Abstract

Phase-contrast X-ray computed laminography is demonstrated for the volume reconstruction of extended flat objects, not suitable to the usual tomographic scan. Using a Talbot interferometer, differential phase measurements are obtained and used to reconstruct the real part of the complex refractive index. The specific geometry of laminography leads to unsampled frequencies in a double cone in the reciprocal space, which degrades the spatial resolution in the direction normal to the object plane. First, the filtered backprojection formula from differential measurements is derived. Then, reconstruction is improved by the use of prior information of compact support and limited range, included in an iterative filtered backprojection algorithm. An implementation on GPU hardware was required to handle the reconstruction of volumes within a reasonable time. A synchrotron radiation experiment on polymer meshes is reported and results of the iterative reconstruction are compared with the simpler filtered backprojection.

© 2011 OSA

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2011

L. Helfen, A. Myagotin, P. Mikulík, P. Pernot, A. Voropaev, M. Elyyan, M. Di Michiel, J. Baruchel, and T. Baumbach, “On the implementation of computed laminography using synchrotron radiation,” Rev. Sci. Instrum. 82, 063702 (2011).
[CrossRef] [PubMed]

J. Als-Nielsen and D. McMorrow, Elements of Modern X-ray Physics , 2nd ed. (John Wiley and Sons, 2011).
[CrossRef]

2010

R. Guedouar and B. Zarrad, “A comparative study between matched and mis-matched projection/back projection pairs used with ASIRT reconstruction method,” Nucl. Instrum. Methods Phys. Res. A 619(1–3), 225–229 (2010).
[CrossRef]

F. Xu, L. Helfen, A. J. Moffat, G. Johnson, I. Sinclair, and T. Baumbach, “Synchrotron radiation computed laminography for polymer composite failure studies,” J. Synchrotron Radiat. 17(2), 222–226 (2010).
[CrossRef] [PubMed]

2009

L. Helfen, T. Baumbach, P. Cloetens, and J. Baruchel, “Phase-contrast and holographic computed laminography,” Appl. Phys. Lett. 94(10), 104103 (2009).
[CrossRef]

2008

E. Y. Sidky and X. Pan, “Image reconstruction in circular cone-beam computed tomography by constrained, total-variation minimization,” Phys. Med. Biol. 53, 4777 (2008).
[CrossRef] [PubMed]

K. Krug, L. Porra, P. Coan, A. Wallert, J. Dik, A. Coerdt, A. Bravin, M. Elyyan, P. Reischig, L. Helfen, and T. Baumbach, “Relics in medieval altarpieces? Combining X-ray tomographic, laminographic and phase-contrast imaging to visualize thin organic objects in paintings,” J. Synchrotron Radiat. 15, 55–61 (2008).
[CrossRef]

F. Pfeiffer, M. Bech, O. Bunk, P. Kraft, E. F. Eikenberry, Ch. Brönnimann, C. Grünzweig, and C. David, “Hard-X-ray dark-field imaging using a grating interferometer,” Nat. Mater. 7, 134–137 (2008).
[CrossRef] [PubMed]

2007

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]

2006

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]

2003

A. Momose, S. Kawamoto, I. Koyama, Y. Hamaishi, K. Takai, and Y. Suzuki, “Demonstration of X-ray Talbot interferometry,” Jpn. J. Appl. Phys. 42(7B), 866–868 (2003).
[CrossRef]

2001

F. Natterer, The Mathematics of Computerized Tomography (Society for Industrial and Applied Mathematics, 2001).
[CrossRef]

G. M. Stevens, R. Fahrig, and N. J. Pelc, “Filtered backprojection for modifying the impulse response of circular tomosynthesis,” Med. Phys. 28, 372–380 (2001).
[CrossRef] [PubMed]

2000

G. L. Zeng and G. T. Gullberg, “Unmatched projector/backprojector pairs in an iterative reconstruction algorithm,” IEEE Trans. Med. Imaging 19, 548–555 (2000).
[CrossRef] [PubMed]

1998

G. Lauritsch and W. H. Härer, “A theoretical framework for filtered backprojection in tomosynthesis,” Proc. SPIE 3338, 1127–1137 (1998).
[CrossRef]

1994

D. Lalush and B. Tsui, “Improving the convergence of iterative filtered backprojection algorithms,” Med. Phys. 21(8), 1283–1286 (1994).
[CrossRef] [PubMed]

1993

H. Matsuo, A. Iwata, I. Horiba, and N. Suzumura, “Three-dimensional image reconstruction by digital tomosynthesis using inverse filtering,” IEEE Trans. Med. Imaging 12(2), 307–313 (1993).
[CrossRef] [PubMed]

1988

1984

1974

1972

D. G. Grant, “Tomosynthesis: a three-dimensional radiographic imaging technique,” IEEE Trans. Biomed. Eng. 19(1), 20–28 (1972).
[CrossRef] [PubMed]

Als-Nielsen, J.

J. Als-Nielsen and D. McMorrow, Elements of Modern X-ray Physics , 2nd ed. (John Wiley and Sons, 2011).
[CrossRef]

Baruchel, J.

L. Helfen, A. Myagotin, P. Mikulík, P. Pernot, A. Voropaev, M. Elyyan, M. Di Michiel, J. Baruchel, and T. Baumbach, “On the implementation of computed laminography using synchrotron radiation,” Rev. Sci. Instrum. 82, 063702 (2011).
[CrossRef] [PubMed]

L. Helfen, T. Baumbach, P. Cloetens, and J. Baruchel, “Phase-contrast and holographic computed laminography,” Appl. Phys. Lett. 94(10), 104103 (2009).
[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]

Baumbach, T.

L. Helfen, A. Myagotin, P. Mikulík, P. Pernot, A. Voropaev, M. Elyyan, M. Di Michiel, J. Baruchel, and T. Baumbach, “On the implementation of computed laminography using synchrotron radiation,” Rev. Sci. Instrum. 82, 063702 (2011).
[CrossRef] [PubMed]

F. Xu, L. Helfen, A. J. Moffat, G. Johnson, I. Sinclair, and T. Baumbach, “Synchrotron radiation computed laminography for polymer composite failure studies,” J. Synchrotron Radiat. 17(2), 222–226 (2010).
[CrossRef] [PubMed]

L. Helfen, T. Baumbach, P. Cloetens, and J. Baruchel, “Phase-contrast and holographic computed laminography,” Appl. Phys. Lett. 94(10), 104103 (2009).
[CrossRef]

K. Krug, L. Porra, P. Coan, A. Wallert, J. Dik, A. Coerdt, A. Bravin, M. Elyyan, P. Reischig, L. Helfen, and T. Baumbach, “Relics in medieval altarpieces? Combining X-ray tomographic, laminographic and phase-contrast imaging to visualize thin organic objects in paintings,” J. Synchrotron Radiat. 15, 55–61 (2008).
[CrossRef]

Bech, M.

F. Pfeiffer, M. Bech, O. Bunk, P. Kraft, E. F. Eikenberry, Ch. Brönnimann, C. Grünzweig, and C. David, “Hard-X-ray dark-field imaging using a grating interferometer,” Nat. Mater. 7, 134–137 (2008).
[CrossRef] [PubMed]

Brangaccio, D. J.

Bravin, A.

K. Krug, L. Porra, P. Coan, A. Wallert, J. Dik, A. Coerdt, A. Bravin, M. Elyyan, P. Reischig, L. Helfen, and T. Baumbach, “Relics in medieval altarpieces? Combining X-ray tomographic, laminographic and phase-contrast imaging to visualize thin organic objects in paintings,” J. Synchrotron Radiat. 15, 55–61 (2008).
[CrossRef]

Brönnimann, Ch.

F. Pfeiffer, M. Bech, O. Bunk, P. Kraft, E. F. Eikenberry, Ch. Brönnimann, C. Grünzweig, and C. David, “Hard-X-ray dark-field imaging using a grating interferometer,” Nat. Mater. 7, 134–137 (2008).
[CrossRef] [PubMed]

Bruning, J. H.

Bunk, O.

F. Pfeiffer, M. Bech, O. Bunk, P. Kraft, E. F. Eikenberry, Ch. Brönnimann, C. Grünzweig, and C. David, “Hard-X-ray dark-field imaging using a grating interferometer,” Nat. Mater. 7, 134–137 (2008).
[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]

Byer, R. L.

Cloetens, P.

L. Helfen, T. Baumbach, P. Cloetens, and J. Baruchel, “Phase-contrast and holographic computed laminography,” Appl. Phys. Lett. 94(10), 104103 (2009).
[CrossRef]

Coan, P.

K. Krug, L. Porra, P. Coan, A. Wallert, J. Dik, A. Coerdt, A. Bravin, M. Elyyan, P. Reischig, L. Helfen, and T. Baumbach, “Relics in medieval altarpieces? Combining X-ray tomographic, laminographic and phase-contrast imaging to visualize thin organic objects in paintings,” J. Synchrotron Radiat. 15, 55–61 (2008).
[CrossRef]

Coerdt, A.

K. Krug, L. Porra, P. Coan, A. Wallert, J. Dik, A. Coerdt, A. Bravin, M. Elyyan, P. Reischig, L. Helfen, and T. Baumbach, “Relics in medieval altarpieces? Combining X-ray tomographic, laminographic and phase-contrast imaging to visualize thin organic objects in paintings,” J. Synchrotron Radiat. 15, 55–61 (2008).
[CrossRef]

David, C.

F. Pfeiffer, M. Bech, O. Bunk, P. Kraft, E. F. Eikenberry, Ch. Brönnimann, C. Grünzweig, and C. David, “Hard-X-ray dark-field imaging using a grating interferometer,” Nat. Mater. 7, 134–137 (2008).
[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]

Davis, L. C.

Di Michiel, M.

L. Helfen, A. Myagotin, P. Mikulík, P. Pernot, A. Voropaev, M. Elyyan, M. Di Michiel, J. Baruchel, and T. Baumbach, “On the implementation of computed laminography using synchrotron radiation,” Rev. Sci. Instrum. 82, 063702 (2011).
[CrossRef] [PubMed]

Dik, J.

K. Krug, L. Porra, P. Coan, A. Wallert, J. Dik, A. Coerdt, A. Bravin, M. Elyyan, P. Reischig, L. Helfen, and T. Baumbach, “Relics in medieval altarpieces? Combining X-ray tomographic, laminographic and phase-contrast imaging to visualize thin organic objects in paintings,” J. Synchrotron Radiat. 15, 55–61 (2008).
[CrossRef]

Eikenberry, E. F.

F. Pfeiffer, M. Bech, O. Bunk, P. Kraft, E. F. Eikenberry, Ch. Brönnimann, C. Grünzweig, and C. David, “Hard-X-ray dark-field imaging using a grating interferometer,” Nat. Mater. 7, 134–137 (2008).
[CrossRef] [PubMed]

Elyyan, M.

L. Helfen, A. Myagotin, P. Mikulík, P. Pernot, A. Voropaev, M. Elyyan, M. Di Michiel, J. Baruchel, and T. Baumbach, “On the implementation of computed laminography using synchrotron radiation,” Rev. Sci. Instrum. 82, 063702 (2011).
[CrossRef] [PubMed]

K. Krug, L. Porra, P. Coan, A. Wallert, J. Dik, A. Coerdt, A. Bravin, M. Elyyan, P. Reischig, L. Helfen, and T. Baumbach, “Relics in medieval altarpieces? Combining X-ray tomographic, laminographic and phase-contrast imaging to visualize thin organic objects in paintings,” J. Synchrotron Radiat. 15, 55–61 (2008).
[CrossRef]

Engelhardt, 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]

Fahrig, R.

G. M. Stevens, R. Fahrig, and N. J. Pelc, “Filtered backprojection for modifying the impulse response of circular tomosynthesis,” Med. Phys. 28, 372–380 (2001).
[CrossRef] [PubMed]

Faris, G. W.

Feldkamp, L. A.

Gallagher, J. E.

Grant, D. G.

D. G. Grant, “Tomosynthesis: a three-dimensional radiographic imaging technique,” IEEE Trans. Biomed. Eng. 19(1), 20–28 (1972).
[CrossRef] [PubMed]

Grünzweig, C.

F. Pfeiffer, M. Bech, O. Bunk, P. Kraft, E. F. Eikenberry, Ch. Brönnimann, C. Grünzweig, and C. David, “Hard-X-ray dark-field imaging using a grating interferometer,” Nat. Mater. 7, 134–137 (2008).
[CrossRef] [PubMed]

Guedouar, R.

R. Guedouar and B. Zarrad, “A comparative study between matched and mis-matched projection/back projection pairs used with ASIRT reconstruction method,” Nucl. Instrum. Methods Phys. Res. A 619(1–3), 225–229 (2010).
[CrossRef]

Gullberg, G. T.

G. L. Zeng and G. T. Gullberg, “Unmatched projector/backprojector pairs in an iterative reconstruction algorithm,” IEEE Trans. Med. Imaging 19, 548–555 (2000).
[CrossRef] [PubMed]

Hamaishi, Y.

A. Momose, S. Kawamoto, I. Koyama, Y. Hamaishi, K. Takai, and Y. Suzuki, “Demonstration of X-ray Talbot interferometry,” Jpn. J. Appl. Phys. 42(7B), 866–868 (2003).
[CrossRef]

Härer, W. H.

G. Lauritsch and W. H. Härer, “A theoretical framework for filtered backprojection in tomosynthesis,” Proc. SPIE 3338, 1127–1137 (1998).
[CrossRef]

Hattori, T.

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]

Helfen, L.

L. Helfen, A. Myagotin, P. Mikulík, P. Pernot, A. Voropaev, M. Elyyan, M. Di Michiel, J. Baruchel, and T. Baumbach, “On the implementation of computed laminography using synchrotron radiation,” Rev. Sci. Instrum. 82, 063702 (2011).
[CrossRef] [PubMed]

F. Xu, L. Helfen, A. J. Moffat, G. Johnson, I. Sinclair, and T. Baumbach, “Synchrotron radiation computed laminography for polymer composite failure studies,” J. Synchrotron Radiat. 17(2), 222–226 (2010).
[CrossRef] [PubMed]

L. Helfen, T. Baumbach, P. Cloetens, and J. Baruchel, “Phase-contrast and holographic computed laminography,” Appl. Phys. Lett. 94(10), 104103 (2009).
[CrossRef]

K. Krug, L. Porra, P. Coan, A. Wallert, J. Dik, A. Coerdt, A. Bravin, M. Elyyan, P. Reischig, L. Helfen, and T. Baumbach, “Relics in medieval altarpieces? Combining X-ray tomographic, laminographic and phase-contrast imaging to visualize thin organic objects in paintings,” J. Synchrotron Radiat. 15, 55–61 (2008).
[CrossRef]

Herriott, D. R.

Horiba, I.

H. Matsuo, A. Iwata, I. Horiba, and N. Suzumura, “Three-dimensional image reconstruction by digital tomosynthesis using inverse filtering,” IEEE Trans. Med. Imaging 12(2), 307–313 (1993).
[CrossRef] [PubMed]

Iwata, A.

H. Matsuo, A. Iwata, I. Horiba, and N. Suzumura, “Three-dimensional image reconstruction by digital tomosynthesis using inverse filtering,” IEEE Trans. Med. Imaging 12(2), 307–313 (1993).
[CrossRef] [PubMed]

Johnson, G.

F. Xu, L. Helfen, A. J. Moffat, G. Johnson, I. Sinclair, and T. Baumbach, “Synchrotron radiation computed laminography for polymer composite failure studies,” J. Synchrotron Radiat. 17(2), 222–226 (2010).
[CrossRef] [PubMed]

Kawamoto, S.

A. Momose, S. Kawamoto, I. Koyama, Y. Hamaishi, K. Takai, and Y. Suzuki, “Demonstration of X-ray Talbot interferometry,” Jpn. J. Appl. Phys. 42(7B), 866–868 (2003).
[CrossRef]

Kawata, S.

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]

Koyama, I.

A. Momose, S. Kawamoto, I. Koyama, Y. Hamaishi, K. Takai, and Y. Suzuki, “Demonstration of X-ray Talbot interferometry,” Jpn. J. Appl. Phys. 42(7B), 866–868 (2003).
[CrossRef]

Kraft, P.

F. Pfeiffer, M. Bech, O. Bunk, P. Kraft, E. F. Eikenberry, Ch. Brönnimann, C. Grünzweig, and C. David, “Hard-X-ray dark-field imaging using a grating interferometer,” Nat. Mater. 7, 134–137 (2008).
[CrossRef] [PubMed]

Kress, J. W.

Krug, K.

K. Krug, L. Porra, P. Coan, A. Wallert, J. Dik, A. Coerdt, A. Bravin, M. Elyyan, P. Reischig, L. Helfen, and T. Baumbach, “Relics in medieval altarpieces? Combining X-ray tomographic, laminographic and phase-contrast imaging to visualize thin organic objects in paintings,” J. Synchrotron Radiat. 15, 55–61 (2008).
[CrossRef]

Lalush, D.

D. Lalush and B. Tsui, “Improving the convergence of iterative filtered backprojection algorithms,” Med. Phys. 21(8), 1283–1286 (1994).
[CrossRef] [PubMed]

Lauritsch, G.

G. Lauritsch and W. H. Härer, “A theoretical framework for filtered backprojection in tomosynthesis,” Proc. SPIE 3338, 1127–1137 (1998).
[CrossRef]

Matsuo, H.

H. Matsuo, A. Iwata, I. Horiba, and N. Suzumura, “Three-dimensional image reconstruction by digital tomosynthesis using inverse filtering,” IEEE Trans. Med. Imaging 12(2), 307–313 (1993).
[CrossRef] [PubMed]

McMorrow, D.

J. Als-Nielsen and D. McMorrow, Elements of Modern X-ray Physics , 2nd ed. (John Wiley and Sons, 2011).
[CrossRef]

Mikulík, P.

L. Helfen, A. Myagotin, P. Mikulík, P. Pernot, A. Voropaev, M. Elyyan, M. Di Michiel, J. Baruchel, and T. Baumbach, “On the implementation of computed laminography using synchrotron radiation,” Rev. Sci. Instrum. 82, 063702 (2011).
[CrossRef] [PubMed]

Minami, S.

Moffat, A. J.

F. Xu, L. Helfen, A. J. Moffat, G. Johnson, I. Sinclair, and T. Baumbach, “Synchrotron radiation computed laminography for polymer composite failure studies,” J. Synchrotron Radiat. 17(2), 222–226 (2010).
[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, S. Kawamoto, I. Koyama, Y. Hamaishi, K. Takai, and Y. Suzuki, “Demonstration of X-ray Talbot interferometry,” Jpn. J. Appl. Phys. 42(7B), 866–868 (2003).
[CrossRef]

Myagotin, A.

L. Helfen, A. Myagotin, P. Mikulík, P. Pernot, A. Voropaev, M. Elyyan, M. Di Michiel, J. Baruchel, and T. Baumbach, “On the implementation of computed laminography using synchrotron radiation,” Rev. Sci. Instrum. 82, 063702 (2011).
[CrossRef] [PubMed]

Nakamura, O.

Natterer, F.

F. Natterer, The Mathematics of Computerized Tomography (Society for Industrial and Applied Mathematics, 2001).
[CrossRef]

Pan, X.

E. Y. Sidky and X. Pan, “Image reconstruction in circular cone-beam computed tomography by constrained, total-variation minimization,” Phys. Med. Biol. 53, 4777 (2008).
[CrossRef] [PubMed]

Pelc, N. J.

G. M. Stevens, R. Fahrig, and N. J. Pelc, “Filtered backprojection for modifying the impulse response of circular tomosynthesis,” Med. Phys. 28, 372–380 (2001).
[CrossRef] [PubMed]

Pernot, P.

L. Helfen, A. Myagotin, P. Mikulík, P. Pernot, A. Voropaev, M. Elyyan, M. Di Michiel, J. Baruchel, and T. Baumbach, “On the implementation of computed laminography using synchrotron radiation,” Rev. Sci. Instrum. 82, 063702 (2011).
[CrossRef] [PubMed]

Pfeiffer, F.

F. Pfeiffer, M. Bech, O. Bunk, P. Kraft, E. F. Eikenberry, Ch. Brönnimann, C. Grünzweig, and C. David, “Hard-X-ray dark-field imaging using a grating interferometer,” Nat. Mater. 7, 134–137 (2008).
[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]

Porra, L.

K. Krug, L. Porra, P. Coan, A. Wallert, J. Dik, A. Coerdt, A. Bravin, M. Elyyan, P. Reischig, L. Helfen, and T. Baumbach, “Relics in medieval altarpieces? Combining X-ray tomographic, laminographic and phase-contrast imaging to visualize thin organic objects in paintings,” J. Synchrotron Radiat. 15, 55–61 (2008).
[CrossRef]

Reischig, P.

K. Krug, L. Porra, P. Coan, A. Wallert, J. Dik, A. Coerdt, A. Bravin, M. Elyyan, P. Reischig, L. Helfen, and T. Baumbach, “Relics in medieval altarpieces? Combining X-ray tomographic, laminographic and phase-contrast imaging to visualize thin organic objects in paintings,” J. Synchrotron Radiat. 15, 55–61 (2008).
[CrossRef]

Rosenfeld, D. P.

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]

Sidky, E. Y.

E. Y. Sidky and X. Pan, “Image reconstruction in circular cone-beam computed tomography by constrained, total-variation minimization,” Phys. Med. Biol. 53, 4777 (2008).
[CrossRef] [PubMed]

Sinclair, I.

F. Xu, L. Helfen, A. J. Moffat, G. Johnson, I. Sinclair, and T. Baumbach, “Synchrotron radiation computed laminography for polymer composite failure studies,” J. Synchrotron Radiat. 17(2), 222–226 (2010).
[CrossRef] [PubMed]

Stevens, G. M.

G. M. Stevens, R. Fahrig, and N. J. Pelc, “Filtered backprojection for modifying the impulse response of circular tomosynthesis,” Med. Phys. 28, 372–380 (2001).
[CrossRef] [PubMed]

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]

A. Momose, S. Kawamoto, I. Koyama, Y. Hamaishi, K. Takai, and Y. Suzuki, “Demonstration of X-ray Talbot interferometry,” Jpn. J. Appl. Phys. 42(7B), 866–868 (2003).
[CrossRef]

Suzumura, N.

H. Matsuo, A. Iwata, I. Horiba, and N. Suzumura, “Three-dimensional image reconstruction by digital tomosynthesis using inverse filtering,” IEEE Trans. Med. Imaging 12(2), 307–313 (1993).
[CrossRef] [PubMed]

Takai, K.

A. Momose, S. Kawamoto, I. Koyama, Y. Hamaishi, K. Takai, and Y. Suzuki, “Demonstration of X-ray Talbot interferometry,” Jpn. J. Appl. Phys. 42(7B), 866–868 (2003).
[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]

Tsui, B.

D. Lalush and B. Tsui, “Improving the convergence of iterative filtered backprojection algorithms,” Med. Phys. 21(8), 1283–1286 (1994).
[CrossRef] [PubMed]

Voropaev, A.

L. Helfen, A. Myagotin, P. Mikulík, P. Pernot, A. Voropaev, M. Elyyan, M. Di Michiel, J. Baruchel, and T. Baumbach, “On the implementation of computed laminography using synchrotron radiation,” Rev. Sci. Instrum. 82, 063702 (2011).
[CrossRef] [PubMed]

Wallert, A.

K. Krug, L. Porra, P. Coan, A. Wallert, J. Dik, A. Coerdt, A. Bravin, M. Elyyan, P. Reischig, L. Helfen, and T. Baumbach, “Relics in medieval altarpieces? Combining X-ray tomographic, laminographic and phase-contrast imaging to visualize thin organic objects in paintings,” J. Synchrotron Radiat. 15, 55–61 (2008).
[CrossRef]

White, A. D.

Xu, F.

F. Xu, L. Helfen, A. J. Moffat, G. Johnson, I. Sinclair, and T. Baumbach, “Synchrotron radiation computed laminography for polymer composite failure studies,” J. Synchrotron Radiat. 17(2), 222–226 (2010).
[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]

Zarrad, B.

R. Guedouar and B. Zarrad, “A comparative study between matched and mis-matched projection/back projection pairs used with ASIRT reconstruction method,” Nucl. Instrum. Methods Phys. Res. A 619(1–3), 225–229 (2010).
[CrossRef]

Zeng, G. L.

G. L. Zeng and G. T. Gullberg, “Unmatched projector/backprojector pairs in an iterative reconstruction algorithm,” IEEE Trans. Med. Imaging 19, 548–555 (2000).
[CrossRef] [PubMed]

Appl. Opt.

Appl. Phys. Lett.

L. Helfen, T. Baumbach, P. Cloetens, and J. Baruchel, “Phase-contrast and holographic computed laminography,” Appl. Phys. Lett. 94(10), 104103 (2009).
[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]

IEEE Trans. Biomed. Eng.

D. G. Grant, “Tomosynthesis: a three-dimensional radiographic imaging technique,” IEEE Trans. Biomed. Eng. 19(1), 20–28 (1972).
[CrossRef] [PubMed]

IEEE Trans. Med. Imaging

H. Matsuo, A. Iwata, I. Horiba, and N. Suzumura, “Three-dimensional image reconstruction by digital tomosynthesis using inverse filtering,” IEEE Trans. Med. Imaging 12(2), 307–313 (1993).
[CrossRef] [PubMed]

G. L. Zeng and G. T. Gullberg, “Unmatched projector/backprojector pairs in an iterative reconstruction algorithm,” IEEE Trans. Med. Imaging 19, 548–555 (2000).
[CrossRef] [PubMed]

J. Opt. Soc. Am. A

J. Synchrotron Radiat.

K. Krug, L. Porra, P. Coan, A. Wallert, J. Dik, A. Coerdt, A. Bravin, M. Elyyan, P. Reischig, L. Helfen, and T. Baumbach, “Relics in medieval altarpieces? Combining X-ray tomographic, laminographic and phase-contrast imaging to visualize thin organic objects in paintings,” J. Synchrotron Radiat. 15, 55–61 (2008).
[CrossRef]

F. Xu, L. Helfen, A. J. Moffat, G. Johnson, I. Sinclair, and T. Baumbach, “Synchrotron radiation computed laminography for polymer composite failure studies,” J. Synchrotron Radiat. 17(2), 222–226 (2010).
[CrossRef] [PubMed]

Jpn. J. Appl. Phys.

A. Momose, S. Kawamoto, I. Koyama, Y. Hamaishi, K. Takai, and Y. Suzuki, “Demonstration of X-ray Talbot interferometry,” Jpn. J. Appl. Phys. 42(7B), 866–868 (2003).
[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]

Med. Phys.

G. M. Stevens, R. Fahrig, and N. J. Pelc, “Filtered backprojection for modifying the impulse response of circular tomosynthesis,” Med. Phys. 28, 372–380 (2001).
[CrossRef] [PubMed]

D. Lalush and B. Tsui, “Improving the convergence of iterative filtered backprojection algorithms,” Med. Phys. 21(8), 1283–1286 (1994).
[CrossRef] [PubMed]

Nat. Mater.

F. Pfeiffer, M. Bech, O. Bunk, P. Kraft, E. F. Eikenberry, Ch. Brönnimann, C. Grünzweig, and C. David, “Hard-X-ray dark-field imaging using a grating interferometer,” Nat. Mater. 7, 134–137 (2008).
[CrossRef] [PubMed]

Nucl. Instrum. Methods Phys. Res. A

R. Guedouar and B. Zarrad, “A comparative study between matched and mis-matched projection/back projection pairs used with ASIRT reconstruction method,” Nucl. Instrum. Methods Phys. Res. A 619(1–3), 225–229 (2010).
[CrossRef]

Phys. Med. Biol.

E. Y. Sidky and X. Pan, “Image reconstruction in circular cone-beam computed tomography by constrained, total-variation minimization,” Phys. Med. Biol. 53, 4777 (2008).
[CrossRef] [PubMed]

Proc. SPIE

G. Lauritsch and W. H. Härer, “A theoretical framework for filtered backprojection in tomosynthesis,” Proc. SPIE 3338, 1127–1137 (1998).
[CrossRef]

Rev. Sci. Instrum.

L. Helfen, A. Myagotin, P. Mikulík, P. Pernot, A. Voropaev, M. Elyyan, M. Di Michiel, J. Baruchel, and T. Baumbach, “On the implementation of computed laminography using synchrotron radiation,” Rev. Sci. Instrum. 82, 063702 (2011).
[CrossRef] [PubMed]

Other

F. Natterer, The Mathematics of Computerized Tomography (Society for Industrial and Applied Mathematics, 2001).
[CrossRef]

J. Als-Nielsen and D. McMorrow, Elements of Modern X-ray Physics , 2nd ed. (John Wiley and Sons, 2011).
[CrossRef]

Supplementary Material (1)

» Media 1: AVI (3664 KB)     

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Figures (8)

Fig. 1
Fig. 1

Geometry of the computed laminography scan. Angle α is fixed between 0 and 90°, whereas angle θ ranges from 0 to 360°. Base vectors (u⃗, v⃗, w⃗) are represented at the detector position for clarity, although the coordinate system origin is point O, at the object position.

Fig. 2
Fig. 2

Reciprocal space of the object, showing the unsampled region as a double cone of aperture 2α and axis ωy . Two projection planes (yellow and green) are also displayed, for different angles θ 1, θ 2. Point A is one of the points at the intersection of the two planes, thus acquired two times during the scan.

Fig. 3
Fig. 3

(a) An optical microscope image of the polypropylene mesh, showing the weave pattern. The wire diameter is 215 μm and the opening is 250 μm. (b) Setup of the mesh sample during CL scan. The X-ray beam transmits through the mesh, acrylic plate support and cylindrical acrylic holder.

Fig. 4
Fig. 4

Beam deflection angle image of the mesh sample obtained by fringe scanning

Fig. 5
Fig. 5

Comparison of FBP and iterative reconstruction methods. (a) Horizontal slice normal to y⃗ obtained by FBP. (b) Same slice obtained by iterative reconstruction. (c) Vertical slice normal to z⃗ obtained by FBP. (d) Same vertical slice obtained by iterative reconstruction, showing fewer artifacts. Red dotted lines show the position of horizontal h and vertical v profiles in Fig. 6. The vertical extent of image (d) matches the prior support of the object.

Fig. 6
Fig. 6

Estimated refractive index decrement δ along the horizontal and vertical profiles defined in Fig. 5(d). The iterative reconstruction (red solid line) has no negative values and square shaped profiles as we can expect from the mesh sample. The filtered backprojection (black dotted line) shows unrealistic negative values as well as strong distortion of the ideal square shape in the vertical direction.

Fig. 7
Fig. 7

Magnified region of a reconstructed horizontal slice after 30 iterations, with and without Hann filtering before backprojection. (a) Without Hann filtering, undesirable high frequency lines are present in the image. (b) With Hann filtering, these artefacts are successfully reduced.

Fig. 8
Fig. 8

Volume rendering of the polypropylene mesh sample, reconstructed by the iterative method. A video is also available (Media 1).

Equations (27)

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

M θ = [ cos θ 0 sin θ sin α sin θ cos α sin α cos θ cos α sin θ sin α cos α cos θ ]
q = M θ r
φ u ( θ , u , v ) = d 2 π z T arg [ k = 1 S I k ( u , v ) exp ( 2 π i k S ) ]
z T = d 2 2 λ
φ u ( θ , u , v ) = λ 2 π Φ ( u , v ) u
φ u ( θ , u , v ) = u δ ( M θ T [ u v w ] ) d w
φ u ˜ ( θ , ω u , ω v ) = 2 π i ω u δ ˜ ( M θ T [ ω u ω v 0 ] )
{ φ ˜ ( θ , ω u , ω v ) = 1 2 π i ω u φ u ˜ ( θ , ω u , ω v ) if ω u 0 φ ˜ ( θ , 0 , ω v ) = 0
b ( x , y , z ) = 0 2 π φ ( θ , u , v ) d θ
= 0 2 π + + φ ˜ ( θ , ω u , ω v ) exp ( 2 π i ( ω u u + ω v v ) ) d ω u d ω v d θ
b ( x , y , z ) = 0 2 π + * φ u ˜ ( θ , ω u , ω v ) 2 π i ω u exp ( 2 π i ( ω u u + ω v v ) ) d ω u d ω v d θ
b ( x , y , z ) = 0 2 π + * δ ˜ ( M θ T [ ω u ω v 0 ] ) exp ( 2 π i ( ω u u + ω v v ) ) d ω u d ω v d θ
[ ω x ω y ω z ] = M θ T [ ω u ω v 0 ]
ω x 2 + ω z 2 > ω y 2 tan 2 α
b ( x , y , z ) = C δ ˜ ( [ ω x ω y ω z ] ) 2 exp ( 2 π i ( ω x x + ω y y + ω z z ) ) cos α ω x 2 + ω z 2 ω y 2 tan 2 α d ω x d ω y d ω z
{ H ( ω x , ω y , ω z ) = 2 cos α ω x 2 + ω z 2 ω y 2 tan 2 α if ω x 2 + ω z 2 > ω y 2 tan 2 α H ( ω x , ω y , ω z ) = 0 else
G ( θ , ω u , ω v ) = H ( M θ [ ω u ω v 0 ] ) = 2 cos α | ω u |
b fbp ( x , y , z ) = 0 2 π + * cos α sgn ( ω u ) 4 π i φ u ˜ ( θ , ω u , ω v ) exp ( 2 π i ( ω u u + ω v v ) ) d ω u d ω v d θ
F inv ( ω u ) = cos α sgn ( ω u ) 4 π i
( s k + 1 s k + λ k h k with h k = B ( b P s k ) s k + 1 𝒞 ( s k + 1 )
{ F ( ω u ) = cos α sgn ( ω u ) 4 π i 1 2 ( 1 + cos ( 2 π ω u a ) ) if ω u 0 F ( 0 ) = 0
{ 𝒞 ( s ) [ i ] = s [ i ] if δ min < = s [ i ] < = δ max and i 𝒮 𝒞 ( s ) [ i ] = δ min if s [ i ] < δ min and i 𝒮 𝒞 ( s ) [ i ] = δ max if s [ i ] > δ max and i 𝒮 𝒞 ( s ) [ i ] = 0 else
f ( u ) = 1 / 2 a 1 / 2 a F ( ω u ) exp ( 2 π i ω u u ) d ω u
{ f ( n a ) = n cos α 4 π 2 ( n 2 1 ) a for n even f ( n a ) = cos α 4 π 2 n a for n odd
h k = s k R ( s k ) = s k [ 1 2 s k T B P s k s k T B b ]
R ( s k + 1 ) λ k = R ( s k + λ k h k ) λ k = h k T B P s k + λ k h k T B P h k h k T B b
λ k = h k T h k h k T B P h k

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