Abstract

Coherent diffraction imaging methods enable imaging beyond lens-imposed resolution limits. In these methods, the object can be recovered by minimizing an error metric that quantifies the difference between diffraction patterns as observed, and those calculated from a present guess of the object. Efficient minimization methods require analytical calculation of the derivatives of the error metric, which is not always straightforward. This limits our ability to explore variations of basic imaging approaches. In this paper, we propose to substitute analytical derivative expressions with the automatic differentiation method, whereby we can achieve object reconstruction by specifying only the physics-based experimental forward model. We demonstrate the generality of the proposed method through straightforward object reconstruction for a variety of complex ptychographic experimental models.

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

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  1. W. Hoppe, “Beugung im inhomogenen Primarstrahlwellenfeld. III. Amplituden-und Phasenbestimmung bei unperiodischen Objekten,” Acta Crystallogr. A 25, 508–514 (1969).
    [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. H. M. L. Faulkner and J. Rodenburg, “Movable aperture lensless transmission microscopy: A novel phase retrieval algorithm,” Phys. Rev. Lett. 93, 023903 (2004).
    [Crossref] [PubMed]
  4. M. Guizar-Sicairos and J. Fienup, “Phase retrieval with transverse translation diversity: a nonlinear optimization approach,” Opt. Express 16, 7264–7278 (2008).
    [Crossref] [PubMed]
  5. A. M. Maiden and J. M. Rodenburg, “An improved ptychographical phase retrieval algorithm for diffractive imaging,” Ultramicroscopy 109, 1256–1262 (2009).
    [Crossref] [PubMed]
  6. 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]
  7. M. Stockmar, P. Cloetens, I. Zanette, B. Enders, M. Dierolf, F. Pfeiffer, and P. Thibault, “Near-field ptychography: phase retrieval for inline holography using a structured illumination,” Sci. Rep. 3, 1927 (2013).
    [Crossref]
  8. P. Godard, G. Carbone, M. Allain, F. Mastropietro, G. Chen, L. Capello, A. Diaz, T. Metzger, J. Stangl, and V. Chamard, “Three-dimensional high-resolution quantitative microscopy of extended crystals,” Nat. Commun. 2, 568 (2011).
    [Crossref] [PubMed]
  9. A. M. Maiden, M. J. Humphry, and J. M. Rodenburg, “Ptychographic transmission microscopy in three dimensions using a multi-slice approach,” J. Opt. Soc. Am. A 29, 1606–1614 (2012).
    [Crossref]
  10. M. Dierolf, A. Menzel, P. Thibault, P. Schneider, C. M. Kewish, R. Wepf, O. Bunk, and F. Pfeiffer, “Ptychographic x-ray computed tomography at the nanoscale,” Nature 467, 436–439 (2010).
    [Crossref] [PubMed]
  11. M. A. Gilles, Y. S. G. Nashed, M. Du, C. Jacobsen, and S. M. Wild, “3D x-ray imaging of continuous objects beyond the depth of focus limit,” Optica 5, 1078–1085 (2018).
    [Crossref] [PubMed]
  12. R. H. T. Bates, “Fourier phase problems are uniquely solvable in more than one dimension. I. Underlying theory,” Optik 61, 247–262 (1982).
  13. R. W. Gerchberg and W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik 35, 237–246 (1972).
  14. J. R. Fienup, “Phase retrieval algorithms: a comparison,” Appl. Opt. 21, 2758–2769 (1982).
    [Crossref] [PubMed]
  15. V. Elser, “Phase retrieval by iterated projections,” J. Opt. Soc. Am. A 20, 40–55 (2003).
    [Crossref]
  16. S. Marchesini, “A unified evaluation of iterative projection algorithms for phase retrieval,” Rev. Sci. Instruments 78, 011301 (2007).
    [Crossref]
  17. J. Clark, X. Huang, R. Harder, and I. Robinson, “High-resolution three-dimensional partially coherent diffraction imaging,” Nat. Commun. 3, 993 (2012).
    [Crossref] [PubMed]
  18. P. Thibault and A. Menzel, “Reconstructing state mixtures from diffraction measurements,” Nature 494, 68–71 (2013).
    [Crossref] [PubMed]
  19. A. Tripathi, I. McNulty, and O. G. Shpyrko, “Ptychographic overlap constraint errors and the limits of their numerical recovery using conjugate gradient descent methods,” Opt. Express 22, 1452–1466 (2014).
    [Crossref] [PubMed]
  20. P. Dwivedi, A. Konijnenberg, S. Pereira, and H. Urbach, “Lateral position correction in ptychography using the gradient of intensity patterns,” Ultramicroscopy 192, 29–36 (2018).
    [Crossref] [PubMed]
  21. P. Thibault and M. Guizar-Sicairos, “Maximum-likelihood refinement for coherent diffractive imaging,” New J. Phys. 14, 063004 (2012).
    [Crossref]
  22. 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]
  23. A. S. Jurling and J. R. Fienup, “Applications of algorithmic differentiation to phase retrieval algorithms,” J. Opt. Soc. Am. A 31, 1348–1359 (2014).
    [Crossref] [PubMed]
  24. E. J. Candes, X. Li, and M. Soltanolkotabi, “Phase retrieval via Wirtinger flow: theory and algorithms,” IEEE Trans. Inf. Theory 61, 1985–2007 (2015).
    [Crossref]
  25. H. Zhang and Y. Liang, “Reshaped wirtinger flow for solving quadratic system of equations,” Adv. Neural. Inf. Process. Syst. 29, 2622–2630 (2016).
  26. Z. Wei, W. Chen, C.-W. Qiu, and X. Chen, “Conjugate gradient method for phase retrieval based on the Wirtinger derivative,” J. Opt. Soc. Am. A 34, 708 (2017).
    [Crossref]
  27. J. Zhong, L. Tian, P. Varma, and L. Waller, “Nonlinear optimization algorithm for partially coherent phase retrieval and source recovery,” IEEE Trans. Comput. Imaging 2, 310–322 (2016).
    [Crossref]
  28. J. Li and T. Zhou, “Numerical optimization algorithms for wavefront phase retrieval from multiple measurements,” Inverse Probl. & Imaging 11, 721–743 (2017).
    [Crossref]
  29. J. Qian, C. Yang, A. Schirotzek, F. Maia, and S. Marchesini, “Efficient algorithms for ptychographic phase retrieval,” Inverse Probl. Appli. Contemp. Math 615, 261–280 (2014).
  30. A. W. Yan, A. J. D’Alfonso, A. J. Morgan, C. T. Putkunz, and L. J. Allen, “Fast deterministic ptychographic imaging using x-rays,” Microsc. Microanal. 20, 1090–1099 (2014).
    [Crossref] [PubMed]
  31. L.-H. Yeh, “Analysis and comparison of fourier ptychographic phase retrieval algorithms,” Technical Report No. UCB/EECS-2016-86 (University of California, 2016).
  32. A. M. Maiden, D. Johnson, and P. Li, “Further improvements to the ptychographical iterative engine,” Optica 4, 736–745 (2017).
    [Crossref]
  33. 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]
  34. A. Griewank and A. Walther, Evaluating derivatives: principles and techniques of algorithmic differentiation, vol. 105 (SIAM, 2008).
    [Crossref]
  35. M. Abadi, A. Agarwal, P. Barham, E. Brevdo, Z. Chen, C. Citro, G. S. Corrado, A. Davis, J. Dean, M. Devin, S. Ghemawat, I. J. Goodfellow, A. Harp, G. Irving, M. Isard, Y. Jia, R. Józefowicz, L. Kaiser, M. Kudlur, J. Levenberg, D. Mané, R. Monga, S. Moore, D. G. Murray, C. Olah, M. Schuster, J. Shlens, B. Steiner, I. Sutskever, K. Talwar, P. A. Tucker, V. Vanhoucke, V. Vasudevan, F. B. Viégas, O. Vinyals, P. Warden, M. Wattenberg, M. Wicke, Y. Yu, and X. Zheng, “Tensorflow: Large-scale machine learning on heterogeneous distributed systems,” arXiv 1603.04467 (2016).
  36. A. Paszke, S. Gross, S. Chintala, G. Chanan, E. Yang, Z. DeVito, Z. Lin, A. Desmaison, L. Antiga, and A. Lerer, “Automatic differentiation in pytorch,” in 31st Conference on Neural Information Processing Systems (NIPS, 2017).
  37. D. Maclaurin, “Modeling, inference and optimization with composable differentiable procedures,” Ph.D. thesis, Harvard University (2016).
  38. Y. S. G. Nashed, T. Peterka, J. Deng, and C. Jacobsen, “Distributed automatic differentiation for ptychography,” Procedia Comput. Sci. 108, 404–414 (2017).
    [Crossref]
  39. S. Ghosh, Y. S. Nashed, O. Cossairt, and A. Katsaggelos, “ADP : Automatic differentiation ptychography,” IEEE International Conference on Computational Photography (ICCP) (IEEE, 2018), pp.1–10.
  40. A. G. Baydin, B. A. Pearlmutter, A. A. Radul, and J. M. Siskind, “Automatic differentiation in machine learning: a survey,” J. Mach. Learn. Res. 18, 1–43 (2015).
  41. P. H. Hoffmann, “A hitchhiker’s guide to automatic differentiation,” Numer. Algorithms 72, 775–811 (2016).
    [Crossref]
  42. L. Bian, J. Suo, G. Zheng, K. Guo, F. Chen, and Q. Dai, “Fourier ptychographic reconstruction using wirtinger flow optimization,” Opt. Express 23, 4856–4866 (2015).
    [Crossref] [PubMed]
  43. Z. Wen, C. Yang, X. Liu, and S. Marchesini, “Alternating direction methods for classical and ptychographic phase retrieval,” Inverse Probl. 28, 115010 (2012).
    [Crossref]
  44. J. R. Fienup, T. R. Crimmins, and W. Holsztynski, “Reconstruction of the support of an object from the support of its autocorrelation,” J. Opt. Soc. Am. 72, 610–624 (1982).
    [Crossref]
  45. D. Brandwood, “A complex gradient operator and its application in adaptive array theory,” IEE Proc. F-Communications, Radar Signal Process. 130, 11–16 (1983).
    [Crossref]
  46. K. Kreutz-Delgado, “The complex gradient operator and the cr-calculus,” arXiv preprint arXiv:0906.4835 (2009).
  47. L. Sorber, M. V. Barel, and L. D. Lathauwer, “Unconstrained optimization of real functions in complex variables,” SIAM J. Optim. 22, 879–898 (2012).
    [Crossref]
  48. D. P. Kingma and J. Ba, “Adam: A method for stochastic optimization,” arXiv 1412.6980 (2014).
  49. S. Ahn and J. A. Fessler, “Globally convergent image reconstruction for emission tomography using relaxed ordered subsets algorithms,” IEEE Trans. Med. Imaging 22, 613–626 (2003).
    [Crossref] [PubMed]
  50. D. P. Bertsekas, Nonlinear programming (Athena Scientific, 1999).
  51. Y. A. LeCun, L. Bottou, G. B. Orr, and K.-R. Müller, “Efficient backprop,” in Neural networks: Tricks of the trade, (Springer, 2012), pp. 9–48.
  52. S. Ruder, “An overview of gradient descent optimization algorithms,” arXiv preprint arXiv:1609.04747 (2016).
  53. E. J. R. Pauwels, A. Beck, Y. C. Eldar, and S. Sabach, “On fienup methods for sparse phase retrieval,” IEEE Trans. Signal Process. 66, 982–991 (2018).
    [Crossref]
  54. R. Xu, M. Soltanolkotabi, J. P. Haldar, W. Unglaub, J. Zusman, A. F. Levi, and R. M. Leahy, “Accelerated wirtinger flow: A fast algorithm for ptychography,” arXiv preprint arXiv:1806.05546 (2018).
  55. M. Guizar-Sicairos, S. T. Thurman, and J. R. Fienup, “Efficient subpixel image registration algorithms,” Opt. Lett. 33, 156–158 (2008).
    [Crossref] [PubMed]
  56. Y. S. Nashed, D. J. Vine, T. Peterka, J. Deng, R. Ross, and C. Jacobsen, “Parallel ptychographic reconstruction,” Opt. Express 22, 223015 (2014).
    [Crossref]
  57. J. Bergstra and Y. Bengio, “Random search for hyper-parameter optimization,” J. Mach. Learn. Res. 13, 281–305 (2012).
  58. Y. Yao, L. Rosasco, and A. Caponnetto, “On early stopping in gradient descent learning,” Constr. Approx. 26, 289–315 (2007).
    [Crossref]
  59. J. W. Goodman, Introduction to Fourier Optics (W.H. Freeman, 2017).
  60. M. Stockmar, I. Zanette, M. Dierolf, B. Enders, R. Clare, F. Pfeiffer, P. Cloetens, A. Bonnin, and P. Thibault, “X-ray near-field ptychography for optically thick specimens,” Phys. Rev. Appl. 3, 1–6 (2015).
    [Crossref]
  61. R. M. Clare, M. Stockmar, M. Dierolf, I. Zanette, and F. Pfeiffer, “Characterization of near-field ptychography,” Opt. Express 23, 19728 (2015).
    [Crossref] [PubMed]
  62. A. L. Robisch, K. Kröger, A. Rack, and T. Salditt, “Near-field ptychography using lateral and longitudinal shifts,” New J. Phys. 17, 073033 (2015).
    [Crossref]
  63. M. O. Hill, I. Calvo-Almazan, M. Allain, M. V. Holt, A. Ulvestad, J. Treu, G. Koblmuller, C. Huang, X. Huang, H. Yan, E. Nazaretski, Y. S. Chu, G. B. Stephenson, V. Chamard, L. J. Lauhon, and S. O. Hruszkewycz, “Measuring three-dimensional strain and structural defects in a single InGaAs nanowire using coherent x-ray multiangle Bragg projection ptychography,” Nano Lett. 18, 811–819 (2018).
    [Crossref] [PubMed]
  64. S. O. Hruszkewycz, M. Allain, M. V. Holt, C. E. Murray, J. R. Holt, P. H. Fuoss, and V. Chamard, “High-resolution three-dimensional structural microscopy by single-angle Bragg ptychography,” Nat. Mater. 16, 244–251 (2017).
    [Crossref]
  65. A. Fannjiang, “Absolute uniqueness of phase retrieval with random illumination,” Inverse Probl. 28, 075008 (2012).
    [Crossref]
  66. T. B. Edo, D. J. Batey, A. M. Maiden, C. Rau, U. Wagner, Z. D. Pešić, T. A. Waigh, and J. M. Rodenburg, “Sampling in x-ray ptychography,” Phys. Rev. A 87, 053850 (2013).
    [Crossref]
  67. V. Chamard, M. Allain, P. Godard, A. Talneau, G. Patriarche, and M. Burghammer, “Strain in a silicon-on-insulator nanostructure revealed by 3D x-ray Bragg ptychography,” Sci. Rep. 5, 9827 (2015).
    [Crossref]

2018 (4)

M. A. Gilles, Y. S. G. Nashed, M. Du, C. Jacobsen, and S. M. Wild, “3D x-ray imaging of continuous objects beyond the depth of focus limit,” Optica 5, 1078–1085 (2018).
[Crossref] [PubMed]

P. Dwivedi, A. Konijnenberg, S. Pereira, and H. Urbach, “Lateral position correction in ptychography using the gradient of intensity patterns,” Ultramicroscopy 192, 29–36 (2018).
[Crossref] [PubMed]

E. J. R. Pauwels, A. Beck, Y. C. Eldar, and S. Sabach, “On fienup methods for sparse phase retrieval,” IEEE Trans. Signal Process. 66, 982–991 (2018).
[Crossref]

M. O. Hill, I. Calvo-Almazan, M. Allain, M. V. Holt, A. Ulvestad, J. Treu, G. Koblmuller, C. Huang, X. Huang, H. Yan, E. Nazaretski, Y. S. Chu, G. B. Stephenson, V. Chamard, L. J. Lauhon, and S. O. Hruszkewycz, “Measuring three-dimensional strain and structural defects in a single InGaAs nanowire using coherent x-ray multiangle Bragg projection ptychography,” Nano Lett. 18, 811–819 (2018).
[Crossref] [PubMed]

2017 (5)

S. O. Hruszkewycz, M. Allain, M. V. Holt, C. E. Murray, J. R. Holt, P. H. Fuoss, and V. Chamard, “High-resolution three-dimensional structural microscopy by single-angle Bragg ptychography,” Nat. Mater. 16, 244–251 (2017).
[Crossref]

Y. S. G. Nashed, T. Peterka, J. Deng, and C. Jacobsen, “Distributed automatic differentiation for ptychography,” Procedia Comput. Sci. 108, 404–414 (2017).
[Crossref]

Z. Wei, W. Chen, C.-W. Qiu, and X. Chen, “Conjugate gradient method for phase retrieval based on the Wirtinger derivative,” J. Opt. Soc. Am. A 34, 708 (2017).
[Crossref]

J. Li and T. Zhou, “Numerical optimization algorithms for wavefront phase retrieval from multiple measurements,” Inverse Probl. & Imaging 11, 721–743 (2017).
[Crossref]

A. M. Maiden, D. Johnson, and P. Li, “Further improvements to the ptychographical iterative engine,” Optica 4, 736–745 (2017).
[Crossref]

2016 (3)

P. H. Hoffmann, “A hitchhiker’s guide to automatic differentiation,” Numer. Algorithms 72, 775–811 (2016).
[Crossref]

J. Zhong, L. Tian, P. Varma, and L. Waller, “Nonlinear optimization algorithm for partially coherent phase retrieval and source recovery,” IEEE Trans. Comput. Imaging 2, 310–322 (2016).
[Crossref]

H. Zhang and Y. Liang, “Reshaped wirtinger flow for solving quadratic system of equations,” Adv. Neural. Inf. Process. Syst. 29, 2622–2630 (2016).

2015 (8)

A. G. Baydin, B. A. Pearlmutter, A. A. Radul, and J. M. Siskind, “Automatic differentiation in machine learning: a survey,” J. Mach. Learn. Res. 18, 1–43 (2015).

V. Chamard, M. Allain, P. Godard, A. Talneau, G. Patriarche, and M. Burghammer, “Strain in a silicon-on-insulator nanostructure revealed by 3D x-ray Bragg ptychography,” Sci. Rep. 5, 9827 (2015).
[Crossref]

M. Stockmar, I. Zanette, M. Dierolf, B. Enders, R. Clare, F. Pfeiffer, P. Cloetens, A. Bonnin, and P. Thibault, “X-ray near-field ptychography for optically thick specimens,” Phys. Rev. Appl. 3, 1–6 (2015).
[Crossref]

R. M. Clare, M. Stockmar, M. Dierolf, I. Zanette, and F. Pfeiffer, “Characterization of near-field ptychography,” Opt. Express 23, 19728 (2015).
[Crossref] [PubMed]

A. L. Robisch, K. Kröger, A. Rack, and T. Salditt, “Near-field ptychography using lateral and longitudinal shifts,” New J. Phys. 17, 073033 (2015).
[Crossref]

E. J. Candes, X. Li, and M. Soltanolkotabi, “Phase retrieval via Wirtinger flow: theory and algorithms,” IEEE Trans. Inf. Theory 61, 1985–2007 (2015).
[Crossref]

L. Bian, J. Suo, G. Zheng, K. Guo, F. Chen, and Q. Dai, “Fourier ptychographic reconstruction using wirtinger flow optimization,” Opt. Express 23, 4856–4866 (2015).
[Crossref] [PubMed]

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 (5)

J. Qian, C. Yang, A. Schirotzek, F. Maia, and S. Marchesini, “Efficient algorithms for ptychographic phase retrieval,” Inverse Probl. Appli. Contemp. Math 615, 261–280 (2014).

A. W. Yan, A. J. D’Alfonso, A. J. Morgan, C. T. Putkunz, and L. J. Allen, “Fast deterministic ptychographic imaging using x-rays,” Microsc. Microanal. 20, 1090–1099 (2014).
[Crossref] [PubMed]

A. S. Jurling and J. R. Fienup, “Applications of algorithmic differentiation to phase retrieval algorithms,” J. Opt. Soc. Am. A 31, 1348–1359 (2014).
[Crossref] [PubMed]

A. Tripathi, I. McNulty, and O. G. Shpyrko, “Ptychographic overlap constraint errors and the limits of their numerical recovery using conjugate gradient descent methods,” Opt. Express 22, 1452–1466 (2014).
[Crossref] [PubMed]

Y. S. Nashed, D. J. Vine, T. Peterka, J. Deng, R. Ross, and C. Jacobsen, “Parallel ptychographic reconstruction,” Opt. Express 22, 223015 (2014).
[Crossref]

2013 (3)

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

P. Thibault and A. Menzel, “Reconstructing state mixtures from diffraction measurements,” Nature 494, 68–71 (2013).
[Crossref] [PubMed]

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

2012 (8)

A. M. Maiden, M. J. Humphry, and J. M. Rodenburg, “Ptychographic transmission microscopy in three dimensions using a multi-slice approach,” J. Opt. Soc. Am. A 29, 1606–1614 (2012).
[Crossref]

J. Clark, X. Huang, R. Harder, and I. Robinson, “High-resolution three-dimensional partially coherent diffraction imaging,” Nat. Commun. 3, 993 (2012).
[Crossref] [PubMed]

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

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

A. Fannjiang, “Absolute uniqueness of phase retrieval with random illumination,” Inverse Probl. 28, 075008 (2012).
[Crossref]

J. Bergstra and Y. Bengio, “Random search for hyper-parameter optimization,” J. Mach. Learn. Res. 13, 281–305 (2012).

L. Sorber, M. V. Barel, and L. D. Lathauwer, “Unconstrained optimization of real functions in complex variables,” SIAM J. Optim. 22, 879–898 (2012).
[Crossref]

2011 (1)

P. Godard, G. Carbone, M. Allain, F. Mastropietro, G. Chen, L. Capello, A. Diaz, T. Metzger, J. Stangl, and V. Chamard, “Three-dimensional high-resolution quantitative microscopy of extended crystals,” Nat. Commun. 2, 568 (2011).
[Crossref] [PubMed]

2010 (1)

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

2009 (2)

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

2008 (2)

2007 (2)

Y. Yao, L. Rosasco, and A. Caponnetto, “On early stopping in gradient descent learning,” Constr. Approx. 26, 289–315 (2007).
[Crossref]

S. Marchesini, “A unified evaluation of iterative projection algorithms for phase retrieval,” Rev. Sci. Instruments 78, 011301 (2007).
[Crossref]

2004 (2)

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

H. M. L. Faulkner and J. Rodenburg, “Movable aperture lensless transmission microscopy: A novel phase retrieval algorithm,” Phys. Rev. Lett. 93, 023903 (2004).
[Crossref] [PubMed]

2003 (2)

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

S. Ahn and J. A. Fessler, “Globally convergent image reconstruction for emission tomography using relaxed ordered subsets algorithms,” IEEE Trans. Med. Imaging 22, 613–626 (2003).
[Crossref] [PubMed]

1983 (1)

D. Brandwood, “A complex gradient operator and its application in adaptive array theory,” IEE Proc. F-Communications, Radar Signal Process. 130, 11–16 (1983).
[Crossref]

1982 (3)

1972 (1)

R. W. Gerchberg and W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik 35, 237–246 (1972).

1969 (1)

W. Hoppe, “Beugung im inhomogenen Primarstrahlwellenfeld. III. Amplituden-und Phasenbestimmung bei unperiodischen Objekten,” Acta Crystallogr. A 25, 508–514 (1969).
[Crossref]

Abadi, M.

M. Abadi, A. Agarwal, P. Barham, E. Brevdo, Z. Chen, C. Citro, G. S. Corrado, A. Davis, J. Dean, M. Devin, S. Ghemawat, I. J. Goodfellow, A. Harp, G. Irving, M. Isard, Y. Jia, R. Józefowicz, L. Kaiser, M. Kudlur, J. Levenberg, D. Mané, R. Monga, S. Moore, D. G. Murray, C. Olah, M. Schuster, J. Shlens, B. Steiner, I. Sutskever, K. Talwar, P. A. Tucker, V. Vanhoucke, V. Vasudevan, F. B. Viégas, O. Vinyals, P. Warden, M. Wattenberg, M. Wicke, Y. Yu, and X. Zheng, “Tensorflow: Large-scale machine learning on heterogeneous distributed systems,” arXiv 1603.04467 (2016).

Agarwal, A.

M. Abadi, A. Agarwal, P. Barham, E. Brevdo, Z. Chen, C. Citro, G. S. Corrado, A. Davis, J. Dean, M. Devin, S. Ghemawat, I. J. Goodfellow, A. Harp, G. Irving, M. Isard, Y. Jia, R. Józefowicz, L. Kaiser, M. Kudlur, J. Levenberg, D. Mané, R. Monga, S. Moore, D. G. Murray, C. Olah, M. Schuster, J. Shlens, B. Steiner, I. Sutskever, K. Talwar, P. A. Tucker, V. Vanhoucke, V. Vasudevan, F. B. Viégas, O. Vinyals, P. Warden, M. Wattenberg, M. Wicke, Y. Yu, and X. Zheng, “Tensorflow: Large-scale machine learning on heterogeneous distributed systems,” arXiv 1603.04467 (2016).

Ahn, S.

S. Ahn and J. A. Fessler, “Globally convergent image reconstruction for emission tomography using relaxed ordered subsets algorithms,” IEEE Trans. Med. Imaging 22, 613–626 (2003).
[Crossref] [PubMed]

Allain, M.

M. O. Hill, I. Calvo-Almazan, M. Allain, M. V. Holt, A. Ulvestad, J. Treu, G. Koblmuller, C. Huang, X. Huang, H. Yan, E. Nazaretski, Y. S. Chu, G. B. Stephenson, V. Chamard, L. J. Lauhon, and S. O. Hruszkewycz, “Measuring three-dimensional strain and structural defects in a single InGaAs nanowire using coherent x-ray multiangle Bragg projection ptychography,” Nano Lett. 18, 811–819 (2018).
[Crossref] [PubMed]

S. O. Hruszkewycz, M. Allain, M. V. Holt, C. E. Murray, J. R. Holt, P. H. Fuoss, and V. Chamard, “High-resolution three-dimensional structural microscopy by single-angle Bragg ptychography,” Nat. Mater. 16, 244–251 (2017).
[Crossref]

V. Chamard, M. Allain, P. Godard, A. Talneau, G. Patriarche, and M. Burghammer, “Strain in a silicon-on-insulator nanostructure revealed by 3D x-ray Bragg ptychography,” Sci. Rep. 5, 9827 (2015).
[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. Godard, G. Carbone, M. Allain, F. Mastropietro, G. Chen, L. Capello, A. Diaz, T. Metzger, J. Stangl, and V. Chamard, “Three-dimensional high-resolution quantitative microscopy of extended crystals,” Nat. Commun. 2, 568 (2011).
[Crossref] [PubMed]

Allen, L. J.

A. W. Yan, A. J. D’Alfonso, A. J. Morgan, C. T. Putkunz, and L. J. Allen, “Fast deterministic ptychographic imaging using x-rays,” Microsc. Microanal. 20, 1090–1099 (2014).
[Crossref] [PubMed]

Antiga, L.

A. Paszke, S. Gross, S. Chintala, G. Chanan, E. Yang, Z. DeVito, Z. Lin, A. Desmaison, L. Antiga, and A. Lerer, “Automatic differentiation in pytorch,” in 31st Conference on Neural Information Processing Systems (NIPS, 2017).

Ba, J.

D. P. Kingma and J. Ba, “Adam: A method for stochastic optimization,” arXiv 1412.6980 (2014).

Barel, M. V.

L. Sorber, M. V. Barel, and L. D. Lathauwer, “Unconstrained optimization of real functions in complex variables,” SIAM J. Optim. 22, 879–898 (2012).
[Crossref]

Barham, P.

M. Abadi, A. Agarwal, P. Barham, E. Brevdo, Z. Chen, C. Citro, G. S. Corrado, A. Davis, J. Dean, M. Devin, S. Ghemawat, I. J. Goodfellow, A. Harp, G. Irving, M. Isard, Y. Jia, R. Józefowicz, L. Kaiser, M. Kudlur, J. Levenberg, D. Mané, R. Monga, S. Moore, D. G. Murray, C. Olah, M. Schuster, J. Shlens, B. Steiner, I. Sutskever, K. Talwar, P. A. Tucker, V. Vanhoucke, V. Vasudevan, F. B. Viégas, O. Vinyals, P. Warden, M. Wattenberg, M. Wicke, Y. Yu, and X. Zheng, “Tensorflow: Large-scale machine learning on heterogeneous distributed systems,” arXiv 1603.04467 (2016).

Bates, R. H. T.

R. H. T. Bates, “Fourier phase problems are uniquely solvable in more than one dimension. I. Underlying theory,” Optik 61, 247–262 (1982).

Batey, D. J.

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

Baydin, A. G.

A. G. Baydin, B. A. Pearlmutter, A. A. Radul, and J. M. Siskind, “Automatic differentiation in machine learning: a survey,” J. Mach. Learn. Res. 18, 1–43 (2015).

Beck, A.

E. J. R. Pauwels, A. Beck, Y. C. Eldar, and S. Sabach, “On fienup methods for sparse phase retrieval,” IEEE Trans. Signal Process. 66, 982–991 (2018).
[Crossref]

Bengio, Y.

J. Bergstra and Y. Bengio, “Random search for hyper-parameter optimization,” J. Mach. Learn. Res. 13, 281–305 (2012).

Bergstra, J.

J. Bergstra and Y. Bengio, “Random search for hyper-parameter optimization,” J. Mach. Learn. Res. 13, 281–305 (2012).

Bertsekas, D. P.

D. P. Bertsekas, Nonlinear programming (Athena Scientific, 1999).

Bian, L.

Bonnin, A.

M. Stockmar, I. Zanette, M. Dierolf, B. Enders, R. Clare, F. Pfeiffer, P. Cloetens, A. Bonnin, and P. Thibault, “X-ray near-field ptychography for optically thick specimens,” Phys. Rev. Appl. 3, 1–6 (2015).
[Crossref]

Bottou, L.

Y. A. LeCun, L. Bottou, G. B. Orr, and K.-R. Müller, “Efficient backprop,” in Neural networks: Tricks of the trade, (Springer, 2012), pp. 9–48.

Brandwood, D.

D. Brandwood, “A complex gradient operator and its application in adaptive array theory,” IEE Proc. F-Communications, Radar Signal Process. 130, 11–16 (1983).
[Crossref]

Brevdo, E.

M. Abadi, A. Agarwal, P. Barham, E. Brevdo, Z. Chen, C. Citro, G. S. Corrado, A. Davis, J. Dean, M. Devin, S. Ghemawat, I. J. Goodfellow, A. Harp, G. Irving, M. Isard, Y. Jia, R. Józefowicz, L. Kaiser, M. Kudlur, J. Levenberg, D. Mané, R. Monga, S. Moore, D. G. Murray, C. Olah, M. Schuster, J. Shlens, B. Steiner, I. Sutskever, K. Talwar, P. A. Tucker, V. Vanhoucke, V. Vasudevan, F. B. Viégas, O. Vinyals, P. Warden, M. Wattenberg, M. Wicke, Y. Yu, and X. Zheng, “Tensorflow: Large-scale machine learning on heterogeneous distributed systems,” arXiv 1603.04467 (2016).

Bunk, O.

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

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]

Burghammer, M.

V. Chamard, M. Allain, P. Godard, A. Talneau, G. Patriarche, and M. Burghammer, “Strain in a silicon-on-insulator nanostructure revealed by 3D x-ray Bragg ptychography,” Sci. Rep. 5, 9827 (2015).
[Crossref]

Calvo-Almazan, I.

M. O. Hill, I. Calvo-Almazan, M. Allain, M. V. Holt, A. Ulvestad, J. Treu, G. Koblmuller, C. Huang, X. Huang, H. Yan, E. Nazaretski, Y. S. Chu, G. B. Stephenson, V. Chamard, L. J. Lauhon, and S. O. Hruszkewycz, “Measuring three-dimensional strain and structural defects in a single InGaAs nanowire using coherent x-ray multiangle Bragg projection ptychography,” Nano Lett. 18, 811–819 (2018).
[Crossref] [PubMed]

Candes, E. J.

E. J. Candes, X. Li, and M. Soltanolkotabi, “Phase retrieval via Wirtinger flow: theory and algorithms,” IEEE Trans. Inf. Theory 61, 1985–2007 (2015).
[Crossref]

Capello, L.

P. Godard, G. Carbone, M. Allain, F. Mastropietro, G. Chen, L. Capello, A. Diaz, T. Metzger, J. Stangl, and V. Chamard, “Three-dimensional high-resolution quantitative microscopy of extended crystals,” Nat. Commun. 2, 568 (2011).
[Crossref] [PubMed]

Caponnetto, A.

Y. Yao, L. Rosasco, and A. Caponnetto, “On early stopping in gradient descent learning,” Constr. Approx. 26, 289–315 (2007).
[Crossref]

Carbone, G.

P. Godard, G. Carbone, M. Allain, F. Mastropietro, G. Chen, L. Capello, A. Diaz, T. Metzger, J. Stangl, and V. Chamard, “Three-dimensional high-resolution quantitative microscopy of extended crystals,” Nat. Commun. 2, 568 (2011).
[Crossref] [PubMed]

Chamard, V.

M. O. Hill, I. Calvo-Almazan, M. Allain, M. V. Holt, A. Ulvestad, J. Treu, G. Koblmuller, C. Huang, X. Huang, H. Yan, E. Nazaretski, Y. S. Chu, G. B. Stephenson, V. Chamard, L. J. Lauhon, and S. O. Hruszkewycz, “Measuring three-dimensional strain and structural defects in a single InGaAs nanowire using coherent x-ray multiangle Bragg projection ptychography,” Nano Lett. 18, 811–819 (2018).
[Crossref] [PubMed]

S. O. Hruszkewycz, M. Allain, M. V. Holt, C. E. Murray, J. R. Holt, P. H. Fuoss, and V. Chamard, “High-resolution three-dimensional structural microscopy by single-angle Bragg ptychography,” Nat. Mater. 16, 244–251 (2017).
[Crossref]

V. Chamard, M. Allain, P. Godard, A. Talneau, G. Patriarche, and M. Burghammer, “Strain in a silicon-on-insulator nanostructure revealed by 3D x-ray Bragg ptychography,” Sci. Rep. 5, 9827 (2015).
[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. Godard, G. Carbone, M. Allain, F. Mastropietro, G. Chen, L. Capello, A. Diaz, T. Metzger, J. Stangl, and V. Chamard, “Three-dimensional high-resolution quantitative microscopy of extended crystals,” Nat. Commun. 2, 568 (2011).
[Crossref] [PubMed]

Chanan, G.

A. Paszke, S. Gross, S. Chintala, G. Chanan, E. Yang, Z. DeVito, Z. Lin, A. Desmaison, L. Antiga, and A. Lerer, “Automatic differentiation in pytorch,” in 31st Conference on Neural Information Processing Systems (NIPS, 2017).

Chen, F.

Chen, G.

P. Godard, G. Carbone, M. Allain, F. Mastropietro, G. Chen, L. Capello, A. Diaz, T. Metzger, J. Stangl, and V. Chamard, “Three-dimensional high-resolution quantitative microscopy of extended crystals,” Nat. Commun. 2, 568 (2011).
[Crossref] [PubMed]

Chen, W.

Chen, X.

Chen, Z.

M. Abadi, A. Agarwal, P. Barham, E. Brevdo, Z. Chen, C. Citro, G. S. Corrado, A. Davis, J. Dean, M. Devin, S. Ghemawat, I. J. Goodfellow, A. Harp, G. Irving, M. Isard, Y. Jia, R. Józefowicz, L. Kaiser, M. Kudlur, J. Levenberg, D. Mané, R. Monga, S. Moore, D. G. Murray, C. Olah, M. Schuster, J. Shlens, B. Steiner, I. Sutskever, K. Talwar, P. A. Tucker, V. Vanhoucke, V. Vasudevan, F. B. Viégas, O. Vinyals, P. Warden, M. Wattenberg, M. Wicke, Y. Yu, and X. Zheng, “Tensorflow: Large-scale machine learning on heterogeneous distributed systems,” arXiv 1603.04467 (2016).

Chintala, S.

A. Paszke, S. Gross, S. Chintala, G. Chanan, E. Yang, Z. DeVito, Z. Lin, A. Desmaison, L. Antiga, and A. Lerer, “Automatic differentiation in pytorch,” in 31st Conference on Neural Information Processing Systems (NIPS, 2017).

Chu, Y. S.

M. O. Hill, I. Calvo-Almazan, M. Allain, M. V. Holt, A. Ulvestad, J. Treu, G. Koblmuller, C. Huang, X. Huang, H. Yan, E. Nazaretski, Y. S. Chu, G. B. Stephenson, V. Chamard, L. J. Lauhon, and S. O. Hruszkewycz, “Measuring three-dimensional strain and structural defects in a single InGaAs nanowire using coherent x-ray multiangle Bragg projection ptychography,” Nano Lett. 18, 811–819 (2018).
[Crossref] [PubMed]

Citro, C.

M. Abadi, A. Agarwal, P. Barham, E. Brevdo, Z. Chen, C. Citro, G. S. Corrado, A. Davis, J. Dean, M. Devin, S. Ghemawat, I. J. Goodfellow, A. Harp, G. Irving, M. Isard, Y. Jia, R. Józefowicz, L. Kaiser, M. Kudlur, J. Levenberg, D. Mané, R. Monga, S. Moore, D. G. Murray, C. Olah, M. Schuster, J. Shlens, B. Steiner, I. Sutskever, K. Talwar, P. A. Tucker, V. Vanhoucke, V. Vasudevan, F. B. Viégas, O. Vinyals, P. Warden, M. Wattenberg, M. Wicke, Y. Yu, and X. Zheng, “Tensorflow: Large-scale machine learning on heterogeneous distributed systems,” arXiv 1603.04467 (2016).

Clare, R.

M. Stockmar, I. Zanette, M. Dierolf, B. Enders, R. Clare, F. Pfeiffer, P. Cloetens, A. Bonnin, and P. Thibault, “X-ray near-field ptychography for optically thick specimens,” Phys. Rev. Appl. 3, 1–6 (2015).
[Crossref]

Clare, R. M.

Clark, J.

J. Clark, X. Huang, R. Harder, and I. Robinson, “High-resolution three-dimensional partially coherent diffraction imaging,” Nat. Commun. 3, 993 (2012).
[Crossref] [PubMed]

Cloetens, P.

M. Stockmar, I. Zanette, M. Dierolf, B. Enders, R. Clare, F. Pfeiffer, P. Cloetens, A. Bonnin, and P. Thibault, “X-ray near-field ptychography for optically thick specimens,” Phys. Rev. Appl. 3, 1–6 (2015).
[Crossref]

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

Corrado, G. S.

M. Abadi, A. Agarwal, P. Barham, E. Brevdo, Z. Chen, C. Citro, G. S. Corrado, A. Davis, J. Dean, M. Devin, S. Ghemawat, I. J. Goodfellow, A. Harp, G. Irving, M. Isard, Y. Jia, R. Józefowicz, L. Kaiser, M. Kudlur, J. Levenberg, D. Mané, R. Monga, S. Moore, D. G. Murray, C. Olah, M. Schuster, J. Shlens, B. Steiner, I. Sutskever, K. Talwar, P. A. Tucker, V. Vanhoucke, V. Vasudevan, F. B. Viégas, O. Vinyals, P. Warden, M. Wattenberg, M. Wicke, Y. Yu, and X. Zheng, “Tensorflow: Large-scale machine learning on heterogeneous distributed systems,” arXiv 1603.04467 (2016).

Cossairt, O.

S. Ghosh, Y. S. Nashed, O. Cossairt, and A. Katsaggelos, “ADP : Automatic differentiation ptychography,” IEEE International Conference on Computational Photography (ICCP) (IEEE, 2018), pp.1–10.

Crimmins, T. R.

D’Alfonso, A. J.

A. W. Yan, A. J. D’Alfonso, A. J. Morgan, C. T. Putkunz, and L. J. Allen, “Fast deterministic ptychographic imaging using x-rays,” Microsc. Microanal. 20, 1090–1099 (2014).
[Crossref] [PubMed]

Dai, Q.

Davis, A.

M. Abadi, A. Agarwal, P. Barham, E. Brevdo, Z. Chen, C. Citro, G. S. Corrado, A. Davis, J. Dean, M. Devin, S. Ghemawat, I. J. Goodfellow, A. Harp, G. Irving, M. Isard, Y. Jia, R. Józefowicz, L. Kaiser, M. Kudlur, J. Levenberg, D. Mané, R. Monga, S. Moore, D. G. Murray, C. Olah, M. Schuster, J. Shlens, B. Steiner, I. Sutskever, K. Talwar, P. A. Tucker, V. Vanhoucke, V. Vasudevan, F. B. Viégas, O. Vinyals, P. Warden, M. Wattenberg, M. Wicke, Y. Yu, and X. Zheng, “Tensorflow: Large-scale machine learning on heterogeneous distributed systems,” arXiv 1603.04467 (2016).

Dean, J.

M. Abadi, A. Agarwal, P. Barham, E. Brevdo, Z. Chen, C. Citro, G. S. Corrado, A. Davis, J. Dean, M. Devin, S. Ghemawat, I. J. Goodfellow, A. Harp, G. Irving, M. Isard, Y. Jia, R. Józefowicz, L. Kaiser, M. Kudlur, J. Levenberg, D. Mané, R. Monga, S. Moore, D. G. Murray, C. Olah, M. Schuster, J. Shlens, B. Steiner, I. Sutskever, K. Talwar, P. A. Tucker, V. Vanhoucke, V. Vasudevan, F. B. Viégas, O. Vinyals, P. Warden, M. Wattenberg, M. Wicke, Y. Yu, and X. Zheng, “Tensorflow: Large-scale machine learning on heterogeneous distributed systems,” arXiv 1603.04467 (2016).

Deng, J.

Y. S. G. Nashed, T. Peterka, J. Deng, and C. Jacobsen, “Distributed automatic differentiation for ptychography,” Procedia Comput. Sci. 108, 404–414 (2017).
[Crossref]

Y. S. Nashed, D. J. Vine, T. Peterka, J. Deng, R. Ross, and C. Jacobsen, “Parallel ptychographic reconstruction,” Opt. Express 22, 223015 (2014).
[Crossref]

Desmaison, A.

A. Paszke, S. Gross, S. Chintala, G. Chanan, E. Yang, Z. DeVito, Z. Lin, A. Desmaison, L. Antiga, and A. Lerer, “Automatic differentiation in pytorch,” in 31st Conference on Neural Information Processing Systems (NIPS, 2017).

Devin, M.

M. Abadi, A. Agarwal, P. Barham, E. Brevdo, Z. Chen, C. Citro, G. S. Corrado, A. Davis, J. Dean, M. Devin, S. Ghemawat, I. J. Goodfellow, A. Harp, G. Irving, M. Isard, Y. Jia, R. Józefowicz, L. Kaiser, M. Kudlur, J. Levenberg, D. Mané, R. Monga, S. Moore, D. G. Murray, C. Olah, M. Schuster, J. Shlens, B. Steiner, I. Sutskever, K. Talwar, P. A. Tucker, V. Vanhoucke, V. Vasudevan, F. B. Viégas, O. Vinyals, P. Warden, M. Wattenberg, M. Wicke, Y. Yu, and X. Zheng, “Tensorflow: Large-scale machine learning on heterogeneous distributed systems,” arXiv 1603.04467 (2016).

DeVito, Z.

A. Paszke, S. Gross, S. Chintala, G. Chanan, E. Yang, Z. DeVito, Z. Lin, A. Desmaison, L. Antiga, and A. Lerer, “Automatic differentiation in pytorch,” in 31st Conference on Neural Information Processing Systems (NIPS, 2017).

Diaz, A.

P. Godard, G. Carbone, M. Allain, F. Mastropietro, G. Chen, L. Capello, A. Diaz, T. Metzger, J. Stangl, and V. Chamard, “Three-dimensional high-resolution quantitative microscopy of extended crystals,” Nat. Commun. 2, 568 (2011).
[Crossref] [PubMed]

Dierolf, M.

M. Stockmar, I. Zanette, M. Dierolf, B. Enders, R. Clare, F. Pfeiffer, P. Cloetens, A. Bonnin, and P. Thibault, “X-ray near-field ptychography for optically thick specimens,” Phys. Rev. Appl. 3, 1–6 (2015).
[Crossref]

R. M. Clare, M. Stockmar, M. Dierolf, I. Zanette, and F. Pfeiffer, “Characterization of near-field ptychography,” Opt. Express 23, 19728 (2015).
[Crossref] [PubMed]

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

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

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

Du, M.

Dwivedi, P.

P. Dwivedi, A. Konijnenberg, S. Pereira, and H. Urbach, “Lateral position correction in ptychography using the gradient of intensity patterns,” Ultramicroscopy 192, 29–36 (2018).
[Crossref] [PubMed]

Edo, T. B.

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

Eldar, Y. C.

E. J. R. Pauwels, A. Beck, Y. C. Eldar, and S. Sabach, “On fienup methods for sparse phase retrieval,” IEEE Trans. Signal Process. 66, 982–991 (2018).
[Crossref]

Elser, V.

Enders, B.

M. Stockmar, I. Zanette, M. Dierolf, B. Enders, R. Clare, F. Pfeiffer, P. Cloetens, A. Bonnin, and P. Thibault, “X-ray near-field ptychography for optically thick specimens,” Phys. Rev. Appl. 3, 1–6 (2015).
[Crossref]

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

Fannjiang, A.

A. Fannjiang, “Absolute uniqueness of phase retrieval with random illumination,” Inverse Probl. 28, 075008 (2012).
[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]

Faulkner, H. M. L.

H. M. L. Faulkner and J. Rodenburg, “Movable aperture lensless transmission microscopy: A novel phase retrieval algorithm,” Phys. Rev. Lett. 93, 023903 (2004).
[Crossref] [PubMed]

Fessler, J. A.

S. Ahn and J. A. Fessler, “Globally convergent image reconstruction for emission tomography using relaxed ordered subsets algorithms,” IEEE Trans. Med. Imaging 22, 613–626 (2003).
[Crossref] [PubMed]

Fienup, J.

Fienup, J. R.

Fuoss, P. H.

S. O. Hruszkewycz, M. Allain, M. V. Holt, C. E. Murray, J. R. Holt, P. H. Fuoss, and V. Chamard, “High-resolution three-dimensional structural microscopy by single-angle Bragg ptychography,” Nat. Mater. 16, 244–251 (2017).
[Crossref]

Gerchberg, R. W.

R. W. Gerchberg and W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik 35, 237–246 (1972).

Ghemawat, S.

M. Abadi, A. Agarwal, P. Barham, E. Brevdo, Z. Chen, C. Citro, G. S. Corrado, A. Davis, J. Dean, M. Devin, S. Ghemawat, I. J. Goodfellow, A. Harp, G. Irving, M. Isard, Y. Jia, R. Józefowicz, L. Kaiser, M. Kudlur, J. Levenberg, D. Mané, R. Monga, S. Moore, D. G. Murray, C. Olah, M. Schuster, J. Shlens, B. Steiner, I. Sutskever, K. Talwar, P. A. Tucker, V. Vanhoucke, V. Vasudevan, F. B. Viégas, O. Vinyals, P. Warden, M. Wattenberg, M. Wicke, Y. Yu, and X. Zheng, “Tensorflow: Large-scale machine learning on heterogeneous distributed systems,” arXiv 1603.04467 (2016).

Ghosh, S.

S. Ghosh, Y. S. Nashed, O. Cossairt, and A. Katsaggelos, “ADP : Automatic differentiation ptychography,” IEEE International Conference on Computational Photography (ICCP) (IEEE, 2018), pp.1–10.

Gilles, M. A.

Godard, P.

V. Chamard, M. Allain, P. Godard, A. Talneau, G. Patriarche, and M. Burghammer, “Strain in a silicon-on-insulator nanostructure revealed by 3D x-ray Bragg ptychography,” Sci. Rep. 5, 9827 (2015).
[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. Godard, G. Carbone, M. Allain, F. Mastropietro, G. Chen, L. Capello, A. Diaz, T. Metzger, J. Stangl, and V. Chamard, “Three-dimensional high-resolution quantitative microscopy of extended crystals,” Nat. Commun. 2, 568 (2011).
[Crossref] [PubMed]

Goodfellow, I. J.

M. Abadi, A. Agarwal, P. Barham, E. Brevdo, Z. Chen, C. Citro, G. S. Corrado, A. Davis, J. Dean, M. Devin, S. Ghemawat, I. J. Goodfellow, A. Harp, G. Irving, M. Isard, Y. Jia, R. Józefowicz, L. Kaiser, M. Kudlur, J. Levenberg, D. Mané, R. Monga, S. Moore, D. G. Murray, C. Olah, M. Schuster, J. Shlens, B. Steiner, I. Sutskever, K. Talwar, P. A. Tucker, V. Vanhoucke, V. Vasudevan, F. B. Viégas, O. Vinyals, P. Warden, M. Wattenberg, M. Wicke, Y. Yu, and X. Zheng, “Tensorflow: Large-scale machine learning on heterogeneous distributed systems,” arXiv 1603.04467 (2016).

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics (W.H. Freeman, 2017).

Griewank, A.

A. Griewank and A. Walther, Evaluating derivatives: principles and techniques of algorithmic differentiation, vol. 105 (SIAM, 2008).
[Crossref]

Gross, S.

A. Paszke, S. Gross, S. Chintala, G. Chanan, E. Yang, Z. DeVito, Z. Lin, A. Desmaison, L. Antiga, and A. Lerer, “Automatic differentiation in pytorch,” in 31st Conference on Neural Information Processing Systems (NIPS, 2017).

Guizar-Sicairos, M.

Guo, K.

Haldar, J. P.

R. Xu, M. Soltanolkotabi, J. P. Haldar, W. Unglaub, J. Zusman, A. F. Levi, and R. M. Leahy, “Accelerated wirtinger flow: A fast algorithm for ptychography,” arXiv preprint arXiv:1806.05546 (2018).

Harder, R.

J. Clark, X. Huang, R. Harder, and I. Robinson, “High-resolution three-dimensional partially coherent diffraction imaging,” Nat. Commun. 3, 993 (2012).
[Crossref] [PubMed]

Harp, A.

M. Abadi, A. Agarwal, P. Barham, E. Brevdo, Z. Chen, C. Citro, G. S. Corrado, A. Davis, J. Dean, M. Devin, S. Ghemawat, I. J. Goodfellow, A. Harp, G. Irving, M. Isard, Y. Jia, R. Józefowicz, L. Kaiser, M. Kudlur, J. Levenberg, D. Mané, R. Monga, S. Moore, D. G. Murray, C. Olah, M. Schuster, J. Shlens, B. Steiner, I. Sutskever, K. Talwar, P. A. Tucker, V. Vanhoucke, V. Vasudevan, F. B. Viégas, O. Vinyals, P. Warden, M. Wattenberg, M. Wicke, Y. Yu, and X. Zheng, “Tensorflow: Large-scale machine learning on heterogeneous distributed systems,” arXiv 1603.04467 (2016).

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]

Hill, M. O.

M. O. Hill, I. Calvo-Almazan, M. Allain, M. V. Holt, A. Ulvestad, J. Treu, G. Koblmuller, C. Huang, X. Huang, H. Yan, E. Nazaretski, Y. S. Chu, G. B. Stephenson, V. Chamard, L. J. Lauhon, and S. O. Hruszkewycz, “Measuring three-dimensional strain and structural defects in a single InGaAs nanowire using coherent x-ray multiangle Bragg projection ptychography,” Nano Lett. 18, 811–819 (2018).
[Crossref] [PubMed]

Hoffmann, P. H.

P. H. Hoffmann, “A hitchhiker’s guide to automatic differentiation,” Numer. Algorithms 72, 775–811 (2016).
[Crossref]

Holsztynski, W.

Holt, J. R.

S. O. Hruszkewycz, M. Allain, M. V. Holt, C. E. Murray, J. R. Holt, P. H. Fuoss, and V. Chamard, “High-resolution three-dimensional structural microscopy by single-angle Bragg ptychography,” Nat. Mater. 16, 244–251 (2017).
[Crossref]

Holt, M. V.

M. O. Hill, I. Calvo-Almazan, M. Allain, M. V. Holt, A. Ulvestad, J. Treu, G. Koblmuller, C. Huang, X. Huang, H. Yan, E. Nazaretski, Y. S. Chu, G. B. Stephenson, V. Chamard, L. J. Lauhon, and S. O. Hruszkewycz, “Measuring three-dimensional strain and structural defects in a single InGaAs nanowire using coherent x-ray multiangle Bragg projection ptychography,” Nano Lett. 18, 811–819 (2018).
[Crossref] [PubMed]

S. O. Hruszkewycz, M. Allain, M. V. Holt, C. E. Murray, J. R. Holt, P. H. Fuoss, and V. Chamard, “High-resolution three-dimensional structural microscopy by single-angle Bragg ptychography,” Nat. Mater. 16, 244–251 (2017).
[Crossref]

Hoppe, W.

W. Hoppe, “Beugung im inhomogenen Primarstrahlwellenfeld. III. Amplituden-und Phasenbestimmung bei unperiodischen Objekten,” Acta Crystallogr. A 25, 508–514 (1969).
[Crossref]

Hruszkewycz, S. O.

M. O. Hill, I. Calvo-Almazan, M. Allain, M. V. Holt, A. Ulvestad, J. Treu, G. Koblmuller, C. Huang, X. Huang, H. Yan, E. Nazaretski, Y. S. Chu, G. B. Stephenson, V. Chamard, L. J. Lauhon, and S. O. Hruszkewycz, “Measuring three-dimensional strain and structural defects in a single InGaAs nanowire using coherent x-ray multiangle Bragg projection ptychography,” Nano Lett. 18, 811–819 (2018).
[Crossref] [PubMed]

S. O. Hruszkewycz, M. Allain, M. V. Holt, C. E. Murray, J. R. Holt, P. H. Fuoss, and V. Chamard, “High-resolution three-dimensional structural microscopy by single-angle Bragg ptychography,” Nat. Mater. 16, 244–251 (2017).
[Crossref]

Huang, C.

M. O. Hill, I. Calvo-Almazan, M. Allain, M. V. Holt, A. Ulvestad, J. Treu, G. Koblmuller, C. Huang, X. Huang, H. Yan, E. Nazaretski, Y. S. Chu, G. B. Stephenson, V. Chamard, L. J. Lauhon, and S. O. Hruszkewycz, “Measuring three-dimensional strain and structural defects in a single InGaAs nanowire using coherent x-ray multiangle Bragg projection ptychography,” Nano Lett. 18, 811–819 (2018).
[Crossref] [PubMed]

Huang, X.

M. O. Hill, I. Calvo-Almazan, M. Allain, M. V. Holt, A. Ulvestad, J. Treu, G. Koblmuller, C. Huang, X. Huang, H. Yan, E. Nazaretski, Y. S. Chu, G. B. Stephenson, V. Chamard, L. J. Lauhon, and S. O. Hruszkewycz, “Measuring three-dimensional strain and structural defects in a single InGaAs nanowire using coherent x-ray multiangle Bragg projection ptychography,” Nano Lett. 18, 811–819 (2018).
[Crossref] [PubMed]

J. Clark, X. Huang, R. Harder, and I. Robinson, “High-resolution three-dimensional partially coherent diffraction imaging,” Nat. Commun. 3, 993 (2012).
[Crossref] [PubMed]

Humphry, M. J.

Irving, G.

M. Abadi, A. Agarwal, P. Barham, E. Brevdo, Z. Chen, C. Citro, G. S. Corrado, A. Davis, J. Dean, M. Devin, S. Ghemawat, I. J. Goodfellow, A. Harp, G. Irving, M. Isard, Y. Jia, R. Józefowicz, L. Kaiser, M. Kudlur, J. Levenberg, D. Mané, R. Monga, S. Moore, D. G. Murray, C. Olah, M. Schuster, J. Shlens, B. Steiner, I. Sutskever, K. Talwar, P. A. Tucker, V. Vanhoucke, V. Vasudevan, F. B. Viégas, O. Vinyals, P. Warden, M. Wattenberg, M. Wicke, Y. Yu, and X. Zheng, “Tensorflow: Large-scale machine learning on heterogeneous distributed systems,” arXiv 1603.04467 (2016).

Isard, M.

M. Abadi, A. Agarwal, P. Barham, E. Brevdo, Z. Chen, C. Citro, G. S. Corrado, A. Davis, J. Dean, M. Devin, S. Ghemawat, I. J. Goodfellow, A. Harp, G. Irving, M. Isard, Y. Jia, R. Józefowicz, L. Kaiser, M. Kudlur, J. Levenberg, D. Mané, R. Monga, S. Moore, D. G. Murray, C. Olah, M. Schuster, J. Shlens, B. Steiner, I. Sutskever, K. Talwar, P. A. Tucker, V. Vanhoucke, V. Vasudevan, F. B. Viégas, O. Vinyals, P. Warden, M. Wattenberg, M. Wicke, Y. Yu, and X. Zheng, “Tensorflow: Large-scale machine learning on heterogeneous distributed systems,” arXiv 1603.04467 (2016).

Jacobsen, C.

M. A. Gilles, Y. S. G. Nashed, M. Du, C. Jacobsen, and S. M. Wild, “3D x-ray imaging of continuous objects beyond the depth of focus limit,” Optica 5, 1078–1085 (2018).
[Crossref] [PubMed]

Y. S. G. Nashed, T. Peterka, J. Deng, and C. Jacobsen, “Distributed automatic differentiation for ptychography,” Procedia Comput. Sci. 108, 404–414 (2017).
[Crossref]

Y. S. Nashed, D. J. Vine, T. Peterka, J. Deng, R. Ross, and C. Jacobsen, “Parallel ptychographic reconstruction,” Opt. Express 22, 223015 (2014).
[Crossref]

Jia, Y.

M. Abadi, A. Agarwal, P. Barham, E. Brevdo, Z. Chen, C. Citro, G. S. Corrado, A. Davis, J. Dean, M. Devin, S. Ghemawat, I. J. Goodfellow, A. Harp, G. Irving, M. Isard, Y. Jia, R. Józefowicz, L. Kaiser, M. Kudlur, J. Levenberg, D. Mané, R. Monga, S. Moore, D. G. Murray, C. Olah, M. Schuster, J. Shlens, B. Steiner, I. Sutskever, K. Talwar, P. A. Tucker, V. Vanhoucke, V. Vasudevan, F. B. Viégas, O. Vinyals, P. Warden, M. Wattenberg, M. Wicke, Y. Yu, and X. Zheng, “Tensorflow: Large-scale machine learning on heterogeneous distributed systems,” arXiv 1603.04467 (2016).

Johnson, D.

Józefowicz, R.

M. Abadi, A. Agarwal, P. Barham, E. Brevdo, Z. Chen, C. Citro, G. S. Corrado, A. Davis, J. Dean, M. Devin, S. Ghemawat, I. J. Goodfellow, A. Harp, G. Irving, M. Isard, Y. Jia, R. Józefowicz, L. Kaiser, M. Kudlur, J. Levenberg, D. Mané, R. Monga, S. Moore, D. G. Murray, C. Olah, M. Schuster, J. Shlens, B. Steiner, I. Sutskever, K. Talwar, P. A. Tucker, V. Vanhoucke, V. Vasudevan, F. B. Viégas, O. Vinyals, P. Warden, M. Wattenberg, M. Wicke, Y. Yu, and X. Zheng, “Tensorflow: Large-scale machine learning on heterogeneous distributed systems,” arXiv 1603.04467 (2016).

Jurling, A. S.

Kaiser, L.

M. Abadi, A. Agarwal, P. Barham, E. Brevdo, Z. Chen, C. Citro, G. S. Corrado, A. Davis, J. Dean, M. Devin, S. Ghemawat, I. J. Goodfellow, A. Harp, G. Irving, M. Isard, Y. Jia, R. Józefowicz, L. Kaiser, M. Kudlur, J. Levenberg, D. Mané, R. Monga, S. Moore, D. G. Murray, C. Olah, M. Schuster, J. Shlens, B. Steiner, I. Sutskever, K. Talwar, P. A. Tucker, V. Vanhoucke, V. Vasudevan, F. B. Viégas, O. Vinyals, P. Warden, M. Wattenberg, M. Wicke, Y. Yu, and X. Zheng, “Tensorflow: Large-scale machine learning on heterogeneous distributed systems,” arXiv 1603.04467 (2016).

Katsaggelos, A.

S. Ghosh, Y. S. Nashed, O. Cossairt, and A. Katsaggelos, “ADP : Automatic differentiation ptychography,” IEEE International Conference on Computational Photography (ICCP) (IEEE, 2018), pp.1–10.

Kewish, C. M.

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

Kingma, D. P.

D. P. Kingma and J. Ba, “Adam: A method for stochastic optimization,” arXiv 1412.6980 (2014).

Koblmuller, G.

M. O. Hill, I. Calvo-Almazan, M. Allain, M. V. Holt, A. Ulvestad, J. Treu, G. Koblmuller, C. Huang, X. Huang, H. Yan, E. Nazaretski, Y. S. Chu, G. B. Stephenson, V. Chamard, L. J. Lauhon, and S. O. Hruszkewycz, “Measuring three-dimensional strain and structural defects in a single InGaAs nanowire using coherent x-ray multiangle Bragg projection ptychography,” Nano Lett. 18, 811–819 (2018).
[Crossref] [PubMed]

Konijnenberg, A.

P. Dwivedi, A. Konijnenberg, S. Pereira, and H. Urbach, “Lateral position correction in ptychography using the gradient of intensity patterns,” Ultramicroscopy 192, 29–36 (2018).
[Crossref] [PubMed]

Kreutz-Delgado, K.

K. Kreutz-Delgado, “The complex gradient operator and the cr-calculus,” arXiv preprint arXiv:0906.4835 (2009).

Kröger, K.

A. L. Robisch, K. Kröger, A. Rack, and T. Salditt, “Near-field ptychography using lateral and longitudinal shifts,” New J. Phys. 17, 073033 (2015).
[Crossref]

Kudlur, M.

M. Abadi, A. Agarwal, P. Barham, E. Brevdo, Z. Chen, C. Citro, G. S. Corrado, A. Davis, J. Dean, M. Devin, S. Ghemawat, I. J. Goodfellow, A. Harp, G. Irving, M. Isard, Y. Jia, R. Józefowicz, L. Kaiser, M. Kudlur, J. Levenberg, D. Mané, R. Monga, S. Moore, D. G. Murray, C. Olah, M. Schuster, J. Shlens, B. Steiner, I. Sutskever, K. Talwar, P. A. Tucker, V. Vanhoucke, V. Vasudevan, F. B. Viégas, O. Vinyals, P. Warden, M. Wattenberg, M. Wicke, Y. Yu, and X. Zheng, “Tensorflow: Large-scale machine learning on heterogeneous distributed systems,” arXiv 1603.04467 (2016).

Lathauwer, L. D.

L. Sorber, M. V. Barel, and L. D. Lathauwer, “Unconstrained optimization of real functions in complex variables,” SIAM J. Optim. 22, 879–898 (2012).
[Crossref]

Lauhon, L. J.

M. O. Hill, I. Calvo-Almazan, M. Allain, M. V. Holt, A. Ulvestad, J. Treu, G. Koblmuller, C. Huang, X. Huang, H. Yan, E. Nazaretski, Y. S. Chu, G. B. Stephenson, V. Chamard, L. J. Lauhon, and S. O. Hruszkewycz, “Measuring three-dimensional strain and structural defects in a single InGaAs nanowire using coherent x-ray multiangle Bragg projection ptychography,” Nano Lett. 18, 811–819 (2018).
[Crossref] [PubMed]

Leahy, R. M.

R. Xu, M. Soltanolkotabi, J. P. Haldar, W. Unglaub, J. Zusman, A. F. Levi, and R. M. Leahy, “Accelerated wirtinger flow: A fast algorithm for ptychography,” arXiv preprint arXiv:1806.05546 (2018).

LeCun, Y. A.

Y. A. LeCun, L. Bottou, G. B. Orr, and K.-R. Müller, “Efficient backprop,” in Neural networks: Tricks of the trade, (Springer, 2012), pp. 9–48.

Lerer, A.

A. Paszke, S. Gross, S. Chintala, G. Chanan, E. Yang, Z. DeVito, Z. Lin, A. Desmaison, L. Antiga, and A. Lerer, “Automatic differentiation in pytorch,” in 31st Conference on Neural Information Processing Systems (NIPS, 2017).

Levenberg, J.

M. Abadi, A. Agarwal, P. Barham, E. Brevdo, Z. Chen, C. Citro, G. S. Corrado, A. Davis, J. Dean, M. Devin, S. Ghemawat, I. J. Goodfellow, A. Harp, G. Irving, M. Isard, Y. Jia, R. Józefowicz, L. Kaiser, M. Kudlur, J. Levenberg, D. Mané, R. Monga, S. Moore, D. G. Murray, C. Olah, M. Schuster, J. Shlens, B. Steiner, I. Sutskever, K. Talwar, P. A. Tucker, V. Vanhoucke, V. Vasudevan, F. B. Viégas, O. Vinyals, P. Warden, M. Wattenberg, M. Wicke, Y. Yu, and X. Zheng, “Tensorflow: Large-scale machine learning on heterogeneous distributed systems,” arXiv 1603.04467 (2016).

Levi, A. F.

R. Xu, M. Soltanolkotabi, J. P. Haldar, W. Unglaub, J. Zusman, A. F. Levi, and R. M. Leahy, “Accelerated wirtinger flow: A fast algorithm for ptychography,” arXiv preprint arXiv:1806.05546 (2018).

Li, J.

J. Li and T. Zhou, “Numerical optimization algorithms for wavefront phase retrieval from multiple measurements,” Inverse Probl. & Imaging 11, 721–743 (2017).
[Crossref]

Li, P.

Li, X.

E. J. Candes, X. Li, and M. Soltanolkotabi, “Phase retrieval via Wirtinger flow: theory and algorithms,” IEEE Trans. Inf. Theory 61, 1985–2007 (2015).
[Crossref]

Liang, Y.

H. Zhang and Y. Liang, “Reshaped wirtinger flow for solving quadratic system of equations,” Adv. Neural. Inf. Process. Syst. 29, 2622–2630 (2016).

Lin, Z.

A. Paszke, S. Gross, S. Chintala, G. Chanan, E. Yang, Z. DeVito, Z. Lin, A. Desmaison, L. Antiga, and A. Lerer, “Automatic differentiation in pytorch,” in 31st Conference on Neural Information Processing Systems (NIPS, 2017).

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]

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]

Maclaurin, D.

D. Maclaurin, “Modeling, inference and optimization with composable differentiable procedures,” Ph.D. thesis, Harvard University (2016).

Maia, F.

J. Qian, C. Yang, A. Schirotzek, F. Maia, and S. Marchesini, “Efficient algorithms for ptychographic phase retrieval,” Inverse Probl. Appli. Contemp. Math 615, 261–280 (2014).

Maiden, A. M.

A. M. Maiden, D. Johnson, and P. Li, “Further improvements to the ptychographical iterative engine,” Optica 4, 736–745 (2017).
[Crossref]

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

A. M. Maiden, M. J. Humphry, and J. M. Rodenburg, “Ptychographic transmission microscopy in three dimensions using a multi-slice approach,” J. Opt. Soc. Am. A 29, 1606–1614 (2012).
[Crossref]

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

Mané, D.

M. Abadi, A. Agarwal, P. Barham, E. Brevdo, Z. Chen, C. Citro, G. S. Corrado, A. Davis, J. Dean, M. Devin, S. Ghemawat, I. J. Goodfellow, A. Harp, G. Irving, M. Isard, Y. Jia, R. Józefowicz, L. Kaiser, M. Kudlur, J. Levenberg, D. Mané, R. Monga, S. Moore, D. G. Murray, C. Olah, M. Schuster, J. Shlens, B. Steiner, I. Sutskever, K. Talwar, P. A. Tucker, V. Vanhoucke, V. Vasudevan, F. B. Viégas, O. Vinyals, P. Warden, M. Wattenberg, M. Wicke, Y. Yu, and X. Zheng, “Tensorflow: Large-scale machine learning on heterogeneous distributed systems,” arXiv 1603.04467 (2016).

Marchesini, S.

J. Qian, C. Yang, A. Schirotzek, F. Maia, and S. Marchesini, “Efficient algorithms for ptychographic phase retrieval,” Inverse Probl. Appli. Contemp. Math 615, 261–280 (2014).

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

S. Marchesini, “A unified evaluation of iterative projection algorithms for phase retrieval,” Rev. Sci. Instruments 78, 011301 (2007).
[Crossref]

Mastropietro, F.

P. Godard, G. Carbone, M. Allain, F. Mastropietro, G. Chen, L. Capello, A. Diaz, T. Metzger, J. Stangl, and V. Chamard, “Three-dimensional high-resolution quantitative microscopy of extended crystals,” Nat. Commun. 2, 568 (2011).
[Crossref] [PubMed]

McNulty, I.

Menzel, A.

P. Thibault and A. Menzel, “Reconstructing state mixtures from diffraction measurements,” Nature 494, 68–71 (2013).
[Crossref] [PubMed]

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

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]

Metzger, T.

P. Godard, G. Carbone, M. Allain, F. Mastropietro, G. Chen, L. Capello, A. Diaz, T. Metzger, J. Stangl, and V. Chamard, “Three-dimensional high-resolution quantitative microscopy of extended crystals,” Nat. Commun. 2, 568 (2011).
[Crossref] [PubMed]

Monga, R.

M. Abadi, A. Agarwal, P. Barham, E. Brevdo, Z. Chen, C. Citro, G. S. Corrado, A. Davis, J. Dean, M. Devin, S. Ghemawat, I. J. Goodfellow, A. Harp, G. Irving, M. Isard, Y. Jia, R. Józefowicz, L. Kaiser, M. Kudlur, J. Levenberg, D. Mané, R. Monga, S. Moore, D. G. Murray, C. Olah, M. Schuster, J. Shlens, B. Steiner, I. Sutskever, K. Talwar, P. A. Tucker, V. Vanhoucke, V. Vasudevan, F. B. Viégas, O. Vinyals, P. Warden, M. Wattenberg, M. Wicke, Y. Yu, and X. Zheng, “Tensorflow: Large-scale machine learning on heterogeneous distributed systems,” arXiv 1603.04467 (2016).

Moore, S.

M. Abadi, A. Agarwal, P. Barham, E. Brevdo, Z. Chen, C. Citro, G. S. Corrado, A. Davis, J. Dean, M. Devin, S. Ghemawat, I. J. Goodfellow, A. Harp, G. Irving, M. Isard, Y. Jia, R. Józefowicz, L. Kaiser, M. Kudlur, J. Levenberg, D. Mané, R. Monga, S. Moore, D. G. Murray, C. Olah, M. Schuster, J. Shlens, B. Steiner, I. Sutskever, K. Talwar, P. A. Tucker, V. Vanhoucke, V. Vasudevan, F. B. Viégas, O. Vinyals, P. Warden, M. Wattenberg, M. Wicke, Y. Yu, and X. Zheng, “Tensorflow: Large-scale machine learning on heterogeneous distributed systems,” arXiv 1603.04467 (2016).

Morgan, A. J.

A. W. Yan, A. J. D’Alfonso, A. J. Morgan, C. T. Putkunz, and L. J. Allen, “Fast deterministic ptychographic imaging using x-rays,” Microsc. Microanal. 20, 1090–1099 (2014).
[Crossref] [PubMed]

Müller, K.-R.

Y. A. LeCun, L. Bottou, G. B. Orr, and K.-R. Müller, “Efficient backprop,” in Neural networks: Tricks of the trade, (Springer, 2012), pp. 9–48.

Murray, C. E.

S. O. Hruszkewycz, M. Allain, M. V. Holt, C. E. Murray, J. R. Holt, P. H. Fuoss, and V. Chamard, “High-resolution three-dimensional structural microscopy by single-angle Bragg ptychography,” Nat. Mater. 16, 244–251 (2017).
[Crossref]

Murray, D. G.

M. Abadi, A. Agarwal, P. Barham, E. Brevdo, Z. Chen, C. Citro, G. S. Corrado, A. Davis, J. Dean, M. Devin, S. Ghemawat, I. J. Goodfellow, A. Harp, G. Irving, M. Isard, Y. Jia, R. Józefowicz, L. Kaiser, M. Kudlur, J. Levenberg, D. Mané, R. Monga, S. Moore, D. G. Murray, C. Olah, M. Schuster, J. Shlens, B. Steiner, I. Sutskever, K. Talwar, P. A. Tucker, V. Vanhoucke, V. Vasudevan, F. B. Viégas, O. Vinyals, P. Warden, M. Wattenberg, M. Wicke, Y. Yu, and X. Zheng, “Tensorflow: Large-scale machine learning on heterogeneous distributed systems,” arXiv 1603.04467 (2016).

Nashed, Y. S.

Y. S. Nashed, D. J. Vine, T. Peterka, J. Deng, R. Ross, and C. Jacobsen, “Parallel ptychographic reconstruction,” Opt. Express 22, 223015 (2014).
[Crossref]

S. Ghosh, Y. S. Nashed, O. Cossairt, and A. Katsaggelos, “ADP : Automatic differentiation ptychography,” IEEE International Conference on Computational Photography (ICCP) (IEEE, 2018), pp.1–10.

Nashed, Y. S. G.

M. A. Gilles, Y. S. G. Nashed, M. Du, C. Jacobsen, and S. M. Wild, “3D x-ray imaging of continuous objects beyond the depth of focus limit,” Optica 5, 1078–1085 (2018).
[Crossref] [PubMed]

Y. S. G. Nashed, T. Peterka, J. Deng, and C. Jacobsen, “Distributed automatic differentiation for ptychography,” Procedia Comput. Sci. 108, 404–414 (2017).
[Crossref]

Nazaretski, E.

M. O. Hill, I. Calvo-Almazan, M. Allain, M. V. Holt, A. Ulvestad, J. Treu, G. Koblmuller, C. Huang, X. Huang, H. Yan, E. Nazaretski, Y. S. Chu, G. B. Stephenson, V. Chamard, L. J. Lauhon, and S. O. Hruszkewycz, “Measuring three-dimensional strain and structural defects in a single InGaAs nanowire using coherent x-ray multiangle Bragg projection ptychography,” Nano Lett. 18, 811–819 (2018).
[Crossref] [PubMed]

Olah, C.

M. Abadi, A. Agarwal, P. Barham, E. Brevdo, Z. Chen, C. Citro, G. S. Corrado, A. Davis, J. Dean, M. Devin, S. Ghemawat, I. J. Goodfellow, A. Harp, G. Irving, M. Isard, Y. Jia, R. Józefowicz, L. Kaiser, M. Kudlur, J. Levenberg, D. Mané, R. Monga, S. Moore, D. G. Murray, C. Olah, M. Schuster, J. Shlens, B. Steiner, I. Sutskever, K. Talwar, P. A. Tucker, V. Vanhoucke, V. Vasudevan, F. B. Viégas, O. Vinyals, P. Warden, M. Wattenberg, M. Wicke, Y. Yu, and X. Zheng, “Tensorflow: Large-scale machine learning on heterogeneous distributed systems,” arXiv 1603.04467 (2016).

Orr, G. B.

Y. A. LeCun, L. Bottou, G. B. Orr, and K.-R. Müller, “Efficient backprop,” in Neural networks: Tricks of the trade, (Springer, 2012), pp. 9–48.

Paszke, A.

A. Paszke, S. Gross, S. Chintala, G. Chanan, E. Yang, Z. DeVito, Z. Lin, A. Desmaison, L. Antiga, and A. Lerer, “Automatic differentiation in pytorch,” in 31st Conference on Neural Information Processing Systems (NIPS, 2017).

Patriarche, G.

V. Chamard, M. Allain, P. Godard, A. Talneau, G. Patriarche, and M. Burghammer, “Strain in a silicon-on-insulator nanostructure revealed by 3D x-ray Bragg ptychography,” Sci. Rep. 5, 9827 (2015).
[Crossref]

Pauwels, E. J. R.

E. J. R. Pauwels, A. Beck, Y. C. Eldar, and S. Sabach, “On fienup methods for sparse phase retrieval,” IEEE Trans. Signal Process. 66, 982–991 (2018).
[Crossref]

Pearlmutter, B. A.

A. G. Baydin, B. A. Pearlmutter, A. A. Radul, and J. M. Siskind, “Automatic differentiation in machine learning: a survey,” J. Mach. Learn. Res. 18, 1–43 (2015).

Pereira, S.

P. Dwivedi, A. Konijnenberg, S. Pereira, and H. Urbach, “Lateral position correction in ptychography using the gradient of intensity patterns,” Ultramicroscopy 192, 29–36 (2018).
[Crossref] [PubMed]

Pešic, Z. D.

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

Peterka, T.

Y. S. G. Nashed, T. Peterka, J. Deng, and C. Jacobsen, “Distributed automatic differentiation for ptychography,” Procedia Comput. Sci. 108, 404–414 (2017).
[Crossref]

Y. S. Nashed, D. J. Vine, T. Peterka, J. Deng, R. Ross, and C. Jacobsen, “Parallel ptychographic reconstruction,” Opt. Express 22, 223015 (2014).
[Crossref]

Pfeiffer, F.

M. Stockmar, I. Zanette, M. Dierolf, B. Enders, R. Clare, F. Pfeiffer, P. Cloetens, A. Bonnin, and P. Thibault, “X-ray near-field ptychography for optically thick specimens,” Phys. Rev. Appl. 3, 1–6 (2015).
[Crossref]

R. M. Clare, M. Stockmar, M. Dierolf, I. Zanette, and F. Pfeiffer, “Characterization of near-field ptychography,” Opt. Express 23, 19728 (2015).
[Crossref] [PubMed]

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

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

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

Putkunz, C. T.

A. W. Yan, A. J. D’Alfonso, A. J. Morgan, C. T. Putkunz, and L. J. Allen, “Fast deterministic ptychographic imaging using x-rays,” Microsc. Microanal. 20, 1090–1099 (2014).
[Crossref] [PubMed]

Qian, J.

J. Qian, C. Yang, A. Schirotzek, F. Maia, and S. Marchesini, “Efficient algorithms for ptychographic phase retrieval,” Inverse Probl. Appli. Contemp. Math 615, 261–280 (2014).

Qiu, C.-W.

Rack, A.

A. L. Robisch, K. Kröger, A. Rack, and T. Salditt, “Near-field ptychography using lateral and longitudinal shifts,” New J. Phys. 17, 073033 (2015).
[Crossref]

Radul, A. A.

A. G. Baydin, B. A. Pearlmutter, A. A. Radul, and J. M. Siskind, “Automatic differentiation in machine learning: a survey,” J. Mach. Learn. Res. 18, 1–43 (2015).

Rau, C.

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

Robinson, I.

J. Clark, X. Huang, R. Harder, and I. Robinson, “High-resolution three-dimensional partially coherent diffraction imaging,” Nat. Commun. 3, 993 (2012).
[Crossref] [PubMed]

Robisch, A. L.

A. L. Robisch, K. Kröger, A. Rack, and T. Salditt, “Near-field ptychography using lateral and longitudinal shifts,” New J. Phys. 17, 073033 (2015).
[Crossref]

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]

H. M. L. Faulkner and J. Rodenburg, “Movable aperture lensless transmission microscopy: A novel phase retrieval algorithm,” Phys. Rev. Lett. 93, 023903 (2004).
[Crossref] [PubMed]

Rodenburg, J. M.

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

A. M. Maiden, M. J. Humphry, and J. M. Rodenburg, “Ptychographic transmission microscopy in three dimensions using a multi-slice approach,” J. Opt. Soc. Am. A 29, 1606–1614 (2012).
[Crossref]

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]

Rosasco, L.

Y. Yao, L. Rosasco, and A. Caponnetto, “On early stopping in gradient descent learning,” Constr. Approx. 26, 289–315 (2007).
[Crossref]

Ross, R.

Y. S. Nashed, D. J. Vine, T. Peterka, J. Deng, R. Ross, and C. Jacobsen, “Parallel ptychographic reconstruction,” Opt. Express 22, 223015 (2014).
[Crossref]

Ruder, S.

S. Ruder, “An overview of gradient descent optimization algorithms,” arXiv preprint arXiv:1609.04747 (2016).

Sabach, S.

E. J. R. Pauwels, A. Beck, Y. C. Eldar, and S. Sabach, “On fienup methods for sparse phase retrieval,” IEEE Trans. Signal Process. 66, 982–991 (2018).
[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]

Salditt, T.

A. L. Robisch, K. Kröger, A. Rack, and T. Salditt, “Near-field ptychography using lateral and longitudinal shifts,” New J. Phys. 17, 073033 (2015).
[Crossref]

Saxton, W. O.

R. W. Gerchberg and W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik 35, 237–246 (1972).

Schirotzek, A.

J. Qian, C. Yang, A. Schirotzek, F. Maia, and S. Marchesini, “Efficient algorithms for ptychographic phase retrieval,” Inverse Probl. Appli. Contemp. Math 615, 261–280 (2014).

Schneider, P.

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

Schuster, M.

M. Abadi, A. Agarwal, P. Barham, E. Brevdo, Z. Chen, C. Citro, G. S. Corrado, A. Davis, J. Dean, M. Devin, S. Ghemawat, I. J. Goodfellow, A. Harp, G. Irving, M. Isard, Y. Jia, R. Józefowicz, L. Kaiser, M. Kudlur, J. Levenberg, D. Mané, R. Monga, S. Moore, D. G. Murray, C. Olah, M. Schuster, J. Shlens, B. Steiner, I. Sutskever, K. Talwar, P. A. Tucker, V. Vanhoucke, V. Vasudevan, F. B. Viégas, O. Vinyals, P. Warden, M. Wattenberg, M. Wicke, Y. Yu, and X. Zheng, “Tensorflow: Large-scale machine learning on heterogeneous distributed systems,” arXiv 1603.04467 (2016).

Shlens, J.

M. Abadi, A. Agarwal, P. Barham, E. Brevdo, Z. Chen, C. Citro, G. S. Corrado, A. Davis, J. Dean, M. Devin, S. Ghemawat, I. J. Goodfellow, A. Harp, G. Irving, M. Isard, Y. Jia, R. Józefowicz, L. Kaiser, M. Kudlur, J. Levenberg, D. Mané, R. Monga, S. Moore, D. G. Murray, C. Olah, M. Schuster, J. Shlens, B. Steiner, I. Sutskever, K. Talwar, P. A. Tucker, V. Vanhoucke, V. Vasudevan, F. B. Viégas, O. Vinyals, P. Warden, M. Wattenberg, M. Wicke, Y. Yu, and X. Zheng, “Tensorflow: Large-scale machine learning on heterogeneous distributed systems,” arXiv 1603.04467 (2016).

Shpyrko, O. G.

Siskind, J. M.

A. G. Baydin, B. A. Pearlmutter, A. A. Radul, and J. M. Siskind, “Automatic differentiation in machine learning: a survey,” J. Mach. Learn. Res. 18, 1–43 (2015).

Soltanolkotabi, M.

E. J. Candes, X. Li, and M. Soltanolkotabi, “Phase retrieval via Wirtinger flow: theory and algorithms,” IEEE Trans. Inf. Theory 61, 1985–2007 (2015).
[Crossref]

R. Xu, M. Soltanolkotabi, J. P. Haldar, W. Unglaub, J. Zusman, A. F. Levi, and R. M. Leahy, “Accelerated wirtinger flow: A fast algorithm for ptychography,” arXiv preprint arXiv:1806.05546 (2018).

Sorber, L.

L. Sorber, M. V. Barel, and L. D. Lathauwer, “Unconstrained optimization of real functions in complex variables,” SIAM J. Optim. 22, 879–898 (2012).
[Crossref]

Stangl, J.

P. Godard, G. Carbone, M. Allain, F. Mastropietro, G. Chen, L. Capello, A. Diaz, T. Metzger, J. Stangl, and V. Chamard, “Three-dimensional high-resolution quantitative microscopy of extended crystals,” Nat. Commun. 2, 568 (2011).
[Crossref] [PubMed]

Steiner, B.

M. Abadi, A. Agarwal, P. Barham, E. Brevdo, Z. Chen, C. Citro, G. S. Corrado, A. Davis, J. Dean, M. Devin, S. Ghemawat, I. J. Goodfellow, A. Harp, G. Irving, M. Isard, Y. Jia, R. Józefowicz, L. Kaiser, M. Kudlur, J. Levenberg, D. Mané, R. Monga, S. Moore, D. G. Murray, C. Olah, M. Schuster, J. Shlens, B. Steiner, I. Sutskever, K. Talwar, P. A. Tucker, V. Vanhoucke, V. Vasudevan, F. B. Viégas, O. Vinyals, P. Warden, M. Wattenberg, M. Wicke, Y. Yu, and X. Zheng, “Tensorflow: Large-scale machine learning on heterogeneous distributed systems,” arXiv 1603.04467 (2016).

Stephenson, G. B.

M. O. Hill, I. Calvo-Almazan, M. Allain, M. V. Holt, A. Ulvestad, J. Treu, G. Koblmuller, C. Huang, X. Huang, H. Yan, E. Nazaretski, Y. S. Chu, G. B. Stephenson, V. Chamard, L. J. Lauhon, and S. O. Hruszkewycz, “Measuring three-dimensional strain and structural defects in a single InGaAs nanowire using coherent x-ray multiangle Bragg projection ptychography,” Nano Lett. 18, 811–819 (2018).
[Crossref] [PubMed]

Stockmar, M.

M. Stockmar, I. Zanette, M. Dierolf, B. Enders, R. Clare, F. Pfeiffer, P. Cloetens, A. Bonnin, and P. Thibault, “X-ray near-field ptychography for optically thick specimens,” Phys. Rev. Appl. 3, 1–6 (2015).
[Crossref]

R. M. Clare, M. Stockmar, M. Dierolf, I. Zanette, and F. Pfeiffer, “Characterization of near-field ptychography,” Opt. Express 23, 19728 (2015).
[Crossref] [PubMed]

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

Suo, J.

Sutskever, I.

M. Abadi, A. Agarwal, P. Barham, E. Brevdo, Z. Chen, C. Citro, G. S. Corrado, A. Davis, J. Dean, M. Devin, S. Ghemawat, I. J. Goodfellow, A. Harp, G. Irving, M. Isard, Y. Jia, R. Józefowicz, L. Kaiser, M. Kudlur, J. Levenberg, D. Mané, R. Monga, S. Moore, D. G. Murray, C. Olah, M. Schuster, J. Shlens, B. Steiner, I. Sutskever, K. Talwar, P. A. Tucker, V. Vanhoucke, V. Vasudevan, F. B. Viégas, O. Vinyals, P. Warden, M. Wattenberg, M. Wicke, Y. Yu, and X. Zheng, “Tensorflow: Large-scale machine learning on heterogeneous distributed systems,” arXiv 1603.04467 (2016).

Talneau, A.

V. Chamard, M. Allain, P. Godard, A. Talneau, G. Patriarche, and M. Burghammer, “Strain in a silicon-on-insulator nanostructure revealed by 3D x-ray Bragg ptychography,” Sci. Rep. 5, 9827 (2015).
[Crossref]

Talwar, K.

M. Abadi, A. Agarwal, P. Barham, E. Brevdo, Z. Chen, C. Citro, G. S. Corrado, A. Davis, J. Dean, M. Devin, S. Ghemawat, I. J. Goodfellow, A. Harp, G. Irving, M. Isard, Y. Jia, R. Józefowicz, L. Kaiser, M. Kudlur, J. Levenberg, D. Mané, R. Monga, S. Moore, D. G. Murray, C. Olah, M. Schuster, J. Shlens, B. Steiner, I. Sutskever, K. Talwar, P. A. Tucker, V. Vanhoucke, V. Vasudevan, F. B. Viégas, O. Vinyals, P. Warden, M. Wattenberg, M. Wicke, Y. Yu, and X. Zheng, “Tensorflow: Large-scale machine learning on heterogeneous distributed systems,” arXiv 1603.04467 (2016).

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]

Thibault, P.

M. Stockmar, I. Zanette, M. Dierolf, B. Enders, R. Clare, F. Pfeiffer, P. Cloetens, A. Bonnin, and P. Thibault, “X-ray near-field ptychography for optically thick specimens,” Phys. Rev. Appl. 3, 1–6 (2015).
[Crossref]

P. Thibault and A. Menzel, “Reconstructing state mixtures from diffraction measurements,” Nature 494, 68–71 (2013).
[Crossref] [PubMed]

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

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

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

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

Thurman, S. T.

Tian, L.

J. Zhong, L. Tian, P. Varma, and L. Waller, “Nonlinear optimization algorithm for partially coherent phase retrieval and source recovery,” IEEE Trans. Comput. Imaging 2, 310–322 (2016).
[Crossref]

Treu, J.

M. O. Hill, I. Calvo-Almazan, M. Allain, M. V. Holt, A. Ulvestad, J. Treu, G. Koblmuller, C. Huang, X. Huang, H. Yan, E. Nazaretski, Y. S. Chu, G. B. Stephenson, V. Chamard, L. J. Lauhon, and S. O. Hruszkewycz, “Measuring three-dimensional strain and structural defects in a single InGaAs nanowire using coherent x-ray multiangle Bragg projection ptychography,” Nano Lett. 18, 811–819 (2018).
[Crossref] [PubMed]

Tripathi, A.

Tucker, P. A.

M. Abadi, A. Agarwal, P. Barham, E. Brevdo, Z. Chen, C. Citro, G. S. Corrado, A. Davis, J. Dean, M. Devin, S. Ghemawat, I. J. Goodfellow, A. Harp, G. Irving, M. Isard, Y. Jia, R. Józefowicz, L. Kaiser, M. Kudlur, J. Levenberg, D. Mané, R. Monga, S. Moore, D. G. Murray, C. Olah, M. Schuster, J. Shlens, B. Steiner, I. Sutskever, K. Talwar, P. A. Tucker, V. Vanhoucke, V. Vasudevan, F. B. Viégas, O. Vinyals, P. Warden, M. Wattenberg, M. Wicke, Y. Yu, and X. Zheng, “Tensorflow: Large-scale machine learning on heterogeneous distributed systems,” arXiv 1603.04467 (2016).

Ulvestad, A.

M. O. Hill, I. Calvo-Almazan, M. Allain, M. V. Holt, A. Ulvestad, J. Treu, G. Koblmuller, C. Huang, X. Huang, H. Yan, E. Nazaretski, Y. S. Chu, G. B. Stephenson, V. Chamard, L. J. Lauhon, and S. O. Hruszkewycz, “Measuring three-dimensional strain and structural defects in a single InGaAs nanowire using coherent x-ray multiangle Bragg projection ptychography,” Nano Lett. 18, 811–819 (2018).
[Crossref] [PubMed]

Unglaub, W.

R. Xu, M. Soltanolkotabi, J. P. Haldar, W. Unglaub, J. Zusman, A. F. Levi, and R. M. Leahy, “Accelerated wirtinger flow: A fast algorithm for ptychography,” arXiv preprint arXiv:1806.05546 (2018).

Urbach, H.

P. Dwivedi, A. Konijnenberg, S. Pereira, and H. Urbach, “Lateral position correction in ptychography using the gradient of intensity patterns,” Ultramicroscopy 192, 29–36 (2018).
[Crossref] [PubMed]

Vanhoucke, V.

M. Abadi, A. Agarwal, P. Barham, E. Brevdo, Z. Chen, C. Citro, G. S. Corrado, A. Davis, J. Dean, M. Devin, S. Ghemawat, I. J. Goodfellow, A. Harp, G. Irving, M. Isard, Y. Jia, R. Józefowicz, L. Kaiser, M. Kudlur, J. Levenberg, D. Mané, R. Monga, S. Moore, D. G. Murray, C. Olah, M. Schuster, J. Shlens, B. Steiner, I. Sutskever, K. Talwar, P. A. Tucker, V. Vanhoucke, V. Vasudevan, F. B. Viégas, O. Vinyals, P. Warden, M. Wattenberg, M. Wicke, Y. Yu, and X. Zheng, “Tensorflow: Large-scale machine learning on heterogeneous distributed systems,” arXiv 1603.04467 (2016).

Varma, P.

J. Zhong, L. Tian, P. Varma, and L. Waller, “Nonlinear optimization algorithm for partially coherent phase retrieval and source recovery,” IEEE Trans. Comput. Imaging 2, 310–322 (2016).
[Crossref]

Vasudevan, V.

M. Abadi, A. Agarwal, P. Barham, E. Brevdo, Z. Chen, C. Citro, G. S. Corrado, A. Davis, J. Dean, M. Devin, S. Ghemawat, I. J. Goodfellow, A. Harp, G. Irving, M. Isard, Y. Jia, R. Józefowicz, L. Kaiser, M. Kudlur, J. Levenberg, D. Mané, R. Monga, S. Moore, D. G. Murray, C. Olah, M. Schuster, J. Shlens, B. Steiner, I. Sutskever, K. Talwar, P. A. Tucker, V. Vanhoucke, V. Vasudevan, F. B. Viégas, O. Vinyals, P. Warden, M. Wattenberg, M. Wicke, Y. Yu, and X. Zheng, “Tensorflow: Large-scale machine learning on heterogeneous distributed systems,” arXiv 1603.04467 (2016).

Viégas, F. B.

M. Abadi, A. Agarwal, P. Barham, E. Brevdo, Z. Chen, C. Citro, G. S. Corrado, A. Davis, J. Dean, M. Devin, S. Ghemawat, I. J. Goodfellow, A. Harp, G. Irving, M. Isard, Y. Jia, R. Józefowicz, L. Kaiser, M. Kudlur, J. Levenberg, D. Mané, R. Monga, S. Moore, D. G. Murray, C. Olah, M. Schuster, J. Shlens, B. Steiner, I. Sutskever, K. Talwar, P. A. Tucker, V. Vanhoucke, V. Vasudevan, F. B. Viégas, O. Vinyals, P. Warden, M. Wattenberg, M. Wicke, Y. Yu, and X. Zheng, “Tensorflow: Large-scale machine learning on heterogeneous distributed systems,” arXiv 1603.04467 (2016).

Vine, D. J.

Y. S. Nashed, D. J. Vine, T. Peterka, J. Deng, R. Ross, and C. Jacobsen, “Parallel ptychographic reconstruction,” Opt. Express 22, 223015 (2014).
[Crossref]

Vinyals, O.

M. Abadi, A. Agarwal, P. Barham, E. Brevdo, Z. Chen, C. Citro, G. S. Corrado, A. Davis, J. Dean, M. Devin, S. Ghemawat, I. J. Goodfellow, A. Harp, G. Irving, M. Isard, Y. Jia, R. Józefowicz, L. Kaiser, M. Kudlur, J. Levenberg, D. Mané, R. Monga, S. Moore, D. G. Murray, C. Olah, M. Schuster, J. Shlens, B. Steiner, I. Sutskever, K. Talwar, P. A. Tucker, V. Vanhoucke, V. Vasudevan, F. B. Viégas, O. Vinyals, P. Warden, M. Wattenberg, M. Wicke, Y. Yu, and X. Zheng, “Tensorflow: Large-scale machine learning on heterogeneous distributed systems,” arXiv 1603.04467 (2016).

Wagner, U.

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

Waigh, T. A.

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

Waller, L.

J. Zhong, L. Tian, P. Varma, and L. Waller, “Nonlinear optimization algorithm for partially coherent phase retrieval and source recovery,” IEEE Trans. Comput. Imaging 2, 310–322 (2016).
[Crossref]

Walther, A.

A. Griewank and A. Walther, Evaluating derivatives: principles and techniques of algorithmic differentiation, vol. 105 (SIAM, 2008).
[Crossref]

Warden, P.

M. Abadi, A. Agarwal, P. Barham, E. Brevdo, Z. Chen, C. Citro, G. S. Corrado, A. Davis, J. Dean, M. Devin, S. Ghemawat, I. J. Goodfellow, A. Harp, G. Irving, M. Isard, Y. Jia, R. Józefowicz, L. Kaiser, M. Kudlur, J. Levenberg, D. Mané, R. Monga, S. Moore, D. G. Murray, C. Olah, M. Schuster, J. Shlens, B. Steiner, I. Sutskever, K. Talwar, P. A. Tucker, V. Vanhoucke, V. Vasudevan, F. B. Viégas, O. Vinyals, P. Warden, M. Wattenberg, M. Wicke, Y. Yu, and X. Zheng, “Tensorflow: Large-scale machine learning on heterogeneous distributed systems,” arXiv 1603.04467 (2016).

Wattenberg, M.

M. Abadi, A. Agarwal, P. Barham, E. Brevdo, Z. Chen, C. Citro, G. S. Corrado, A. Davis, J. Dean, M. Devin, S. Ghemawat, I. J. Goodfellow, A. Harp, G. Irving, M. Isard, Y. Jia, R. Józefowicz, L. Kaiser, M. Kudlur, J. Levenberg, D. Mané, R. Monga, S. Moore, D. G. Murray, C. Olah, M. Schuster, J. Shlens, B. Steiner, I. Sutskever, K. Talwar, P. A. Tucker, V. Vanhoucke, V. Vasudevan, F. B. Viégas, O. Vinyals, P. Warden, M. Wattenberg, M. Wicke, Y. Yu, and X. Zheng, “Tensorflow: Large-scale machine learning on heterogeneous distributed systems,” arXiv 1603.04467 (2016).

Wei, Z.

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]

Wepf, R.

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

Wicke, M.

M. Abadi, A. Agarwal, P. Barham, E. Brevdo, Z. Chen, C. Citro, G. S. Corrado, A. Davis, J. Dean, M. Devin, S. Ghemawat, I. J. Goodfellow, A. Harp, G. Irving, M. Isard, Y. Jia, R. Józefowicz, L. Kaiser, M. Kudlur, J. Levenberg, D. Mané, R. Monga, S. Moore, D. G. Murray, C. Olah, M. Schuster, J. Shlens, B. Steiner, I. Sutskever, K. Talwar, P. A. Tucker, V. Vanhoucke, V. Vasudevan, F. B. Viégas, O. Vinyals, P. Warden, M. Wattenberg, M. Wicke, Y. Yu, and X. Zheng, “Tensorflow: Large-scale machine learning on heterogeneous distributed systems,” arXiv 1603.04467 (2016).

Wild, S. M.

Xu, R.

R. Xu, M. Soltanolkotabi, J. P. Haldar, W. Unglaub, J. Zusman, A. F. Levi, and R. M. Leahy, “Accelerated wirtinger flow: A fast algorithm for ptychography,” arXiv preprint arXiv:1806.05546 (2018).

Yan, A. W.

A. W. Yan, A. J. D’Alfonso, A. J. Morgan, C. T. Putkunz, and L. J. Allen, “Fast deterministic ptychographic imaging using x-rays,” Microsc. Microanal. 20, 1090–1099 (2014).
[Crossref] [PubMed]

Yan, H.

M. O. Hill, I. Calvo-Almazan, M. Allain, M. V. Holt, A. Ulvestad, J. Treu, G. Koblmuller, C. Huang, X. Huang, H. Yan, E. Nazaretski, Y. S. Chu, G. B. Stephenson, V. Chamard, L. J. Lauhon, and S. O. Hruszkewycz, “Measuring three-dimensional strain and structural defects in a single InGaAs nanowire using coherent x-ray multiangle Bragg projection ptychography,” Nano Lett. 18, 811–819 (2018).
[Crossref] [PubMed]

Yang, C.

J. Qian, C. Yang, A. Schirotzek, F. Maia, and S. Marchesini, “Efficient algorithms for ptychographic phase retrieval,” Inverse Probl. Appli. Contemp. Math 615, 261–280 (2014).

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, E.

A. Paszke, S. Gross, S. Chintala, G. Chanan, E. Yang, Z. DeVito, Z. Lin, A. Desmaison, L. Antiga, and A. Lerer, “Automatic differentiation in pytorch,” in 31st Conference on Neural Information Processing Systems (NIPS, 2017).

Yao, Y.

Y. Yao, L. Rosasco, and A. Caponnetto, “On early stopping in gradient descent learning,” Constr. Approx. 26, 289–315 (2007).
[Crossref]

Yeh, L.-H.

L.-H. Yeh, “Analysis and comparison of fourier ptychographic phase retrieval algorithms,” Technical Report No. UCB/EECS-2016-86 (University of California, 2016).

Yu, Y.

M. Abadi, A. Agarwal, P. Barham, E. Brevdo, Z. Chen, C. Citro, G. S. Corrado, A. Davis, J. Dean, M. Devin, S. Ghemawat, I. J. Goodfellow, A. Harp, G. Irving, M. Isard, Y. Jia, R. Józefowicz, L. Kaiser, M. Kudlur, J. Levenberg, D. Mané, R. Monga, S. Moore, D. G. Murray, C. Olah, M. Schuster, J. Shlens, B. Steiner, I. Sutskever, K. Talwar, P. A. Tucker, V. Vanhoucke, V. Vasudevan, F. B. Viégas, O. Vinyals, P. Warden, M. Wattenberg, M. Wicke, Y. Yu, and X. Zheng, “Tensorflow: Large-scale machine learning on heterogeneous distributed systems,” arXiv 1603.04467 (2016).

Zanette, I.

M. Stockmar, I. Zanette, M. Dierolf, B. Enders, R. Clare, F. Pfeiffer, P. Cloetens, A. Bonnin, and P. Thibault, “X-ray near-field ptychography for optically thick specimens,” Phys. Rev. Appl. 3, 1–6 (2015).
[Crossref]

R. M. Clare, M. Stockmar, M. Dierolf, I. Zanette, and F. Pfeiffer, “Characterization of near-field ptychography,” Opt. Express 23, 19728 (2015).
[Crossref] [PubMed]

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

Zhang, H.

H. Zhang and Y. Liang, “Reshaped wirtinger flow for solving quadratic system of equations,” Adv. Neural. Inf. Process. Syst. 29, 2622–2630 (2016).

Zheng, G.

Zheng, X.

M. Abadi, A. Agarwal, P. Barham, E. Brevdo, Z. Chen, C. Citro, G. S. Corrado, A. Davis, J. Dean, M. Devin, S. Ghemawat, I. J. Goodfellow, A. Harp, G. Irving, M. Isard, Y. Jia, R. Józefowicz, L. Kaiser, M. Kudlur, J. Levenberg, D. Mané, R. Monga, S. Moore, D. G. Murray, C. Olah, M. Schuster, J. Shlens, B. Steiner, I. Sutskever, K. Talwar, P. A. Tucker, V. Vanhoucke, V. Vasudevan, F. B. Viégas, O. Vinyals, P. Warden, M. Wattenberg, M. Wicke, Y. Yu, and X. Zheng, “Tensorflow: Large-scale machine learning on heterogeneous distributed systems,” arXiv 1603.04467 (2016).

Zhong, J.

J. Zhong, L. Tian, P. Varma, and L. Waller, “Nonlinear optimization algorithm for partially coherent phase retrieval and source recovery,” IEEE Trans. Comput. Imaging 2, 310–322 (2016).
[Crossref]

Zhou, T.

J. Li and T. Zhou, “Numerical optimization algorithms for wavefront phase retrieval from multiple measurements,” Inverse Probl. & Imaging 11, 721–743 (2017).
[Crossref]

Zusman, J.

R. Xu, M. Soltanolkotabi, J. P. Haldar, W. Unglaub, J. Zusman, A. F. Levi, and R. M. Leahy, “Accelerated wirtinger flow: A fast algorithm for ptychography,” arXiv preprint arXiv:1806.05546 (2018).

Acta Crystallogr. A (1)

W. Hoppe, “Beugung im inhomogenen Primarstrahlwellenfeld. III. Amplituden-und Phasenbestimmung bei unperiodischen Objekten,” Acta Crystallogr. A 25, 508–514 (1969).
[Crossref]

Adv. Neural. Inf. Process. Syst. (1)

H. Zhang and Y. Liang, “Reshaped wirtinger flow for solving quadratic system of equations,” Adv. Neural. Inf. Process. Syst. 29, 2622–2630 (2016).

Appl. Opt. (1)

Appl. Phys. Lett. (1)

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

Constr. Approx. (1)

Y. Yao, L. Rosasco, and A. Caponnetto, “On early stopping in gradient descent learning,” Constr. Approx. 26, 289–315 (2007).
[Crossref]

IEE Proc. F-Communications, Radar Signal Process. (1)

D. Brandwood, “A complex gradient operator and its application in adaptive array theory,” IEE Proc. F-Communications, Radar Signal Process. 130, 11–16 (1983).
[Crossref]

IEEE Trans. Comput. Imaging (1)

J. Zhong, L. Tian, P. Varma, and L. Waller, “Nonlinear optimization algorithm for partially coherent phase retrieval and source recovery,” IEEE Trans. Comput. Imaging 2, 310–322 (2016).
[Crossref]

IEEE Trans. Inf. Theory (1)

E. J. Candes, X. Li, and M. Soltanolkotabi, “Phase retrieval via Wirtinger flow: theory and algorithms,” IEEE Trans. Inf. Theory 61, 1985–2007 (2015).
[Crossref]

IEEE Trans. Med. Imaging (1)

S. Ahn and J. A. Fessler, “Globally convergent image reconstruction for emission tomography using relaxed ordered subsets algorithms,” IEEE Trans. Med. Imaging 22, 613–626 (2003).
[Crossref] [PubMed]

IEEE Trans. Signal Process. (1)

E. J. R. Pauwels, A. Beck, Y. C. Eldar, and S. Sabach, “On fienup methods for sparse phase retrieval,” IEEE Trans. Signal Process. 66, 982–991 (2018).
[Crossref]

Inverse Probl. (2)

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

A. Fannjiang, “Absolute uniqueness of phase retrieval with random illumination,” Inverse Probl. 28, 075008 (2012).
[Crossref]

Inverse Probl. & Imaging (1)

J. Li and T. Zhou, “Numerical optimization algorithms for wavefront phase retrieval from multiple measurements,” Inverse Probl. & Imaging 11, 721–743 (2017).
[Crossref]

Inverse Probl. Appli. Contemp. Math (1)

J. Qian, C. Yang, A. Schirotzek, F. Maia, and S. Marchesini, “Efficient algorithms for ptychographic phase retrieval,” Inverse Probl. Appli. Contemp. Math 615, 261–280 (2014).

J. Mach. Learn. Res. (2)

J. Bergstra and Y. Bengio, “Random search for hyper-parameter optimization,” J. Mach. Learn. Res. 13, 281–305 (2012).

A. G. Baydin, B. A. Pearlmutter, A. A. Radul, and J. M. Siskind, “Automatic differentiation in machine learning: a survey,” J. Mach. Learn. Res. 18, 1–43 (2015).

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (4)

Microsc. Microanal. (1)

A. W. Yan, A. J. D’Alfonso, A. J. Morgan, C. T. Putkunz, and L. J. Allen, “Fast deterministic ptychographic imaging using x-rays,” Microsc. Microanal. 20, 1090–1099 (2014).
[Crossref] [PubMed]

Nano Lett. (1)

M. O. Hill, I. Calvo-Almazan, M. Allain, M. V. Holt, A. Ulvestad, J. Treu, G. Koblmuller, C. Huang, X. Huang, H. Yan, E. Nazaretski, Y. S. Chu, G. B. Stephenson, V. Chamard, L. J. Lauhon, and S. O. Hruszkewycz, “Measuring three-dimensional strain and structural defects in a single InGaAs nanowire using coherent x-ray multiangle Bragg projection ptychography,” Nano Lett. 18, 811–819 (2018).
[Crossref] [PubMed]

Nat. Commun. (2)

P. Godard, G. Carbone, M. Allain, F. Mastropietro, G. Chen, L. Capello, A. Diaz, T. Metzger, J. Stangl, and V. Chamard, “Three-dimensional high-resolution quantitative microscopy of extended crystals,” Nat. Commun. 2, 568 (2011).
[Crossref] [PubMed]

J. Clark, X. Huang, R. Harder, and I. Robinson, “High-resolution three-dimensional partially coherent diffraction imaging,” Nat. Commun. 3, 993 (2012).
[Crossref] [PubMed]

Nat. Mater. (1)

S. O. Hruszkewycz, M. Allain, M. V. Holt, C. E. Murray, J. R. Holt, P. H. Fuoss, and V. Chamard, “High-resolution three-dimensional structural microscopy by single-angle Bragg ptychography,” Nat. Mater. 16, 244–251 (2017).
[Crossref]

Nature (2)

P. Thibault and A. Menzel, “Reconstructing state mixtures from diffraction measurements,” Nature 494, 68–71 (2013).
[Crossref] [PubMed]

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

New J. Phys. (2)

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

A. L. Robisch, K. Kröger, A. Rack, and T. Salditt, “Near-field ptychography using lateral and longitudinal shifts,” New J. Phys. 17, 073033 (2015).
[Crossref]

Numer. Algorithms (1)

P. H. Hoffmann, “A hitchhiker’s guide to automatic differentiation,” Numer. Algorithms 72, 775–811 (2016).
[Crossref]

Opt. Express (6)

Opt. Lett. (1)

Optica (2)

Optik (2)

R. H. T. Bates, “Fourier phase problems are uniquely solvable in more than one dimension. I. Underlying theory,” Optik 61, 247–262 (1982).

R. W. Gerchberg and W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik 35, 237–246 (1972).

Phys. Rev. A (1)

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

Phys. Rev. Appl. (1)

M. Stockmar, I. Zanette, M. Dierolf, B. Enders, R. Clare, F. Pfeiffer, P. Cloetens, A. Bonnin, and P. Thibault, “X-ray near-field ptychography for optically thick specimens,” Phys. Rev. Appl. 3, 1–6 (2015).
[Crossref]

Phys. Rev. Lett. (1)

H. M. L. Faulkner and J. Rodenburg, “Movable aperture lensless transmission microscopy: A novel phase retrieval algorithm,” Phys. Rev. Lett. 93, 023903 (2004).
[Crossref] [PubMed]

Procedia Comput. Sci. (1)

Y. S. G. Nashed, T. Peterka, J. Deng, and C. Jacobsen, “Distributed automatic differentiation for ptychography,” Procedia Comput. Sci. 108, 404–414 (2017).
[Crossref]

Rev. Sci. Instruments (1)

S. Marchesini, “A unified evaluation of iterative projection algorithms for phase retrieval,” Rev. Sci. Instruments 78, 011301 (2007).
[Crossref]

Sci. Rep. (2)

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

V. Chamard, M. Allain, P. Godard, A. Talneau, G. Patriarche, and M. Burghammer, “Strain in a silicon-on-insulator nanostructure revealed by 3D x-ray Bragg ptychography,” Sci. Rep. 5, 9827 (2015).
[Crossref]

SIAM J. Imaging Sci. (1)

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]

SIAM J. Optim. (1)

L. Sorber, M. V. Barel, and L. D. Lathauwer, “Unconstrained optimization of real functions in complex variables,” SIAM J. Optim. 22, 879–898 (2012).
[Crossref]

Ultramicroscopy (3)

P. Dwivedi, A. Konijnenberg, S. Pereira, and H. Urbach, “Lateral position correction in ptychography using the gradient of intensity patterns,” Ultramicroscopy 192, 29–36 (2018).
[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]

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

Other (13)

A. Griewank and A. Walther, Evaluating derivatives: principles and techniques of algorithmic differentiation, vol. 105 (SIAM, 2008).
[Crossref]

M. Abadi, A. Agarwal, P. Barham, E. Brevdo, Z. Chen, C. Citro, G. S. Corrado, A. Davis, J. Dean, M. Devin, S. Ghemawat, I. J. Goodfellow, A. Harp, G. Irving, M. Isard, Y. Jia, R. Józefowicz, L. Kaiser, M. Kudlur, J. Levenberg, D. Mané, R. Monga, S. Moore, D. G. Murray, C. Olah, M. Schuster, J. Shlens, B. Steiner, I. Sutskever, K. Talwar, P. A. Tucker, V. Vanhoucke, V. Vasudevan, F. B. Viégas, O. Vinyals, P. Warden, M. Wattenberg, M. Wicke, Y. Yu, and X. Zheng, “Tensorflow: Large-scale machine learning on heterogeneous distributed systems,” arXiv 1603.04467 (2016).

A. Paszke, S. Gross, S. Chintala, G. Chanan, E. Yang, Z. DeVito, Z. Lin, A. Desmaison, L. Antiga, and A. Lerer, “Automatic differentiation in pytorch,” in 31st Conference on Neural Information Processing Systems (NIPS, 2017).

D. Maclaurin, “Modeling, inference and optimization with composable differentiable procedures,” Ph.D. thesis, Harvard University (2016).

S. Ghosh, Y. S. Nashed, O. Cossairt, and A. Katsaggelos, “ADP : Automatic differentiation ptychography,” IEEE International Conference on Computational Photography (ICCP) (IEEE, 2018), pp.1–10.

L.-H. Yeh, “Analysis and comparison of fourier ptychographic phase retrieval algorithms,” Technical Report No. UCB/EECS-2016-86 (University of California, 2016).

D. P. Kingma and J. Ba, “Adam: A method for stochastic optimization,” arXiv 1412.6980 (2014).

K. Kreutz-Delgado, “The complex gradient operator and the cr-calculus,” arXiv preprint arXiv:0906.4835 (2009).

R. Xu, M. Soltanolkotabi, J. P. Haldar, W. Unglaub, J. Zusman, A. F. Levi, and R. M. Leahy, “Accelerated wirtinger flow: A fast algorithm for ptychography,” arXiv preprint arXiv:1806.05546 (2018).

D. P. Bertsekas, Nonlinear programming (Athena Scientific, 1999).

Y. A. LeCun, L. Bottou, G. B. Orr, and K.-R. Müller, “Efficient backprop,” in Neural networks: Tricks of the trade, (Springer, 2012), pp. 9–48.

S. Ruder, “An overview of gradient descent optimization algorithms,” arXiv preprint arXiv:1609.04747 (2016).

J. W. Goodman, Introduction to Fourier Optics (W.H. Freeman, 2017).

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Figures (6)

Fig. 1
Fig. 1 Simplified representation of the forward and backward passes for the object update for far-field ptychography. The solid blue arrows indicate the forward pass direction. The dashed orange arrows indicate the backward pass direction.
Fig. 2
Fig. 2 The value of the (a) error metric g ( O , P ) and (d) the normalized reconstruction error for the object for the ePIE method, the AD-ePIE method, and the Adam method for the far-field ptychography experiment. Adam reconstructions for the (b) object magnitude, (c) object phase, (e) probe magnitude, and (f) probe phase. The normalized root-mean-squared reconstruction error (NRMSE) was 0.03 for the object and 0.02 for the probe.
Fig. 3
Fig. 3 (a) Reconstruction errors (NRMSE) obtained after 1500 iterations of the Adam algorithm (with b = 100) as a function of initial object and probe update step sizes, for incident probe integrated intensities of 103 (left), 106 (mid), and 109 (right) respectively. For clarity, the NRMSEs plotted are capped at 0.6. (b) Final object magnitudes and (c) phases obtained for α   O A = 0.01 and α   P = 1.0 . For the low photon count of 103 (left), the reconstructed structures are deteriorated due to the raster grid artifact.
Fig. 4
Fig. 4 Near-field ptychographic reconstruction with reverse-mode AD. Shown here are the successfully reconstructed (a) object magnitude, (b) object phase, (c) probe magnitude, and (d) probe phase. The object and probe were reconstructed with overall NRMSEs of 0.01 and 0.12 respectively.
Fig. 5
Fig. 5 (a) Simulated experimental geometry for multi-angle Bragg ptychography. The incident (ki) and exit (kf) beams define a scanning angle. (b) At each scanning angle, the incident beam is shifted along an overlapping raster grid in the yz plane to generate a set of 2D coherent diffraction patterns. (c) For angular sampling around the Bragg peak (say with 2θ = 60°), the sample is rotated through small angles Δθ, and the incident and exit beams are modulated correspondingly.
Fig. 6
Fig. 6 Multi-angle Bragg ptychographic reconstruction with reverse-mode AD. (a) True and (d) reconstructed object structure with surface phase variation. YZ cross-section of the (b) probe magnitude and (e) phase at x = 32. (c) XY cross-section of the error in the reconstructions of the object (c) magnitude and (f) phase at z = 41. For the sake of clarity, the error in the phase is set to 0 wherever O r e c o n s < 0.01 . The normalized error (NRMSE) for the 3D reconstruction is 0.09.

Tables (4)

Tables Icon

Table 1 Schema for reverse-mode automatic differentiation.

Tables Icon

Algorithm 1 Generalized ePIE gradient descent iteration

Tables Icon

Algorithm 2a Adam object initialization

Tables Icon

Algorithm 2b Adam object update (at step k)

Equations (27)

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

d f d x | x = c = d ϕ 3 d x | x = ϕ 2 ( ϕ 1 ( c ) ) d ϕ 2 d x | x = ϕ 1 ( c ) d ϕ 1 d x | x = c .
d v i + 1 d v i = d ϕ i + 1 d x x = v i ,
ops ( f ) = N
ops ( f ) = k ops ( f ) = k N , with 0 < k < 6 a constant ,
f ( z ) = f z * = 1 2 ( f [ z ] + i f [ z ] ) ,
ψ j = P ( S j O ) for j = 1 , 2 , ... , J ;
h j = | F ψ j | 2 + ν j
y j 1 / 2 = h j 1 / 2 + ε j
( O , P ) argmin   O , P g ( O , P )
g = j = 1 J g j with g j ( O , P ) : = | | y j 1 / 2 h j 1 / 2 ( O , P ) | | 2
  O g k : = j = b ( k 1 ) + 1 b k   O g ( O j k 1 ) and   P g k : = j = b ( k 1 ) + 1 b k   P g ( P j k 1 )
α   O k : = 1 / P k 1   2 max and α   P k : = 1 / j = b ( k 1 ) + 1 b k S j * O k 1   2 max .
O k = O k 1 α   O k   O g k and P k = P k 1 α   P k   P g k
  O g k : = 1 2 ( g [ O ] | O k 1 + i g [ O ] | O = O k 1 )
  P g k : = 1 2 ( g [ P ] | P = P k 1 + i g [ P ] | P = P k 1 ) .
W 2 / λ z 1 ,
h j = D z { ψ j } 2 + ν j
D z { ψ j } = F 1 [ ( F ψ j ) exp  i z λ 4 π ( q x 2 + q y 2 ) ]
P 0 = F 1 [ ( F y avg ) exp  i z λ 4 π ( q x 2 + q y 2 ) ] .
ψ j = R P ( S j O )
ψ j = R Q j ( P ( S j O ) ) ,
h j = F ψ j   2 + ν j = F R Q j ( P ( S j O ) ) 2 + ν j ,
  [ O ] g k = 1 2 g k [ O ] and   [ O ] g k = 1 2 g k [ O ]
m [ O ] 0 0 m J [ O ] 0 0 } First moments,  m [ O ] , m J [ O ] N v [ O ] 0 0 v J [ O ] 0 0 } Second moments,  v [ O ] , v J [ O ] N
β 1 = 0.9 ( Decay rate for m   [ O ] , m   [ O ] ) β 2 = 0.999 ( Decay rate for v   [ O ] , v   [ O ] ) ϵ = 10 8 ( Constant used to avoid division by zero )
m [ O ] k = β 1 m   [ O ] k 1 + ( 1 β 1 )   [ O ] g k 1 ( β 1 ) k v [ O ] k = β 2 v   [ O ] k 1 + ( 1 β 2 )   [ O ] g k   2 1 ( β 2 ) k
[ O ] k = [ O ] k 1 α   O A m   [ O ] k v   [ O ] k + ϵ [ O ] k = [ O ] k 1 α   O A m   [ O ] k v   [ O ] k + ϵ

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