Abstract

We illustrate the errors inherent in the conventional empty beam correction of full field X-ray propagation imaging, i.e. the division of intensities in the detection plane measured with an object in the beam by the intensity pattern measured without the object, i.e. the empty beam intensity pattern. The error of this conventional approximation is controlled by the ratio of the source size to the smallest feature in the object, as is shown by numerical simulation. In a second step, we investigate how to overcome the flawed empty beam division by simultaneous reconstruction of the probing wavefront (probe) and of the object, based on measurements in several detection planes (multi-projection approach). The algorithmic scheme is demonstrated numerically and experimentally, using the defocus wavefront of the hard X-ray nanoprobe setup at the European Synchrotron Radiation Facility (ESRF).

© 2014 Optical Society of America

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2013

M. Krenkel, M. Bartels, T. Salditt, “Transport of intensity phase reconstruction to solve the twin image problem in holographic x-ray imaging,” Opt. Express 21, 2220–2235 (2013).
[CrossRef] [PubMed]

B. D. Metscher, “Biological applications of x-ray microtomography: Imaging microanatomy, molecular expression and organismal diversity,” Microsc. Anal. (Am Ed) 27, 13–16 (2013).

J. Moosmann, A. Ershov, V. Altapova, T. Baumbach, M. S. Prasad, C. LaBonne, X. Xiao, J. Kashef, R. Hofmann, “X-ray phase-contrast in vivo microtomography probes new aspects of xenopus gastrulation,” Nature (London) 497, 374–377 (2013).
[CrossRef]

M. Bartels, V. H. Hernandez, M. Krenkel, T. Moser, T. Salditt, “Phase contrast tomography of the mouse cochlea at microfocus x-ray sources,” Appl. Phys. Lett. 103, 083703 (2013).
[CrossRef]

M. Stockmar, P. Cloetens, I. Zanette, B. Enders, M. Dierolf, F. Pfeiffer, P. Thibault, “Near-field ptychography: Phase retrieval for inline holography using a structured illumination,” Sci. Rep. 3, 1–6 (2013).
[CrossRef]

F. Döring, A.-L. Robisch, C. Eberl, M. Osterhoff, A. Ruhlandt, T. Liese, F. Schlenkrich, S. Hoffmann, M. Bartels, T. Salditt, H. U. Krebs, “Sub-5 nm hard x-ray point focusing by a combined Kirkpatrick-Baez mirror and multilayer zone plate,” Opt. Express 21, 19311–19323 (2013).
[CrossRef] [PubMed]

T. B. Edo, D. J. Batey, A. M. Maiden, C. Rau, U. Wagner, Z. D. Pešić, T. A. Waigh, J. M. Rodenburg, “Sampling in x-ray ptychography,” Phys. Rev. A 87, 053850 (2013).
[CrossRef]

A.-L. Robisch, T. Salditt, “Phase retrieval for object and probe using a series of defocus near-field images,” Opt. Express 21, 23345–23357 (2013).
[CrossRef] [PubMed]

2012

M. Bartels, M. Priebe, R. Wilke, S. Kruger, K. Giewekemeyer, S. Kalbfleisch, C. Olendrowitz, M. Sprung, T. Salditt, “Low-dose three-dimensional hard x-ray imaging of bacterial cells,” Opt. Nanoscopy, 1,1–10 (2012).

C. Olendrowitz, M. Bartels, M. Krenkel, A. Beerlink, R. Mokso, M. Sprung, T. Salditt, “Phase-contrast x-ray imaging and tomography of the nematode Caenorhabditis elegans,” Phys. Med. Biol. 57, 5309–5323 (2012).
[CrossRef] [PubMed]

G. Martinez-Criado, R. Tucoulou, P. Cloetens, P. Bleuet, S. Bohic, J. Cauzid, I. Kieffer, E. Kosior, S. Laboure, S. Petitgirard, A. Rack, J. A. Sans, J. Segura-Ruiz, H. Suhonen, J. Susini, J. Villanova, “Status of the hard x-ray microprobe beamline ID22 of the European Synchrotron Radiation Facility,” J. Synchrotron Radiat. 19, 10–18 (2012).
[CrossRef]

2011

S. Kalbfleisch, H. Neubauer, S. P. Krüger, M. Bartels, M. Osterhoff, D. D. Mai, K. Giewekemeyer, B. Hartmann, M. Sprung, T. Salditt, “The Göttingen holography endstation of beamline P10 at PETRA III/DESY,” AIP Conf. Proc. 1365, 96–99 (2011).
[CrossRef]

T. van de Kamp, P. Vagovič, T. Baumbach, A. Riedel, “A biological screw in a beetle’s leg,” Science 333, 52 (2011).
[CrossRef]

K. A. Nugent, “The measurement of phase through the propagation of intensity: An introduction,” Contemp. Phys. 52, 55–69 (2011).
[CrossRef]

K. Giewekemeyer, S. Krüger, S. Kalbfleisch, M. Bartels, C. Beta, T. Salditt, “X-ray propagation microscopy of biological cells using waveguides as a quasipoint source,” Phys. Rev. A 83, 023804 (2011).
[CrossRef]

2010

2009

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

A. M. Maiden, J. M. Rodenburg, “An improved ptychographical phase retrieval algorithm for diffractive imaging,” Ultramicroscopy 109, 1256–1262 (2009).
[CrossRef] [PubMed]

2006

P. Tafforeau, R. Boistel, E. Boller, A. Bravin, M. Brunet, Y. Chaimanee, P. Cloetens, M. Feist, J. Hoszowska, J.-J. Jaeger, “Applications of x-ray synchrotron microtomography for non-destructive 3D studies of paleontological specimens,” Appl. Phys. A Mater. 83, 195–202 (2006).
[CrossRef]

P. Cloetens, R. Mache, M. Schlenker, S. Lerbs-Mache, “Quantitative phase tomography of Arabidopsis seeds reveals intercellular void network,” Proc. Natl. Acad. Sci. USA 103, 14626–14630 (2006).
[CrossRef] [PubMed]

H. M. Quiney, A. G. Peele, Z. Cai, D. Paterson, K. A. Nugent, “Diffractive imaging of highly focused x-ray fields,” Nature (London) 2, 101–104 (2006).

2005

D. R. Luke, “Relaxed averaged alternating reflections for diffraction imaging,” Inverse Probl. 21, 37–50 (2005).
[CrossRef]

2004

B. M. Hanser, M. G. L. Gustafsson, D. A. Agard, J. W. Sedat, “Phase-retrieved pupil functions in wide-field fluorescence microscopy,” J. Microsc. 216, 32–48 (2004).
[CrossRef] [PubMed]

L. J. Allen, W. McBride, M. P. Oxley, “Exit wave reconstruction using soft x-rays,” Opt. Commun. 233, 77–82 (2004).
[CrossRef]

L. J. Allen, W. McBride, N. O’Leary, M. P. Oxley, “Exit wave reconstruction at atomic resolution,” Ultramicroscopy 100, 91–104 (2004).
[CrossRef] [PubMed]

2002

D. R. Luke, J. V. Burke, R. G. Lyon, “Optical wavefront reconstruction: Theory and numerical methods,” SIAM Rev. 44, 169–224 (2002).
[CrossRef]

2001

E. Maire, J.-Y. Buffière, L. Salvo, J. J. Blandin, W. Ludwig, J. M. Letang, “On the application of x-ray microtomography in the field of materials science,” Adv. Eng. Mater. 3, 539–546 (2001).
[CrossRef]

L. J. Allen, H. M. L. Faulkner, K. A. Nugent, M. P. Oxley, D. Paganin, “Phase retrieval from images in the presence of first-order vortices,” Phys. Rev. E 63, 037602 (2001).
[CrossRef]

L. J. Allen, M. P. Oxley, “Phase retrieval from series of images obtained by defocus variation,” Opt. Commun. 199, 65–75 (2001).
[CrossRef]

1999

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

1997

T. E. Gureyev, K. A. Nugent, “Rapid quantitative phase imaging using the transport of intensity equation,” Opt. Commun. 133, 339–346 (1997).
[CrossRef]

A. Pogany, D. Gao, S. W. Wilkins, “Contrast and resolution in imaging with a microfocus x-ray source,” Rev. Sci. Instrum. 68, 2774–2782 (1997).
[CrossRef]

1973

D. L. Misell, “An examination of an iterative method for the solution of the phase problem in optics and electron optics: I. Test calculations,” J. Phys. D: Appl. Phys. 6, 2200–2216 (1973).
[CrossRef]

Agard, D. A.

B. M. Hanser, M. G. L. Gustafsson, D. A. Agard, J. W. Sedat, “Phase-retrieved pupil functions in wide-field fluorescence microscopy,” J. Microsc. 216, 32–48 (2004).
[CrossRef] [PubMed]

Allen, L. J.

L. J. Allen, W. McBride, N. O’Leary, M. P. Oxley, “Exit wave reconstruction at atomic resolution,” Ultramicroscopy 100, 91–104 (2004).
[CrossRef] [PubMed]

L. J. Allen, W. McBride, M. P. Oxley, “Exit wave reconstruction using soft x-rays,” Opt. Commun. 233, 77–82 (2004).
[CrossRef]

L. J. Allen, H. M. L. Faulkner, K. A. Nugent, M. P. Oxley, D. Paganin, “Phase retrieval from images in the presence of first-order vortices,” Phys. Rev. E 63, 037602 (2001).
[CrossRef]

L. J. Allen, M. P. Oxley, “Phase retrieval from series of images obtained by defocus variation,” Opt. Commun. 199, 65–75 (2001).
[CrossRef]

Altapova, V.

J. Moosmann, A. Ershov, V. Altapova, T. Baumbach, M. S. Prasad, C. LaBonne, X. Xiao, J. Kashef, R. Hofmann, “X-ray phase-contrast in vivo microtomography probes new aspects of xenopus gastrulation,” Nature (London) 497, 374–377 (2013).
[CrossRef]

Baker, R.

R. Barrett, R. Baker, P. Cloetens, Y. Dabin, C. Morawe, H. Suhonen, R. Tucoulou, A. Vivo, L. Zhang, “Dynamically-figured mirror system for high-energy nanofocusing at the ESRF,” Proc. SPIE pp. 813904–813912 (2011).
[CrossRef]

Barrett, R.

R. Barrett, R. Baker, P. Cloetens, Y. Dabin, C. Morawe, H. Suhonen, R. Tucoulou, A. Vivo, L. Zhang, “Dynamically-figured mirror system for high-energy nanofocusing at the ESRF,” Proc. SPIE pp. 813904–813912 (2011).
[CrossRef]

Bartels, M.

F. Döring, A.-L. Robisch, C. Eberl, M. Osterhoff, A. Ruhlandt, T. Liese, F. Schlenkrich, S. Hoffmann, M. Bartels, T. Salditt, H. U. Krebs, “Sub-5 nm hard x-ray point focusing by a combined Kirkpatrick-Baez mirror and multilayer zone plate,” Opt. Express 21, 19311–19323 (2013).
[CrossRef] [PubMed]

M. Bartels, V. H. Hernandez, M. Krenkel, T. Moser, T. Salditt, “Phase contrast tomography of the mouse cochlea at microfocus x-ray sources,” Appl. Phys. Lett. 103, 083703 (2013).
[CrossRef]

M. Krenkel, M. Bartels, T. Salditt, “Transport of intensity phase reconstruction to solve the twin image problem in holographic x-ray imaging,” Opt. Express 21, 2220–2235 (2013).
[CrossRef] [PubMed]

M. Bartels, M. Priebe, R. Wilke, S. Kruger, K. Giewekemeyer, S. Kalbfleisch, C. Olendrowitz, M. Sprung, T. Salditt, “Low-dose three-dimensional hard x-ray imaging of bacterial cells,” Opt. Nanoscopy, 1,1–10 (2012).

C. Olendrowitz, M. Bartels, M. Krenkel, A. Beerlink, R. Mokso, M. Sprung, T. Salditt, “Phase-contrast x-ray imaging and tomography of the nematode Caenorhabditis elegans,” Phys. Med. Biol. 57, 5309–5323 (2012).
[CrossRef] [PubMed]

K. Giewekemeyer, S. Krüger, S. Kalbfleisch, M. Bartels, C. Beta, T. Salditt, “X-ray propagation microscopy of biological cells using waveguides as a quasipoint source,” Phys. Rev. A 83, 023804 (2011).
[CrossRef]

S. Kalbfleisch, H. Neubauer, S. P. Krüger, M. Bartels, M. Osterhoff, D. D. Mai, K. Giewekemeyer, B. Hartmann, M. Sprung, T. Salditt, “The Göttingen holography endstation of beamline P10 at PETRA III/DESY,” AIP Conf. Proc. 1365, 96–99 (2011).
[CrossRef]

Baruchel, J.

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

Batey, D. J.

T. B. Edo, D. J. Batey, A. M. Maiden, C. Rau, U. Wagner, Z. D. Pešić, T. A. Waigh, J. M. Rodenburg, “Sampling in x-ray ptychography,” Phys. Rev. A 87, 053850 (2013).
[CrossRef]

Baumbach, T.

J. Moosmann, A. Ershov, V. Altapova, T. Baumbach, M. S. Prasad, C. LaBonne, X. Xiao, J. Kashef, R. Hofmann, “X-ray phase-contrast in vivo microtomography probes new aspects of xenopus gastrulation,” Nature (London) 497, 374–377 (2013).
[CrossRef]

T. van de Kamp, P. Vagovič, T. Baumbach, A. Riedel, “A biological screw in a beetle’s leg,” Science 333, 52 (2011).
[CrossRef]

Beerlink, A.

C. Olendrowitz, M. Bartels, M. Krenkel, A. Beerlink, R. Mokso, M. Sprung, T. Salditt, “Phase-contrast x-ray imaging and tomography of the nematode Caenorhabditis elegans,” Phys. Med. Biol. 57, 5309–5323 (2012).
[CrossRef] [PubMed]

Beta, C.

K. Giewekemeyer, S. Krüger, S. Kalbfleisch, M. Bartels, C. Beta, T. Salditt, “X-ray propagation microscopy of biological cells using waveguides as a quasipoint source,” Phys. Rev. A 83, 023804 (2011).
[CrossRef]

Blandin, J. J.

E. Maire, J.-Y. Buffière, L. Salvo, J. J. Blandin, W. Ludwig, J. M. Letang, “On the application of x-ray microtomography in the field of materials science,” Adv. Eng. Mater. 3, 539–546 (2001).
[CrossRef]

Bleuet, P.

G. Martinez-Criado, R. Tucoulou, P. Cloetens, P. Bleuet, S. Bohic, J. Cauzid, I. Kieffer, E. Kosior, S. Laboure, S. Petitgirard, A. Rack, J. A. Sans, J. Segura-Ruiz, H. Suhonen, J. Susini, J. Villanova, “Status of the hard x-ray microprobe beamline ID22 of the European Synchrotron Radiation Facility,” J. Synchrotron Radiat. 19, 10–18 (2012).
[CrossRef]

Bohic, S.

G. Martinez-Criado, R. Tucoulou, P. Cloetens, P. Bleuet, S. Bohic, J. Cauzid, I. Kieffer, E. Kosior, S. Laboure, S. Petitgirard, A. Rack, J. A. Sans, J. Segura-Ruiz, H. Suhonen, J. Susini, J. Villanova, “Status of the hard x-ray microprobe beamline ID22 of the European Synchrotron Radiation Facility,” J. Synchrotron Radiat. 19, 10–18 (2012).
[CrossRef]

Boistel, R.

P. Tafforeau, R. Boistel, E. Boller, A. Bravin, M. Brunet, Y. Chaimanee, P. Cloetens, M. Feist, J. Hoszowska, J.-J. Jaeger, “Applications of x-ray synchrotron microtomography for non-destructive 3D studies of paleontological specimens,” Appl. Phys. A Mater. 83, 195–202 (2006).
[CrossRef]

Boller, E.

P. Tafforeau, R. Boistel, E. Boller, A. Bravin, M. Brunet, Y. Chaimanee, P. Cloetens, M. Feist, J. Hoszowska, J.-J. Jaeger, “Applications of x-ray synchrotron microtomography for non-destructive 3D studies of paleontological specimens,” Appl. Phys. A Mater. 83, 195–202 (2006).
[CrossRef]

Bravin, A.

P. Tafforeau, R. Boistel, E. Boller, A. Bravin, M. Brunet, Y. Chaimanee, P. Cloetens, M. Feist, J. Hoszowska, J.-J. Jaeger, “Applications of x-ray synchrotron microtomography for non-destructive 3D studies of paleontological specimens,” Appl. Phys. A Mater. 83, 195–202 (2006).
[CrossRef]

Brunet, M.

P. Tafforeau, R. Boistel, E. Boller, A. Bravin, M. Brunet, Y. Chaimanee, P. Cloetens, M. Feist, J. Hoszowska, J.-J. Jaeger, “Applications of x-ray synchrotron microtomography for non-destructive 3D studies of paleontological specimens,” Appl. Phys. A Mater. 83, 195–202 (2006).
[CrossRef]

Buffière, J.-Y.

E. Maire, J.-Y. Buffière, L. Salvo, J. J. Blandin, W. Ludwig, J. M. Letang, “On the application of x-ray microtomography in the field of materials science,” Adv. Eng. Mater. 3, 539–546 (2001).
[CrossRef]

Bunk, O.

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

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D. R. Luke, J. V. Burke, R. G. Lyon, “Optical wavefront reconstruction: Theory and numerical methods,” SIAM Rev. 44, 169–224 (2002).
[CrossRef]

Cai, Z.

H. M. Quiney, A. G. Peele, Z. Cai, D. Paterson, K. A. Nugent, “Diffractive imaging of highly focused x-ray fields,” Nature (London) 2, 101–104 (2006).

Cauzid, J.

G. Martinez-Criado, R. Tucoulou, P. Cloetens, P. Bleuet, S. Bohic, J. Cauzid, I. Kieffer, E. Kosior, S. Laboure, S. Petitgirard, A. Rack, J. A. Sans, J. Segura-Ruiz, H. Suhonen, J. Susini, J. Villanova, “Status of the hard x-ray microprobe beamline ID22 of the European Synchrotron Radiation Facility,” J. Synchrotron Radiat. 19, 10–18 (2012).
[CrossRef]

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P. Tafforeau, R. Boistel, E. Boller, A. Bravin, M. Brunet, Y. Chaimanee, P. Cloetens, M. Feist, J. Hoszowska, J.-J. Jaeger, “Applications of x-ray synchrotron microtomography for non-destructive 3D studies of paleontological specimens,” Appl. Phys. A Mater. 83, 195–202 (2006).
[CrossRef]

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Clark, J. N.

Cloetens, P.

M. Stockmar, P. Cloetens, I. Zanette, B. Enders, M. Dierolf, F. Pfeiffer, P. Thibault, “Near-field ptychography: Phase retrieval for inline holography using a structured illumination,” Sci. Rep. 3, 1–6 (2013).
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[CrossRef]

P. Tafforeau, R. Boistel, E. Boller, A. Bravin, M. Brunet, Y. Chaimanee, P. Cloetens, M. Feist, J. Hoszowska, J.-J. Jaeger, “Applications of x-ray synchrotron microtomography for non-destructive 3D studies of paleontological specimens,” Appl. Phys. A Mater. 83, 195–202 (2006).
[CrossRef]

P. Cloetens, R. Mache, M. Schlenker, S. Lerbs-Mache, “Quantitative phase tomography of Arabidopsis seeds reveals intercellular void network,” Proc. Natl. Acad. Sci. USA 103, 14626–14630 (2006).
[CrossRef] [PubMed]

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

R. Barrett, R. Baker, P. Cloetens, Y. Dabin, C. Morawe, H. Suhonen, R. Tucoulou, A. Vivo, L. Zhang, “Dynamically-figured mirror system for high-energy nanofocusing at the ESRF,” Proc. SPIE pp. 813904–813912 (2011).
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R. Barrett, R. Baker, P. Cloetens, Y. Dabin, C. Morawe, H. Suhonen, R. Tucoulou, A. Vivo, L. Zhang, “Dynamically-figured mirror system for high-energy nanofocusing at the ESRF,” Proc. SPIE pp. 813904–813912 (2011).
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M. Stockmar, P. Cloetens, I. Zanette, B. Enders, M. Dierolf, F. Pfeiffer, P. Thibault, “Near-field ptychography: Phase retrieval for inline holography using a structured illumination,” Sci. Rep. 3, 1–6 (2013).
[CrossRef]

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

Döring, F.

Eberl, C.

Edo, T. B.

T. B. Edo, D. J. Batey, A. M. Maiden, C. Rau, U. Wagner, Z. D. Pešić, T. A. Waigh, J. M. Rodenburg, “Sampling in x-ray ptychography,” Phys. Rev. A 87, 053850 (2013).
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M. Stockmar, P. Cloetens, I. Zanette, B. Enders, M. Dierolf, F. Pfeiffer, P. Thibault, “Near-field ptychography: Phase retrieval for inline holography using a structured illumination,” Sci. Rep. 3, 1–6 (2013).
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P. Tafforeau, R. Boistel, E. Boller, A. Bravin, M. Brunet, Y. Chaimanee, P. Cloetens, M. Feist, J. Hoszowska, J.-J. Jaeger, “Applications of x-ray synchrotron microtomography for non-destructive 3D studies of paleontological specimens,” Appl. Phys. A Mater. 83, 195–202 (2006).
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K. Giewekemeyer, S. Krüger, S. Kalbfleisch, M. Bartels, C. Beta, T. Salditt, “X-ray propagation microscopy of biological cells using waveguides as a quasipoint source,” Phys. Rev. A 83, 023804 (2011).
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T. E. Gureyev, K. A. Nugent, “Rapid quantitative phase imaging using the transport of intensity equation,” Opt. Commun. 133, 339–346 (1997).
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B. M. Hanser, M. G. L. Gustafsson, D. A. Agard, J. W. Sedat, “Phase-retrieved pupil functions in wide-field fluorescence microscopy,” J. Microsc. 216, 32–48 (2004).
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S. Kalbfleisch, H. Neubauer, S. P. Krüger, M. Bartels, M. Osterhoff, D. D. Mai, K. Giewekemeyer, B. Hartmann, M. Sprung, T. Salditt, “The Göttingen holography endstation of beamline P10 at PETRA III/DESY,” AIP Conf. Proc. 1365, 96–99 (2011).
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M. Bartels, V. H. Hernandez, M. Krenkel, T. Moser, T. Salditt, “Phase contrast tomography of the mouse cochlea at microfocus x-ray sources,” Appl. Phys. Lett. 103, 083703 (2013).
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Hofmann, R.

J. Moosmann, A. Ershov, V. Altapova, T. Baumbach, M. S. Prasad, C. LaBonne, X. Xiao, J. Kashef, R. Hofmann, “X-ray phase-contrast in vivo microtomography probes new aspects of xenopus gastrulation,” Nature (London) 497, 374–377 (2013).
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P. Tafforeau, R. Boistel, E. Boller, A. Bravin, M. Brunet, Y. Chaimanee, P. Cloetens, M. Feist, J. Hoszowska, J.-J. Jaeger, “Applications of x-ray synchrotron microtomography for non-destructive 3D studies of paleontological specimens,” Appl. Phys. A Mater. 83, 195–202 (2006).
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P. Tafforeau, R. Boistel, E. Boller, A. Bravin, M. Brunet, Y. Chaimanee, P. Cloetens, M. Feist, J. Hoszowska, J.-J. Jaeger, “Applications of x-ray synchrotron microtomography for non-destructive 3D studies of paleontological specimens,” Appl. Phys. A Mater. 83, 195–202 (2006).
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M. Bartels, M. Priebe, R. Wilke, S. Kruger, K. Giewekemeyer, S. Kalbfleisch, C. Olendrowitz, M. Sprung, T. Salditt, “Low-dose three-dimensional hard x-ray imaging of bacterial cells,” Opt. Nanoscopy, 1,1–10 (2012).

K. Giewekemeyer, S. Krüger, S. Kalbfleisch, M. Bartels, C. Beta, T. Salditt, “X-ray propagation microscopy of biological cells using waveguides as a quasipoint source,” Phys. Rev. A 83, 023804 (2011).
[CrossRef]

S. Kalbfleisch, H. Neubauer, S. P. Krüger, M. Bartels, M. Osterhoff, D. D. Mai, K. Giewekemeyer, B. Hartmann, M. Sprung, T. Salditt, “The Göttingen holography endstation of beamline P10 at PETRA III/DESY,” AIP Conf. Proc. 1365, 96–99 (2011).
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J. Moosmann, A. Ershov, V. Altapova, T. Baumbach, M. S. Prasad, C. LaBonne, X. Xiao, J. Kashef, R. Hofmann, “X-ray phase-contrast in vivo microtomography probes new aspects of xenopus gastrulation,” Nature (London) 497, 374–377 (2013).
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G. Martinez-Criado, R. Tucoulou, P. Cloetens, P. Bleuet, S. Bohic, J. Cauzid, I. Kieffer, E. Kosior, S. Laboure, S. Petitgirard, A. Rack, J. A. Sans, J. Segura-Ruiz, H. Suhonen, J. Susini, J. Villanova, “Status of the hard x-ray microprobe beamline ID22 of the European Synchrotron Radiation Facility,” J. Synchrotron Radiat. 19, 10–18 (2012).
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Kosior, E.

G. Martinez-Criado, R. Tucoulou, P. Cloetens, P. Bleuet, S. Bohic, J. Cauzid, I. Kieffer, E. Kosior, S. Laboure, S. Petitgirard, A. Rack, J. A. Sans, J. Segura-Ruiz, H. Suhonen, J. Susini, J. Villanova, “Status of the hard x-ray microprobe beamline ID22 of the European Synchrotron Radiation Facility,” J. Synchrotron Radiat. 19, 10–18 (2012).
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S. C. Kramer, J. Hagemann, “SciPAL: Expression templates and composition closure objects for high performance computational physics with CUDA and OpenMP,” ACM Trans. on Parallel Comput. (submitted).

Krebs, H. U.

Krenkel, M.

M. Bartels, V. H. Hernandez, M. Krenkel, T. Moser, T. Salditt, “Phase contrast tomography of the mouse cochlea at microfocus x-ray sources,” Appl. Phys. Lett. 103, 083703 (2013).
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M. Krenkel, M. Bartels, T. Salditt, “Transport of intensity phase reconstruction to solve the twin image problem in holographic x-ray imaging,” Opt. Express 21, 2220–2235 (2013).
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C. Olendrowitz, M. Bartels, M. Krenkel, A. Beerlink, R. Mokso, M. Sprung, T. Salditt, “Phase-contrast x-ray imaging and tomography of the nematode Caenorhabditis elegans,” Phys. Med. Biol. 57, 5309–5323 (2012).
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Kruger, S.

M. Bartels, M. Priebe, R. Wilke, S. Kruger, K. Giewekemeyer, S. Kalbfleisch, C. Olendrowitz, M. Sprung, T. Salditt, “Low-dose three-dimensional hard x-ray imaging of bacterial cells,” Opt. Nanoscopy, 1,1–10 (2012).

Krüger, S.

K. Giewekemeyer, S. Krüger, S. Kalbfleisch, M. Bartels, C. Beta, T. Salditt, “X-ray propagation microscopy of biological cells using waveguides as a quasipoint source,” Phys. Rev. A 83, 023804 (2011).
[CrossRef]

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S. Kalbfleisch, H. Neubauer, S. P. Krüger, M. Bartels, M. Osterhoff, D. D. Mai, K. Giewekemeyer, B. Hartmann, M. Sprung, T. Salditt, “The Göttingen holography endstation of beamline P10 at PETRA III/DESY,” AIP Conf. Proc. 1365, 96–99 (2011).
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J. Moosmann, A. Ershov, V. Altapova, T. Baumbach, M. S. Prasad, C. LaBonne, X. Xiao, J. Kashef, R. Hofmann, “X-ray phase-contrast in vivo microtomography probes new aspects of xenopus gastrulation,” Nature (London) 497, 374–377 (2013).
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G. Martinez-Criado, R. Tucoulou, P. Cloetens, P. Bleuet, S. Bohic, J. Cauzid, I. Kieffer, E. Kosior, S. Laboure, S. Petitgirard, A. Rack, J. A. Sans, J. Segura-Ruiz, H. Suhonen, J. Susini, J. Villanova, “Status of the hard x-ray microprobe beamline ID22 of the European Synchrotron Radiation Facility,” J. Synchrotron Radiat. 19, 10–18 (2012).
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P. Cloetens, R. Mache, M. Schlenker, S. Lerbs-Mache, “Quantitative phase tomography of Arabidopsis seeds reveals intercellular void network,” Proc. Natl. Acad. Sci. USA 103, 14626–14630 (2006).
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Ludwig, W.

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P. Cloetens, W. Ludwig, J. Baruchel, D. van Dyck, J. van Landuyt, J. Guigay, 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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D. R. Luke, J. V. Burke, R. G. Lyon, “Optical wavefront reconstruction: Theory and numerical methods,” SIAM Rev. 44, 169–224 (2002).
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P. Cloetens, R. Mache, M. Schlenker, S. Lerbs-Mache, “Quantitative phase tomography of Arabidopsis seeds reveals intercellular void network,” Proc. Natl. Acad. Sci. USA 103, 14626–14630 (2006).
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S. Kalbfleisch, H. Neubauer, S. P. Krüger, M. Bartels, M. Osterhoff, D. D. Mai, K. Giewekemeyer, B. Hartmann, M. Sprung, T. Salditt, “The Göttingen holography endstation of beamline P10 at PETRA III/DESY,” AIP Conf. Proc. 1365, 96–99 (2011).
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T. B. Edo, D. J. Batey, A. M. Maiden, C. Rau, U. Wagner, Z. D. Pešić, T. A. Waigh, J. M. Rodenburg, “Sampling in x-ray ptychography,” Phys. Rev. A 87, 053850 (2013).
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E. Maire, J.-Y. Buffière, L. Salvo, J. J. Blandin, W. Ludwig, J. M. Letang, “On the application of x-ray microtomography in the field of materials science,” Adv. Eng. Mater. 3, 539–546 (2001).
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G. Martinez-Criado, R. Tucoulou, P. Cloetens, P. Bleuet, S. Bohic, J. Cauzid, I. Kieffer, E. Kosior, S. Laboure, S. Petitgirard, A. Rack, J. A. Sans, J. Segura-Ruiz, H. Suhonen, J. Susini, J. Villanova, “Status of the hard x-ray microprobe beamline ID22 of the European Synchrotron Radiation Facility,” J. Synchrotron Radiat. 19, 10–18 (2012).
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L. J. Allen, W. McBride, N. O’Leary, M. P. Oxley, “Exit wave reconstruction at atomic resolution,” Ultramicroscopy 100, 91–104 (2004).
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L. J. Allen, W. McBride, M. P. Oxley, “Exit wave reconstruction using soft x-rays,” Opt. Commun. 233, 77–82 (2004).
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P. Thibault, M. Dierolf, O. Bunk, A. Menzel, F. Pfeiffer, “Probe retrieval in ptychographic coherent diffractive imaging,” Ultramicroscopy 109, 338–343 (2009).
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Moosmann, J.

J. Moosmann, A. Ershov, V. Altapova, T. Baumbach, M. S. Prasad, C. LaBonne, X. Xiao, J. Kashef, R. Hofmann, “X-ray phase-contrast in vivo microtomography probes new aspects of xenopus gastrulation,” Nature (London) 497, 374–377 (2013).
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R. Barrett, R. Baker, P. Cloetens, Y. Dabin, C. Morawe, H. Suhonen, R. Tucoulou, A. Vivo, L. Zhang, “Dynamically-figured mirror system for high-energy nanofocusing at the ESRF,” Proc. SPIE pp. 813904–813912 (2011).
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M. Bartels, V. H. Hernandez, M. Krenkel, T. Moser, T. Salditt, “Phase contrast tomography of the mouse cochlea at microfocus x-ray sources,” Appl. Phys. Lett. 103, 083703 (2013).
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S. Kalbfleisch, H. Neubauer, S. P. Krüger, M. Bartels, M. Osterhoff, D. D. Mai, K. Giewekemeyer, B. Hartmann, M. Sprung, T. Salditt, “The Göttingen holography endstation of beamline P10 at PETRA III/DESY,” AIP Conf. Proc. 1365, 96–99 (2011).
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T. E. Gureyev, K. A. Nugent, “Rapid quantitative phase imaging using the transport of intensity equation,” Opt. Commun. 133, 339–346 (1997).
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L. J. Allen, W. McBride, N. O’Leary, M. P. Oxley, “Exit wave reconstruction at atomic resolution,” Ultramicroscopy 100, 91–104 (2004).
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M. Bartels, M. Priebe, R. Wilke, S. Kruger, K. Giewekemeyer, S. Kalbfleisch, C. Olendrowitz, M. Sprung, T. Salditt, “Low-dose three-dimensional hard x-ray imaging of bacterial cells,” Opt. Nanoscopy, 1,1–10 (2012).

C. Olendrowitz, M. Bartels, M. Krenkel, A. Beerlink, R. Mokso, M. Sprung, T. Salditt, “Phase-contrast x-ray imaging and tomography of the nematode Caenorhabditis elegans,” Phys. Med. Biol. 57, 5309–5323 (2012).
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L. J. Allen, W. McBride, N. O’Leary, M. P. Oxley, “Exit wave reconstruction at atomic resolution,” Ultramicroscopy 100, 91–104 (2004).
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L. J. Allen, W. McBride, M. P. Oxley, “Exit wave reconstruction using soft x-rays,” Opt. Commun. 233, 77–82 (2004).
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L. J. Allen, H. M. L. Faulkner, K. A. Nugent, M. P. Oxley, D. Paganin, “Phase retrieval from images in the presence of first-order vortices,” Phys. Rev. E 63, 037602 (2001).
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L. J. Allen, H. M. L. Faulkner, K. A. Nugent, M. P. Oxley, D. Paganin, “Phase retrieval from images in the presence of first-order vortices,” Phys. Rev. E 63, 037602 (2001).
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H. M. Quiney, A. G. Peele, Z. Cai, D. Paterson, K. A. Nugent, “Diffractive imaging of highly focused x-ray fields,” Nature (London) 2, 101–104 (2006).

Peele, A. G.

Pešic, Z. D.

T. B. Edo, D. J. Batey, A. M. Maiden, C. Rau, U. Wagner, Z. D. Pešić, T. A. Waigh, J. M. Rodenburg, “Sampling in x-ray ptychography,” Phys. Rev. A 87, 053850 (2013).
[CrossRef]

Petitgirard, S.

G. Martinez-Criado, R. Tucoulou, P. Cloetens, P. Bleuet, S. Bohic, J. Cauzid, I. Kieffer, E. Kosior, S. Laboure, S. Petitgirard, A. Rack, J. A. Sans, J. Segura-Ruiz, H. Suhonen, J. Susini, J. Villanova, “Status of the hard x-ray microprobe beamline ID22 of the European Synchrotron Radiation Facility,” J. Synchrotron Radiat. 19, 10–18 (2012).
[CrossRef]

Pfeifer, M. A.

Pfeiffer, F.

M. Stockmar, P. Cloetens, I. Zanette, B. Enders, M. Dierolf, F. Pfeiffer, P. Thibault, “Near-field ptychography: Phase retrieval for inline holography using a structured illumination,” Sci. Rep. 3, 1–6 (2013).
[CrossRef]

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

Pogany, A.

A. Pogany, D. Gao, S. W. Wilkins, “Contrast and resolution in imaging with a microfocus x-ray source,” Rev. Sci. Instrum. 68, 2774–2782 (1997).
[CrossRef]

Prasad, M. S.

J. Moosmann, A. Ershov, V. Altapova, T. Baumbach, M. S. Prasad, C. LaBonne, X. Xiao, J. Kashef, R. Hofmann, “X-ray phase-contrast in vivo microtomography probes new aspects of xenopus gastrulation,” Nature (London) 497, 374–377 (2013).
[CrossRef]

Priebe, M.

M. Bartels, M. Priebe, R. Wilke, S. Kruger, K. Giewekemeyer, S. Kalbfleisch, C. Olendrowitz, M. Sprung, T. Salditt, “Low-dose three-dimensional hard x-ray imaging of bacterial cells,” Opt. Nanoscopy, 1,1–10 (2012).

Putkunz, C. T.

Quiney, H. M.

H. M. Quiney, “Coherent diffractive imaging using short wavelength light sources,” J. Mod. Opt. 57, 1109–1149 (2010).
[CrossRef]

H. M. Quiney, A. G. Peele, Z. Cai, D. Paterson, K. A. Nugent, “Diffractive imaging of highly focused x-ray fields,” Nature (London) 2, 101–104 (2006).

Rack, A.

G. Martinez-Criado, R. Tucoulou, P. Cloetens, P. Bleuet, S. Bohic, J. Cauzid, I. Kieffer, E. Kosior, S. Laboure, S. Petitgirard, A. Rack, J. A. Sans, J. Segura-Ruiz, H. Suhonen, J. Susini, J. Villanova, “Status of the hard x-ray microprobe beamline ID22 of the European Synchrotron Radiation Facility,” J. Synchrotron Radiat. 19, 10–18 (2012).
[CrossRef]

Rau, C.

T. B. Edo, D. J. Batey, A. M. Maiden, C. Rau, U. Wagner, Z. D. Pešić, T. A. Waigh, J. M. Rodenburg, “Sampling in x-ray ptychography,” Phys. Rev. A 87, 053850 (2013).
[CrossRef]

Riedel, A.

T. van de Kamp, P. Vagovič, T. Baumbach, A. Riedel, “A biological screw in a beetle’s leg,” Science 333, 52 (2011).
[CrossRef]

Robisch, A.-L.

Rodenburg, J. M.

T. B. Edo, D. J. Batey, A. M. Maiden, C. Rau, U. Wagner, Z. D. Pešić, T. A. Waigh, J. M. Rodenburg, “Sampling in x-ray ptychography,” Phys. Rev. A 87, 053850 (2013).
[CrossRef]

A. M. Maiden, J. M. Rodenburg, “An improved ptychographical phase retrieval algorithm for diffractive imaging,” Ultramicroscopy 109, 1256–1262 (2009).
[CrossRef] [PubMed]

Ruhlandt, A.

Salditt, T.

F. Döring, A.-L. Robisch, C. Eberl, M. Osterhoff, A. Ruhlandt, T. Liese, F. Schlenkrich, S. Hoffmann, M. Bartels, T. Salditt, H. U. Krebs, “Sub-5 nm hard x-ray point focusing by a combined Kirkpatrick-Baez mirror and multilayer zone plate,” Opt. Express 21, 19311–19323 (2013).
[CrossRef] [PubMed]

M. Bartels, V. H. Hernandez, M. Krenkel, T. Moser, T. Salditt, “Phase contrast tomography of the mouse cochlea at microfocus x-ray sources,” Appl. Phys. Lett. 103, 083703 (2013).
[CrossRef]

M. Krenkel, M. Bartels, T. Salditt, “Transport of intensity phase reconstruction to solve the twin image problem in holographic x-ray imaging,” Opt. Express 21, 2220–2235 (2013).
[CrossRef] [PubMed]

A.-L. Robisch, T. Salditt, “Phase retrieval for object and probe using a series of defocus near-field images,” Opt. Express 21, 23345–23357 (2013).
[CrossRef] [PubMed]

M. Bartels, M. Priebe, R. Wilke, S. Kruger, K. Giewekemeyer, S. Kalbfleisch, C. Olendrowitz, M. Sprung, T. Salditt, “Low-dose three-dimensional hard x-ray imaging of bacterial cells,” Opt. Nanoscopy, 1,1–10 (2012).

C. Olendrowitz, M. Bartels, M. Krenkel, A. Beerlink, R. Mokso, M. Sprung, T. Salditt, “Phase-contrast x-ray imaging and tomography of the nematode Caenorhabditis elegans,” Phys. Med. Biol. 57, 5309–5323 (2012).
[CrossRef] [PubMed]

K. Giewekemeyer, S. Krüger, S. Kalbfleisch, M. Bartels, C. Beta, T. Salditt, “X-ray propagation microscopy of biological cells using waveguides as a quasipoint source,” Phys. Rev. A 83, 023804 (2011).
[CrossRef]

S. Kalbfleisch, H. Neubauer, S. P. Krüger, M. Bartels, M. Osterhoff, D. D. Mai, K. Giewekemeyer, B. Hartmann, M. Sprung, T. Salditt, “The Göttingen holography endstation of beamline P10 at PETRA III/DESY,” AIP Conf. Proc. 1365, 96–99 (2011).
[CrossRef]

C. Homann, J. Hagemann, T. Salditt, T. Hohage, “Remarks on the product approximation in the detector plane,” (in prepartion).

Salvo, L.

E. Maire, J.-Y. Buffière, L. Salvo, J. J. Blandin, W. Ludwig, J. M. Letang, “On the application of x-ray microtomography in the field of materials science,” Adv. Eng. Mater. 3, 539–546 (2001).
[CrossRef]

Sans, J. A.

G. Martinez-Criado, R. Tucoulou, P. Cloetens, P. Bleuet, S. Bohic, J. Cauzid, I. Kieffer, E. Kosior, S. Laboure, S. Petitgirard, A. Rack, J. A. Sans, J. Segura-Ruiz, H. Suhonen, J. Susini, J. Villanova, “Status of the hard x-ray microprobe beamline ID22 of the European Synchrotron Radiation Facility,” J. Synchrotron Radiat. 19, 10–18 (2012).
[CrossRef]

Schlenker, M.

P. Cloetens, R. Mache, M. Schlenker, S. Lerbs-Mache, “Quantitative phase tomography of Arabidopsis seeds reveals intercellular void network,” Proc. Natl. Acad. Sci. USA 103, 14626–14630 (2006).
[CrossRef] [PubMed]

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

Schlenkrich, F.

Sedat, J. W.

B. M. Hanser, M. G. L. Gustafsson, D. A. Agard, J. W. Sedat, “Phase-retrieved pupil functions in wide-field fluorescence microscopy,” J. Microsc. 216, 32–48 (2004).
[CrossRef] [PubMed]

Segura-Ruiz, J.

G. Martinez-Criado, R. Tucoulou, P. Cloetens, P. Bleuet, S. Bohic, J. Cauzid, I. Kieffer, E. Kosior, S. Laboure, S. Petitgirard, A. Rack, J. A. Sans, J. Segura-Ruiz, H. Suhonen, J. Susini, J. Villanova, “Status of the hard x-ray microprobe beamline ID22 of the European Synchrotron Radiation Facility,” J. Synchrotron Radiat. 19, 10–18 (2012).
[CrossRef]

Sprung, M.

C. Olendrowitz, M. Bartels, M. Krenkel, A. Beerlink, R. Mokso, M. Sprung, T. Salditt, “Phase-contrast x-ray imaging and tomography of the nematode Caenorhabditis elegans,” Phys. Med. Biol. 57, 5309–5323 (2012).
[CrossRef] [PubMed]

M. Bartels, M. Priebe, R. Wilke, S. Kruger, K. Giewekemeyer, S. Kalbfleisch, C. Olendrowitz, M. Sprung, T. Salditt, “Low-dose three-dimensional hard x-ray imaging of bacterial cells,” Opt. Nanoscopy, 1,1–10 (2012).

S. Kalbfleisch, H. Neubauer, S. P. Krüger, M. Bartels, M. Osterhoff, D. D. Mai, K. Giewekemeyer, B. Hartmann, M. Sprung, T. Salditt, “The Göttingen holography endstation of beamline P10 at PETRA III/DESY,” AIP Conf. Proc. 1365, 96–99 (2011).
[CrossRef]

Stockmar, M.

M. Stockmar, P. Cloetens, I. Zanette, B. Enders, M. Dierolf, F. Pfeiffer, P. Thibault, “Near-field ptychography: Phase retrieval for inline holography using a structured illumination,” Sci. Rep. 3, 1–6 (2013).
[CrossRef]

Suhonen, H.

G. Martinez-Criado, R. Tucoulou, P. Cloetens, P. Bleuet, S. Bohic, J. Cauzid, I. Kieffer, E. Kosior, S. Laboure, S. Petitgirard, A. Rack, J. A. Sans, J. Segura-Ruiz, H. Suhonen, J. Susini, J. Villanova, “Status of the hard x-ray microprobe beamline ID22 of the European Synchrotron Radiation Facility,” J. Synchrotron Radiat. 19, 10–18 (2012).
[CrossRef]

R. Barrett, R. Baker, P. Cloetens, Y. Dabin, C. Morawe, H. Suhonen, R. Tucoulou, A. Vivo, L. Zhang, “Dynamically-figured mirror system for high-energy nanofocusing at the ESRF,” Proc. SPIE pp. 813904–813912 (2011).
[CrossRef]

Susini, J.

G. Martinez-Criado, R. Tucoulou, P. Cloetens, P. Bleuet, S. Bohic, J. Cauzid, I. Kieffer, E. Kosior, S. Laboure, S. Petitgirard, A. Rack, J. A. Sans, J. Segura-Ruiz, H. Suhonen, J. Susini, J. Villanova, “Status of the hard x-ray microprobe beamline ID22 of the European Synchrotron Radiation Facility,” J. Synchrotron Radiat. 19, 10–18 (2012).
[CrossRef]

Tafforeau, P.

P. Tafforeau, R. Boistel, E. Boller, A. Bravin, M. Brunet, Y. Chaimanee, P. Cloetens, M. Feist, J. Hoszowska, J.-J. Jaeger, “Applications of x-ray synchrotron microtomography for non-destructive 3D studies of paleontological specimens,” Appl. Phys. A Mater. 83, 195–202 (2006).
[CrossRef]

Thibault, P.

M. Stockmar, P. Cloetens, I. Zanette, B. Enders, M. Dierolf, F. Pfeiffer, P. Thibault, “Near-field ptychography: Phase retrieval for inline holography using a structured illumination,” Sci. Rep. 3, 1–6 (2013).
[CrossRef]

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

Tucoulou, R.

G. Martinez-Criado, R. Tucoulou, P. Cloetens, P. Bleuet, S. Bohic, J. Cauzid, I. Kieffer, E. Kosior, S. Laboure, S. Petitgirard, A. Rack, J. A. Sans, J. Segura-Ruiz, H. Suhonen, J. Susini, J. Villanova, “Status of the hard x-ray microprobe beamline ID22 of the European Synchrotron Radiation Facility,” J. Synchrotron Radiat. 19, 10–18 (2012).
[CrossRef]

R. Barrett, R. Baker, P. Cloetens, Y. Dabin, C. Morawe, H. Suhonen, R. Tucoulou, A. Vivo, L. Zhang, “Dynamically-figured mirror system for high-energy nanofocusing at the ESRF,” Proc. SPIE pp. 813904–813912 (2011).
[CrossRef]

Vagovic, P.

T. van de Kamp, P. Vagovič, T. Baumbach, A. Riedel, “A biological screw in a beetle’s leg,” Science 333, 52 (2011).
[CrossRef]

van de Kamp, T.

T. van de Kamp, P. Vagovič, T. Baumbach, A. Riedel, “A biological screw in a beetle’s leg,” Science 333, 52 (2011).
[CrossRef]

van Dyck, D.

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

Villanova, J.

G. Martinez-Criado, R. Tucoulou, P. Cloetens, P. Bleuet, S. Bohic, J. Cauzid, I. Kieffer, E. Kosior, S. Laboure, S. Petitgirard, A. Rack, J. A. Sans, J. Segura-Ruiz, H. Suhonen, J. Susini, J. Villanova, “Status of the hard x-ray microprobe beamline ID22 of the European Synchrotron Radiation Facility,” J. Synchrotron Radiat. 19, 10–18 (2012).
[CrossRef]

Vivo, A.

R. Barrett, R. Baker, P. Cloetens, Y. Dabin, C. Morawe, H. Suhonen, R. Tucoulou, A. Vivo, L. Zhang, “Dynamically-figured mirror system for high-energy nanofocusing at the ESRF,” Proc. SPIE pp. 813904–813912 (2011).
[CrossRef]

Wagner, U.

T. B. Edo, D. J. Batey, A. M. Maiden, C. Rau, U. Wagner, Z. D. Pešić, T. A. Waigh, J. M. Rodenburg, “Sampling in x-ray ptychography,” Phys. Rev. A 87, 053850 (2013).
[CrossRef]

Waigh, T. A.

T. B. Edo, D. J. Batey, A. M. Maiden, C. Rau, U. Wagner, Z. D. Pešić, T. A. Waigh, J. M. Rodenburg, “Sampling in x-ray ptychography,” Phys. Rev. A 87, 053850 (2013).
[CrossRef]

Wilke, R.

M. Bartels, M. Priebe, R. Wilke, S. Kruger, K. Giewekemeyer, S. Kalbfleisch, C. Olendrowitz, M. Sprung, T. Salditt, “Low-dose three-dimensional hard x-ray imaging of bacterial cells,” Opt. Nanoscopy, 1,1–10 (2012).

Wilkins, S. W.

A. Pogany, D. Gao, S. W. Wilkins, “Contrast and resolution in imaging with a microfocus x-ray source,” Rev. Sci. Instrum. 68, 2774–2782 (1997).
[CrossRef]

Williams, G. J.

Xiao, X.

J. Moosmann, A. Ershov, V. Altapova, T. Baumbach, M. S. Prasad, C. LaBonne, X. Xiao, J. Kashef, R. Hofmann, “X-ray phase-contrast in vivo microtomography probes new aspects of xenopus gastrulation,” Nature (London) 497, 374–377 (2013).
[CrossRef]

Zanette, I.

M. Stockmar, P. Cloetens, I. Zanette, B. Enders, M. Dierolf, F. Pfeiffer, P. Thibault, “Near-field ptychography: Phase retrieval for inline holography using a structured illumination,” Sci. Rep. 3, 1–6 (2013).
[CrossRef]

Zhang, L.

R. Barrett, R. Baker, P. Cloetens, Y. Dabin, C. Morawe, H. Suhonen, R. Tucoulou, A. Vivo, L. Zhang, “Dynamically-figured mirror system for high-energy nanofocusing at the ESRF,” Proc. SPIE pp. 813904–813912 (2011).
[CrossRef]

Adv. Eng. Mater.

E. Maire, J.-Y. Buffière, L. Salvo, J. J. Blandin, W. Ludwig, J. M. Letang, “On the application of x-ray microtomography in the field of materials science,” Adv. Eng. Mater. 3, 539–546 (2001).
[CrossRef]

AIP Conf. Proc.

S. Kalbfleisch, H. Neubauer, S. P. Krüger, M. Bartels, M. Osterhoff, D. D. Mai, K. Giewekemeyer, B. Hartmann, M. Sprung, T. Salditt, “The Göttingen holography endstation of beamline P10 at PETRA III/DESY,” AIP Conf. Proc. 1365, 96–99 (2011).
[CrossRef]

Appl. Phys. A Mater.

P. Tafforeau, R. Boistel, E. Boller, A. Bravin, M. Brunet, Y. Chaimanee, P. Cloetens, M. Feist, J. Hoszowska, J.-J. Jaeger, “Applications of x-ray synchrotron microtomography for non-destructive 3D studies of paleontological specimens,” Appl. Phys. A Mater. 83, 195–202 (2006).
[CrossRef]

Appl. Phys. Lett.

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

M. Bartels, V. H. Hernandez, M. Krenkel, T. Moser, T. Salditt, “Phase contrast tomography of the mouse cochlea at microfocus x-ray sources,” Appl. Phys. Lett. 103, 083703 (2013).
[CrossRef]

Contemp. Phys.

K. A. Nugent, “The measurement of phase through the propagation of intensity: An introduction,” Contemp. Phys. 52, 55–69 (2011).
[CrossRef]

Inverse Probl.

D. R. Luke, “Relaxed averaged alternating reflections for diffraction imaging,” Inverse Probl. 21, 37–50 (2005).
[CrossRef]

J. Microsc.

B. M. Hanser, M. G. L. Gustafsson, D. A. Agard, J. W. Sedat, “Phase-retrieved pupil functions in wide-field fluorescence microscopy,” J. Microsc. 216, 32–48 (2004).
[CrossRef] [PubMed]

J. Mod. Opt.

H. M. Quiney, “Coherent diffractive imaging using short wavelength light sources,” J. Mod. Opt. 57, 1109–1149 (2010).
[CrossRef]

J. Phys. D: Appl. Phys.

D. L. Misell, “An examination of an iterative method for the solution of the phase problem in optics and electron optics: I. Test calculations,” J. Phys. D: Appl. Phys. 6, 2200–2216 (1973).
[CrossRef]

J. Synchrotron Radiat.

G. Martinez-Criado, R. Tucoulou, P. Cloetens, P. Bleuet, S. Bohic, J. Cauzid, I. Kieffer, E. Kosior, S. Laboure, S. Petitgirard, A. Rack, J. A. Sans, J. Segura-Ruiz, H. Suhonen, J. Susini, J. Villanova, “Status of the hard x-ray microprobe beamline ID22 of the European Synchrotron Radiation Facility,” J. Synchrotron Radiat. 19, 10–18 (2012).
[CrossRef]

Microsc. Anal. (Am Ed)

B. D. Metscher, “Biological applications of x-ray microtomography: Imaging microanatomy, molecular expression and organismal diversity,” Microsc. Anal. (Am Ed) 27, 13–16 (2013).

Nature (London)

J. Moosmann, A. Ershov, V. Altapova, T. Baumbach, M. S. Prasad, C. LaBonne, X. Xiao, J. Kashef, R. Hofmann, “X-ray phase-contrast in vivo microtomography probes new aspects of xenopus gastrulation,” Nature (London) 497, 374–377 (2013).
[CrossRef]

H. M. Quiney, A. G. Peele, Z. Cai, D. Paterson, K. A. Nugent, “Diffractive imaging of highly focused x-ray fields,” Nature (London) 2, 101–104 (2006).

Opt. Commun.

L. J. Allen, W. McBride, M. P. Oxley, “Exit wave reconstruction using soft x-rays,” Opt. Commun. 233, 77–82 (2004).
[CrossRef]

T. E. Gureyev, K. A. Nugent, “Rapid quantitative phase imaging using the transport of intensity equation,” Opt. Commun. 133, 339–346 (1997).
[CrossRef]

L. J. Allen, M. P. Oxley, “Phase retrieval from series of images obtained by defocus variation,” Opt. Commun. 199, 65–75 (2001).
[CrossRef]

Opt. Express

Opt. Nanoscopy

M. Bartels, M. Priebe, R. Wilke, S. Kruger, K. Giewekemeyer, S. Kalbfleisch, C. Olendrowitz, M. Sprung, T. Salditt, “Low-dose three-dimensional hard x-ray imaging of bacterial cells,” Opt. Nanoscopy, 1,1–10 (2012).

Phys. Med. Biol.

C. Olendrowitz, M. Bartels, M. Krenkel, A. Beerlink, R. Mokso, M. Sprung, T. Salditt, “Phase-contrast x-ray imaging and tomography of the nematode Caenorhabditis elegans,” Phys. Med. Biol. 57, 5309–5323 (2012).
[CrossRef] [PubMed]

Phys. Rev. A

T. B. Edo, D. J. Batey, A. M. Maiden, C. Rau, U. Wagner, Z. D. Pešić, T. A. Waigh, J. M. Rodenburg, “Sampling in x-ray ptychography,” Phys. Rev. A 87, 053850 (2013).
[CrossRef]

K. Giewekemeyer, S. Krüger, S. Kalbfleisch, M. Bartels, C. Beta, T. Salditt, “X-ray propagation microscopy of biological cells using waveguides as a quasipoint source,” Phys. Rev. A 83, 023804 (2011).
[CrossRef]

Phys. Rev. E

L. J. Allen, H. M. L. Faulkner, K. A. Nugent, M. P. Oxley, D. Paganin, “Phase retrieval from images in the presence of first-order vortices,” Phys. Rev. E 63, 037602 (2001).
[CrossRef]

Proc. Natl. Acad. Sci. USA

P. Cloetens, R. Mache, M. Schlenker, S. Lerbs-Mache, “Quantitative phase tomography of Arabidopsis seeds reveals intercellular void network,” Proc. Natl. Acad. Sci. USA 103, 14626–14630 (2006).
[CrossRef] [PubMed]

Rev. Sci. Instrum.

A. Pogany, D. Gao, S. W. Wilkins, “Contrast and resolution in imaging with a microfocus x-ray source,” Rev. Sci. Instrum. 68, 2774–2782 (1997).
[CrossRef]

Sci. Rep.

M. Stockmar, P. Cloetens, I. Zanette, B. Enders, M. Dierolf, F. Pfeiffer, P. Thibault, “Near-field ptychography: Phase retrieval for inline holography using a structured illumination,” Sci. Rep. 3, 1–6 (2013).
[CrossRef]

Science

T. van de Kamp, P. Vagovič, T. Baumbach, A. Riedel, “A biological screw in a beetle’s leg,” Science 333, 52 (2011).
[CrossRef]

SIAM Rev.

D. R. Luke, J. V. Burke, R. G. Lyon, “Optical wavefront reconstruction: Theory and numerical methods,” SIAM Rev. 44, 169–224 (2002).
[CrossRef]

Ultramicroscopy

L. J. Allen, W. McBride, N. O’Leary, M. P. Oxley, “Exit wave reconstruction at atomic resolution,” Ultramicroscopy 100, 91–104 (2004).
[CrossRef] [PubMed]

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

A. M. Maiden, J. M. Rodenburg, “An improved ptychographical phase retrieval algorithm for diffractive imaging,” Ultramicroscopy 109, 1256–1262 (2009).
[CrossRef] [PubMed]

Other

R. Barrett, R. Baker, P. Cloetens, Y. Dabin, C. Morawe, H. Suhonen, R. Tucoulou, A. Vivo, L. Zhang, “Dynamically-figured mirror system for high-energy nanofocusing at the ESRF,” Proc. SPIE pp. 813904–813912 (2011).
[CrossRef]

K. Giewekemeyer, “A study on new approaches in coherent x-ray microscopy of biological specimens,” Ph.D. thesis, Universität Göttingen (2011).

S. C. Kramer, J. Hagemann, “SciPAL: Expression templates and composition closure objects for high performance computational physics with CUDA and OpenMP,” ACM Trans. on Parallel Comput. (submitted).

D. M. Paganin, Coherent x-ray optics (Oxford University Press,2006).
[CrossRef]

C. Homann, J. Hagemann, T. Salditt, T. Hohage, “Remarks on the product approximation in the detector plane,” (in prepartion).

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Figures (11)

Fig. 1
Fig. 1

Illustration of the flat field problem, for a 10μm (diameter) polystyrene sphere positioned at a defocus distance of z1 = 10 mm behind the focal plane of the KB mirror system at beamline ID22NI (ESRF, Grenoble), recorded at a detector distance of z1 + z2 = 525 mm and photon energy of E = 17 keV. (a) Empty beam recording showing the artifacts stemming from mirror imperfections. (b) Raw hologram of the sphere. (c) Corrected hologram as obtained by division of (b) by (a), i.e. the standard flat field or empty beam correction scheme. All data sets are normalized to the respective mean intensity. Scale bar of 2μm indicates the corresponding distance in the object plane.

Fig. 2
Fig. 2

(a) Sketch of the imaging scheme with the mirrors, the focus, the object and the detector. The probing beam P is generated by a pair of KB-mirrors at the distance f in front of the focal plane. The sample is placed at a distance z1 behind the focal plane, and the sample to detector distance is z2. (b) The equivalent parallel beam geometry corresponding to (a) with zeff = z1z2/(z1 + z2) and a geometrically reduced detection plane with M = (z1 + z2)/z1. (c) The phantom of P used in the numerical simulations. A mandrill monkey image serves as the input for the phases, Dürer’s Melencolia I as input for the amplitudes, both images are taken from MATLAB. The phases are in the range of [−0.4 0.4] rad and the amplitudes in [0.8 1.2]. The image size is 512 × 512 pixel. (d) The object O used for the numerical simulations consisting of a phase shifting 2D sine grating, here shown for a periodicity of 32 px and a phase shift of −0.3 rad. (e) The test object used for multiple plane reconstructions. The image of a stone sculpture serves as phases. The phases are in the range [−0.4 0] rad, the amplitudes are uniformly 1, thus we have a purely phase shifting object.

Fig. 3
Fig. 3

The error δ̄ as function of the normalized cut-off frequency ξ/ξmax in the filter applied to the probe P, for different periodicities ΔxO (grid spacings). The alternative x-label on the top indicates the effective source size a in units of pixels, corresponding to ξ. The simulation can be carried out in unit-less pixel variables and Fresnel numbers.

Fig. 4
Fig. 4

Examples of images recorded for a gold grid test pattern with 1.5 μm spacing at a defocus distance z1 = 26 mm. (a) Empty beam measurement P. (b) Recorded (raw) hologram of P · O. (c) Standard empty beam correction obtained by division of (b) by (a). (d), (e), (f) same as (a), (b), (c), but with a pinhole of 1.4 μm (diameter) inserted in the focal plane, which cuts the tails of the KB-beam, leading to a filtering of P. All data sets are shown after normalization to the average intensity. Scale bar indicates 2 μm in the sample plane.

Fig. 5
Fig. 5

Examples from a data set using the same object as in Fig. 4, but recorded in the detector scan (z2 variation at fixed z1). (a) Raw hologram of P · O for z2 = 510 mm. (b) Corresponding empty beam P. (c) Empty beam correction: (a) divided by (b). (d) Phases of a reconstruction based on GS using (c) as measurement constraint and the projection on amplitude 1 and negative phases as sample constraint (cf. Section 4). (a), (b), (c) share the same colorbar shown below (c). (a) and (b) show the average of 100 measurements of the object and the empty beam, respectively, normalized by the mean intensity. Scale bar indicates 4 μm in the sample plane.

Fig. 6
Fig. 6

Algorithm sketch for divide&update (d&u) and simulated reconstructions. (a) Principal sketch of d&u. Further explanations in Section 4. (b) Phases of the reconstructed mandrill-Dürer-probe and (c) corresponding amplitudes. (d) Phases of the first guess of O as obtained by the GS algorithm applied to the empty beam corrected hologram. (e) Phases of O after 200 iterations of d&u, showing the improvement by the nested update. (f) and (g) show reconstructions from a pure phase probe, i.e. a beam with phases represented by the mandrill image, and amplitudes equal to one (denoted as mandrill-1-probe). Besides the negative phase constraint also the projection to uniform amplitude is used. (f) Phases of P, amplitudes of P not shown. (g) Phases of O after 200 iterations of d&u. Scale bar indicates 4 μm.

Fig. 7
Fig. 7

Reconstructions of experimental data with the divide&update algorithm. (a) Phases and (b) amplitudes of the reconstructed P. (c) Phases of the first guess of O (gold grid) as obtained by the GS algorithm applied to the empty beam corrected hologram. (d) Phases of O after 200 iterations of d&u, clearly showing the improvement over (c). Scale bar indicates 4 μm in the sample plane.

Fig. 8
Fig. 8

Detector distance variations. In this approach z2 is varied and z1 is kept constant.

Fig. 9
Fig. 9

Reconstruction of the simulated detector scan experiment with 4 distances using the multiple magnitude projection algorithm (Eq. (9)). The left column shows phase and the right column shows amplitude. (a), (b) Reconstructed P. (c), (d) Reconstructed P · O. (e), (f) Reconstructed O, from division of P · O by P. The Δ in the title denotes the reconstruction error as defined in Eq. (10). Scale bar indicates 4 μm in the sample plane.

Fig. 10
Fig. 10

Reconstruction of the experimental detector scan with the multiple magnitude projection algorithm (Eq. (9)). The left column shows phase and the right column shows amplitude of the reconstruction. (a), (b) Reconstructed P. (c), (d) Reconstructed P · O. (e), (f) Reconstructed O, from division of P · O by P. The pixel size is 36 nm. The colorbar below the last image in a column applies to all images of the column. Scale bar indicates 4 μm in the sample plane.

Fig. 11
Fig. 11

The mean error of Eq. (11) as a function of the normalized cut-off frequency ξ/ξmax in the filter applied to the probe P, for different periodicities ΔxO (grid spacings). The alternative x-label on the top indicates the effective source size a in units of pixels, corresponding to ξ.

Tables (1)

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Table 1 Main parameters for the datasets shown in Figs. 1, 4, 5.

Equations (12)

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| 𝒟 z ( O ) | 2 | 𝒟 z ( P O ) | 2 | 𝒟 z ( P ) | 2 ,
𝒟 z ( Ψ ) = exp ( i 2 π / λ z ) 1 [ exp ( i π λ z ( k x 2 + k y 2 ) ) ( Ψ ) ] ,
δ = | 𝒟 z eff ( O ) | 2 | 𝒟 z eff ( P O ) | 2 | 𝒟 z eff ( P ) | 2 .
δ ¯ = 1 N 2 all pixel δ ,
a ( Δ x P ) = z 1 λ 2 Δ x P .
H ( k x , k y ) = { 1 if max ( k x , k y ) < ξ 0 if max ( k x , k y ) ξ ,
P M ( Ψ ) 𝒟 z eff ( 𝒟 z eff ( Ψ ) | 𝒟 z eff ( Ψ ) | I ) ,
P S ( Ψ ) { Ψ ( x ) ϕ ( x ) 0 A ( x ) exp ( i 0 ) ϕ ( x ) > 0 ,
Ψ n + 1 = P M 4 ( P M 3 ( P M 2 ( P M 1 ( Ψ n ) ) ) ) ,
Δ = 1 N 2 pixel | original reconstruction | ,
δ = 𝒟 z eff ( P O ) exp ( i 2 π / λ z eff ) 𝒟 z eff ( O ) 𝒟 z eff ( P ) 𝒟 z eff ( P ) 𝒟 z eff ( O ) ,
2 z eff λ 8 π Δ x P Δ x O 2 | ( P ) | d k 2 | ( O ) | d k 𝒟 z eff ( P ) | 𝒟 z eff ( O ) .

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