoz, M. Endrizzi, and A. Olivo, “The effect of the spatial sampling rate on quantitative phase information extracted from planar and tomographic edge illumination x-ray phase contrast images,” J. Phys. D. Appl. Phys. 47, 455401 (2014).
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P. C. Diemoz, M. Endrizzi, C. E. Zapata, Z. D. Pesic, C. Rau, A. Bravin, I. K. Robinson, and A. Olivo, “X-ray phase-contrast imaging with nanoradian angular resolution,” Phys. Rev. Lett. 110, 138105 (2013).
[Crossref] [PubMed]

F. A. Vittoria, P. C. Diemoz, M. Endrizzi, L. Rigon, F. C. Lopez, D. Dreossi, P. R. T. Munro, and A. Olivo, “Strategies for efficient and fast wave optics simulation of coded-aperture and other x-ray phase-contrast imaging methods,” Appl. Opt. 52, 6940–6947 (2013).
[Crossref] [PubMed]

P. C. Diemoz, C. K. Hagen, M. Endrizzi, M. Minuti, R. Bellazzini, L. Urbani, P. De Coppi, and A. Olivo, “Single-shot X-ray phase-contrast computed tomography with non-microfocal laboratory sources,” Phys. Rev. Appl. (2017, in press).

Dreossi, D.

M. Endrizzi, P. C. Diemoz, C. K. Hagen, F. A. Vittoria, P. R. T. Munro, L. Rigon, D. Dreossi, F. Arfelli, F. C. M. Lopez, R. Longo, M. Marenzana, P. Delogu, A. Vincenzi, L. D. Ruvo, G. Spandre, A. Brez, R. Bellazzini, and A. Olivo, “Edge-illumination X-ray phase contrast imaging: matching the imaging method to the detector technology,” J. Instrum. 9, C11004 (2014).
[Crossref]

F. A. Vittoria, P. C. Diemoz, M. Endrizzi, L. Rigon, F. C. Lopez, D. Dreossi, P. R. T. Munro, and A. Olivo, “Strategies for efficient and fast wave optics simulation of coded-aperture and other x-ray phase-contrast imaging methods,” Appl. Opt. 52, 6940–6947 (2013).
[Crossref] [PubMed]

Endrizzi, M.

P. C. Diemoz, F. A. Vittoria, C. K. Hagen, M. Endrizzi, P. Coan, E. Brun, U. H. Wagner, C. Rau, I. K. Robinson, A. Bravin, and A. Olivo, “Single-image phase retrieval using an edge illumination x-ray phase-contrast imaging setup,” J. Synchrotron Radiat. 22, 1072–1077 (2015).
[Crossref] [PubMed]

M. Endrizzi, F. A. Vittoria, G. Kallon, D. Basta, P. C. Diemoz, A. Vincenzi, P. Delogu, R. Bellazzini, and A. Olivo, “Achromatic approach to phase-based multi-modal imaging with conventional x-ray sources,” Opt. Exp. 23, 16473–16480 (2015).
[Crossref]

C. K. Hagen, P. R. T. Munro, M. Endrizzi, P. C. Diemoz, and A. Olivo, “Low-dose phase contrast tomography with conventional x-ray sources,” Med. Phys. 41, 070701 (2014).
[Crossref] [PubMed]

M. Endrizzi, P. C. Diemoz, C. K. Hagen, F. A. Vittoria, P. R. T. Munro, L. Rigon, D. Dreossi, F. Arfelli, F. C. M. Lopez, R. Longo, M. Marenzana, P. Delogu, A. Vincenzi, L. D. Ruvo, G. Spandre, A. Brez, R. Bellazzini, and A. Olivo, “Edge-illumination X-ray phase contrast imaging: matching the imaging method to the detector technology,” J. Instrum. 9, C11004 (2014).
[Crossref]

C. K. Hagen, P. C. Diemoz, M. Endrizzi, and A. Olivo, “The effect of the spatial sampling rate on quantitative phase information extracted from planar and tomographic edge illumination x-ray phase contrast images,” J. Phys. D. Appl. Phys. 47, 455401 (2014).
[Crossref]

P. C. Diemoz, M. Endrizzi, C. E. Zapata, Z. D. Pesic, C. Rau, A. Bravin, I. K. Robinson, and A. Olivo, “X-ray phase-contrast imaging with nanoradian angular resolution,” Phys. Rev. Lett. 110, 138105 (2013).
[Crossref] [PubMed]

F. A. Vittoria, P. C. Diemoz, M. Endrizzi, L. Rigon, F. C. Lopez, D. Dreossi, P. R. T. Munro, and A. Olivo, “Strategies for efficient and fast wave optics simulation of coded-aperture and other x-ray phase-contrast imaging methods,” Appl. Opt. 52, 6940–6947 (2013).
[Crossref] [PubMed]

P. C. Diemoz, C. K. Hagen, M. Endrizzi, M. Minuti, R. Bellazzini, L. Urbani, P. De Coppi, and A. Olivo, “Single-shot X-ray phase-contrast computed tomography with non-microfocal laboratory sources,” Phys. Rev. Appl. (2017, in press).

Fouras, A.

D. Macindoe, M. J. Kitchen, S. C. Irvine, A. Fouras, and K. S. Morgan, “Requirements for dynamical differential phase contrast x-ray imaging with a laboratory source,” Phys. Med. Biol. 61, 8720–8735 (2016).
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M. A. Beltran, D. M. Paganin, K. K. W. Siu, A. Fouras, S. B. Hooper, D. H. Reser, and M. J. Kitchen, “Interface-specific x-ray phase retrieval tomography of complex biological organs,” Phys. Med. Biol. 56, 7353–7369 (2011).
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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, 335–338 (1996).
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Gasilov, S.

A. Horng, E. Brun, A. Mittone, S. Gasilov, L. Weber, T. Geith, S. Adam-neumair, S. D. Auweter, A. Bravin, M. F. Reiser, and P. Coan, “Cartilage and soft tissue imaging using x-rays,” Invest. Radiol. 49, 627–634 (2014).
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Geith, T.

A. Horng, E. Brun, A. Mittone, S. Gasilov, L. Weber, T. Geith, S. Adam-neumair, S. D. Auweter, A. Bravin, M. F. Reiser, and P. Coan, “Cartilage and soft tissue imaging using x-rays,” Invest. Radiol. 49, 627–634 (2014).
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Goldschmidt, J.

M. Topperwien, M. Krenkel, D. Vincenz, F. Stober, A. M. Oelschlegel, J. Goldschmidt, and T. Salditt, “Three-dimensional mouse brain cytoarchitecture revealed by laboratory-based x-ray phase-contrast tomography,” Sci. Rep. 7, 42847 (2017).
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Gureyev, T. E

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

S. W. Wilkins, Y. I. Nesterets, T. E. Gureyev, S. C. Mayo, A. Pogany, and A. W. Stevenson, “On the evolution and relative merits of hard X-ray phase-contrast imaging methods,” Phil. Trans. R. Soc. 372, 20130021 (2014).
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T. E. Gureyev, Y. I. Nesterets, A. W. Stevenson, P. R. Miller, A. Pogany, and S. W. Wilkins, “Some simple rules for contrast, signal-to-noise and resolution in in-line x-ray phase-contrast imaging,” Opt. Express 16, 3223–3241 (2008).
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D. Paganin, S. C. Mayo, T. E. Gureyev, P. R. Miller, and S. W. Wilkins, “Simultaneous phase and amplitude extraction from a single defocused image of a homogeneous object,” J. Microsc. 206, 33–40 (2002).
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Hagen, C. K.

P. C. Diemoz, F. A. Vittoria, C. K. Hagen, M. Endrizzi, P. Coan, E. Brun, U. H. Wagner, C. Rau, I. K. Robinson, A. Bravin, and A. Olivo, “Single-image phase retrieval using an edge illumination x-ray phase-contrast imaging setup,” J. Synchrotron Radiat. 22, 1072–1077 (2015).
[Crossref] [PubMed]

C. K. Hagen, P. R. T. Munro, M. Endrizzi, P. C. Diemoz, and A. Olivo, “Low-dose phase contrast tomography with conventional x-ray sources,” Med. Phys. 41, 070701 (2014).
[Crossref] [PubMed]

M. Endrizzi, P. C. Diemoz, C. K. Hagen, F. A. Vittoria, P. R. T. Munro, L. Rigon, D. Dreossi, F. Arfelli, F. C. M. Lopez, R. Longo, M. Marenzana, P. Delogu, A. Vincenzi, L. D. Ruvo, G. Spandre, A. Brez, R. Bellazzini, and A. Olivo, “Edge-illumination X-ray phase contrast imaging: matching the imaging method to the detector technology,” J. Instrum. 9, C11004 (2014).
[Crossref]

C. K. Hagen, P. C. Diemoz, M. Endrizzi, and A. Olivo, “The effect of the spatial sampling rate on quantitative phase information extracted from planar and tomographic edge illumination x-ray phase contrast images,” J. Phys. D. Appl. Phys. 47, 455401 (2014).
[Crossref]

P. C. Diemoz, C. K. Hagen, M. Endrizzi, M. Minuti, R. Bellazzini, L. Urbani, P. De Coppi, and A. Olivo, “Single-shot X-ray phase-contrast computed tomography with non-microfocal laboratory sources,” Phys. Rev. Appl. (2017, in press).

Hamaishi, Y.

A. Momose, S. Kawamoto, I. Koyama, Y. Hamaishi, K. Takai, and Y. Suzuki, “Demonstration of x-ray talbot interferometry,” Jpn. J. Appl. Phys. 42, L866–L868 (2003).
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Hooper, S. B.

M. A. Beltran, D. M. Paganin, K. K. W. Siu, A. Fouras, S. B. Hooper, D. H. Reser, and M. J. Kitchen, “Interface-specific x-ray phase retrieval tomography of complex biological organs,” Phys. Med. Biol. 56, 7353–7369 (2011).
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Horng, A.

A. Horng, E. Brun, A. Mittone, S. Gasilov, L. Weber, T. Geith, S. Adam-neumair, S. D. Auweter, A. Bravin, M. F. Reiser, and P. Coan, “Cartilage and soft tissue imaging using x-rays,” Invest. Radiol. 49, 627–634 (2014).
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Ignatyev, K.

P. R. T. Munro, K. Ignatyev, R. D. Speller, and A. Olivo, “Phase and absorption retrieval using incoherent x-ray sources,” Proc. Natl. Acad. Sci. 109, 13922–13927 (2012).
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K. Ignatyev, P. R. T. Munro, R. D. Speller, and A. Olivo, “Effects of signal diffusion on x-ray phase contrast images,” Rev. Sci. Instrum. 82, 073702 (2011).
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P. R. T. Munro, K. Ignatyev, R. D. Speller, and A. Olivo, “Design of a novel phase contrast x-ray imaging system for mammography,” Phys. Med. Biol. 55, 4169–4185 (2010).
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Irvine, S. C.

D. Macindoe, M. J. Kitchen, S. C. Irvine, A. Fouras, and K. S. Morgan, “Requirements for dynamical differential phase contrast x-ray imaging with a laboratory source,” Phys. Med. Biol. 61, 8720–8735 (2016).
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Kallon, G.

M. Endrizzi, F. A. Vittoria, G. Kallon, D. Basta, P. C. Diemoz, A. Vincenzi, P. Delogu, R. Bellazzini, and A. Olivo, “Achromatic approach to phase-based multi-modal imaging with conventional x-ray sources,” Opt. Exp. 23, 16473–16480 (2015).
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Kawamoto, S.

A. Momose, S. Kawamoto, I. Koyama, Y. Hamaishi, K. Takai, and Y. Suzuki, “Demonstration of x-ray talbot interferometry,” Jpn. J. Appl. Phys. 42, L866–L868 (2003).
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Kitchen, M. J.

D. Macindoe, M. J. Kitchen, S. C. Irvine, A. Fouras, and K. S. Morgan, “Requirements for dynamical differential phase contrast x-ray imaging with a laboratory source,” Phys. Med. Biol. 61, 8720–8735 (2016).
[Crossref] [PubMed]

M. A. Beltran, D. M. Paganin, K. K. W. Siu, A. Fouras, S. B. Hooper, D. H. Reser, and M. J. Kitchen, “Interface-specific x-ray phase retrieval tomography of complex biological organs,” Phys. Med. Biol. 56, 7353–7369 (2011).
[Crossref] [PubMed]

M. A. Beltran, D. M. Paganin, K. Uesugi, and M. J. Kitchen, “2D and 3D X-ray phase retrieval of multi-material objects using a single defocus distance,” Opt. Express 18, 6423–6436 (2010).
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Kohn, V.

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

F. Pfeiffer, C. Kottler, O. Bunk, and C. David, “Hard x-ray phase tomography with low-brilliance sources,” Phys. Rev. Lett. 98, 108105 (2007).
[Crossref] [PubMed]

Koyama, I.

A. Momose, S. Kawamoto, I. Koyama, Y. Hamaishi, K. Takai, and Y. Suzuki, “Demonstration of x-ray talbot interferometry,” Jpn. J. Appl. Phys. 42, L866–L868 (2003).
[Crossref]

Krenkel, M.

M. Topperwien, M. Krenkel, D. Vincenz, F. Stober, A. M. Oelschlegel, J. Goldschmidt, and T. Salditt, “Three-dimensional mouse brain cytoarchitecture revealed by laboratory-based x-ray phase-contrast tomography,” Sci. Rep. 7, 42847 (2017).
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Kuznetsov, S.

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

M. Endrizzi, P. C. Diemoz, C. K. Hagen, F. A. Vittoria, P. R. T. Munro, L. Rigon, D. Dreossi, F. Arfelli, F. C. M. Lopez, R. Longo, M. Marenzana, P. Delogu, A. Vincenzi, L. D. Ruvo, G. Spandre, A. Brez, R. Bellazzini, and A. Olivo, “Edge-illumination X-ray phase contrast imaging: matching the imaging method to the detector technology,” J. Instrum. 9, C11004 (2014).
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A. Olivo, F. Arfelli, G. Cantatore, R. Longo, R. H. Menk, S. Pani, M. Prest, P. Poropat, L. Rigon, G. Tromba, E. Vallazza, and E. Castelli, “An innovative digital imaging set-up allowing a low-dose approach to phase contrast applications in the medical field,” Med. Phys. 28, 1610–1619 (2001).
[Crossref] [PubMed]

Lopez, F. C.

Lopez, F. C. M.

M. Endrizzi, P. C. Diemoz, C. K. Hagen, F. A. Vittoria, P. R. T. Munro, L. Rigon, D. Dreossi, F. Arfelli, F. C. M. Lopez, R. Longo, M. Marenzana, P. Delogu, A. Vincenzi, L. D. Ruvo, G. Spandre, A. Brez, R. Bellazzini, and A. Olivo, “Edge-illumination X-ray phase contrast imaging: matching the imaging method to the detector technology,” J. Instrum. 9, C11004 (2014).
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Macindoe, D.

D. Macindoe, M. J. Kitchen, S. C. Irvine, A. Fouras, and K. S. Morgan, “Requirements for dynamical differential phase contrast x-ray imaging with a laboratory source,” Phys. Med. Biol. 61, 8720–8735 (2016).
[Crossref] [PubMed]

Marenzana, M.

M. Endrizzi, P. C. Diemoz, C. K. Hagen, F. A. Vittoria, P. R. T. Munro, L. Rigon, D. Dreossi, F. Arfelli, F. C. M. Lopez, R. Longo, M. Marenzana, P. Delogu, A. Vincenzi, L. D. Ruvo, G. Spandre, A. Brez, R. Bellazzini, and A. Olivo, “Edge-illumination X-ray phase contrast imaging: matching the imaging method to the detector technology,” J. Instrum. 9, C11004 (2014).
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Mayo, S. C.

S. W. Wilkins, Y. I. Nesterets, T. E. Gureyev, S. C. Mayo, A. Pogany, and A. W. Stevenson, “On the evolution and relative merits of hard X-ray phase-contrast imaging methods,” Phil. Trans. R. Soc. 372, 20130021 (2014).
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S. C. Mayo, A. W. Stevenson, and S. W. Wilkins, “In-line phase-contrast x-ray imaging and tomography for materials science,” Materials 5, 937–965 (2012).
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D. Paganin, S. C. Mayo, T. E. Gureyev, P. R. Miller, and S. W. Wilkins, “Simultaneous phase and amplitude extraction from a single defocused image of a homogeneous object,” J. Microsc. 206, 33–40 (2002).
[Crossref] [PubMed]

Menk, R. H.

A. Olivo, F. Arfelli, G. Cantatore, R. Longo, R. H. Menk, S. Pani, M. Prest, P. Poropat, L. Rigon, G. Tromba, E. Vallazza, and E. Castelli, “An innovative digital imaging set-up allowing a low-dose approach to phase contrast applications in the medical field,” Med. Phys. 28, 1610–1619 (2001).
[Crossref] [PubMed]

Miller, P. R.

T. E. Gureyev, Y. I. Nesterets, A. W. Stevenson, P. R. Miller, A. Pogany, and S. W. Wilkins, “Some simple rules for contrast, signal-to-noise and resolution in in-line x-ray phase-contrast imaging,” Opt. Express 16, 3223–3241 (2008).
[Crossref] [PubMed]

D. Paganin, S. C. Mayo, T. E. Gureyev, P. R. Miller, and S. W. Wilkins, “Simultaneous phase and amplitude extraction from a single defocused image of a homogeneous object,” J. Microsc. 206, 33–40 (2002).
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Minuti, M.

R. Bellazzini, G. Spandre, A. Brez, M. Minuti, M. Pinchera, and P. Mozzo, “Chromatic X-ray imaging with a fine pitch CdTe sensor coupled to a large area photon counting pixel ASIC,” J. Instrum. 8, C02028 (2013).
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P. C. Diemoz, C. K. Hagen, M. Endrizzi, M. Minuti, R. Bellazzini, L. Urbani, P. De Coppi, and A. Olivo, “Single-shot X-ray phase-contrast computed tomography with non-microfocal laboratory sources,” Phys. Rev. Appl. (2017, in press).

Mittone, A.

A. Horng, E. Brun, A. Mittone, S. Gasilov, L. Weber, T. Geith, S. Adam-neumair, S. D. Auweter, A. Bravin, M. F. Reiser, and P. Coan, “Cartilage and soft tissue imaging using x-rays,” Invest. Radiol. 49, 627–634 (2014).
[Crossref] [PubMed]

Momose, A.

A. Momose, S. Kawamoto, I. Koyama, Y. Hamaishi, K. Takai, and Y. Suzuki, “Demonstration of x-ray talbot interferometry,” Jpn. J. Appl. Phys. 42, L866–L868 (2003).
[Crossref]

Morgan, K. S.

D. Macindoe, M. J. Kitchen, S. C. Irvine, A. Fouras, and K. S. Morgan, “Requirements for dynamical differential phase contrast x-ray imaging with a laboratory source,” Phys. Med. Biol. 61, 8720–8735 (2016).
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K. S. Morgan, D. M. Paganin, and K. K.W. Siu, “Quantitative single-exposure x-ray phase contrast imaging using a single attenuation grid,” Opt. Exp. 19, 19781–19789 (2011).
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Mozzo, P.

R. Bellazzini, G. Spandre, A. Brez, M. Minuti, M. Pinchera, and P. Mozzo, “Chromatic X-ray imaging with a fine pitch CdTe sensor coupled to a large area photon counting pixel ASIC,” J. Instrum. 8, C02028 (2013).
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P. Munro and A. Olivo, “X-ray phase-contrast imaging with polychromatic sources and the concept of effective energy,” Phys. Rev. A 87, 053838 (2013).
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Munro, P. R. T.

M. Endrizzi, P. C. Diemoz, C. K. Hagen, F. A. Vittoria, P. R. T. Munro, L. Rigon, D. Dreossi, F. Arfelli, F. C. M. Lopez, R. Longo, M. Marenzana, P. Delogu, A. Vincenzi, L. D. Ruvo, G. Spandre, A. Brez, R. Bellazzini, and A. Olivo, “Edge-illumination X-ray phase contrast imaging: matching the imaging method to the detector technology,” J. Instrum. 9, C11004 (2014).
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C. K. Hagen, P. R. T. Munro, M. Endrizzi, P. C. Diemoz, and A. Olivo, “Low-dose phase contrast tomography with conventional x-ray sources,” Med. Phys. 41, 070701 (2014).
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F. A. Vittoria, P. C. Diemoz, M. Endrizzi, L. Rigon, F. C. Lopez, D. Dreossi, P. R. T. Munro, and A. Olivo, “Strategies for efficient and fast wave optics simulation of coded-aperture and other x-ray phase-contrast imaging methods,” Appl. Opt. 52, 6940–6947 (2013).
[Crossref] [PubMed]

P. R. T. Munro, K. Ignatyev, R. D. Speller, and A. Olivo, “Phase and absorption retrieval using incoherent x-ray sources,” Proc. Natl. Acad. Sci. 109, 13922–13927 (2012).
[Crossref] [PubMed]

K. Ignatyev, P. R. T. Munro, R. D. Speller, and A. Olivo, “Effects of signal diffusion on x-ray phase contrast images,” Rev. Sci. Instrum. 82, 073702 (2011).
[Crossref] [PubMed]

P. R. T. Munro, K. Ignatyev, R. D. Speller, and A. Olivo, “Design of a novel phase contrast x-ray imaging system for mammography,” Phys. Med. Biol. 55, 4169–4185 (2010).
[Crossref] [PubMed]

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S. W. Wilkins, Y. I. Nesterets, T. E. Gureyev, S. C. Mayo, A. Pogany, and A. W. Stevenson, “On the evolution and relative merits of hard X-ray phase-contrast imaging methods,” Phil. Trans. R. Soc. 372, 20130021 (2014).
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T. E. Gureyev, Y. I. Nesterets, A. W. Stevenson, P. R. Miller, A. Pogany, and S. W. Wilkins, “Some simple rules for contrast, signal-to-noise and resolution in in-line x-ray phase-contrast imaging,” Opt. Express 16, 3223–3241 (2008).
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M. Topperwien, M. Krenkel, D. Vincenz, F. Stober, A. M. Oelschlegel, J. Goldschmidt, and T. Salditt, “Three-dimensional mouse brain cytoarchitecture revealed by laboratory-based x-ray phase-contrast tomography,” Sci. Rep. 7, 42847 (2017).
[Crossref] [PubMed]

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M. Endrizzi, F. A. Vittoria, G. Kallon, D. Basta, P. C. Diemoz, A. Vincenzi, P. Delogu, R. Bellazzini, and A. Olivo, “Achromatic approach to phase-based multi-modal imaging with conventional x-ray sources,” Opt. Exp. 23, 16473–16480 (2015).
[Crossref]

P. C. Diemoz, F. A. Vittoria, C. K. Hagen, M. Endrizzi, P. Coan, E. Brun, U. H. Wagner, C. Rau, I. K. Robinson, A. Bravin, and A. Olivo, “Single-image phase retrieval using an edge illumination x-ray phase-contrast imaging setup,” J. Synchrotron Radiat. 22, 1072–1077 (2015).
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P. C. Diemoz and A. Olivo, “On the origin of contrast in edge illumination x-ray phase-contrast imaging,” Opt. Exp. 22, 28199–28214 (2014).
[Crossref]

C. K. Hagen, P. R. T. Munro, M. Endrizzi, P. C. Diemoz, and A. Olivo, “Low-dose phase contrast tomography with conventional x-ray sources,” Med. Phys. 41, 070701 (2014).
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C. K. Hagen, P. C. Diemoz, M. Endrizzi, and A. Olivo, “The effect of the spatial sampling rate on quantitative phase information extracted from planar and tomographic edge illumination x-ray phase contrast images,” J. Phys. D. Appl. Phys. 47, 455401 (2014).
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P. Diemoz, F. Vittoria, and A. Olivo, “Spatial resolution of edge illumination x-ray phase-contrast imaging,” Opt. Express 22, 15514–29 (2014).
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M. Endrizzi, P. C. Diemoz, C. K. Hagen, F. A. Vittoria, P. R. T. Munro, L. Rigon, D. Dreossi, F. Arfelli, F. C. M. Lopez, R. Longo, M. Marenzana, P. Delogu, A. Vincenzi, L. D. Ruvo, G. Spandre, A. Brez, R. Bellazzini, and A. Olivo, “Edge-illumination X-ray phase contrast imaging: matching the imaging method to the detector technology,” J. Instrum. 9, C11004 (2014).
[Crossref]

P. Munro and A. Olivo, “X-ray phase-contrast imaging with polychromatic sources and the concept of effective energy,” Phys. Rev. A 87, 053838 (2013).
[Crossref]

F. A. Vittoria, P. C. Diemoz, M. Endrizzi, L. Rigon, F. C. Lopez, D. Dreossi, P. R. T. Munro, and A. Olivo, “Strategies for efficient and fast wave optics simulation of coded-aperture and other x-ray phase-contrast imaging methods,” Appl. Opt. 52, 6940–6947 (2013).
[Crossref] [PubMed]

P. C. Diemoz, M. Endrizzi, C. E. Zapata, Z. D. Pesic, C. Rau, A. Bravin, I. K. Robinson, and A. Olivo, “X-ray phase-contrast imaging with nanoradian angular resolution,” Phys. Rev. Lett. 110, 138105 (2013).
[Crossref] [PubMed]

P. R. T. Munro, K. Ignatyev, R. D. Speller, and A. Olivo, “Phase and absorption retrieval using incoherent x-ray sources,” Proc. Natl. Acad. Sci. 109, 13922–13927 (2012).
[Crossref] [PubMed]

K. Ignatyev, P. R. T. Munro, R. D. Speller, and A. Olivo, “Effects of signal diffusion on x-ray phase contrast images,” Rev. Sci. Instrum. 82, 073702 (2011).
[Crossref] [PubMed]

P. R. T. Munro, K. Ignatyev, R. D. Speller, and A. Olivo, “Design of a novel phase contrast x-ray imaging system for mammography,” Phys. Med. Biol. 55, 4169–4185 (2010).
[Crossref] [PubMed]

A. Olivo and R. Speller, “Image formation principles in coded-aperture based x-ray phase contrast imaging,” Phys. Med. Biol. 53, 6461–6474 (2008).
[Crossref] [PubMed]

A. Olivo and R. Speller, “A coded-aperture technique allowing x-ray phase contrast imaging with conventional sources,” Appl. Phys. Lett. 91, 074106 (2007).
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A. Olivo and R. Speller, “Modelling of a novel x-ray phase contrast imaging technique based on coded apertures,” Phys. Med. Biol. 52, 6555–6573 (2007).
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A. Olivo, F. Arfelli, G. Cantatore, R. Longo, R. H. Menk, S. Pani, M. Prest, P. Poropat, L. Rigon, G. Tromba, E. Vallazza, and E. Castelli, “An innovative digital imaging set-up allowing a low-dose approach to phase contrast applications in the medical field,” Med. Phys. 28, 1610–1619 (2001).
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P. C. Diemoz, C. K. Hagen, M. Endrizzi, M. Minuti, R. Bellazzini, L. Urbani, P. De Coppi, and A. Olivo, “Single-shot X-ray phase-contrast computed tomography with non-microfocal laboratory sources,” Phys. Rev. Appl. (2017, in press).

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P. C. Diemoz, M. Endrizzi, C. E. Zapata, Z. D. Pesic, C. Rau, A. Bravin, I. K. Robinson, and A. Olivo, “X-ray phase-contrast imaging with nanoradian angular resolution,” Phys. Rev. Lett. 110, 138105 (2013).
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S. W. Wilkins, Y. I. Nesterets, T. E. Gureyev, S. C. Mayo, A. Pogany, and A. W. Stevenson, “On the evolution and relative merits of hard X-ray phase-contrast imaging methods,” Phil. Trans. R. Soc. 372, 20130021 (2014).
[Crossref]

T. E. Gureyev, Y. I. Nesterets, A. W. Stevenson, P. R. Miller, A. Pogany, and S. W. Wilkins, “Some simple rules for contrast, signal-to-noise and resolution in in-line x-ray phase-contrast imaging,” Opt. Express 16, 3223–3241 (2008).
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A. Olivo, F. Arfelli, G. Cantatore, R. Longo, R. H. Menk, S. Pani, M. Prest, P. Poropat, L. Rigon, G. Tromba, E. Vallazza, and E. Castelli, “An innovative digital imaging set-up allowing a low-dose approach to phase contrast applications in the medical field,” Med. Phys. 28, 1610–1619 (2001).
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M. Endrizzi, P. C. Diemoz, C. K. Hagen, F. A. Vittoria, P. R. T. Munro, L. Rigon, D. Dreossi, F. Arfelli, F. C. M. Lopez, R. Longo, M. Marenzana, P. Delogu, A. Vincenzi, L. D. Ruvo, G. Spandre, A. Brez, R. Bellazzini, and A. Olivo, “Edge-illumination X-ray phase contrast imaging: matching the imaging method to the detector technology,” J. Instrum. 9, C11004 (2014).
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F. A. Vittoria, P. C. Diemoz, M. Endrizzi, L. Rigon, F. C. Lopez, D. Dreossi, P. R. T. Munro, and A. Olivo, “Strategies for efficient and fast wave optics simulation of coded-aperture and other x-ray phase-contrast imaging methods,” Appl. Opt. 52, 6940–6947 (2013).
[Crossref] [PubMed]

A. Olivo, F. Arfelli, G. Cantatore, R. Longo, R. H. Menk, S. Pani, M. Prest, P. Poropat, L. Rigon, G. Tromba, E. Vallazza, and E. Castelli, “An innovative digital imaging set-up allowing a low-dose approach to phase contrast applications in the medical field,” Med. Phys. 28, 1610–1619 (2001).
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P. C. Diemoz, F. A. Vittoria, C. K. Hagen, M. Endrizzi, P. Coan, E. Brun, U. H. Wagner, C. Rau, I. K. Robinson, A. Bravin, and A. Olivo, “Single-image phase retrieval using an edge illumination x-ray phase-contrast imaging setup,” J. Synchrotron Radiat. 22, 1072–1077 (2015).
[Crossref] [PubMed]

P. C. Diemoz, M. Endrizzi, C. E. Zapata, Z. D. Pesic, C. Rau, A. Bravin, I. K. Robinson, and A. Olivo, “X-ray phase-contrast imaging with nanoradian angular resolution,” Phys. Rev. Lett. 110, 138105 (2013).
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M. Endrizzi, P. C. Diemoz, C. K. Hagen, F. A. Vittoria, P. R. T. Munro, L. Rigon, D. Dreossi, F. Arfelli, F. C. M. Lopez, R. Longo, M. Marenzana, P. Delogu, A. Vincenzi, L. D. Ruvo, G. Spandre, A. Brez, R. Bellazzini, and A. Olivo, “Edge-illumination X-ray phase contrast imaging: matching the imaging method to the detector technology,” J. Instrum. 9, C11004 (2014).
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M. Topperwien, M. Krenkel, D. Vincenz, F. Stober, A. M. Oelschlegel, J. Goldschmidt, and T. Salditt, “Three-dimensional mouse brain cytoarchitecture revealed by laboratory-based x-ray phase-contrast tomography,” Sci. Rep. 7, 42847 (2017).
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K. S. Morgan, D. M. Paganin, and K. K.W. Siu, “Quantitative single-exposure x-ray phase contrast imaging using a single attenuation grid,” Opt. Exp. 19, 19781–19789 (2011).
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M. Endrizzi, P. C. Diemoz, C. K. Hagen, F. A. Vittoria, P. R. T. Munro, L. Rigon, D. Dreossi, F. Arfelli, F. C. M. Lopez, R. Longo, M. Marenzana, P. Delogu, A. Vincenzi, L. D. Ruvo, G. Spandre, A. Brez, R. Bellazzini, and A. Olivo, “Edge-illumination X-ray phase contrast imaging: matching the imaging method to the detector technology,” J. Instrum. 9, C11004 (2014).
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R. Bellazzini, G. Spandre, A. Brez, M. Minuti, M. Pinchera, and P. Mozzo, “Chromatic X-ray imaging with a fine pitch CdTe sensor coupled to a large area photon counting pixel ASIC,” J. Instrum. 8, C02028 (2013).
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A. Olivo and R. Speller, “Image formation principles in coded-aperture based x-ray phase contrast imaging,” Phys. Med. Biol. 53, 6461–6474 (2008).
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A. Olivo and R. Speller, “A coded-aperture technique allowing x-ray phase contrast imaging with conventional sources,” Appl. Phys. Lett. 91, 074106 (2007).
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A. Olivo and R. Speller, “Modelling of a novel x-ray phase contrast imaging technique based on coded apertures,” Phys. Med. Biol. 52, 6555–6573 (2007).
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P. R. T. Munro, K. Ignatyev, R. D. Speller, and A. Olivo, “Phase and absorption retrieval using incoherent x-ray sources,” Proc. Natl. Acad. Sci. 109, 13922–13927 (2012).
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K. Ignatyev, P. R. T. Munro, R. D. Speller, and A. Olivo, “Effects of signal diffusion on x-ray phase contrast images,” Rev. Sci. Instrum. 82, 073702 (2011).
[Crossref] [PubMed]

P. R. T. Munro, K. Ignatyev, R. D. Speller, and A. Olivo, “Design of a novel phase contrast x-ray imaging system for mammography,” Phys. Med. Biol. 55, 4169–4185 (2010).
[Crossref] [PubMed]

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S. W. Wilkins, Y. I. Nesterets, T. E. Gureyev, S. C. Mayo, A. Pogany, and A. W. Stevenson, “On the evolution and relative merits of hard X-ray phase-contrast imaging methods,” Phil. Trans. R. Soc. 372, 20130021 (2014).
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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, 335–338 (1996).
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M. Topperwien, M. Krenkel, D. Vincenz, F. Stober, A. M. Oelschlegel, J. Goldschmidt, and T. Salditt, “Three-dimensional mouse brain cytoarchitecture revealed by laboratory-based x-ray phase-contrast tomography,” Sci. Rep. 7, 42847 (2017).
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A. Olivo, F. Arfelli, G. Cantatore, R. Longo, R. H. Menk, S. Pani, M. Prest, P. Poropat, L. Rigon, G. Tromba, E. Vallazza, and E. Castelli, “An innovative digital imaging set-up allowing a low-dose approach to phase contrast applications in the medical field,” Med. Phys. 28, 1610–1619 (2001).
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Urbani, L.

P. C. Diemoz, C. K. Hagen, M. Endrizzi, M. Minuti, R. Bellazzini, L. Urbani, P. De Coppi, and A. Olivo, “Single-shot X-ray phase-contrast computed tomography with non-microfocal laboratory sources,” Phys. Rev. Appl. (2017, in press).

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A. Olivo, F. Arfelli, G. Cantatore, R. Longo, R. H. Menk, S. Pani, M. Prest, P. Poropat, L. Rigon, G. Tromba, E. Vallazza, and E. Castelli, “An innovative digital imaging set-up allowing a low-dose approach to phase contrast applications in the medical field,” Med. Phys. 28, 1610–1619 (2001).
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M. Topperwien, M. Krenkel, D. Vincenz, F. Stober, A. M. Oelschlegel, J. Goldschmidt, and T. Salditt, “Three-dimensional mouse brain cytoarchitecture revealed by laboratory-based x-ray phase-contrast tomography,” Sci. Rep. 7, 42847 (2017).
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M. Endrizzi, F. A. Vittoria, G. Kallon, D. Basta, P. C. Diemoz, A. Vincenzi, P. Delogu, R. Bellazzini, and A. Olivo, “Achromatic approach to phase-based multi-modal imaging with conventional x-ray sources,” Opt. Exp. 23, 16473–16480 (2015).
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Vittoria, F. A.

P. C. Diemoz, F. A. Vittoria, C. K. Hagen, M. Endrizzi, P. Coan, E. Brun, U. H. Wagner, C. Rau, I. K. Robinson, A. Bravin, and A. Olivo, “Single-image phase retrieval using an edge illumination x-ray phase-contrast imaging setup,” J. Synchrotron Radiat. 22, 1072–1077 (2015).
[Crossref] [PubMed]

M. Endrizzi, F. A. Vittoria, G. Kallon, D. Basta, P. C. Diemoz, A. Vincenzi, P. Delogu, R. Bellazzini, and A. Olivo, “Achromatic approach to phase-based multi-modal imaging with conventional x-ray sources,” Opt. Exp. 23, 16473–16480 (2015).
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M. Endrizzi, P. C. Diemoz, C. K. Hagen, F. A. Vittoria, P. R. T. Munro, L. Rigon, D. Dreossi, F. Arfelli, F. C. M. Lopez, R. Longo, M. Marenzana, P. Delogu, A. Vincenzi, L. D. Ruvo, G. Spandre, A. Brez, R. Bellazzini, and A. Olivo, “Edge-illumination X-ray phase contrast imaging: matching the imaging method to the detector technology,” J. Instrum. 9, C11004 (2014).
[Crossref]

F. A. Vittoria, P. C. Diemoz, M. Endrizzi, L. Rigon, F. C. Lopez, D. Dreossi, P. R. T. Munro, and A. Olivo, “Strategies for efficient and fast wave optics simulation of coded-aperture and other x-ray phase-contrast imaging methods,” Appl. Opt. 52, 6940–6947 (2013).
[Crossref] [PubMed]

Wagner, U. H.

P. C. Diemoz, F. A. Vittoria, C. K. Hagen, M. Endrizzi, P. Coan, E. Brun, U. H. Wagner, C. Rau, I. K. Robinson, A. Bravin, and A. Olivo, “Single-image phase retrieval using an edge illumination x-ray phase-contrast imaging setup,” J. Synchrotron Radiat. 22, 1072–1077 (2015).
[Crossref] [PubMed]

Weber, L.

A. Horng, E. Brun, A. Mittone, S. Gasilov, L. Weber, T. Geith, S. Adam-neumair, S. D. Auweter, A. Bravin, M. F. Reiser, and P. Coan, “Cartilage and soft tissue imaging using x-rays,” Invest. Radiol. 49, 627–634 (2014).
[Crossref] [PubMed]

Weitkamp, T.

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

Wilkins, S. W.

S. W. Wilkins, Y. I. Nesterets, T. E. Gureyev, S. C. Mayo, A. Pogany, and A. W. Stevenson, “On the evolution and relative merits of hard X-ray phase-contrast imaging methods,” Phil. Trans. R. Soc. 372, 20130021 (2014).
[Crossref]

S. C. Mayo, A. W. Stevenson, and S. W. Wilkins, “In-line phase-contrast x-ray imaging and tomography for materials science,” Materials 5, 937–965 (2012).
[Crossref]

T. E. Gureyev, Y. I. Nesterets, A. W. Stevenson, P. R. Miller, A. Pogany, and S. W. Wilkins, “Some simple rules for contrast, signal-to-noise and resolution in in-line x-ray phase-contrast imaging,” Opt. Express 16, 3223–3241 (2008).
[Crossref] [PubMed]

D. Paganin, S. C. Mayo, T. E. Gureyev, P. R. Miller, and S. W. Wilkins, “Simultaneous phase and amplitude extraction from a single defocused image of a homogeneous object,” J. Microsc. 206, 33–40 (2002).
[Crossref] [PubMed]

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

Zapata, C. E.

P. C. Diemoz, M. Endrizzi, C. E. Zapata, Z. D. Pesic, C. Rau, A. Bravin, I. K. Robinson, and A. Olivo, “X-ray phase-contrast imaging with nanoradian angular resolution,” Phys. Rev. Lett. 110, 138105 (2013).
[Crossref] [PubMed]

Appl. Opt. (1)

Appl. Phys. Lett. (1)

A. Olivo and R. Speller, “A coded-aperture technique allowing x-ray phase contrast imaging with conventional sources,” Appl. Phys. Lett. 91, 074106 (2007).
[Crossref]

IEEE Trans. Syst. Man. Cybern. (1)

N. Otsu, “A threshold selection method from gray-level histograms,” IEEE Trans. Syst. Man. Cybern. 9, 62–66 (1979).
[Crossref]

Invest. Radiol. (1)

A. Horng, E. Brun, A. Mittone, S. Gasilov, L. Weber, T. Geith, S. Adam-neumair, S. D. Auweter, A. Bravin, M. F. Reiser, and P. Coan, “Cartilage and soft tissue imaging using x-rays,” Invest. Radiol. 49, 627–634 (2014).
[Crossref] [PubMed]

J. Instrum. (2)

R. Bellazzini, G. Spandre, A. Brez, M. Minuti, M. Pinchera, and P. Mozzo, “Chromatic X-ray imaging with a fine pitch CdTe sensor coupled to a large area photon counting pixel ASIC,” J. Instrum. 8, C02028 (2013).
[Crossref]

M. Endrizzi, P. C. Diemoz, C. K. Hagen, F. A. Vittoria, P. R. T. Munro, L. Rigon, D. Dreossi, F. Arfelli, F. C. M. Lopez, R. Longo, M. Marenzana, P. Delogu, A. Vincenzi, L. D. Ruvo, G. Spandre, A. Brez, R. Bellazzini, and A. Olivo, “Edge-illumination X-ray phase contrast imaging: matching the imaging method to the detector technology,” J. Instrum. 9, C11004 (2014).
[Crossref]

J. Microsc. (1)

D. Paganin, S. C. Mayo, T. E. Gureyev, P. R. Miller, and S. W. Wilkins, “Simultaneous phase and amplitude extraction from a single defocused image of a homogeneous object,” J. Microsc. 206, 33–40 (2002).
[Crossref] [PubMed]

J. Opt. Soc. Am. (1)

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

C. K. Hagen, P. C. Diemoz, M. Endrizzi, and A. Olivo, “The effect of the spatial sampling rate on quantitative phase information extracted from planar and tomographic edge illumination x-ray phase contrast images,” J. Phys. D. Appl. Phys. 47, 455401 (2014).
[Crossref]

J. Synchrotron Radiat. (1)

P. C. Diemoz, F. A. Vittoria, C. K. Hagen, M. Endrizzi, P. Coan, E. Brun, U. H. Wagner, C. Rau, I. K. Robinson, A. Bravin, and A. Olivo, “Single-image phase retrieval using an edge illumination x-ray phase-contrast imaging setup,” J. Synchrotron Radiat. 22, 1072–1077 (2015).
[Crossref] [PubMed]

Jpn. J. Appl. Phys. (1)

A. Momose, S. Kawamoto, I. Koyama, Y. Hamaishi, K. Takai, and Y. Suzuki, “Demonstration of x-ray talbot interferometry,” Jpn. J. Appl. Phys. 42, L866–L868 (2003).
[Crossref]

Materials (1)

S. C. Mayo, A. W. Stevenson, and S. W. Wilkins, “In-line phase-contrast x-ray imaging and tomography for materials science,” Materials 5, 937–965 (2012).
[Crossref]

Med. Phys. (2)

A. Olivo, F. Arfelli, G. Cantatore, R. Longo, R. H. Menk, S. Pani, M. Prest, P. Poropat, L. Rigon, G. Tromba, E. Vallazza, and E. Castelli, “An innovative digital imaging set-up allowing a low-dose approach to phase contrast applications in the medical field,” Med. Phys. 28, 1610–1619 (2001).
[Crossref] [PubMed]

C. K. Hagen, P. R. T. Munro, M. Endrizzi, P. C. Diemoz, and A. Olivo, “Low-dose phase contrast tomography with conventional x-ray sources,” Med. Phys. 41, 070701 (2014).
[Crossref] [PubMed]

Nature (1)

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

Nature Phys. (1)

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

Opt. Exp. (3)

K. S. Morgan, D. M. Paganin, and K. K.W. Siu, “Quantitative single-exposure x-ray phase contrast imaging using a single attenuation grid,” Opt. Exp. 19, 19781–19789 (2011).
[Crossref]

M. Endrizzi, F. A. Vittoria, G. Kallon, D. Basta, P. C. Diemoz, A. Vincenzi, P. Delogu, R. Bellazzini, and A. Olivo, “Achromatic approach to phase-based multi-modal imaging with conventional x-ray sources,” Opt. Exp. 23, 16473–16480 (2015).
[Crossref]

P. C. Diemoz and A. Olivo, “On the origin of contrast in edge illumination x-ray phase-contrast imaging,” Opt. Exp. 22, 28199–28214 (2014).
[Crossref]

Opt. Express (3)

Phil. Trans. R. Soc. (1)

S. W. Wilkins, Y. I. Nesterets, T. E. Gureyev, S. C. Mayo, A. Pogany, and A. W. Stevenson, “On the evolution and relative merits of hard X-ray phase-contrast imaging methods,” Phil. Trans. R. Soc. 372, 20130021 (2014).
[Crossref]

Phys. Med. Biol. (6)

A. Bravin, P. Coan, and P. Suortti, “X-ray phase-contrast imaging: from pre-clinical applications towards clinics,” Phys. Med. Biol. 58, R1–R35 (2013).
[Crossref]

D. Macindoe, M. J. Kitchen, S. C. Irvine, A. Fouras, and K. S. Morgan, “Requirements for dynamical differential phase contrast x-ray imaging with a laboratory source,” Phys. Med. Biol. 61, 8720–8735 (2016).
[Crossref] [PubMed]

A. Olivo and R. Speller, “Image formation principles in coded-aperture based x-ray phase contrast imaging,” Phys. Med. Biol. 53, 6461–6474 (2008).
[Crossref] [PubMed]

A. Olivo and R. Speller, “Modelling of a novel x-ray phase contrast imaging technique based on coded apertures,” Phys. Med. Biol. 52, 6555–6573 (2007).
[Crossref] [PubMed]

M. A. Beltran, D. M. Paganin, K. K. W. Siu, A. Fouras, S. B. Hooper, D. H. Reser, and M. J. Kitchen, “Interface-specific x-ray phase retrieval tomography of complex biological organs,” Phys. Med. Biol. 56, 7353–7369 (2011).
[Crossref] [PubMed]

P. R. T. Munro, K. Ignatyev, R. D. Speller, and A. Olivo, “Design of a novel phase contrast x-ray imaging system for mammography,” Phys. Med. Biol. 55, 4169–4185 (2010).
[Crossref] [PubMed]

Phys. Rev. A (1)

P. Munro and A. Olivo, “X-ray phase-contrast imaging with polychromatic sources and the concept of effective energy,” Phys. Rev. A 87, 053838 (2013).
[Crossref]

Phys. Rev. Lett. (2)

P. C. Diemoz, M. Endrizzi, C. E. Zapata, Z. D. Pesic, C. Rau, A. Bravin, I. K. Robinson, and A. Olivo, “X-ray phase-contrast imaging with nanoradian angular resolution,” Phys. Rev. Lett. 110, 138105 (2013).
[Crossref] [PubMed]

F. Pfeiffer, C. Kottler, O. Bunk, and C. David, “Hard x-ray phase tomography with low-brilliance sources,” Phys. Rev. Lett. 98, 108105 (2007).
[Crossref] [PubMed]

Proc. Natl. Acad. Sci. (1)

P. R. T. Munro, K. Ignatyev, R. D. Speller, and A. Olivo, “Phase and absorption retrieval using incoherent x-ray sources,” Proc. Natl. Acad. Sci. 109, 13922–13927 (2012).
[Crossref] [PubMed]

Rev. Sci. Instrum. (2)

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

K. Ignatyev, P. R. T. Munro, R. D. Speller, and A. Olivo, “Effects of signal diffusion on x-ray phase contrast images,” Rev. Sci. Instrum. 82, 073702 (2011).
[Crossref] [PubMed]

Sci. Rep. (1)

M. Topperwien, M. Krenkel, D. Vincenz, F. Stober, A. M. Oelschlegel, J. Goldschmidt, and T. Salditt, “Three-dimensional mouse brain cytoarchitecture revealed by laboratory-based x-ray phase-contrast tomography,” Sci. Rep. 7, 42847 (2017).
[Crossref] [PubMed]

Other (3)

P. C. Diemoz, C. K. Hagen, M. Endrizzi, M. Minuti, R. Bellazzini, L. Urbani, P. De Coppi, and A. Olivo, “Single-shot X-ray phase-contrast computed tomography with non-microfocal laboratory sources,” Phys. Rev. Appl. (2017, in press).

International Commission on Radiation Units and Measurements, “Tissue substitutes in radiation dosimetry and measurement,” ICRU Report44 (1989).

D. M. Paganin, Coherent X-Ray Optics, (Oxford University, 2009).

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Figures (4)

Fig. 1
Fig. 1 A top-view schematic of the laboratory implementation of an EI setup. The sample’s rotation axis is aligned with the y-direction.
Fig. 2
Fig. 2 Retrieved projected thicknesses for water, aluminium and LDPE, using values reported in tables 1 and 2. The red arrows point at the material of focus in each case. The experimental profiles were averaged over 5 pixel rows.
Fig. 3
Fig. 3 Simulated CT slices focusing on the water-air (a), aluminium-water (b) and LDPE-water interfaces, using E=17.5 keV. The white arrows indicate artefacts arising from locally incorrect choices of refractive index values. Simulated (d) and experimental (e) spliced slices along with a plot of profiles through them (f), in the position indicated by the red line in (d). The difference in the retrieved values of the aluminium rod in (f, see arrow) is expected and is due to the difference in effective energies (see tables 1 and 2).
Fig. 4
Fig. 4 Axial slices of a chicken bone sample reconstructed using values optimized for the cylinder-air interface (a), intra-soft tissue contrast (b) and bone-soft tissue interface (c). Panel (d) shows a composite slice obtained by splicing slices (a–c).

Tables (2)

Tables Icon

Table 1 Input Refractive Index Values - Phantom, Simulation

Tables Icon

Table 2 Input Refractive Index Values - Phantom, Experimental

Equations (15)

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S n = { T T C ( y e ) C ( y e ) k 1 z 2 y ϕ k 1 z 2 x [ T x ϕ ] } * LSF y ,
T ( x , y ) = exp ( 2 k β t ( x , y ) ) ,
ϕ ( x , y ) = k δ t ( x , y ) .
t = 1 μ log [ F 1 { F { S n } M T F y ( k y ) [ 1 + i J E I δ μ 1 k y + z 2 δ μ 1 k x 2 ] } ] ,
T ( x , y ) = exp [ ( μ 1 t 1 ( x , y ) + μ j t j ( x , y ) ) ] ,
ϕ ( x , y ) = k ( δ 1 t 1 ( x , y ) + δ j t j ( x , y ) ) ,
T = exp [ μ 1 A ] exp [ ( μ j μ 1 ) t j ] ,
x , y ϕ = k [ δ 1 x , y A + ( δ j δ 1 ) x , y t j ] .
T C ( y e ) C ( y e ) k 1 z 2 y ϕ = ( δ j δ 1 ) J E I exp [ μ 1 A ] exp [ ( μ j μ 1 ) t j ] y t j = δ j δ 1 μ j μ 1 J E I exp [ μ 1 A ] y exp [ ( μ j μ 1 ) t j ] ,
exp [ ( μ j μ 1 ) t j ] y t j = 1 μ j μ 1 y exp [ ( μ j μ 1 ) t j ] ,
k 1 z 2 x [ T x ϕ ] = z 2 ( δ j δ 1 ) exp [ μ 1 A ] x { exp [ ( μ j μ 1 ) t j ] x t j } = δ j δ 1 μ j μ 1 z 2 exp [ μ 1 A ] x 2 exp [ ( μ j μ 1 ) t j ] ,
x { exp ( ( μ j μ 1 ) t j ) x t j } = 1 μ j μ 1 x 2 exp [ ( μ j μ 1 ) t j ] .
S n exp [ μ 1 A ] = [ L S F y * ( 1 γ J E I y γ z 2 x 2 ) ] exp [ Δ μ t j ] ,
F { S n exp [ μ 1 A ] } = [ MTF y ( f y ) ( 1 2 π i γ J E I f y + 4 π 2 γ z 2 f x 2 ) ] F { exp [ Δ μ t j ] } .
t j = 1 Δ μ log [ F 1 { F { S n / exp [ μ 1 A ] } MTF y ( f y ) ( 1 2 π i γ J E I f y + 4 π 2 γ z 2 f x 2 ) } ] .

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