Abstract

In propagation based phase contrast imaging, intensity patterns are recorded on a x-ray detector at one or multiple propagation distances, called in-line holograms. They form the input of an inversion algorithm that aims at retrieving the phase shift induced by the object. The problem of phase retrieval in in-line holography is an ill-posed inverse problem. Consequently an adequate solution requires some form of regularization with the most commonly applied being the classical Tikhonov regularization. While generally satisfying this method suffers from a few issues such as the choice of the regularization parameter. Here, we offer an alternative to the established method by applying the principles of Bayesian inference. We construct an iterative optimization algorithm capable of both retrieving the unknown phase and determining a multi-dimensional regularization parameter. In the end, we highlight the advantages of the introduced algorithm, chief among them being the unsupervised determination of the regularization parameter(s). The proposed approach is tested on both simulated and experimental data and is found to provide robust solutions, with improved response to typical issues like low frequency noise and the twin-image problem.

© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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References

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2017 (1)

2016 (3)

2015 (1)

C. Morawe, R. Barrett, P. Cloetens, B. Lantelme, J.-C. Peffen, and A. Vivo, “Graded multilayers for figured kirkpatrick-baez mirrors on the new esrf end station id16a,” Proc. SPIE 9588, 958803 (2015).
[Crossref]

2014 (1)

A. Mirone, E. Brun, E. Gouillart, P. Tafforeau, and J. Kieffer, “The pyhst2 hybrid distributed code for high speed tomographic reconstruction with iterative reconstruction and a priori knowledge capabilities,” Nucl. Instrum. Meth. Phys. Res. Sect. B 324, 41–48 (2014).
[Crossref]

2013 (2)

M. Stockmar, P. Cloetens, I. Zanette, B. Enders, M. Dierolf, F. Pfeiffer, and P. Thibault, “Near-field ptychography: phase retrieval for inline holography using a structured illumination,” Sci. Rep. 3, 1927 (2013).
[Crossref] [PubMed]

A. Kostenko, K. J. Batenburg, H. Suhonen, S. E. Offerman, and L. J. Van Vliet, “Phase retrieval in in-line x-ray phase contrast imaging based on total variation minimization,” Opt. Express 21, 710–723 (2013).
[Crossref] [PubMed]

2012 (2)

M. Guizar-Sicairos and J. R. Fienup, “Understanding the twin-image problem in phase retrieval,” J. Opt. Soc. Am. A 29, 2367–2375 (2012).
[Crossref]

A. Mohammad-Djafari, “Bayesian approach with prior models which enforce sparsity in signal and image processing,” EURASIP J. on Adv. Signal Process.  2012, 52 (2012).
[Crossref]

2011 (1)

2010 (2)

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

K. A. Nugent, “Coherent methods in the x-ray sciences,” Adv. Phys. 59, 1–99 (2010).
[Crossref]

2009 (1)

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]

2007 (2)

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]

J. P. Guigay, M. Langer, R. Boistel, and P. Cloetens, “Mixed transfer function and transport of intensity approach for phase retrieval in the fresnel region,” Opt. Lett. 32, 1617–1619 (2007).
[Crossref] [PubMed]

2005 (1)

S. Zabler, P. Cloetens, J.-P. Guigay, J. Baruchel, and M. Schlenker, “Optimization of phase contrast imaging using hard x rays,” Rev. Sci. Instrum. 76, 073705 (2005).
[Crossref]

2004 (2)

2002 (1)

D. Paganin, S. Mayo, T. E. Gureyev, P. R. Miller, and S. W. Wilkins, “Simultaneous phase and amplitude extraction from a single defocused image of a homogeneous object,” J. Microsc. 206, 33–40 (2002).
[Crossref] [PubMed]

1996 (2)

P. Cloetens, R. Barrett, J. Baruchel, J.-P. Guigay, and M. Schlenker, “Phase objects in synchrotron radiation hard x-ray imaging,” J. Phys. D: Appl. Phys. 29, 133 (1996).
[Crossref]

K. Nugent, T. Gureyev, D. Cookson, D. Paganin, and Z. Barnea, “Quantitative phase imaging using hard x rays,” Phys. Rev. Lett. 77, 2961 (1996).
[Crossref] [PubMed]

1995 (3)

T. Gureyev, A. Roberts, and K. Nugent, “Partially coherent fields, the transport-of-intensity equation, and phase uniqueness,” J. Opt. Soc. Am. A 12, 1942–1946 (1995).
[Crossref]

T. Davis, D. Gao, T. Gureyev, A. Stevenson, and S. Wilkins, “Phase-contrast imaging of weakly absorbing materials using hard x-rays,” Nature 373, 595 (1995).
[Crossref]

A. Snigirev, I. Snigireva, V. Kohn, S. Kuznetsov, and I. Schelokov, “On the possibilities of x-ray phase contrast microimaging by coherent high-energy synchrotron radiation,” Rev. Sci. Instrum. 66, 5486–5492 (1995).
[Crossref]

1992 (1)

P. C. Hansen, “Analysis of discrete ill-posed problems by means of the l-curve,” SIAM review 34, 561–580 (1992).
[Crossref]

1977 (1)

J.-P. Guigay, “Fourier-transform analysis of fresnel diffraction patterns and in-line holograms,” Optik 49, 121–125 (1977).

1948 (1)

Als-Nielsen, J.

J. Als-Nielsen and D. McMorrow, Elements of modern X-ray physics (John Wiley & Sons, 2011).
[Crossref]

Baez, A. V.

Barnea, Z.

K. Nugent, T. Gureyev, D. Cookson, D. Paganin, and Z. Barnea, “Quantitative phase imaging using hard x rays,” Phys. Rev. Lett. 77, 2961 (1996).
[Crossref] [PubMed]

Barrett, R.

J. C. da Silva, A. Pacureanu, Y. Yang, S. Bohic, C. Morawe, R. Barrett, and P. Cloetens, “Efficient concentration of high-energy x-rays for diffraction-limited imaging resolution,” Optica 4, 492–495 (2017).
[Crossref]

C. Morawe, R. Barrett, P. Cloetens, B. Lantelme, J.-C. Peffen, and A. Vivo, “Graded multilayers for figured kirkpatrick-baez mirrors on the new esrf end station id16a,” Proc. SPIE 9588, 958803 (2015).
[Crossref]

P. Cloetens, R. Barrett, J. Baruchel, J.-P. Guigay, and M. Schlenker, “Phase objects in synchrotron radiation hard x-ray imaging,” J. Phys. D: Appl. Phys. 29, 133 (1996).
[Crossref]

Bartels, M.

Baruchel, J.

S. Zabler, P. Cloetens, J.-P. Guigay, J. Baruchel, and M. Schlenker, “Optimization of phase contrast imaging using hard x rays,” Rev. Sci. Instrum. 76, 073705 (2005).
[Crossref]

P. Cloetens, R. Barrett, J. Baruchel, J.-P. Guigay, and M. Schlenker, “Phase objects in synchrotron radiation hard x-ray imaging,” J. Phys. D: Appl. Phys. 29, 133 (1996).
[Crossref]

Batenburg, K. J.

Baumbach, T.

Benesty, J.

J. Benesty, J. Chen, Y. A. Huang, and S. Doclo, “Study of the wiener filter for noise reduction,” in Speech Enhancement (Springer, 2005), pp. 9–41.
[Crossref]

Bohic, S.

Boistel, R.

Box, G. E.

G. E. Box and G. C. Tiao, Bayesian inference in statistical analysis, vol. 40 (John Wiley & Sons, 2011).

Brun, E.

A. Mirone, E. Brun, E. Gouillart, P. Tafforeau, and J. Kieffer, “The pyhst2 hybrid distributed code for high speed tomographic reconstruction with iterative reconstruction and a priori knowledge capabilities,” Nucl. Instrum. Meth. Phys. Res. Sect. B 324, 41–48 (2014).
[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, P. Thibault, A. Menzel, C. M. Kewish, K. Jefimovs, I. Schlichting, K. Von Koenig, 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]

Chambolle, A.

A. Chambolle, “An algorithm for total variation minimization and applications,” J. Math. Imag. Vis. 20, 89–97 (2004).
[Crossref]

Chen, J.

J. Benesty, J. Chen, Y. A. Huang, and S. Doclo, “Study of the wiener filter for noise reduction,” in Speech Enhancement (Springer, 2005), pp. 9–41.
[Crossref]

Cloetens, P.

J. C. da Silva, A. Pacureanu, Y. Yang, S. Bohic, C. Morawe, R. Barrett, and P. Cloetens, “Efficient concentration of high-energy x-rays for diffraction-limited imaging resolution,” Optica 4, 492–495 (2017).
[Crossref]

C. Morawe, R. Barrett, P. Cloetens, B. Lantelme, J.-C. Peffen, and A. Vivo, “Graded multilayers for figured kirkpatrick-baez mirrors on the new esrf end station id16a,” Proc. SPIE 9588, 958803 (2015).
[Crossref]

M. Stockmar, P. Cloetens, I. Zanette, B. Enders, M. Dierolf, F. Pfeiffer, and P. Thibault, “Near-field ptychography: phase retrieval for inline holography using a structured illumination,” Sci. Rep. 3, 1927 (2013).
[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]

J. P. Guigay, M. Langer, R. Boistel, and P. Cloetens, “Mixed transfer function and transport of intensity approach for phase retrieval in the fresnel region,” Opt. Lett. 32, 1617–1619 (2007).
[Crossref] [PubMed]

S. Zabler, P. Cloetens, J.-P. Guigay, J. Baruchel, and M. Schlenker, “Optimization of phase contrast imaging using hard x rays,” Rev. Sci. Instrum. 76, 073705 (2005).
[Crossref]

P. Cloetens, R. Barrett, J. Baruchel, J.-P. Guigay, and M. Schlenker, “Phase objects in synchrotron radiation hard x-ray imaging,” J. Phys. D: Appl. Phys. 29, 133 (1996).
[Crossref]

Cookson, D.

K. Nugent, T. Gureyev, D. Cookson, D. Paganin, and Z. Barnea, “Quantitative phase imaging using hard x rays,” Phys. Rev. Lett. 77, 2961 (1996).
[Crossref] [PubMed]

Courbin, F.

da Silva, J. C.

Davis, T.

T. Davis, D. Gao, T. Gureyev, A. Stevenson, and S. Wilkins, “Phase-contrast imaging of weakly absorbing materials using hard x-rays,” Nature 373, 595 (1995).
[Crossref]

Dhal, B.

Dierolf, M.

M. Stockmar, P. Cloetens, I. Zanette, B. Enders, M. Dierolf, F. Pfeiffer, and P. Thibault, “Near-field ptychography: phase retrieval for inline holography using a structured illumination,” Sci. Rep. 3, 1927 (2013).
[Crossref] [PubMed]

M. Dierolf, P. Thibault, A. Menzel, C. M. Kewish, K. Jefimovs, I. Schlichting, K. Von Koenig, 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]

Doclo, S.

J. Benesty, J. Chen, Y. A. Huang, and S. Doclo, “Study of the wiener filter for noise reduction,” in Speech Enhancement (Springer, 2005), pp. 9–41.
[Crossref]

Enders, B.

M. Stockmar, P. Cloetens, I. Zanette, B. Enders, M. Dierolf, F. Pfeiffer, and P. Thibault, “Near-field ptychography: phase retrieval for inline holography using a structured illumination,” Sci. Rep. 3, 1927 (2013).
[Crossref] [PubMed]

Ferrari, A.

Fienup, J. R.

Gao, D.

T. Davis, D. Gao, T. Gureyev, A. Stevenson, and S. Wilkins, “Phase-contrast imaging of weakly absorbing materials using hard x-rays,” Nature 373, 595 (1995).
[Crossref]

Goncharsky, A.

A. N. Tikhonov, A. Goncharsky, V. Stepanov, and A. G. Yagola, Numerical methods for the solution of ill-posed problems, vol. 328 (Springer Science & Business Media, 2013).

Goodman, J. W.

J. W. Goodman, Introduction to Fourier optics (Roberts and Company Publishers, 2005).

Gouillart, E.

A. Mirone, E. Brun, E. Gouillart, P. Tafforeau, and J. Kieffer, “The pyhst2 hybrid distributed code for high speed tomographic reconstruction with iterative reconstruction and a priori knowledge capabilities,” Nucl. Instrum. Meth. Phys. Res. Sect. B 324, 41–48 (2014).
[Crossref]

Guigay, J. P.

Guigay, J.-P.

S. Zabler, P. Cloetens, J.-P. Guigay, J. Baruchel, and M. Schlenker, “Optimization of phase contrast imaging using hard x rays,” Rev. Sci. Instrum. 76, 073705 (2005).
[Crossref]

P. Cloetens, R. Barrett, J. Baruchel, J.-P. Guigay, and M. Schlenker, “Phase objects in synchrotron radiation hard x-ray imaging,” J. Phys. D: Appl. Phys. 29, 133 (1996).
[Crossref]

J.-P. Guigay, “Fourier-transform analysis of fresnel diffraction patterns and in-line holograms,” Optik 49, 121–125 (1977).

Guizar-Sicairos, M.

Gureyev, T.

K. Nugent, T. Gureyev, D. Cookson, D. Paganin, and Z. Barnea, “Quantitative phase imaging using hard x rays,” Phys. Rev. Lett. 77, 2961 (1996).
[Crossref] [PubMed]

T. Davis, D. Gao, T. Gureyev, A. Stevenson, and S. Wilkins, “Phase-contrast imaging of weakly absorbing materials using hard x-rays,” Nature 373, 595 (1995).
[Crossref]

T. Gureyev, A. Roberts, and K. Nugent, “Partially coherent fields, the transport-of-intensity equation, and phase uniqueness,” J. Opt. Soc. Am. A 12, 1942–1946 (1995).
[Crossref]

Gureyev, T. E.

D. Paganin, S. Mayo, T. E. Gureyev, P. R. Miller, and S. W. Wilkins, “Simultaneous phase and amplitude extraction from a single defocused image of a homogeneous object,” J. Microsc. 206, 33–40 (2002).
[Crossref] [PubMed]

Hansen, P. C.

P. C. Hansen, “Analysis of discrete ill-posed problems by means of the l-curve,” SIAM review 34, 561–580 (1992).
[Crossref]

Hayes, J.

Hershey, J. R.

J. R. Hershey and P. A. Olsen, “Approximating the kullback leibler divergence between gaussian mixture models,” in IEEE International Conference on Acoustics, Speech and Signal Processing, ICASSP (IEEE, 2007), vol. 4, pp. IV–317.

Hofmann, R.

Hohage, T.

Huang, Y. A.

J. Benesty, J. Chen, Y. A. Huang, and S. Doclo, “Study of the wiener filter for noise reduction,” in Speech Enhancement (Springer, 2005), pp. 9–41.
[Crossref]

Jefimovs, K.

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

Kewish, C. M.

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

Kieffer, J.

A. Mirone, E. Brun, E. Gouillart, P. Tafforeau, and J. Kieffer, “The pyhst2 hybrid distributed code for high speed tomographic reconstruction with iterative reconstruction and a priori knowledge capabilities,” Nucl. Instrum. Meth. Phys. Res. Sect. B 324, 41–48 (2014).
[Crossref]

Kirkpatrick, P.

Kohn, V.

A. Snigirev, I. Snigireva, V. Kohn, S. Kuznetsov, and I. Schelokov, “On the possibilities of x-ray phase contrast microimaging by coherent high-energy synchrotron radiation,” Rev. Sci. Instrum. 66, 5486–5492 (1995).
[Crossref]

Kostenko, A.

Krenkel, M.

Kuznetsov, S.

A. Snigirev, I. Snigireva, V. Kohn, S. Kuznetsov, and I. Schelokov, “On the possibilities of x-ray phase contrast microimaging by coherent high-energy synchrotron radiation,” Rev. Sci. Instrum. 66, 5486–5492 (1995).
[Crossref]

Langer, M.

Lantelme, B.

C. Morawe, R. Barrett, P. Cloetens, B. Lantelme, J.-C. Peffen, and A. Vivo, “Graded multilayers for figured kirkpatrick-baez mirrors on the new esrf end station id16a,” Proc. SPIE 9588, 958803 (2015).
[Crossref]

Loock, S.

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]

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]

Mancuso, A.

Maretzke, S.

Mayo, S.

D. Paganin, S. Mayo, T. E. Gureyev, P. R. Miller, and S. W. Wilkins, “Simultaneous phase and amplitude extraction from a single defocused image of a homogeneous object,” J. Microsc. 206, 33–40 (2002).
[Crossref] [PubMed]

McMorrow, D.

J. Als-Nielsen and D. McMorrow, Elements of modern X-ray physics (John Wiley & Sons, 2011).
[Crossref]

Menzel, A.

M. Dierolf, P. Thibault, A. Menzel, C. M. Kewish, K. Jefimovs, I. Schlichting, K. Von Koenig, 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]

Miller, P. R.

D. Paganin, S. Mayo, T. E. Gureyev, P. R. Miller, and S. W. Wilkins, “Simultaneous phase and amplitude extraction from a single defocused image of a homogeneous object,” J. Microsc. 206, 33–40 (2002).
[Crossref] [PubMed]

Mirone, A.

A. Mirone, E. Brun, E. Gouillart, P. Tafforeau, and J. Kieffer, “The pyhst2 hybrid distributed code for high speed tomographic reconstruction with iterative reconstruction and a priori knowledge capabilities,” Nucl. Instrum. Meth. Phys. Res. Sect. B 324, 41–48 (2014).
[Crossref]

Mohammad-Djafari, A.

A. Mohammad-Djafari, “Bayesian approach with prior models which enforce sparsity in signal and image processing,” EURASIP J. on Adv. Signal Process.  2012, 52 (2012).
[Crossref]

A. Mohammad-Djafari, “A full bayesian approach for inverse problems,” in Maximum Entropy and Bayesian Methods (Springer, 1996), pp. 135–144.
[Crossref]

Mokso, R.

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]

Moosmann, J.

Morawe, C.

J. C. da Silva, A. Pacureanu, Y. Yang, S. Bohic, C. Morawe, R. Barrett, and P. Cloetens, “Efficient concentration of high-energy x-rays for diffraction-limited imaging resolution,” Optica 4, 492–495 (2017).
[Crossref]

C. Morawe, R. Barrett, P. Cloetens, B. Lantelme, J.-C. Peffen, and A. Vivo, “Graded multilayers for figured kirkpatrick-baez mirrors on the new esrf end station id16a,” Proc. SPIE 9588, 958803 (2015).
[Crossref]

Nugent, K.

Nugent, K. A.

K. A. Nugent, “Coherent methods in the x-ray sciences,” Adv. Phys. 59, 1–99 (2010).
[Crossref]

Offerman, S. E.

Olsen, P. A.

J. R. Hershey and P. A. Olsen, “Approximating the kullback leibler divergence between gaussian mixture models,” in IEEE International Conference on Acoustics, Speech and Signal Processing, ICASSP (IEEE, 2007), vol. 4, pp. IV–317.

Pacureanu, A.

Paganin, D.

D. Paganin, S. Mayo, T. E. Gureyev, P. R. Miller, and S. W. Wilkins, “Simultaneous phase and amplitude extraction from a single defocused image of a homogeneous object,” J. Microsc. 206, 33–40 (2002).
[Crossref] [PubMed]

K. Nugent, T. Gureyev, D. Cookson, D. Paganin, and Z. Barnea, “Quantitative phase imaging using hard x rays,” Phys. Rev. Lett. 77, 2961 (1996).
[Crossref] [PubMed]

Paterson, D.

Peele, A.

Peffen, J.-C.

C. Morawe, R. Barrett, P. Cloetens, B. Lantelme, J.-C. Peffen, and A. Vivo, “Graded multilayers for figured kirkpatrick-baez mirrors on the new esrf end station id16a,” Proc. SPIE 9588, 958803 (2015).
[Crossref]

Pein, A.

Pfeiffer, F.

M. Stockmar, P. Cloetens, I. Zanette, B. Enders, M. Dierolf, F. Pfeiffer, and P. Thibault, “Near-field ptychography: phase retrieval for inline holography using a structured illumination,” Sci. Rep. 3, 1927 (2013).
[Crossref] [PubMed]

M. Dierolf, P. Thibault, A. Menzel, C. M. Kewish, K. Jefimovs, I. Schlichting, K. Von Koenig, 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]

Plonka, G.

Roberts, A.

Salditt, T.

Schelokov, I.

A. Snigirev, I. Snigireva, V. Kohn, S. Kuznetsov, and I. Schelokov, “On the possibilities of x-ray phase contrast microimaging by coherent high-energy synchrotron radiation,” Rev. Sci. Instrum. 66, 5486–5492 (1995).
[Crossref]

Schlenker, M.

S. Zabler, P. Cloetens, J.-P. Guigay, J. Baruchel, and M. Schlenker, “Optimization of phase contrast imaging using hard x rays,” Rev. Sci. Instrum. 76, 073705 (2005).
[Crossref]

P. Cloetens, R. Barrett, J. Baruchel, J.-P. Guigay, and M. Schlenker, “Phase objects in synchrotron radiation hard x-ray imaging,” J. Phys. D: Appl. Phys. 29, 133 (1996).
[Crossref]

Schlichting, I.

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

Scholten, R.

Schutz, A.

Snigirev, A.

A. Snigirev, I. Snigireva, V. Kohn, S. Kuznetsov, and I. Schelokov, “On the possibilities of x-ray phase contrast microimaging by coherent high-energy synchrotron radiation,” Rev. Sci. Instrum. 66, 5486–5492 (1995).
[Crossref]

Snigireva, I.

A. Snigirev, I. Snigireva, V. Kohn, S. Kuznetsov, and I. Schelokov, “On the possibilities of x-ray phase contrast microimaging by coherent high-energy synchrotron radiation,” Rev. Sci. Instrum. 66, 5486–5492 (1995).
[Crossref]

Soulez, F.

Stepanov, V.

A. N. Tikhonov, A. Goncharsky, V. Stepanov, and A. G. Yagola, Numerical methods for the solution of ill-posed problems, vol. 328 (Springer Science & Business Media, 2013).

Stevenson, A.

T. Davis, D. Gao, T. Gureyev, A. Stevenson, and S. Wilkins, “Phase-contrast imaging of weakly absorbing materials using hard x-rays,” Nature 373, 595 (1995).
[Crossref]

Stockmar, M.

M. Stockmar, P. Cloetens, I. Zanette, B. Enders, M. Dierolf, F. Pfeiffer, and P. Thibault, “Near-field ptychography: phase retrieval for inline holography using a structured illumination,” Sci. Rep. 3, 1927 (2013).
[Crossref] [PubMed]

Suhonen, H.

Tafforeau, P.

A. Mirone, E. Brun, E. Gouillart, P. Tafforeau, and J. Kieffer, “The pyhst2 hybrid distributed code for high speed tomographic reconstruction with iterative reconstruction and a priori knowledge capabilities,” Nucl. Instrum. Meth. Phys. Res. Sect. B 324, 41–48 (2014).
[Crossref]

Thibault, P.

M. Stockmar, P. Cloetens, I. Zanette, B. Enders, M. Dierolf, F. Pfeiffer, and P. Thibault, “Near-field ptychography: phase retrieval for inline holography using a structured illumination,” Sci. Rep. 3, 1927 (2013).
[Crossref] [PubMed]

M. Dierolf, P. Thibault, A. Menzel, C. M. Kewish, K. Jefimovs, I. Schlichting, K. Von Koenig, 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]

Thiébaut, Éric

Tiao, G. C.

G. E. Box and G. C. Tiao, Bayesian inference in statistical analysis, vol. 40 (John Wiley & Sons, 2011).

Tikhonov, A. N.

A. N. Tikhonov, A. Goncharsky, V. Stepanov, and A. G. Yagola, Numerical methods for the solution of ill-posed problems, vol. 328 (Springer Science & Business Media, 2013).

Tran, C.

Turner, L.

Unser, M.

Van Vliet, L. J.

Vivo, A.

C. Morawe, R. Barrett, P. Cloetens, B. Lantelme, J.-C. Peffen, and A. Vivo, “Graded multilayers for figured kirkpatrick-baez mirrors on the new esrf end station id16a,” Proc. SPIE 9588, 958803 (2015).
[Crossref]

Von Koenig, K.

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

Wilkins, S.

T. Davis, D. Gao, T. Gureyev, A. Stevenson, and S. Wilkins, “Phase-contrast imaging of weakly absorbing materials using hard x-rays,” Nature 373, 595 (1995).
[Crossref]

Wilkins, S. W.

D. Paganin, S. Mayo, T. E. Gureyev, P. R. Miller, and S. W. Wilkins, “Simultaneous phase and amplitude extraction from a single defocused image of a homogeneous object,” J. Microsc. 206, 33–40 (2002).
[Crossref] [PubMed]

Yagola, A. G.

A. N. Tikhonov, A. Goncharsky, V. Stepanov, and A. G. Yagola, Numerical methods for the solution of ill-posed problems, vol. 328 (Springer Science & Business Media, 2013).

Yang, Y.

Zabler, S.

S. Zabler, P. Cloetens, J.-P. Guigay, J. Baruchel, and M. Schlenker, “Optimization of phase contrast imaging using hard x rays,” Rev. Sci. Instrum. 76, 073705 (2005).
[Crossref]

Zanette, I.

M. Stockmar, P. Cloetens, I. Zanette, B. Enders, M. Dierolf, F. Pfeiffer, and P. Thibault, “Near-field ptychography: phase retrieval for inline holography using a structured illumination,” Sci. Rep. 3, 1927 (2013).
[Crossref] [PubMed]

Adv. Phys. (1)

K. A. Nugent, “Coherent methods in the x-ray sciences,” Adv. Phys. 59, 1–99 (2010).
[Crossref]

Appl. Opt. (1)

Appl. Phys. Lett. (1)

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]

EURASIP J. on Adv. Signal Process (1)

A. Mohammad-Djafari, “Bayesian approach with prior models which enforce sparsity in signal and image processing,” EURASIP J. on Adv. Signal Process.  2012, 52 (2012).
[Crossref]

J. Math. Imag. Vis. (1)

A. Chambolle, “An algorithm for total variation minimization and applications,” J. Math. Imag. Vis. 20, 89–97 (2004).
[Crossref]

J. Microsc. (1)

D. Paganin, S. Mayo, T. E. Gureyev, P. R. Miller, and S. W. Wilkins, “Simultaneous phase and amplitude extraction from a single defocused image of a homogeneous object,” J. Microsc. 206, 33–40 (2002).
[Crossref] [PubMed]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (2)

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

P. Cloetens, R. Barrett, J. Baruchel, J.-P. Guigay, and M. Schlenker, “Phase objects in synchrotron radiation hard x-ray imaging,” J. Phys. D: Appl. Phys. 29, 133 (1996).
[Crossref]

Nature (1)

T. Davis, D. Gao, T. Gureyev, A. Stevenson, and S. Wilkins, “Phase-contrast imaging of weakly absorbing materials using hard x-rays,” Nature 373, 595 (1995).
[Crossref]

New J. Phys. (1)

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

Nucl. Instrum. Meth. Phys. Res. Sect. B (1)

A. Mirone, E. Brun, E. Gouillart, P. Tafforeau, and J. Kieffer, “The pyhst2 hybrid distributed code for high speed tomographic reconstruction with iterative reconstruction and a priori knowledge capabilities,” Nucl. Instrum. Meth. Phys. Res. Sect. B 324, 41–48 (2014).
[Crossref]

Opt. Express (5)

Opt. Lett. (1)

Optica (1)

Optik (1)

J.-P. Guigay, “Fourier-transform analysis of fresnel diffraction patterns and in-line holograms,” Optik 49, 121–125 (1977).

Phys. Rev. Lett. (1)

K. Nugent, T. Gureyev, D. Cookson, D. Paganin, and Z. Barnea, “Quantitative phase imaging using hard x rays,” Phys. Rev. Lett. 77, 2961 (1996).
[Crossref] [PubMed]

Proc. SPIE (1)

C. Morawe, R. Barrett, P. Cloetens, B. Lantelme, J.-C. Peffen, and A. Vivo, “Graded multilayers for figured kirkpatrick-baez mirrors on the new esrf end station id16a,” Proc. SPIE 9588, 958803 (2015).
[Crossref]

Rev. Sci. Instrum. (2)

S. Zabler, P. Cloetens, J.-P. Guigay, J. Baruchel, and M. Schlenker, “Optimization of phase contrast imaging using hard x rays,” Rev. Sci. Instrum. 76, 073705 (2005).
[Crossref]

A. Snigirev, I. Snigireva, V. Kohn, S. Kuznetsov, and I. Schelokov, “On the possibilities of x-ray phase contrast microimaging by coherent high-energy synchrotron radiation,” Rev. Sci. Instrum. 66, 5486–5492 (1995).
[Crossref]

Sci. Rep. (1)

M. Stockmar, P. Cloetens, I. Zanette, B. Enders, M. Dierolf, F. Pfeiffer, and P. Thibault, “Near-field ptychography: phase retrieval for inline holography using a structured illumination,” Sci. Rep. 3, 1927 (2013).
[Crossref] [PubMed]

SIAM review (1)

P. C. Hansen, “Analysis of discrete ill-posed problems by means of the l-curve,” SIAM review 34, 561–580 (1992).
[Crossref]

Ultramicroscopy (1)

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]

Other (7)

G. E. Box and G. C. Tiao, Bayesian inference in statistical analysis, vol. 40 (John Wiley & Sons, 2011).

A. Mohammad-Djafari, “A full bayesian approach for inverse problems,” in Maximum Entropy and Bayesian Methods (Springer, 1996), pp. 135–144.
[Crossref]

A. N. Tikhonov, A. Goncharsky, V. Stepanov, and A. G. Yagola, Numerical methods for the solution of ill-posed problems, vol. 328 (Springer Science & Business Media, 2013).

J. Benesty, J. Chen, Y. A. Huang, and S. Doclo, “Study of the wiener filter for noise reduction,” in Speech Enhancement (Springer, 2005), pp. 9–41.
[Crossref]

J. Als-Nielsen and D. McMorrow, Elements of modern X-ray physics (John Wiley & Sons, 2011).
[Crossref]

J. W. Goodman, Introduction to Fourier optics (Roberts and Company Publishers, 2005).

J. R. Hershey and P. A. Olsen, “Approximating the kullback leibler divergence between gaussian mixture models,” in IEEE International Conference on Acoustics, Speech and Signal Processing, ICASSP (IEEE, 2007), vol. 4, pp. IV–317.

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Figures (9)

Fig. 1
Fig. 1 The value of the Contrast Transfer Function for the amplitude, phase and phase of a homogeneous object with δ/β = 20. They are expressed as combinations of sine/cosine functions of the square modulus of the spatial frequency coordinate.
Fig. 2
Fig. 2 Phase component of the constructed object in radians.
Fig. 3
Fig. 3 Simulated holograms of the constructed object for different propagation distances: a) D1 = 16.8 mm, b) D2 = 17.8 mm, c) D3 = 21.4 mm, d) D4 = 30.4 mm.
Fig. 4
Fig. 4 Phase map obtained through a) standard regularization and b) Bayesian inference.
Fig. 5
Fig. 5 Pre-processed holograms of a red blood cell at four propagation distances a) D1 = 4 mm, b) D2 = 4.16 mm, c) D3 = 4.82 mm, d) D4 = 6.17 mm.
Fig. 6
Fig. 6 Phase map of a red blood cell obtained through a) standard regularization and b) Bayesian inference.
Fig. 7
Fig. 7 Line profiles (along the yellow lines of Fig. 6) through the retrieved phase maps obtained by classical regularization (blue) vs. Bayesian inference (green).
Fig. 8
Fig. 8 Solutions (retrieved phase maps) provided by the Bayesian algorithm at each iteration. The norm of these solutions calculated in Fourier space indicate quick convergence (for convenience, a factor of 10−2 is omitted from the table).
Fig. 9
Fig. 9 Plot of the radially averaged regularization parameter|/j| versus the amplitude of the spatial frequency, normalized by the sampling frequency fs.

Tables (1)

Tables Icon

Table 1 Numerical appreciation of the quality of phase retrieval through the standard and the Bayesian algorithms by means of standard deviation and signal-to-noise evaluation.

Equations (47)

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

n ( r ) = 1 δ ( r ) + i β ( r ) ,
u 0 ( r T ) = T ( r T ) u i n c ( r T ) = T ( r T ) , r T = ( x , y )
T ( r T ) = exp { B ( r T ) } exp { i ϕ ( r T ) } , B ( r T ) = 2 π λ β ( x , y , z ) d z ϕ ( r T ) = 2 π λ 1 δ ( x , y , z ) d z
I 0 ( r T ) = | u 0 ( r T ) | 2 = | T ( r T ) | 2 = exp { 2 B ( r T ) }
u D ( r T ) = exp { i 2 π D / λ } i λ D u 0 ( r T 0 ) exp { i π λ D | r T r T 0 | 2 } d r T 0 ,
real space : u D ( r T ) = P D ( r T ) * u 0 ( r T ) , P D ( r T ) = 1 i λ D exp { i π λ D r T 2 } reciprocal space : u ˜ D ( f ) = { u D ( r T ) } = P ˜ D ( f ) u ˜ 0 ( f ) , P ˜ D ( f ) = { P D ( r T ) } = exp { i π λ D | f | 2 } ,
u ˜ D ( f ) = { u D ( r T ) } = u D ( r T ) exp { 2 i π f r T } d r T u D ( r T ) = 1 { u ˜ D ( f ) } = u ˜ D ( f ) exp { 2 i π r T f } d f
I D ( r T ) = | u D ( r T ) | 2 I ˜ D ( f ) = u ˜ D ( η ) u ˜ D * ( η f ) d η
I ˜ D ( f ) = T ( r T λ D f 2 ) T * ( r T + λ D f 2 ) exp { i 2 π r T f } d r T
T ( r T ) 1 B ( r T ) + i ϕ ( r T )
I ˜ D ( f ) = δ ( f ) 2 cos ( π λ D | f | 2 ) B ˜ ( f ) + 2 sin ( π λ D | f | 2 ) ϕ ˜ ( f ) ,
I ˜ D ( f ) = δ ( f ) + 2 sin ( π λ D | f | 2 ) ϕ ˜ ( f ) ,
I ˜ D ( f ) = δ ( f ) + [ 2 sin ( π λ D | f | 2 ) + 2 β δ cos ( π λ D | f | 2 ) ] ϕ ˜ ( f )
I ˜ D ( f ) = I ˜ 0 ( f ) + 2 sin ( π λ D | f | 2 ) Ψ ˜ ( f ) + λ D 2 π cos ( π λ D | f | 2 ) { ( Ψ ln I 0 ) } ,
x ^ = arg min x y Hx 2 = arg min x J LS ( x ) ,
J LS ( x ) x = 0 2 H * ( y Hx ) = 0 x ^ = [ H * H ] 1 H * y
J ( x ) = y Hx 2 + λ R Δ ( x , x 0 ) ,
J ( x ) = y Hx 2 + λ R x x 0 2 ,
x ^ = [ H * H + λ R I ] 1 ( H * y + λ R x 0 ) .
x ^ = [ H * H + λ R I ] 1 H * y .
{ ( 11 ) : ϕ ^ = 1 2 Δ + λ R [ k I ˜ k sin ( π λ D k | f | 2 ) 𝔄 k I ˜ k cos ( π λ D k | f | 2 ) ] ( 12 ) : ϕ ^ = k I D k 2 sin ( π λ D k | f | 2 ) 4 sin 2 ( π λ D k | f | 2 + λ R ( 13 ) : ϕ ^ k I D k 2 ( sin ( π λ D k | f | 2 ) + β δ cos ( π λ D k | f | 2 ) ) 4 ( sin ( π λ D k | f | 2 + β δ ( π λ D k | f | 2 ) ) 2 + λ R
Δ ( y , Hx ) = y Hx 2 = i = 1 M | y i [ Hx ] i | 2 , Δ ( x , x 0 ) = x x 0 2 = j = 1 N | x j x 0 j | 2
Δ ( y , Hx ) = y Hx Q 1 2 = ( y Hx ) * Q 1 1 ( y Hx ) , Δ ( x , x 0 ) = x x 0 Q 2 2 = ( x x 0 ) * Q 2 1 ( x x 0 )
Δ ( y , Hx ) = y Hx p 1 = i = 1 M | y i [ Hx ] i | p 1 , Δ ( x , x 0 ) = x x 0 p 2 = j = 1 N | x j x 0 j | p 2
Δ ( y , Hx ) = i = 1 M y i ln y i [ Hx ] i , Δ ( x , x 0 ) = j = 1 N x j ln x j x 0 j
x ( k + 1 ) = x ( k ) + α ( k ) δ ( x ( k ) )
x ( k + 1 ) = x ( k ) α J ( x ( k ) )
x ( k + 1 ) = x ( k ) + α [ H * ( y H x ( k ) ) λ R x ( k ) ] ,
y = Hx + ,
p ( x | y ) = p ( y | x ) p ( x ) p ( y ) p ( y | x ) p ( x ) ,
p ( y | x ) = 𝒩 ( y | Hx , v I ) v M 2 exp { 1 2 v y Hx 2 } .
p ( v ) = 𝒢 ( v | α , β ) v ( α + 1 ) exp { β / v }
p ( x ) = 𝒩 ( x | x 0 , V x ) exp { 1 2 ( x x 0 ) * V x 1 ( x x 0 ) } p ( x ) = j 𝒩 ( x j | x 0 j , v j ) j v j 1 2 exp { j 1 2 v j | x x 0 j | 2 }
x ^ = arg max x p ( x , θ / y ) = arg min x J ( x ) , where J ( x ) = ln p ( x , θ | y ) , θ ^ = arg max θ p ( x , θ | y ) = arg min θ J ( x )
KL ( q 1 ( x ) q 2 ( θ ) : p ( x , θ | y ) ) = q 1 ( x ) q 2 ( θ ) ln q 1 ( x ) q 2 ( θ ) p ( x , θ | y ) d x d θ
{ q 1 ( x ) exp ln p ( x , θ | y ) q 2 = q 1 ( x | θ ˜ ) , with ln p ( x , θ | y ) q 2 = ln p ( x , θ | y ) q 2 ( θ ) d θ q 2 ( θ ) exp ln p ( x , θ | y ) q 1 = q 1 ( θ | x ˜ ) , with ln p ( x , θ | y ) q 1 = ln p ( x , θ | y ) q 1 ( x ) d x
p ( x , v ) = p ( x | v ) p ( v ) = j = 1 N p ( x j | v j ) j = 1 N p ( v j ) = j = 1 N 𝒩 ( x j | 0 , v j ) j = 1 N 𝒢 ( v j | α v , β v ) .
p ( x , v , v | y ) = p ( y | x , v ) p ( x | v ) p ( v ) p ( v )
p ( x , v , v | y ) = 𝒩 ( y | Hx , v I ) j = 1 N 𝒩 ( x j | 0 , v j ) j = 1 N 𝒢 ( v j | α v , β v ) 𝒢 ( v | α , β )
ln p ( x , v , v | y ) = M 2 ln v y Hx 2 2 v 1 2 j = 1 N ln v j 1 2 j = 1 N 1 v j x j 2 ( α v + 1 ) j = 1 N ln v j β v j = 1 N 1 v j ( α + 1 ) ln v β v
q ( x , v , v ) = q 1 ( x ) q 2 ( x ) q 3 ( v ) = j = 1 N q 1 , j ( x j ) j = 1 N q 2 , j ( v j ) q 3 ( v ) ,
q ( x , v , v ) = q 1 ( x ) q 2 ( v ) q 3 ( v ) = q 1 ( x ) j = 1 N q 2 , j ( v j ) q 3 ( v ) ,
{ q 1 ( x ) exp q 2 q 3 or q 1 , j ( x j ) exp q 1 , j q 2 q 3 q 2 , j ( v j ) exp q 1 q 2 , j q 3 , j { 1 , , N } q 3 ( v ) exp q 1 q 2 ,
{ q 1 ( x ) exp { 1 2 ( x μ ˜ ) * Σ ˜ 1 ( x μ ˜ ) } 𝒩 ( x | μ ˜ , Σ ˜ ) or q 1 , j ( x j ) exp { ( x j μ j ) 2 2 [ Σ ˜ ] j j } q 2 , j ( v j ) v j ( α ˜ v + 1 ) exp { β ˜ v , j / v j } 𝒢 ( v j | α ˜ v , β ˜ v , j ) q 3 ( v ) v ( α ˜ + 1 ) exp { β ˜ / v } 𝒢 ( v | α ˜ , β ˜ )
{ x ^ = μ ˜ = ( H * H + v ^ V ˜ * V ˜ ) 1 H * y , V ˜ = diag { 1 / v ^ j } , j { 1 , , N } or x ^ j = [ i = 1 M h i j 2 + v ^ / v ^ j ] 1 i = 1 M h i j y i Σ ˜ = v ^ ( H * H v ^ V ˜ * V ˜ ) 1 , [ Σ ˜ ] j j = v ^ [ i = 1 M h i j 2 + v ^ / v ^ j ] 1 v ^ j = β ˜ v , j / α ˜ v , α ˜ v = α v + 1 2 , β ˜ v , j = β v + 1 2 ( μ ˜ j 2 + [ Σ ˜ ] j j ) v ^ = β ˜ / α ˜ , α ˜ = α + M 2 , β ˜ = β + 1 2 y H μ ˜ 2
[ I ˜ 1 ( f 11 ) I ˜ 1 ( f n n ) I ˜ 2 ( f 11 ) I ˜ 2 ( f n n ) I ˜ 3 ( f 11 ) I ˜ 3 ( f n n ) I ˜ 4 ( f 11 ) I ˜ 4 ( f n n ) ] = [ s 1 ( f 11 2 ) 0 c 1 ( f 11 2 ) 0 0 s 1 ( f n n 2 ) 0 c 1 ( f n n 2 ) s 2 ( f 11 2 ) 0 c 2 ( f 11 2 ) 0 0 s 2 ( f n n 2 ) 0 c 2 ( f n n 2 ) s 3 ( f 11 2 ) 0 c 3 ( f 11 2 ) 0 0 s 3 ( f n n 2 ) 0 c 3 ( f n n 2 ) s 4 ( f 11 2 ) 0 c 4 ( f 11 2 ) 0 0 s 4 ( f n n 2 ) 0 c 4 ( f n n 2 ) ] [ ϕ ˜ ( f 11 ) ϕ ˜ ( f n n ) B ˜ ( f 11 ) B ˜ ( f n n ) ] y [ 4 K n 2 , 1 ] H [ 4 K n 2 , 8 n 2 ] x [ 8 n 2 , 1 ]
s k ( f i j 2 ) = 2 sin ( π λ D k f i j 2 ) ; c k ( f i j 2 ) = 2 cos ( π λ D k f i j 2 )

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