Abstract

We present the extension of ptychography for three-dimensional object reconstruction in a tomography setting. We describe the alternating direction method of multipliers (ADMM) as a generic reconstruction framework to efficiently solve the nonlinear optimization problem. In this framework, the ADMM breaks the joint reconstruction problem into two well-defined subproblems: ptychographic phase retrieval and tomographic reconstruction. In this paper, we use the gradient descent algorithm to solve both problems and demonstrate the efficiency of the proposed approach through numerical simulations. Further, we show that the proposed joint approach relaxes existing requirements for lateral probe overlap in conventional ptychography. Thus, it can allow more flexible data acquisition.

© 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 primärstrahlwellenfeld, I. prinzip einer phasenmessung,” Acta Crystallogr. A 25, 495–501 (1969).
    [Crossref]
  2. D. Gürsoy, “Direct coupling of tomography and ptychography,” Opt. Lett. 42, 3169–3172 (2017).
    [Crossref] [PubMed]
  3. J. M. Rodenburg, A. C. Hurst, A. G. Cullis, B. R. Dobson, F. Pfeiffer, O. Bunk, C. David, K. Jefimovs, and I. Johnson, “Hard-x-ray lensless imaging of extended objects,” Phys. Rev. Lett. 98, 034801 (2007).
    [Crossref] [PubMed]
  4. J. M. Rodenburg, A. C. Hurst, and A. G. Cullis, “Transmission microscopy without lenses for objects of unlimited size,” Ultramicroscopy 107, 227–231 (2007).
    [Crossref]
  5. P. Thibault, M. Dierolf, A. Menzel, O. Bunk, C. David, and F. Pfeiffer, “High-resolution scanning x-ray diffraction microscopy,” Science 321, 379–382 (2008).
    [Crossref] [PubMed]
  6. M. Dierolf, O. Bunk, S. Kynde, P. Thibault, I. Johnson, A. Menzel, K. Jefimovs, C. David, O. Marti, and F. Pfeiffer, “Ptychography & lensless x-ray imaging,” Europhys. News 39, 22–24 (2008).
    [Crossref]
  7. 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]
  8. M. Holler, J. Raabe, A. Diaz, M. Guizar-Sicairos, C. Quitmann, A. Menzel, and O. Bunk, “An instrument for 3D x-ray nano-imaging,” Rev. Sci. Instrum. 83, 073703 (2012).
    [Crossref] [PubMed]
  9. A. Diaz, P. Trtik, M. Guizar-Sicairos, A. Menzel, P. Thibault, and O. Bunk, “Quantitative x-ray phase nanotomography,” Phys. Rev. B 85, 020104 (2012).
    [Crossref]
  10. K. Giewekemeyera, P. Thibault, S. Kalbfleisch, A. Beerlinka, C. M. Kewishc, M. Dierolf, F. Pfeiffer, and T. Salditt, “Quantitative biological imaging by ptychographic x-ray diffraction microscopy,” PNAS 107, 529–534 (2009).
    [Crossref]
  11. R. N. Wilke, M. Priebe, M. Bartels, K. Giewekemeyer, A. Diaz, P. Karvinen, and T. Salditt, “Hard x-ray imaging of bacterial cells: nano-diffraction and ptychographic reconstruction,” Opt. Express 20, 19232–19254 (2012).
    [Crossref] [PubMed]
  12. E. Lima, A. Diaz, M. Guizar-Sicairos, S. Gorelick, P. Pernot, T. Schleirer, and A. Menzel, “Cryo-scanning x-ray diffraction microscopy of frozen-hydrated yeast,” J. Microsc. 249, 1–7 (2013).
    [Crossref]
  13. A. Diaz, B. Malkova, M. Holler, M. Guizar-Sicairos, E. Lima, V. Panneels, G. Pigino, A. G. Bittermann, L. Wettstein, T. Tomizaki, O. Bunk, G. Schertler, T. Ishikawa, R. Wepf, and A. Menzel, “Three-dimensional mass density mapping of cellular ultrastructure by ptychographic x-ray nanotomography,” J. Struct. Biol. 192, 461–469 (2015).
    [Crossref] [PubMed]
  14. J. Deng, Y. S. G. Nashed, S. Chen, N. W. Phillips, T. Peterka, R. Ross, S. Vogt, C. Jacobsen, and D. J. Vine, “Continuous motion scan ptychography: characterization for increased speed in coherent x-ray imaging,” Opt. Express 235438 (2015).
    [Crossref]
  15. A. Schropp, P. Boye, A. Goldschmidt, S. Honig, R. Hoppe, J. Patommel, C. Rakete, D. Samberg, S. Stephan, S. Schoder, M. Burghammer, and C. G. Schroer, “Non-destructive and quantitative imaging of a nano-structured microchip by ptychographic hard x-ray scanning microscopy,” J. Microsc. 241, 9–12 (2011).
    [Crossref]
  16. B. Mike, S. Tobias, G. Thomas, R. Michael, G. Klaus, G. Sophie-Charlotte, S. Tim, and R. Axel, “Chemical contrast in soft x-ray ptychography,” Phys. Rev. Lett. 107, 208101 (2011).
    [Crossref]
  17. C. L. Hoy, N. J. Durr, and A. Ben-Yakar, “Fast-updating and nonrepeating Lissajous image reconstruction method for capturing increased dynamic information,” Appl. Opt. 50, 2376–2382 (2011).
    [Crossref] [PubMed]
  18. P. Trtik, A. Diaz, M. Guizar-Sicairos, A. Menzel, and O. Bunk, “Density mapping of hardened cement paste using ptychographic x-ray computed tomography,” Cem. & Concr. Compos. 36, 71–77 (2013).
    [Crossref]
  19. B. Chen, M. Guizar-Sicairos, G. Xiong, L. Shemilt, A. Diaz, J. Nutter, N. Burdet, S. Huo, J. Mancuso, A. Monteith, F. Vergeer, A. Burgess, and I. Robinson, “Three-dimensional structure analysis and percolation properties of a barrier marine coating,” Sci. Rep. 3, 1177 (2013).
    [Crossref] [PubMed]
  20. M. Holler, A. Diaz, M. Guizar-Sicairos, P. Karvinen, E. Färm, E. Härkönen, M. Ritala, A. Menzel, J. Raabe, and O. Bunk, “X-ray ptychographic computed tomography at 16 nm isotropic 3D resolution,” Sci. Reports 4, 3857 (2014).
    [Crossref]
  21. K. Høydalsvik, J. B. Fløystad, T. Zhao, M. Esmaeili, A. Diaz, J. W. Andreasen, R. H. Mathiesen, M. Rønning, and D. W. Breiby, “In situ x-ray ptychography imaging of high-temperature CO2 acceptor particle agglomerates,” Appl. Phys. Lett. 104, 241909 (2014).
    [Crossref]
  22. J. N. Weker and M. F. Toney, “Emerging in situ and operando nanoscale x-ray imaging techniques for energy storage materials,” Adv. Funct. Mater. 25, 1622–1637 (2015).
    [Crossref]
  23. J. Deng, D. J. Vine, S. Chen, Y. S. G. Nashed, Q. Jin, N. W. Phillips, T. Peterka, R. Ross, S. Vogt, and C. J. Jacobsen, “Simultaneous cryo x-ray ptychographic and fluorescence microscopy of green algae,” PNAS 112, 2314–2319 (2015).
    [Crossref] [PubMed]
  24. M. W. M. Jones, K. D. Elgass, M. D. Junker, M. D. de Jonge, and G. A. van Riessen, “Molar concentration from sequential 2-D water-window x-ray ptychography and x-ray fluorescence in hydrated cells,” Sci. Rep. 6, 24280 (2016).
    [Crossref] [PubMed]
  25. M. Holler, M. Guizar-Sicairos, E. H. R. Tsai, R. Dinapoli, E. Müller, O. Bunk, J. Raabe, and G. Aeppli, “High-resolution non-destructive three-dimensional imaging of integrated circuits,” Nature 543, 402–406 (2017).
    [Crossref] [PubMed]
  26. J. Deng, Y. P. Hong, S. Chen, Y. S. G. Nashed, T. Peterka, A. J. F. Levi, J. Damoulakis, S. Saha, T. Eiles, and C. Jacobsen, “Nanoscale x-ray imaging of circuit features without wafer etching,” Phys. Rev. B 95, 104111 (2017).
    [Crossref] [PubMed]
  27. F. Pfeiffer, “X-ray ptychography,” Nat. Photonics 12, 9–17 (2018).
    [Crossref]
  28. 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).
  29. J. R. Fienup, “Reconstruction of an object from the modulus of its Fourier transform,” Opt. Lett. 3, 27–29 (1978).
    [Crossref] [PubMed]
  30. J. R. Fienup, “Phase retrieval algorithms: a comparison,” Appl. Opt. 21, 2758–2769 (1982).
    [Crossref] [PubMed]
  31. M. Guizar-Sicairos and J. R. Fienup, “Phase retrieval with transverse translation diversity: a nonlinear optimization approach,” Opt. Express 16, 7264–7278 (2008).
    [Crossref] [PubMed]
  32. V. Elser, “Phase retrieval by iterated projections,” J. Opt. Soc. Am. A 20, 40–55 (2003).
    [Crossref]
  33. J. M. Rodenburg and H. M. L. Faulkner, “A phase retrieval algorithm for shifting illumination,” Appl. Phys. Lett. 85, 4795–4797 (2004).
    [Crossref]
  34. Z. Wen, C. Yang, X. Liu, and S. Marchesini, “Alternating direction methods for classical and ptychographic phase retrieval,” Inverse Probl. 28, 115010 (2012).
    [Crossref]
  35. P. Thibault and M. Guizar-Sicairos, “Maximum-likelihood refinement for coherent diffractive imaging,” New J. Phys. 14, 063004 (2012).
    [Crossref]
  36. E. J. Candès, Y. C. Eldar, T. Strohmer, and V. Voroninski, “Phase retrieval via matrix completion,” SIAM J. on Imaging Sci. 6, 199–225 (2013).
    [Crossref]
  37. S. Marchesini, A. Schirotzek, C. Yang, H. Wu, and F. Maia, “Augmented projections for ptychographic imaging,” Inverse Probl. 29, 115009 (2013).
    [Crossref]
  38. J. Bolte, S. Sabach, and M. Teboulle, “Proximal alternating linearized minimization for nonconvex and nonsmooth problems,” Math. Program. 146, 459–494 (2014).
    [Crossref]
  39. E. J. Candès, X. Li, and M. Soltanolkotabi, “Phase retrieval via Wirtinger flow: theory and algorithms,” IEEE Transactions on Inf. Theory 61, 1985–2007 (2015).
    [Crossref]
  40. 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. on Imaging Sci. 8, 426–457 (2015).
    [Crossref]
  41. 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]
  42. A. Maiden, D. Johnson, and P. Li, “Further improvements to the ptychographical iterative engine,” Optica 4, 736–745 (2017).
    [Crossref]
  43. P. Li and A. M. Maiden, “Ten implementations of ptychography,” J. Microsc. 269, 187–194 (2018).
    [Crossref]
  44. H. Chang, Y. Lou, Y. Duan, and S. Marchesini, “Total variation-based phase retrieval for Poisson noise removal,” SIAM J. on Imaging Sci. 11, 24–55 (2018).
    [Crossref]
  45. H. Chang, S. Marchesini, Y. Lou, and T. Zeng, “Variational phase retrieval with globally convergent preconditioned proximal algorithm,” SIAM J. on Imaging Sci. 11, 56–93 (2018).
    [Crossref]
  46. T. Ramos, B. E. Grønager, M. S. Andersen, and J. W. Andreasen, “Direct 3D tomographic reconstruction and phase-retrieval of far-field coherent diffraction patterns,” arXiv preprint arXiv:1808.02109 (2018).
  47. P. Li and A. Maiden, “Multi-slice ptychographic tomography,” Sci. Reports 82049 (2018).
  48. M. A. Gilles, Y. S. G. Nashed, C. J. M. Du, and S. M. Wild, “3D x-ray imaging of continuous objects beyond the depth of focus limit,” Optica 5, 1078–1086 (2018).
    [Crossref] [PubMed]
  49. K. Shimomura, M. Hirose, T. Higashino, and Y. Takahashi, “Three-dimensional iterative multislice reconstruction for ptychographic x-ray computed tomography,” Opt. Exp. 26, 31199 (2018).
    [Crossref]
  50. H. Öztürk, H. Yan, Y. He, M. Ge, Z. Dong, M. Lin, E. Nazaretski, I. K. Robinson, Y. S. Chu, and X. Huang, “Multi-slice ptychography with large numerical aperture multilayer laue lenses,” Optica 5, 601 (2018).
    [Crossref]
  51. S. Boyd, N. Parikh, E. Chu, B. Peleato, and J. Eckstein, “Distributed optimization and statistical learning via the alternating direction method of multipliers,” Foundations Trends Mach. Learn. 3, 1–122 (2011).
    [Crossref]
  52. R. Glowinski and A. Marroco, “Sur l’approximation, par éléments finis d’ordre un, et la résolution, par pénalisation-dualité d’une classe de problémes de Dirichlet non linéaires,” RAIRO Anal. Numer 9, 41–76 (1975).
  53. D. Gabay and B. Mercier, “A dual algorithm for the solution of nonlinear variational problems via finite element approximation,” Optik 2, 17–40 (1976).
  54. B. He and X. Yuan, “On the O(1/n) convergence rate of the Douglas-Rachford alternating direction method,” SIAM J. Numer. Anal. 50, 700–709 (2012).
    [Crossref]
  55. M. Hong, Z. Luo, and M. Razaviyayn, “Convergence analysis of alternating direction method of multipliers for a family of nonconvex problems,” SIAM J. on Optim. 26, 337–364 (2016).
    [Crossref]
  56. Y. Wang, W. Yin, and J. Zeng, “Global convergence of ADMM in nonconvex nonsmooth optimization,” J. Sci. Comput. 7829–63 (2015).
  57. D. Davis and W. Yin, “Convergence rate analysis of several splitting schemes,” in Splitting Methods in Communication, Imaging, Science, and Engineering, R. Glowinski, S. J. Osher, and W. Yin, eds. (Springer International Publishing, Cham, 2016), pp. 115–163.
    [Crossref]
  58. C. Yang, J. Qian, A. Schirotzek, F. Maia, and S. Marchesini, “Iterative algorithms for ptychographic phase retrieval,” arXiv preprint arXiv:1105.5628 (2011).
  59. L.-H. Yeh, J. Dong, J. Zhong, L. Tian, M. Chen, G. Tang, M. Soltanolkotabi, and L. Waller, “Experimental robustness of Fourier ptychography phase retrieval algorithms,” Opt. Express 23, 33214–33240 (2015).
    [Crossref]
  60. J. Douglas and H. H. Rachford, “On the numerical solution of heat conduction problems in two and three space variables,” Transactions Am. Math. Soc. 82, 421–439 (1956).
    [Crossref]
  61. M. R. Hestenes, “Multiplier and gradient methods,” J. Optim. Theory Appl. 4, 303–320 (1969).
    [Crossref]
  62. M. J. Powell, “A method for nonlinear constraints in minimization problems,” Optimization 1969283–298 (1969).
  63. L. Li, X. Wang, and G. Wang, “Alternating direction method of multipliers for separable convex optimization of real functions in complex variables,” Math. Probl. Eng. 2015, 104531 (2015).
    [Crossref]
  64. R. Remmert, Theory of Complex Functions (Springer, 1991).
    [Crossref]
  65. K. Kreutz-Delgado, “The complex gradient operator and the CR-calculus,” arXiv preprint arXiv:0906.4835 (2009).
  66. G. Beylkin, “On the fast Fourier transform of functions with singularities,” Appl. Comput. Harmon. Analysis 2, 363–381 (1995).
    [Crossref]
  67. D. J. Ching and D. Gürsoy, “XDesign: an open-source software package for designing x-ray imaging phantoms and experiments,” J. synchrotron radiation 24, 537–544 (2017).
    [Crossref]
  68. B. L. Henke, E. M. Gullikson, and J. C. Davis, “X-ray interactions: photoabsorption, scattering, transmission, and reflection at E=50–30,000 eV, Z=1–92,” At. Data Nucl. Data Tables 54, 181–342 (1993).
    [Crossref]
  69. D. J. Ching, M. Hidayetoğlu, T. Bicer, and D. Gürsoy, “Rotation-as-fast-axis scanning-probe x-ray tomography: the importance of angular diversity for fly-scan modes,” Appl. Opt. 57, 8780–8789 (2018).
    [Crossref] [PubMed]
  70. T. Bicer, D. Gürsoy, R. Kettimuthu, I. T. Foster, B. Ren, V. D. Andrede, and F. De Carlo, “Real-time data analysis and autonomous steering of synchrotron light source experiments,” in 2017 IEEE 13th International Conference on e-Science, (2017), pp. 59–68.
    [Crossref]
  71. T. Bicer, D. Gürsoy, V. D. Andrade, R. Kettimuthu, W. Scullin, F. D. Carlo, and I. T. Foster, “Trace: a high-throughput tomographic reconstruction engine for large-scale datasets,” Adv. Struct. Chem. Imaging 3, 6 (2017).
    [Crossref] [PubMed]
  72. T. Bicer, D. Gürsoy, R. Kettimuthu, F. De Carlo, G. Agrawal, and I. T. Foster, “Rapid tomographic image reconstruction via large-scale parallelization,” in Euro-Par 2015: Parallel Processing, J. L. Träff, S. Hunold, and F. Versaci, eds. (Springer, Berlin, Heidelberg, 2015), pp. 289–302.
  73. Y. S. G. Nashed, D. J. Vine, T. Peterka, J. Deng, R. Ross, and C. Jacobsen, “Parallel ptychographic reconstruction,” Opt. Express 2, 32082–32097 (2014).
    [Crossref]
  74. S. Marchesini, H. Krishnan, B. J. Daurer, D. A. Shapiro, T. Perciano, J. A. Sethian, and F. R. N. C. Maia, “SHARP: a distributed GPU-based ptychographic solver,” J. Appl. Cryst. 49, 1245–1252 (2016).
    [Crossref]
  75. F. Andersson, M. Carlsson, and V. V. Nikitin, “Fast algorithms and efficient GPU implementations for the Radon transform and the back-projection operator represented as convolution operators,” SIAM J. on Imaging Sci. 9, 637–664 (2016).
    [Crossref]
  76. D. Gürsoy, X. Xiao, F. D. Carlo, and C. Jacobsen, “Tomopy: a framework for the analysis of synchrotron tomographic data,” J. Synchrotron Rad. 21, 1188–1193 (2014).
    [Crossref]
  77. 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]
  78. E. Wolf, “New theory of partial coherence in the space-frequency domain. Part I: spectra and cross spectra of steady-state sources,” JOSA 72, 343–351 (1982).
    [Crossref]
  79. H. Chang, P. Enfedaque, Y. Lou, and S. Marchesini, “Partially coherent ptychography by gradient decomposition of the probe,” Acta Crystallogr. Sect. A: Foundations Adv. 74, 157–169 (2018).
    [Crossref]

2018 (10)

F. Pfeiffer, “X-ray ptychography,” Nat. Photonics 12, 9–17 (2018).
[Crossref]

P. Li and A. M. Maiden, “Ten implementations of ptychography,” J. Microsc. 269, 187–194 (2018).
[Crossref]

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

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

P. Li and A. Maiden, “Multi-slice ptychographic tomography,” Sci. Reports 82049 (2018).

M. A. Gilles, Y. S. G. Nashed, C. J. M. Du, and S. M. Wild, “3D x-ray imaging of continuous objects beyond the depth of focus limit,” Optica 5, 1078–1086 (2018).
[Crossref] [PubMed]

K. Shimomura, M. Hirose, T. Higashino, and Y. Takahashi, “Three-dimensional iterative multislice reconstruction for ptychographic x-ray computed tomography,” Opt. Exp. 26, 31199 (2018).
[Crossref]

H. Öztürk, H. Yan, Y. He, M. Ge, Z. Dong, M. Lin, E. Nazaretski, I. K. Robinson, Y. S. Chu, and X. Huang, “Multi-slice ptychography with large numerical aperture multilayer laue lenses,” Optica 5, 601 (2018).
[Crossref]

D. J. Ching, M. Hidayetoğlu, T. Bicer, and D. Gürsoy, “Rotation-as-fast-axis scanning-probe x-ray tomography: the importance of angular diversity for fly-scan modes,” Appl. Opt. 57, 8780–8789 (2018).
[Crossref] [PubMed]

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

2017 (6)

T. Bicer, D. Gürsoy, V. D. Andrade, R. Kettimuthu, W. Scullin, F. D. Carlo, and I. T. Foster, “Trace: a high-throughput tomographic reconstruction engine for large-scale datasets,” Adv. Struct. Chem. Imaging 3, 6 (2017).
[Crossref] [PubMed]

D. J. Ching and D. Gürsoy, “XDesign: an open-source software package for designing x-ray imaging phantoms and experiments,” J. synchrotron radiation 24, 537–544 (2017).
[Crossref]

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

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

J. Deng, Y. P. Hong, S. Chen, Y. S. G. Nashed, T. Peterka, A. J. F. Levi, J. Damoulakis, S. Saha, T. Eiles, and C. Jacobsen, “Nanoscale x-ray imaging of circuit features without wafer etching,” Phys. Rev. B 95, 104111 (2017).
[Crossref] [PubMed]

D. Gürsoy, “Direct coupling of tomography and ptychography,” Opt. Lett. 42, 3169–3172 (2017).
[Crossref] [PubMed]

2016 (4)

M. W. M. Jones, K. D. Elgass, M. D. Junker, M. D. de Jonge, and G. A. van Riessen, “Molar concentration from sequential 2-D water-window x-ray ptychography and x-ray fluorescence in hydrated cells,” Sci. Rep. 6, 24280 (2016).
[Crossref] [PubMed]

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

F. Andersson, M. Carlsson, and V. V. Nikitin, “Fast algorithms and efficient GPU implementations for the Radon transform and the back-projection operator represented as convolution operators,” SIAM J. on Imaging Sci. 9, 637–664 (2016).
[Crossref]

M. Hong, Z. Luo, and M. Razaviyayn, “Convergence analysis of alternating direction method of multipliers for a family of nonconvex problems,” SIAM J. on Optim. 26, 337–364 (2016).
[Crossref]

2015 (10)

Y. Wang, W. Yin, and J. Zeng, “Global convergence of ADMM in nonconvex nonsmooth optimization,” J. Sci. Comput. 7829–63 (2015).

L.-H. Yeh, J. Dong, J. Zhong, L. Tian, M. Chen, G. Tang, M. Soltanolkotabi, and L. Waller, “Experimental robustness of Fourier ptychography phase retrieval algorithms,” Opt. Express 23, 33214–33240 (2015).
[Crossref]

L. Li, X. Wang, and G. Wang, “Alternating direction method of multipliers for separable convex optimization of real functions in complex variables,” Math. Probl. Eng. 2015, 104531 (2015).
[Crossref]

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

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

J. N. Weker and M. F. Toney, “Emerging in situ and operando nanoscale x-ray imaging techniques for energy storage materials,” Adv. Funct. Mater. 25, 1622–1637 (2015).
[Crossref]

J. Deng, D. J. Vine, S. Chen, Y. S. G. Nashed, Q. Jin, N. W. Phillips, T. Peterka, R. Ross, S. Vogt, and C. J. Jacobsen, “Simultaneous cryo x-ray ptychographic and fluorescence microscopy of green algae,” PNAS 112, 2314–2319 (2015).
[Crossref] [PubMed]

A. Diaz, B. Malkova, M. Holler, M. Guizar-Sicairos, E. Lima, V. Panneels, G. Pigino, A. G. Bittermann, L. Wettstein, T. Tomizaki, O. Bunk, G. Schertler, T. Ishikawa, R. Wepf, and A. Menzel, “Three-dimensional mass density mapping of cellular ultrastructure by ptychographic x-ray nanotomography,” J. Struct. Biol. 192, 461–469 (2015).
[Crossref] [PubMed]

J. Deng, Y. S. G. Nashed, S. Chen, N. W. Phillips, T. Peterka, R. Ross, S. Vogt, C. Jacobsen, and D. J. Vine, “Continuous motion scan ptychography: characterization for increased speed in coherent x-ray imaging,” Opt. Express 235438 (2015).
[Crossref]

2014 (5)

M. Holler, A. Diaz, M. Guizar-Sicairos, P. Karvinen, E. Färm, E. Härkönen, M. Ritala, A. Menzel, J. Raabe, and O. Bunk, “X-ray ptychographic computed tomography at 16 nm isotropic 3D resolution,” Sci. Reports 4, 3857 (2014).
[Crossref]

K. Høydalsvik, J. B. Fløystad, T. Zhao, M. Esmaeili, A. Diaz, J. W. Andreasen, R. H. Mathiesen, M. Rønning, and D. W. Breiby, “In situ x-ray ptychography imaging of high-temperature CO2 acceptor particle agglomerates,” Appl. Phys. Lett. 104, 241909 (2014).
[Crossref]

J. Bolte, S. Sabach, and M. Teboulle, “Proximal alternating linearized minimization for nonconvex and nonsmooth problems,” Math. Program. 146, 459–494 (2014).
[Crossref]

D. Gürsoy, X. Xiao, F. D. Carlo, and C. Jacobsen, “Tomopy: a framework for the analysis of synchrotron tomographic data,” J. Synchrotron Rad. 21, 1188–1193 (2014).
[Crossref]

Y. S. G. Nashed, D. J. Vine, T. Peterka, J. Deng, R. Ross, and C. Jacobsen, “Parallel ptychographic reconstruction,” Opt. Express 2, 32082–32097 (2014).
[Crossref]

2013 (5)

E. J. Candès, Y. C. Eldar, T. Strohmer, and V. Voroninski, “Phase retrieval via matrix completion,” SIAM J. on Imaging Sci. 6, 199–225 (2013).
[Crossref]

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

E. Lima, A. Diaz, M. Guizar-Sicairos, S. Gorelick, P. Pernot, T. Schleirer, and A. Menzel, “Cryo-scanning x-ray diffraction microscopy of frozen-hydrated yeast,” J. Microsc. 249, 1–7 (2013).
[Crossref]

P. Trtik, A. Diaz, M. Guizar-Sicairos, A. Menzel, and O. Bunk, “Density mapping of hardened cement paste using ptychographic x-ray computed tomography,” Cem. & Concr. Compos. 36, 71–77 (2013).
[Crossref]

B. Chen, M. Guizar-Sicairos, G. Xiong, L. Shemilt, A. Diaz, J. Nutter, N. Burdet, S. Huo, J. Mancuso, A. Monteith, F. Vergeer, A. Burgess, and I. Robinson, “Three-dimensional structure analysis and percolation properties of a barrier marine coating,” Sci. Rep. 3, 1177 (2013).
[Crossref] [PubMed]

2012 (6)

R. N. Wilke, M. Priebe, M. Bartels, K. Giewekemeyer, A. Diaz, P. Karvinen, and T. Salditt, “Hard x-ray imaging of bacterial cells: nano-diffraction and ptychographic reconstruction,” Opt. Express 20, 19232–19254 (2012).
[Crossref] [PubMed]

M. Holler, J. Raabe, A. Diaz, M. Guizar-Sicairos, C. Quitmann, A. Menzel, and O. Bunk, “An instrument for 3D x-ray nano-imaging,” Rev. Sci. Instrum. 83, 073703 (2012).
[Crossref] [PubMed]

A. Diaz, P. Trtik, M. Guizar-Sicairos, A. Menzel, P. Thibault, and O. Bunk, “Quantitative x-ray phase nanotomography,” Phys. Rev. B 85, 020104 (2012).
[Crossref]

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

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

B. He and X. Yuan, “On the O(1/n) convergence rate of the Douglas-Rachford alternating direction method,” SIAM J. Numer. Anal. 50, 700–709 (2012).
[Crossref]

2011 (4)

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

A. Schropp, P. Boye, A. Goldschmidt, S. Honig, R. Hoppe, J. Patommel, C. Rakete, D. Samberg, S. Stephan, S. Schoder, M. Burghammer, and C. G. Schroer, “Non-destructive and quantitative imaging of a nano-structured microchip by ptychographic hard x-ray scanning microscopy,” J. Microsc. 241, 9–12 (2011).
[Crossref]

B. Mike, S. Tobias, G. Thomas, R. Michael, G. Klaus, G. Sophie-Charlotte, S. Tim, and R. Axel, “Chemical contrast in soft x-ray ptychography,” Phys. Rev. Lett. 107, 208101 (2011).
[Crossref]

C. L. Hoy, N. J. Durr, and A. Ben-Yakar, “Fast-updating and nonrepeating Lissajous image reconstruction method for capturing increased dynamic information,” Appl. Opt. 50, 2376–2382 (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)

K. Giewekemeyera, P. Thibault, S. Kalbfleisch, A. Beerlinka, C. M. Kewishc, M. Dierolf, F. Pfeiffer, and T. Salditt, “Quantitative biological imaging by ptychographic x-ray diffraction microscopy,” PNAS 107, 529–534 (2009).
[Crossref]

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

2008 (3)

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

M. Dierolf, O. Bunk, S. Kynde, P. Thibault, I. Johnson, A. Menzel, K. Jefimovs, C. David, O. Marti, and F. Pfeiffer, “Ptychography & lensless x-ray imaging,” Europhys. News 39, 22–24 (2008).
[Crossref]

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

2007 (2)

J. M. Rodenburg, A. C. Hurst, A. G. Cullis, B. R. Dobson, F. Pfeiffer, O. Bunk, C. David, K. Jefimovs, and I. Johnson, “Hard-x-ray lensless imaging of extended objects,” Phys. Rev. Lett. 98, 034801 (2007).
[Crossref] [PubMed]

J. M. Rodenburg, A. C. Hurst, and A. G. Cullis, “Transmission microscopy without lenses for objects of unlimited size,” Ultramicroscopy 107, 227–231 (2007).
[Crossref]

2004 (1)

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

2003 (1)

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

1995 (1)

G. Beylkin, “On the fast Fourier transform of functions with singularities,” Appl. Comput. Harmon. Analysis 2, 363–381 (1995).
[Crossref]

1993 (1)

B. L. Henke, E. M. Gullikson, and J. C. Davis, “X-ray interactions: photoabsorption, scattering, transmission, and reflection at E=50–30,000 eV, Z=1–92,” At. Data Nucl. Data Tables 54, 181–342 (1993).
[Crossref]

1982 (2)

E. Wolf, “New theory of partial coherence in the space-frequency domain. Part I: spectra and cross spectra of steady-state sources,” JOSA 72, 343–351 (1982).
[Crossref]

J. R. Fienup, “Phase retrieval algorithms: a comparison,” Appl. Opt. 21, 2758–2769 (1982).
[Crossref] [PubMed]

1978 (1)

J. R. Fienup, “Reconstruction of an object from the modulus of its Fourier transform,” Opt. Lett. 3, 27–29 (1978).
[Crossref] [PubMed]

1976 (1)

D. Gabay and B. Mercier, “A dual algorithm for the solution of nonlinear variational problems via finite element approximation,” Optik 2, 17–40 (1976).

1975 (1)

R. Glowinski and A. Marroco, “Sur l’approximation, par éléments finis d’ordre un, et la résolution, par pénalisation-dualité d’une classe de problémes de Dirichlet non linéaires,” RAIRO Anal. Numer 9, 41–76 (1975).

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 (3)

W. Hoppe, “Beugung im inhomogenen primärstrahlwellenfeld, I. prinzip einer phasenmessung,” Acta Crystallogr. A 25, 495–501 (1969).
[Crossref]

M. R. Hestenes, “Multiplier and gradient methods,” J. Optim. Theory Appl. 4, 303–320 (1969).
[Crossref]

M. J. Powell, “A method for nonlinear constraints in minimization problems,” Optimization 1969283–298 (1969).

1956 (1)

J. Douglas and H. H. Rachford, “On the numerical solution of heat conduction problems in two and three space variables,” Transactions Am. Math. Soc. 82, 421–439 (1956).
[Crossref]

Aeppli, G.

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

Agrawal, G.

T. Bicer, D. Gürsoy, R. Kettimuthu, F. De Carlo, G. Agrawal, and I. T. Foster, “Rapid tomographic image reconstruction via large-scale parallelization,” in Euro-Par 2015: Parallel Processing, J. L. Träff, S. Hunold, and F. Versaci, eds. (Springer, Berlin, Heidelberg, 2015), pp. 289–302.

Andersen, M. S.

T. Ramos, B. E. Grønager, M. S. Andersen, and J. W. Andreasen, “Direct 3D tomographic reconstruction and phase-retrieval of far-field coherent diffraction patterns,” arXiv preprint arXiv:1808.02109 (2018).

Andersson, F.

F. Andersson, M. Carlsson, and V. V. Nikitin, “Fast algorithms and efficient GPU implementations for the Radon transform and the back-projection operator represented as convolution operators,” SIAM J. on Imaging Sci. 9, 637–664 (2016).
[Crossref]

Andrade, V. D.

T. Bicer, D. Gürsoy, V. D. Andrade, R. Kettimuthu, W. Scullin, F. D. Carlo, and I. T. Foster, “Trace: a high-throughput tomographic reconstruction engine for large-scale datasets,” Adv. Struct. Chem. Imaging 3, 6 (2017).
[Crossref] [PubMed]

Andreasen, J. W.

K. Høydalsvik, J. B. Fløystad, T. Zhao, M. Esmaeili, A. Diaz, J. W. Andreasen, R. H. Mathiesen, M. Rønning, and D. W. Breiby, “In situ x-ray ptychography imaging of high-temperature CO2 acceptor particle agglomerates,” Appl. Phys. Lett. 104, 241909 (2014).
[Crossref]

T. Ramos, B. E. Grønager, M. S. Andersen, and J. W. Andreasen, “Direct 3D tomographic reconstruction and phase-retrieval of far-field coherent diffraction patterns,” arXiv preprint arXiv:1808.02109 (2018).

Andrede, V. D.

T. Bicer, D. Gürsoy, R. Kettimuthu, I. T. Foster, B. Ren, V. D. Andrede, and F. De Carlo, “Real-time data analysis and autonomous steering of synchrotron light source experiments,” in 2017 IEEE 13th International Conference on e-Science, (2017), pp. 59–68.
[Crossref]

Axel, R.

B. Mike, S. Tobias, G. Thomas, R. Michael, G. Klaus, G. Sophie-Charlotte, S. Tim, and R. Axel, “Chemical contrast in soft x-ray ptychography,” Phys. Rev. Lett. 107, 208101 (2011).
[Crossref]

Bartels, M.

R. N. Wilke, M. Priebe, M. Bartels, K. Giewekemeyer, A. Diaz, P. Karvinen, and T. Salditt, “Hard x-ray imaging of bacterial cells: nano-diffraction and ptychographic reconstruction,” Opt. Express 20, 19232–19254 (2012).
[Crossref] [PubMed]

Beerlinka, A.

K. Giewekemeyera, P. Thibault, S. Kalbfleisch, A. Beerlinka, C. M. Kewishc, M. Dierolf, F. Pfeiffer, and T. Salditt, “Quantitative biological imaging by ptychographic x-ray diffraction microscopy,” PNAS 107, 529–534 (2009).
[Crossref]

Ben-Yakar, A.

C. L. Hoy, N. J. Durr, and A. Ben-Yakar, “Fast-updating and nonrepeating Lissajous image reconstruction method for capturing increased dynamic information,” Appl. Opt. 50, 2376–2382 (2011).
[Crossref] [PubMed]

Beylkin, G.

G. Beylkin, “On the fast Fourier transform of functions with singularities,” Appl. Comput. Harmon. Analysis 2, 363–381 (1995).
[Crossref]

Bian, L.

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]

Bicer, T.

D. J. Ching, M. Hidayetoğlu, T. Bicer, and D. Gürsoy, “Rotation-as-fast-axis scanning-probe x-ray tomography: the importance of angular diversity for fly-scan modes,” Appl. Opt. 57, 8780–8789 (2018).
[Crossref] [PubMed]

T. Bicer, D. Gürsoy, V. D. Andrade, R. Kettimuthu, W. Scullin, F. D. Carlo, and I. T. Foster, “Trace: a high-throughput tomographic reconstruction engine for large-scale datasets,” Adv. Struct. Chem. Imaging 3, 6 (2017).
[Crossref] [PubMed]

T. Bicer, D. Gürsoy, R. Kettimuthu, F. De Carlo, G. Agrawal, and I. T. Foster, “Rapid tomographic image reconstruction via large-scale parallelization,” in Euro-Par 2015: Parallel Processing, J. L. Träff, S. Hunold, and F. Versaci, eds. (Springer, Berlin, Heidelberg, 2015), pp. 289–302.

T. Bicer, D. Gürsoy, R. Kettimuthu, I. T. Foster, B. Ren, V. D. Andrede, and F. De Carlo, “Real-time data analysis and autonomous steering of synchrotron light source experiments,” in 2017 IEEE 13th International Conference on e-Science, (2017), pp. 59–68.
[Crossref]

Bittermann, A. G.

A. Diaz, B. Malkova, M. Holler, M. Guizar-Sicairos, E. Lima, V. Panneels, G. Pigino, A. G. Bittermann, L. Wettstein, T. Tomizaki, O. Bunk, G. Schertler, T. Ishikawa, R. Wepf, and A. Menzel, “Three-dimensional mass density mapping of cellular ultrastructure by ptychographic x-ray nanotomography,” J. Struct. Biol. 192, 461–469 (2015).
[Crossref] [PubMed]

Bolte, J.

J. Bolte, S. Sabach, and M. Teboulle, “Proximal alternating linearized minimization for nonconvex and nonsmooth problems,” Math. Program. 146, 459–494 (2014).
[Crossref]

Boyd, S.

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

Boye, P.

A. Schropp, P. Boye, A. Goldschmidt, S. Honig, R. Hoppe, J. Patommel, C. Rakete, D. Samberg, S. Stephan, S. Schoder, M. Burghammer, and C. G. Schroer, “Non-destructive and quantitative imaging of a nano-structured microchip by ptychographic hard x-ray scanning microscopy,” J. Microsc. 241, 9–12 (2011).
[Crossref]

Breiby, D. W.

K. Høydalsvik, J. B. Fløystad, T. Zhao, M. Esmaeili, A. Diaz, J. W. Andreasen, R. H. Mathiesen, M. Rønning, and D. W. Breiby, “In situ x-ray ptychography imaging of high-temperature CO2 acceptor particle agglomerates,” Appl. Phys. Lett. 104, 241909 (2014).
[Crossref]

Bunk, O.

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

A. Diaz, B. Malkova, M. Holler, M. Guizar-Sicairos, E. Lima, V. Panneels, G. Pigino, A. G. Bittermann, L. Wettstein, T. Tomizaki, O. Bunk, G. Schertler, T. Ishikawa, R. Wepf, and A. Menzel, “Three-dimensional mass density mapping of cellular ultrastructure by ptychographic x-ray nanotomography,” J. Struct. Biol. 192, 461–469 (2015).
[Crossref] [PubMed]

M. Holler, A. Diaz, M. Guizar-Sicairos, P. Karvinen, E. Färm, E. Härkönen, M. Ritala, A. Menzel, J. Raabe, and O. Bunk, “X-ray ptychographic computed tomography at 16 nm isotropic 3D resolution,” Sci. Reports 4, 3857 (2014).
[Crossref]

P. Trtik, A. Diaz, M. Guizar-Sicairos, A. Menzel, and O. Bunk, “Density mapping of hardened cement paste using ptychographic x-ray computed tomography,” Cem. & Concr. Compos. 36, 71–77 (2013).
[Crossref]

M. Holler, J. Raabe, A. Diaz, M. Guizar-Sicairos, C. Quitmann, A. Menzel, and O. Bunk, “An instrument for 3D x-ray nano-imaging,” Rev. Sci. Instrum. 83, 073703 (2012).
[Crossref] [PubMed]

A. Diaz, P. Trtik, M. Guizar-Sicairos, A. Menzel, P. Thibault, and O. Bunk, “Quantitative x-ray phase nanotomography,” Phys. Rev. B 85, 020104 (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]

M. Dierolf, O. Bunk, S. Kynde, P. Thibault, I. Johnson, A. Menzel, K. Jefimovs, C. David, O. Marti, and F. Pfeiffer, “Ptychography & lensless x-ray imaging,” Europhys. News 39, 22–24 (2008).
[Crossref]

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

J. M. Rodenburg, A. C. Hurst, A. G. Cullis, B. R. Dobson, F. Pfeiffer, O. Bunk, C. David, K. Jefimovs, and I. Johnson, “Hard-x-ray lensless imaging of extended objects,” Phys. Rev. Lett. 98, 034801 (2007).
[Crossref] [PubMed]

Burdet, N.

B. Chen, M. Guizar-Sicairos, G. Xiong, L. Shemilt, A. Diaz, J. Nutter, N. Burdet, S. Huo, J. Mancuso, A. Monteith, F. Vergeer, A. Burgess, and I. Robinson, “Three-dimensional structure analysis and percolation properties of a barrier marine coating,” Sci. Rep. 3, 1177 (2013).
[Crossref] [PubMed]

Burgess, A.

B. Chen, M. Guizar-Sicairos, G. Xiong, L. Shemilt, A. Diaz, J. Nutter, N. Burdet, S. Huo, J. Mancuso, A. Monteith, F. Vergeer, A. Burgess, and I. Robinson, “Three-dimensional structure analysis and percolation properties of a barrier marine coating,” Sci. Rep. 3, 1177 (2013).
[Crossref] [PubMed]

Burghammer, M.

A. Schropp, P. Boye, A. Goldschmidt, S. Honig, R. Hoppe, J. Patommel, C. Rakete, D. Samberg, S. Stephan, S. Schoder, M. Burghammer, and C. G. Schroer, “Non-destructive and quantitative imaging of a nano-structured microchip by ptychographic hard x-ray scanning microscopy,” J. Microsc. 241, 9–12 (2011).
[Crossref]

Candès, E. J.

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

E. J. Candès, Y. C. Eldar, T. Strohmer, and V. Voroninski, “Phase retrieval via matrix completion,” SIAM J. on Imaging Sci. 6, 199–225 (2013).
[Crossref]

Carlo, F. D.

T. Bicer, D. Gürsoy, V. D. Andrade, R. Kettimuthu, W. Scullin, F. D. Carlo, and I. T. Foster, “Trace: a high-throughput tomographic reconstruction engine for large-scale datasets,” Adv. Struct. Chem. Imaging 3, 6 (2017).
[Crossref] [PubMed]

D. Gürsoy, X. Xiao, F. D. Carlo, and C. Jacobsen, “Tomopy: a framework for the analysis of synchrotron tomographic data,” J. Synchrotron Rad. 21, 1188–1193 (2014).
[Crossref]

Carlsson, M.

F. Andersson, M. Carlsson, and V. V. Nikitin, “Fast algorithms and efficient GPU implementations for the Radon transform and the back-projection operator represented as convolution operators,” SIAM J. on Imaging Sci. 9, 637–664 (2016).
[Crossref]

Chang, H.

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

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

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

Chen, B.

B. Chen, M. Guizar-Sicairos, G. Xiong, L. Shemilt, A. Diaz, J. Nutter, N. Burdet, S. Huo, J. Mancuso, A. Monteith, F. Vergeer, A. Burgess, and I. Robinson, “Three-dimensional structure analysis and percolation properties of a barrier marine coating,” Sci. Rep. 3, 1177 (2013).
[Crossref] [PubMed]

Chen, F.

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]

Chen, M.

L.-H. Yeh, J. Dong, J. Zhong, L. Tian, M. Chen, G. Tang, M. Soltanolkotabi, and L. Waller, “Experimental robustness of Fourier ptychography phase retrieval algorithms,” Opt. Express 23, 33214–33240 (2015).
[Crossref]

Chen, S.

J. Deng, Y. P. Hong, S. Chen, Y. S. G. Nashed, T. Peterka, A. J. F. Levi, J. Damoulakis, S. Saha, T. Eiles, and C. Jacobsen, “Nanoscale x-ray imaging of circuit features without wafer etching,” Phys. Rev. B 95, 104111 (2017).
[Crossref] [PubMed]

J. Deng, D. J. Vine, S. Chen, Y. S. G. Nashed, Q. Jin, N. W. Phillips, T. Peterka, R. Ross, S. Vogt, and C. J. Jacobsen, “Simultaneous cryo x-ray ptychographic and fluorescence microscopy of green algae,” PNAS 112, 2314–2319 (2015).
[Crossref] [PubMed]

J. Deng, Y. S. G. Nashed, S. Chen, N. W. Phillips, T. Peterka, R. Ross, S. Vogt, C. Jacobsen, and D. J. Vine, “Continuous motion scan ptychography: characterization for increased speed in coherent x-ray imaging,” Opt. Express 235438 (2015).
[Crossref]

Ching, D. J.

D. J. Ching, M. Hidayetoğlu, T. Bicer, and D. Gürsoy, “Rotation-as-fast-axis scanning-probe x-ray tomography: the importance of angular diversity for fly-scan modes,” Appl. Opt. 57, 8780–8789 (2018).
[Crossref] [PubMed]

D. J. Ching and D. Gürsoy, “XDesign: an open-source software package for designing x-ray imaging phantoms and experiments,” J. synchrotron radiation 24, 537–544 (2017).
[Crossref]

Chu, E.

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

Chu, Y. S.

H. Öztürk, H. Yan, Y. He, M. Ge, Z. Dong, M. Lin, E. Nazaretski, I. K. Robinson, Y. S. Chu, and X. Huang, “Multi-slice ptychography with large numerical aperture multilayer laue lenses,” Optica 5, 601 (2018).
[Crossref]

Cullis, A. G.

J. M. Rodenburg, A. C. Hurst, A. G. Cullis, B. R. Dobson, F. Pfeiffer, O. Bunk, C. David, K. Jefimovs, and I. Johnson, “Hard-x-ray lensless imaging of extended objects,” Phys. Rev. Lett. 98, 034801 (2007).
[Crossref] [PubMed]

J. M. Rodenburg, A. C. Hurst, and A. G. Cullis, “Transmission microscopy without lenses for objects of unlimited size,” Ultramicroscopy 107, 227–231 (2007).
[Crossref]

Dai, Q.

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]

Damoulakis, J.

J. Deng, Y. P. Hong, S. Chen, Y. S. G. Nashed, T. Peterka, A. J. F. Levi, J. Damoulakis, S. Saha, T. Eiles, and C. Jacobsen, “Nanoscale x-ray imaging of circuit features without wafer etching,” Phys. Rev. B 95, 104111 (2017).
[Crossref] [PubMed]

Daurer, B. J.

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

David, C.

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

M. Dierolf, O. Bunk, S. Kynde, P. Thibault, I. Johnson, A. Menzel, K. Jefimovs, C. David, O. Marti, and F. Pfeiffer, “Ptychography & lensless x-ray imaging,” Europhys. News 39, 22–24 (2008).
[Crossref]

J. M. Rodenburg, A. C. Hurst, A. G. Cullis, B. R. Dobson, F. Pfeiffer, O. Bunk, C. David, K. Jefimovs, and I. Johnson, “Hard-x-ray lensless imaging of extended objects,” Phys. Rev. Lett. 98, 034801 (2007).
[Crossref] [PubMed]

Davis, D.

D. Davis and W. Yin, “Convergence rate analysis of several splitting schemes,” in Splitting Methods in Communication, Imaging, Science, and Engineering, R. Glowinski, S. J. Osher, and W. Yin, eds. (Springer International Publishing, Cham, 2016), pp. 115–163.
[Crossref]

Davis, J. C.

B. L. Henke, E. M. Gullikson, and J. C. Davis, “X-ray interactions: photoabsorption, scattering, transmission, and reflection at E=50–30,000 eV, Z=1–92,” At. Data Nucl. Data Tables 54, 181–342 (1993).
[Crossref]

De Carlo, F.

T. Bicer, D. Gürsoy, R. Kettimuthu, F. De Carlo, G. Agrawal, and I. T. Foster, “Rapid tomographic image reconstruction via large-scale parallelization,” in Euro-Par 2015: Parallel Processing, J. L. Träff, S. Hunold, and F. Versaci, eds. (Springer, Berlin, Heidelberg, 2015), pp. 289–302.

T. Bicer, D. Gürsoy, R. Kettimuthu, I. T. Foster, B. Ren, V. D. Andrede, and F. De Carlo, “Real-time data analysis and autonomous steering of synchrotron light source experiments,” in 2017 IEEE 13th International Conference on e-Science, (2017), pp. 59–68.
[Crossref]

de Jonge, M. D.

M. W. M. Jones, K. D. Elgass, M. D. Junker, M. D. de Jonge, and G. A. van Riessen, “Molar concentration from sequential 2-D water-window x-ray ptychography and x-ray fluorescence in hydrated cells,” Sci. Rep. 6, 24280 (2016).
[Crossref] [PubMed]

Deng, J.

J. Deng, Y. P. Hong, S. Chen, Y. S. G. Nashed, T. Peterka, A. J. F. Levi, J. Damoulakis, S. Saha, T. Eiles, and C. Jacobsen, “Nanoscale x-ray imaging of circuit features without wafer etching,” Phys. Rev. B 95, 104111 (2017).
[Crossref] [PubMed]

J. Deng, D. J. Vine, S. Chen, Y. S. G. Nashed, Q. Jin, N. W. Phillips, T. Peterka, R. Ross, S. Vogt, and C. J. Jacobsen, “Simultaneous cryo x-ray ptychographic and fluorescence microscopy of green algae,” PNAS 112, 2314–2319 (2015).
[Crossref] [PubMed]

J. Deng, Y. S. G. Nashed, S. Chen, N. W. Phillips, T. Peterka, R. Ross, S. Vogt, C. Jacobsen, and D. J. Vine, “Continuous motion scan ptychography: characterization for increased speed in coherent x-ray imaging,” Opt. Express 235438 (2015).
[Crossref]

Y. S. G. Nashed, D. J. Vine, T. Peterka, J. Deng, R. Ross, and C. Jacobsen, “Parallel ptychographic reconstruction,” Opt. Express 2, 32082–32097 (2014).
[Crossref]

Diaz, A.

A. Diaz, B. Malkova, M. Holler, M. Guizar-Sicairos, E. Lima, V. Panneels, G. Pigino, A. G. Bittermann, L. Wettstein, T. Tomizaki, O. Bunk, G. Schertler, T. Ishikawa, R. Wepf, and A. Menzel, “Three-dimensional mass density mapping of cellular ultrastructure by ptychographic x-ray nanotomography,” J. Struct. Biol. 192, 461–469 (2015).
[Crossref] [PubMed]

M. Holler, A. Diaz, M. Guizar-Sicairos, P. Karvinen, E. Färm, E. Härkönen, M. Ritala, A. Menzel, J. Raabe, and O. Bunk, “X-ray ptychographic computed tomography at 16 nm isotropic 3D resolution,” Sci. Reports 4, 3857 (2014).
[Crossref]

K. Høydalsvik, J. B. Fløystad, T. Zhao, M. Esmaeili, A. Diaz, J. W. Andreasen, R. H. Mathiesen, M. Rønning, and D. W. Breiby, “In situ x-ray ptychography imaging of high-temperature CO2 acceptor particle agglomerates,” Appl. Phys. Lett. 104, 241909 (2014).
[Crossref]

B. Chen, M. Guizar-Sicairos, G. Xiong, L. Shemilt, A. Diaz, J. Nutter, N. Burdet, S. Huo, J. Mancuso, A. Monteith, F. Vergeer, A. Burgess, and I. Robinson, “Three-dimensional structure analysis and percolation properties of a barrier marine coating,” Sci. Rep. 3, 1177 (2013).
[Crossref] [PubMed]

P. Trtik, A. Diaz, M. Guizar-Sicairos, A. Menzel, and O. Bunk, “Density mapping of hardened cement paste using ptychographic x-ray computed tomography,” Cem. & Concr. Compos. 36, 71–77 (2013).
[Crossref]

E. Lima, A. Diaz, M. Guizar-Sicairos, S. Gorelick, P. Pernot, T. Schleirer, and A. Menzel, “Cryo-scanning x-ray diffraction microscopy of frozen-hydrated yeast,” J. Microsc. 249, 1–7 (2013).
[Crossref]

R. N. Wilke, M. Priebe, M. Bartels, K. Giewekemeyer, A. Diaz, P. Karvinen, and T. Salditt, “Hard x-ray imaging of bacterial cells: nano-diffraction and ptychographic reconstruction,” Opt. Express 20, 19232–19254 (2012).
[Crossref] [PubMed]

A. Diaz, P. Trtik, M. Guizar-Sicairos, A. Menzel, P. Thibault, and O. Bunk, “Quantitative x-ray phase nanotomography,” Phys. Rev. B 85, 020104 (2012).
[Crossref]

M. Holler, J. Raabe, A. Diaz, M. Guizar-Sicairos, C. Quitmann, A. Menzel, and O. Bunk, “An instrument for 3D x-ray nano-imaging,” Rev. Sci. Instrum. 83, 073703 (2012).
[Crossref] [PubMed]

Dierolf, M.

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

K. Giewekemeyera, P. Thibault, S. Kalbfleisch, A. Beerlinka, C. M. Kewishc, M. Dierolf, F. Pfeiffer, and T. Salditt, “Quantitative biological imaging by ptychographic x-ray diffraction microscopy,” PNAS 107, 529–534 (2009).
[Crossref]

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

M. Dierolf, O. Bunk, S. Kynde, P. Thibault, I. Johnson, A. Menzel, K. Jefimovs, C. David, O. Marti, and F. Pfeiffer, “Ptychography & lensless x-ray imaging,” Europhys. News 39, 22–24 (2008).
[Crossref]

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

Dinapoli, R.

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

Dobson, B. R.

J. M. Rodenburg, A. C. Hurst, A. G. Cullis, B. R. Dobson, F. Pfeiffer, O. Bunk, C. David, K. Jefimovs, and I. Johnson, “Hard-x-ray lensless imaging of extended objects,” Phys. Rev. Lett. 98, 034801 (2007).
[Crossref] [PubMed]

Dong, J.

L.-H. Yeh, J. Dong, J. Zhong, L. Tian, M. Chen, G. Tang, M. Soltanolkotabi, and L. Waller, “Experimental robustness of Fourier ptychography phase retrieval algorithms,” Opt. Express 23, 33214–33240 (2015).
[Crossref]

Dong, Z.

H. Öztürk, H. Yan, Y. He, M. Ge, Z. Dong, M. Lin, E. Nazaretski, I. K. Robinson, Y. S. Chu, and X. Huang, “Multi-slice ptychography with large numerical aperture multilayer laue lenses,” Optica 5, 601 (2018).
[Crossref]

Douglas, J.

J. Douglas and H. H. Rachford, “On the numerical solution of heat conduction problems in two and three space variables,” Transactions Am. Math. Soc. 82, 421–439 (1956).
[Crossref]

Du, C. J. M.

M. A. Gilles, Y. S. G. Nashed, C. J. M. Du, and S. M. Wild, “3D x-ray imaging of continuous objects beyond the depth of focus limit,” Optica 5, 1078–1086 (2018).
[Crossref] [PubMed]

Duan, Y.

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

Durr, N. J.

C. L. Hoy, N. J. Durr, and A. Ben-Yakar, “Fast-updating and nonrepeating Lissajous image reconstruction method for capturing increased dynamic information,” Appl. Opt. 50, 2376–2382 (2011).
[Crossref] [PubMed]

Eckstein, J.

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

Eiles, T.

J. Deng, Y. P. Hong, S. Chen, Y. S. G. Nashed, T. Peterka, A. J. F. Levi, J. Damoulakis, S. Saha, T. Eiles, and C. Jacobsen, “Nanoscale x-ray imaging of circuit features without wafer etching,” Phys. Rev. B 95, 104111 (2017).
[Crossref] [PubMed]

Eldar, Y. C.

E. J. Candès, Y. C. Eldar, T. Strohmer, and V. Voroninski, “Phase retrieval via matrix completion,” SIAM J. on Imaging Sci. 6, 199–225 (2013).
[Crossref]

Elgass, K. D.

M. W. M. Jones, K. D. Elgass, M. D. Junker, M. D. de Jonge, and G. A. van Riessen, “Molar concentration from sequential 2-D water-window x-ray ptychography and x-ray fluorescence in hydrated cells,” Sci. Rep. 6, 24280 (2016).
[Crossref] [PubMed]

Elser, V.

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

Enfedaque, P.

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

Esmaeili, M.

K. Høydalsvik, J. B. Fløystad, T. Zhao, M. Esmaeili, A. Diaz, J. W. Andreasen, R. H. Mathiesen, M. Rønning, and D. W. Breiby, “In situ x-ray ptychography imaging of high-temperature CO2 acceptor particle agglomerates,” Appl. Phys. Lett. 104, 241909 (2014).
[Crossref]

Färm, E.

M. Holler, A. Diaz, M. Guizar-Sicairos, P. Karvinen, E. Färm, E. Härkönen, M. Ritala, A. Menzel, J. Raabe, and O. Bunk, “X-ray ptychographic computed tomography at 16 nm isotropic 3D resolution,” Sci. Reports 4, 3857 (2014).
[Crossref]

Faulkner, H. M. L.

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

Fienup, J. R.

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

J. R. Fienup, “Phase retrieval algorithms: a comparison,” Appl. Opt. 21, 2758–2769 (1982).
[Crossref] [PubMed]

J. R. Fienup, “Reconstruction of an object from the modulus of its Fourier transform,” Opt. Lett. 3, 27–29 (1978).
[Crossref] [PubMed]

Fløystad, J. B.

K. Høydalsvik, J. B. Fløystad, T. Zhao, M. Esmaeili, A. Diaz, J. W. Andreasen, R. H. Mathiesen, M. Rønning, and D. W. Breiby, “In situ x-ray ptychography imaging of high-temperature CO2 acceptor particle agglomerates,” Appl. Phys. Lett. 104, 241909 (2014).
[Crossref]

Foster, I. T.

T. Bicer, D. Gürsoy, V. D. Andrade, R. Kettimuthu, W. Scullin, F. D. Carlo, and I. T. Foster, “Trace: a high-throughput tomographic reconstruction engine for large-scale datasets,” Adv. Struct. Chem. Imaging 3, 6 (2017).
[Crossref] [PubMed]

T. Bicer, D. Gürsoy, R. Kettimuthu, F. De Carlo, G. Agrawal, and I. T. Foster, “Rapid tomographic image reconstruction via large-scale parallelization,” in Euro-Par 2015: Parallel Processing, J. L. Träff, S. Hunold, and F. Versaci, eds. (Springer, Berlin, Heidelberg, 2015), pp. 289–302.

T. Bicer, D. Gürsoy, R. Kettimuthu, I. T. Foster, B. Ren, V. D. Andrede, and F. De Carlo, “Real-time data analysis and autonomous steering of synchrotron light source experiments,” in 2017 IEEE 13th International Conference on e-Science, (2017), pp. 59–68.
[Crossref]

Gabay, D.

D. Gabay and B. Mercier, “A dual algorithm for the solution of nonlinear variational problems via finite element approximation,” Optik 2, 17–40 (1976).

Ge, M.

H. Öztürk, H. Yan, Y. He, M. Ge, Z. Dong, M. Lin, E. Nazaretski, I. K. Robinson, Y. S. Chu, and X. Huang, “Multi-slice ptychography with large numerical aperture multilayer laue lenses,” Optica 5, 601 (2018).
[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).

Giewekemeyer, K.

R. N. Wilke, M. Priebe, M. Bartels, K. Giewekemeyer, A. Diaz, P. Karvinen, and T. Salditt, “Hard x-ray imaging of bacterial cells: nano-diffraction and ptychographic reconstruction,” Opt. Express 20, 19232–19254 (2012).
[Crossref] [PubMed]

Giewekemeyera, K.

K. Giewekemeyera, P. Thibault, S. Kalbfleisch, A. Beerlinka, C. M. Kewishc, M. Dierolf, F. Pfeiffer, and T. Salditt, “Quantitative biological imaging by ptychographic x-ray diffraction microscopy,” PNAS 107, 529–534 (2009).
[Crossref]

Gilles, M. A.

M. A. Gilles, Y. S. G. Nashed, C. J. M. Du, and S. M. Wild, “3D x-ray imaging of continuous objects beyond the depth of focus limit,” Optica 5, 1078–1086 (2018).
[Crossref] [PubMed]

Glowinski, R.

R. Glowinski and A. Marroco, “Sur l’approximation, par éléments finis d’ordre un, et la résolution, par pénalisation-dualité d’une classe de problémes de Dirichlet non linéaires,” RAIRO Anal. Numer 9, 41–76 (1975).

Goldschmidt, A.

A. Schropp, P. Boye, A. Goldschmidt, S. Honig, R. Hoppe, J. Patommel, C. Rakete, D. Samberg, S. Stephan, S. Schoder, M. Burghammer, and C. G. Schroer, “Non-destructive and quantitative imaging of a nano-structured microchip by ptychographic hard x-ray scanning microscopy,” J. Microsc. 241, 9–12 (2011).
[Crossref]

Gorelick, S.

E. Lima, A. Diaz, M. Guizar-Sicairos, S. Gorelick, P. Pernot, T. Schleirer, and A. Menzel, “Cryo-scanning x-ray diffraction microscopy of frozen-hydrated yeast,” J. Microsc. 249, 1–7 (2013).
[Crossref]

Grønager, B. E.

T. Ramos, B. E. Grønager, M. S. Andersen, and J. W. Andreasen, “Direct 3D tomographic reconstruction and phase-retrieval of far-field coherent diffraction patterns,” arXiv preprint arXiv:1808.02109 (2018).

Guizar-Sicairos, M.

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

A. Diaz, B. Malkova, M. Holler, M. Guizar-Sicairos, E. Lima, V. Panneels, G. Pigino, A. G. Bittermann, L. Wettstein, T. Tomizaki, O. Bunk, G. Schertler, T. Ishikawa, R. Wepf, and A. Menzel, “Three-dimensional mass density mapping of cellular ultrastructure by ptychographic x-ray nanotomography,” J. Struct. Biol. 192, 461–469 (2015).
[Crossref] [PubMed]

M. Holler, A. Diaz, M. Guizar-Sicairos, P. Karvinen, E. Färm, E. Härkönen, M. Ritala, A. Menzel, J. Raabe, and O. Bunk, “X-ray ptychographic computed tomography at 16 nm isotropic 3D resolution,” Sci. Reports 4, 3857 (2014).
[Crossref]

P. Trtik, A. Diaz, M. Guizar-Sicairos, A. Menzel, and O. Bunk, “Density mapping of hardened cement paste using ptychographic x-ray computed tomography,” Cem. & Concr. Compos. 36, 71–77 (2013).
[Crossref]

B. Chen, M. Guizar-Sicairos, G. Xiong, L. Shemilt, A. Diaz, J. Nutter, N. Burdet, S. Huo, J. Mancuso, A. Monteith, F. Vergeer, A. Burgess, and I. Robinson, “Three-dimensional structure analysis and percolation properties of a barrier marine coating,” Sci. Rep. 3, 1177 (2013).
[Crossref] [PubMed]

E. Lima, A. Diaz, M. Guizar-Sicairos, S. Gorelick, P. Pernot, T. Schleirer, and A. Menzel, “Cryo-scanning x-ray diffraction microscopy of frozen-hydrated yeast,” J. Microsc. 249, 1–7 (2013).
[Crossref]

M. Holler, J. Raabe, A. Diaz, M. Guizar-Sicairos, C. Quitmann, A. Menzel, and O. Bunk, “An instrument for 3D x-ray nano-imaging,” Rev. Sci. Instrum. 83, 073703 (2012).
[Crossref] [PubMed]

A. Diaz, P. Trtik, M. Guizar-Sicairos, A. Menzel, P. Thibault, and O. Bunk, “Quantitative x-ray phase nanotomography,” Phys. Rev. B 85, 020104 (2012).
[Crossref]

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

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

Gullikson, E. M.

B. L. Henke, E. M. Gullikson, and J. C. Davis, “X-ray interactions: photoabsorption, scattering, transmission, and reflection at E=50–30,000 eV, Z=1–92,” At. Data Nucl. Data Tables 54, 181–342 (1993).
[Crossref]

Guo, K.

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]

Gürsoy, D.

D. J. Ching, M. Hidayetoğlu, T. Bicer, and D. Gürsoy, “Rotation-as-fast-axis scanning-probe x-ray tomography: the importance of angular diversity for fly-scan modes,” Appl. Opt. 57, 8780–8789 (2018).
[Crossref] [PubMed]

D. J. Ching and D. Gürsoy, “XDesign: an open-source software package for designing x-ray imaging phantoms and experiments,” J. synchrotron radiation 24, 537–544 (2017).
[Crossref]

T. Bicer, D. Gürsoy, V. D. Andrade, R. Kettimuthu, W. Scullin, F. D. Carlo, and I. T. Foster, “Trace: a high-throughput tomographic reconstruction engine for large-scale datasets,” Adv. Struct. Chem. Imaging 3, 6 (2017).
[Crossref] [PubMed]

D. Gürsoy, “Direct coupling of tomography and ptychography,” Opt. Lett. 42, 3169–3172 (2017).
[Crossref] [PubMed]

D. Gürsoy, X. Xiao, F. D. Carlo, and C. Jacobsen, “Tomopy: a framework for the analysis of synchrotron tomographic data,” J. Synchrotron Rad. 21, 1188–1193 (2014).
[Crossref]

T. Bicer, D. Gürsoy, R. Kettimuthu, F. De Carlo, G. Agrawal, and I. T. Foster, “Rapid tomographic image reconstruction via large-scale parallelization,” in Euro-Par 2015: Parallel Processing, J. L. Träff, S. Hunold, and F. Versaci, eds. (Springer, Berlin, Heidelberg, 2015), pp. 289–302.

T. Bicer, D. Gürsoy, R. Kettimuthu, I. T. Foster, B. Ren, V. D. Andrede, and F. De Carlo, “Real-time data analysis and autonomous steering of synchrotron light source experiments,” in 2017 IEEE 13th International Conference on e-Science, (2017), pp. 59–68.
[Crossref]

Härkönen, E.

M. Holler, A. Diaz, M. Guizar-Sicairos, P. Karvinen, E. Färm, E. Härkönen, M. Ritala, A. Menzel, J. Raabe, and O. Bunk, “X-ray ptychographic computed tomography at 16 nm isotropic 3D resolution,” Sci. Reports 4, 3857 (2014).
[Crossref]

He, B.

B. He and X. Yuan, “On the O(1/n) convergence rate of the Douglas-Rachford alternating direction method,” SIAM J. Numer. Anal. 50, 700–709 (2012).
[Crossref]

He, Y.

H. Öztürk, H. Yan, Y. He, M. Ge, Z. Dong, M. Lin, E. Nazaretski, I. K. Robinson, Y. S. Chu, and X. Huang, “Multi-slice ptychography with large numerical aperture multilayer laue lenses,” Optica 5, 601 (2018).
[Crossref]

Henke, B. L.

B. L. Henke, E. M. Gullikson, and J. C. Davis, “X-ray interactions: photoabsorption, scattering, transmission, and reflection at E=50–30,000 eV, Z=1–92,” At. Data Nucl. Data Tables 54, 181–342 (1993).
[Crossref]

Hesse, R.

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

Hestenes, M. R.

M. R. Hestenes, “Multiplier and gradient methods,” J. Optim. Theory Appl. 4, 303–320 (1969).
[Crossref]

Hidayetoglu, M.

D. J. Ching, M. Hidayetoğlu, T. Bicer, and D. Gürsoy, “Rotation-as-fast-axis scanning-probe x-ray tomography: the importance of angular diversity for fly-scan modes,” Appl. Opt. 57, 8780–8789 (2018).
[Crossref] [PubMed]

Higashino, T.

K. Shimomura, M. Hirose, T. Higashino, and Y. Takahashi, “Three-dimensional iterative multislice reconstruction for ptychographic x-ray computed tomography,” Opt. Exp. 26, 31199 (2018).
[Crossref]

Hirose, M.

K. Shimomura, M. Hirose, T. Higashino, and Y. Takahashi, “Three-dimensional iterative multislice reconstruction for ptychographic x-ray computed tomography,” Opt. Exp. 26, 31199 (2018).
[Crossref]

Holler, M.

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

A. Diaz, B. Malkova, M. Holler, M. Guizar-Sicairos, E. Lima, V. Panneels, G. Pigino, A. G. Bittermann, L. Wettstein, T. Tomizaki, O. Bunk, G. Schertler, T. Ishikawa, R. Wepf, and A. Menzel, “Three-dimensional mass density mapping of cellular ultrastructure by ptychographic x-ray nanotomography,” J. Struct. Biol. 192, 461–469 (2015).
[Crossref] [PubMed]

M. Holler, A. Diaz, M. Guizar-Sicairos, P. Karvinen, E. Färm, E. Härkönen, M. Ritala, A. Menzel, J. Raabe, and O. Bunk, “X-ray ptychographic computed tomography at 16 nm isotropic 3D resolution,” Sci. Reports 4, 3857 (2014).
[Crossref]

M. Holler, J. Raabe, A. Diaz, M. Guizar-Sicairos, C. Quitmann, A. Menzel, and O. Bunk, “An instrument for 3D x-ray nano-imaging,” Rev. Sci. Instrum. 83, 073703 (2012).
[Crossref] [PubMed]

Hong, M.

M. Hong, Z. Luo, and M. Razaviyayn, “Convergence analysis of alternating direction method of multipliers for a family of nonconvex problems,” SIAM J. on Optim. 26, 337–364 (2016).
[Crossref]

Hong, Y. P.

J. Deng, Y. P. Hong, S. Chen, Y. S. G. Nashed, T. Peterka, A. J. F. Levi, J. Damoulakis, S. Saha, T. Eiles, and C. Jacobsen, “Nanoscale x-ray imaging of circuit features without wafer etching,” Phys. Rev. B 95, 104111 (2017).
[Crossref] [PubMed]

Honig, S.

A. Schropp, P. Boye, A. Goldschmidt, S. Honig, R. Hoppe, J. Patommel, C. Rakete, D. Samberg, S. Stephan, S. Schoder, M. Burghammer, and C. G. Schroer, “Non-destructive and quantitative imaging of a nano-structured microchip by ptychographic hard x-ray scanning microscopy,” J. Microsc. 241, 9–12 (2011).
[Crossref]

Hoppe, R.

A. Schropp, P. Boye, A. Goldschmidt, S. Honig, R. Hoppe, J. Patommel, C. Rakete, D. Samberg, S. Stephan, S. Schoder, M. Burghammer, and C. G. Schroer, “Non-destructive and quantitative imaging of a nano-structured microchip by ptychographic hard x-ray scanning microscopy,” J. Microsc. 241, 9–12 (2011).
[Crossref]

Hoppe, W.

W. Hoppe, “Beugung im inhomogenen primärstrahlwellenfeld, I. prinzip einer phasenmessung,” Acta Crystallogr. A 25, 495–501 (1969).
[Crossref]

Hoy, C. L.

C. L. Hoy, N. J. Durr, and A. Ben-Yakar, “Fast-updating and nonrepeating Lissajous image reconstruction method for capturing increased dynamic information,” Appl. Opt. 50, 2376–2382 (2011).
[Crossref] [PubMed]

Høydalsvik, K.

K. Høydalsvik, J. B. Fløystad, T. Zhao, M. Esmaeili, A. Diaz, J. W. Andreasen, R. H. Mathiesen, M. Rønning, and D. W. Breiby, “In situ x-ray ptychography imaging of high-temperature CO2 acceptor particle agglomerates,” Appl. Phys. Lett. 104, 241909 (2014).
[Crossref]

Huang, X.

H. Öztürk, H. Yan, Y. He, M. Ge, Z. Dong, M. Lin, E. Nazaretski, I. K. Robinson, Y. S. Chu, and X. Huang, “Multi-slice ptychography with large numerical aperture multilayer laue lenses,” Optica 5, 601 (2018).
[Crossref]

Huo, S.

B. Chen, M. Guizar-Sicairos, G. Xiong, L. Shemilt, A. Diaz, J. Nutter, N. Burdet, S. Huo, J. Mancuso, A. Monteith, F. Vergeer, A. Burgess, and I. Robinson, “Three-dimensional structure analysis and percolation properties of a barrier marine coating,” Sci. Rep. 3, 1177 (2013).
[Crossref] [PubMed]

Hurst, A. C.

J. M. Rodenburg, A. C. Hurst, A. G. Cullis, B. R. Dobson, F. Pfeiffer, O. Bunk, C. David, K. Jefimovs, and I. Johnson, “Hard-x-ray lensless imaging of extended objects,” Phys. Rev. Lett. 98, 034801 (2007).
[Crossref] [PubMed]

J. M. Rodenburg, A. C. Hurst, and A. G. Cullis, “Transmission microscopy without lenses for objects of unlimited size,” Ultramicroscopy 107, 227–231 (2007).
[Crossref]

Ishikawa, T.

A. Diaz, B. Malkova, M. Holler, M. Guizar-Sicairos, E. Lima, V. Panneels, G. Pigino, A. G. Bittermann, L. Wettstein, T. Tomizaki, O. Bunk, G. Schertler, T. Ishikawa, R. Wepf, and A. Menzel, “Three-dimensional mass density mapping of cellular ultrastructure by ptychographic x-ray nanotomography,” J. Struct. Biol. 192, 461–469 (2015).
[Crossref] [PubMed]

Jacobsen, C.

J. Deng, Y. P. Hong, S. Chen, Y. S. G. Nashed, T. Peterka, A. J. F. Levi, J. Damoulakis, S. Saha, T. Eiles, and C. Jacobsen, “Nanoscale x-ray imaging of circuit features without wafer etching,” Phys. Rev. B 95, 104111 (2017).
[Crossref] [PubMed]

J. Deng, Y. S. G. Nashed, S. Chen, N. W. Phillips, T. Peterka, R. Ross, S. Vogt, C. Jacobsen, and D. J. Vine, “Continuous motion scan ptychography: characterization for increased speed in coherent x-ray imaging,” Opt. Express 235438 (2015).
[Crossref]

Y. S. G. Nashed, D. J. Vine, T. Peterka, J. Deng, R. Ross, and C. Jacobsen, “Parallel ptychographic reconstruction,” Opt. Express 2, 32082–32097 (2014).
[Crossref]

D. Gürsoy, X. Xiao, F. D. Carlo, and C. Jacobsen, “Tomopy: a framework for the analysis of synchrotron tomographic data,” J. Synchrotron Rad. 21, 1188–1193 (2014).
[Crossref]

Jacobsen, C. J.

J. Deng, D. J. Vine, S. Chen, Y. S. G. Nashed, Q. Jin, N. W. Phillips, T. Peterka, R. Ross, S. Vogt, and C. J. Jacobsen, “Simultaneous cryo x-ray ptychographic and fluorescence microscopy of green algae,” PNAS 112, 2314–2319 (2015).
[Crossref] [PubMed]

Jefimovs, K.

M. Dierolf, O. Bunk, S. Kynde, P. Thibault, I. Johnson, A. Menzel, K. Jefimovs, C. David, O. Marti, and F. Pfeiffer, “Ptychography & lensless x-ray imaging,” Europhys. News 39, 22–24 (2008).
[Crossref]

J. M. Rodenburg, A. C. Hurst, A. G. Cullis, B. R. Dobson, F. Pfeiffer, O. Bunk, C. David, K. Jefimovs, and I. Johnson, “Hard-x-ray lensless imaging of extended objects,” Phys. Rev. Lett. 98, 034801 (2007).
[Crossref] [PubMed]

Jin, Q.

J. Deng, D. J. Vine, S. Chen, Y. S. G. Nashed, Q. Jin, N. W. Phillips, T. Peterka, R. Ross, S. Vogt, and C. J. Jacobsen, “Simultaneous cryo x-ray ptychographic and fluorescence microscopy of green algae,” PNAS 112, 2314–2319 (2015).
[Crossref] [PubMed]

Johnson, D.

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

Johnson, I.

M. Dierolf, O. Bunk, S. Kynde, P. Thibault, I. Johnson, A. Menzel, K. Jefimovs, C. David, O. Marti, and F. Pfeiffer, “Ptychography & lensless x-ray imaging,” Europhys. News 39, 22–24 (2008).
[Crossref]

J. M. Rodenburg, A. C. Hurst, A. G. Cullis, B. R. Dobson, F. Pfeiffer, O. Bunk, C. David, K. Jefimovs, and I. Johnson, “Hard-x-ray lensless imaging of extended objects,” Phys. Rev. Lett. 98, 034801 (2007).
[Crossref] [PubMed]

Jones, M. W. M.

M. W. M. Jones, K. D. Elgass, M. D. Junker, M. D. de Jonge, and G. A. van Riessen, “Molar concentration from sequential 2-D water-window x-ray ptychography and x-ray fluorescence in hydrated cells,” Sci. Rep. 6, 24280 (2016).
[Crossref] [PubMed]

Junker, M. D.

M. W. M. Jones, K. D. Elgass, M. D. Junker, M. D. de Jonge, and G. A. van Riessen, “Molar concentration from sequential 2-D water-window x-ray ptychography and x-ray fluorescence in hydrated cells,” Sci. Rep. 6, 24280 (2016).
[Crossref] [PubMed]

Kalbfleisch, S.

K. Giewekemeyera, P. Thibault, S. Kalbfleisch, A. Beerlinka, C. M. Kewishc, M. Dierolf, F. Pfeiffer, and T. Salditt, “Quantitative biological imaging by ptychographic x-ray diffraction microscopy,” PNAS 107, 529–534 (2009).
[Crossref]

Karvinen, P.

M. Holler, A. Diaz, M. Guizar-Sicairos, P. Karvinen, E. Färm, E. Härkönen, M. Ritala, A. Menzel, J. Raabe, and O. Bunk, “X-ray ptychographic computed tomography at 16 nm isotropic 3D resolution,” Sci. Reports 4, 3857 (2014).
[Crossref]

R. N. Wilke, M. Priebe, M. Bartels, K. Giewekemeyer, A. Diaz, P. Karvinen, and T. Salditt, “Hard x-ray imaging of bacterial cells: nano-diffraction and ptychographic reconstruction,” Opt. Express 20, 19232–19254 (2012).
[Crossref] [PubMed]

Kettimuthu, R.

T. Bicer, D. Gürsoy, V. D. Andrade, R. Kettimuthu, W. Scullin, F. D. Carlo, and I. T. Foster, “Trace: a high-throughput tomographic reconstruction engine for large-scale datasets,” Adv. Struct. Chem. Imaging 3, 6 (2017).
[Crossref] [PubMed]

T. Bicer, D. Gürsoy, R. Kettimuthu, F. De Carlo, G. Agrawal, and I. T. Foster, “Rapid tomographic image reconstruction via large-scale parallelization,” in Euro-Par 2015: Parallel Processing, J. L. Träff, S. Hunold, and F. Versaci, eds. (Springer, Berlin, Heidelberg, 2015), pp. 289–302.

T. Bicer, D. Gürsoy, R. Kettimuthu, I. T. Foster, B. Ren, V. D. Andrede, and F. De Carlo, “Real-time data analysis and autonomous steering of synchrotron light source experiments,” in 2017 IEEE 13th International Conference on e-Science, (2017), pp. 59–68.
[Crossref]

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]

Kewishc, C. M.

K. Giewekemeyera, P. Thibault, S. Kalbfleisch, A. Beerlinka, C. M. Kewishc, M. Dierolf, F. Pfeiffer, and T. Salditt, “Quantitative biological imaging by ptychographic x-ray diffraction microscopy,” PNAS 107, 529–534 (2009).
[Crossref]

Klaus, G.

B. Mike, S. Tobias, G. Thomas, R. Michael, G. Klaus, G. Sophie-Charlotte, S. Tim, and R. Axel, “Chemical contrast in soft x-ray ptychography,” Phys. Rev. Lett. 107, 208101 (2011).
[Crossref]

Kreutz-Delgado, K.

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

Krishnan, H.

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

Kynde, S.

M. Dierolf, O. Bunk, S. Kynde, P. Thibault, I. Johnson, A. Menzel, K. Jefimovs, C. David, O. Marti, and F. Pfeiffer, “Ptychography & lensless x-ray imaging,” Europhys. News 39, 22–24 (2008).
[Crossref]

Levi, A. J. F.

J. Deng, Y. P. Hong, S. Chen, Y. S. G. Nashed, T. Peterka, A. J. F. Levi, J. Damoulakis, S. Saha, T. Eiles, and C. Jacobsen, “Nanoscale x-ray imaging of circuit features without wafer etching,” Phys. Rev. B 95, 104111 (2017).
[Crossref] [PubMed]

Li, L.

L. Li, X. Wang, and G. Wang, “Alternating direction method of multipliers for separable convex optimization of real functions in complex variables,” Math. Probl. Eng. 2015, 104531 (2015).
[Crossref]

Li, P.

P. Li and A. M. Maiden, “Ten implementations of ptychography,” J. Microsc. 269, 187–194 (2018).
[Crossref]

P. Li and A. Maiden, “Multi-slice ptychographic tomography,” Sci. Reports 82049 (2018).

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

Li, X.

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

Lima, E.

A. Diaz, B. Malkova, M. Holler, M. Guizar-Sicairos, E. Lima, V. Panneels, G. Pigino, A. G. Bittermann, L. Wettstein, T. Tomizaki, O. Bunk, G. Schertler, T. Ishikawa, R. Wepf, and A. Menzel, “Three-dimensional mass density mapping of cellular ultrastructure by ptychographic x-ray nanotomography,” J. Struct. Biol. 192, 461–469 (2015).
[Crossref] [PubMed]

E. Lima, A. Diaz, M. Guizar-Sicairos, S. Gorelick, P. Pernot, T. Schleirer, and A. Menzel, “Cryo-scanning x-ray diffraction microscopy of frozen-hydrated yeast,” J. Microsc. 249, 1–7 (2013).
[Crossref]

Lin, M.

H. Öztürk, H. Yan, Y. He, M. Ge, Z. Dong, M. Lin, E. Nazaretski, I. K. Robinson, Y. S. Chu, and X. Huang, “Multi-slice ptychography with large numerical aperture multilayer laue lenses,” Optica 5, 601 (2018).
[Crossref]

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]

Lou, Y.

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

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

H. Chang, P. Enfedaque, Y. Lou, and S. Marchesini, “Partially coherent ptychography by gradient decomposition of the probe,” Acta Crystallogr. Sect. A: Foundations Adv. 74, 157–169 (2018).
[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. on Imaging Sci. 8, 426–457 (2015).
[Crossref]

Luo, Z.

M. Hong, Z. Luo, and M. Razaviyayn, “Convergence analysis of alternating direction method of multipliers for a family of nonconvex problems,” SIAM J. on Optim. 26, 337–364 (2016).
[Crossref]

Maia, F.

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

C. Yang, J. Qian, A. Schirotzek, F. Maia, and S. Marchesini, “Iterative algorithms for ptychographic phase retrieval,” arXiv preprint arXiv:1105.5628 (2011).

Maia, F. R. N. C.

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

Maiden, A.

P. Li and A. Maiden, “Multi-slice ptychographic tomography,” Sci. Reports 82049 (2018).

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

Maiden, A. M.

P. Li and A. M. Maiden, “Ten implementations of ptychography,” J. Microsc. 269, 187–194 (2018).
[Crossref]

Malkova, B.

A. Diaz, B. Malkova, M. Holler, M. Guizar-Sicairos, E. Lima, V. Panneels, G. Pigino, A. G. Bittermann, L. Wettstein, T. Tomizaki, O. Bunk, G. Schertler, T. Ishikawa, R. Wepf, and A. Menzel, “Three-dimensional mass density mapping of cellular ultrastructure by ptychographic x-ray nanotomography,” J. Struct. Biol. 192, 461–469 (2015).
[Crossref] [PubMed]

Mancuso, J.

B. Chen, M. Guizar-Sicairos, G. Xiong, L. Shemilt, A. Diaz, J. Nutter, N. Burdet, S. Huo, J. Mancuso, A. Monteith, F. Vergeer, A. Burgess, and I. Robinson, “Three-dimensional structure analysis and percolation properties of a barrier marine coating,” Sci. Rep. 3, 1177 (2013).
[Crossref] [PubMed]

Marchesini, S.

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

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

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

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

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

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

C. Yang, J. Qian, A. Schirotzek, F. Maia, and S. Marchesini, “Iterative algorithms for ptychographic phase retrieval,” arXiv preprint arXiv:1105.5628 (2011).

Marroco, A.

R. Glowinski and A. Marroco, “Sur l’approximation, par éléments finis d’ordre un, et la résolution, par pénalisation-dualité d’une classe de problémes de Dirichlet non linéaires,” RAIRO Anal. Numer 9, 41–76 (1975).

Marti, O.

M. Dierolf, O. Bunk, S. Kynde, P. Thibault, I. Johnson, A. Menzel, K. Jefimovs, C. David, O. Marti, and F. Pfeiffer, “Ptychography & lensless x-ray imaging,” Europhys. News 39, 22–24 (2008).
[Crossref]

Mathiesen, R. H.

K. Høydalsvik, J. B. Fløystad, T. Zhao, M. Esmaeili, A. Diaz, J. W. Andreasen, R. H. Mathiesen, M. Rønning, and D. W. Breiby, “In situ x-ray ptychography imaging of high-temperature CO2 acceptor particle agglomerates,” Appl. Phys. Lett. 104, 241909 (2014).
[Crossref]

Menzel, A.

A. Diaz, B. Malkova, M. Holler, M. Guizar-Sicairos, E. Lima, V. Panneels, G. Pigino, A. G. Bittermann, L. Wettstein, T. Tomizaki, O. Bunk, G. Schertler, T. Ishikawa, R. Wepf, and A. Menzel, “Three-dimensional mass density mapping of cellular ultrastructure by ptychographic x-ray nanotomography,” J. Struct. Biol. 192, 461–469 (2015).
[Crossref] [PubMed]

M. Holler, A. Diaz, M. Guizar-Sicairos, P. Karvinen, E. Färm, E. Härkönen, M. Ritala, A. Menzel, J. Raabe, and O. Bunk, “X-ray ptychographic computed tomography at 16 nm isotropic 3D resolution,” Sci. Reports 4, 3857 (2014).
[Crossref]

P. Trtik, A. Diaz, M. Guizar-Sicairos, A. Menzel, and O. Bunk, “Density mapping of hardened cement paste using ptychographic x-ray computed tomography,” Cem. & Concr. Compos. 36, 71–77 (2013).
[Crossref]

E. Lima, A. Diaz, M. Guizar-Sicairos, S. Gorelick, P. Pernot, T. Schleirer, and A. Menzel, “Cryo-scanning x-ray diffraction microscopy of frozen-hydrated yeast,” J. Microsc. 249, 1–7 (2013).
[Crossref]

M. Holler, J. Raabe, A. Diaz, M. Guizar-Sicairos, C. Quitmann, A. Menzel, and O. Bunk, “An instrument for 3D x-ray nano-imaging,” Rev. Sci. Instrum. 83, 073703 (2012).
[Crossref] [PubMed]

A. Diaz, P. Trtik, M. Guizar-Sicairos, A. Menzel, P. Thibault, and O. Bunk, “Quantitative x-ray phase nanotomography,” Phys. Rev. B 85, 020104 (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]

M. Dierolf, O. Bunk, S. Kynde, P. Thibault, I. Johnson, A. Menzel, K. Jefimovs, C. David, O. Marti, and F. Pfeiffer, “Ptychography & lensless x-ray imaging,” Europhys. News 39, 22–24 (2008).
[Crossref]

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

Mercier, B.

D. Gabay and B. Mercier, “A dual algorithm for the solution of nonlinear variational problems via finite element approximation,” Optik 2, 17–40 (1976).

Michael, R.

B. Mike, S. Tobias, G. Thomas, R. Michael, G. Klaus, G. Sophie-Charlotte, S. Tim, and R. Axel, “Chemical contrast in soft x-ray ptychography,” Phys. Rev. Lett. 107, 208101 (2011).
[Crossref]

Mike, B.

B. Mike, S. Tobias, G. Thomas, R. Michael, G. Klaus, G. Sophie-Charlotte, S. Tim, and R. Axel, “Chemical contrast in soft x-ray ptychography,” Phys. Rev. Lett. 107, 208101 (2011).
[Crossref]

Monteith, A.

B. Chen, M. Guizar-Sicairos, G. Xiong, L. Shemilt, A. Diaz, J. Nutter, N. Burdet, S. Huo, J. Mancuso, A. Monteith, F. Vergeer, A. Burgess, and I. Robinson, “Three-dimensional structure analysis and percolation properties of a barrier marine coating,” Sci. Rep. 3, 1177 (2013).
[Crossref] [PubMed]

Müller, E.

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

Nashed, Y. S. G.

M. A. Gilles, Y. S. G. Nashed, C. J. M. Du, and S. M. Wild, “3D x-ray imaging of continuous objects beyond the depth of focus limit,” Optica 5, 1078–1086 (2018).
[Crossref] [PubMed]

J. Deng, Y. P. Hong, S. Chen, Y. S. G. Nashed, T. Peterka, A. J. F. Levi, J. Damoulakis, S. Saha, T. Eiles, and C. Jacobsen, “Nanoscale x-ray imaging of circuit features without wafer etching,” Phys. Rev. B 95, 104111 (2017).
[Crossref] [PubMed]

J. Deng, D. J. Vine, S. Chen, Y. S. G. Nashed, Q. Jin, N. W. Phillips, T. Peterka, R. Ross, S. Vogt, and C. J. Jacobsen, “Simultaneous cryo x-ray ptychographic and fluorescence microscopy of green algae,” PNAS 112, 2314–2319 (2015).
[Crossref] [PubMed]

J. Deng, Y. S. G. Nashed, S. Chen, N. W. Phillips, T. Peterka, R. Ross, S. Vogt, C. Jacobsen, and D. J. Vine, “Continuous motion scan ptychography: characterization for increased speed in coherent x-ray imaging,” Opt. Express 235438 (2015).
[Crossref]

Y. S. G. Nashed, D. J. Vine, T. Peterka, J. Deng, R. Ross, and C. Jacobsen, “Parallel ptychographic reconstruction,” Opt. Express 2, 32082–32097 (2014).
[Crossref]

Nazaretski, E.

H. Öztürk, H. Yan, Y. He, M. Ge, Z. Dong, M. Lin, E. Nazaretski, I. K. Robinson, Y. S. Chu, and X. Huang, “Multi-slice ptychography with large numerical aperture multilayer laue lenses,” Optica 5, 601 (2018).
[Crossref]

Nikitin, V. V.

F. Andersson, M. Carlsson, and V. V. Nikitin, “Fast algorithms and efficient GPU implementations for the Radon transform and the back-projection operator represented as convolution operators,” SIAM J. on Imaging Sci. 9, 637–664 (2016).
[Crossref]

Nutter, J.

B. Chen, M. Guizar-Sicairos, G. Xiong, L. Shemilt, A. Diaz, J. Nutter, N. Burdet, S. Huo, J. Mancuso, A. Monteith, F. Vergeer, A. Burgess, and I. Robinson, “Three-dimensional structure analysis and percolation properties of a barrier marine coating,” Sci. Rep. 3, 1177 (2013).
[Crossref] [PubMed]

Öztürk, H.

H. Öztürk, H. Yan, Y. He, M. Ge, Z. Dong, M. Lin, E. Nazaretski, I. K. Robinson, Y. S. Chu, and X. Huang, “Multi-slice ptychography with large numerical aperture multilayer laue lenses,” Optica 5, 601 (2018).
[Crossref]

Panneels, V.

A. Diaz, B. Malkova, M. Holler, M. Guizar-Sicairos, E. Lima, V. Panneels, G. Pigino, A. G. Bittermann, L. Wettstein, T. Tomizaki, O. Bunk, G. Schertler, T. Ishikawa, R. Wepf, and A. Menzel, “Three-dimensional mass density mapping of cellular ultrastructure by ptychographic x-ray nanotomography,” J. Struct. Biol. 192, 461–469 (2015).
[Crossref] [PubMed]

Parikh, N.

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

Patommel, J.

A. Schropp, P. Boye, A. Goldschmidt, S. Honig, R. Hoppe, J. Patommel, C. Rakete, D. Samberg, S. Stephan, S. Schoder, M. Burghammer, and C. G. Schroer, “Non-destructive and quantitative imaging of a nano-structured microchip by ptychographic hard x-ray scanning microscopy,” J. Microsc. 241, 9–12 (2011).
[Crossref]

Peleato, B.

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

Perciano, T.

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

Pernot, P.

E. Lima, A. Diaz, M. Guizar-Sicairos, S. Gorelick, P. Pernot, T. Schleirer, and A. Menzel, “Cryo-scanning x-ray diffraction microscopy of frozen-hydrated yeast,” J. Microsc. 249, 1–7 (2013).
[Crossref]

Peterka, T.

J. Deng, Y. P. Hong, S. Chen, Y. S. G. Nashed, T. Peterka, A. J. F. Levi, J. Damoulakis, S. Saha, T. Eiles, and C. Jacobsen, “Nanoscale x-ray imaging of circuit features without wafer etching,” Phys. Rev. B 95, 104111 (2017).
[Crossref] [PubMed]

J. Deng, D. J. Vine, S. Chen, Y. S. G. Nashed, Q. Jin, N. W. Phillips, T. Peterka, R. Ross, S. Vogt, and C. J. Jacobsen, “Simultaneous cryo x-ray ptychographic and fluorescence microscopy of green algae,” PNAS 112, 2314–2319 (2015).
[Crossref] [PubMed]

J. Deng, Y. S. G. Nashed, S. Chen, N. W. Phillips, T. Peterka, R. Ross, S. Vogt, C. Jacobsen, and D. J. Vine, “Continuous motion scan ptychography: characterization for increased speed in coherent x-ray imaging,” Opt. Express 235438 (2015).
[Crossref]

Y. S. G. Nashed, D. J. Vine, T. Peterka, J. Deng, R. Ross, and C. Jacobsen, “Parallel ptychographic reconstruction,” Opt. Express 2, 32082–32097 (2014).
[Crossref]

Pfeiffer, F.

F. Pfeiffer, “X-ray ptychography,” Nat. Photonics 12, 9–17 (2018).
[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]

K. Giewekemeyera, P. Thibault, S. Kalbfleisch, A. Beerlinka, C. M. Kewishc, M. Dierolf, F. Pfeiffer, and T. Salditt, “Quantitative biological imaging by ptychographic x-ray diffraction microscopy,” PNAS 107, 529–534 (2009).
[Crossref]

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

M. Dierolf, O. Bunk, S. Kynde, P. Thibault, I. Johnson, A. Menzel, K. Jefimovs, C. David, O. Marti, and F. Pfeiffer, “Ptychography & lensless x-ray imaging,” Europhys. News 39, 22–24 (2008).
[Crossref]

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

J. M. Rodenburg, A. C. Hurst, A. G. Cullis, B. R. Dobson, F. Pfeiffer, O. Bunk, C. David, K. Jefimovs, and I. Johnson, “Hard-x-ray lensless imaging of extended objects,” Phys. Rev. Lett. 98, 034801 (2007).
[Crossref] [PubMed]

Phillips, N. W.

J. Deng, Y. S. G. Nashed, S. Chen, N. W. Phillips, T. Peterka, R. Ross, S. Vogt, C. Jacobsen, and D. J. Vine, “Continuous motion scan ptychography: characterization for increased speed in coherent x-ray imaging,” Opt. Express 235438 (2015).
[Crossref]

J. Deng, D. J. Vine, S. Chen, Y. S. G. Nashed, Q. Jin, N. W. Phillips, T. Peterka, R. Ross, S. Vogt, and C. J. Jacobsen, “Simultaneous cryo x-ray ptychographic and fluorescence microscopy of green algae,” PNAS 112, 2314–2319 (2015).
[Crossref] [PubMed]

Pigino, G.

A. Diaz, B. Malkova, M. Holler, M. Guizar-Sicairos, E. Lima, V. Panneels, G. Pigino, A. G. Bittermann, L. Wettstein, T. Tomizaki, O. Bunk, G. Schertler, T. Ishikawa, R. Wepf, and A. Menzel, “Three-dimensional mass density mapping of cellular ultrastructure by ptychographic x-ray nanotomography,” J. Struct. Biol. 192, 461–469 (2015).
[Crossref] [PubMed]

Powell, M. J.

M. J. Powell, “A method for nonlinear constraints in minimization problems,” Optimization 1969283–298 (1969).

Priebe, M.

R. N. Wilke, M. Priebe, M. Bartels, K. Giewekemeyer, A. Diaz, P. Karvinen, and T. Salditt, “Hard x-ray imaging of bacterial cells: nano-diffraction and ptychographic reconstruction,” Opt. Express 20, 19232–19254 (2012).
[Crossref] [PubMed]

Qian, J.

C. Yang, J. Qian, A. Schirotzek, F. Maia, and S. Marchesini, “Iterative algorithms for ptychographic phase retrieval,” arXiv preprint arXiv:1105.5628 (2011).

Quitmann, C.

M. Holler, J. Raabe, A. Diaz, M. Guizar-Sicairos, C. Quitmann, A. Menzel, and O. Bunk, “An instrument for 3D x-ray nano-imaging,” Rev. Sci. Instrum. 83, 073703 (2012).
[Crossref] [PubMed]

Raabe, J.

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

M. Holler, A. Diaz, M. Guizar-Sicairos, P. Karvinen, E. Färm, E. Härkönen, M. Ritala, A. Menzel, J. Raabe, and O. Bunk, “X-ray ptychographic computed tomography at 16 nm isotropic 3D resolution,” Sci. Reports 4, 3857 (2014).
[Crossref]

M. Holler, J. Raabe, A. Diaz, M. Guizar-Sicairos, C. Quitmann, A. Menzel, and O. Bunk, “An instrument for 3D x-ray nano-imaging,” Rev. Sci. Instrum. 83, 073703 (2012).
[Crossref] [PubMed]

Rachford, H. H.

J. Douglas and H. H. Rachford, “On the numerical solution of heat conduction problems in two and three space variables,” Transactions Am. Math. Soc. 82, 421–439 (1956).
[Crossref]

Rakete, C.

A. Schropp, P. Boye, A. Goldschmidt, S. Honig, R. Hoppe, J. Patommel, C. Rakete, D. Samberg, S. Stephan, S. Schoder, M. Burghammer, and C. G. Schroer, “Non-destructive and quantitative imaging of a nano-structured microchip by ptychographic hard x-ray scanning microscopy,” J. Microsc. 241, 9–12 (2011).
[Crossref]

Ramos, T.

T. Ramos, B. E. Grønager, M. S. Andersen, and J. W. Andreasen, “Direct 3D tomographic reconstruction and phase-retrieval of far-field coherent diffraction patterns,” arXiv preprint arXiv:1808.02109 (2018).

Razaviyayn, M.

M. Hong, Z. Luo, and M. Razaviyayn, “Convergence analysis of alternating direction method of multipliers for a family of nonconvex problems,” SIAM J. on Optim. 26, 337–364 (2016).
[Crossref]

Remmert, R.

R. Remmert, Theory of Complex Functions (Springer, 1991).
[Crossref]

Ren, B.

T. Bicer, D. Gürsoy, R. Kettimuthu, I. T. Foster, B. Ren, V. D. Andrede, and F. De Carlo, “Real-time data analysis and autonomous steering of synchrotron light source experiments,” in 2017 IEEE 13th International Conference on e-Science, (2017), pp. 59–68.
[Crossref]

Ritala, M.

M. Holler, A. Diaz, M. Guizar-Sicairos, P. Karvinen, E. Färm, E. Härkönen, M. Ritala, A. Menzel, J. Raabe, and O. Bunk, “X-ray ptychographic computed tomography at 16 nm isotropic 3D resolution,” Sci. Reports 4, 3857 (2014).
[Crossref]

Robinson, I.

B. Chen, M. Guizar-Sicairos, G. Xiong, L. Shemilt, A. Diaz, J. Nutter, N. Burdet, S. Huo, J. Mancuso, A. Monteith, F. Vergeer, A. Burgess, and I. Robinson, “Three-dimensional structure analysis and percolation properties of a barrier marine coating,” Sci. Rep. 3, 1177 (2013).
[Crossref] [PubMed]

Robinson, I. K.

H. Öztürk, H. Yan, Y. He, M. Ge, Z. Dong, M. Lin, E. Nazaretski, I. K. Robinson, Y. S. Chu, and X. Huang, “Multi-slice ptychography with large numerical aperture multilayer laue lenses,” Optica 5, 601 (2018).
[Crossref]

Rodenburg, J. M.

J. M. Rodenburg, A. C. Hurst, and A. G. Cullis, “Transmission microscopy without lenses for objects of unlimited size,” Ultramicroscopy 107, 227–231 (2007).
[Crossref]

J. M. Rodenburg, A. C. Hurst, A. G. Cullis, B. R. Dobson, F. Pfeiffer, O. Bunk, C. David, K. Jefimovs, and I. Johnson, “Hard-x-ray lensless imaging of extended objects,” Phys. Rev. Lett. 98, 034801 (2007).
[Crossref] [PubMed]

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

Rønning, M.

K. Høydalsvik, J. B. Fløystad, T. Zhao, M. Esmaeili, A. Diaz, J. W. Andreasen, R. H. Mathiesen, M. Rønning, and D. W. Breiby, “In situ x-ray ptychography imaging of high-temperature CO2 acceptor particle agglomerates,” Appl. Phys. Lett. 104, 241909 (2014).
[Crossref]

Ross, R.

J. Deng, D. J. Vine, S. Chen, Y. S. G. Nashed, Q. Jin, N. W. Phillips, T. Peterka, R. Ross, S. Vogt, and C. J. Jacobsen, “Simultaneous cryo x-ray ptychographic and fluorescence microscopy of green algae,” PNAS 112, 2314–2319 (2015).
[Crossref] [PubMed]

J. Deng, Y. S. G. Nashed, S. Chen, N. W. Phillips, T. Peterka, R. Ross, S. Vogt, C. Jacobsen, and D. J. Vine, “Continuous motion scan ptychography: characterization for increased speed in coherent x-ray imaging,” Opt. Express 235438 (2015).
[Crossref]

Y. S. G. Nashed, D. J. Vine, T. Peterka, J. Deng, R. Ross, and C. Jacobsen, “Parallel ptychographic reconstruction,” Opt. Express 2, 32082–32097 (2014).
[Crossref]

Sabach, S.

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

J. Bolte, S. Sabach, and M. Teboulle, “Proximal alternating linearized minimization for nonconvex and nonsmooth problems,” Math. Program. 146, 459–494 (2014).
[Crossref]

Saha, S.

J. Deng, Y. P. Hong, S. Chen, Y. S. G. Nashed, T. Peterka, A. J. F. Levi, J. Damoulakis, S. Saha, T. Eiles, and C. Jacobsen, “Nanoscale x-ray imaging of circuit features without wafer etching,” Phys. Rev. B 95, 104111 (2017).
[Crossref] [PubMed]

Salditt, T.

R. N. Wilke, M. Priebe, M. Bartels, K. Giewekemeyer, A. Diaz, P. Karvinen, and T. Salditt, “Hard x-ray imaging of bacterial cells: nano-diffraction and ptychographic reconstruction,” Opt. Express 20, 19232–19254 (2012).
[Crossref] [PubMed]

K. Giewekemeyera, P. Thibault, S. Kalbfleisch, A. Beerlinka, C. M. Kewishc, M. Dierolf, F. Pfeiffer, and T. Salditt, “Quantitative biological imaging by ptychographic x-ray diffraction microscopy,” PNAS 107, 529–534 (2009).
[Crossref]

Samberg, D.

A. Schropp, P. Boye, A. Goldschmidt, S. Honig, R. Hoppe, J. Patommel, C. Rakete, D. Samberg, S. Stephan, S. Schoder, M. Burghammer, and C. G. Schroer, “Non-destructive and quantitative imaging of a nano-structured microchip by ptychographic hard x-ray scanning microscopy,” J. Microsc. 241, 9–12 (2011).
[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).

Schertler, G.

A. Diaz, B. Malkova, M. Holler, M. Guizar-Sicairos, E. Lima, V. Panneels, G. Pigino, A. G. Bittermann, L. Wettstein, T. Tomizaki, O. Bunk, G. Schertler, T. Ishikawa, R. Wepf, and A. Menzel, “Three-dimensional mass density mapping of cellular ultrastructure by ptychographic x-ray nanotomography,” J. Struct. Biol. 192, 461–469 (2015).
[Crossref] [PubMed]

Schirotzek, A.

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

C. Yang, J. Qian, A. Schirotzek, F. Maia, and S. Marchesini, “Iterative algorithms for ptychographic phase retrieval,” arXiv preprint arXiv:1105.5628 (2011).

Schleirer, T.

E. Lima, A. Diaz, M. Guizar-Sicairos, S. Gorelick, P. Pernot, T. Schleirer, and A. Menzel, “Cryo-scanning x-ray diffraction microscopy of frozen-hydrated yeast,” J. Microsc. 249, 1–7 (2013).
[Crossref]

Schneider, P.

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

Schoder, S.

A. Schropp, P. Boye, A. Goldschmidt, S. Honig, R. Hoppe, J. Patommel, C. Rakete, D. Samberg, S. Stephan, S. Schoder, M. Burghammer, and C. G. Schroer, “Non-destructive and quantitative imaging of a nano-structured microchip by ptychographic hard x-ray scanning microscopy,” J. Microsc. 241, 9–12 (2011).
[Crossref]

Schroer, C. G.

A. Schropp, P. Boye, A. Goldschmidt, S. Honig, R. Hoppe, J. Patommel, C. Rakete, D. Samberg, S. Stephan, S. Schoder, M. Burghammer, and C. G. Schroer, “Non-destructive and quantitative imaging of a nano-structured microchip by ptychographic hard x-ray scanning microscopy,” J. Microsc. 241, 9–12 (2011).
[Crossref]

Schropp, A.

A. Schropp, P. Boye, A. Goldschmidt, S. Honig, R. Hoppe, J. Patommel, C. Rakete, D. Samberg, S. Stephan, S. Schoder, M. Burghammer, and C. G. Schroer, “Non-destructive and quantitative imaging of a nano-structured microchip by ptychographic hard x-ray scanning microscopy,” J. Microsc. 241, 9–12 (2011).
[Crossref]

Scullin, W.

T. Bicer, D. Gürsoy, V. D. Andrade, R. Kettimuthu, W. Scullin, F. D. Carlo, and I. T. Foster, “Trace: a high-throughput tomographic reconstruction engine for large-scale datasets,” Adv. Struct. Chem. Imaging 3, 6 (2017).
[Crossref] [PubMed]

Sethian, J. A.

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

Shapiro, D. A.

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

Shemilt, L.

B. Chen, M. Guizar-Sicairos, G. Xiong, L. Shemilt, A. Diaz, J. Nutter, N. Burdet, S. Huo, J. Mancuso, A. Monteith, F. Vergeer, A. Burgess, and I. Robinson, “Three-dimensional structure analysis and percolation properties of a barrier marine coating,” Sci. Rep. 3, 1177 (2013).
[Crossref] [PubMed]

Shimomura, K.

K. Shimomura, M. Hirose, T. Higashino, and Y. Takahashi, “Three-dimensional iterative multislice reconstruction for ptychographic x-ray computed tomography,” Opt. Exp. 26, 31199 (2018).
[Crossref]

Soltanolkotabi, M.

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

L.-H. Yeh, J. Dong, J. Zhong, L. Tian, M. Chen, G. Tang, M. Soltanolkotabi, and L. Waller, “Experimental robustness of Fourier ptychography phase retrieval algorithms,” Opt. Express 23, 33214–33240 (2015).
[Crossref]

Sophie-Charlotte, G.

B. Mike, S. Tobias, G. Thomas, R. Michael, G. Klaus, G. Sophie-Charlotte, S. Tim, and R. Axel, “Chemical contrast in soft x-ray ptychography,” Phys. Rev. Lett. 107, 208101 (2011).
[Crossref]

Stephan, S.

A. Schropp, P. Boye, A. Goldschmidt, S. Honig, R. Hoppe, J. Patommel, C. Rakete, D. Samberg, S. Stephan, S. Schoder, M. Burghammer, and C. G. Schroer, “Non-destructive and quantitative imaging of a nano-structured microchip by ptychographic hard x-ray scanning microscopy,” J. Microsc. 241, 9–12 (2011).
[Crossref]

Strohmer, T.

E. J. Candès, Y. C. Eldar, T. Strohmer, and V. Voroninski, “Phase retrieval via matrix completion,” SIAM J. on Imaging Sci. 6, 199–225 (2013).
[Crossref]

Suo, J.

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]

Takahashi, Y.

K. Shimomura, M. Hirose, T. Higashino, and Y. Takahashi, “Three-dimensional iterative multislice reconstruction for ptychographic x-ray computed tomography,” Opt. Exp. 26, 31199 (2018).
[Crossref]

Tam, M. K.

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

Tang, G.

L.-H. Yeh, J. Dong, J. Zhong, L. Tian, M. Chen, G. Tang, M. Soltanolkotabi, and L. Waller, “Experimental robustness of Fourier ptychography phase retrieval algorithms,” Opt. Express 23, 33214–33240 (2015).
[Crossref]

Teboulle, M.

J. Bolte, S. Sabach, and M. Teboulle, “Proximal alternating linearized minimization for nonconvex and nonsmooth problems,” Math. Program. 146, 459–494 (2014).
[Crossref]

Thibault, P.

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

A. Diaz, P. Trtik, M. Guizar-Sicairos, A. Menzel, P. Thibault, and O. Bunk, “Quantitative x-ray phase nanotomography,” Phys. Rev. B 85, 020104 (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]

K. Giewekemeyera, P. Thibault, S. Kalbfleisch, A. Beerlinka, C. M. Kewishc, M. Dierolf, F. Pfeiffer, and T. Salditt, “Quantitative biological imaging by ptychographic x-ray diffraction microscopy,” PNAS 107, 529–534 (2009).
[Crossref]

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

M. Dierolf, O. Bunk, S. Kynde, P. Thibault, I. Johnson, A. Menzel, K. Jefimovs, C. David, O. Marti, and F. Pfeiffer, “Ptychography & lensless x-ray imaging,” Europhys. News 39, 22–24 (2008).
[Crossref]

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

Thomas, G.

B. Mike, S. Tobias, G. Thomas, R. Michael, G. Klaus, G. Sophie-Charlotte, S. Tim, and R. Axel, “Chemical contrast in soft x-ray ptychography,” Phys. Rev. Lett. 107, 208101 (2011).
[Crossref]

Tian, L.

L.-H. Yeh, J. Dong, J. Zhong, L. Tian, M. Chen, G. Tang, M. Soltanolkotabi, and L. Waller, “Experimental robustness of Fourier ptychography phase retrieval algorithms,” Opt. Express 23, 33214–33240 (2015).
[Crossref]

Tim, S.

B. Mike, S. Tobias, G. Thomas, R. Michael, G. Klaus, G. Sophie-Charlotte, S. Tim, and R. Axel, “Chemical contrast in soft x-ray ptychography,” Phys. Rev. Lett. 107, 208101 (2011).
[Crossref]

Tobias, S.

B. Mike, S. Tobias, G. Thomas, R. Michael, G. Klaus, G. Sophie-Charlotte, S. Tim, and R. Axel, “Chemical contrast in soft x-ray ptychography,” Phys. Rev. Lett. 107, 208101 (2011).
[Crossref]

Tomizaki, T.

A. Diaz, B. Malkova, M. Holler, M. Guizar-Sicairos, E. Lima, V. Panneels, G. Pigino, A. G. Bittermann, L. Wettstein, T. Tomizaki, O. Bunk, G. Schertler, T. Ishikawa, R. Wepf, and A. Menzel, “Three-dimensional mass density mapping of cellular ultrastructure by ptychographic x-ray nanotomography,” J. Struct. Biol. 192, 461–469 (2015).
[Crossref] [PubMed]

Toney, M. F.

J. N. Weker and M. F. Toney, “Emerging in situ and operando nanoscale x-ray imaging techniques for energy storage materials,” Adv. Funct. Mater. 25, 1622–1637 (2015).
[Crossref]

Trtik, P.

P. Trtik, A. Diaz, M. Guizar-Sicairos, A. Menzel, and O. Bunk, “Density mapping of hardened cement paste using ptychographic x-ray computed tomography,” Cem. & Concr. Compos. 36, 71–77 (2013).
[Crossref]

A. Diaz, P. Trtik, M. Guizar-Sicairos, A. Menzel, P. Thibault, and O. Bunk, “Quantitative x-ray phase nanotomography,” Phys. Rev. B 85, 020104 (2012).
[Crossref]

Tsai, E. H. R.

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

van Riessen, G. A.

M. W. M. Jones, K. D. Elgass, M. D. Junker, M. D. de Jonge, and G. A. van Riessen, “Molar concentration from sequential 2-D water-window x-ray ptychography and x-ray fluorescence in hydrated cells,” Sci. Rep. 6, 24280 (2016).
[Crossref] [PubMed]

Vergeer, F.

B. Chen, M. Guizar-Sicairos, G. Xiong, L. Shemilt, A. Diaz, J. Nutter, N. Burdet, S. Huo, J. Mancuso, A. Monteith, F. Vergeer, A. Burgess, and I. Robinson, “Three-dimensional structure analysis and percolation properties of a barrier marine coating,” Sci. Rep. 3, 1177 (2013).
[Crossref] [PubMed]

Vine, D. J.

J. Deng, Y. S. G. Nashed, S. Chen, N. W. Phillips, T. Peterka, R. Ross, S. Vogt, C. Jacobsen, and D. J. Vine, “Continuous motion scan ptychography: characterization for increased speed in coherent x-ray imaging,” Opt. Express 235438 (2015).
[Crossref]

J. Deng, D. J. Vine, S. Chen, Y. S. G. Nashed, Q. Jin, N. W. Phillips, T. Peterka, R. Ross, S. Vogt, and C. J. Jacobsen, “Simultaneous cryo x-ray ptychographic and fluorescence microscopy of green algae,” PNAS 112, 2314–2319 (2015).
[Crossref] [PubMed]

Y. S. G. Nashed, D. J. Vine, T. Peterka, J. Deng, R. Ross, and C. Jacobsen, “Parallel ptychographic reconstruction,” Opt. Express 2, 32082–32097 (2014).
[Crossref]

Vogt, S.

J. Deng, D. J. Vine, S. Chen, Y. S. G. Nashed, Q. Jin, N. W. Phillips, T. Peterka, R. Ross, S. Vogt, and C. J. Jacobsen, “Simultaneous cryo x-ray ptychographic and fluorescence microscopy of green algae,” PNAS 112, 2314–2319 (2015).
[Crossref] [PubMed]

J. Deng, Y. S. G. Nashed, S. Chen, N. W. Phillips, T. Peterka, R. Ross, S. Vogt, C. Jacobsen, and D. J. Vine, “Continuous motion scan ptychography: characterization for increased speed in coherent x-ray imaging,” Opt. Express 235438 (2015).
[Crossref]

Voroninski, V.

E. J. Candès, Y. C. Eldar, T. Strohmer, and V. Voroninski, “Phase retrieval via matrix completion,” SIAM J. on Imaging Sci. 6, 199–225 (2013).
[Crossref]

Waller, L.

L.-H. Yeh, J. Dong, J. Zhong, L. Tian, M. Chen, G. Tang, M. Soltanolkotabi, and L. Waller, “Experimental robustness of Fourier ptychography phase retrieval algorithms,” Opt. Express 23, 33214–33240 (2015).
[Crossref]

Wang, G.

L. Li, X. Wang, and G. Wang, “Alternating direction method of multipliers for separable convex optimization of real functions in complex variables,” Math. Probl. Eng. 2015, 104531 (2015).
[Crossref]

Wang, X.

L. Li, X. Wang, and G. Wang, “Alternating direction method of multipliers for separable convex optimization of real functions in complex variables,” Math. Probl. Eng. 2015, 104531 (2015).
[Crossref]

Wang, Y.

Y. Wang, W. Yin, and J. Zeng, “Global convergence of ADMM in nonconvex nonsmooth optimization,” J. Sci. Comput. 7829–63 (2015).

Weker, J. N.

J. N. Weker and M. F. Toney, “Emerging in situ and operando nanoscale x-ray imaging techniques for energy storage materials,” Adv. Funct. Mater. 25, 1622–1637 (2015).
[Crossref]

Wen, Z.

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

Wepf, R.

A. Diaz, B. Malkova, M. Holler, M. Guizar-Sicairos, E. Lima, V. Panneels, G. Pigino, A. G. Bittermann, L. Wettstein, T. Tomizaki, O. Bunk, G. Schertler, T. Ishikawa, R. Wepf, and A. Menzel, “Three-dimensional mass density mapping of cellular ultrastructure by ptychographic x-ray nanotomography,” J. Struct. Biol. 192, 461–469 (2015).
[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]

Wettstein, L.

A. Diaz, B. Malkova, M. Holler, M. Guizar-Sicairos, E. Lima, V. Panneels, G. Pigino, A. G. Bittermann, L. Wettstein, T. Tomizaki, O. Bunk, G. Schertler, T. Ishikawa, R. Wepf, and A. Menzel, “Three-dimensional mass density mapping of cellular ultrastructure by ptychographic x-ray nanotomography,” J. Struct. Biol. 192, 461–469 (2015).
[Crossref] [PubMed]

Wild, S. M.

M. A. Gilles, Y. S. G. Nashed, C. J. M. Du, and S. M. Wild, “3D x-ray imaging of continuous objects beyond the depth of focus limit,” Optica 5, 1078–1086 (2018).
[Crossref] [PubMed]

Wilke, R. N.

R. N. Wilke, M. Priebe, M. Bartels, K. Giewekemeyer, A. Diaz, P. Karvinen, and T. Salditt, “Hard x-ray imaging of bacterial cells: nano-diffraction and ptychographic reconstruction,” Opt. Express 20, 19232–19254 (2012).
[Crossref] [PubMed]

Wolf, E.

E. Wolf, “New theory of partial coherence in the space-frequency domain. Part I: spectra and cross spectra of steady-state sources,” JOSA 72, 343–351 (1982).
[Crossref]

Wu, H.

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

Xiao, X.

D. Gürsoy, X. Xiao, F. D. Carlo, and C. Jacobsen, “Tomopy: a framework for the analysis of synchrotron tomographic data,” J. Synchrotron Rad. 21, 1188–1193 (2014).
[Crossref]

Xiong, G.

B. Chen, M. Guizar-Sicairos, G. Xiong, L. Shemilt, A. Diaz, J. Nutter, N. Burdet, S. Huo, J. Mancuso, A. Monteith, F. Vergeer, A. Burgess, and I. Robinson, “Three-dimensional structure analysis and percolation properties of a barrier marine coating,” Sci. Rep. 3, 1177 (2013).
[Crossref] [PubMed]

Yan, H.

H. Öztürk, H. Yan, Y. He, M. Ge, Z. Dong, M. Lin, E. Nazaretski, I. K. Robinson, Y. S. Chu, and X. Huang, “Multi-slice ptychography with large numerical aperture multilayer laue lenses,” Optica 5, 601 (2018).
[Crossref]

Yang, C.

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

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

C. Yang, J. Qian, A. Schirotzek, F. Maia, and S. Marchesini, “Iterative algorithms for ptychographic phase retrieval,” arXiv preprint arXiv:1105.5628 (2011).

Yeh, L.-H.

L.-H. Yeh, J. Dong, J. Zhong, L. Tian, M. Chen, G. Tang, M. Soltanolkotabi, and L. Waller, “Experimental robustness of Fourier ptychography phase retrieval algorithms,” Opt. Express 23, 33214–33240 (2015).
[Crossref]

Yin, W.

Y. Wang, W. Yin, and J. Zeng, “Global convergence of ADMM in nonconvex nonsmooth optimization,” J. Sci. Comput. 7829–63 (2015).

D. Davis and W. Yin, “Convergence rate analysis of several splitting schemes,” in Splitting Methods in Communication, Imaging, Science, and Engineering, R. Glowinski, S. J. Osher, and W. Yin, eds. (Springer International Publishing, Cham, 2016), pp. 115–163.
[Crossref]

Yuan, X.

B. He and X. Yuan, “On the O(1/n) convergence rate of the Douglas-Rachford alternating direction method,” SIAM J. Numer. Anal. 50, 700–709 (2012).
[Crossref]

Zeng, J.

Y. Wang, W. Yin, and J. Zeng, “Global convergence of ADMM in nonconvex nonsmooth optimization,” J. Sci. Comput. 7829–63 (2015).

Zeng, T.

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

Zhao, T.

K. Høydalsvik, J. B. Fløystad, T. Zhao, M. Esmaeili, A. Diaz, J. W. Andreasen, R. H. Mathiesen, M. Rønning, and D. W. Breiby, “In situ x-ray ptychography imaging of high-temperature CO2 acceptor particle agglomerates,” Appl. Phys. Lett. 104, 241909 (2014).
[Crossref]

Zheng, G.

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]

Zhong, J.

L.-H. Yeh, J. Dong, J. Zhong, L. Tian, M. Chen, G. Tang, M. Soltanolkotabi, and L. Waller, “Experimental robustness of Fourier ptychography phase retrieval algorithms,” Opt. Express 23, 33214–33240 (2015).
[Crossref]

Acta Crystallogr. A (1)

W. Hoppe, “Beugung im inhomogenen primärstrahlwellenfeld, I. prinzip einer phasenmessung,” Acta Crystallogr. A 25, 495–501 (1969).
[Crossref]

Acta Crystallogr. Sect. A: Foundations Adv. (1)

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

Adv. Funct. Mater. (1)

J. N. Weker and M. F. Toney, “Emerging in situ and operando nanoscale x-ray imaging techniques for energy storage materials,” Adv. Funct. Mater. 25, 1622–1637 (2015).
[Crossref]

Adv. Struct. Chem. Imaging (1)

T. Bicer, D. Gürsoy, V. D. Andrade, R. Kettimuthu, W. Scullin, F. D. Carlo, and I. T. Foster, “Trace: a high-throughput tomographic reconstruction engine for large-scale datasets,” Adv. Struct. Chem. Imaging 3, 6 (2017).
[Crossref] [PubMed]

Appl. Comput. Harmon. Analysis (1)

G. Beylkin, “On the fast Fourier transform of functions with singularities,” Appl. Comput. Harmon. Analysis 2, 363–381 (1995).
[Crossref]

Appl. Opt. (3)

D. J. Ching, M. Hidayetoğlu, T. Bicer, and D. Gürsoy, “Rotation-as-fast-axis scanning-probe x-ray tomography: the importance of angular diversity for fly-scan modes,” Appl. Opt. 57, 8780–8789 (2018).
[Crossref] [PubMed]

J. R. Fienup, “Phase retrieval algorithms: a comparison,” Appl. Opt. 21, 2758–2769 (1982).
[Crossref] [PubMed]

C. L. Hoy, N. J. Durr, and A. Ben-Yakar, “Fast-updating and nonrepeating Lissajous image reconstruction method for capturing increased dynamic information,” Appl. Opt. 50, 2376–2382 (2011).
[Crossref] [PubMed]

Appl. Phys. Lett. (2)

K. Høydalsvik, J. B. Fløystad, T. Zhao, M. Esmaeili, A. Diaz, J. W. Andreasen, R. H. Mathiesen, M. Rønning, and D. W. Breiby, “In situ x-ray ptychography imaging of high-temperature CO2 acceptor particle agglomerates,” Appl. Phys. Lett. 104, 241909 (2014).
[Crossref]

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

At. Data Nucl. Data Tables (1)

B. L. Henke, E. M. Gullikson, and J. C. Davis, “X-ray interactions: photoabsorption, scattering, transmission, and reflection at E=50–30,000 eV, Z=1–92,” At. Data Nucl. Data Tables 54, 181–342 (1993).
[Crossref]

Cem. & Concr. Compos. (1)

P. Trtik, A. Diaz, M. Guizar-Sicairos, A. Menzel, and O. Bunk, “Density mapping of hardened cement paste using ptychographic x-ray computed tomography,” Cem. & Concr. Compos. 36, 71–77 (2013).
[Crossref]

Europhys. News (1)

M. Dierolf, O. Bunk, S. Kynde, P. Thibault, I. Johnson, A. Menzel, K. Jefimovs, C. David, O. Marti, and F. Pfeiffer, “Ptychography & lensless x-ray imaging,” Europhys. News 39, 22–24 (2008).
[Crossref]

Foundations Trends Mach. Learn. (1)

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

IEEE Transactions on Inf. Theory (1)

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

Inverse Probl. (2)

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

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

J. Appl. Cryst. (1)

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

J. Microsc. (3)

A. Schropp, P. Boye, A. Goldschmidt, S. Honig, R. Hoppe, J. Patommel, C. Rakete, D. Samberg, S. Stephan, S. Schoder, M. Burghammer, and C. G. Schroer, “Non-destructive and quantitative imaging of a nano-structured microchip by ptychographic hard x-ray scanning microscopy,” J. Microsc. 241, 9–12 (2011).
[Crossref]

E. Lima, A. Diaz, M. Guizar-Sicairos, S. Gorelick, P. Pernot, T. Schleirer, and A. Menzel, “Cryo-scanning x-ray diffraction microscopy of frozen-hydrated yeast,” J. Microsc. 249, 1–7 (2013).
[Crossref]

P. Li and A. M. Maiden, “Ten implementations of ptychography,” J. Microsc. 269, 187–194 (2018).
[Crossref]

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

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

J. Optim. Theory Appl. (1)

M. R. Hestenes, “Multiplier and gradient methods,” J. Optim. Theory Appl. 4, 303–320 (1969).
[Crossref]

J. Sci. Comput. (1)

Y. Wang, W. Yin, and J. Zeng, “Global convergence of ADMM in nonconvex nonsmooth optimization,” J. Sci. Comput. 7829–63 (2015).

J. Struct. Biol. (1)

A. Diaz, B. Malkova, M. Holler, M. Guizar-Sicairos, E. Lima, V. Panneels, G. Pigino, A. G. Bittermann, L. Wettstein, T. Tomizaki, O. Bunk, G. Schertler, T. Ishikawa, R. Wepf, and A. Menzel, “Three-dimensional mass density mapping of cellular ultrastructure by ptychographic x-ray nanotomography,” J. Struct. Biol. 192, 461–469 (2015).
[Crossref] [PubMed]

J. Synchrotron Rad. (1)

D. Gürsoy, X. Xiao, F. D. Carlo, and C. Jacobsen, “Tomopy: a framework for the analysis of synchrotron tomographic data,” J. Synchrotron Rad. 21, 1188–1193 (2014).
[Crossref]

J. synchrotron radiation (1)

D. J. Ching and D. Gürsoy, “XDesign: an open-source software package for designing x-ray imaging phantoms and experiments,” J. synchrotron radiation 24, 537–544 (2017).
[Crossref]

JOSA (1)

E. Wolf, “New theory of partial coherence in the space-frequency domain. Part I: spectra and cross spectra of steady-state sources,” JOSA 72, 343–351 (1982).
[Crossref]

Math. Probl. Eng. (1)

L. Li, X. Wang, and G. Wang, “Alternating direction method of multipliers for separable convex optimization of real functions in complex variables,” Math. Probl. Eng. 2015, 104531 (2015).
[Crossref]

Math. Program. (1)

J. Bolte, S. Sabach, and M. Teboulle, “Proximal alternating linearized minimization for nonconvex and nonsmooth problems,” Math. Program. 146, 459–494 (2014).
[Crossref]

Nat. Photonics (1)

F. Pfeiffer, “X-ray ptychography,” Nat. Photonics 12, 9–17 (2018).
[Crossref]

Nature (2)

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

M. 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. (1)

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

Opt. Exp. (1)

K. Shimomura, M. Hirose, T. Higashino, and Y. Takahashi, “Three-dimensional iterative multislice reconstruction for ptychographic x-ray computed tomography,” Opt. Exp. 26, 31199 (2018).
[Crossref]

Opt. Express (6)

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]

L.-H. Yeh, J. Dong, J. Zhong, L. Tian, M. Chen, G. Tang, M. Soltanolkotabi, and L. Waller, “Experimental robustness of Fourier ptychography phase retrieval algorithms,” Opt. Express 23, 33214–33240 (2015).
[Crossref]

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

J. Deng, Y. S. G. Nashed, S. Chen, N. W. Phillips, T. Peterka, R. Ross, S. Vogt, C. Jacobsen, and D. J. Vine, “Continuous motion scan ptychography: characterization for increased speed in coherent x-ray imaging,” Opt. Express 235438 (2015).
[Crossref]

R. N. Wilke, M. Priebe, M. Bartels, K. Giewekemeyer, A. Diaz, P. Karvinen, and T. Salditt, “Hard x-ray imaging of bacterial cells: nano-diffraction and ptychographic reconstruction,” Opt. Express 20, 19232–19254 (2012).
[Crossref] [PubMed]

Y. S. G. Nashed, D. J. Vine, T. Peterka, J. Deng, R. Ross, and C. Jacobsen, “Parallel ptychographic reconstruction,” Opt. Express 2, 32082–32097 (2014).
[Crossref]

Opt. Lett. (2)

D. Gürsoy, “Direct coupling of tomography and ptychography,” Opt. Lett. 42, 3169–3172 (2017).
[Crossref] [PubMed]

J. R. Fienup, “Reconstruction of an object from the modulus of its Fourier transform,” Opt. Lett. 3, 27–29 (1978).
[Crossref] [PubMed]

Optica (3)

M. A. Gilles, Y. S. G. Nashed, C. J. M. Du, and S. M. Wild, “3D x-ray imaging of continuous objects beyond the depth of focus limit,” Optica 5, 1078–1086 (2018).
[Crossref] [PubMed]

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

H. Öztürk, H. Yan, Y. He, M. Ge, Z. Dong, M. Lin, E. Nazaretski, I. K. Robinson, Y. S. Chu, and X. Huang, “Multi-slice ptychography with large numerical aperture multilayer laue lenses,” Optica 5, 601 (2018).
[Crossref]

Optik (2)

D. Gabay and B. Mercier, “A dual algorithm for the solution of nonlinear variational problems via finite element approximation,” Optik 2, 17–40 (1976).

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).

Optimization (1)

M. J. Powell, “A method for nonlinear constraints in minimization problems,” Optimization 1969283–298 (1969).

Phys. Rev. B (2)

J. Deng, Y. P. Hong, S. Chen, Y. S. G. Nashed, T. Peterka, A. J. F. Levi, J. Damoulakis, S. Saha, T. Eiles, and C. Jacobsen, “Nanoscale x-ray imaging of circuit features without wafer etching,” Phys. Rev. B 95, 104111 (2017).
[Crossref] [PubMed]

A. Diaz, P. Trtik, M. Guizar-Sicairos, A. Menzel, P. Thibault, and O. Bunk, “Quantitative x-ray phase nanotomography,” Phys. Rev. B 85, 020104 (2012).
[Crossref]

Phys. Rev. Lett. (2)

J. M. Rodenburg, A. C. Hurst, A. G. Cullis, B. R. Dobson, F. Pfeiffer, O. Bunk, C. David, K. Jefimovs, and I. Johnson, “Hard-x-ray lensless imaging of extended objects,” Phys. Rev. Lett. 98, 034801 (2007).
[Crossref] [PubMed]

B. Mike, S. Tobias, G. Thomas, R. Michael, G. Klaus, G. Sophie-Charlotte, S. Tim, and R. Axel, “Chemical contrast in soft x-ray ptychography,” Phys. Rev. Lett. 107, 208101 (2011).
[Crossref]

PNAS (2)

J. Deng, D. J. Vine, S. Chen, Y. S. G. Nashed, Q. Jin, N. W. Phillips, T. Peterka, R. Ross, S. Vogt, and C. J. Jacobsen, “Simultaneous cryo x-ray ptychographic and fluorescence microscopy of green algae,” PNAS 112, 2314–2319 (2015).
[Crossref] [PubMed]

K. Giewekemeyera, P. Thibault, S. Kalbfleisch, A. Beerlinka, C. M. Kewishc, M. Dierolf, F. Pfeiffer, and T. Salditt, “Quantitative biological imaging by ptychographic x-ray diffraction microscopy,” PNAS 107, 529–534 (2009).
[Crossref]

RAIRO Anal. Numer (1)

R. Glowinski and A. Marroco, “Sur l’approximation, par éléments finis d’ordre un, et la résolution, par pénalisation-dualité d’une classe de problémes de Dirichlet non linéaires,” RAIRO Anal. Numer 9, 41–76 (1975).

Rev. Sci. Instrum. (1)

M. Holler, J. Raabe, A. Diaz, M. Guizar-Sicairos, C. Quitmann, A. Menzel, and O. Bunk, “An instrument for 3D x-ray nano-imaging,” Rev. Sci. Instrum. 83, 073703 (2012).
[Crossref] [PubMed]

Sci. Rep. (2)

B. Chen, M. Guizar-Sicairos, G. Xiong, L. Shemilt, A. Diaz, J. Nutter, N. Burdet, S. Huo, J. Mancuso, A. Monteith, F. Vergeer, A. Burgess, and I. Robinson, “Three-dimensional structure analysis and percolation properties of a barrier marine coating,” Sci. Rep. 3, 1177 (2013).
[Crossref] [PubMed]

M. W. M. Jones, K. D. Elgass, M. D. Junker, M. D. de Jonge, and G. A. van Riessen, “Molar concentration from sequential 2-D water-window x-ray ptychography and x-ray fluorescence in hydrated cells,” Sci. Rep. 6, 24280 (2016).
[Crossref] [PubMed]

Sci. Reports (2)

M. Holler, A. Diaz, M. Guizar-Sicairos, P. Karvinen, E. Färm, E. Härkönen, M. Ritala, A. Menzel, J. Raabe, and O. Bunk, “X-ray ptychographic computed tomography at 16 nm isotropic 3D resolution,” Sci. Reports 4, 3857 (2014).
[Crossref]

P. Li and A. Maiden, “Multi-slice ptychographic tomography,” Sci. Reports 82049 (2018).

Science (1)

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

SIAM J. Numer. Anal. (1)

B. He and X. Yuan, “On the O(1/n) convergence rate of the Douglas-Rachford alternating direction method,” SIAM J. Numer. Anal. 50, 700–709 (2012).
[Crossref]

SIAM J. on Imaging Sci. (5)

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

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

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

E. J. Candès, Y. C. Eldar, T. Strohmer, and V. Voroninski, “Phase retrieval via matrix completion,” SIAM J. on Imaging Sci. 6, 199–225 (2013).
[Crossref]

F. Andersson, M. Carlsson, and V. V. Nikitin, “Fast algorithms and efficient GPU implementations for the Radon transform and the back-projection operator represented as convolution operators,” SIAM J. on Imaging Sci. 9, 637–664 (2016).
[Crossref]

SIAM J. on Optim. (1)

M. Hong, Z. Luo, and M. Razaviyayn, “Convergence analysis of alternating direction method of multipliers for a family of nonconvex problems,” SIAM J. on Optim. 26, 337–364 (2016).
[Crossref]

Transactions Am. Math. Soc. (1)

J. Douglas and H. H. Rachford, “On the numerical solution of heat conduction problems in two and three space variables,” Transactions Am. Math. Soc. 82, 421–439 (1956).
[Crossref]

Ultramicroscopy (2)

J. M. Rodenburg, A. C. Hurst, and A. G. Cullis, “Transmission microscopy without lenses for objects of unlimited size,” Ultramicroscopy 107, 227–231 (2007).
[Crossref]

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

Other (7)

T. Bicer, D. Gürsoy, R. Kettimuthu, F. De Carlo, G. Agrawal, and I. T. Foster, “Rapid tomographic image reconstruction via large-scale parallelization,” in Euro-Par 2015: Parallel Processing, J. L. Träff, S. Hunold, and F. Versaci, eds. (Springer, Berlin, Heidelberg, 2015), pp. 289–302.

D. Davis and W. Yin, “Convergence rate analysis of several splitting schemes,” in Splitting Methods in Communication, Imaging, Science, and Engineering, R. Glowinski, S. J. Osher, and W. Yin, eds. (Springer International Publishing, Cham, 2016), pp. 115–163.
[Crossref]

C. Yang, J. Qian, A. Schirotzek, F. Maia, and S. Marchesini, “Iterative algorithms for ptychographic phase retrieval,” arXiv preprint arXiv:1105.5628 (2011).

T. Bicer, D. Gürsoy, R. Kettimuthu, I. T. Foster, B. Ren, V. D. Andrede, and F. De Carlo, “Real-time data analysis and autonomous steering of synchrotron light source experiments,” in 2017 IEEE 13th International Conference on e-Science, (2017), pp. 59–68.
[Crossref]

R. Remmert, Theory of Complex Functions (Springer, 1991).
[Crossref]

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

T. Ramos, B. E. Grønager, M. S. Andersen, and J. W. Andreasen, “Direct 3D tomographic reconstruction and phase-retrieval of far-field coherent diffraction patterns,” arXiv preprint arXiv:1808.02109 (2018).

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Figures (6)

Fig. 1
Fig. 1 Layout of a ptychographic data collection setup [2]. An object is raster-scanned with a finite-extent coherent beam (painted in pink) to obtain a set of diffraction data recorded by a pixelated detector located downstream of the object. A full tomography dataset can be obtained by repeating the planar scanning process for different angular positions of the object around a vertical rotation axis.
Fig. 2
Fig. 2 (a) 3D object; (b) its 2D cross-section and the material composition; (c) samples of simulated measurement data for the noise-free and noise-added simulations with σ, where σ is the square-root of the corresponding peak intensity (standard deviation).
Fig. 3
Fig. 3 Convergence plot of the reconstruction error of x from solving tomography and ptychographic phase retrieval problems using the joint approach with noise applied. The object is rotated 400 times at regular intervals from 0 to 2π and has approximately 30% probe overlap between adjacent probe positions. The inner and outer iterations are 500 and 40, respectively. The l2 norm of the ground truth model is 1.1e-2.
Fig. 4
Fig. 4 The primal Eq. (18) and dual residual Eq. (19) decay values for five simulations for the noise-free model and for the model with Poisson noise. The object is rotated 400 times at regular intervals from 0 to 2π and has approximately 30% probe overlap between adjacent probe positions. The number of inner iterations for ptychographic phase retrieval (P) and tomography (T) are shown in legends as P/T and the outer iterations is fixed to 200.
Fig. 5
Fig. 5 Reconstructions obtained by using the conventional sequential (standard) and the proposed joint approach for noise-free and noise-added simulations with σ = 46 (see Fig. 2c). The object is rotated 100 and 400 times at regular intervals from 0 to 2π and has approximately 30% and 70% probe overlap between adjacent probe positions for each simulation.
Fig. 6
Fig. 6 Reconstructions obtained by using the coupled approach [2] and the proposed joint approach for noise-added simulations with σ = 46 (see Fig. 2c). The object is rotated 100 and 400 times at regular intervals from 0 to 2π and has approximately 30% probe overlap between adjacent probe positions for each simulation.

Tables (2)

Tables Icon

Algorithm 1 Joint ptycho-tomography reconstruction via ADMM

Tables Icon

Table 1 Primal and dual residual values at the last ADMM outer iteration for twelve simulations with and without applying Poisson noise; see Figure 2c. P and T correspond to ptychographic phase retrieval and tomography problems, respectively. The object is rotated 400 times at regular intervals from 0 to 2π and has approximately 30% probe overlap between adjacent probe positions.

Equations (35)

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min x 1 2 θ = 1 M i = 1 N f θ , i ( x ) d θ , i 2 2 ,
min ψ 1 , , ψ M θ = 1 M ( 1 2 i = 1 N g i ( ψ θ ) d θ , i 2 2 ) ,
subject to ψ θ = h θ ( x ) , θ = 1 , 2 , , M ,
ρ ( ψ 1 , , ψ M , x , λ 1 , , λ M ) = θ = 1 M ( 1 2 i = 1 N g i ( ψ θ ) d θ , i 2 2 + 2 Re { λ θ * ( ψ θ h θ ( x ) ) } + ρ ψ θ h θ ( x ) 2 2 ) ,
ψ θ k + 1 = arg min ψ θ ρ ( ψ θ , x k , λ θ k ) , θ = 1 , 2 , , M ,
x k + 1 = arg min x ρ ( ψ 1 k + 1 , , ψ M k + 1 , x , λ 1 k , , λ M k ) ,
λ θ k + 1 = λ θ k + ρ ( ψ θ k + 1 h θ ( x k + 1 ) ) , θ = 1 , 2 , , M ,
ψ θ k + 1 = arg min ψ θ 1 2 i = 1 N g i ( ψ θ ) d θ , i 2 2 + ρ ψ θ h θ ( x k ) + λ θ k / ρ 2 2 , θ = 1 , 2 , , M
x k + 1 = arg min x θ = 1 M ρ h θ ( x ) ψ θ k + 1 λ θ k / ρ 2 2 ,
λ θ k + 1 = λ θ k + ρ ( ψ θ k + 1 h θ ( x k + 1 ) ) , θ = 1 , 2 , , M .
0 = ψ θ h θ ( x ) , θ = 1 , 2 , , M ,
0 = ψ θ 0 ( ψ θ , x , λ θ ) = 1 2 ψ θ ( i = 1 N g i ( ψ θ ) d θ , i 2 2 ) + λ θ θ = 1 , 2 , , M ,
0 = x 0 ( ψ 1 , , ψ M , x , λ 1 , , λ M ) = 2 θ = 1 M x Re { h θ ( x ) } T λ θ ,
0 = 1 2 ψ θ ( i = 1 N g i ( ψ θ ) d θ , i 2 2 ) + ρ ( ψ θ k + 1 h θ ( x k ) + λ θ k / ρ )
= 1 2 ψ θ ( i = 1 N g i ( ψ θ ) d θ , i 2 2 ) + λ θ k + 1 + ρ ( h θ ( x k + 1 ) h θ ( x k ) ) .
0 = θ = 1 M 2 ρ x Re { h θ ( x k + 1 ) } T ( h θ ( x k + 1 ) ψ θ k + 1 λ θ k / ρ )
= θ = 1 M 2 x Re { h θ ( x k + 1 ) } T λ θ k + 1 .
r θ k + 1 = ψ θ k + 1 h θ ( x k + 1 ) , θ = 1 , 2 , , M ,
s θ k + 1 = ρ ( h θ ( x k + 1 ) h θ ( x k ) ) , θ = 1 , 2 , , M .
s 2 k + 1 = 1 2 ψ θ ( i = 1 N g i ( ψ θ ) d θ , i 2 2 ) + λ θ k + 1 , θ = 1 , 2 , , M .
s 3 k + 1 = θ = 1 M 2 x Re { h θ ( x k + 1 ) } T λ θ k + 1 ,
r k 2 pri , and s k 2 dual , s 2 k 2 dual 2 , and s 3 k 2 dual 3 ,
ρ k + 1 = { τ incr ρ k if r k 2 > μ s k 2 ρ k / τ decr if s k 2 > μ r k 2 ρ k otherwise ,
ψ θ m + 1 = ψ θ m γ ψ θ ρ ( ψ θ , x k , λ θ k ) ,
d θ , i 2 = | F Q i ψ θ | 2 ,
ψ θ ρ ( ψ θ , x k , λ θ k ) = 1 2 i = 1 N ( Q i * Q i ψ θ Q i * F * Diag ( F Q i ψ θ | F Q i ψ θ | ) d θ , i ) + ρ ( ψ θ h θ ( x k ) + λ θ k / ρ ) , θ = 1 , 2 , , M ,
ψ θ m + 1 = ( 1 γ ρ ) ψ θ m + γ ρ ( h θ ( x k ) λ θ k / ρ ) γ 2 i = 1 N Q i * ( Q i ψ θ m F * Diag ( F Q i ψ θ m | F Q i ψ θ m | ) d θ , i ) , θ = 1 , 2 , , M .
h θ ( x ) = exp ( ι 2 π ν R θ x ) ,
min x θ = 1 M R θ x ϕ θ 2 2 ,
R T ( R x + ι ν 2 π log ( ψ k + 1 + λ k / ρ ) ) = 0 .
( R x + ι ν 2 π log ( ψ k + 1 + λ k / ρ ) ) = 0 ,
exp ( ι 2 π ν R x ) ψ k + 1 λ k / ρ = 0 .
x k + 1 = arg min x θ = 1 M R θ x ϕ θ 2 2 .
s 3 k + 1 = θ = 1 M R θ T ( R θ x k + 1 ϕ θ ) .
x n + 1 = x n + η θ = 1 M R θ T ( ϕ θ R θ x n ) ,

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