Abstract

A method is presented for quantitative polychromatic cone-beam phase-contrast tomographic imaging of a single-material object from few projections. This algorithm exploits the natural combination of binary tomography with a phase-retrieval method that makes explicit use of the single-material nature of the sample. Such consistent use of a priori knowledge reduces the number of required projections, implying significantly reduced dose and scanning time when compared to existing phase-contrast tomography methods. Reconstructions from simulated data sets are used to investigate the effects of noise and establish a minimum required number of projections. An experimental demonstration is then given, using data from a point-projection X-ray microscope. Here, the complex distribution of refractive index in a sample containing several nylon fibers with diameters between 100 µm and 420 µm is reconstructed at a spatial resolution of ~4 µm from 20 polychromatic phase-contrast projection images with a mean photon energy of 8.4 keV.

© 2008 Optical Society of America

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2007 (7)

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]

G. R. Myers, T. E. Gureyev, and D. M. Paganin, "Stability of phase-contrast tomography," J. Opt. Soc. Am. A 24, 2516-2526 (2007).
[CrossRef]

E. M. Friis, P. R. Crane, K. R. Pedersen, S. Bengtson, P. C. J. Donoghue, G. W. Grimm, and M. Stampanoni, "Phase-contrast X-ray microtomography links Cretaceous seeds with Gnetales and Bennettitales," Nature 450, 549-553 (2007).
[CrossRef] [PubMed]

2006 (9)

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]

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

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]

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

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]

2005 (3)

2004 (4)

T. E. Gureyev, D. M. Paganin, A. W. Stevenson, S. C. Mayo, and S. W. Wilkins, "Generalized eikonal of partially coherent beams and its use in quantitative imaging," Phys. Rev. Lett. 93, 068103 (2004).
[CrossRef] [PubMed]

L. D. Turner, K. P. Weber, D. Paganin, and R. E. Scholten, "Off-resonant defocus-contrast imaging of cold atoms," Opt. Lett. 29, 232-234 (2004).
[CrossRef] [PubMed]

T. E. Gureyev, A. Pogany, D. M. Paganin, and S. W. Wilkins, "Linear algorithms for phase retrieval in the Fresnel region," Opt. Commun. 2531, 53-70 (2004).
[CrossRef]

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

2003 (1)

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 (3)

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]

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

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

2000 (3)

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]

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

R. Fitzgerald, "Phase-sensitive x-ray imaging," Phys. Today 53(7), 23-26 (2000).
[CrossRef]

1999 (3)

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]

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. Imag. 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. Molec. Biol. 65, 133-169 (1996).
[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]

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]

1994 (1)

D. S. Lalush and B. M. W. Tsui, "Improving the convergence of iterative filtered back projection algorithms," Med. Phys. 21, 1283-1286 (1994).
[CrossRef] [PubMed]

1991 (1)

X. Yan and R. M. Leahy, "Derivation and analysis of a filtered backprojection algorithm for cone beam projection data," IEEE Trans. Med. Im. 10, 462-472 (1991).
[CrossRef]

1986 (1)

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

1984 (1)

1969 (1)

E. Wolf, "Three-dimensional structure determination of semi-transparent objects from holographic data," Opt. Commun. 1, 153-156 (1969).
[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. Sci. 2243, 175-186 (2001).
[CrossRef]

Anastasio, M. A.

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.Sci. 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]

Bengtson, S.

E. M. Friis, P. R. Crane, K. R. Pedersen, S. Bengtson, P. C. J. Donoghue, G. W. Grimm, and M. Stampanoni, "Phase-contrast X-ray microtomography links Cretaceous seeds with Gnetales and Bennettitales," Nature 450, 549-553 (2007).
[CrossRef] [PubMed]

Bonse, U.

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

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]

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]

Busch, F.

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

Chapman, L. D.

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]

Chen, C.-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]

Cloetens, P.

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]

Crane, P. R.

E. M. Friis, P. R. Crane, K. R. Pedersen, S. Bengtson, P. C. J. Donoghue, G. W. Grimm, and M. Stampanoni, "Phase-contrast X-ray microtomography links Cretaceous seeds with Gnetales and Bennettitales," Nature 450, 549-553 (2007).
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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).
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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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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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Davis, T. J.

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E. M. Friis, P. R. Crane, K. R. Pedersen, S. Bengtson, P. C. J. Donoghue, G. W. Grimm, and M. Stampanoni, "Phase-contrast X-ray microtomography links Cretaceous seeds with Gnetales and Bennettitales," Nature 450, 549-553 (2007).
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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).
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J. Řeháček, Z. Hradil, J. Peřina, S. Pascazio, P. Facchi, and M. Zawisky, "Advanced neutron imaging and sensing," Adv. Imaging Electron. Phys. 142, 53-157 (2006).
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Fishburn, P.

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E. M. Friis, P. R. Crane, K. R. Pedersen, S. Bengtson, P. C. J. Donoghue, G. W. Grimm, and M. Stampanoni, "Phase-contrast X-ray microtomography links Cretaceous seeds with Gnetales and Bennettitales," Nature 450, 549-553 (2007).
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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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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).
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Grimm, G. W.

E. M. Friis, P. R. Crane, K. R. Pedersen, S. Bengtson, P. C. J. Donoghue, G. W. Grimm, and M. Stampanoni, "Phase-contrast X-ray microtomography links Cretaceous seeds with Gnetales and Bennettitales," Nature 450, 549-553 (2007).
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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).
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A. Alpers, P. Gritzmann, and L. Thorens, "Stability and instability in discrete tomography," Lecture Notes in Comput. Sci. 2243, 175-186 (2001).
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P. Gritzmann, S. de Vries, and M. Wiegelmann, "Approximating binary images from discrete x-rays," SIAM J. Optimization 11, 522-546 (2000).
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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).
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P. Gritzmann, D. Prangenberg, S. de Vries, and M. Wiegelmann, "Success and failure of certain reconstruction and uniqueness algorithms in discrete tomography," Int. J. Imag. Syst. Technol. 9, 101-109 (1998).
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R. J. Gardner and P. Gritzmann, "Discrete tomography: Determination of finite sets by x-rays," Trans. Amer. Math. Soc. 349, 2271-2295 (1997).
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Guigay, J. P.

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).
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Gureyev, T. E.

G. R. Myers, T. E. Gureyev, and D. M. Paganin, "Stability of phase-contrast tomography," J. Opt. Soc. Am. A 24, 2516-2526 (2007).
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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).
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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).
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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).
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T. E. Gureyev, A. Pogany, D. M. Paganin, and S. W. Wilkins, "Linear algorithms for phase retrieval in the Fresnel region," Opt. Commun. 2531, 53-70 (2004).
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T. E. Gureyev, D. M. Paganin, A. W. Stevenson, S. C. Mayo, and S. W. Wilkins, "Generalized eikonal of partially coherent beams and its use in quantitative imaging," Phys. Rev. Lett. 93, 068103 (2004).
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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).
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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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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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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).
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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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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).
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J. Řeháček, Z. Hradil, J. Peřina, S. Pascazio, P. Facchi, and M. Zawisky, "Advanced neutron imaging and sensing," Adv. Imaging Electron. Phys. 142, 53-157 (2006).
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Huang, Y.

Ishikawa, T.

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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Itai, Y.

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).
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Kewish, C. M.

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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Knudsen, E.

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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Kohmura, Y.

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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Kohn, V.

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).
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Kottler, C.

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]

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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Lalush, D. S.

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X. Yan and R. M. Leahy, "Derivation and analysis of a filtered backprojection algorithm for cone beam projection data," IEEE Trans. Med. Im. 10, 462-472 (1991).
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Lee, T.-K.

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]

Liu, H.

Ludwig, W.

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]

Mayo, S. C.

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, A. W. Stevenson, S. C. Mayo, and S. W. Wilkins, "Generalized eikonal of partially coherent beams and its use in quantitative imaging," Phys. Rev. Lett. 93, 068103 (2004).
[CrossRef] [PubMed]

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]

Miao, J.

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]

Miller, P. R.

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]

Momose, A.

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, T. E. Gureyev, and D. M. Paganin, "Stability of phase-contrast tomography," J. Opt. Soc. Am. A 24, 2516-2526 (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, 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]

Nesterets, Y. I.

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]

Nishino, Y.

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]

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.

Paganin, D. M.

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]

G. R. Myers, T. E. Gureyev, and D. M. Paganin, "Stability of phase-contrast tomography," J. Opt. Soc. Am. A 24, 2516-2526 (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]

T. E. Gureyev, A. Pogany, D. M. Paganin, and S. W. Wilkins, "Linear algorithms for phase retrieval in the Fresnel region," Opt. Commun. 2531, 53-70 (2004).
[CrossRef]

T. E. Gureyev, D. M. Paganin, A. W. Stevenson, S. C. Mayo, and S. W. Wilkins, "Generalized eikonal of partially coherent beams and its use in quantitative imaging," Phys. Rev. Lett. 93, 068103 (2004).
[CrossRef] [PubMed]

Pascazio, S.

J. Řeháček, Z. Hradil, J. Peřina, 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]

Pedersen, K. R.

E. M. Friis, P. R. Crane, K. R. Pedersen, S. Bengtson, P. C. J. Donoghue, G. W. Grimm, and M. Stampanoni, "Phase-contrast X-ray microtomography links Cretaceous seeds with Gnetales and Bennettitales," Nature 450, 549-553 (2007).
[CrossRef] [PubMed]

Perina, J.

J. Řeháček, Z. Hradil, J. Peřina, 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, 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]

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]

Pogany, A.

T. E. Gureyev, A. Pogany, D. M. Paganin, and S. W. Wilkins, "Linear algorithms for phase retrieval in the Fresnel region," Opt. Commun. 2531, 53-70 (2004).
[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]

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. Imag. 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]

Rehácek, J.

J. Řeháček, Z. Hradil, J. Peřina, S. Pascazio, P. Facchi, and M. Zawisky, "Advanced neutron imaging and sensing," Adv. Imaging Electron. Phys. 142, 53-157 (2006).
[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. Sci. 3322, 38-51 (2004)
[CrossRef]

Scholten, R. E.

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. Sci. 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]

Shi, D.

Snigirev, 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]

Snigireva, I.

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.

E. M. Friis, P. R. Crane, K. R. Pedersen, S. Bengtson, P. C. J. Donoghue, G. W. Grimm, and M. Stampanoni, "Phase-contrast X-ray microtomography links Cretaceous seeds with Gnetales and Bennettitales," Nature 450, 549-553 (2007).
[CrossRef] [PubMed]

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]

Stevenson, A. W.

T. E. Gureyev, D. M. Paganin, A. W. Stevenson, S. C. Mayo, and S. W. Wilkins, "Generalized eikonal of partially coherent beams and its use in quantitative imaging," Phys. Rev. Lett. 93, 068103 (2004).
[CrossRef] [PubMed]

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]

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]

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. Sci. 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]

Tsui, B. M. W.

D. S. Lalush and B. M. W. Tsui, "Improving the convergence of iterative filtered back projection algorithms," Med. Phys. 21, 1283-1286 (1994).
[CrossRef] [PubMed]

Turner, L. D.

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, K. P.

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. Sci. 3322, 38-51 (2004)
[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. Imag. 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]

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]

T. E. Gureyev, A. Pogany, D. M. Paganin, and S. W. Wilkins, "Linear algorithms for phase retrieval in the Fresnel region," Opt. Commun. 2531, 53-70 (2004).
[CrossRef]

T. E. Gureyev, D. M. Paganin, A. W. Stevenson, S. C. Mayo, and S. W. Wilkins, "Generalized eikonal of partially coherent beams and its use in quantitative imaging," Phys. Rev. Lett. 93, 068103 (2004).
[CrossRef] [PubMed]

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]

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]

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]

Yan, X.

X. Yan and R. M. Leahy, "Derivation and analysis of a filtered backprojection algorithm for cone beam projection data," IEEE Trans. Med. Im. 10, 462-472 (1991).
[CrossRef]

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

Appl. Phys. Lett. (4)

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]

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. Med. Im. (1)

X. Yan and R. M. Leahy, "Derivation and analysis of a filtered backprojection algorithm for cone beam projection data," IEEE Trans. Med. Im. 10, 462-472 (1991).
[CrossRef]

Int. J. Imag. 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. Imag. Syst. Technol. 9, 101-109 (1998).
[CrossRef]

J. Appl. Cryst. (1)

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]

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. A (4)

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

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]

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).
[CrossRef]

Med. Phys. (1)

D. S. Lalush and B. M. W. Tsui, "Improving the convergence of iterative filtered back projection algorithms," Med. Phys. 21, 1283-1286 (1994).
[CrossRef] [PubMed]

Nature (2)

E. M. Friis, P. R. Crane, K. R. Pedersen, S. Bengtson, P. C. J. Donoghue, G. W. Grimm, and M. Stampanoni, "Phase-contrast X-ray microtomography links Cretaceous seeds with Gnetales and Bennettitales," Nature 450, 549-553 (2007).
[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]

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. (3)

T. E. Gureyev, A. Pogany, D. M. Paganin, and S. W. Wilkins, "Linear algorithms for phase retrieval in the Fresnel region," Opt. Commun. 2531, 53-70 (2004).
[CrossRef]

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 (3)

Opt. Lett. (1)

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]

T. E. Gureyev, D. M. Paganin, A. W. Stevenson, S. C. Mayo, and S. W. Wilkins, "Generalized eikonal of partially coherent beams and its use in quantitative imaging," Phys. Rev. Lett. 93, 068103 (2004).
[CrossRef] [PubMed]

Phys. Today (1)

R. Fitzgerald, "Phase-sensitive x-ray imaging," Phys. Today 53(7), 23-26 (2000).
[CrossRef]

Problems (1)

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

Proc. of SPIE (1)

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

Prog. Biophys. Molec. Biol. (1)

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

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]

Sci. (3)

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

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

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

SIAM J. Discrete Math. (1)

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 (6)

A. Alpers and P. Gritzmann, "On the degree of ill-posedness in discrete tomography," preprint, 2004.

P. Gritzmann and S. de Vries, "Reconstructing crystalline structures from few images under high resolution transmission electron microscopy," in Mathematics: Key Technology for the Future, W. Jäger and H.-J. Krebs, eds, (Springer-Verlag, Berlin, 2003) pp. 441-459.
[CrossRef]

F. Natterer, The Mathematics of Computerized Tomography (Wiley, New York, 1986).

D. M. Paganin, Coherent X-Ray Optics (Oxford University Press, New York, 2006).

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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Figures (8)

Fig 1.
Fig 1.

The experimental setup.

Fig. 2.
Fig. 2.

Phantoms used for the numerical simulations: a vascular structure (left), a splatter pattern (centre), and a nylon phantom (right). Each phantom is 2 mm×2 mm=600×600 pixels in size.

Fig. 3.
Fig. 3.

Reconstructions with 0 (left), 5 (centre) and 10 (right) percent noise for each of the three phantoms. From left to right, the errors in the reconstructions for the nylon phantom were 0.0079, 0.0095, 0.012. The errors in the vascular reconstructions are 0.014, 0.027 and 0.037, and the errors in the sparse splatter reconstructions are 0.0015, 0.0047 and 0.0073, again reading left to right.

Fig 4.
Fig 4.

Error vs. number of projections. The nylon, vascular and splatter phantoms are represented by the solid, dashed, and dotted lines respectively.

Fig. 5.
Fig. 5.

A phase-contrast image typical of those in the experimental data set.

Fig. 6.
Fig. 6.

Reconstruction from experimental data using the PACT algorithm (720 projections).

Fig. 7.
Fig. 7.

Reconstruction from experimental data using Binary PACT (20 projections).

Fig. 8.
Fig. 8.

Cross-sections through both the PACT reconstruction from 720 projections (solid line), and the Binary PACT reconstruction from 20 images (dotted line).

Equations (10)

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

ϕ 0 ( x , θ , λ ) = k Δ ( s ( θ ) + p p t , λ ) dt .
Δ ( r , λ ) = R 1 ρ 2 k 0 2 π 1 p 2 2 F 1 1 ( ξ 1 F 1 [ ϕ 0 ( x , θ , λ ) ] ) d θ k 1 ( B ϕ 0 ) ( r , λ ) ,
M 2 S R 2 ( M x , θ , λ ) = S in ( λ ) { 1 + 2 F 2 1 [ { π λ R ξ 2 + ε ( λ ) } F 2 [ P M ( M x ) ] F 2 [ ϕ 0 ( x , θ , λ ) ] ] } .
Δ S ( r ) = R 1 ρ M 2 8 π I in 0 2 π 1 p 2 2 F 2 1 ( ξ 1 F 2 [ I R 2 ( M x , θ ) ] [ π R ξ 2 + ε ( λ 0 ) λ 0 ] F 2 [ P M ( M x ) ] ) d θ .
T ( x , θ ) = F 2 1 ( 1 { 4 π 2 Δ ¯ R ξ 2 + μ ¯ } F 2 [ P M ( M x ) ] F 2 [ I in M 2 I R 2 ( M x , θ ) I in ] ) ,
μ ¯ = 2 0 S in ( λ ) β ( λ ) D ( λ ) kd λ I in ,
Δ ¯ = 0 S in ( λ ) Δ ( λ ) D ( λ ) d λ I in .
Δ s , 0 ( r ) = ( T Δ ~ 0 ) ( r ) { 0 , if Δ ~ 0 ( r ) < M , Δ ¯ , if Δ ~ 0 ( r ) M ,
{ Δ ~ i + 1 ( r ) = Δ ~ i ( r ) + γ i Δ ¯ B [ T ( x , θ ) ( P Δ S , i ) ( x , θ ) ] , Δ S , i + 1 ( r ) = ( T Δ ~ i + 1 ) ( r ) ,
l , j , k T ( x l , y j , θ k ) ( P Δ S , i + 1 ) ( x l , y j , θ k ) < l , j , k T ( x l , y j , θ k ) ( P Δ S , i ) ( x l , y j , θ k ) ,

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