Abstract

Coherent diffractive imaging provides accurate phase projections that can be tomographically combined to yield detailed quantitative 3D reconstructions with a resolution that is not limited by imaging optics. We present robust algorithms for post-processing and alignment of these tomographic phase projections. A simple method to remove undesired constant and linear phase terms on the reconstructions is given. Also, we provide an algorithm for automatic alignment of projections that has good performance even for samples with no fiducial markers. Currently applied to phase projections, this alignment algorithm has proven to be robust and should also be useful for lens-based tomography techniques that pursue nanoscale 3D imaging. Lastly, we provide a method for tomographic reconstruction that works on phase projections that are known modulo 2π, such that the phase unwrapping step is avoided. We demonstrate the performance of these algorithms by 3D imaging of bacteria population in legume root-nodule cells.

© 2011 OSA

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    [Crossref]

2011 (1)

C. T. Putkunz, J. N. Clark, D. J. Vine, G. J. Williams, M. A. Pfeifer, E. Balaur, I. McNulty, K. A. Nugent, and A. G. Peele, “Phase-diverse coherent diffractive imaging: High sensitivity with low dose,” Phys. Rev. Lett. 106, 013903 (2011).
[Crossref] [PubMed]

2010 (5)

M. Dierolf, P. Thibault, A. Menzel, C. M. Kewish, K. Jefimovs, I. Schlichting, K. von König, O. Bunk, and F. Pfeiffer, “Ptychographic coherent diffractive imaging of weakly scattering specimens,” New J. Phys. 12, 035017 (2010).
[Crossref]

M. Stampanoni, R. Mokso, F. Marone, J. Vila-Comamala, S. Gorelick, P. Trik, K. Jefimovs, and C. David, “Phase-contrast tomography at the nanoscale using hard x-rays,” Phys. Rev. B 81, 140105 (2010).
[Crossref]

H. Jiang, C. Song, C.-C. Chen, R. Xu, K. S. Raines, B. P. Fahimian, C.-H. Lu, T.-K. Lee, A. Nakashima, J. Urano, T. Ishikawa, F. Tamanoi, and J. Miao, “Quantitative 3D imaging of whole, unstained cells by using x-ray diffraction microscopy,” Proc. Natl. Acad. Sci. U.S.A. 107, 11234–11239 (2010).
[Crossref] [PubMed]

M. Dierolf, A. Menzel, P. Thibault, P. Schneider, C. M. Kewish, R. Wepf, O. Bunk, and F. Pfeiffer, “Ptychographic x-ray computed tomography at the nanoscale,” Nature 467, 436–439 (2010).
[Crossref] [PubMed]

C. T. Putkunz, M. A. Pfeifer, A. G. Peele, G. J. Williams, H. M. Quiney, B. Abbey, K. A. Nugent, and I. McNulty, “Fresnel coherent diffraction tomography,” Opt. Express 18, 11746–11753 (2010).
[Crossref] [PubMed]

2009 (2)

C.-C. Chen, J. Miao, and T. K. Lee, “Tomographic image alignment in three-dimensional coherent diffraction microscopy,” Phys. Rev. B 79, 052102 (2009).
[Crossref]

P. Thibault, M. Dierolf, O. Bunk, A. Menzel, and F. Pfeiffer, “Probe retrieval in ptychographic coherent diffractive imaging,” Ultramicroscopy 109, 338–343 (2009).
[Crossref] [PubMed]

2008 (4)

P. Thibault, M. Dierolf, A. Menzel, O. Bunk, C. David, and F. Pfeiffer, “High-resolution scanning x-ray diffraction microscopy,” Science 321, 379–382 (2008).
[Crossref] [PubMed]

O. Bunk, M. Dierolf, S. Kynde, I. Johnson, O. Marti, and F. Pfeiffer, “Influence of the overlap parameter on the convergence of the ptychographical iterative engine,” Ultramicroscopy 108, 481–487 (2008).
[Crossref]

M. Guizar-Sicairos, S. T. Thurman, and J. R. Fienup, “Efficient subpixel image registration algorithms,” Opt. Lett. 33, 156–158 (2008).
[Crossref] [PubMed]

M. Guizar-Sicairos and J. R. Fienup, “Phase retrieval with transverse translation diversity: a nonlinear optimization approach,” Opt. Express 16, 7264–7278 (2008).
[Crossref] [PubMed]

2007 (4)

R. Soummer, L. Pueyo, A. Sivaramakrishnan, and R. J. Vanderbei, “Fast computation of Lyot-style coronagraph propagation,” Opt. Express 15, 15935–15951 (2007).
[Crossref] [PubMed]

R. Mokso, P. Cloetens, E. Maire, W. Ludwig, and J.-Y. Buffière, “Nanoscale zoom tomography with hard x rays using Kirkpatrick-Baez optics,” Appl. Phys. Lett. 90, 144104 (2007).
[Crossref]

F. Pfeiffer, O. Bunk, C. David, M. Bech, G. Le Duc, A. Bravin, and P. Cloetens, “High-resolution brain tumor visualization using three-dimensional x-ray phase contrast tomography,” Phys. Med. Biol. 52, 6923–6930 (2007).
[Crossref] [PubMed]

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

2006 (2)

2004 (2)

C. A. Larabell and M. A. Le Gros, “X-ray tomography generates 3-D reconstructions of the yeast, Saccharomyces cerevisiae, at 60-nm resolution,” Molec. Biol. Cell 15, 957–962 (2004).
[Crossref]

J. M. Rodenburg and H. M. L. Faulkner, “A phase retrieval algorithm for shifting illumination,” Appl. Phys. Lett. 85, 4795–4797 (2004).
[Crossref]

2003 (2)

S. S. Biel, K. Kawaschinski, K.-P. Wittern, U. Hintze, and R. Wepf, “From tissue to cellular ultrastructure: closing the gap between micro- and nanostructural imaging,” J. Microsc. 212, 91–99 (2003).
[Crossref] [PubMed]

V. Elser, “Phase retrieval by iterated projections,” J. Opt. Soc. Am. A 20, 40–55 (2003).
[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]

J. Miao, P. Charalambous, J. Kirz, and D. Sayre, “Extending the methodology of x-ray crystallography to allow imaging of micrometre-sized non-crystalline specimens,” Nature 400, 342–344 (1999).
[Crossref]

D. Ress, M. L. Harlow, M. Schwarz, R. M. Marshall, and U. J. McMahan, “Automatic acquisition of fiducial markers and alignment of images in tilt series for electron tomography,” J. Electron Microsc. 48, 277–287 (1999).

1997 (1)

1991 (1)

Z. Liang, “Implementation of linear filters for iterative penalized maximum likelihood SPECT reconstruction,” IEEE Trans. Nucl. Sci. 38, 606–611 (1991).
[Crossref]

1988 (2)

R. M. Goldstein, H. A. Zebker, and C. L. Werner, “Satellite radar interferometry: Two-dimensional phase unwrapping,” Radio Sci. 23, 713–720 (1988).
[Crossref]

G. W. Faris and R. L. Byer, “Three-dimensional beam-deflection optical tomography of a supersonic jet,” Appl. Opt. 27, 5202–5212 (1988).
[Crossref] [PubMed]

1982 (1)

Abbey, B.

Balaur, E.

C. T. Putkunz, J. N. Clark, D. J. Vine, G. J. Williams, M. A. Pfeifer, E. Balaur, I. McNulty, K. A. Nugent, and A. G. Peele, “Phase-diverse coherent diffractive imaging: High sensitivity with low dose,” Phys. Rev. Lett. 106, 013903 (2011).
[Crossref] [PubMed]

Barty, 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]

Bech, M.

F. Pfeiffer, O. Bunk, C. David, M. Bech, G. Le Duc, A. Bravin, and P. Cloetens, “High-resolution brain tumor visualization using three-dimensional x-ray phase contrast tomography,” Phys. Med. Biol. 52, 6923–6930 (2007).
[Crossref] [PubMed]

Beetz, T.

Bergamaschi, A.

P. Kraft, A. Bergamaschi, C. Bronnimann, R. Dinapoli, E. F. Eikenberry, H. Graafsma, B. Henrich, I. Johnson, M. Kobas, A. Mozzanica, C. M. Schleputz, and B. Schmitt, “Characterization and calibration of PILATUS detectors,” IEEE Trans. Nucl. Sci.56, 758–764 (2009).
[Crossref]

Biel, S. S.

S. S. Biel, K. Kawaschinski, K.-P. Wittern, U. Hintze, and R. Wepf, “From tissue to cellular ultrastructure: closing the gap between micro- and nanostructural imaging,” J. Microsc. 212, 91–99 (2003).
[Crossref] [PubMed]

Bravin, A.

F. Pfeiffer, O. Bunk, C. David, M. Bech, G. Le Duc, A. Bravin, and P. Cloetens, “High-resolution brain tumor visualization using three-dimensional x-ray phase contrast tomography,” Phys. Med. Biol. 52, 6923–6930 (2007).
[Crossref] [PubMed]

Bronnimann, C.

P. Kraft, A. Bergamaschi, C. Bronnimann, R. Dinapoli, E. F. Eikenberry, H. Graafsma, B. Henrich, I. Johnson, M. Kobas, A. Mozzanica, C. M. Schleputz, and B. Schmitt, “Characterization and calibration of PILATUS detectors,” IEEE Trans. Nucl. Sci.56, 758–764 (2009).
[Crossref]

Buffière, J.-Y.

R. Mokso, P. Cloetens, E. Maire, W. Ludwig, and J.-Y. Buffière, “Nanoscale zoom tomography with hard x rays using Kirkpatrick-Baez optics,” Appl. Phys. Lett. 90, 144104 (2007).
[Crossref]

Bunk, O.

M. Dierolf, A. Menzel, P. Thibault, P. Schneider, C. M. Kewish, R. Wepf, O. Bunk, and F. Pfeiffer, “Ptychographic x-ray computed tomography at the nanoscale,” Nature 467, 436–439 (2010).
[Crossref] [PubMed]

M. Dierolf, P. Thibault, A. Menzel, C. M. Kewish, K. Jefimovs, I. Schlichting, K. von König, O. Bunk, and F. Pfeiffer, “Ptychographic coherent diffractive imaging of weakly scattering specimens,” New J. Phys. 12, 035017 (2010).
[Crossref]

P. Thibault, M. Dierolf, O. Bunk, A. Menzel, and F. Pfeiffer, “Probe retrieval in ptychographic coherent diffractive imaging,” Ultramicroscopy 109, 338–343 (2009).
[Crossref] [PubMed]

P. Thibault, M. Dierolf, A. Menzel, O. Bunk, C. David, and F. Pfeiffer, “High-resolution scanning x-ray diffraction microscopy,” Science 321, 379–382 (2008).
[Crossref] [PubMed]

O. Bunk, M. Dierolf, S. Kynde, I. Johnson, O. Marti, and F. Pfeiffer, “Influence of the overlap parameter on the convergence of the ptychographical iterative engine,” Ultramicroscopy 108, 481–487 (2008).
[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]

F. Pfeiffer, O. Bunk, C. David, M. Bech, G. Le Duc, A. Bravin, and P. Cloetens, “High-resolution brain tumor visualization using three-dimensional x-ray phase contrast tomography,” Phys. Med. Biol. 52, 6923–6930 (2007).
[Crossref] [PubMed]

A. Diaz, P. Trtik, M. Guizar-Sicairos, A. Menzel, P. Thibault, and O. Bunk, “Quantitative x-ray phase tomography with resolution in the 100 nm range,” manuscript in preparation (2011).

Byer, R. L.

Chapman, H. N.

Charalambous, P.

J. Miao, P. Charalambous, J. Kirz, and D. Sayre, “Extending the methodology of x-ray crystallography to allow imaging of micrometre-sized non-crystalline specimens,” Nature 400, 342–344 (1999).
[Crossref]

Chen, C.-C.

H. Jiang, C. Song, C.-C. Chen, R. Xu, K. S. Raines, B. P. Fahimian, C.-H. Lu, T.-K. Lee, A. Nakashima, J. Urano, T. Ishikawa, F. Tamanoi, and J. Miao, “Quantitative 3D imaging of whole, unstained cells by using x-ray diffraction microscopy,” Proc. Natl. Acad. Sci. U.S.A. 107, 11234–11239 (2010).
[Crossref] [PubMed]

C.-C. Chen, J. Miao, and T. K. Lee, “Tomographic image alignment in three-dimensional coherent diffraction microscopy,” Phys. Rev. B 79, 052102 (2009).
[Crossref]

Clark, J. N.

C. T. Putkunz, J. N. Clark, D. J. Vine, G. J. Williams, M. A. Pfeifer, E. Balaur, I. McNulty, K. A. Nugent, and A. G. Peele, “Phase-diverse coherent diffractive imaging: High sensitivity with low dose,” Phys. Rev. Lett. 106, 013903 (2011).
[Crossref] [PubMed]

Cloetens, P.

F. Pfeiffer, O. Bunk, C. David, M. Bech, G. Le Duc, A. Bravin, and P. Cloetens, “High-resolution brain tumor visualization using three-dimensional x-ray phase contrast tomography,” Phys. Med. Biol. 52, 6923–6930 (2007).
[Crossref] [PubMed]

R. Mokso, P. Cloetens, E. Maire, W. Ludwig, and J.-Y. Buffière, “Nanoscale zoom tomography with hard x rays using Kirkpatrick-Baez optics,” Appl. Phys. Lett. 90, 144104 (2007).
[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]

Cui, C.

David, C.

M. Stampanoni, R. Mokso, F. Marone, J. Vila-Comamala, S. Gorelick, P. Trik, K. Jefimovs, and C. David, “Phase-contrast tomography at the nanoscale using hard x-rays,” Phys. Rev. B 81, 140105 (2010).
[Crossref]

P. Thibault, M. Dierolf, A. Menzel, O. Bunk, C. David, and F. Pfeiffer, “High-resolution scanning x-ray diffraction microscopy,” Science 321, 379–382 (2008).
[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]

F. Pfeiffer, O. Bunk, C. David, M. Bech, G. Le Duc, A. Bravin, and P. Cloetens, “High-resolution brain tumor visualization using three-dimensional x-ray phase contrast tomography,” Phys. Med. Biol. 52, 6923–6930 (2007).
[Crossref] [PubMed]

Diaz, A.

A. Diaz, P. Trtik, M. Guizar-Sicairos, A. Menzel, P. Thibault, and O. Bunk, “Quantitative x-ray phase tomography with resolution in the 100 nm range,” manuscript in preparation (2011).

Dierolf, M.

M. Dierolf, A. Menzel, P. Thibault, P. Schneider, C. M. Kewish, R. Wepf, O. Bunk, and F. Pfeiffer, “Ptychographic x-ray computed tomography at the nanoscale,” Nature 467, 436–439 (2010).
[Crossref] [PubMed]

M. Dierolf, P. Thibault, A. Menzel, C. M. Kewish, K. Jefimovs, I. Schlichting, K. von König, O. Bunk, and F. Pfeiffer, “Ptychographic coherent diffractive imaging of weakly scattering specimens,” New J. Phys. 12, 035017 (2010).
[Crossref]

P. Thibault, M. Dierolf, O. Bunk, A. Menzel, and F. Pfeiffer, “Probe retrieval in ptychographic coherent diffractive imaging,” Ultramicroscopy 109, 338–343 (2009).
[Crossref] [PubMed]

O. Bunk, M. Dierolf, S. Kynde, I. Johnson, O. Marti, and F. Pfeiffer, “Influence of the overlap parameter on the convergence of the ptychographical iterative engine,” Ultramicroscopy 108, 481–487 (2008).
[Crossref]

P. Thibault, M. Dierolf, A. Menzel, O. Bunk, C. David, and F. Pfeiffer, “High-resolution scanning x-ray diffraction microscopy,” Science 321, 379–382 (2008).
[Crossref] [PubMed]

Dinapoli, R.

P. Kraft, A. Bergamaschi, C. Bronnimann, R. Dinapoli, E. F. Eikenberry, H. Graafsma, B. Henrich, I. Johnson, M. Kobas, A. Mozzanica, C. M. Schleputz, and B. Schmitt, “Characterization and calibration of PILATUS detectors,” IEEE Trans. Nucl. Sci.56, 758–764 (2009).
[Crossref]

Eikenberry, E. F.

P. Kraft, A. Bergamaschi, C. Bronnimann, R. Dinapoli, E. F. Eikenberry, H. Graafsma, B. Henrich, I. Johnson, M. Kobas, A. Mozzanica, C. M. Schleputz, and B. Schmitt, “Characterization and calibration of PILATUS detectors,” IEEE Trans. Nucl. Sci.56, 758–764 (2009).
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Elser, V.

Fahimian, B. P.

H. Jiang, C. Song, C.-C. Chen, R. Xu, K. S. Raines, B. P. Fahimian, C.-H. Lu, T.-K. Lee, A. Nakashima, J. Urano, T. Ishikawa, F. Tamanoi, and J. Miao, “Quantitative 3D imaging of whole, unstained cells by using x-ray diffraction microscopy,” Proc. Natl. Acad. Sci. U.S.A. 107, 11234–11239 (2010).
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Faulkner, H. M. L.

J. M. Rodenburg and H. M. L. Faulkner, “A phase retrieval algorithm for shifting illumination,” Appl. Phys. Lett. 85, 4795–4797 (2004).
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Fienup, J. R.

Frank, J.

J. Frank and B. F. McEwen, “Alignment by cross-correlation,” in Electron Tomography: Three-Dimensional Imaging with the Transmission Electron Microscope, J. Frank, ed. (Springer, 1992), chapter 9.

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D. C. Ghiglia and M. D. Pritt, Two-Dimensional Phase Unwrapping: Theory, Algorithms and Software, (Wiley, 1998).

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R. M. Goldstein, H. A. Zebker, and C. L. Werner, “Satellite radar interferometry: Two-dimensional phase unwrapping,” Radio Sci. 23, 713–720 (1988).
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Gorelick, S.

M. Stampanoni, R. Mokso, F. Marone, J. Vila-Comamala, S. Gorelick, P. Trik, K. Jefimovs, and C. David, “Phase-contrast tomography at the nanoscale using hard x-rays,” Phys. Rev. B 81, 140105 (2010).
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Graafsma, H.

P. Kraft, A. Bergamaschi, C. Bronnimann, R. Dinapoli, E. F. Eikenberry, H. Graafsma, B. Henrich, I. Johnson, M. Kobas, A. Mozzanica, C. M. Schleputz, and B. Schmitt, “Characterization and calibration of PILATUS detectors,” IEEE Trans. Nucl. Sci.56, 758–764 (2009).
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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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Guizar-Sicairos, M.

Harder, R.

M. A. Pfeifer, G. J. Williams, I. A. Vartanyants, R. Harder, and I. K. Robinson, “Three-dimensional mapping of a deformation field inside a nanocrystal,” Nature 442, 63–66 (2006).
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Harlow, M. L.

D. Ress, M. L. Harlow, M. Schwarz, R. M. Marshall, and U. J. McMahan, “Automatic acquisition of fiducial markers and alignment of images in tilt series for electron tomography,” J. Electron Microsc. 48, 277–287 (1999).

Hau-Riege, S. P.

He, H.

Henrich, B.

P. Kraft, A. Bergamaschi, C. Bronnimann, R. Dinapoli, E. F. Eikenberry, H. Graafsma, B. Henrich, I. Johnson, M. Kobas, A. Mozzanica, C. M. Schleputz, and B. Schmitt, “Characterization and calibration of PILATUS detectors,” IEEE Trans. Nucl. Sci.56, 758–764 (2009).
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S. S. Biel, K. Kawaschinski, K.-P. Wittern, U. Hintze, and R. Wepf, “From tissue to cellular ultrastructure: closing the gap between micro- and nanostructural imaging,” J. Microsc. 212, 91–99 (2003).
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Ishikawa, T.

H. Jiang, C. Song, C.-C. Chen, R. Xu, K. S. Raines, B. P. Fahimian, C.-H. Lu, T.-K. Lee, A. Nakashima, J. Urano, T. Ishikawa, F. Tamanoi, and J. Miao, “Quantitative 3D imaging of whole, unstained cells by using x-ray diffraction microscopy,” Proc. Natl. Acad. Sci. U.S.A. 107, 11234–11239 (2010).
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Jacobsen, C.

Jefimovs, K.

M. Dierolf, P. Thibault, A. Menzel, C. M. Kewish, K. Jefimovs, I. Schlichting, K. von König, O. Bunk, and F. Pfeiffer, “Ptychographic coherent diffractive imaging of weakly scattering specimens,” New J. Phys. 12, 035017 (2010).
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M. Stampanoni, R. Mokso, F. Marone, J. Vila-Comamala, S. Gorelick, P. Trik, K. Jefimovs, and C. David, “Phase-contrast tomography at the nanoscale using hard x-rays,” Phys. Rev. B 81, 140105 (2010).
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Jiang, H.

H. Jiang, C. Song, C.-C. Chen, R. Xu, K. S. Raines, B. P. Fahimian, C.-H. Lu, T.-K. Lee, A. Nakashima, J. Urano, T. Ishikawa, F. Tamanoi, and J. Miao, “Quantitative 3D imaging of whole, unstained cells by using x-ray diffraction microscopy,” Proc. Natl. Acad. Sci. U.S.A. 107, 11234–11239 (2010).
[Crossref] [PubMed]

Johnson, I.

O. Bunk, M. Dierolf, S. Kynde, I. Johnson, O. Marti, and F. Pfeiffer, “Influence of the overlap parameter on the convergence of the ptychographical iterative engine,” Ultramicroscopy 108, 481–487 (2008).
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P. Kraft, A. Bergamaschi, C. Bronnimann, R. Dinapoli, E. F. Eikenberry, H. Graafsma, B. Henrich, I. Johnson, M. Kobas, A. Mozzanica, C. M. Schleputz, and B. Schmitt, “Characterization and calibration of PILATUS detectors,” IEEE Trans. Nucl. Sci.56, 758–764 (2009).
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Kak, A. C.

A. C. Kak and M. Slaney, Principles of Computerized Tomographic Imaging, (IEEE1989).

Kawaschinski, K.

S. S. Biel, K. Kawaschinski, K.-P. Wittern, U. Hintze, and R. Wepf, “From tissue to cellular ultrastructure: closing the gap between micro- and nanostructural imaging,” J. Microsc. 212, 91–99 (2003).
[Crossref] [PubMed]

Kewish, C. M.

M. Dierolf, P. Thibault, A. Menzel, C. M. Kewish, K. Jefimovs, I. Schlichting, K. von König, O. Bunk, and F. Pfeiffer, “Ptychographic coherent diffractive imaging of weakly scattering specimens,” New J. Phys. 12, 035017 (2010).
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M. Dierolf, A. Menzel, P. Thibault, P. Schneider, C. M. Kewish, R. Wepf, O. Bunk, and F. Pfeiffer, “Ptychographic x-ray computed tomography at the nanoscale,” Nature 467, 436–439 (2010).
[Crossref] [PubMed]

Kirz, J.

J. Miao, P. Charalambous, J. Kirz, and D. Sayre, “Extending the methodology of x-ray crystallography to allow imaging of micrometre-sized non-crystalline specimens,” Nature 400, 342–344 (1999).
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Kobas, M.

P. Kraft, A. Bergamaschi, C. Bronnimann, R. Dinapoli, E. F. Eikenberry, H. Graafsma, B. Henrich, I. Johnson, M. Kobas, A. Mozzanica, C. M. Schleputz, and B. Schmitt, “Characterization and calibration of PILATUS detectors,” IEEE Trans. Nucl. Sci.56, 758–764 (2009).
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Kottler, C.

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

P. Kraft, A. Bergamaschi, C. Bronnimann, R. Dinapoli, E. F. Eikenberry, H. Graafsma, B. Henrich, I. Johnson, M. Kobas, A. Mozzanica, C. M. Schleputz, and B. Schmitt, “Characterization and calibration of PILATUS detectors,” IEEE Trans. Nucl. Sci.56, 758–764 (2009).
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Kynde, S.

O. Bunk, M. Dierolf, S. Kynde, I. Johnson, O. Marti, and F. Pfeiffer, “Influence of the overlap parameter on the convergence of the ptychographical iterative engine,” Ultramicroscopy 108, 481–487 (2008).
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Larabell, C. A.

C. A. Larabell and M. A. Le Gros, “X-ray tomography generates 3-D reconstructions of the yeast, Saccharomyces cerevisiae, at 60-nm resolution,” Molec. Biol. Cell 15, 957–962 (2004).
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Le Duc, G.

F. Pfeiffer, O. Bunk, C. David, M. Bech, G. Le Duc, A. Bravin, and P. Cloetens, “High-resolution brain tumor visualization using three-dimensional x-ray phase contrast tomography,” Phys. Med. Biol. 52, 6923–6930 (2007).
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Le Gros, M. A.

C. A. Larabell and M. A. Le Gros, “X-ray tomography generates 3-D reconstructions of the yeast, Saccharomyces cerevisiae, at 60-nm resolution,” Molec. Biol. Cell 15, 957–962 (2004).
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Lee, T. K.

C.-C. Chen, J. Miao, and T. K. Lee, “Tomographic image alignment in three-dimensional coherent diffraction microscopy,” Phys. Rev. B 79, 052102 (2009).
[Crossref]

Lee, T.-K.

H. Jiang, C. Song, C.-C. Chen, R. Xu, K. S. Raines, B. P. Fahimian, C.-H. Lu, T.-K. Lee, A. Nakashima, J. Urano, T. Ishikawa, F. Tamanoi, and J. Miao, “Quantitative 3D imaging of whole, unstained cells by using x-ray diffraction microscopy,” Proc. Natl. Acad. Sci. U.S.A. 107, 11234–11239 (2010).
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Liang, Z.

Z. Liang, “Implementation of linear filters for iterative penalized maximum likelihood SPECT reconstruction,” IEEE Trans. Nucl. Sci. 38, 606–611 (1991).
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Lu, C.-H.

H. Jiang, C. Song, C.-C. Chen, R. Xu, K. S. Raines, B. P. Fahimian, C.-H. Lu, T.-K. Lee, A. Nakashima, J. Urano, T. Ishikawa, F. Tamanoi, and J. Miao, “Quantitative 3D imaging of whole, unstained cells by using x-ray diffraction microscopy,” Proc. Natl. Acad. Sci. U.S.A. 107, 11234–11239 (2010).
[Crossref] [PubMed]

Ludwig, W.

R. Mokso, P. Cloetens, E. Maire, W. Ludwig, and J.-Y. Buffière, “Nanoscale zoom tomography with hard x rays using Kirkpatrick-Baez optics,” Appl. Phys. Lett. 90, 144104 (2007).
[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]

Maire, E.

R. Mokso, P. Cloetens, E. Maire, W. Ludwig, and J.-Y. Buffière, “Nanoscale zoom tomography with hard x rays using Kirkpatrick-Baez optics,” Appl. Phys. Lett. 90, 144104 (2007).
[Crossref]

Marchesini, S.

Marone, F.

M. Stampanoni, R. Mokso, F. Marone, J. Vila-Comamala, S. Gorelick, P. Trik, K. Jefimovs, and C. David, “Phase-contrast tomography at the nanoscale using hard x-rays,” Phys. Rev. B 81, 140105 (2010).
[Crossref]

Marshall, R. M.

D. Ress, M. L. Harlow, M. Schwarz, R. M. Marshall, and U. J. McMahan, “Automatic acquisition of fiducial markers and alignment of images in tilt series for electron tomography,” J. Electron Microsc. 48, 277–287 (1999).

Marti, O.

O. Bunk, M. Dierolf, S. Kynde, I. Johnson, O. Marti, and F. Pfeiffer, “Influence of the overlap parameter on the convergence of the ptychographical iterative engine,” Ultramicroscopy 108, 481–487 (2008).
[Crossref]

McEwen, B. F.

J. Frank and B. F. McEwen, “Alignment by cross-correlation,” in Electron Tomography: Three-Dimensional Imaging with the Transmission Electron Microscope, J. Frank, ed. (Springer, 1992), chapter 9.

McMahan, U. J.

D. Ress, M. L. Harlow, M. Schwarz, R. M. Marshall, and U. J. McMahan, “Automatic acquisition of fiducial markers and alignment of images in tilt series for electron tomography,” J. Electron Microsc. 48, 277–287 (1999).

McNulty, I.

C. T. Putkunz, J. N. Clark, D. J. Vine, G. J. Williams, M. A. Pfeifer, E. Balaur, I. McNulty, K. A. Nugent, and A. G. Peele, “Phase-diverse coherent diffractive imaging: High sensitivity with low dose,” Phys. Rev. Lett. 106, 013903 (2011).
[Crossref] [PubMed]

C. T. Putkunz, M. A. Pfeifer, A. G. Peele, G. J. Williams, H. M. Quiney, B. Abbey, K. A. Nugent, and I. McNulty, “Fresnel coherent diffraction tomography,” Opt. Express 18, 11746–11753 (2010).
[Crossref] [PubMed]

Menzel, A.

M. Dierolf, P. Thibault, A. Menzel, C. M. Kewish, K. Jefimovs, I. Schlichting, K. von König, O. Bunk, and F. Pfeiffer, “Ptychographic coherent diffractive imaging of weakly scattering specimens,” New J. Phys. 12, 035017 (2010).
[Crossref]

M. Dierolf, A. Menzel, P. Thibault, P. Schneider, C. M. Kewish, R. Wepf, O. Bunk, and F. Pfeiffer, “Ptychographic x-ray computed tomography at the nanoscale,” Nature 467, 436–439 (2010).
[Crossref] [PubMed]

P. Thibault, M. Dierolf, O. Bunk, A. Menzel, and F. Pfeiffer, “Probe retrieval in ptychographic coherent diffractive imaging,” Ultramicroscopy 109, 338–343 (2009).
[Crossref] [PubMed]

P. Thibault, M. Dierolf, A. Menzel, O. Bunk, C. David, and F. Pfeiffer, “High-resolution scanning x-ray diffraction microscopy,” Science 321, 379–382 (2008).
[Crossref] [PubMed]

A. Diaz, P. Trtik, M. Guizar-Sicairos, A. Menzel, P. Thibault, and O. Bunk, “Quantitative x-ray phase tomography with resolution in the 100 nm range,” manuscript in preparation (2011).

Miao, J.

H. Jiang, C. Song, C.-C. Chen, R. Xu, K. S. Raines, B. P. Fahimian, C.-H. Lu, T.-K. Lee, A. Nakashima, J. Urano, T. Ishikawa, F. Tamanoi, and J. Miao, “Quantitative 3D imaging of whole, unstained cells by using x-ray diffraction microscopy,” Proc. Natl. Acad. Sci. U.S.A. 107, 11234–11239 (2010).
[Crossref] [PubMed]

C.-C. Chen, J. Miao, and T. K. Lee, “Tomographic image alignment in three-dimensional coherent diffraction microscopy,” Phys. Rev. B 79, 052102 (2009).
[Crossref]

J. Miao, P. Charalambous, J. Kirz, and D. Sayre, “Extending the methodology of x-ray crystallography to allow imaging of micrometre-sized non-crystalline specimens,” Nature 400, 342–344 (1999).
[Crossref]

Mokso, R.

M. Stampanoni, R. Mokso, F. Marone, J. Vila-Comamala, S. Gorelick, P. Trik, K. Jefimovs, and C. David, “Phase-contrast tomography at the nanoscale using hard x-rays,” Phys. Rev. B 81, 140105 (2010).
[Crossref]

R. Mokso, P. Cloetens, E. Maire, W. Ludwig, and J.-Y. Buffière, “Nanoscale zoom tomography with hard x rays using Kirkpatrick-Baez optics,” Appl. Phys. Lett. 90, 144104 (2007).
[Crossref]

Mozzanica, A.

P. Kraft, A. Bergamaschi, C. Bronnimann, R. Dinapoli, E. F. Eikenberry, H. Graafsma, B. Henrich, I. Johnson, M. Kobas, A. Mozzanica, C. M. Schleputz, and B. Schmitt, “Characterization and calibration of PILATUS detectors,” IEEE Trans. Nucl. Sci.56, 758–764 (2009).
[Crossref]

Nakashima, A.

H. Jiang, C. Song, C.-C. Chen, R. Xu, K. S. Raines, B. P. Fahimian, C.-H. Lu, T.-K. Lee, A. Nakashima, J. Urano, T. Ishikawa, F. Tamanoi, and J. Miao, “Quantitative 3D imaging of whole, unstained cells by using x-ray diffraction microscopy,” Proc. Natl. Acad. Sci. U.S.A. 107, 11234–11239 (2010).
[Crossref] [PubMed]

Noy, A.

Nugent, K. A.

C. T. Putkunz, J. N. Clark, D. J. Vine, G. J. Williams, M. A. Pfeifer, E. Balaur, I. McNulty, K. A. Nugent, and A. G. Peele, “Phase-diverse coherent diffractive imaging: High sensitivity with low dose,” Phys. Rev. Lett. 106, 013903 (2011).
[Crossref] [PubMed]

C. T. Putkunz, M. A. Pfeifer, A. G. Peele, G. J. Williams, H. M. Quiney, B. Abbey, K. A. Nugent, and I. McNulty, “Fresnel coherent diffraction tomography,” Opt. Express 18, 11746–11753 (2010).
[Crossref] [PubMed]

Peele, A. G.

C. T. Putkunz, J. N. Clark, D. J. Vine, G. J. Williams, M. A. Pfeifer, E. Balaur, I. McNulty, K. A. Nugent, and A. G. Peele, “Phase-diverse coherent diffractive imaging: High sensitivity with low dose,” Phys. Rev. Lett. 106, 013903 (2011).
[Crossref] [PubMed]

C. T. Putkunz, M. A. Pfeifer, A. G. Peele, G. J. Williams, H. M. Quiney, B. Abbey, K. A. Nugent, and I. McNulty, “Fresnel coherent diffraction tomography,” Opt. Express 18, 11746–11753 (2010).
[Crossref] [PubMed]

Pfeifer, M. A.

C. T. Putkunz, J. N. Clark, D. J. Vine, G. J. Williams, M. A. Pfeifer, E. Balaur, I. McNulty, K. A. Nugent, and A. G. Peele, “Phase-diverse coherent diffractive imaging: High sensitivity with low dose,” Phys. Rev. Lett. 106, 013903 (2011).
[Crossref] [PubMed]

C. T. Putkunz, M. A. Pfeifer, A. G. Peele, G. J. Williams, H. M. Quiney, B. Abbey, K. A. Nugent, and I. McNulty, “Fresnel coherent diffraction tomography,” Opt. Express 18, 11746–11753 (2010).
[Crossref] [PubMed]

M. A. Pfeifer, G. J. Williams, I. A. Vartanyants, R. Harder, and I. K. Robinson, “Three-dimensional mapping of a deformation field inside a nanocrystal,” Nature 442, 63–66 (2006).
[Crossref] [PubMed]

Pfeiffer, F.

M. Dierolf, A. Menzel, P. Thibault, P. Schneider, C. M. Kewish, R. Wepf, O. Bunk, and F. Pfeiffer, “Ptychographic x-ray computed tomography at the nanoscale,” Nature 467, 436–439 (2010).
[Crossref] [PubMed]

M. Dierolf, P. Thibault, A. Menzel, C. M. Kewish, K. Jefimovs, I. Schlichting, K. von König, O. Bunk, and F. Pfeiffer, “Ptychographic coherent diffractive imaging of weakly scattering specimens,” New J. Phys. 12, 035017 (2010).
[Crossref]

P. Thibault, M. Dierolf, O. Bunk, A. Menzel, and F. Pfeiffer, “Probe retrieval in ptychographic coherent diffractive imaging,” Ultramicroscopy 109, 338–343 (2009).
[Crossref] [PubMed]

P. Thibault, M. Dierolf, A. Menzel, O. Bunk, C. David, and F. Pfeiffer, “High-resolution scanning x-ray diffraction microscopy,” Science 321, 379–382 (2008).
[Crossref] [PubMed]

O. Bunk, M. Dierolf, S. Kynde, I. Johnson, O. Marti, and F. Pfeiffer, “Influence of the overlap parameter on the convergence of the ptychographical iterative engine,” Ultramicroscopy 108, 481–487 (2008).
[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]

F. Pfeiffer, O. Bunk, C. David, M. Bech, G. Le Duc, A. Bravin, and P. Cloetens, “High-resolution brain tumor visualization using three-dimensional x-ray phase contrast tomography,” Phys. Med. Biol. 52, 6923–6930 (2007).
[Crossref] [PubMed]

Pritt, M. D.

D. C. Ghiglia and M. D. Pritt, Two-Dimensional Phase Unwrapping: Theory, Algorithms and Software, (Wiley, 1998).

Pueyo, L.

Putkunz, C. T.

C. T. Putkunz, J. N. Clark, D. J. Vine, G. J. Williams, M. A. Pfeifer, E. Balaur, I. McNulty, K. A. Nugent, and A. G. Peele, “Phase-diverse coherent diffractive imaging: High sensitivity with low dose,” Phys. Rev. Lett. 106, 013903 (2011).
[Crossref] [PubMed]

C. T. Putkunz, M. A. Pfeifer, A. G. Peele, G. J. Williams, H. M. Quiney, B. Abbey, K. A. Nugent, and I. McNulty, “Fresnel coherent diffraction tomography,” Opt. Express 18, 11746–11753 (2010).
[Crossref] [PubMed]

Quiney, H. M.

Raines, K. S.

H. Jiang, C. Song, C.-C. Chen, R. Xu, K. S. Raines, B. P. Fahimian, C.-H. Lu, T.-K. Lee, A. Nakashima, J. Urano, T. Ishikawa, F. Tamanoi, and J. Miao, “Quantitative 3D imaging of whole, unstained cells by using x-ray diffraction microscopy,” Proc. Natl. Acad. Sci. U.S.A. 107, 11234–11239 (2010).
[Crossref] [PubMed]

Ress, D.

D. Ress, M. L. Harlow, M. Schwarz, R. M. Marshall, and U. J. McMahan, “Automatic acquisition of fiducial markers and alignment of images in tilt series for electron tomography,” J. Electron Microsc. 48, 277–287 (1999).

Robinson, I. K.

M. A. Pfeifer, G. J. Williams, I. A. Vartanyants, R. Harder, and I. K. Robinson, “Three-dimensional mapping of a deformation field inside a nanocrystal,” Nature 442, 63–66 (2006).
[Crossref] [PubMed]

Rodenburg, J. M.

J. M. Rodenburg and H. M. L. Faulkner, “A phase retrieval algorithm for shifting illumination,” Appl. Phys. Lett. 85, 4795–4797 (2004).
[Crossref]

Rosen, R.

Sayre, D.

J. Miao, P. Charalambous, J. Kirz, and D. Sayre, “Extending the methodology of x-ray crystallography to allow imaging of micrometre-sized non-crystalline specimens,” Nature 400, 342–344 (1999).
[Crossref]

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]

Schleputz, C. M.

P. Kraft, A. Bergamaschi, C. Bronnimann, R. Dinapoli, E. F. Eikenberry, H. Graafsma, B. Henrich, I. Johnson, M. Kobas, A. Mozzanica, C. M. Schleputz, and B. Schmitt, “Characterization and calibration of PILATUS detectors,” IEEE Trans. Nucl. Sci.56, 758–764 (2009).
[Crossref]

Schlichting, I.

M. Dierolf, P. Thibault, A. Menzel, C. M. Kewish, K. Jefimovs, I. Schlichting, K. von König, O. Bunk, and F. Pfeiffer, “Ptychographic coherent diffractive imaging of weakly scattering specimens,” New J. Phys. 12, 035017 (2010).
[Crossref]

Schmitt, B.

P. Kraft, A. Bergamaschi, C. Bronnimann, R. Dinapoli, E. F. Eikenberry, H. Graafsma, B. Henrich, I. Johnson, M. Kobas, A. Mozzanica, C. M. Schleputz, and B. Schmitt, “Characterization and calibration of PILATUS detectors,” IEEE Trans. Nucl. Sci.56, 758–764 (2009).
[Crossref]

Schneider, P.

M. Dierolf, A. Menzel, P. Thibault, P. Schneider, C. M. Kewish, R. Wepf, O. Bunk, and F. Pfeiffer, “Ptychographic x-ray computed tomography at the nanoscale,” Nature 467, 436–439 (2010).
[Crossref] [PubMed]

Schwarz, M.

D. Ress, M. L. Harlow, M. Schwarz, R. M. Marshall, and U. J. McMahan, “Automatic acquisition of fiducial markers and alignment of images in tilt series for electron tomography,” J. Electron Microsc. 48, 277–287 (1999).

Shapiro, D.

Sivaramakrishnan, A.

Slaney, M.

A. C. Kak and M. Slaney, Principles of Computerized Tomographic Imaging, (IEEE1989).

Song, C.

H. Jiang, C. Song, C.-C. Chen, R. Xu, K. S. Raines, B. P. Fahimian, C.-H. Lu, T.-K. Lee, A. Nakashima, J. Urano, T. Ishikawa, F. Tamanoi, and J. Miao, “Quantitative 3D imaging of whole, unstained cells by using x-ray diffraction microscopy,” Proc. Natl. Acad. Sci. U.S.A. 107, 11234–11239 (2010).
[Crossref] [PubMed]

Soummer, R.

Spence, J. C. H.

Stampanoni, M.

M. Stampanoni, R. Mokso, F. Marone, J. Vila-Comamala, S. Gorelick, P. Trik, K. Jefimovs, and C. David, “Phase-contrast tomography at the nanoscale using hard x-rays,” Phys. Rev. B 81, 140105 (2010).
[Crossref]

Tamanoi, F.

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[Crossref] [PubMed]

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[Crossref]

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[Crossref] [PubMed]

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[Crossref]

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H. Jiang, C. Song, C.-C. Chen, R. Xu, K. S. Raines, B. P. Fahimian, C.-H. Lu, T.-K. Lee, A. Nakashima, J. Urano, T. Ishikawa, F. Tamanoi, and J. Miao, “Quantitative 3D imaging of whole, unstained cells by using x-ray diffraction microscopy,” Proc. Natl. Acad. Sci. U.S.A. 107, 11234–11239 (2010).
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S. S. Biel, K. Kawaschinski, K.-P. Wittern, U. Hintze, and R. Wepf, “From tissue to cellular ultrastructure: closing the gap between micro- and nanostructural imaging,” J. Microsc. 212, 91–99 (2003).
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H. Jiang, C. Song, C.-C. Chen, R. Xu, K. S. Raines, B. P. Fahimian, C.-H. Lu, T.-K. Lee, A. Nakashima, J. Urano, T. Ishikawa, F. Tamanoi, and J. Miao, “Quantitative 3D imaging of whole, unstained cells by using x-ray diffraction microscopy,” Proc. Natl. Acad. Sci. U.S.A. 107, 11234–11239 (2010).
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S. S. Biel, K. Kawaschinski, K.-P. Wittern, U. Hintze, and R. Wepf, “From tissue to cellular ultrastructure: closing the gap between micro- and nanostructural imaging,” J. Microsc. 212, 91–99 (2003).
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New J. Phys. (1)

M. Dierolf, P. Thibault, A. Menzel, C. M. Kewish, K. Jefimovs, I. Schlichting, K. von König, O. Bunk, and F. Pfeiffer, “Ptychographic coherent diffractive imaging of weakly scattering specimens,” New J. Phys. 12, 035017 (2010).
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Opt. Express (3)

Opt. Lett. (1)

Phys. Med. Biol. (1)

F. Pfeiffer, O. Bunk, C. David, M. Bech, G. Le Duc, A. Bravin, and P. Cloetens, “High-resolution brain tumor visualization using three-dimensional x-ray phase contrast tomography,” Phys. Med. Biol. 52, 6923–6930 (2007).
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Phys. Rev. B (2)

M. Stampanoni, R. Mokso, F. Marone, J. Vila-Comamala, S. Gorelick, P. Trik, K. Jefimovs, and C. David, “Phase-contrast tomography at the nanoscale using hard x-rays,” Phys. Rev. B 81, 140105 (2010).
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C.-C. Chen, J. Miao, and T. K. Lee, “Tomographic image alignment in three-dimensional coherent diffraction microscopy,” Phys. Rev. B 79, 052102 (2009).
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C. T. Putkunz, J. N. Clark, D. J. Vine, G. J. Williams, M. A. Pfeifer, E. Balaur, I. McNulty, K. A. Nugent, and A. G. Peele, “Phase-diverse coherent diffractive imaging: High sensitivity with low dose,” Phys. Rev. Lett. 106, 013903 (2011).
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Proc. Natl. Acad. Sci. U.S.A. (1)

H. Jiang, C. Song, C.-C. Chen, R. Xu, K. S. Raines, B. P. Fahimian, C.-H. Lu, T.-K. Lee, A. Nakashima, J. Urano, T. Ishikawa, F. Tamanoi, and J. Miao, “Quantitative 3D imaging of whole, unstained cells by using x-ray diffraction microscopy,” Proc. Natl. Acad. Sci. U.S.A. 107, 11234–11239 (2010).
[Crossref] [PubMed]

Radio Sci. (1)

R. M. Goldstein, H. A. Zebker, and C. L. Werner, “Satellite radar interferometry: Two-dimensional phase unwrapping,” Radio Sci. 23, 713–720 (1988).
[Crossref]

Science (1)

P. Thibault, M. Dierolf, A. Menzel, O. Bunk, C. David, and F. Pfeiffer, “High-resolution scanning x-ray diffraction microscopy,” Science 321, 379–382 (2008).
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[Crossref]

P. Thibault, M. Dierolf, O. Bunk, A. Menzel, and F. Pfeiffer, “Probe retrieval in ptychographic coherent diffractive imaging,” Ultramicroscopy 109, 338–343 (2009).
[Crossref] [PubMed]

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A. Diaz, P. Trtik, M. Guizar-Sicairos, A. Menzel, P. Thibault, and O. Bunk, “Quantitative x-ray phase tomography with resolution in the 100 nm range,” manuscript in preparation (2011).

Supplementary Material (3)

» Media 1: AVI (10587 KB)     
» Media 2: AVI (6827 KB)     
» Media 3: AVI (8528 KB)     

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Figures (5)

Fig. 1
Fig. 1

A three-dimensional quantitative map of the real part of the sample refractive index can be obtained by tomographic combination of phase projections.

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Fig. 2
Fig. 2

After removing constant and linear phase terms, a well-behaved area is selected for unwrapping. This area is used for alignment of the phase projections. The obtained translation offsets (Δx, Δy) are then used along with the wrapped phase to obtain a 3D reconstruction of the sample index of refraction.

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Fig. 3
Fig. 3

(a) Phase projection (in radians) of the sample with a “well-behaved” region unwrapped for alignment. Red rectangle denotes the region used for alignment. The projection alignment functions ψθ (y) are shown (b) before and (d) after the alignment. (c) Sample position offsets vs. projection number.

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Fig. 4
Fig. 4

Comparison of vertical alignment of projections and interferometer for two independent measurements. Lower subplots show the difference with RMS values of (a) 55 nm and (b) 40 nm, respectively.

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Fig. 5
Fig. 5

Three-dimensional renderings of the mung bean tissue sample are shown in (a) and (c). (b) Horizontal and (d) vertical tomographic sections revealing a platinum compound layer (P), bacteria population (B), cellular walls (W), round vesicles (V) and intercellular spaces. The location of the tomographic sections is indicated in (a) as intersecting planes and in (b) and (d) with white lines. Tomographic slices and an animated rendering are provided as multimedia files ( Media 1, Media 2, Media 3) for visualization of the full reconstructed volume.

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Equations (19)

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t ( x , y , θ ) = exp { i 2 π λ ∫ [ n ( r ′ ) − 1 ] d z } ,
o ( x , y , θ ) = t ( x , y , θ ) exp [ i ( a θ + b θ x + c θ y ) ] ,
σ 2 = ∑ x , y w ( x , y ) | 1 − exp ( i arg [ o ( x , y ) ] ) exp ( − i a ) exp [ − i 2 π ( u x M + v y N ) ] | 2 ∑ x , y w ( x , y )
σ 2 ≈ ∑ x , y w ( x , y ) { arg [ o ( x , y ) ] − a − u x M − v y N } 2 ∑ x , y w ( x , y ) ,
min a , u , v σ 2 = 2 − 2 max u , v | R ( u , v ) | ∑ x , y w ( x , y ) ,
R ( u , v ) = ∑ x , y w ( x , y ) exp { i arg [ o ( x , y ) ] } exp [ − i 2 π ( u x M + v y N ) ] .
a = arg [ R ( u , v ) ] .
ϕ ( x , y , θ ) = arg [ t ( x , y , θ ) ] = 2 π λ ∫ δ ( r ′ ) d z ,
M θ ( y ) = ∫ ϕ ( x , y , θ ) d x = − 2 π λ ∬ δ ( x cos θ − z sin θ , y , z cos θ + x sin θ ) d x d z .
E 2 = ∑ y , θ [ ψ ( y − Δ y θ ( n + 1 ) , θ ) − 〈 ψ ( y − Δ y θ ( n ) , θ ) 〉 ] 2 ,
− 2 π Δ λ δ ( r ) = ∫ 0 π ℱ u − 1 { | u | ℱ x ′ { ϕ ( x ′ , y , θ ) } } d θ ,
− 2 π Δ λ δ ( r ) = ∫ 0 π ℱ u − 1 { 1 i 2 π sgn ( u ) ℱ x { ∂ ϕ ( x , y , θ ) ∂ x } } d θ .
∂ ϕ ( x , y , θ ) ∂ x ≈ 1 h arg [ g ( x + h / 2 , y , θ ) g * ( x − h / 2 , y , θ ) ] ,
g ( x , y , θ ) = exp [ i ϕ ( x , y , θ ) ] ,
f n ( x , y ) = o ( x − x n , y − y n ) p ( x , y ) .
I n ( u , v ) = | F n ( u , v ) | 2 ,
o ′ ( x , y ) = o ( x , y ) exp [ i 2 π ( b x + c y ) ] ,
p ′ ( x , y ) = p ( x , y ) exp [ − i 2 π ( b x + c y ) ] ,
f n ′ ( x , y ) = o ′ ( x − x n , y − y n ) p ′ ( x , y ) = f n ( x , y ) exp [ − i 2 π ( b x n + c y n ) ] .

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