Abstract

In X-ray computed tomography (CT) increased information requirements (e.g. increased resolution) typically lead to a concurrent increase in the required number of viewing angles, scanning time and delivered dose. We demonstrate that using phase-contrast imaging it is possible to “dissect” two- and three-material objects into their component materials, which in combination with binary tomographic techniques allows us to satisfy increased information requirements without taking the usual images at additional viewing angles. This imaging scheme reduces the scanning time and dose delivered to samples by at least an order of magnitude when compared to conventional X-ray CT. The effects of noise on our reconstruction scheme are investigated for simulated data. Finally, a slice through a glass tube filled with silica and water is reconstructed from 18 projection images taken on an X-ray ultra Microscope (XuM).

© 2008 Optical Society of America

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  57. T. E. Gureyev, A. W. Stevenson, D. M. Paganin, T. Weitkamp, A. Snigirev, I. Snigireva, and S. W. Wilkins, "Quantitative analysis of two component samples using in-line hard X-ray images," J. Synchrotron Rad. 9, 148-153 (2002).
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2008 (1)

2007 (5)

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]

M. Engelhardt, J. Baumann, M. Schuster, C. Kottler, F. Pfeiffer, O. Bunk, and C. David, "High-resolution differential phase contrast imaging using a magnifying projection geometry with a microfocus x-ray source," Appl. Phys. Lett. 90, 224101 (2007).
[CrossRef]

G. R. Myers, S. C. Mayo, T. E. Gureyev, D. M. Paganin, and S. W. Wilkins, "Polychromatic cone-beam phase-contrast tomography," Phys. Rev. A 76, 045804 (2007).
[CrossRef]

T. E. Gureyev, Y. I. Nesterets, K. M. Pavlov, and S. W. Wilkins, "Computed tomography with linear shift-invariant optical systems," J. Opt. Soc. Am. A 24, 2230-2241 (2007).
[CrossRef]

F. Pfeiffer, O. Bunk, C. Kottler, and C. David, "Tomographic reconstruction of three-dimensional objects from hard X-ray differential phase contrast projection images," Nucl. Instrum. Methods Phys. Res. A 580, 925-928 (2007).
[CrossRef]

2006 (10)

A. Groso, R. Abela, and M. Stampanoni, "Implementation of a fast method for high resolution phase contrast tomography," Opt. Express 14, 8103-8110 (2006).
[CrossRef] [PubMed]

J. Miao, C.-C. Chen, C. Song, Y. Nishino, Y. Kohmura, T. Ishikawa, D. Ramunno-Johnson, T.-K. Lee, and S. H. Risbud, "Three-dimensional GaN-Ga2O3 core shell structure revealed by X-ray diffraction microscopy," Phys. Rev. Lett. 97, 215503 (2006).
[CrossRef] [PubMed]

A. V. Bronnikov, "Phase contrast CT: Fundamental theorem and fast image reconstruction algorithms," Proc. SPIE 6318, 63180Q (2006).
[CrossRef]

J. ?ehá?ek, Z. Hradil, J. Pe?ina, S. Pascazio, P. Facchi, and M. Zawisky, "Advanced neutron imaging and sensing," Adv. Imaging Electron. Phys. 142, 53-157 (2006).
[CrossRef]

Y. I. Nesterets, T. E. Gureyev, and S. W. Wilkins, "General reconstruction formulas for analyzer-based computed tomography," Appl. Phys. Lett. 89, 264103 (2006).
[CrossRef]

A. Momose, W. Yashiro, Y. Takeda, Y. Suzuki, and T. Hattori, "Phase tomography by X-ray Talbot interferometry for biological imaging," Jpn. J. Appl. Phys. 45, 5254-5262 (2006).
[CrossRef]

T. E. Gureyev, D. M. Paganin, G. R. Myers, Ya. I. Nesterets, and S. W. Wilkins, "Phase-and-amplitude computer tomography," Appl. Phys. Lett. 89, 034102 (2006).
[CrossRef]

A. Alpers and P. Gritzmann, "On stability, error correction, and noise compensation in discrete tomography," SIAM J. Discrete Math. 20, 227-239 (2006).
[CrossRef]

K. J. Batenburg, "A network flow algorithm for binary image reconstruction from few projections," Lecture Notes in Comput.Science 4245, 86-97 (2006).
[CrossRef]

A. Alpers, H. F. Poulsen, E. Knudsen, and G. T. Herman, "A discrete tomography algorithm for improving the quality of 3DXRD grain maps," J. Appl. Cryst. 39, 582-588 (2006).
[CrossRef]

2005 (3)

2004 (2)

S. Weber, T. Schüle, J. Hornegger, and C. Schnörr, "Binary tomography by iterating linear programs from noisy projections," Lecture Notes in Comput.Science 3322, 38-51 (2004).
[CrossRef]

D. M. Paganin, A. Barty, P. J. McMahon, and K. A. Nugent, "Quantitative phase-amplitude microscopy. III. The effects of noise," J. Microsc. 214, 51-61 (2004).
[CrossRef] [PubMed]

2003 (1)

2002 (2)

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]

T. E. Gureyev, A. W. Stevenson, D. M. Paganin, T. Weitkamp, A. Snigirev, I. Snigireva, and S. W. Wilkins, "Quantitative analysis of two component samples using in-line hard X-ray images," J. Synchrotron Rad. 9, 148-153 (2002).
[CrossRef]

2001 (3)

A. Alpers, P. Gritzmann, and L. Thorens, "Stability and instability in discrete tomography," Lecture Notes in Comput.Science 2243, 175-186 (2001).
[CrossRef]

L. Hajdu and R. Tijdeman, "An algorithm for discrete tomography," Linear Algebra Appl. 339, 119-128 (2001).
[CrossRef]

K. M. Pavlov, C. M. Kewish, J. R. Davis, and M. J. Morgan, "A variant on the geometrical optics approximation in diffraction enhanced tomography," J. Phys. D: Appl. Phys. 34, A168-A172 (2001).
[CrossRef]

2000 (3)

P. Gritzmann, S. de Vries, and M. Wiegelmann, "Approximating binary images from discrete x-rays," SIAM J. Optimization 11, 522-546 (2000).
[CrossRef]

F. A. Dilmanian, Z. Zhong, B. Ren, X. Y. Wu, L. D. Chapman, I. Orion, and W. C. Thomlinson, "Computed tomography of x-ray index of refraction using the diffraction enhanced imaging method," Phys. Med. Biol. 45, 933-946 (2000).
[CrossRef] [PubMed]

R. Fitzgerald, "Phase-sensitive x-ray imaging," Phys. Today 53, 23-26 (2000).
[CrossRef]

1999 (4)

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

A. V. Bronnikov, "Reconstruction formulas in phase-contrast tomography," Opt. Commun. 171, 239-244 (1999).
[CrossRef]

R. J. Gardner, P. Gritzmann, and D. Prangenberg, "On the computational complexity of reconstructing lattice sets from their X-rays," Discrete Math. 202, 45-71 (1999).
[CrossRef]

T. E. Gureyev, "Transport of intensity equation for beams in an arbitrary state of temporal and spatial coherence," Optik 110, 263-266 (1999).

1998 (1)

P. Gritzmann, D. Prangenberg, S. de Vries, and M. Wiegelmann, "Success and failure of certain reconstruction and uniqueness algorithms in discrete tomography," Int. J. Imaging Syst. Technol. 9, 101-109 (1998).
[CrossRef]

1997 (2)

P. Fishburn, P. Schwander, L. Shepp, and R. J. Vanderbei, "The discrete Radon transform and its approximate inversion via linear programming," Discrete Appl. Math. 75, 39-61 (1997).
[CrossRef]

R. J. Gardner and P. Gritzmann, "Discrete tomography: Determination of finite sets by x-rays," Trans. Amer. Math. Soc. 349, 2271-2295 (1997).
[CrossRef]

1996 (3)

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]

U. Bonse and F. Busch, "X-ray computed microtomography (µCT) using synchrotron radiation (SR)," Prog. Biophys. Mol. Biol. 65, 133-169 (1996).
[CrossRef] [PubMed]

C. Raven, A. Snigirev, I. Snigireva, P. Spanne, A. Souvorov, and V. Kohn, "Phase-contrast microtomography with coherent high-energy synchrotron x rays," Appl. Phys. Lett. 69, 1826-1828 (1996).
[CrossRef]

1995 (1)

A. Momose, T. Takeda, and Y. Itai, "Phase-contrast X-ray computed tomography for observing biological specimens and organic materials," Rev. Sci. Instrum. 66, 1434-1436 (1995).
[CrossRef]

1988 (1)

P. Besl, "Active, optical range imaging sensors", Machine Vision and Applications 1, 127-152 (1988).
[CrossRef]

1986 (1)

A. J. Devaney, "Reconstructive tomography with diffracting wave-fields," Inv.Problems 2, 161-183 (1986).
[CrossRef]

1983 (1)

1980 (1)

W. R. Brody, G. Butt, A. Hall, and A. Macovski, "A method for selective tissue and bone visualization using dual energy scanned projection radiography," Med. Phys. 8, 353-357 (1980).
[CrossRef]

1974 (1)

L. A. Shepp and B. F. Logan, "Reconstructing interior head tissue from X-ray transmissions," IEEE Trans. Nucl. Sci. 21, 228-236 (1974).
[CrossRef]

1969 (1)

E. Wolf, "Three-dimensional structure determination of semi-transparent objects from holographic data," Opt. Commun. 1, 153-156 (1969).
[CrossRef]

1968 (3)

J. R. Shewell and E. Wolf, "Inverse diffraction and a new reciprocity theorem," J. Opt. Soc. Am. A 58, 1596-1603 (1968).
[CrossRef]

E. Lalor, "Inverse wave propagator," J. Math. Phys. 9, 2001-2006 (1968).
[CrossRef]

G. C. Sherman, "Diffracted wave fields expressible by plane-wave expansions containing only homogenous components," Phys. Rev. Lett. 21, 761-764 (1968).
[CrossRef]

1967 (1)

E. Wolf and J. R. Shewell, "The inverse wave propagator," Phys. Lett. A 25, 417-418 (1967); see also E. Wolf and J. R. Shewell, "Errata," Phys. Lett. A 26, 104 (1967).
[CrossRef]

?ehá?ek, J.

J. ?ehá?ek, Z. Hradil, J. Pe?ina, S. Pascazio, P. Facchi, and M. Zawisky, "Advanced neutron imaging and sensing," Adv. Imaging Electron. Phys. 142, 53-157 (2006).
[CrossRef]

Abela, R.

Alpers, A.

A. Alpers, H. F. Poulsen, E. Knudsen, and G. T. Herman, "A discrete tomography algorithm for improving the quality of 3DXRD grain maps," J. Appl. Cryst. 39, 582-588 (2006).
[CrossRef]

A. Alpers and P. Gritzmann, "On stability, error correction, and noise compensation in discrete tomography," SIAM J. Discrete Math. 20, 227-239 (2006).
[CrossRef]

A. Alpers, P. Gritzmann, and L. Thorens, "Stability and instability in discrete tomography," Lecture Notes in Comput.Science 2243, 175-186 (2001).
[CrossRef]

Anastasio, M. A.

Barty, A.

D. M. Paganin, A. Barty, P. J. McMahon, and K. A. Nugent, "Quantitative phase-amplitude microscopy. III. The effects of noise," J. Microsc. 214, 51-61 (2004).
[CrossRef] [PubMed]

Baruchel, J.

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

Batenburg, K. J.

K. J. Batenburg, "A network flow algorithm for binary image reconstruction from few projections," Lecture Notes in Comput.Science 4245, 86-97 (2006).
[CrossRef]

Baumann, J.

M. Engelhardt, J. Baumann, M. Schuster, C. Kottler, F. Pfeiffer, O. Bunk, and C. David, "High-resolution differential phase contrast imaging using a magnifying projection geometry with a microfocus x-ray source," Appl. Phys. Lett. 90, 224101 (2007).
[CrossRef]

Besl, P.

P. Besl, "Active, optical range imaging sensors", Machine Vision and Applications 1, 127-152 (1988).
[CrossRef]

Bonse, U.

U. Bonse and F. Busch, "X-ray computed microtomography (µCT) using synchrotron radiation (SR)," Prog. Biophys. Mol. Biol. 65, 133-169 (1996).
[CrossRef] [PubMed]

Brody, W. R.

W. R. Brody, G. Butt, A. Hall, and A. Macovski, "A method for selective tissue and bone visualization using dual energy scanned projection radiography," Med. Phys. 8, 353-357 (1980).
[CrossRef]

Bronnikov, A. V.

A. V. Bronnikov, "Phase contrast CT: Fundamental theorem and fast image reconstruction algorithms," Proc. SPIE 6318, 63180Q (2006).
[CrossRef]

A. V. Bronnikov, "Reconstruction formulas in phase-contrast tomography," Opt. Commun. 171, 239-244 (1999).
[CrossRef]

Bunk, O.

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G. R. Myers, D. M. Paganin, T. E. Gureyev, and S. C. Mayo, "Phase-contrast tomography of single-material objects from few projections," Opt. Express 16, 908-919 (2008).
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J. Miao, C.-C. Chen, C. Song, Y. Nishino, Y. Kohmura, T. Ishikawa, D. Ramunno-Johnson, T.-K. Lee, and S. H. Risbud, "Three-dimensional GaN-Ga2O3 core shell structure revealed by X-ray diffraction microscopy," Phys. Rev. Lett. 97, 215503 (2006).
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K. M. Pavlov, C. M. Kewish, J. R. Davis, and M. J. Morgan, "A variant on the geometrical optics approximation in diffraction enhanced tomography," J. Phys. D: Appl. Phys. 34, A168-A172 (2001).
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A. Alpers, H. F. Poulsen, E. Knudsen, and G. T. Herman, "A discrete tomography algorithm for improving the quality of 3DXRD grain maps," J. Appl. Cryst. 39, 582-588 (2006).
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J. Miao, C.-C. Chen, C. Song, Y. Nishino, Y. Kohmura, T. Ishikawa, D. Ramunno-Johnson, T.-K. Lee, and S. H. Risbud, "Three-dimensional GaN-Ga2O3 core shell structure revealed by X-ray diffraction microscopy," Phys. Rev. Lett. 97, 215503 (2006).
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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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W. R. Brody, G. Butt, A. Hall, and A. Macovski, "A method for selective tissue and bone visualization using dual energy scanned projection radiography," Med. Phys. 8, 353-357 (1980).
[CrossRef]

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G. R. Myers, D. M. Paganin, T. E. Gureyev, and S. C. Mayo, "Phase-contrast tomography of single-material objects from few projections," Opt. Express 16, 908-919 (2008).
[CrossRef] [PubMed]

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[CrossRef]

S. C. Mayo, T. J. Davis, T. E. Gureyev, P. R. Miller, D. Paganin, A. Pogany, A. W. Stevenson, and S. W. Wilkins, "X-ray phase-contrast microscopy and microtomography," Opt. Express 11, 2289-2302 (2003).
[CrossRef] [PubMed]

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D. M. Paganin, A. Barty, P. J. McMahon, and K. A. Nugent, "Quantitative phase-amplitude microscopy. III. The effects of noise," J. Microsc. 214, 51-61 (2004).
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J. Miao, C.-C. Chen, C. Song, Y. Nishino, Y. Kohmura, T. Ishikawa, D. Ramunno-Johnson, T.-K. Lee, and S. H. Risbud, "Three-dimensional GaN-Ga2O3 core shell structure revealed by X-ray diffraction microscopy," Phys. Rev. Lett. 97, 215503 (2006).
[CrossRef] [PubMed]

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S. C. Mayo, T. J. Davis, T. E. Gureyev, P. R. Miller, D. Paganin, A. Pogany, A. W. Stevenson, and S. W. Wilkins, "X-ray phase-contrast microscopy and microtomography," Opt. Express 11, 2289-2302 (2003).
[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]

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A. Momose, W. Yashiro, Y. Takeda, Y. Suzuki, and T. Hattori, "Phase tomography by X-ray Talbot interferometry for biological imaging," Jpn. J. Appl. Phys. 45, 5254-5262 (2006).
[CrossRef]

A. Momose, T. Takeda, and Y. Itai, "Phase-contrast X-ray computed tomography for observing biological specimens and organic materials," Rev. Sci. Instrum. 66, 1434-1436 (1995).
[CrossRef]

Morgan, M. J.

K. M. Pavlov, C. M. Kewish, J. R. Davis, and M. J. Morgan, "A variant on the geometrical optics approximation in diffraction enhanced tomography," J. Phys. D: Appl. Phys. 34, A168-A172 (2001).
[CrossRef]

Myers, G. R.

G. R. Myers, D. M. Paganin, T. E. Gureyev, and S. C. Mayo, "Phase-contrast tomography of single-material objects from few projections," Opt. Express 16, 908-919 (2008).
[CrossRef] [PubMed]

G. R. Myers, S. C. Mayo, T. E. Gureyev, D. M. Paganin, and S. W. Wilkins, "Polychromatic cone-beam phase-contrast tomography," Phys. Rev. A 76, 045804 (2007).
[CrossRef]

T. E. Gureyev, D. M. Paganin, G. R. Myers, Ya. I. Nesterets, and S. W. Wilkins, "Phase-and-amplitude computer tomography," Appl. Phys. Lett. 89, 034102 (2006).
[CrossRef]

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T. E. Gureyev, Y. I. Nesterets, K. M. Pavlov, and S. W. Wilkins, "Computed tomography with linear shift-invariant optical systems," J. Opt. Soc. Am. A 24, 2230-2241 (2007).
[CrossRef]

Y. I. Nesterets, T. E. Gureyev, and S. W. Wilkins, "General reconstruction formulas for analyzer-based computed tomography," Appl. Phys. Lett. 89, 264103 (2006).
[CrossRef]

Nesterets, Ya. I.

T. E. Gureyev, D. M. Paganin, G. R. Myers, Ya. I. Nesterets, and S. W. Wilkins, "Phase-and-amplitude computer tomography," Appl. Phys. Lett. 89, 034102 (2006).
[CrossRef]

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J. Miao, C.-C. Chen, C. Song, Y. Nishino, Y. Kohmura, T. Ishikawa, D. Ramunno-Johnson, T.-K. Lee, and S. H. Risbud, "Three-dimensional GaN-Ga2O3 core shell structure revealed by X-ray diffraction microscopy," Phys. Rev. Lett. 97, 215503 (2006).
[CrossRef] [PubMed]

Nugent, K. A.

D. M. Paganin, A. Barty, P. J. McMahon, and K. A. Nugent, "Quantitative phase-amplitude microscopy. III. The effects of noise," J. Microsc. 214, 51-61 (2004).
[CrossRef] [PubMed]

Orion, I.

F. A. Dilmanian, Z. Zhong, B. Ren, X. Y. Wu, L. D. Chapman, I. Orion, and W. C. Thomlinson, "Computed tomography of x-ray index of refraction using the diffraction enhanced imaging method," Phys. Med. Biol. 45, 933-946 (2000).
[CrossRef] [PubMed]

Paganin, D.

S. C. Mayo, T. J. Davis, T. E. Gureyev, P. R. Miller, D. Paganin, A. Pogany, A. W. Stevenson, and S. W. Wilkins, "X-ray phase-contrast microscopy and microtomography," Opt. Express 11, 2289-2302 (2003).
[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]

Paganin, D. M.

G. R. Myers, D. M. Paganin, T. E. Gureyev, and S. C. Mayo, "Phase-contrast tomography of single-material objects from few projections," Opt. Express 16, 908-919 (2008).
[CrossRef] [PubMed]

G. R. Myers, S. C. Mayo, T. E. Gureyev, D. M. Paganin, and S. W. Wilkins, "Polychromatic cone-beam phase-contrast tomography," Phys. Rev. A 76, 045804 (2007).
[CrossRef]

T. E. Gureyev, D. M. Paganin, G. R. Myers, Ya. I. Nesterets, and S. W. Wilkins, "Phase-and-amplitude computer tomography," Appl. Phys. Lett. 89, 034102 (2006).
[CrossRef]

D. M. Paganin, A. Barty, P. J. McMahon, and K. A. Nugent, "Quantitative phase-amplitude microscopy. III. The effects of noise," J. Microsc. 214, 51-61 (2004).
[CrossRef] [PubMed]

T. E. Gureyev, A. W. Stevenson, D. M. Paganin, T. Weitkamp, A. Snigirev, I. Snigireva, and S. W. Wilkins, "Quantitative analysis of two component samples using in-line hard X-ray images," J. Synchrotron Rad. 9, 148-153 (2002).
[CrossRef]

Pascazio, S.

J. ?ehá?ek, Z. Hradil, J. Pe?ina, S. Pascazio, P. Facchi, and M. Zawisky, "Advanced neutron imaging and sensing," Adv. Imaging Electron. Phys. 142, 53-157 (2006).
[CrossRef]

Pavlov, K. M.

T. E. Gureyev, Y. I. Nesterets, K. M. Pavlov, and S. W. Wilkins, "Computed tomography with linear shift-invariant optical systems," J. Opt. Soc. Am. A 24, 2230-2241 (2007).
[CrossRef]

K. M. Pavlov, C. M. Kewish, J. R. Davis, and M. J. Morgan, "A variant on the geometrical optics approximation in diffraction enhanced tomography," J. Phys. D: Appl. Phys. 34, A168-A172 (2001).
[CrossRef]

Pe?ina, J.

J. ?ehá?ek, Z. Hradil, J. Pe?ina, S. Pascazio, P. Facchi, and M. Zawisky, "Advanced neutron imaging and sensing," Adv. Imaging Electron. Phys. 142, 53-157 (2006).
[CrossRef]

Pfeiffer, F.

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]

M. Engelhardt, J. Baumann, M. Schuster, C. Kottler, F. Pfeiffer, O. Bunk, and C. David, "High-resolution differential phase contrast imaging using a magnifying projection geometry with a microfocus x-ray source," Appl. Phys. Lett. 90, 224101 (2007).
[CrossRef]

F. Pfeiffer, O. Bunk, C. Kottler, and C. David, "Tomographic reconstruction of three-dimensional objects from hard X-ray differential phase contrast projection images," Nucl. Instrum. Methods Phys. Res. A 580, 925-928 (2007).
[CrossRef]

Pogany, A.

S. C. Mayo, T. J. Davis, T. E. Gureyev, P. R. Miller, D. Paganin, A. Pogany, A. W. Stevenson, and S. W. Wilkins, "X-ray phase-contrast microscopy and microtomography," Opt. Express 11, 2289-2302 (2003).
[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]

Poulsen, H. F.

A. Alpers, H. F. Poulsen, E. Knudsen, and G. T. Herman, "A discrete tomography algorithm for improving the quality of 3DXRD grain maps," J. Appl. Cryst. 39, 582-588 (2006).
[CrossRef]

Prangenberg, D.

R. J. Gardner, P. Gritzmann, and D. Prangenberg, "On the computational complexity of reconstructing lattice sets from their X-rays," Discrete Math. 202, 45-71 (1999).
[CrossRef]

P. Gritzmann, D. Prangenberg, S. de Vries, and M. Wiegelmann, "Success and failure of certain reconstruction and uniqueness algorithms in discrete tomography," Int. J. Imaging Syst. Technol. 9, 101-109 (1998).
[CrossRef]

Ramunno-Johnson, D.

J. Miao, C.-C. Chen, C. Song, Y. Nishino, Y. Kohmura, T. Ishikawa, D. Ramunno-Johnson, T.-K. Lee, and S. H. Risbud, "Three-dimensional GaN-Ga2O3 core shell structure revealed by X-ray diffraction microscopy," Phys. Rev. Lett. 97, 215503 (2006).
[CrossRef] [PubMed]

Raven, C.

C. Raven, A. Snigirev, I. Snigireva, P. Spanne, A. Souvorov, and V. Kohn, "Phase-contrast microtomography with coherent high-energy synchrotron x rays," Appl. Phys. Lett. 69, 1826-1828 (1996).
[CrossRef]

Ren, B.

F. A. Dilmanian, Z. Zhong, B. Ren, X. Y. Wu, L. D. Chapman, I. Orion, and W. C. Thomlinson, "Computed tomography of x-ray index of refraction using the diffraction enhanced imaging method," Phys. Med. Biol. 45, 933-946 (2000).
[CrossRef] [PubMed]

Risbud, S. H.

J. Miao, C.-C. Chen, C. Song, Y. Nishino, Y. Kohmura, T. Ishikawa, D. Ramunno-Johnson, T.-K. Lee, and S. H. Risbud, "Three-dimensional GaN-Ga2O3 core shell structure revealed by X-ray diffraction microscopy," Phys. Rev. Lett. 97, 215503 (2006).
[CrossRef] [PubMed]

Schlenker, M.

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

Schnörr, C.

T. Schüle, C. Schnörr, S. Weber, and J. Hornegger, "Discrete tomography by convex-concave regularization and D.C. programming," Discrete Appl. Math. 151, 229-243 (2005).
[CrossRef]

S. Weber, T. Schüle, J. Hornegger, and C. Schnörr, "Binary tomography by iterating linear programs from noisy projections," Lecture Notes in Comput.Science 3322, 38-51 (2004).
[CrossRef]

Schüle, T.

T. Schüle, C. Schnörr, S. Weber, and J. Hornegger, "Discrete tomography by convex-concave regularization and D.C. programming," Discrete Appl. Math. 151, 229-243 (2005).
[CrossRef]

S. Weber, T. Schüle, J. Hornegger, and C. Schnörr, "Binary tomography by iterating linear programs from noisy projections," Lecture Notes in Comput.Science 3322, 38-51 (2004).
[CrossRef]

Schuster, M.

M. Engelhardt, J. Baumann, M. Schuster, C. Kottler, F. Pfeiffer, O. Bunk, and C. David, "High-resolution differential phase contrast imaging using a magnifying projection geometry with a microfocus x-ray source," Appl. Phys. Lett. 90, 224101 (2007).
[CrossRef]

Schwander, P.

P. Fishburn, P. Schwander, L. Shepp, and R. J. Vanderbei, "The discrete Radon transform and its approximate inversion via linear programming," Discrete Appl. Math. 75, 39-61 (1997).
[CrossRef]

Shepp, L.

P. Fishburn, P. Schwander, L. Shepp, and R. J. Vanderbei, "The discrete Radon transform and its approximate inversion via linear programming," Discrete Appl. Math. 75, 39-61 (1997).
[CrossRef]

Shepp, L. A.

L. A. Shepp and B. F. Logan, "Reconstructing interior head tissue from X-ray transmissions," IEEE Trans. Nucl. Sci. 21, 228-236 (1974).
[CrossRef]

Sherman, G. C.

G. C. Sherman, "Diffracted wave fields expressible by plane-wave expansions containing only homogenous components," Phys. Rev. Lett. 21, 761-764 (1968).
[CrossRef]

Shewell, J. R.

J. R. Shewell and E. Wolf, "Inverse diffraction and a new reciprocity theorem," J. Opt. Soc. Am. A 58, 1596-1603 (1968).
[CrossRef]

E. Wolf and J. R. Shewell, "The inverse wave propagator," Phys. Lett. A 25, 417-418 (1967); see also E. Wolf and J. R. Shewell, "Errata," Phys. Lett. A 26, 104 (1967).
[CrossRef]

Shi, D.

Snigirev, A.

T. E. Gureyev, A. W. Stevenson, D. M. Paganin, T. Weitkamp, A. Snigirev, I. Snigireva, and S. W. Wilkins, "Quantitative analysis of two component samples using in-line hard X-ray images," J. Synchrotron Rad. 9, 148-153 (2002).
[CrossRef]

C. Raven, A. Snigirev, I. Snigireva, P. Spanne, A. Souvorov, and V. Kohn, "Phase-contrast microtomography with coherent high-energy synchrotron x rays," Appl. Phys. Lett. 69, 1826-1828 (1996).
[CrossRef]

Snigireva, I.

T. E. Gureyev, A. W. Stevenson, D. M. Paganin, T. Weitkamp, A. Snigirev, I. Snigireva, and S. W. Wilkins, "Quantitative analysis of two component samples using in-line hard X-ray images," J. Synchrotron Rad. 9, 148-153 (2002).
[CrossRef]

C. Raven, A. Snigirev, I. Snigireva, P. Spanne, A. Souvorov, and V. Kohn, "Phase-contrast microtomography with coherent high-energy synchrotron x rays," Appl. Phys. Lett. 69, 1826-1828 (1996).
[CrossRef]

Song, C.

J. Miao, C.-C. Chen, C. Song, Y. Nishino, Y. Kohmura, T. Ishikawa, D. Ramunno-Johnson, T.-K. Lee, and S. H. Risbud, "Three-dimensional GaN-Ga2O3 core shell structure revealed by X-ray diffraction microscopy," Phys. Rev. Lett. 97, 215503 (2006).
[CrossRef] [PubMed]

Souvorov, A.

C. Raven, A. Snigirev, I. Snigireva, P. Spanne, A. Souvorov, and V. Kohn, "Phase-contrast microtomography with coherent high-energy synchrotron x rays," Appl. Phys. Lett. 69, 1826-1828 (1996).
[CrossRef]

Spanne, P.

C. Raven, A. Snigirev, I. Snigireva, P. Spanne, A. Souvorov, and V. Kohn, "Phase-contrast microtomography with coherent high-energy synchrotron x rays," Appl. Phys. Lett. 69, 1826-1828 (1996).
[CrossRef]

Stampanoni, M.

Stevenson, A. W.

S. C. Mayo, T. J. Davis, T. E. Gureyev, P. R. Miller, D. Paganin, A. Pogany, A. W. Stevenson, and S. W. Wilkins, "X-ray phase-contrast microscopy and microtomography," Opt. Express 11, 2289-2302 (2003).
[CrossRef] [PubMed]

T. E. Gureyev, A. W. Stevenson, D. M. Paganin, T. Weitkamp, A. Snigirev, I. Snigireva, and S. W. Wilkins, "Quantitative analysis of two component samples using in-line hard X-ray images," J. Synchrotron Rad. 9, 148-153 (2002).
[CrossRef]

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

Suzuki, Y.

A. Momose, W. Yashiro, Y. Takeda, Y. Suzuki, and T. Hattori, "Phase tomography by X-ray Talbot interferometry for biological imaging," Jpn. J. Appl. Phys. 45, 5254-5262 (2006).
[CrossRef]

Takeda, T.

A. Momose, T. Takeda, and Y. Itai, "Phase-contrast X-ray computed tomography for observing biological specimens and organic materials," Rev. Sci. Instrum. 66, 1434-1436 (1995).
[CrossRef]

Takeda, Y.

A. Momose, W. Yashiro, Y. Takeda, Y. Suzuki, and T. Hattori, "Phase tomography by X-ray Talbot interferometry for biological imaging," Jpn. J. Appl. Phys. 45, 5254-5262 (2006).
[CrossRef]

Teague, M. R.

Thomlinson, W. C.

F. A. Dilmanian, Z. Zhong, B. Ren, X. Y. Wu, L. D. Chapman, I. Orion, and W. C. Thomlinson, "Computed tomography of x-ray index of refraction using the diffraction enhanced imaging method," Phys. Med. Biol. 45, 933-946 (2000).
[CrossRef] [PubMed]

Thorens, L.

A. Alpers, P. Gritzmann, and L. Thorens, "Stability and instability in discrete tomography," Lecture Notes in Comput.Science 2243, 175-186 (2001).
[CrossRef]

Tijdeman, R.

L. Hajdu and R. Tijdeman, "An algorithm for discrete tomography," Linear Algebra Appl. 339, 119-128 (2001).
[CrossRef]

Van Dyck, D.

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

Van Landuyt, J.

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

Vanderbei, R. J.

P. Fishburn, P. Schwander, L. Shepp, and R. J. Vanderbei, "The discrete Radon transform and its approximate inversion via linear programming," Discrete Appl. Math. 75, 39-61 (1997).
[CrossRef]

Weber, S.

T. Schüle, C. Schnörr, S. Weber, and J. Hornegger, "Discrete tomography by convex-concave regularization and D.C. programming," Discrete Appl. Math. 151, 229-243 (2005).
[CrossRef]

S. Weber, T. Schüle, J. Hornegger, and C. Schnörr, "Binary tomography by iterating linear programs from noisy projections," Lecture Notes in Comput.Science 3322, 38-51 (2004).
[CrossRef]

Weitkamp, T.

T. E. Gureyev, A. W. Stevenson, D. M. Paganin, T. Weitkamp, A. Snigirev, I. Snigireva, and S. W. Wilkins, "Quantitative analysis of two component samples using in-line hard X-ray images," J. Synchrotron Rad. 9, 148-153 (2002).
[CrossRef]

Wiegelmann, M.

P. Gritzmann, S. de Vries, and M. Wiegelmann, "Approximating binary images from discrete x-rays," SIAM J. Optimization 11, 522-546 (2000).
[CrossRef]

P. Gritzmann, D. Prangenberg, S. de Vries, and M. Wiegelmann, "Success and failure of certain reconstruction and uniqueness algorithms in discrete tomography," Int. J. Imaging Syst. Technol. 9, 101-109 (1998).
[CrossRef]

Wilkins, S. W.

G. R. Myers, S. C. Mayo, T. E. Gureyev, D. M. Paganin, and S. W. Wilkins, "Polychromatic cone-beam phase-contrast tomography," Phys. Rev. A 76, 045804 (2007).
[CrossRef]

T. E. Gureyev, Y. I. Nesterets, K. M. Pavlov, and S. W. Wilkins, "Computed tomography with linear shift-invariant optical systems," J. Opt. Soc. Am. A 24, 2230-2241 (2007).
[CrossRef]

Y. I. Nesterets, T. E. Gureyev, and S. W. Wilkins, "General reconstruction formulas for analyzer-based computed tomography," Appl. Phys. Lett. 89, 264103 (2006).
[CrossRef]

T. E. Gureyev, D. M. Paganin, G. R. Myers, Ya. I. Nesterets, and S. W. Wilkins, "Phase-and-amplitude computer tomography," Appl. Phys. Lett. 89, 034102 (2006).
[CrossRef]

S. C. Mayo, T. J. Davis, T. E. Gureyev, P. R. Miller, D. Paganin, A. Pogany, A. W. Stevenson, and S. W. Wilkins, "X-ray phase-contrast microscopy and microtomography," Opt. Express 11, 2289-2302 (2003).
[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]

T. E. Gureyev, A. W. Stevenson, D. M. Paganin, T. Weitkamp, A. Snigirev, I. Snigireva, and S. W. Wilkins, "Quantitative analysis of two component samples using in-line hard X-ray images," J. Synchrotron Rad. 9, 148-153 (2002).
[CrossRef]

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

Wolf, E.

E. Wolf, "Three-dimensional structure determination of semi-transparent objects from holographic data," Opt. Commun. 1, 153-156 (1969).
[CrossRef]

J. R. Shewell and E. Wolf, "Inverse diffraction and a new reciprocity theorem," J. Opt. Soc. Am. A 58, 1596-1603 (1968).
[CrossRef]

E. Wolf and J. R. Shewell, "The inverse wave propagator," Phys. Lett. A 25, 417-418 (1967); see also E. Wolf and J. R. Shewell, "Errata," Phys. Lett. A 26, 104 (1967).
[CrossRef]

Wu, X.

Wu, X. Y.

F. A. Dilmanian, Z. Zhong, B. Ren, X. Y. Wu, L. D. Chapman, I. Orion, and W. C. Thomlinson, "Computed tomography of x-ray index of refraction using the diffraction enhanced imaging method," Phys. Med. Biol. 45, 933-946 (2000).
[CrossRef] [PubMed]

Yashiro, W.

A. Momose, W. Yashiro, Y. Takeda, Y. Suzuki, and T. Hattori, "Phase tomography by X-ray Talbot interferometry for biological imaging," Jpn. J. Appl. Phys. 45, 5254-5262 (2006).
[CrossRef]

Zawisky, M.

J. ?ehá?ek, Z. Hradil, J. Pe?ina, S. Pascazio, P. Facchi, and M. Zawisky, "Advanced neutron imaging and sensing," Adv. Imaging Electron. Phys. 142, 53-157 (2006).
[CrossRef]

Zhong, Z.

F. A. Dilmanian, Z. Zhong, B. Ren, X. Y. Wu, L. D. Chapman, I. Orion, and W. C. Thomlinson, "Computed tomography of x-ray index of refraction using the diffraction enhanced imaging method," Phys. Med. Biol. 45, 933-946 (2000).
[CrossRef] [PubMed]

Adv. Imaging Electron. Phys. (1)

J. ?ehá?ek, Z. Hradil, J. Pe?ina, S. Pascazio, P. Facchi, and M. Zawisky, "Advanced neutron imaging and sensing," Adv. Imaging Electron. Phys. 142, 53-157 (2006).
[CrossRef]

Appl. Phys. Lett. (5)

C. Raven, A. Snigirev, I. Snigireva, P. Spanne, A. Souvorov, and V. Kohn, "Phase-contrast microtomography with coherent high-energy synchrotron x rays," Appl. Phys. Lett. 69, 1826-1828 (1996).
[CrossRef]

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

Y. I. Nesterets, T. E. Gureyev, and S. W. Wilkins, "General reconstruction formulas for analyzer-based computed tomography," Appl. Phys. Lett. 89, 264103 (2006).
[CrossRef]

T. E. Gureyev, D. M. Paganin, G. R. Myers, Ya. I. Nesterets, and S. W. Wilkins, "Phase-and-amplitude computer tomography," Appl. Phys. Lett. 89, 034102 (2006).
[CrossRef]

M. Engelhardt, J. Baumann, M. Schuster, C. Kottler, F. Pfeiffer, O. Bunk, and C. David, "High-resolution differential phase contrast imaging using a magnifying projection geometry with a microfocus x-ray source," Appl. Phys. Lett. 90, 224101 (2007).
[CrossRef]

Discrete Appl. Math. (2)

P. Fishburn, P. Schwander, L. Shepp, and R. J. Vanderbei, "The discrete Radon transform and its approximate inversion via linear programming," Discrete Appl. Math. 75, 39-61 (1997).
[CrossRef]

T. Schüle, C. Schnörr, S. Weber, and J. Hornegger, "Discrete tomography by convex-concave regularization and D.C. programming," Discrete Appl. Math. 151, 229-243 (2005).
[CrossRef]

Discrete Math. (1)

R. J. Gardner, P. Gritzmann, and D. Prangenberg, "On the computational complexity of reconstructing lattice sets from their X-rays," Discrete Math. 202, 45-71 (1999).
[CrossRef]

IEEE Trans. Nucl. Sci. (1)

L. A. Shepp and B. F. Logan, "Reconstructing interior head tissue from X-ray transmissions," IEEE Trans. Nucl. Sci. 21, 228-236 (1974).
[CrossRef]

Int. J. Imaging Syst. Technol. (1)

P. Gritzmann, D. Prangenberg, S. de Vries, and M. Wiegelmann, "Success and failure of certain reconstruction and uniqueness algorithms in discrete tomography," Int. J. Imaging Syst. Technol. 9, 101-109 (1998).
[CrossRef]

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A. Alpers, H. F. Poulsen, E. Knudsen, and G. T. Herman, "A discrete tomography algorithm for improving the quality of 3DXRD grain maps," J. Appl. Cryst. 39, 582-588 (2006).
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D. M. Paganin, A. Barty, P. J. McMahon, and K. A. Nugent, "Quantitative phase-amplitude microscopy. III. The effects of noise," J. Microsc. 214, 51-61 (2004).
[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]

J. Opt. Soc. Am. (1)

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

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K. M. Pavlov, C. M. Kewish, J. R. Davis, and M. J. Morgan, "A variant on the geometrical optics approximation in diffraction enhanced tomography," J. Phys. D: Appl. Phys. 34, A168-A172 (2001).
[CrossRef]

J. Synchrotron Rad. (1)

T. E. Gureyev, A. W. Stevenson, D. M. Paganin, T. Weitkamp, A. Snigirev, I. Snigireva, and S. W. Wilkins, "Quantitative analysis of two component samples using in-line hard X-ray images," J. Synchrotron Rad. 9, 148-153 (2002).
[CrossRef]

Jpn. J. Appl. Phys. (1)

A. Momose, W. Yashiro, Y. Takeda, Y. Suzuki, and T. Hattori, "Phase tomography by X-ray Talbot interferometry for biological imaging," Jpn. J. Appl. Phys. 45, 5254-5262 (2006).
[CrossRef]

Linear Algebra Appl. (1)

L. Hajdu and R. Tijdeman, "An algorithm for discrete tomography," Linear Algebra Appl. 339, 119-128 (2001).
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P. Besl, "Active, optical range imaging sensors", Machine Vision and Applications 1, 127-152 (1988).
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W. R. Brody, G. Butt, A. Hall, and A. Macovski, "A method for selective tissue and bone visualization using dual energy scanned projection radiography," Med. Phys. 8, 353-357 (1980).
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Nature (1)

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

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

F. Pfeiffer, O. Bunk, C. Kottler, and C. David, "Tomographic reconstruction of three-dimensional objects from hard X-ray differential phase contrast projection images," Nucl. Instrum. Methods Phys. Res. A 580, 925-928 (2007).
[CrossRef]

Opt. Commun. (2)

A. V. Bronnikov, "Reconstruction formulas in phase-contrast tomography," Opt. Commun. 171, 239-244 (1999).
[CrossRef]

E. Wolf, "Three-dimensional structure determination of semi-transparent objects from holographic data," Opt. Commun. 1, 153-156 (1969).
[CrossRef]

Opt. Express (4)

Optik (1)

T. E. Gureyev, "Transport of intensity equation for beams in an arbitrary state of temporal and spatial coherence," Optik 110, 263-266 (1999).

Phys. Lett. A (1)

E. Wolf and J. R. Shewell, "The inverse wave propagator," Phys. Lett. A 25, 417-418 (1967); see also E. Wolf and J. R. Shewell, "Errata," Phys. Lett. A 26, 104 (1967).
[CrossRef]

Phys. Med. Biol. (1)

F. A. Dilmanian, Z. Zhong, B. Ren, X. Y. Wu, L. D. Chapman, I. Orion, and W. C. Thomlinson, "Computed tomography of x-ray index of refraction using the diffraction enhanced imaging method," Phys. Med. Biol. 45, 933-946 (2000).
[CrossRef] [PubMed]

Phys. Rev. A (1)

G. R. Myers, S. C. Mayo, T. E. Gureyev, D. M. Paganin, and S. W. Wilkins, "Polychromatic cone-beam phase-contrast tomography," Phys. Rev. A 76, 045804 (2007).
[CrossRef]

Phys. Rev. Lett. (3)

J. Miao, C.-C. Chen, C. Song, Y. Nishino, Y. Kohmura, T. Ishikawa, D. Ramunno-Johnson, T.-K. Lee, and S. H. Risbud, "Three-dimensional GaN-Ga2O3 core shell structure revealed by X-ray diffraction microscopy," Phys. Rev. Lett. 97, 215503 (2006).
[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]

G. C. Sherman, "Diffracted wave fields expressible by plane-wave expansions containing only homogenous components," Phys. Rev. Lett. 21, 761-764 (1968).
[CrossRef]

Phys. Today (1)

R. Fitzgerald, "Phase-sensitive x-ray imaging," Phys. Today 53, 23-26 (2000).
[CrossRef]

Problems (1)

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[CrossRef]

Proc. SPIE (1)

A. V. Bronnikov, "Phase contrast CT: Fundamental theorem and fast image reconstruction algorithms," Proc. SPIE 6318, 63180Q (2006).
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U. Bonse and F. Busch, "X-ray computed microtomography (µCT) using synchrotron radiation (SR)," Prog. Biophys. Mol. Biol. 65, 133-169 (1996).
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Rev. Sci. Instrum. (1)

A. Momose, T. Takeda, and Y. Itai, "Phase-contrast X-ray computed tomography for observing biological specimens and organic materials," Rev. Sci. Instrum. 66, 1434-1436 (1995).
[CrossRef]

Science (3)

A. Alpers, P. Gritzmann, and L. Thorens, "Stability and instability in discrete tomography," Lecture Notes in Comput.Science 2243, 175-186 (2001).
[CrossRef]

S. Weber, T. Schüle, J. Hornegger, and C. Schnörr, "Binary tomography by iterating linear programs from noisy projections," Lecture Notes in Comput.Science 3322, 38-51 (2004).
[CrossRef]

K. J. Batenburg, "A network flow algorithm for binary image reconstruction from few projections," Lecture Notes in Comput.Science 4245, 86-97 (2006).
[CrossRef]

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A. Alpers and P. Gritzmann, "On stability, error correction, and noise compensation in discrete tomography," SIAM J. Discrete Math. 20, 227-239 (2006).
[CrossRef]

SIAM J. Optimization (1)

P. Gritzmann, S. de Vries, and M. Wiegelmann, "Approximating binary images from discrete x-rays," SIAM J. Optimization 11, 522-546 (2000).
[CrossRef]

Trans. Amer. Math. Soc. (1)

R. J. Gardner and P. Gritzmann, "Discrete tomography: Determination of finite sets by x-rays," Trans. Amer. Math. Soc. 349, 2271-2295 (1997).
[CrossRef]

Other (8)

G. T. Herman and A. Kuba, eds., Discrete Tomography: Foundations, Algorithms and Applications (Birkhäuser, Boston, 1999).

F. Natterer and F. Wübbeling, Mathematical Methods in Image Reconstruction (SIAM, Philadelphia, 2001).
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D. M. Paganin, Coherent X-Ray Optics (Oxford University Press, New York, 2006).
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Figures (7)

Fig. 1.
Fig. 1.

Sample (a) binary (one material and void), (b) ternary (two materials and void), and (c) quaternary (three materials and void) objects. Different gray levels denote different materials.

Fig. 2.
Fig. 2.

Imaging geometry and coordinate systems.

Fig. 3.
Fig. 3.

The phantoms used for our (a) ternary (section 3.1) and (b) quaternary (section 3.2) reconstructions. Gray levels in both images are false color and not indicative of the relative magnitude of refractive indices.

Fig. 4.
Fig. 4.

Slices through a ternary object, reconstructed from simulated data. Projected thicknesses were retrieved using Eq. (20). The phantom compositions are: carbon mass with water inclusions (top row), aluminum mass with Mylar inclusions (middle row), and Mylar mass with aluminum inclusions (bottom row), and the noise levels are 0%, 5% and 10% from left to right. See Table 1 for associated errors.

Fig. 5.
Fig. 5.

Slices through a “skull-like” quaternary object, reconstructed from simulated data. The noise levels are 0%, 5% and 10% from left to right, with corresponding errors of 0.010, 0.029 and 0.041.

Fig. 6.
Fig. 6.

Sample image of a glass tube filled with silica and water dosed with 10% potassium iodide by weight. Fringing due to propagation-based phase contrast is visible around the edges of the glass tube and the silica grains. An air bubble can be seen approximately halfway up the tube.

Fig. 7.
Fig. 7.

Reconstructions of a slice through a glass tube containing silica (dark gray), water (light gray) and air (white) from (a) 180 projections and (b) 18 projections. The slice on the left is the result of a qualitative reconstruction performed using filtered backprojection, and the slice on the right is reconstructed using Eqs. (20) and (9)-(13) for ternary phase-and-amplitude computed tomography. As the diameter and composition of the glass tube containing the sample was known, it was artificially removed from each image, during the ternary tomographic reconstruction; hence its absence in Fig. 7(b).

Tables (1)

Tables Icon

Table 1. Calculated errors for ternary object reconstructions shown in Fig. 4.

Equations (27)

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T 1 ( x , θ ) = 1 Δ 1 Δ ( s cos θ x 1 sin θ , x 1 cos θ + s sin θ , x 2 ) ds 1 Δ 1 ( P Δ ) ( x , θ ) ,
I 0 ( x , θ ) = I in exp [ 2 k β 1 T 1 ( x , θ ) ] ,
ϕ 0 ( x , θ ) = k Δ 1 T 1 ( x , θ ) .
T 1 ( x , θ ) = ln [ I 0 ( x , θ ) I in ] 2 k β 1 .
( 2 ) n U 0 ( x , θ ) ( 2 k R ) n U 0 ( x , θ ) , n = 1 , 2 , 3 . . . ,
I R ( x , θ ) = I 0 ( x , θ ) R k 1 [ I 0 ( x , θ ) ϕ 0 ( x , θ ) ] .
I R ( x , θ ) I in = ( 1 R Δ 1 2 k β 1 2 ) exp [ 2 k β 1 T 1 ( x , θ ) ] ,
T 1 ( x , θ ) = 1 2 k β 1 ln ( F 2 1 [ ( 1 + R Δ 1 2 k β 1 ξ 2 ) 1 F 2 [ I R ( x , θ ) I in ] ] ) ,
Δ ˘ 0 ( r ) = Δ 1 0 π F 1 1 ( ξ 1 F 1 [ T 1 ( x , θ ) ] ) ( x 1 , x 2 ) ( r 2 cos θ r 1 sin θ , r 3 ) d θ Δ 1 ( B T 1 ) ( r ) ,
Δ ˘ T , 0 ( r ) = ( T Δ ˘ 0 ) ( r ) { 0 , if Δ ˘ 0 ( r ) < M , Δ 1 , if Δ ˘ 0 ( r ) M ,
Δ 1 1 R 3 Δ ˘ T , 0 ( r ) d r = R 2 T 1 ( x , θ ) d x
{ Δ ˘ i + 1 ( r ) = Δ ˘ i ( r ) + γ i Δ 1 B [ T 1 ( x , θ ) ( P Δ ˘ T , i ) ( x , θ ) ] , Δ ˘ T , i + 1 ( r ) = ( T Δ ˘ i + 1 ) ( r ) .
x R 2 T 1 ( x , θ ) ( P Δ ˘ T , i + 1 ) ( x , θ ) d x < x R 2 T 1 ( x , θ ) ( P Δ ˘ T , i ) ( x , θ ) d x ,
n ( r ) = n 1 ( r ) n 2 ( r ) , n 1 ( r ) = { 1 Δ 1 + i β 1 1 , n 2 ( r ) = { 1 Δ 2 + i β 2 1 .
ln [ I 0 ( x , θ ) I in ] = 2 k β 1 T 1 ( x , θ ) 2 k β 2 T 2 ( x , θ ) ,
ϕ 0 ( x , θ ) = k Δ 1 T 1 ( x , θ ) k Δ 2 T 2 ( x , θ ) .
A ( x , θ ) = T 1 ( x , θ ) + T 2 ( x , θ )
ln [ I 0 ( x , θ ) I in ] = 2 k ( β 1 β 2 ) T 1 ( x , θ ) 2 k β 2 A ( x , θ ) ,
ϕ 0 ( x , θ ) = k ( Δ 1 Δ 2 ) T 1 ( x , θ ) k Δ 2 A ( x , θ ) ,
I R ( x , θ ) I in = B ( x , θ ) C ( x , θ ) R 2 k κ [ B ( x , θ ) C ( x , θ ) ] ,
B ( x , θ ) = exp [ 2 k ( β 2 Δ 2 κ ) A ( x , θ ) ] ,
C ( x , θ ) = exp [ 2 k ( β 1 β 2 ) T 1 ( x , θ ) 2 k Δ 2 κ A ( x , θ ) ] ,
T 1 ( x , θ ) = ln [ C ( x , θ ) ] + 2 k Δ 2 κ A ( x , θ ) 2 k ( β 2 β 1 ) .
n ( r ) = n 1 ( r ) n 2 ( r ) n 3 ( r ) , n 1 ( r ) = { 1 Δ 1 + i β 1 1 , n 2 ( r ) = { 1 Δ 2 + i β 2 1 , n 3 ( r ) = { 1 Δ 3 + i β 3 1 .
ln [ I 0 ( x , θ ) I in ] = 2 k β 1 T 1 ( x , θ ) 2 k β 2 T 2 ( x , θ ) 2 k β 3 T 3 ( x , θ ) ,
ϕ 0 ( x , θ ) = k Δ 1 T 1 ( x , θ ) k Δ 2 T 2 ( x , θ ) k Δ 3 T 3 ( x , θ ) .
A ( x , θ ) = T 1 ( x , θ ) + T 2 ( x , θ ) + T 3 ( x , θ ) .

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