Abstract

The success of ptychographic imaging experiments strongly depends on achieving high signal-to-noise ratio. This is particularly important in nanoscale imaging experiments when diffraction signals are very weak and the experiments are accompanied by significant parasitic scattering (background), outliers or correlated noise sources. It is also critical when rare events, such as cosmic rays, or bad frames caused by electronic glitches or shutter timing malfunction take place. In this paper, we propose a novel iterative algorithm with rigorous analysis that exploits the direct forward model for parasitic noise and sample smoothness to achieve a thorough characterization and removal of structured and random noise. We present a formal description of the proposed algorithm and prove its convergence under mild conditions. Numerical experiments from simulations and real data (both soft and hard X-ray beamlines) demonstrate that the proposed algorithms produce better results when compared to state-of-the-art methods.

© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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  1. P. Nellist, B. McCallum, and J. Rodenburg, “Resolution beyond the ‘information limit’ in transmission electron microscopy,” Nature 374, 630 (1995).
    [Crossref]
  2. J. M. Rodenburg and H. M. Faulkner, “A phase retrieval algorithm for shifting illumination,” Appl. Phys. Lett. 85, 4795–4797 (2004).
    [Crossref]
  3. A. M. Maiden and J. M. Rodenburg, “An improved ptychographical phase retrieval algorithm for diffractive imaging,” Ultramicroscopy 109, 1256–1262 (2009).
    [Crossref] [PubMed]
  4. Y. Jiang, Z. Chen, Y. Han, P. Deb, H. Gao, S. Xie, P. Purohit, M. W. Tate, J. Park, S. M. Gruner, V. Elser, and D. A. Muller, “Electron ptychography of 2d materials to deep sub-Ångström resolution,” Nature 559, 343 (2018).
    [Crossref] [PubMed]
  5. X. Shi, P. Fischer, V. Neu, D. Elefant, J. Lee, D. Shapiro, M. Farmand, T. Tyliszczak, H.-W. Shiu, S. Marchesini, and S. Roy, “Soft x-ray ptychography studies of nanoscale magnetic and structural correlations in thin SmCo5 films,” Appl. Phys. Lett. 108, 094103 (2016).
    [Crossref]
  6. K. Giewekemeyer, P. Thibault, S. Kalbfleisch, A. Beerlink, C. M. Kewish, M. Dierolf, F. Pfeiffer, and T. Salditt, “Quantitative biological imaging by ptychographic x-ray diffraction microscopy,” Proc. Natl. Acad. Sci. U.S.A. 107, 529–534 (2010).
    [Crossref]
  7. D. A. Shapiro, Y.-S. Yu, T. Tyliszczak, J. Cabana, R. Celestre, W. Chao, K. Kaznatcheev, A. D. Kilcoyne, F. Maia, S. Marchesini, Y. S. Meng, T. Warwick, L. L. Yang, and H. Padmore, “Chemical composition mapping with nanometre resolution by soft x-ray microscopy,” Nat. Photonics 8, 765–769 (2014).
    [Crossref]
  8. Y.-S. Yu, M. Farmand, C. Kim, Y. Liu, C. P. Grey, F. C. Strobridge, T. Tyliszczak, R. Celestre, P. Denes, J. Joseph, H. Krishnan, F. R. N. C. Maia, A. L. D. Kilcoyne, S. Marchesini, T. P. C. Leite, T. Warwick, H. Padmore, J. Cabana, and D. A. Shapiro, “Three-dimensional localization of nanoscale battery reactions using soft x-ray tomography,” Nat. Commun. 9, 921 (2018).
    [Crossref] [PubMed]
  9. M. Holler, M. Guizar-Sicairos, E. H. Tsai, R. Dinapoli, E. Müller, O. Bunk, J. Raabe, and G. Aeppli, “High-resolution non-destructive three-dimensional imaging of integrated circuits,” Nature 543, 402–406 (2017).
    [Crossref] [PubMed]
  10. M. O. Wiedorn, S. Awel, A. J. Morgan, M. Barthelmess, R. Bean, K. R. Beyerlein, L. M. G. Chavas, N. Eckerskorn, H. Fleckenstein, M. Heymann, D. A. Horke, J. Knoška, V. Mariani, D. Oberthür, N. Roth, O. Yefanov, A. Barty, S. Bajt, J. Küpper, A. V. Rode, R. A. Kirian, and H. N. Chapman, “Post-sample aperture for low background diffraction experiments at x-ray free-electron lasers,” J. Synchrotron Radiat. 24, 1296–1298 (2017).
    [Crossref] [PubMed]
  11. J. Reinhardt, R. Hoppe, G. Hofmann, C. D. Damsgaard, J. Patommel, C. Baumbach, S. Baier, A. Rochet, J.-D. Grunwaldt, G. Falkenberg, and C. G. Schroer, “Beamstop-based low-background ptychography to image weakly scattering objects,” Ultramicroscopy 173, 52–57 (2017).
    [Crossref]
  12. C. Wang, Z. Xu, H. Liu, Y. Wang, J. Wang, and R. Tai, “Background noise removal in x-ray ptychography,” Appl. Opt. 56, 2099–2111 (2017).
    [Crossref] [PubMed]
  13. 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]
  14. R. Hesse, D. R. Luke, S. Sabach, and M. K. Tam, “Proximal heterogeneous block implicit-explicit method and application to blind ptychographic diffraction imaging,” SIAM J. Imaging Sci. 8, 426–457 (2015).
    [Crossref]
  15. H. Chang, P. Enfedaque, and S. Marchesini, “Blind ptychographic phase retrieval via convergent alternating direction method of multipliers,” SIAM J. Imaging Sci. 12, 153–185 (2019).
    [Crossref]
  16. V. Elser, “Phase retrieval by iterated projections,” J. Opt. Soc. Am. A 20, 40–55 (2003).
    [Crossref]
  17. P. Thibault and M. Guizar-Sicairos, “Maximum-likelihood refinement for coherent diffractive imaging,” New J. Phys. 14, 063004 (2012).
    [Crossref]
  18. D. R. Luke, “Relaxed averaged alternating reflections for diffraction imaging,” Inverse Probl. 21, 37–50 (2005).
    [Crossref]
  19. S. Marchesini, H. Krishnan, D. A. Shapiro, T. Perciano, J. A. Sethian, B. J. Daurer, and F. R. Maia, “SHARP: a distributed, GPU-based ptychographic solver,” J. Appl. Crystallogr. 49, 1245–1252 (2016).
    [Crossref]
  20. R. Glowinski and P. Le Tallec, Augmented Lagrangian and operator-splitting methods in nonlinear mechanics (SIAM, 1989).
    [Crossref]
  21. C. Wu and X.-C. Tai, “Augmented Lagrangian method, dual methods and split-Bregman iterations for ROF, vectorial TV and higher order models,” SIAM J. Imaging Sci. 3, 300–339 (2010).
    [Crossref]
  22. S. Boyd, N. Parikh, E. Chu, B. Peleato, and J. Eckstein, “Distributed optimization and statistical learning via the alternating direction method of multipliers,” Found. Trends Mach. Learn. 3, 1–122 (2011).
    [Crossref]
  23. P. Godard, M. Allain, V. Chamard, and J. Rodenburg, “Noise models for low counting rate coherent diffraction imaging,” Opt. Express 20, 25914–25934 (2012).
    [Crossref] [PubMed]
  24. M. Odstrčil, A. Menzel, and M. Guizar-Sicairos, “Iterative least-squares solver for generalized maximum-likelihood ptychography,” Opt. Express 26, 3108–3123 (2018).
    [Crossref]
  25. B. Enders, M. Dierolf, P. Cloetens, M. Stockmar, F. Pfeiffer, and P. Thibault, “Ptychography with broad-bandwidth radiation,” Appl. Phys. Lett. 104, 171104 (2014).
    [Crossref]
  26. S. Marchesini, A. Schirotzek, C. Yang, H.-T. Wu, and F. Maia, “Augmented projections for ptychographic imaging,” Inverse Probl. 29, 115009 (2013).
    [Crossref]
  27. B. J. Daurer, H. Krishnan, T. Perciano, F. R. Maia, D. A. Shapiro, J. A. Sethian, and S. Marchesini, “Nanosurveyor: a framework for real-time data processing,” Adv. Struct. Chem. Imaging 3, 7 (2017).
    [Crossref] [PubMed]
  28. H. Chang, Y. Lou, Y. Duan, and S. Marchesini, “Total variation–based phase retrieval for Poisson noise removal,” SIAM J. Imaging Sci. 11, 24–55 (2018).
    [Crossref]
  29. H. Chang and S. Marchesini, “Denoising Poisson phaseless measurements via orthogonal dictionary learning,” Opt. Express 26, 19773–19796 (2018).
    [Crossref] [PubMed]
  30. F. Murtagh, J.-L. Starck, and A. Bijaoui, “Image restoration with noise suppression using a multiresolution support,” Astron. Astrophys., Suppl. Ser. 112, 179 (1995).
  31. A. Chakrabarti and T. Zickler, “Image restoration with signal-dependent camera noise,” arXiv 1204.2994 (2012).
  32. J. Li, Z. Shen, R. Yin, and X. Zhang, “A reweighted l2 method for image restoration with poisson and mixed poisson-gaussian noise,” Inverse Probl. Imaging (Springfield) 9, 875–894 (2015).
    [Crossref]
  33. N. M. Kirby, S. T. Mudie, A. M. Hawley, D. J. Cookson, H. D. Mertens, N. Cowieson, and V. Samardzic-Boban, “A low-background-intensity focusing small-angle x-ray scattering undulator beamline,” J. Appl. Crystallogr. 46, 1670–1680 (2013).
    [Crossref]
  34. H. Chang, S. Marchesini, Y. Lou, and T. Zeng, “Variational phase retrieval with globally convergent preconditioned proximal algorithm,” SIAM J. Imaging Sci. 11, 56–93 (2018).
    [Crossref]
  35. Z. Wen, C. Yang, X. Liu, and S. Marchesini, “Alternating direction methods for classical and ptychographic phase retrieval,” Inverse Probl. 28, 115010 (2012).
    [Crossref]
  36. H. Chang, Y. Lou, M. K. Ng, and T. Zeng, “Phase retrieval from incomplete magnitude information via total variation regularization,” SIAM J. Sci. Comput. 38, A3672–A3695 (2016).
    [Crossref]
  37. H. Chang, P. Enfedaque, Y. Lou, and S. Marchesini, “Partially coherent ptychography by gradient decomposition of the probe,” Acta Crystallogr., Sect. A: Found. Adv. 74, 157–169 (2018).
    [Crossref]
  38. H. Chang and S. Marchesini, “A general framework for denoising phaseless diffraction measurements,” arXiv 1611.01417 (2016).
  39. M. J. Cherukara, Y. S. Nashed, and R. J. Harder, “Real-time coherent diffraction inversion using deep generative networks,” arXiv 1806.03992 (2018).

2019 (1)

H. Chang, P. Enfedaque, and S. Marchesini, “Blind ptychographic phase retrieval via convergent alternating direction method of multipliers,” SIAM J. Imaging Sci. 12, 153–185 (2019).
[Crossref]

2018 (7)

Y. Jiang, Z. Chen, Y. Han, P. Deb, H. Gao, S. Xie, P. Purohit, M. W. Tate, J. Park, S. M. Gruner, V. Elser, and D. A. Muller, “Electron ptychography of 2d materials to deep sub-Ångström resolution,” Nature 559, 343 (2018).
[Crossref] [PubMed]

Y.-S. Yu, M. Farmand, C. Kim, Y. Liu, C. P. Grey, F. C. Strobridge, T. Tyliszczak, R. Celestre, P. Denes, J. Joseph, H. Krishnan, F. R. N. C. Maia, A. L. D. Kilcoyne, S. Marchesini, T. P. C. Leite, T. Warwick, H. Padmore, J. Cabana, and D. A. Shapiro, “Three-dimensional localization of nanoscale battery reactions using soft x-ray tomography,” Nat. Commun. 9, 921 (2018).
[Crossref] [PubMed]

M. Odstrčil, A. Menzel, and M. Guizar-Sicairos, “Iterative least-squares solver for generalized maximum-likelihood ptychography,” Opt. Express 26, 3108–3123 (2018).
[Crossref]

H. Chang, Y. Lou, Y. Duan, and S. Marchesini, “Total variation–based phase retrieval for Poisson noise removal,” SIAM J. Imaging Sci. 11, 24–55 (2018).
[Crossref]

H. Chang and S. Marchesini, “Denoising Poisson phaseless measurements via orthogonal dictionary learning,” Opt. Express 26, 19773–19796 (2018).
[Crossref] [PubMed]

H. Chang, S. Marchesini, Y. Lou, and T. Zeng, “Variational phase retrieval with globally convergent preconditioned proximal algorithm,” SIAM J. Imaging Sci. 11, 56–93 (2018).
[Crossref]

H. Chang, P. Enfedaque, Y. Lou, and S. Marchesini, “Partially coherent ptychography by gradient decomposition of the probe,” Acta Crystallogr., Sect. A: Found. Adv. 74, 157–169 (2018).
[Crossref]

2017 (5)

B. J. Daurer, H. Krishnan, T. Perciano, F. R. Maia, D. A. Shapiro, J. A. Sethian, and S. Marchesini, “Nanosurveyor: a framework for real-time data processing,” Adv. Struct. Chem. Imaging 3, 7 (2017).
[Crossref] [PubMed]

M. Holler, M. Guizar-Sicairos, E. H. Tsai, R. Dinapoli, E. Müller, O. Bunk, J. Raabe, and G. Aeppli, “High-resolution non-destructive three-dimensional imaging of integrated circuits,” Nature 543, 402–406 (2017).
[Crossref] [PubMed]

M. O. Wiedorn, S. Awel, A. J. Morgan, M. Barthelmess, R. Bean, K. R. Beyerlein, L. M. G. Chavas, N. Eckerskorn, H. Fleckenstein, M. Heymann, D. A. Horke, J. Knoška, V. Mariani, D. Oberthür, N. Roth, O. Yefanov, A. Barty, S. Bajt, J. Küpper, A. V. Rode, R. A. Kirian, and H. N. Chapman, “Post-sample aperture for low background diffraction experiments at x-ray free-electron lasers,” J. Synchrotron Radiat. 24, 1296–1298 (2017).
[Crossref] [PubMed]

J. Reinhardt, R. Hoppe, G. Hofmann, C. D. Damsgaard, J. Patommel, C. Baumbach, S. Baier, A. Rochet, J.-D. Grunwaldt, G. Falkenberg, and C. G. Schroer, “Beamstop-based low-background ptychography to image weakly scattering objects,” Ultramicroscopy 173, 52–57 (2017).
[Crossref]

C. Wang, Z. Xu, H. Liu, Y. Wang, J. Wang, and R. Tai, “Background noise removal in x-ray ptychography,” Appl. Opt. 56, 2099–2111 (2017).
[Crossref] [PubMed]

2016 (3)

X. Shi, P. Fischer, V. Neu, D. Elefant, J. Lee, D. Shapiro, M. Farmand, T. Tyliszczak, H.-W. Shiu, S. Marchesini, and S. Roy, “Soft x-ray ptychography studies of nanoscale magnetic and structural correlations in thin SmCo5 films,” Appl. Phys. Lett. 108, 094103 (2016).
[Crossref]

S. Marchesini, H. Krishnan, D. A. Shapiro, T. Perciano, J. A. Sethian, B. J. Daurer, and F. R. Maia, “SHARP: a distributed, GPU-based ptychographic solver,” J. Appl. Crystallogr. 49, 1245–1252 (2016).
[Crossref]

H. Chang, Y. Lou, M. K. Ng, and T. Zeng, “Phase retrieval from incomplete magnitude information via total variation regularization,” SIAM J. Sci. Comput. 38, A3672–A3695 (2016).
[Crossref]

2015 (2)

J. Li, Z. Shen, R. Yin, and X. Zhang, “A reweighted l2 method for image restoration with poisson and mixed poisson-gaussian noise,” Inverse Probl. Imaging (Springfield) 9, 875–894 (2015).
[Crossref]

R. Hesse, D. R. Luke, S. Sabach, and M. K. Tam, “Proximal heterogeneous block implicit-explicit method and application to blind ptychographic diffraction imaging,” SIAM J. Imaging Sci. 8, 426–457 (2015).
[Crossref]

2014 (2)

B. Enders, M. Dierolf, P. Cloetens, M. Stockmar, F. Pfeiffer, and P. Thibault, “Ptychography with broad-bandwidth radiation,” Appl. Phys. Lett. 104, 171104 (2014).
[Crossref]

D. A. Shapiro, Y.-S. Yu, T. Tyliszczak, J. Cabana, R. Celestre, W. Chao, K. Kaznatcheev, A. D. Kilcoyne, F. Maia, S. Marchesini, Y. S. Meng, T. Warwick, L. L. Yang, and H. Padmore, “Chemical composition mapping with nanometre resolution by soft x-ray microscopy,” Nat. Photonics 8, 765–769 (2014).
[Crossref]

2013 (2)

S. Marchesini, A. Schirotzek, C. Yang, H.-T. Wu, and F. Maia, “Augmented projections for ptychographic imaging,” Inverse Probl. 29, 115009 (2013).
[Crossref]

N. M. Kirby, S. T. Mudie, A. M. Hawley, D. J. Cookson, H. D. Mertens, N. Cowieson, and V. Samardzic-Boban, “A low-background-intensity focusing small-angle x-ray scattering undulator beamline,” J. Appl. Crystallogr. 46, 1670–1680 (2013).
[Crossref]

2012 (3)

Z. Wen, C. Yang, X. Liu, and S. Marchesini, “Alternating direction methods for classical and ptychographic phase retrieval,” Inverse Probl. 28, 115010 (2012).
[Crossref]

P. Godard, M. Allain, V. Chamard, and J. Rodenburg, “Noise models for low counting rate coherent diffraction imaging,” Opt. Express 20, 25914–25934 (2012).
[Crossref] [PubMed]

P. Thibault and M. Guizar-Sicairos, “Maximum-likelihood refinement for coherent diffractive imaging,” New J. Phys. 14, 063004 (2012).
[Crossref]

2011 (1)

S. Boyd, N. Parikh, E. Chu, B. Peleato, and J. Eckstein, “Distributed optimization and statistical learning via the alternating direction method of multipliers,” Found. Trends Mach. Learn. 3, 1–122 (2011).
[Crossref]

2010 (2)

C. Wu and X.-C. Tai, “Augmented Lagrangian method, dual methods and split-Bregman iterations for ROF, vectorial TV and higher order models,” SIAM J. Imaging Sci. 3, 300–339 (2010).
[Crossref]

K. Giewekemeyer, P. Thibault, S. Kalbfleisch, A. Beerlink, C. M. Kewish, M. Dierolf, F. Pfeiffer, and T. Salditt, “Quantitative biological imaging by ptychographic x-ray diffraction microscopy,” Proc. Natl. Acad. Sci. U.S.A. 107, 529–534 (2010).
[Crossref]

2009 (2)

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]

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

2005 (1)

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

2004 (1)

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

2003 (1)

1995 (2)

P. Nellist, B. McCallum, and J. Rodenburg, “Resolution beyond the ‘information limit’ in transmission electron microscopy,” Nature 374, 630 (1995).
[Crossref]

F. Murtagh, J.-L. Starck, and A. Bijaoui, “Image restoration with noise suppression using a multiresolution support,” Astron. Astrophys., Suppl. Ser. 112, 179 (1995).

Aeppli, G.

M. Holler, M. Guizar-Sicairos, E. H. Tsai, R. Dinapoli, E. Müller, O. Bunk, J. Raabe, and G. Aeppli, “High-resolution non-destructive three-dimensional imaging of integrated circuits,” Nature 543, 402–406 (2017).
[Crossref] [PubMed]

Allain, M.

Awel, S.

M. O. Wiedorn, S. Awel, A. J. Morgan, M. Barthelmess, R. Bean, K. R. Beyerlein, L. M. G. Chavas, N. Eckerskorn, H. Fleckenstein, M. Heymann, D. A. Horke, J. Knoška, V. Mariani, D. Oberthür, N. Roth, O. Yefanov, A. Barty, S. Bajt, J. Küpper, A. V. Rode, R. A. Kirian, and H. N. Chapman, “Post-sample aperture for low background diffraction experiments at x-ray free-electron lasers,” J. Synchrotron Radiat. 24, 1296–1298 (2017).
[Crossref] [PubMed]

Baier, S.

J. Reinhardt, R. Hoppe, G. Hofmann, C. D. Damsgaard, J. Patommel, C. Baumbach, S. Baier, A. Rochet, J.-D. Grunwaldt, G. Falkenberg, and C. G. Schroer, “Beamstop-based low-background ptychography to image weakly scattering objects,” Ultramicroscopy 173, 52–57 (2017).
[Crossref]

Bajt, S.

M. O. Wiedorn, S. Awel, A. J. Morgan, M. Barthelmess, R. Bean, K. R. Beyerlein, L. M. G. Chavas, N. Eckerskorn, H. Fleckenstein, M. Heymann, D. A. Horke, J. Knoška, V. Mariani, D. Oberthür, N. Roth, O. Yefanov, A. Barty, S. Bajt, J. Küpper, A. V. Rode, R. A. Kirian, and H. N. Chapman, “Post-sample aperture for low background diffraction experiments at x-ray free-electron lasers,” J. Synchrotron Radiat. 24, 1296–1298 (2017).
[Crossref] [PubMed]

Barthelmess, M.

M. O. Wiedorn, S. Awel, A. J. Morgan, M. Barthelmess, R. Bean, K. R. Beyerlein, L. M. G. Chavas, N. Eckerskorn, H. Fleckenstein, M. Heymann, D. A. Horke, J. Knoška, V. Mariani, D. Oberthür, N. Roth, O. Yefanov, A. Barty, S. Bajt, J. Küpper, A. V. Rode, R. A. Kirian, and H. N. Chapman, “Post-sample aperture for low background diffraction experiments at x-ray free-electron lasers,” J. Synchrotron Radiat. 24, 1296–1298 (2017).
[Crossref] [PubMed]

Barty, A.

M. O. Wiedorn, S. Awel, A. J. Morgan, M. Barthelmess, R. Bean, K. R. Beyerlein, L. M. G. Chavas, N. Eckerskorn, H. Fleckenstein, M. Heymann, D. A. Horke, J. Knoška, V. Mariani, D. Oberthür, N. Roth, O. Yefanov, A. Barty, S. Bajt, J. Küpper, A. V. Rode, R. A. Kirian, and H. N. Chapman, “Post-sample aperture for low background diffraction experiments at x-ray free-electron lasers,” J. Synchrotron Radiat. 24, 1296–1298 (2017).
[Crossref] [PubMed]

Baumbach, C.

J. Reinhardt, R. Hoppe, G. Hofmann, C. D. Damsgaard, J. Patommel, C. Baumbach, S. Baier, A. Rochet, J.-D. Grunwaldt, G. Falkenberg, and C. G. Schroer, “Beamstop-based low-background ptychography to image weakly scattering objects,” Ultramicroscopy 173, 52–57 (2017).
[Crossref]

Bean, R.

M. O. Wiedorn, S. Awel, A. J. Morgan, M. Barthelmess, R. Bean, K. R. Beyerlein, L. M. G. Chavas, N. Eckerskorn, H. Fleckenstein, M. Heymann, D. A. Horke, J. Knoška, V. Mariani, D. Oberthür, N. Roth, O. Yefanov, A. Barty, S. Bajt, J. Küpper, A. V. Rode, R. A. Kirian, and H. N. Chapman, “Post-sample aperture for low background diffraction experiments at x-ray free-electron lasers,” J. Synchrotron Radiat. 24, 1296–1298 (2017).
[Crossref] [PubMed]

Beerlink, A.

K. Giewekemeyer, P. Thibault, S. Kalbfleisch, A. Beerlink, C. M. Kewish, M. Dierolf, F. Pfeiffer, and T. Salditt, “Quantitative biological imaging by ptychographic x-ray diffraction microscopy,” Proc. Natl. Acad. Sci. U.S.A. 107, 529–534 (2010).
[Crossref]

Beyerlein, K. R.

M. O. Wiedorn, S. Awel, A. J. Morgan, M. Barthelmess, R. Bean, K. R. Beyerlein, L. M. G. Chavas, N. Eckerskorn, H. Fleckenstein, M. Heymann, D. A. Horke, J. Knoška, V. Mariani, D. Oberthür, N. Roth, O. Yefanov, A. Barty, S. Bajt, J. Küpper, A. V. Rode, R. A. Kirian, and H. N. Chapman, “Post-sample aperture for low background diffraction experiments at x-ray free-electron lasers,” J. Synchrotron Radiat. 24, 1296–1298 (2017).
[Crossref] [PubMed]

Bijaoui, A.

F. Murtagh, J.-L. Starck, and A. Bijaoui, “Image restoration with noise suppression using a multiresolution support,” Astron. Astrophys., Suppl. Ser. 112, 179 (1995).

Boyd, S.

S. Boyd, N. Parikh, E. Chu, B. Peleato, and J. Eckstein, “Distributed optimization and statistical learning via the alternating direction method of multipliers,” Found. Trends Mach. Learn. 3, 1–122 (2011).
[Crossref]

Bunk, O.

M. Holler, M. Guizar-Sicairos, E. H. Tsai, R. Dinapoli, E. Müller, O. Bunk, J. Raabe, and G. Aeppli, “High-resolution non-destructive three-dimensional imaging of integrated circuits,” Nature 543, 402–406 (2017).
[Crossref] [PubMed]

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

Cabana, J.

Y.-S. Yu, M. Farmand, C. Kim, Y. Liu, C. P. Grey, F. C. Strobridge, T. Tyliszczak, R. Celestre, P. Denes, J. Joseph, H. Krishnan, F. R. N. C. Maia, A. L. D. Kilcoyne, S. Marchesini, T. P. C. Leite, T. Warwick, H. Padmore, J. Cabana, and D. A. Shapiro, “Three-dimensional localization of nanoscale battery reactions using soft x-ray tomography,” Nat. Commun. 9, 921 (2018).
[Crossref] [PubMed]

D. A. Shapiro, Y.-S. Yu, T. Tyliszczak, J. Cabana, R. Celestre, W. Chao, K. Kaznatcheev, A. D. Kilcoyne, F. Maia, S. Marchesini, Y. S. Meng, T. Warwick, L. L. Yang, and H. Padmore, “Chemical composition mapping with nanometre resolution by soft x-ray microscopy,” Nat. Photonics 8, 765–769 (2014).
[Crossref]

Celestre, R.

Y.-S. Yu, M. Farmand, C. Kim, Y. Liu, C. P. Grey, F. C. Strobridge, T. Tyliszczak, R. Celestre, P. Denes, J. Joseph, H. Krishnan, F. R. N. C. Maia, A. L. D. Kilcoyne, S. Marchesini, T. P. C. Leite, T. Warwick, H. Padmore, J. Cabana, and D. A. Shapiro, “Three-dimensional localization of nanoscale battery reactions using soft x-ray tomography,” Nat. Commun. 9, 921 (2018).
[Crossref] [PubMed]

D. A. Shapiro, Y.-S. Yu, T. Tyliszczak, J. Cabana, R. Celestre, W. Chao, K. Kaznatcheev, A. D. Kilcoyne, F. Maia, S. Marchesini, Y. S. Meng, T. Warwick, L. L. Yang, and H. Padmore, “Chemical composition mapping with nanometre resolution by soft x-ray microscopy,” Nat. Photonics 8, 765–769 (2014).
[Crossref]

Chakrabarti, A.

A. Chakrabarti and T. Zickler, “Image restoration with signal-dependent camera noise,” arXiv 1204.2994 (2012).

Chamard, V.

Chang, H.

H. Chang, P. Enfedaque, and S. Marchesini, “Blind ptychographic phase retrieval via convergent alternating direction method of multipliers,” SIAM J. Imaging Sci. 12, 153–185 (2019).
[Crossref]

H. Chang and S. Marchesini, “Denoising Poisson phaseless measurements via orthogonal dictionary learning,” Opt. Express 26, 19773–19796 (2018).
[Crossref] [PubMed]

H. Chang, S. Marchesini, Y. Lou, and T. Zeng, “Variational phase retrieval with globally convergent preconditioned proximal algorithm,” SIAM J. Imaging Sci. 11, 56–93 (2018).
[Crossref]

H. Chang, P. Enfedaque, Y. Lou, and S. Marchesini, “Partially coherent ptychography by gradient decomposition of the probe,” Acta Crystallogr., Sect. A: Found. Adv. 74, 157–169 (2018).
[Crossref]

H. Chang, Y. Lou, Y. Duan, and S. Marchesini, “Total variation–based phase retrieval for Poisson noise removal,” SIAM J. Imaging Sci. 11, 24–55 (2018).
[Crossref]

H. Chang, Y. Lou, M. K. Ng, and T. Zeng, “Phase retrieval from incomplete magnitude information via total variation regularization,” SIAM J. Sci. Comput. 38, A3672–A3695 (2016).
[Crossref]

H. Chang and S. Marchesini, “A general framework for denoising phaseless diffraction measurements,” arXiv 1611.01417 (2016).

Chao, W.

D. A. Shapiro, Y.-S. Yu, T. Tyliszczak, J. Cabana, R. Celestre, W. Chao, K. Kaznatcheev, A. D. Kilcoyne, F. Maia, S. Marchesini, Y. S. Meng, T. Warwick, L. L. Yang, and H. Padmore, “Chemical composition mapping with nanometre resolution by soft x-ray microscopy,” Nat. Photonics 8, 765–769 (2014).
[Crossref]

Chapman, H. N.

M. O. Wiedorn, S. Awel, A. J. Morgan, M. Barthelmess, R. Bean, K. R. Beyerlein, L. M. G. Chavas, N. Eckerskorn, H. Fleckenstein, M. Heymann, D. A. Horke, J. Knoška, V. Mariani, D. Oberthür, N. Roth, O. Yefanov, A. Barty, S. Bajt, J. Küpper, A. V. Rode, R. A. Kirian, and H. N. Chapman, “Post-sample aperture for low background diffraction experiments at x-ray free-electron lasers,” J. Synchrotron Radiat. 24, 1296–1298 (2017).
[Crossref] [PubMed]

Chavas, L. M. G.

M. O. Wiedorn, S. Awel, A. J. Morgan, M. Barthelmess, R. Bean, K. R. Beyerlein, L. M. G. Chavas, N. Eckerskorn, H. Fleckenstein, M. Heymann, D. A. Horke, J. Knoška, V. Mariani, D. Oberthür, N. Roth, O. Yefanov, A. Barty, S. Bajt, J. Küpper, A. V. Rode, R. A. Kirian, and H. N. Chapman, “Post-sample aperture for low background diffraction experiments at x-ray free-electron lasers,” J. Synchrotron Radiat. 24, 1296–1298 (2017).
[Crossref] [PubMed]

Chen, Z.

Y. Jiang, Z. Chen, Y. Han, P. Deb, H. Gao, S. Xie, P. Purohit, M. W. Tate, J. Park, S. M. Gruner, V. Elser, and D. A. Muller, “Electron ptychography of 2d materials to deep sub-Ångström resolution,” Nature 559, 343 (2018).
[Crossref] [PubMed]

Cherukara, M. J.

M. J. Cherukara, Y. S. Nashed, and R. J. Harder, “Real-time coherent diffraction inversion using deep generative networks,” arXiv 1806.03992 (2018).

Chu, E.

S. Boyd, N. Parikh, E. Chu, B. Peleato, and J. Eckstein, “Distributed optimization and statistical learning via the alternating direction method of multipliers,” Found. Trends Mach. Learn. 3, 1–122 (2011).
[Crossref]

Cloetens, P.

B. Enders, M. Dierolf, P. Cloetens, M. Stockmar, F. Pfeiffer, and P. Thibault, “Ptychography with broad-bandwidth radiation,” Appl. Phys. Lett. 104, 171104 (2014).
[Crossref]

Cookson, D. J.

N. M. Kirby, S. T. Mudie, A. M. Hawley, D. J. Cookson, H. D. Mertens, N. Cowieson, and V. Samardzic-Boban, “A low-background-intensity focusing small-angle x-ray scattering undulator beamline,” J. Appl. Crystallogr. 46, 1670–1680 (2013).
[Crossref]

Cowieson, N.

N. M. Kirby, S. T. Mudie, A. M. Hawley, D. J. Cookson, H. D. Mertens, N. Cowieson, and V. Samardzic-Boban, “A low-background-intensity focusing small-angle x-ray scattering undulator beamline,” J. Appl. Crystallogr. 46, 1670–1680 (2013).
[Crossref]

Damsgaard, C. D.

J. Reinhardt, R. Hoppe, G. Hofmann, C. D. Damsgaard, J. Patommel, C. Baumbach, S. Baier, A. Rochet, J.-D. Grunwaldt, G. Falkenberg, and C. G. Schroer, “Beamstop-based low-background ptychography to image weakly scattering objects,” Ultramicroscopy 173, 52–57 (2017).
[Crossref]

Daurer, B. J.

B. J. Daurer, H. Krishnan, T. Perciano, F. R. Maia, D. A. Shapiro, J. A. Sethian, and S. Marchesini, “Nanosurveyor: a framework for real-time data processing,” Adv. Struct. Chem. Imaging 3, 7 (2017).
[Crossref] [PubMed]

S. Marchesini, H. Krishnan, D. A. Shapiro, T. Perciano, J. A. Sethian, B. J. Daurer, and F. R. Maia, “SHARP: a distributed, GPU-based ptychographic solver,” J. Appl. Crystallogr. 49, 1245–1252 (2016).
[Crossref]

Deb, P.

Y. Jiang, Z. Chen, Y. Han, P. Deb, H. Gao, S. Xie, P. Purohit, M. W. Tate, J. Park, S. M. Gruner, V. Elser, and D. A. Muller, “Electron ptychography of 2d materials to deep sub-Ångström resolution,” Nature 559, 343 (2018).
[Crossref] [PubMed]

Denes, P.

Y.-S. Yu, M. Farmand, C. Kim, Y. Liu, C. P. Grey, F. C. Strobridge, T. Tyliszczak, R. Celestre, P. Denes, J. Joseph, H. Krishnan, F. R. N. C. Maia, A. L. D. Kilcoyne, S. Marchesini, T. P. C. Leite, T. Warwick, H. Padmore, J. Cabana, and D. A. Shapiro, “Three-dimensional localization of nanoscale battery reactions using soft x-ray tomography,” Nat. Commun. 9, 921 (2018).
[Crossref] [PubMed]

Dierolf, M.

B. Enders, M. Dierolf, P. Cloetens, M. Stockmar, F. Pfeiffer, and P. Thibault, “Ptychography with broad-bandwidth radiation,” Appl. Phys. Lett. 104, 171104 (2014).
[Crossref]

K. Giewekemeyer, P. Thibault, S. Kalbfleisch, A. Beerlink, C. M. Kewish, M. Dierolf, F. Pfeiffer, and T. Salditt, “Quantitative biological imaging by ptychographic x-ray diffraction microscopy,” Proc. Natl. Acad. Sci. U.S.A. 107, 529–534 (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]

Dinapoli, R.

M. Holler, M. Guizar-Sicairos, E. H. Tsai, R. Dinapoli, E. Müller, O. Bunk, J. Raabe, and G. Aeppli, “High-resolution non-destructive three-dimensional imaging of integrated circuits,” Nature 543, 402–406 (2017).
[Crossref] [PubMed]

Duan, Y.

H. Chang, Y. Lou, Y. Duan, and S. Marchesini, “Total variation–based phase retrieval for Poisson noise removal,” SIAM J. Imaging Sci. 11, 24–55 (2018).
[Crossref]

Eckerskorn, N.

M. O. Wiedorn, S. Awel, A. J. Morgan, M. Barthelmess, R. Bean, K. R. Beyerlein, L. M. G. Chavas, N. Eckerskorn, H. Fleckenstein, M. Heymann, D. A. Horke, J. Knoška, V. Mariani, D. Oberthür, N. Roth, O. Yefanov, A. Barty, S. Bajt, J. Küpper, A. V. Rode, R. A. Kirian, and H. N. Chapman, “Post-sample aperture for low background diffraction experiments at x-ray free-electron lasers,” J. Synchrotron Radiat. 24, 1296–1298 (2017).
[Crossref] [PubMed]

Eckstein, J.

S. Boyd, N. Parikh, E. Chu, B. Peleato, and J. Eckstein, “Distributed optimization and statistical learning via the alternating direction method of multipliers,” Found. Trends Mach. Learn. 3, 1–122 (2011).
[Crossref]

Elefant, D.

X. Shi, P. Fischer, V. Neu, D. Elefant, J. Lee, D. Shapiro, M. Farmand, T. Tyliszczak, H.-W. Shiu, S. Marchesini, and S. Roy, “Soft x-ray ptychography studies of nanoscale magnetic and structural correlations in thin SmCo5 films,” Appl. Phys. Lett. 108, 094103 (2016).
[Crossref]

Elser, V.

Y. Jiang, Z. Chen, Y. Han, P. Deb, H. Gao, S. Xie, P. Purohit, M. W. Tate, J. Park, S. M. Gruner, V. Elser, and D. A. Muller, “Electron ptychography of 2d materials to deep sub-Ångström resolution,” Nature 559, 343 (2018).
[Crossref] [PubMed]

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

Enders, B.

B. Enders, M. Dierolf, P. Cloetens, M. Stockmar, F. Pfeiffer, and P. Thibault, “Ptychography with broad-bandwidth radiation,” Appl. Phys. Lett. 104, 171104 (2014).
[Crossref]

Enfedaque, P.

H. Chang, P. Enfedaque, and S. Marchesini, “Blind ptychographic phase retrieval via convergent alternating direction method of multipliers,” SIAM J. Imaging Sci. 12, 153–185 (2019).
[Crossref]

H. Chang, P. Enfedaque, Y. Lou, and S. Marchesini, “Partially coherent ptychography by gradient decomposition of the probe,” Acta Crystallogr., Sect. A: Found. Adv. 74, 157–169 (2018).
[Crossref]

Falkenberg, G.

J. Reinhardt, R. Hoppe, G. Hofmann, C. D. Damsgaard, J. Patommel, C. Baumbach, S. Baier, A. Rochet, J.-D. Grunwaldt, G. Falkenberg, and C. G. Schroer, “Beamstop-based low-background ptychography to image weakly scattering objects,” Ultramicroscopy 173, 52–57 (2017).
[Crossref]

Farmand, M.

Y.-S. Yu, M. Farmand, C. Kim, Y. Liu, C. P. Grey, F. C. Strobridge, T. Tyliszczak, R. Celestre, P. Denes, J. Joseph, H. Krishnan, F. R. N. C. Maia, A. L. D. Kilcoyne, S. Marchesini, T. P. C. Leite, T. Warwick, H. Padmore, J. Cabana, and D. A. Shapiro, “Three-dimensional localization of nanoscale battery reactions using soft x-ray tomography,” Nat. Commun. 9, 921 (2018).
[Crossref] [PubMed]

X. Shi, P. Fischer, V. Neu, D. Elefant, J. Lee, D. Shapiro, M. Farmand, T. Tyliszczak, H.-W. Shiu, S. Marchesini, and S. Roy, “Soft x-ray ptychography studies of nanoscale magnetic and structural correlations in thin SmCo5 films,” Appl. Phys. Lett. 108, 094103 (2016).
[Crossref]

Faulkner, H. M.

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

Fischer, P.

X. Shi, P. Fischer, V. Neu, D. Elefant, J. Lee, D. Shapiro, M. Farmand, T. Tyliszczak, H.-W. Shiu, S. Marchesini, and S. Roy, “Soft x-ray ptychography studies of nanoscale magnetic and structural correlations in thin SmCo5 films,” Appl. Phys. Lett. 108, 094103 (2016).
[Crossref]

Fleckenstein, H.

M. O. Wiedorn, S. Awel, A. J. Morgan, M. Barthelmess, R. Bean, K. R. Beyerlein, L. M. G. Chavas, N. Eckerskorn, H. Fleckenstein, M. Heymann, D. A. Horke, J. Knoška, V. Mariani, D. Oberthür, N. Roth, O. Yefanov, A. Barty, S. Bajt, J. Küpper, A. V. Rode, R. A. Kirian, and H. N. Chapman, “Post-sample aperture for low background diffraction experiments at x-ray free-electron lasers,” J. Synchrotron Radiat. 24, 1296–1298 (2017).
[Crossref] [PubMed]

Gao, H.

Y. Jiang, Z. Chen, Y. Han, P. Deb, H. Gao, S. Xie, P. Purohit, M. W. Tate, J. Park, S. M. Gruner, V. Elser, and D. A. Muller, “Electron ptychography of 2d materials to deep sub-Ångström resolution,” Nature 559, 343 (2018).
[Crossref] [PubMed]

Giewekemeyer, K.

K. Giewekemeyer, P. Thibault, S. Kalbfleisch, A. Beerlink, C. M. Kewish, M. Dierolf, F. Pfeiffer, and T. Salditt, “Quantitative biological imaging by ptychographic x-ray diffraction microscopy,” Proc. Natl. Acad. Sci. U.S.A. 107, 529–534 (2010).
[Crossref]

Glowinski, R.

R. Glowinski and P. Le Tallec, Augmented Lagrangian and operator-splitting methods in nonlinear mechanics (SIAM, 1989).
[Crossref]

Godard, P.

Grey, C. P.

Y.-S. Yu, M. Farmand, C. Kim, Y. Liu, C. P. Grey, F. C. Strobridge, T. Tyliszczak, R. Celestre, P. Denes, J. Joseph, H. Krishnan, F. R. N. C. Maia, A. L. D. Kilcoyne, S. Marchesini, T. P. C. Leite, T. Warwick, H. Padmore, J. Cabana, and D. A. Shapiro, “Three-dimensional localization of nanoscale battery reactions using soft x-ray tomography,” Nat. Commun. 9, 921 (2018).
[Crossref] [PubMed]

Gruner, S. M.

Y. Jiang, Z. Chen, Y. Han, P. Deb, H. Gao, S. Xie, P. Purohit, M. W. Tate, J. Park, S. M. Gruner, V. Elser, and D. A. Muller, “Electron ptychography of 2d materials to deep sub-Ångström resolution,” Nature 559, 343 (2018).
[Crossref] [PubMed]

Grunwaldt, J.-D.

J. Reinhardt, R. Hoppe, G. Hofmann, C. D. Damsgaard, J. Patommel, C. Baumbach, S. Baier, A. Rochet, J.-D. Grunwaldt, G. Falkenberg, and C. G. Schroer, “Beamstop-based low-background ptychography to image weakly scattering objects,” Ultramicroscopy 173, 52–57 (2017).
[Crossref]

Guizar-Sicairos, M.

M. Odstrčil, A. Menzel, and M. Guizar-Sicairos, “Iterative least-squares solver for generalized maximum-likelihood ptychography,” Opt. Express 26, 3108–3123 (2018).
[Crossref]

M. Holler, M. Guizar-Sicairos, E. H. Tsai, R. Dinapoli, E. Müller, O. Bunk, J. Raabe, and G. Aeppli, “High-resolution non-destructive three-dimensional imaging of integrated circuits,” Nature 543, 402–406 (2017).
[Crossref] [PubMed]

P. Thibault and M. Guizar-Sicairos, “Maximum-likelihood refinement for coherent diffractive imaging,” New J. Phys. 14, 063004 (2012).
[Crossref]

Han, Y.

Y. Jiang, Z. Chen, Y. Han, P. Deb, H. Gao, S. Xie, P. Purohit, M. W. Tate, J. Park, S. M. Gruner, V. Elser, and D. A. Muller, “Electron ptychography of 2d materials to deep sub-Ångström resolution,” Nature 559, 343 (2018).
[Crossref] [PubMed]

Harder, R. J.

M. J. Cherukara, Y. S. Nashed, and R. J. Harder, “Real-time coherent diffraction inversion using deep generative networks,” arXiv 1806.03992 (2018).

Hawley, A. M.

N. M. Kirby, S. T. Mudie, A. M. Hawley, D. J. Cookson, H. D. Mertens, N. Cowieson, and V. Samardzic-Boban, “A low-background-intensity focusing small-angle x-ray scattering undulator beamline,” J. Appl. Crystallogr. 46, 1670–1680 (2013).
[Crossref]

Hesse, R.

R. Hesse, D. R. Luke, S. Sabach, and M. K. Tam, “Proximal heterogeneous block implicit-explicit method and application to blind ptychographic diffraction imaging,” SIAM J. Imaging Sci. 8, 426–457 (2015).
[Crossref]

Heymann, M.

M. O. Wiedorn, S. Awel, A. J. Morgan, M. Barthelmess, R. Bean, K. R. Beyerlein, L. M. G. Chavas, N. Eckerskorn, H. Fleckenstein, M. Heymann, D. A. Horke, J. Knoška, V. Mariani, D. Oberthür, N. Roth, O. Yefanov, A. Barty, S. Bajt, J. Küpper, A. V. Rode, R. A. Kirian, and H. N. Chapman, “Post-sample aperture for low background diffraction experiments at x-ray free-electron lasers,” J. Synchrotron Radiat. 24, 1296–1298 (2017).
[Crossref] [PubMed]

Hofmann, G.

J. Reinhardt, R. Hoppe, G. Hofmann, C. D. Damsgaard, J. Patommel, C. Baumbach, S. Baier, A. Rochet, J.-D. Grunwaldt, G. Falkenberg, and C. G. Schroer, “Beamstop-based low-background ptychography to image weakly scattering objects,” Ultramicroscopy 173, 52–57 (2017).
[Crossref]

Holler, M.

M. Holler, M. Guizar-Sicairos, E. H. Tsai, R. Dinapoli, E. Müller, O. Bunk, J. Raabe, and G. Aeppli, “High-resolution non-destructive three-dimensional imaging of integrated circuits,” Nature 543, 402–406 (2017).
[Crossref] [PubMed]

Hoppe, R.

J. Reinhardt, R. Hoppe, G. Hofmann, C. D. Damsgaard, J. Patommel, C. Baumbach, S. Baier, A. Rochet, J.-D. Grunwaldt, G. Falkenberg, and C. G. Schroer, “Beamstop-based low-background ptychography to image weakly scattering objects,” Ultramicroscopy 173, 52–57 (2017).
[Crossref]

Horke, D. A.

M. O. Wiedorn, S. Awel, A. J. Morgan, M. Barthelmess, R. Bean, K. R. Beyerlein, L. M. G. Chavas, N. Eckerskorn, H. Fleckenstein, M. Heymann, D. A. Horke, J. Knoška, V. Mariani, D. Oberthür, N. Roth, O. Yefanov, A. Barty, S. Bajt, J. Küpper, A. V. Rode, R. A. Kirian, and H. N. Chapman, “Post-sample aperture for low background diffraction experiments at x-ray free-electron lasers,” J. Synchrotron Radiat. 24, 1296–1298 (2017).
[Crossref] [PubMed]

Jiang, Y.

Y. Jiang, Z. Chen, Y. Han, P. Deb, H. Gao, S. Xie, P. Purohit, M. W. Tate, J. Park, S. M. Gruner, V. Elser, and D. A. Muller, “Electron ptychography of 2d materials to deep sub-Ångström resolution,” Nature 559, 343 (2018).
[Crossref] [PubMed]

Joseph, J.

Y.-S. Yu, M. Farmand, C. Kim, Y. Liu, C. P. Grey, F. C. Strobridge, T. Tyliszczak, R. Celestre, P. Denes, J. Joseph, H. Krishnan, F. R. N. C. Maia, A. L. D. Kilcoyne, S. Marchesini, T. P. C. Leite, T. Warwick, H. Padmore, J. Cabana, and D. A. Shapiro, “Three-dimensional localization of nanoscale battery reactions using soft x-ray tomography,” Nat. Commun. 9, 921 (2018).
[Crossref] [PubMed]

Kalbfleisch, S.

K. Giewekemeyer, P. Thibault, S. Kalbfleisch, A. Beerlink, C. M. Kewish, M. Dierolf, F. Pfeiffer, and T. Salditt, “Quantitative biological imaging by ptychographic x-ray diffraction microscopy,” Proc. Natl. Acad. Sci. U.S.A. 107, 529–534 (2010).
[Crossref]

Kaznatcheev, K.

D. A. Shapiro, Y.-S. Yu, T. Tyliszczak, J. Cabana, R. Celestre, W. Chao, K. Kaznatcheev, A. D. Kilcoyne, F. Maia, S. Marchesini, Y. S. Meng, T. Warwick, L. L. Yang, and H. Padmore, “Chemical composition mapping with nanometre resolution by soft x-ray microscopy,” Nat. Photonics 8, 765–769 (2014).
[Crossref]

Kewish, C. M.

K. Giewekemeyer, P. Thibault, S. Kalbfleisch, A. Beerlink, C. M. Kewish, M. Dierolf, F. Pfeiffer, and T. Salditt, “Quantitative biological imaging by ptychographic x-ray diffraction microscopy,” Proc. Natl. Acad. Sci. U.S.A. 107, 529–534 (2010).
[Crossref]

Kilcoyne, A. D.

D. A. Shapiro, Y.-S. Yu, T. Tyliszczak, J. Cabana, R. Celestre, W. Chao, K. Kaznatcheev, A. D. Kilcoyne, F. Maia, S. Marchesini, Y. S. Meng, T. Warwick, L. L. Yang, and H. Padmore, “Chemical composition mapping with nanometre resolution by soft x-ray microscopy,” Nat. Photonics 8, 765–769 (2014).
[Crossref]

Kilcoyne, A. L. D.

Y.-S. Yu, M. Farmand, C. Kim, Y. Liu, C. P. Grey, F. C. Strobridge, T. Tyliszczak, R. Celestre, P. Denes, J. Joseph, H. Krishnan, F. R. N. C. Maia, A. L. D. Kilcoyne, S. Marchesini, T. P. C. Leite, T. Warwick, H. Padmore, J. Cabana, and D. A. Shapiro, “Three-dimensional localization of nanoscale battery reactions using soft x-ray tomography,” Nat. Commun. 9, 921 (2018).
[Crossref] [PubMed]

Kim, C.

Y.-S. Yu, M. Farmand, C. Kim, Y. Liu, C. P. Grey, F. C. Strobridge, T. Tyliszczak, R. Celestre, P. Denes, J. Joseph, H. Krishnan, F. R. N. C. Maia, A. L. D. Kilcoyne, S. Marchesini, T. P. C. Leite, T. Warwick, H. Padmore, J. Cabana, and D. A. Shapiro, “Three-dimensional localization of nanoscale battery reactions using soft x-ray tomography,” Nat. Commun. 9, 921 (2018).
[Crossref] [PubMed]

Kirby, N. M.

N. M. Kirby, S. T. Mudie, A. M. Hawley, D. J. Cookson, H. D. Mertens, N. Cowieson, and V. Samardzic-Boban, “A low-background-intensity focusing small-angle x-ray scattering undulator beamline,” J. Appl. Crystallogr. 46, 1670–1680 (2013).
[Crossref]

Kirian, R. A.

M. O. Wiedorn, S. Awel, A. J. Morgan, M. Barthelmess, R. Bean, K. R. Beyerlein, L. M. G. Chavas, N. Eckerskorn, H. Fleckenstein, M. Heymann, D. A. Horke, J. Knoška, V. Mariani, D. Oberthür, N. Roth, O. Yefanov, A. Barty, S. Bajt, J. Küpper, A. V. Rode, R. A. Kirian, and H. N. Chapman, “Post-sample aperture for low background diffraction experiments at x-ray free-electron lasers,” J. Synchrotron Radiat. 24, 1296–1298 (2017).
[Crossref] [PubMed]

Knoška, J.

M. O. Wiedorn, S. Awel, A. J. Morgan, M. Barthelmess, R. Bean, K. R. Beyerlein, L. M. G. Chavas, N. Eckerskorn, H. Fleckenstein, M. Heymann, D. A. Horke, J. Knoška, V. Mariani, D. Oberthür, N. Roth, O. Yefanov, A. Barty, S. Bajt, J. Küpper, A. V. Rode, R. A. Kirian, and H. N. Chapman, “Post-sample aperture for low background diffraction experiments at x-ray free-electron lasers,” J. Synchrotron Radiat. 24, 1296–1298 (2017).
[Crossref] [PubMed]

Krishnan, H.

Y.-S. Yu, M. Farmand, C. Kim, Y. Liu, C. P. Grey, F. C. Strobridge, T. Tyliszczak, R. Celestre, P. Denes, J. Joseph, H. Krishnan, F. R. N. C. Maia, A. L. D. Kilcoyne, S. Marchesini, T. P. C. Leite, T. Warwick, H. Padmore, J. Cabana, and D. A. Shapiro, “Three-dimensional localization of nanoscale battery reactions using soft x-ray tomography,” Nat. Commun. 9, 921 (2018).
[Crossref] [PubMed]

B. J. Daurer, H. Krishnan, T. Perciano, F. R. Maia, D. A. Shapiro, J. A. Sethian, and S. Marchesini, “Nanosurveyor: a framework for real-time data processing,” Adv. Struct. Chem. Imaging 3, 7 (2017).
[Crossref] [PubMed]

S. Marchesini, H. Krishnan, D. A. Shapiro, T. Perciano, J. A. Sethian, B. J. Daurer, and F. R. Maia, “SHARP: a distributed, GPU-based ptychographic solver,” J. Appl. Crystallogr. 49, 1245–1252 (2016).
[Crossref]

Küpper, J.

M. O. Wiedorn, S. Awel, A. J. Morgan, M. Barthelmess, R. Bean, K. R. Beyerlein, L. M. G. Chavas, N. Eckerskorn, H. Fleckenstein, M. Heymann, D. A. Horke, J. Knoška, V. Mariani, D. Oberthür, N. Roth, O. Yefanov, A. Barty, S. Bajt, J. Küpper, A. V. Rode, R. A. Kirian, and H. N. Chapman, “Post-sample aperture for low background diffraction experiments at x-ray free-electron lasers,” J. Synchrotron Radiat. 24, 1296–1298 (2017).
[Crossref] [PubMed]

Le Tallec, P.

R. Glowinski and P. Le Tallec, Augmented Lagrangian and operator-splitting methods in nonlinear mechanics (SIAM, 1989).
[Crossref]

Lee, J.

X. Shi, P. Fischer, V. Neu, D. Elefant, J. Lee, D. Shapiro, M. Farmand, T. Tyliszczak, H.-W. Shiu, S. Marchesini, and S. Roy, “Soft x-ray ptychography studies of nanoscale magnetic and structural correlations in thin SmCo5 films,” Appl. Phys. Lett. 108, 094103 (2016).
[Crossref]

Leite, T. P. C.

Y.-S. Yu, M. Farmand, C. Kim, Y. Liu, C. P. Grey, F. C. Strobridge, T. Tyliszczak, R. Celestre, P. Denes, J. Joseph, H. Krishnan, F. R. N. C. Maia, A. L. D. Kilcoyne, S. Marchesini, T. P. C. Leite, T. Warwick, H. Padmore, J. Cabana, and D. A. Shapiro, “Three-dimensional localization of nanoscale battery reactions using soft x-ray tomography,” Nat. Commun. 9, 921 (2018).
[Crossref] [PubMed]

Li, J.

J. Li, Z. Shen, R. Yin, and X. Zhang, “A reweighted l2 method for image restoration with poisson and mixed poisson-gaussian noise,” Inverse Probl. Imaging (Springfield) 9, 875–894 (2015).
[Crossref]

Liu, H.

Liu, X.

Z. Wen, C. Yang, X. Liu, and S. Marchesini, “Alternating direction methods for classical and ptychographic phase retrieval,” Inverse Probl. 28, 115010 (2012).
[Crossref]

Liu, Y.

Y.-S. Yu, M. Farmand, C. Kim, Y. Liu, C. P. Grey, F. C. Strobridge, T. Tyliszczak, R. Celestre, P. Denes, J. Joseph, H. Krishnan, F. R. N. C. Maia, A. L. D. Kilcoyne, S. Marchesini, T. P. C. Leite, T. Warwick, H. Padmore, J. Cabana, and D. A. Shapiro, “Three-dimensional localization of nanoscale battery reactions using soft x-ray tomography,” Nat. Commun. 9, 921 (2018).
[Crossref] [PubMed]

Lou, Y.

H. Chang, Y. Lou, Y. Duan, and S. Marchesini, “Total variation–based phase retrieval for Poisson noise removal,” SIAM J. Imaging Sci. 11, 24–55 (2018).
[Crossref]

H. Chang, P. Enfedaque, Y. Lou, and S. Marchesini, “Partially coherent ptychography by gradient decomposition of the probe,” Acta Crystallogr., Sect. A: Found. Adv. 74, 157–169 (2018).
[Crossref]

H. Chang, S. Marchesini, Y. Lou, and T. Zeng, “Variational phase retrieval with globally convergent preconditioned proximal algorithm,” SIAM J. Imaging Sci. 11, 56–93 (2018).
[Crossref]

H. Chang, Y. Lou, M. K. Ng, and T. Zeng, “Phase retrieval from incomplete magnitude information via total variation regularization,” SIAM J. Sci. Comput. 38, A3672–A3695 (2016).
[Crossref]

Luke, D. R.

R. Hesse, D. R. Luke, S. Sabach, and M. K. Tam, “Proximal heterogeneous block implicit-explicit method and application to blind ptychographic diffraction imaging,” SIAM J. Imaging Sci. 8, 426–457 (2015).
[Crossref]

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

Maia, F.

D. A. Shapiro, Y.-S. Yu, T. Tyliszczak, J. Cabana, R. Celestre, W. Chao, K. Kaznatcheev, A. D. Kilcoyne, F. Maia, S. Marchesini, Y. S. Meng, T. Warwick, L. L. Yang, and H. Padmore, “Chemical composition mapping with nanometre resolution by soft x-ray microscopy,” Nat. Photonics 8, 765–769 (2014).
[Crossref]

S. Marchesini, A. Schirotzek, C. Yang, H.-T. Wu, and F. Maia, “Augmented projections for ptychographic imaging,” Inverse Probl. 29, 115009 (2013).
[Crossref]

Maia, F. R.

B. J. Daurer, H. Krishnan, T. Perciano, F. R. Maia, D. A. Shapiro, J. A. Sethian, and S. Marchesini, “Nanosurveyor: a framework for real-time data processing,” Adv. Struct. Chem. Imaging 3, 7 (2017).
[Crossref] [PubMed]

S. Marchesini, H. Krishnan, D. A. Shapiro, T. Perciano, J. A. Sethian, B. J. Daurer, and F. R. Maia, “SHARP: a distributed, GPU-based ptychographic solver,” J. Appl. Crystallogr. 49, 1245–1252 (2016).
[Crossref]

Maia, F. R. N. C.

Y.-S. Yu, M. Farmand, C. Kim, Y. Liu, C. P. Grey, F. C. Strobridge, T. Tyliszczak, R. Celestre, P. Denes, J. Joseph, H. Krishnan, F. R. N. C. Maia, A. L. D. Kilcoyne, S. Marchesini, T. P. C. Leite, T. Warwick, H. Padmore, J. Cabana, and D. A. Shapiro, “Three-dimensional localization of nanoscale battery reactions using soft x-ray tomography,” Nat. Commun. 9, 921 (2018).
[Crossref] [PubMed]

Maiden, A. M.

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

Marchesini, S.

H. Chang, P. Enfedaque, and S. Marchesini, “Blind ptychographic phase retrieval via convergent alternating direction method of multipliers,” SIAM J. Imaging Sci. 12, 153–185 (2019).
[Crossref]

H. Chang and S. Marchesini, “Denoising Poisson phaseless measurements via orthogonal dictionary learning,” Opt. Express 26, 19773–19796 (2018).
[Crossref] [PubMed]

H. Chang, P. Enfedaque, Y. Lou, and S. Marchesini, “Partially coherent ptychography by gradient decomposition of the probe,” Acta Crystallogr., Sect. A: Found. Adv. 74, 157–169 (2018).
[Crossref]

H. Chang, S. Marchesini, Y. Lou, and T. Zeng, “Variational phase retrieval with globally convergent preconditioned proximal algorithm,” SIAM J. Imaging Sci. 11, 56–93 (2018).
[Crossref]

H. Chang, Y. Lou, Y. Duan, and S. Marchesini, “Total variation–based phase retrieval for Poisson noise removal,” SIAM J. Imaging Sci. 11, 24–55 (2018).
[Crossref]

Y.-S. Yu, M. Farmand, C. Kim, Y. Liu, C. P. Grey, F. C. Strobridge, T. Tyliszczak, R. Celestre, P. Denes, J. Joseph, H. Krishnan, F. R. N. C. Maia, A. L. D. Kilcoyne, S. Marchesini, T. P. C. Leite, T. Warwick, H. Padmore, J. Cabana, and D. A. Shapiro, “Three-dimensional localization of nanoscale battery reactions using soft x-ray tomography,” Nat. Commun. 9, 921 (2018).
[Crossref] [PubMed]

B. J. Daurer, H. Krishnan, T. Perciano, F. R. Maia, D. A. Shapiro, J. A. Sethian, and S. Marchesini, “Nanosurveyor: a framework for real-time data processing,” Adv. Struct. Chem. Imaging 3, 7 (2017).
[Crossref] [PubMed]

X. Shi, P. Fischer, V. Neu, D. Elefant, J. Lee, D. Shapiro, M. Farmand, T. Tyliszczak, H.-W. Shiu, S. Marchesini, and S. Roy, “Soft x-ray ptychography studies of nanoscale magnetic and structural correlations in thin SmCo5 films,” Appl. Phys. Lett. 108, 094103 (2016).
[Crossref]

S. Marchesini, H. Krishnan, D. A. Shapiro, T. Perciano, J. A. Sethian, B. J. Daurer, and F. R. Maia, “SHARP: a distributed, GPU-based ptychographic solver,” J. Appl. Crystallogr. 49, 1245–1252 (2016).
[Crossref]

D. A. Shapiro, Y.-S. Yu, T. Tyliszczak, J. Cabana, R. Celestre, W. Chao, K. Kaznatcheev, A. D. Kilcoyne, F. Maia, S. Marchesini, Y. S. Meng, T. Warwick, L. L. Yang, and H. Padmore, “Chemical composition mapping with nanometre resolution by soft x-ray microscopy,” Nat. Photonics 8, 765–769 (2014).
[Crossref]

S. Marchesini, A. Schirotzek, C. Yang, H.-T. Wu, and F. Maia, “Augmented projections for ptychographic imaging,” Inverse Probl. 29, 115009 (2013).
[Crossref]

Z. Wen, C. Yang, X. Liu, and S. Marchesini, “Alternating direction methods for classical and ptychographic phase retrieval,” Inverse Probl. 28, 115010 (2012).
[Crossref]

H. Chang and S. Marchesini, “A general framework for denoising phaseless diffraction measurements,” arXiv 1611.01417 (2016).

Mariani, V.

M. O. Wiedorn, S. Awel, A. J. Morgan, M. Barthelmess, R. Bean, K. R. Beyerlein, L. M. G. Chavas, N. Eckerskorn, H. Fleckenstein, M. Heymann, D. A. Horke, J. Knoška, V. Mariani, D. Oberthür, N. Roth, O. Yefanov, A. Barty, S. Bajt, J. Küpper, A. V. Rode, R. A. Kirian, and H. N. Chapman, “Post-sample aperture for low background diffraction experiments at x-ray free-electron lasers,” J. Synchrotron Radiat. 24, 1296–1298 (2017).
[Crossref] [PubMed]

McCallum, B.

P. Nellist, B. McCallum, and J. Rodenburg, “Resolution beyond the ‘information limit’ in transmission electron microscopy,” Nature 374, 630 (1995).
[Crossref]

Meng, Y. S.

D. A. Shapiro, Y.-S. Yu, T. Tyliszczak, J. Cabana, R. Celestre, W. Chao, K. Kaznatcheev, A. D. Kilcoyne, F. Maia, S. Marchesini, Y. S. Meng, T. Warwick, L. L. Yang, and H. Padmore, “Chemical composition mapping with nanometre resolution by soft x-ray microscopy,” Nat. Photonics 8, 765–769 (2014).
[Crossref]

Menzel, A.

M. Odstrčil, A. Menzel, and M. Guizar-Sicairos, “Iterative least-squares solver for generalized maximum-likelihood ptychography,” Opt. Express 26, 3108–3123 (2018).
[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]

Mertens, H. D.

N. M. Kirby, S. T. Mudie, A. M. Hawley, D. J. Cookson, H. D. Mertens, N. Cowieson, and V. Samardzic-Boban, “A low-background-intensity focusing small-angle x-ray scattering undulator beamline,” J. Appl. Crystallogr. 46, 1670–1680 (2013).
[Crossref]

Morgan, A. J.

M. O. Wiedorn, S. Awel, A. J. Morgan, M. Barthelmess, R. Bean, K. R. Beyerlein, L. M. G. Chavas, N. Eckerskorn, H. Fleckenstein, M. Heymann, D. A. Horke, J. Knoška, V. Mariani, D. Oberthür, N. Roth, O. Yefanov, A. Barty, S. Bajt, J. Küpper, A. V. Rode, R. A. Kirian, and H. N. Chapman, “Post-sample aperture for low background diffraction experiments at x-ray free-electron lasers,” J. Synchrotron Radiat. 24, 1296–1298 (2017).
[Crossref] [PubMed]

Mudie, S. T.

N. M. Kirby, S. T. Mudie, A. M. Hawley, D. J. Cookson, H. D. Mertens, N. Cowieson, and V. Samardzic-Boban, “A low-background-intensity focusing small-angle x-ray scattering undulator beamline,” J. Appl. Crystallogr. 46, 1670–1680 (2013).
[Crossref]

Muller, D. A.

Y. Jiang, Z. Chen, Y. Han, P. Deb, H. Gao, S. Xie, P. Purohit, M. W. Tate, J. Park, S. M. Gruner, V. Elser, and D. A. Muller, “Electron ptychography of 2d materials to deep sub-Ångström resolution,” Nature 559, 343 (2018).
[Crossref] [PubMed]

Müller, E.

M. Holler, M. Guizar-Sicairos, E. H. Tsai, R. Dinapoli, E. Müller, O. Bunk, J. Raabe, and G. Aeppli, “High-resolution non-destructive three-dimensional imaging of integrated circuits,” Nature 543, 402–406 (2017).
[Crossref] [PubMed]

Murtagh, F.

F. Murtagh, J.-L. Starck, and A. Bijaoui, “Image restoration with noise suppression using a multiresolution support,” Astron. Astrophys., Suppl. Ser. 112, 179 (1995).

Nashed, Y. S.

M. J. Cherukara, Y. S. Nashed, and R. J. Harder, “Real-time coherent diffraction inversion using deep generative networks,” arXiv 1806.03992 (2018).

Nellist, P.

P. Nellist, B. McCallum, and J. Rodenburg, “Resolution beyond the ‘information limit’ in transmission electron microscopy,” Nature 374, 630 (1995).
[Crossref]

Neu, V.

X. Shi, P. Fischer, V. Neu, D. Elefant, J. Lee, D. Shapiro, M. Farmand, T. Tyliszczak, H.-W. Shiu, S. Marchesini, and S. Roy, “Soft x-ray ptychography studies of nanoscale magnetic and structural correlations in thin SmCo5 films,” Appl. Phys. Lett. 108, 094103 (2016).
[Crossref]

Ng, M. K.

H. Chang, Y. Lou, M. K. Ng, and T. Zeng, “Phase retrieval from incomplete magnitude information via total variation regularization,” SIAM J. Sci. Comput. 38, A3672–A3695 (2016).
[Crossref]

Oberthür, D.

M. O. Wiedorn, S. Awel, A. J. Morgan, M. Barthelmess, R. Bean, K. R. Beyerlein, L. M. G. Chavas, N. Eckerskorn, H. Fleckenstein, M. Heymann, D. A. Horke, J. Knoška, V. Mariani, D. Oberthür, N. Roth, O. Yefanov, A. Barty, S. Bajt, J. Küpper, A. V. Rode, R. A. Kirian, and H. N. Chapman, “Post-sample aperture for low background diffraction experiments at x-ray free-electron lasers,” J. Synchrotron Radiat. 24, 1296–1298 (2017).
[Crossref] [PubMed]

Odstrcil, M.

Padmore, H.

Y.-S. Yu, M. Farmand, C. Kim, Y. Liu, C. P. Grey, F. C. Strobridge, T. Tyliszczak, R. Celestre, P. Denes, J. Joseph, H. Krishnan, F. R. N. C. Maia, A. L. D. Kilcoyne, S. Marchesini, T. P. C. Leite, T. Warwick, H. Padmore, J. Cabana, and D. A. Shapiro, “Three-dimensional localization of nanoscale battery reactions using soft x-ray tomography,” Nat. Commun. 9, 921 (2018).
[Crossref] [PubMed]

D. A. Shapiro, Y.-S. Yu, T. Tyliszczak, J. Cabana, R. Celestre, W. Chao, K. Kaznatcheev, A. D. Kilcoyne, F. Maia, S. Marchesini, Y. S. Meng, T. Warwick, L. L. Yang, and H. Padmore, “Chemical composition mapping with nanometre resolution by soft x-ray microscopy,” Nat. Photonics 8, 765–769 (2014).
[Crossref]

Parikh, N.

S. Boyd, N. Parikh, E. Chu, B. Peleato, and J. Eckstein, “Distributed optimization and statistical learning via the alternating direction method of multipliers,” Found. Trends Mach. Learn. 3, 1–122 (2011).
[Crossref]

Park, J.

Y. Jiang, Z. Chen, Y. Han, P. Deb, H. Gao, S. Xie, P. Purohit, M. W. Tate, J. Park, S. M. Gruner, V. Elser, and D. A. Muller, “Electron ptychography of 2d materials to deep sub-Ångström resolution,” Nature 559, 343 (2018).
[Crossref] [PubMed]

Patommel, J.

J. Reinhardt, R. Hoppe, G. Hofmann, C. D. Damsgaard, J. Patommel, C. Baumbach, S. Baier, A. Rochet, J.-D. Grunwaldt, G. Falkenberg, and C. G. Schroer, “Beamstop-based low-background ptychography to image weakly scattering objects,” Ultramicroscopy 173, 52–57 (2017).
[Crossref]

Peleato, B.

S. Boyd, N. Parikh, E. Chu, B. Peleato, and J. Eckstein, “Distributed optimization and statistical learning via the alternating direction method of multipliers,” Found. Trends Mach. Learn. 3, 1–122 (2011).
[Crossref]

Perciano, T.

B. J. Daurer, H. Krishnan, T. Perciano, F. R. Maia, D. A. Shapiro, J. A. Sethian, and S. Marchesini, “Nanosurveyor: a framework for real-time data processing,” Adv. Struct. Chem. Imaging 3, 7 (2017).
[Crossref] [PubMed]

S. Marchesini, H. Krishnan, D. A. Shapiro, T. Perciano, J. A. Sethian, B. J. Daurer, and F. R. Maia, “SHARP: a distributed, GPU-based ptychographic solver,” J. Appl. Crystallogr. 49, 1245–1252 (2016).
[Crossref]

Pfeiffer, F.

B. Enders, M. Dierolf, P. Cloetens, M. Stockmar, F. Pfeiffer, and P. Thibault, “Ptychography with broad-bandwidth radiation,” Appl. Phys. Lett. 104, 171104 (2014).
[Crossref]

K. Giewekemeyer, P. Thibault, S. Kalbfleisch, A. Beerlink, C. M. Kewish, M. Dierolf, F. Pfeiffer, and T. Salditt, “Quantitative biological imaging by ptychographic x-ray diffraction microscopy,” Proc. Natl. Acad. Sci. U.S.A. 107, 529–534 (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]

Purohit, P.

Y. Jiang, Z. Chen, Y. Han, P. Deb, H. Gao, S. Xie, P. Purohit, M. W. Tate, J. Park, S. M. Gruner, V. Elser, and D. A. Muller, “Electron ptychography of 2d materials to deep sub-Ångström resolution,” Nature 559, 343 (2018).
[Crossref] [PubMed]

Raabe, J.

M. Holler, M. Guizar-Sicairos, E. H. Tsai, R. Dinapoli, E. Müller, O. Bunk, J. Raabe, and G. Aeppli, “High-resolution non-destructive three-dimensional imaging of integrated circuits,” Nature 543, 402–406 (2017).
[Crossref] [PubMed]

Reinhardt, J.

J. Reinhardt, R. Hoppe, G. Hofmann, C. D. Damsgaard, J. Patommel, C. Baumbach, S. Baier, A. Rochet, J.-D. Grunwaldt, G. Falkenberg, and C. G. Schroer, “Beamstop-based low-background ptychography to image weakly scattering objects,” Ultramicroscopy 173, 52–57 (2017).
[Crossref]

Rochet, A.

J. Reinhardt, R. Hoppe, G. Hofmann, C. D. Damsgaard, J. Patommel, C. Baumbach, S. Baier, A. Rochet, J.-D. Grunwaldt, G. Falkenberg, and C. G. Schroer, “Beamstop-based low-background ptychography to image weakly scattering objects,” Ultramicroscopy 173, 52–57 (2017).
[Crossref]

Rode, A. V.

M. O. Wiedorn, S. Awel, A. J. Morgan, M. Barthelmess, R. Bean, K. R. Beyerlein, L. M. G. Chavas, N. Eckerskorn, H. Fleckenstein, M. Heymann, D. A. Horke, J. Knoška, V. Mariani, D. Oberthür, N. Roth, O. Yefanov, A. Barty, S. Bajt, J. Küpper, A. V. Rode, R. A. Kirian, and H. N. Chapman, “Post-sample aperture for low background diffraction experiments at x-ray free-electron lasers,” J. Synchrotron Radiat. 24, 1296–1298 (2017).
[Crossref] [PubMed]

Rodenburg, J.

P. Godard, M. Allain, V. Chamard, and J. Rodenburg, “Noise models for low counting rate coherent diffraction imaging,” Opt. Express 20, 25914–25934 (2012).
[Crossref] [PubMed]

P. Nellist, B. McCallum, and J. Rodenburg, “Resolution beyond the ‘information limit’ in transmission electron microscopy,” Nature 374, 630 (1995).
[Crossref]

Rodenburg, J. M.

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

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

Roth, N.

M. O. Wiedorn, S. Awel, A. J. Morgan, M. Barthelmess, R. Bean, K. R. Beyerlein, L. M. G. Chavas, N. Eckerskorn, H. Fleckenstein, M. Heymann, D. A. Horke, J. Knoška, V. Mariani, D. Oberthür, N. Roth, O. Yefanov, A. Barty, S. Bajt, J. Küpper, A. V. Rode, R. A. Kirian, and H. N. Chapman, “Post-sample aperture for low background diffraction experiments at x-ray free-electron lasers,” J. Synchrotron Radiat. 24, 1296–1298 (2017).
[Crossref] [PubMed]

Roy, S.

X. Shi, P. Fischer, V. Neu, D. Elefant, J. Lee, D. Shapiro, M. Farmand, T. Tyliszczak, H.-W. Shiu, S. Marchesini, and S. Roy, “Soft x-ray ptychography studies of nanoscale magnetic and structural correlations in thin SmCo5 films,” Appl. Phys. Lett. 108, 094103 (2016).
[Crossref]

Sabach, S.

R. Hesse, D. R. Luke, S. Sabach, and M. K. Tam, “Proximal heterogeneous block implicit-explicit method and application to blind ptychographic diffraction imaging,” SIAM J. Imaging Sci. 8, 426–457 (2015).
[Crossref]

Salditt, T.

K. Giewekemeyer, P. Thibault, S. Kalbfleisch, A. Beerlink, C. M. Kewish, M. Dierolf, F. Pfeiffer, and T. Salditt, “Quantitative biological imaging by ptychographic x-ray diffraction microscopy,” Proc. Natl. Acad. Sci. U.S.A. 107, 529–534 (2010).
[Crossref]

Samardzic-Boban, V.

N. M. Kirby, S. T. Mudie, A. M. Hawley, D. J. Cookson, H. D. Mertens, N. Cowieson, and V. Samardzic-Boban, “A low-background-intensity focusing small-angle x-ray scattering undulator beamline,” J. Appl. Crystallogr. 46, 1670–1680 (2013).
[Crossref]

Schirotzek, A.

S. Marchesini, A. Schirotzek, C. Yang, H.-T. Wu, and F. Maia, “Augmented projections for ptychographic imaging,” Inverse Probl. 29, 115009 (2013).
[Crossref]

Schroer, C. G.

J. Reinhardt, R. Hoppe, G. Hofmann, C. D. Damsgaard, J. Patommel, C. Baumbach, S. Baier, A. Rochet, J.-D. Grunwaldt, G. Falkenberg, and C. G. Schroer, “Beamstop-based low-background ptychography to image weakly scattering objects,” Ultramicroscopy 173, 52–57 (2017).
[Crossref]

Sethian, J. A.

B. J. Daurer, H. Krishnan, T. Perciano, F. R. Maia, D. A. Shapiro, J. A. Sethian, and S. Marchesini, “Nanosurveyor: a framework for real-time data processing,” Adv. Struct. Chem. Imaging 3, 7 (2017).
[Crossref] [PubMed]

S. Marchesini, H. Krishnan, D. A. Shapiro, T. Perciano, J. A. Sethian, B. J. Daurer, and F. R. Maia, “SHARP: a distributed, GPU-based ptychographic solver,” J. Appl. Crystallogr. 49, 1245–1252 (2016).
[Crossref]

Shapiro, D.

X. Shi, P. Fischer, V. Neu, D. Elefant, J. Lee, D. Shapiro, M. Farmand, T. Tyliszczak, H.-W. Shiu, S. Marchesini, and S. Roy, “Soft x-ray ptychography studies of nanoscale magnetic and structural correlations in thin SmCo5 films,” Appl. Phys. Lett. 108, 094103 (2016).
[Crossref]

Shapiro, D. A.

Y.-S. Yu, M. Farmand, C. Kim, Y. Liu, C. P. Grey, F. C. Strobridge, T. Tyliszczak, R. Celestre, P. Denes, J. Joseph, H. Krishnan, F. R. N. C. Maia, A. L. D. Kilcoyne, S. Marchesini, T. P. C. Leite, T. Warwick, H. Padmore, J. Cabana, and D. A. Shapiro, “Three-dimensional localization of nanoscale battery reactions using soft x-ray tomography,” Nat. Commun. 9, 921 (2018).
[Crossref] [PubMed]

B. J. Daurer, H. Krishnan, T. Perciano, F. R. Maia, D. A. Shapiro, J. A. Sethian, and S. Marchesini, “Nanosurveyor: a framework for real-time data processing,” Adv. Struct. Chem. Imaging 3, 7 (2017).
[Crossref] [PubMed]

S. Marchesini, H. Krishnan, D. A. Shapiro, T. Perciano, J. A. Sethian, B. J. Daurer, and F. R. Maia, “SHARP: a distributed, GPU-based ptychographic solver,” J. Appl. Crystallogr. 49, 1245–1252 (2016).
[Crossref]

D. A. Shapiro, Y.-S. Yu, T. Tyliszczak, J. Cabana, R. Celestre, W. Chao, K. Kaznatcheev, A. D. Kilcoyne, F. Maia, S. Marchesini, Y. S. Meng, T. Warwick, L. L. Yang, and H. Padmore, “Chemical composition mapping with nanometre resolution by soft x-ray microscopy,” Nat. Photonics 8, 765–769 (2014).
[Crossref]

Shen, Z.

J. Li, Z. Shen, R. Yin, and X. Zhang, “A reweighted l2 method for image restoration with poisson and mixed poisson-gaussian noise,” Inverse Probl. Imaging (Springfield) 9, 875–894 (2015).
[Crossref]

Shi, X.

X. Shi, P. Fischer, V. Neu, D. Elefant, J. Lee, D. Shapiro, M. Farmand, T. Tyliszczak, H.-W. Shiu, S. Marchesini, and S. Roy, “Soft x-ray ptychography studies of nanoscale magnetic and structural correlations in thin SmCo5 films,” Appl. Phys. Lett. 108, 094103 (2016).
[Crossref]

Shiu, H.-W.

X. Shi, P. Fischer, V. Neu, D. Elefant, J. Lee, D. Shapiro, M. Farmand, T. Tyliszczak, H.-W. Shiu, S. Marchesini, and S. Roy, “Soft x-ray ptychography studies of nanoscale magnetic and structural correlations in thin SmCo5 films,” Appl. Phys. Lett. 108, 094103 (2016).
[Crossref]

Starck, J.-L.

F. Murtagh, J.-L. Starck, and A. Bijaoui, “Image restoration with noise suppression using a multiresolution support,” Astron. Astrophys., Suppl. Ser. 112, 179 (1995).

Stockmar, M.

B. Enders, M. Dierolf, P. Cloetens, M. Stockmar, F. Pfeiffer, and P. Thibault, “Ptychography with broad-bandwidth radiation,” Appl. Phys. Lett. 104, 171104 (2014).
[Crossref]

Strobridge, F. C.

Y.-S. Yu, M. Farmand, C. Kim, Y. Liu, C. P. Grey, F. C. Strobridge, T. Tyliszczak, R. Celestre, P. Denes, J. Joseph, H. Krishnan, F. R. N. C. Maia, A. L. D. Kilcoyne, S. Marchesini, T. P. C. Leite, T. Warwick, H. Padmore, J. Cabana, and D. A. Shapiro, “Three-dimensional localization of nanoscale battery reactions using soft x-ray tomography,” Nat. Commun. 9, 921 (2018).
[Crossref] [PubMed]

Tai, R.

Tai, X.-C.

C. Wu and X.-C. Tai, “Augmented Lagrangian method, dual methods and split-Bregman iterations for ROF, vectorial TV and higher order models,” SIAM J. Imaging Sci. 3, 300–339 (2010).
[Crossref]

Tam, M. K.

R. Hesse, D. R. Luke, S. Sabach, and M. K. Tam, “Proximal heterogeneous block implicit-explicit method and application to blind ptychographic diffraction imaging,” SIAM J. Imaging Sci. 8, 426–457 (2015).
[Crossref]

Tate, M. W.

Y. Jiang, Z. Chen, Y. Han, P. Deb, H. Gao, S. Xie, P. Purohit, M. W. Tate, J. Park, S. M. Gruner, V. Elser, and D. A. Muller, “Electron ptychography of 2d materials to deep sub-Ångström resolution,” Nature 559, 343 (2018).
[Crossref] [PubMed]

Thibault, P.

B. Enders, M. Dierolf, P. Cloetens, M. Stockmar, F. Pfeiffer, and P. Thibault, “Ptychography with broad-bandwidth radiation,” Appl. Phys. Lett. 104, 171104 (2014).
[Crossref]

P. Thibault and M. Guizar-Sicairos, “Maximum-likelihood refinement for coherent diffractive imaging,” New J. Phys. 14, 063004 (2012).
[Crossref]

K. Giewekemeyer, P. Thibault, S. Kalbfleisch, A. Beerlink, C. M. Kewish, M. Dierolf, F. Pfeiffer, and T. Salditt, “Quantitative biological imaging by ptychographic x-ray diffraction microscopy,” Proc. Natl. Acad. Sci. U.S.A. 107, 529–534 (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]

Tsai, E. H.

M. Holler, M. Guizar-Sicairos, E. H. Tsai, R. Dinapoli, E. Müller, O. Bunk, J. Raabe, and G. Aeppli, “High-resolution non-destructive three-dimensional imaging of integrated circuits,” Nature 543, 402–406 (2017).
[Crossref] [PubMed]

Tyliszczak, T.

Y.-S. Yu, M. Farmand, C. Kim, Y. Liu, C. P. Grey, F. C. Strobridge, T. Tyliszczak, R. Celestre, P. Denes, J. Joseph, H. Krishnan, F. R. N. C. Maia, A. L. D. Kilcoyne, S. Marchesini, T. P. C. Leite, T. Warwick, H. Padmore, J. Cabana, and D. A. Shapiro, “Three-dimensional localization of nanoscale battery reactions using soft x-ray tomography,” Nat. Commun. 9, 921 (2018).
[Crossref] [PubMed]

X. Shi, P. Fischer, V. Neu, D. Elefant, J. Lee, D. Shapiro, M. Farmand, T. Tyliszczak, H.-W. Shiu, S. Marchesini, and S. Roy, “Soft x-ray ptychography studies of nanoscale magnetic and structural correlations in thin SmCo5 films,” Appl. Phys. Lett. 108, 094103 (2016).
[Crossref]

D. A. Shapiro, Y.-S. Yu, T. Tyliszczak, J. Cabana, R. Celestre, W. Chao, K. Kaznatcheev, A. D. Kilcoyne, F. Maia, S. Marchesini, Y. S. Meng, T. Warwick, L. L. Yang, and H. Padmore, “Chemical composition mapping with nanometre resolution by soft x-ray microscopy,” Nat. Photonics 8, 765–769 (2014).
[Crossref]

Wang, C.

Wang, J.

Wang, Y.

Warwick, T.

Y.-S. Yu, M. Farmand, C. Kim, Y. Liu, C. P. Grey, F. C. Strobridge, T. Tyliszczak, R. Celestre, P. Denes, J. Joseph, H. Krishnan, F. R. N. C. Maia, A. L. D. Kilcoyne, S. Marchesini, T. P. C. Leite, T. Warwick, H. Padmore, J. Cabana, and D. A. Shapiro, “Three-dimensional localization of nanoscale battery reactions using soft x-ray tomography,” Nat. Commun. 9, 921 (2018).
[Crossref] [PubMed]

D. A. Shapiro, Y.-S. Yu, T. Tyliszczak, J. Cabana, R. Celestre, W. Chao, K. Kaznatcheev, A. D. Kilcoyne, F. Maia, S. Marchesini, Y. S. Meng, T. Warwick, L. L. Yang, and H. Padmore, “Chemical composition mapping with nanometre resolution by soft x-ray microscopy,” Nat. Photonics 8, 765–769 (2014).
[Crossref]

Wen, Z.

Z. Wen, C. Yang, X. Liu, and S. Marchesini, “Alternating direction methods for classical and ptychographic phase retrieval,” Inverse Probl. 28, 115010 (2012).
[Crossref]

Wiedorn, M. O.

M. O. Wiedorn, S. Awel, A. J. Morgan, M. Barthelmess, R. Bean, K. R. Beyerlein, L. M. G. Chavas, N. Eckerskorn, H. Fleckenstein, M. Heymann, D. A. Horke, J. Knoška, V. Mariani, D. Oberthür, N. Roth, O. Yefanov, A. Barty, S. Bajt, J. Küpper, A. V. Rode, R. A. Kirian, and H. N. Chapman, “Post-sample aperture for low background diffraction experiments at x-ray free-electron lasers,” J. Synchrotron Radiat. 24, 1296–1298 (2017).
[Crossref] [PubMed]

Wu, C.

C. Wu and X.-C. Tai, “Augmented Lagrangian method, dual methods and split-Bregman iterations for ROF, vectorial TV and higher order models,” SIAM J. Imaging Sci. 3, 300–339 (2010).
[Crossref]

Wu, H.-T.

S. Marchesini, A. Schirotzek, C. Yang, H.-T. Wu, and F. Maia, “Augmented projections for ptychographic imaging,” Inverse Probl. 29, 115009 (2013).
[Crossref]

Xie, S.

Y. Jiang, Z. Chen, Y. Han, P. Deb, H. Gao, S. Xie, P. Purohit, M. W. Tate, J. Park, S. M. Gruner, V. Elser, and D. A. Muller, “Electron ptychography of 2d materials to deep sub-Ångström resolution,” Nature 559, 343 (2018).
[Crossref] [PubMed]

Xu, Z.

Yang, C.

S. Marchesini, A. Schirotzek, C. Yang, H.-T. Wu, and F. Maia, “Augmented projections for ptychographic imaging,” Inverse Probl. 29, 115009 (2013).
[Crossref]

Z. Wen, C. Yang, X. Liu, and S. Marchesini, “Alternating direction methods for classical and ptychographic phase retrieval,” Inverse Probl. 28, 115010 (2012).
[Crossref]

Yang, L. L.

D. A. Shapiro, Y.-S. Yu, T. Tyliszczak, J. Cabana, R. Celestre, W. Chao, K. Kaznatcheev, A. D. Kilcoyne, F. Maia, S. Marchesini, Y. S. Meng, T. Warwick, L. L. Yang, and H. Padmore, “Chemical composition mapping with nanometre resolution by soft x-ray microscopy,” Nat. Photonics 8, 765–769 (2014).
[Crossref]

Yefanov, O.

M. O. Wiedorn, S. Awel, A. J. Morgan, M. Barthelmess, R. Bean, K. R. Beyerlein, L. M. G. Chavas, N. Eckerskorn, H. Fleckenstein, M. Heymann, D. A. Horke, J. Knoška, V. Mariani, D. Oberthür, N. Roth, O. Yefanov, A. Barty, S. Bajt, J. Küpper, A. V. Rode, R. A. Kirian, and H. N. Chapman, “Post-sample aperture for low background diffraction experiments at x-ray free-electron lasers,” J. Synchrotron Radiat. 24, 1296–1298 (2017).
[Crossref] [PubMed]

Yin, R.

J. Li, Z. Shen, R. Yin, and X. Zhang, “A reweighted l2 method for image restoration with poisson and mixed poisson-gaussian noise,” Inverse Probl. Imaging (Springfield) 9, 875–894 (2015).
[Crossref]

Yu, Y.-S.

Y.-S. Yu, M. Farmand, C. Kim, Y. Liu, C. P. Grey, F. C. Strobridge, T. Tyliszczak, R. Celestre, P. Denes, J. Joseph, H. Krishnan, F. R. N. C. Maia, A. L. D. Kilcoyne, S. Marchesini, T. P. C. Leite, T. Warwick, H. Padmore, J. Cabana, and D. A. Shapiro, “Three-dimensional localization of nanoscale battery reactions using soft x-ray tomography,” Nat. Commun. 9, 921 (2018).
[Crossref] [PubMed]

D. A. Shapiro, Y.-S. Yu, T. Tyliszczak, J. Cabana, R. Celestre, W. Chao, K. Kaznatcheev, A. D. Kilcoyne, F. Maia, S. Marchesini, Y. S. Meng, T. Warwick, L. L. Yang, and H. Padmore, “Chemical composition mapping with nanometre resolution by soft x-ray microscopy,” Nat. Photonics 8, 765–769 (2014).
[Crossref]

Zeng, T.

H. Chang, S. Marchesini, Y. Lou, and T. Zeng, “Variational phase retrieval with globally convergent preconditioned proximal algorithm,” SIAM J. Imaging Sci. 11, 56–93 (2018).
[Crossref]

H. Chang, Y. Lou, M. K. Ng, and T. Zeng, “Phase retrieval from incomplete magnitude information via total variation regularization,” SIAM J. Sci. Comput. 38, A3672–A3695 (2016).
[Crossref]

Zhang, X.

J. Li, Z. Shen, R. Yin, and X. Zhang, “A reweighted l2 method for image restoration with poisson and mixed poisson-gaussian noise,” Inverse Probl. Imaging (Springfield) 9, 875–894 (2015).
[Crossref]

Zickler, T.

A. Chakrabarti and T. Zickler, “Image restoration with signal-dependent camera noise,” arXiv 1204.2994 (2012).

Acta Crystallogr., Sect. A: Found. Adv. (1)

H. Chang, P. Enfedaque, Y. Lou, and S. Marchesini, “Partially coherent ptychography by gradient decomposition of the probe,” Acta Crystallogr., Sect. A: Found. Adv. 74, 157–169 (2018).
[Crossref]

Adv. Struct. Chem. Imaging (1)

B. J. Daurer, H. Krishnan, T. Perciano, F. R. Maia, D. A. Shapiro, J. A. Sethian, and S. Marchesini, “Nanosurveyor: a framework for real-time data processing,” Adv. Struct. Chem. Imaging 3, 7 (2017).
[Crossref] [PubMed]

Appl. Opt. (1)

Appl. Phys. Lett. (3)

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

X. Shi, P. Fischer, V. Neu, D. Elefant, J. Lee, D. Shapiro, M. Farmand, T. Tyliszczak, H.-W. Shiu, S. Marchesini, and S. Roy, “Soft x-ray ptychography studies of nanoscale magnetic and structural correlations in thin SmCo5 films,” Appl. Phys. Lett. 108, 094103 (2016).
[Crossref]

B. Enders, M. Dierolf, P. Cloetens, M. Stockmar, F. Pfeiffer, and P. Thibault, “Ptychography with broad-bandwidth radiation,” Appl. Phys. Lett. 104, 171104 (2014).
[Crossref]

Astron. Astrophys., Suppl. Ser. (1)

F. Murtagh, J.-L. Starck, and A. Bijaoui, “Image restoration with noise suppression using a multiresolution support,” Astron. Astrophys., Suppl. Ser. 112, 179 (1995).

Found. Trends Mach. Learn. (1)

S. Boyd, N. Parikh, E. Chu, B. Peleato, and J. Eckstein, “Distributed optimization and statistical learning via the alternating direction method of multipliers,” Found. Trends Mach. Learn. 3, 1–122 (2011).
[Crossref]

Inverse Probl. (3)

S. Marchesini, A. Schirotzek, C. Yang, H.-T. Wu, and F. Maia, “Augmented projections for ptychographic imaging,” Inverse Probl. 29, 115009 (2013).
[Crossref]

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

Z. Wen, C. Yang, X. Liu, and S. Marchesini, “Alternating direction methods for classical and ptychographic phase retrieval,” Inverse Probl. 28, 115010 (2012).
[Crossref]

Inverse Probl. Imaging (Springfield) (1)

J. Li, Z. Shen, R. Yin, and X. Zhang, “A reweighted l2 method for image restoration with poisson and mixed poisson-gaussian noise,” Inverse Probl. Imaging (Springfield) 9, 875–894 (2015).
[Crossref]

J. Appl. Crystallogr. (2)

N. M. Kirby, S. T. Mudie, A. M. Hawley, D. J. Cookson, H. D. Mertens, N. Cowieson, and V. Samardzic-Boban, “A low-background-intensity focusing small-angle x-ray scattering undulator beamline,” J. Appl. Crystallogr. 46, 1670–1680 (2013).
[Crossref]

S. Marchesini, H. Krishnan, D. A. Shapiro, T. Perciano, J. A. Sethian, B. J. Daurer, and F. R. Maia, “SHARP: a distributed, GPU-based ptychographic solver,” J. Appl. Crystallogr. 49, 1245–1252 (2016).
[Crossref]

J. Opt. Soc. Am. A (1)

J. Synchrotron Radiat. (1)

M. O. Wiedorn, S. Awel, A. J. Morgan, M. Barthelmess, R. Bean, K. R. Beyerlein, L. M. G. Chavas, N. Eckerskorn, H. Fleckenstein, M. Heymann, D. A. Horke, J. Knoška, V. Mariani, D. Oberthür, N. Roth, O. Yefanov, A. Barty, S. Bajt, J. Küpper, A. V. Rode, R. A. Kirian, and H. N. Chapman, “Post-sample aperture for low background diffraction experiments at x-ray free-electron lasers,” J. Synchrotron Radiat. 24, 1296–1298 (2017).
[Crossref] [PubMed]

Nat. Commun. (1)

Y.-S. Yu, M. Farmand, C. Kim, Y. Liu, C. P. Grey, F. C. Strobridge, T. Tyliszczak, R. Celestre, P. Denes, J. Joseph, H. Krishnan, F. R. N. C. Maia, A. L. D. Kilcoyne, S. Marchesini, T. P. C. Leite, T. Warwick, H. Padmore, J. Cabana, and D. A. Shapiro, “Three-dimensional localization of nanoscale battery reactions using soft x-ray tomography,” Nat. Commun. 9, 921 (2018).
[Crossref] [PubMed]

Nat. Photonics (1)

D. A. Shapiro, Y.-S. Yu, T. Tyliszczak, J. Cabana, R. Celestre, W. Chao, K. Kaznatcheev, A. D. Kilcoyne, F. Maia, S. Marchesini, Y. S. Meng, T. Warwick, L. L. Yang, and H. Padmore, “Chemical composition mapping with nanometre resolution by soft x-ray microscopy,” Nat. Photonics 8, 765–769 (2014).
[Crossref]

Nature (3)

M. Holler, M. Guizar-Sicairos, E. H. Tsai, R. Dinapoli, E. Müller, O. Bunk, J. Raabe, and G. Aeppli, “High-resolution non-destructive three-dimensional imaging of integrated circuits,” Nature 543, 402–406 (2017).
[Crossref] [PubMed]

P. Nellist, B. McCallum, and J. Rodenburg, “Resolution beyond the ‘information limit’ in transmission electron microscopy,” Nature 374, 630 (1995).
[Crossref]

Y. Jiang, Z. Chen, Y. Han, P. Deb, H. Gao, S. Xie, P. Purohit, M. W. Tate, J. Park, S. M. Gruner, V. Elser, and D. A. Muller, “Electron ptychography of 2d materials to deep sub-Ångström resolution,” Nature 559, 343 (2018).
[Crossref] [PubMed]

New J. Phys. (1)

P. Thibault and M. Guizar-Sicairos, “Maximum-likelihood refinement for coherent diffractive imaging,” New J. Phys. 14, 063004 (2012).
[Crossref]

Opt. Express (3)

Proc. Natl. Acad. Sci. U.S.A. (1)

K. Giewekemeyer, P. Thibault, S. Kalbfleisch, A. Beerlink, C. M. Kewish, M. Dierolf, F. Pfeiffer, and T. Salditt, “Quantitative biological imaging by ptychographic x-ray diffraction microscopy,” Proc. Natl. Acad. Sci. U.S.A. 107, 529–534 (2010).
[Crossref]

SIAM J. Imaging Sci. (5)

R. Hesse, D. R. Luke, S. Sabach, and M. K. Tam, “Proximal heterogeneous block implicit-explicit method and application to blind ptychographic diffraction imaging,” SIAM J. Imaging Sci. 8, 426–457 (2015).
[Crossref]

H. Chang, P. Enfedaque, and S. Marchesini, “Blind ptychographic phase retrieval via convergent alternating direction method of multipliers,” SIAM J. Imaging Sci. 12, 153–185 (2019).
[Crossref]

C. Wu and X.-C. Tai, “Augmented Lagrangian method, dual methods and split-Bregman iterations for ROF, vectorial TV and higher order models,” SIAM J. Imaging Sci. 3, 300–339 (2010).
[Crossref]

H. Chang, S. Marchesini, Y. Lou, and T. Zeng, “Variational phase retrieval with globally convergent preconditioned proximal algorithm,” SIAM J. Imaging Sci. 11, 56–93 (2018).
[Crossref]

H. Chang, Y. Lou, Y. Duan, and S. Marchesini, “Total variation–based phase retrieval for Poisson noise removal,” SIAM J. Imaging Sci. 11, 24–55 (2018).
[Crossref]

SIAM J. Sci. Comput. (1)

H. Chang, Y. Lou, M. K. Ng, and T. Zeng, “Phase retrieval from incomplete magnitude information via total variation regularization,” SIAM J. Sci. Comput. 38, A3672–A3695 (2016).
[Crossref]

Ultramicroscopy (3)

J. Reinhardt, R. Hoppe, G. Hofmann, C. D. Damsgaard, J. Patommel, C. Baumbach, S. Baier, A. Rochet, J.-D. Grunwaldt, G. Falkenberg, and C. G. Schroer, “Beamstop-based low-background ptychography to image weakly scattering objects,” Ultramicroscopy 173, 52–57 (2017).
[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]

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

Other (4)

A. Chakrabarti and T. Zickler, “Image restoration with signal-dependent camera noise,” arXiv 1204.2994 (2012).

R. Glowinski and P. Le Tallec, Augmented Lagrangian and operator-splitting methods in nonlinear mechanics (SIAM, 1989).
[Crossref]

H. Chang and S. Marchesini, “A general framework for denoising phaseless diffraction measurements,” arXiv 1611.01417 (2016).

M. J. Cherukara, Y. S. Nashed, and R. J. Harder, “Real-time coherent diffraction inversion using deep generative networks,” arXiv 1806.03992 (2018).

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Figures (9)

Fig. 1
Fig. 1 Reconstruction results using RAAR (implemented in SHARP [19]) and the proposed algorithm. The dataset corresponds to one of the 3D ptychography experiments from the results presented in [8]. (a) Measured intensities from a single diffraction pattern. Recovered intensities (b), object amplitude (amp.) (d) and phase (e), respectively, using RAAR without structured noise correction. Recovered intensities (c), object amplitude (f) and phase (g), respectively, using the proposed algorithm (alg.). The figures in the first row are shown in scale (·)0.05 following the colorbar shown on the top right corner; the figures in the second row are shown in linear scale following the colorbars on the left corner (amplitude) and right corner (phase).
Fig. 2
Fig. 2 Simulation experiment. First row: (a)–(b) Amplitude and phase for true illumination (illu., 64 × 64 pixels), with Full Width at Half Max (FWHM) = 7 pixels. Second row: Amplitude of true image (c) with 496 × 496 pixels, and amplitude of recovered images with data contaminated by only parasitic noise, using SHARP (d), SHARP-B (e) and ADP (f). Third row: Amplitude of recovered images with data contaminated as in Eq. (32) and also by additional outliers, using SHARP (g), SHARP-B (h), ADP (i), and ADPr (j). The fourth, and fifth rows show the corresponding phase parts of the images in the second and third rows.
Fig. 3
Fig. 3 Retrieved backgrounds (backg.) for the simulation experiment of Fig. 2. First row: True background (parasitic noise) (a) and results retrieved in a simulation only with parasitic noise for SHARP-B (b) and ADP (c). Second row: Background retrieved in a simulation as in Eq. (32) with additional outliers for SHARP-B (d), ADP (e) and ADPr (f).The figures are shown in scale (·)0.05 with the colorbar shown on the top left corner
Fig. 4
Fig. 4 Soft X-ray experimental results from the data presented in [8]. First row: reconstructed amplitude using SHARP (a), SHARP-B (b), ADP (c) and ADPr (d). Second row: reconstructed phase using SHARP (e), SHARP-B (f), ADP (g) and ADPr (h).
Fig. 5
Fig. 5 Zoom-in view of parts of Fig. 4. First row: reconstructed amplitude using SHARP (a), SHARP-B (b), ADP (c) and ADPr (d). Second row: reconstructed phase using SHARP (e), SHARP-B (f), ADP (g) and ADPr (h).
Fig. 6
Fig. 6 Recovered intensities and background from the experiment presented in Fig. 4. First row: Recovered intensities by SHARP (a), SHARP-B (b), ADP (c), and ADPr (d). Second row: recovered backgrounds by SHARP-B (e), ADP (f), and ADPr (g). The figures are shown in scale (·)0.05 with the colorbar shown on the down left corner.
Fig. 7
Fig. 7 Cutlines of the retrieved amplitude (amp.) (a) and phase part (b) from the center of the reconstruction reported in Fig. 4, using the dataset from [8].
Fig. 8
Fig. 8 Convergence histories of the R-factor for the proposed algorithm (both ADP and ADPr) when performing the experiment reported in Fig. 4, using the dataset from [8].
Fig. 9
Fig. 9 Hard X-ray experimental results from the data presented in [11]. First row: recovered phase using ePIE, with the noBS dataset (a), and with the merged dataset (b). Second row: recovered phase using ADP, with noBS data (c), merged data (d), and BS data (e).Results from (a) to (d) are displayed in the range [−0.1, 0.02] (colorbar shown in the top right corner) while (e) is depicted in the range [−0.8, 0.3] (colorbar shown in the down right corner).

Tables (2)

Tables Icon

Algorithm 2: ADP with regularization (ADPr)

Equations (73)

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

To find ω m ¯ and u n , such that | 𝒜 ( ω , u ) | 2 = I ˜ ,
0 I ^ j = I ˜ j + ϕ ^ 0 j J 1 .
I ^ j = Noi ( I ˜ j + ϕ ^ ) 0 j J 1 ,
I ^ j ( t ) ~ i . i . d Poisson ( | 𝒜 j ( ω , u ) ( t ) | 2 + ϕ ^ ( t ) ) + n j ( t ) ,
min ω , u , ϕ 1 2 j 1 m ¯ , | 𝒜 j ( ω , u ) | 2 + ϕ I j log ( | 𝒜 j ( ω , u ) | 2 + ϕ ) + 𝕀 ( ϕ ) ,
min ω , u , ϕ 1 2 j = 0 J 1 | 𝒜 j ( ω , u ) | 2 + ϕ I j 2 + 𝕀 ( ϕ ) .
min ω , u , ϕ 1 2 j C j , | 𝒜 j ( ω , u ) | 2 + ϕ + ε 2 1 m ¯ ( I j + ε 2 1 m ¯ ) log ( | 𝒜 j ( ω , u ) | 2 + ϕ + ε 2 1 m ¯ ) + λ j TV ( 𝒮 j u ) + 𝕀 ( ϕ ) ,
min ω , u , ϕ 1 2 j C j , | 𝒜 j ( ω , u ) | 2 + ϕ + ε 2 1 m ¯ ( I j + ε 2 1 m ¯ ) log ( | 𝒜 j ( ω , u ) | 2 + ϕ + ε 2 1 m ¯ ) + 𝕀 ( ϕ ) .
min ω , u , ϕ ˜ 1 2 j C j , | 𝒜 j ( ω , u ) | 2 + ϕ ˜ 2 + ε 2 1 m ¯ ( I j + ε 2 1 m ¯ ) log ( | 𝒜 j ( ω , u ) | 2 + ϕ ˜ 2 + ε 2 1 m ¯ ) ,
min ω , u , z , μ , ϕ ˜ 𝒢 ε ( z , μ ) , such that z j 𝒜 j ( ω , u ) = 0 , μ j ϕ ˜ = 0 0 j J 1 ,
𝒢 ε ( z , μ ) : = 1 2 j C j , | z j | 2 + μ j 2 + ε 2 1 m ¯ ( I j + ε 2 1 m ¯ ) log ( | z j | 2 + μ j 2 + ε 2 1 m ¯ ) .
( ω , u , z , μ , ϕ ˜ , Λ 1 , Λ 2 ) : = 𝒢 ε ( z , μ ) + r j Λ 1 , j , z j 𝒜 j ( ω , u ) + r 2 j z j 𝒜 j ( ω , u ) 2 + r j Λ 2 , j , μ j ϕ ˜ + r 2 j μ j ϕ ˜ 2 ,
{ ω k + 1 = arg min ω ( ω , u k , z k , μ k , ϕ ˜ k , Λ 1 k , Λ 2 k ) + α 1 2 ω ω k 2 ; u k + 1 = arg min u ( ω k + 1 , u , z k , μ k , ϕ ˜ k , Λ 1 k , Λ 2 k ) + α 2 2 u u k 2 ; ( z k + 1 , μ k + 1 ) = arg min z , μ ( ω k + 1 , u k + 1 , z , μ , ϕ ˜ k , Λ 1 k , Λ 2 k ) ; ϕ ˜ k + 1 = arg min ϕ ˜ ( ω k + 1 , u k + 1 , z k + 1 , μ k + 1 , ϕ ˜ , Λ 1 k , Λ 2 k ) ; Λ 1 k + 1 = Λ 1 k + z k + 1 𝒜 ( ω k + 1 , u k + 1 ) ; Λ 2 , j k + 1 = Λ 2 , j k + μ j k + 1 ϕ ˜ k + 1 0 j J 1 ,
ω k + 1 = j 𝒮 j ( u k ) * * ( z j k + Λ 1 , j k ) + α 1 ω k j | 𝒮 j u k | 2 + α 1 1 m ¯ .
u k + 1 = j 𝒮 j T ( ( ω k + 1 ) * * ( z j k + Λ 1 , j k ) ) + α 2 u k j 𝒮 j T | ω k + 1 | 2 + α 2 1 n .
( ( z j k + 1 ) T , ( μ j k + 1 ) T ) T = ( ( ρ j k ) T , ( ρ j k ) T ) T sign ( X ¯ j k ) ,
( X ¯ j k ) T : = ( 𝒜 j T ( ω k + 1 , u k + 1 ) ( Λ 1 , j k ) T , ( 1 j j μ j k ) T ( Λ 2 , j k ) T )
ρ j k : = r | X ¯ j k | * + r 2 | X ¯ j k | * 2 + 4 ( C j + r 1 m ¯ ) C j I j 2 ( C j + r 1 m ¯ ) ;
ρ j , l + 1 = max { 0 , ( ( 1 τ r ) 1 m ¯ τ C j + τ C j I j + ε 2 1 m ¯ ρ j , l 2 + ε 2 1 m ¯ ) ρ j , l + τ r | X ¯ j k | * }
Λ 1 , j k + 1 = Λ 1 , j k + z j k + 1 𝒜 j ( ω k + 1 , u k + 1 ) ; Λ 2 , j k + 1 = Λ 2 , j k + μ j k + 1 1 J j μ j k + 1 .
1 J μ j k + 1 : = max { 0 , 1 J j ( I j | 𝒜 ( ω k + 1 , u k + 1 ) | 2 ) } ;
min ω , u , z , μ , ϕ ˜ λ j | p j | + 𝒢 ε ( z , μ ) , such that z j 𝒜 j ( ω , u ) = 0 , μ j ϕ ˜ = 0 , p j 𝒮 j u = 0 .
reg ( ω , u , p , z , μ , ϕ ˜ , Λ 1 , Λ 2 , Λ 3 ) : = 𝒢 ε ( z , μ ) + j ( r Λ 1 , j , z j 𝒜 j ( ω , u ) + r 2 z j 𝒜 j ( ω , u ) 2 + r Λ 2 , j , μ j ϕ ˜ + r 2 μ j ϕ ˜ 2 + λ | p j | + β p j 𝒮 j u , Λ 3 , j + β 2 p j 𝒮 j u 2 ) .
{ ω k + 1 = arg min ω reg ( ω , u k , p k , z k , μ k , ϕ ˜ k , Λ 1 k , Λ 2 k , Λ 3 k ) + α 1 2 ω ω k 2 ; u k + 1 = arg min u reg ( ω k + 1 , u , p k , z k , μ k , ϕ ˜ k , Λ 1 k , Λ 2 k , Λ 3 k ) + α 2 2 u u k 2 ; p k + 1 = arg min p reg ( ω k + 1 , u k + 1 , p , z k , μ k , ϕ ˜ k , Λ 1 k , Λ 2 k , Λ 3 k ) ; ( z k + 1 , μ k + 1 ) = arg min z , μ reg ( ω k + 1 , u k + 1 , p k + 1 , z , μ , ϕ ˜ k , Λ 1 k , Λ 2 k , Λ 3 k ) ; ϕ ˜ k + 1 = arg min ϕ ˜ reg ( ω k + 1 , u k + 1 , p k + 1 , z k + 1 , μ k + 1 , ϕ ˜ , Λ 1 k , Λ 2 k , Λ 3 k ) ; Λ 1 k + 1 = Λ 1 k + z k + 1 𝒜 ( ω k + 1 , u k + 1 ) ; Λ 2 , j k + 1 = Λ 2 , j k + μ j k + 1 ϕ ˜ k + 1 0 j J 1 ; Λ 3 , j k + 1 = Λ 3 , j k + 1 + p j k + 1 𝒮 j u k + 1 0 j J 1 .
j ( diag ( r 𝒮 j T | ω k + 1 | 2 + α 2 1 n ) β 𝒮 j T Δ 𝒮 j ) u k + 1 = j ( r 𝒮 j T ( ( ω k + 1 ) * * ( z j k + Λ 1 , j k ) ) β 𝒮 j T div ( p j k + Λ 3 , j k ) ) + α 2 u k ,
p j k + 1 = max { 0 , | 𝒮 j u k + 1 Λ 3 , j k | λ β 1 m ¯ } 𝒮 j u k + 1 Λ 3 , j k | 𝒮 j u k + 1 Λ 3 , j k | .
Λ 3 , j k + 1 = Λ 3 , j k + p j k + 1 𝒮 j u k + 1 .
C j 2 γ | 1 | z j k + 1 | 2 + | μ j k + 1 | 2 + ε 2 1 m ¯ I j + ε 2 1 m ¯ | 2 γ ,
SNR ( u k , u g ) = 10 log 10 t = 0 n 1 | ζ * u k ( t + T * ) u g ( t ) | 2 / ζ * u k 2 ,
( ζ * , T * ) : = arg min ζ , T t | ζ u k ( t + T ) u g ( t ) | 2 .
R factor k : = j | 𝒜 j ( ω k , u k ) | 2 + ( ϕ ˜ k ) 2 I j 1 I 1 .
I ^ j ( t ) ~ i . i . d Poisson ( | 𝒜 j ( ω , u ) | 2 ( t ) + ϕ ( t ) ) + n j ( t ) , 0 t m ¯ 1 ,
min ϕ , η η ( η , ϕ ) : = 1 2 j η ( I j ϕ ) | z j k | 2 2 ,
η k = arg min η η j η ( I j ϕ k 1 ) | z j k | 2 2 = max { η , j | z j k | 2 , I j ϕ k 1 j | I j ϕ k 1 | 2 } .
ϕ k = ϕ k 1 1 J | η k | 2 ϕ ( η k , ϕ ) = 1 J j ( I j | z j k | 2 ) + ( 1 1 η k ) ( 1 J j | z j k | 2 ) ,
ω opt : = arg min ω ( ω , u , z , μ , ϕ ˜ , Λ 1 , Λ 2 ) + α 1 2 ω ω ^ 2 = arg min ω 1 2 j ω 𝒮 j u * ( z j + Λ 1 , j ) 2 + α 1 2 ω ω ^ 2 ,
u opt : = arg min u ( ω , u , z , μ , ϕ ˜ ; Λ 1 , Λ 2 ) + α 2 2 u u ^ 2 = arg min u 1 2 j ω 𝒮 j u * ( z j + Λ 1 , j ) 2 + α 2 2 u u ^ 2 ,
{ ω opt = j 𝒮 j u * * ( z j + Λ 1 , j ) + α 1 ω ^ j | 𝒮 j u | 2 + α 1 1 m ¯ u opt = j 𝒮 j T ( ω * * ( z j + Λ 1 , j ) ) + α 2 u ^ j 𝒮 j T | ω | 2 + α 2 1 n .
min z j , μ j 𝒢 ε , j ( z j , μ j ) + r 2 z j ( 𝒜 j ( ω , u ) Λ 1 , j ) 2 + r 2 μ j ( ϕ ˜ Λ 2 , j ) 2 ,
𝒢 ε , j ( z j , μ j ) : = 1 2 C j ( | z j | 2 + μ j 2 + ε 2 1 m ¯ I j + ε 2 1 m ¯ ) 2 .
X j opt : = arg min 𝒢 ε , j ( X j ) + r 2 X j X j 0 2 ,
X j opt = ( ( ρ j opt ) T , ( ρ j opt ) T ) T sign ( X j 0 ) ,
sign ( X j 0 ) : = ( 𝒜 j T ( ω , u ) Λ 1 , j T | X j 0 | * T , ϕ ˜ T Λ 2 , j T | X j 0 | * T ) T ,
| X j 0 | * : = | 𝒜 j ( ω , u ) Λ 1 , j | 2 + | ϕ ˜ Λ 2 , j | 2 ,
ρ j opt = arg min ρ j + m ¯ ε ( ρ j ) : = 1 2 C j , ρ j 2 ( I j + ε 2 1 m ¯ ) log ( ρ j 2 + ε 2 1 m ¯ ) + r 2 ρ j | X j 0 | * 2 .
ρ j opt = r | X j 0 | * + r 2 | X j 0 | * 2 + 4 ( C j + r 1 m ¯ ) C j I j 2 ( C j + r 1 m ¯ ) ,
X j opt = r | X j 0 | * + r 2 | X j 0 | * 2 + 4 ( C j + r 1 m ¯ ) C j I j 2 ( C j + r 1 m ¯ ) sign ( X j 0 ) .
ρ j , l + 1 = max { 0 , ρ j , l τ ( ρ j , l ) } = max { 0 , ( ( 1 τ r ) 1 m ¯ τ C j + τ C j I j + ε 2 1 m ¯ ρ j , l 2 + ε 2 1 m ¯ ) ρ j , l + τ r | X j 0 | * } ,
( ρ j ) = ( C j + r 1 m ¯ C j I j + ε 2 1 m ¯ ρ j 2 + ε 2 1 m ¯ ) ρ j r | X j 0 | * .
ϕ ˜ opt = arg min ϕ ˜ ( ω , u , z , μ , ϕ ˜ , Λ 1 , Λ 2 ) = 1 J ( μ j + Λ 2 , j ) .
min r 2 j ω 𝒮 j u * ( z j + Λ 1 , j ) 2 + β 2 j Λ 3 , j + p j 𝒮 j u 2 + α 2 2 u u ^ 2 ,
j ( diag ( r 𝒮 j T | ω | 2 + α 2 1 n ) β 𝒮 j T Δ 𝒮 j ) u = j ( r 𝒮 j T ( ω * * ( z j + Λ 1 , j ) ) β 𝒮 j T div ( p j + Λ 3 , j ) ) + α 2 u ^ ,
( diag ( r 𝒮 j T | ω | 2 + α 2 1 n ) β 𝒮 j T Δ 𝒮 j ) v , v = r 𝒮 j T ω v , 𝒮 j T ω v + α 2 v 2 + β 𝒮 j v , 𝒮 j v = r 𝒮 j T ω v 2 + α 2 v 2 + β 𝒮 j v 2 > 0 , 0 v n ,
p j opt = max { 0 , | 𝒮 j u Λ 3 , j | λ β 1 m ¯ } 𝒮 j u Λ 3 , j | 𝒮 j u Λ 3 , j | .
{ 0 = ω k + 1 j | 𝒮 j u k | 2 j ( 𝒮 j u k ) * * ( z j k + Λ 1 , j k ) + α 1 ( ω k + 1 ω k ) ; 0 = j 𝒮 j T | ω k + 1 | 2 u k + 1 j 𝒮 j T ( ( ω k + 1 ) * * ( z j k + Λ 1 , j k ) ) + α 2 ( u k + 1 u k ) ; 0 = 𝒢 ε ( z k + 1 , μ k + 1 ) + r ( ( Λ 1 k + 1 ) T , ( Λ 2 k + 1 ) T ) T ; 0 = Λ 1 , j k + 1 + Λ 1 , j k + z j k + 1 𝒜 j ( ω k + 1 , u k + 1 ) ; 0 = Λ 2 , j k + 1 + Λ 2 , j k + μ j k + 1 1 J j μ j k + 1 .
| z j k + 1 | 2 + | μ j k + 1 | 2 = I j + r | Y j k | 1 + r I j 1 + r > 0 ,
𝒢 ε ( z 1 , μ 1 ) 𝒢 ε ( z 2 , μ 2 ) L ε ( z 1 z 2 + μ 1 μ 2 ) ,
( Y k ) ( Y k + 1 ) c ε , r Y k + 1 Y k 2 ,
( Y k ) ( ω k + 1 , u k + 1 , z k , μ k , ϕ ˜ k , Λ 1 k , Λ 2 k ) 1 2 𝒜 ( ω k + 1 ω k , u k ) 2 + α 1 2 ω k + 1 ω k 2 + 1 2 𝒜 ( ω k + 1 , u k + 1 u k ) 2 + α 2 2 u k + 1 u k 2 .
( ω k + 1 , u k + 1 , z k , μ k , ϕ ˜ k , Λ 1 k , Λ 2 k ) ( ω k + 1 , u k + 1 , z k + 1 , μ k + 1 , ϕ ˜ k , Λ 1 k , Λ 2 k ) r 3 L ε 2 ( z j k + 1 z j k 2 + μ j k + 1 μ j k 2 ) .
( ω k + 1 , u k + 1 , z k + 1 , μ k + 1 , ϕ ˜ k , Λ 1 k , Λ 2 k ) ( ω k + 1 , u k + 1 , z k + 1 , μ k + 1 , ϕ ˜ k + 1 , Λ 1 k , Λ 2 k ) r J 2 ϕ ˜ k + 1 ϕ ˜ k 2 .
( ω k + 1 , u k + 1 , z k + 1 , μ k + 1 , ϕ ˜ k + 1 , Λ 1 k , Λ 2 k ) ( Y k + 1 ) = Eq . ( 20 ) r j ( Λ 1 , j k + 1 Λ 1 , j k 2 + Λ 2 , j k + 1 Λ 2 , j k 2 ) Eq . ( 55 ) 2 L ε 2 r ( z k + 1 z k 2 ω k + 1 ω k 2 ) ,
( Y k ) ( Y k + 1 ) min { α 1 2 , α 2 2 , r 3 L ε 2 2 L ε 2 r , r J 2 } Y k + 1 Y k 2 .
( Y k + 1 ) = 𝒢 ε ( z k + 1 , μ k + 1 ) z 𝒢 ε , j ( z j k + 1 , μ j k + 1 ) , z j k + 1 𝒜 j ( ω k + 1 , u k + 1 ) μ 𝒢 ε , j ( z j k + 1 , μ j k + 1 ) , μ j k + 1 ϕ ˜ k + 1 + r 2 z j k + 1 𝒜 j ( ω k + 1 , u k + 1 ) 2 + r 2 μ j k + 1 ϕ ˜ k + 1 2 = 𝒢 ε ( z k + 1 , μ k + 1 ) 𝒢 ε ( 𝒜 j ( ω k + 1 , u k + 1 ) , ϕ ˜ k + 1 ) 𝒢 ε , j ( z j k + 1 , μ j k + 1 ) , ( ( z j k + 1 𝒜 j ( ω k + 1 , u k + 1 ) ) T , ( μ j k + 1 ϕ ˜ k + 1 ) T ) T + r 2 z j k + 1 𝒜 j ( ω k + 1 , u k + 1 ) 2 + r 2 μ j k + 1 ϕ ˜ k + 1 2 + 𝒢 ε ( 𝒜 j ( ω k + 1 , u k + 1 ) , ϕ ˜ k + 1 ) r L ε 2 ( z j k + 1 𝒜 j ( ω k + 1 , u k + 1 ) 2 + μ j k + 1 ϕ ˜ k + 1 2 ) + 𝒢 ε ( 𝒜 j ( ω k + 1 , u k + 1 ) , ϕ ˜ k + 1 ) ,
+ > ( Y 0 ) ( Y k ) = l = 0 k ( ( Y l ) ( Y l + 1 ) ) c ε , r l = 0 k Y l + 1 Y l 2 .
lim l + Y l + 1 Y l = 0 .
lim k + Y n k = Y .
lim k + ω n k j | 𝒮 j u n k 1 | 2 = ω j | 𝒮 j u | 2 ,
lim k + j ( 𝒮 j u n k 1 ) * * ( z j n k 1 + Λ 1 , j n k 1 ) = j ( 𝒮 j u ) * * ( z j + Λ 1 , j ) ,
lim k + j 𝒮 j T ( ( ω n k ) * * ( z j n k 1 + Λ 1 , j n k 1 ) ) = j 𝒮 j T ( ( ω ) * * ( z j + Λ 1 , j ) ) .
lim k + 𝒢 ε ( z n k , μ n k ) = 𝒢 ε ( z , μ ) .
0 = ω n k j | 𝒮 j u n k 1 | 2 j ( 𝒮 j u n k 1 ) * * ( z j n k 1 + Λ 1 , j n k 1 ) + α 1 ( ω n k ω n k 1 ) ; 0 = j 𝒮 j T | ω n k | 2 u n k j 𝒮 j T ( ( ω n k ) * * ( z j n k 1 + Λ 1 , j n k 1 ) ) + α 2 ( u n k u n k 1 ) ; 0 = 𝒢 ε ( z n k , μ n k ) + r ( ( Λ 1 n k ) T , ( Λ 2 n k ) T ) T ; 0 = Λ 1 , j n k + Λ 1 , j n k 1 + z j n k 𝒜 j ( ω n k , u n k ) ; 0 = Λ 2 , j n k + Λ 2 , j n k 1 + μ j n k 1 J j μ j n k ,
{ 0 = ω j | 𝒮 j u | 2 j ( 𝒮 j u ) * * ( z j + Λ 1 , j * ) ; 0 = j 𝒮 j T | ω | 2 u j 𝒮 j T ( ( ω ) * * ( z j + Λ 1 , j ) ) ; 0 = 𝒢 ε ( z , μ ) + r ( ( Λ 1 ) T , ( Λ 2 ) T ) T ; 0 = z j 𝒜 j ( ω , u ) ; 0 = μ j 1 J j μ j ,

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