Abstract

In this paper, we report the development of the generalized proximal smoothing (GPS) algorithm for phase retrieval of noisy data. GPS is a optimization-based algorithm, in which we relax both the Fourier magnitudes and support constraint. We relax the support constraint by incorporating the Moreau-Yosida regularization and heat kernel smoothing, and derive the associated proximal mapping. We also relax the magnitude constraint into a least squares fidelity term, whose proximal mapping is available as well. GPS alternatively iterates between the two proximal mappings in primal and dual spaces, respectively. Using both numerical simulation and experimental data, we show that GPS algorithm consistently outperforms the classical phase retrieval algorithms such as hybrid input-output (HIO) and oversampling smoothness (OSS), in terms of the convergence speed, consistency of the phase retrieval, and robustness to noise.

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

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References

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2018 (3)

P. Chen and A. Fannjiang, “Fourier phase retrieval with a single mask by Douglas-Rachford algorithms,” Appl. Comput. Harmon. Anal. 44, 665–699 (2018).
[Crossref] [PubMed]

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

P. Yin, S. Zhang, J. Lyu, S. Osher, Y. Qi, and J. Xin, “Binaryrelax: A relaxation approach for training deep neural networks with quantized weights,” SIAM J. on Imaging Sci. 11, 2205–2223 (2018).
[Crossref]

2016 (1)

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

2015 (5)

E. Candès, X. Li, and M. Soltanolkotabi, “Phase retrieval via wirtinger flow: Theory and algorithms,” IEEE Trans. Info. Theory 61, 1985–2007 (2015).
[Crossref]

I. Waldspurger, A. d’Aspremont, and S. Mallat, “Phase recovery, maxcut and complex semidefinite programming,” Math. Program. 149, 47–81 (2015).
[Crossref]

P. Yin and J. Xin, “Phaseliftoff: an accurate and stable phase retrieval method based on difference of trace and frobenius norms,” Commun. Math. Sci. 13, 1033–1049 (2015).
[Crossref]

J. Miao, T. Ishikawa, I. Robinson, and M. Murnane, “Beyond crystallography: Diffractive imaging using coherent x-ray light sources,” Science 348, 530–535 (2015).
[Crossref] [PubMed]

Y. Shechtman, Y. Eldar, O. Cohen, H. Chapman, J. Miao, and M. Segev, “Phase retrieval with application to optical imaging: a contemporary overview,” IEEE Signal Process. Mag. 32, 87–109 (2015).
[Crossref]

2014 (1)

C. Song, K. Tono, J. Park, T. Ebisu, S. Kim, H. Shimada, S. Kim, M. Gallagher-Jones, D. Nam, T. Sato, T. Togashi, K. Ogawa, Y. Joti, T. Kameshima, S. Ono, T. Hatsui, S. Iwata, M. Yabashi, and T. Ishikawa, “Multiple application X-ray imaging chamber for single-shot diffraction experiments with femtosecond X-ray laser pulses,” J. Appl. Crystallogr. 47, 188–197 (2014).
[Crossref]

2013 (3)

J. Rodriguez, R. Xu, C. Chen, Y. Zou, and J. Miao, “Oversampling smoothness (OSS): an effective algorithm for phase retrieval of noisy diffraction intensities,” J. Appl. Crystallogr. 46, 312–318 (2013).
[Crossref] [PubMed]

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

E. Candès, T. Strohmer, and V. Voroninski, “Phaselift: Exact and stable signal recovery from magnitude measurements via convex programming,” Comm. Pure Appl. Math. 66, 1241–1274 (2013).
[Crossref]

2012 (4)

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

A. V. Martin, F. Wang, N. D. Loh, T. Ekeberg, F. R. N. C. Maia, M. Hantke, G. van der Schot, C. Y. Hampton, R. G. Sierra, A. Aquila, S. Bajt, M. Barthelmess, C. Bostedt, J. D. Bozek, N. Coppola, S. W. Epp, B. Erk, H. Fleckenstein, L. Foucar, M. Frank, H. Graafsma, L. Gumprecht, A. Hartmann, R. Hartmann, G. Hauser, H. Hirsemann, P. Holl, S. Kassemeyer, N. Kimmel, M. Liang, L. Lomb, S. Marchesini, K. Nass, E. Pedersoli, C. Reich, D. Rolles, B. Rudek, A. Rudenko, J. Schulz, R. L. Shoeman, H. Soltau, D. Starodub, J. Steinbrener, F. Stellato, L. Strüder, J. Ullrich, G. Weidenspointner, T. A. White, C. B. Wunderer, A. Barty, I. Schlichting, M. J. Bogan, and H. N. Chapman, “Noise-robust coherent diffractive imaging with a single diffraction pattern,” Opt. Express 20, 16650–16661 (2012).
[Crossref]

S. Kassemeyer, J. Steinbrener, L. Lomb, E. Hartmann, A. Aquila, A. Barty, A. V. Martin, C. Y. Hampton, S. Bajt, M. Barthelmess, T. R. Barends, C. Bostedt, M. Bott, J. D. Bozek, N. Coppola, M. Cryle, D. P. DePonte, R. B. Doak, S. W. Epp, B. Erk, H. Fleckenstein, L. Foucar, H. Graafsma, L. Gumprecht, A. Hartmann, R. Hartmann, G. Hauser, H. Hirsemann, A. Hömke, P. Holl, O. Jönsson, N. Kimmel, F. Krasniqi, M. Liang, F. R. Maia, S. Marchesini, K. Nass, C. Reich, D. Rolles, B. Rudek, A. Rudenko, C. Schmidt, J. Schulz, R. L. Shoeman, R. G. Sierra, H. Soltau, J. C. H. Spence, D. Starodub, F. Stellato, S. Stern, G. Stier, M. Svenda, G. Weidenspointner, U. Weierstall, T. A. White, C. Wunderer, M. Frank, H. N. Chapman, J. Ullrich, L. Strüder, M. J. Bogan, and I. Schlichting, “Femtosecond free-electron laser x-ray diffraction data sets for algorithm development,” Opt. Express 20, 4149–4158 (2012).
[Crossref] [PubMed]

F. R. N. C. Maia, “The Coherent X-ray Imaging Data Bank,” Nat. Methods 9, 854–855 (2012).
[Crossref] [PubMed]

2011 (2)

A. Chai, M. Moscoso, and G. Papanicolaou, “Array imaging using intensity-only measurements,” Inv. Prob.  27, 015005 (2011).
[Crossref]

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]

2010 (5)

A. Chambolle and T. Pock, “A first-order primal-dual algorithm for convex problems with applications to imaging,” J. Math. Imaging Vis. 40, 120–145 (2010).
[Crossref]

E. Esser, X. Zhang, and T. Chan, “A general framework for a class of first order primal-dual algorithms for convex optimization in imaging science,” SIAM J. Imaging Sci. 3, 1015–1046 (2010).
[Crossref]

K. Raines, S. Salha, R. Sandberg, H. Jiang, J. Rodriguez, B. Fahimian, H. Kapteyn, J. Du, and J. Miao, “Threedimensional structure determination from a single view,” Nature 463, 214–217 (2010).
[Crossref]

H. Chapman and K. Nugent, “Coherent lensless x-ray imaging,” Nat. Photon. 4, 833–839 (2010).
[Crossref]

H. Jiang, C. Song, C.-C. Chen, R. Xu, K. S. Raines, B. P. Fahimian, C.-H. Lu, T.-K. Lee, A. Nakashima, J. Urano, T. Ishikawa, F. Tamanoi, and J. Miao, “Quantitative 3d imaging of whole, unstained cells by using x-ray diffraction microscopy,” Proc. Natl. Acad. Sci. USA 107, 11234–11239 (2010).
[Crossref] [PubMed]

2009 (1)

I. Robinson and R. Harder, “Coherent x-ray diffraction imaging of strain at the nanoscale,” Nat. Mater. 8, 291–298 (2009).
[Crossref] [PubMed]

2007 (2)

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

C. Chen, J. Miao, C. Wang, and T. Lee, “Application of optimization technique to noncrystalline x-ray diffraction microscopy: Guided hybrid input-output method,” Phys. Rev. B 76, 064113 (2007).
[Crossref]

2005 (2)

J. Miao, Y. Nishino, Y. Kohmura, B. Johnson, C. Song, S. H. Risbud, and T. Ishikawa, “Quantitative image reconstruction of gan quantum dots from oversampled diffraction intensities alone,” Phys. Rev. Lett. 95, 085503 (2005).
[Crossref] [PubMed]

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

2003 (3)

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

J. Miao, K. Hodgson, T. Ishikawa, C. Larabell, M. Legros, and Y. Nishino, “Imaging whole escherichia coli bacteria by using single-particle x-ray diffraction,” Proc. Natl. Acad. Sci. USA 100, 110–112 (2003).
[Crossref] [PubMed]

H. Bauschke, P. Combettes, and D. Luke, “Hybrid projection-reflection method for phase retrieval,” J. Opt. Soc. Am. A. 20, 1025–1034 (2003).
[Crossref]

2002 (1)

H. Bauschke, P. Combettes, and D. Luke, “Phase retrieval, error reduction algorithm, and fienup variants: a view from convex optimization,” J. Opt. Soc. Am. A. 19, 1334–1345 (2002).
[Crossref]

2000 (1)

J. Miao and D. Sayre, “On possible extensions of x-ray crystallography through diffraction-pattern oversampling,” Acta Crystallogr. A 56, 596–605 (2000).
[Crossref] [PubMed]

1999 (1)

J. Miao, P. Charalambous, J. Kirz, and D. Sayre, “Extending the methodology of x-ray crystallography to allow imaging of micrometre-sized non-crystalline specimens,” Nature 400, 342–344 (1999).
[Crossref]

1998 (1)

J. Miao, D. Sayre, and H. Chapman, “Phase retrieval from the magnitude of the fourier transforms of nonperiodic objects,” J. Opt. Soc. Am. A. 15, 1662–1669 (1998).
[Crossref]

1990 (1)

1982 (1)

1978 (1)

1975 (1)

R. Glowinski and A. Marroco, “Sur l’approximation, par éléments finis d’ordre un, et la résolution, par pénalisationdualité d’une classe de problèmes de dirichlet non linéaires,” Revue française d’automatique, informatique, recherche opérationnelle. Analyse numérique 9, 41–76 (1975).
[Crossref]

1972 (1)

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

1965 (1)

J.-J. Moreau, “Proximité et dualité dans un espace hilbertien,” Bull. de la Société Mathématique de France 93, 273–299 (1965).
[Crossref]

1956 (1)

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

Aquila, A.

A. V. Martin, F. Wang, N. D. Loh, T. Ekeberg, F. R. N. C. Maia, M. Hantke, G. van der Schot, C. Y. Hampton, R. G. Sierra, A. Aquila, S. Bajt, M. Barthelmess, C. Bostedt, J. D. Bozek, N. Coppola, S. W. Epp, B. Erk, H. Fleckenstein, L. Foucar, M. Frank, H. Graafsma, L. Gumprecht, A. Hartmann, R. Hartmann, G. Hauser, H. Hirsemann, P. Holl, S. Kassemeyer, N. Kimmel, M. Liang, L. Lomb, S. Marchesini, K. Nass, E. Pedersoli, C. Reich, D. Rolles, B. Rudek, A. Rudenko, J. Schulz, R. L. Shoeman, H. Soltau, D. Starodub, J. Steinbrener, F. Stellato, L. Strüder, J. Ullrich, G. Weidenspointner, T. A. White, C. B. Wunderer, A. Barty, I. Schlichting, M. J. Bogan, and H. N. Chapman, “Noise-robust coherent diffractive imaging with a single diffraction pattern,” Opt. Express 20, 16650–16661 (2012).
[Crossref]

S. Kassemeyer, J. Steinbrener, L. Lomb, E. Hartmann, A. Aquila, A. Barty, A. V. Martin, C. Y. Hampton, S. Bajt, M. Barthelmess, T. R. Barends, C. Bostedt, M. Bott, J. D. Bozek, N. Coppola, M. Cryle, D. P. DePonte, R. B. Doak, S. W. Epp, B. Erk, H. Fleckenstein, L. Foucar, H. Graafsma, L. Gumprecht, A. Hartmann, R. Hartmann, G. Hauser, H. Hirsemann, A. Hömke, P. Holl, O. Jönsson, N. Kimmel, F. Krasniqi, M. Liang, F. R. Maia, S. Marchesini, K. Nass, C. Reich, D. Rolles, B. Rudek, A. Rudenko, C. Schmidt, J. Schulz, R. L. Shoeman, R. G. Sierra, H. Soltau, J. C. H. Spence, D. Starodub, F. Stellato, S. Stern, G. Stier, M. Svenda, G. Weidenspointner, U. Weierstall, T. A. White, C. Wunderer, M. Frank, H. N. Chapman, J. Ullrich, L. Strüder, M. J. Bogan, and I. Schlichting, “Femtosecond free-electron laser x-ray diffraction data sets for algorithm development,” Opt. Express 20, 4149–4158 (2012).
[Crossref] [PubMed]

Bajt, S.

S. Kassemeyer, J. Steinbrener, L. Lomb, E. Hartmann, A. Aquila, A. Barty, A. V. Martin, C. Y. Hampton, S. Bajt, M. Barthelmess, T. R. Barends, C. Bostedt, M. Bott, J. D. Bozek, N. Coppola, M. Cryle, D. P. DePonte, R. B. Doak, S. W. Epp, B. Erk, H. Fleckenstein, L. Foucar, H. Graafsma, L. Gumprecht, A. Hartmann, R. Hartmann, G. Hauser, H. Hirsemann, A. Hömke, P. Holl, O. Jönsson, N. Kimmel, F. Krasniqi, M. Liang, F. R. Maia, S. Marchesini, K. Nass, C. Reich, D. Rolles, B. Rudek, A. Rudenko, C. Schmidt, J. Schulz, R. L. Shoeman, R. G. Sierra, H. Soltau, J. C. H. Spence, D. Starodub, F. Stellato, S. Stern, G. Stier, M. Svenda, G. Weidenspointner, U. Weierstall, T. A. White, C. Wunderer, M. Frank, H. N. Chapman, J. Ullrich, L. Strüder, M. J. Bogan, and I. Schlichting, “Femtosecond free-electron laser x-ray diffraction data sets for algorithm development,” Opt. Express 20, 4149–4158 (2012).
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A. V. Martin, F. Wang, N. D. Loh, T. Ekeberg, F. R. N. C. Maia, M. Hantke, G. van der Schot, C. Y. Hampton, R. G. Sierra, A. Aquila, S. Bajt, M. Barthelmess, C. Bostedt, J. D. Bozek, N. Coppola, S. W. Epp, B. Erk, H. Fleckenstein, L. Foucar, M. Frank, H. Graafsma, L. Gumprecht, A. Hartmann, R. Hartmann, G. Hauser, H. Hirsemann, P. Holl, S. Kassemeyer, N. Kimmel, M. Liang, L. Lomb, S. Marchesini, K. Nass, E. Pedersoli, C. Reich, D. Rolles, B. Rudek, A. Rudenko, J. Schulz, R. L. Shoeman, H. Soltau, D. Starodub, J. Steinbrener, F. Stellato, L. Strüder, J. Ullrich, G. Weidenspointner, T. A. White, C. B. Wunderer, A. Barty, I. Schlichting, M. J. Bogan, and H. N. Chapman, “Noise-robust coherent diffractive imaging with a single diffraction pattern,” Opt. Express 20, 16650–16661 (2012).
[Crossref]

Barends, T. R.

Barthelmess, M.

S. Kassemeyer, J. Steinbrener, L. Lomb, E. Hartmann, A. Aquila, A. Barty, A. V. Martin, C. Y. Hampton, S. Bajt, M. Barthelmess, T. R. Barends, C. Bostedt, M. Bott, J. D. Bozek, N. Coppola, M. Cryle, D. P. DePonte, R. B. Doak, S. W. Epp, B. Erk, H. Fleckenstein, L. Foucar, H. Graafsma, L. Gumprecht, A. Hartmann, R. Hartmann, G. Hauser, H. Hirsemann, A. Hömke, P. Holl, O. Jönsson, N. Kimmel, F. Krasniqi, M. Liang, F. R. Maia, S. Marchesini, K. Nass, C. Reich, D. Rolles, B. Rudek, A. Rudenko, C. Schmidt, J. Schulz, R. L. Shoeman, R. G. Sierra, H. Soltau, J. C. H. Spence, D. Starodub, F. Stellato, S. Stern, G. Stier, M. Svenda, G. Weidenspointner, U. Weierstall, T. A. White, C. Wunderer, M. Frank, H. N. Chapman, J. Ullrich, L. Strüder, M. J. Bogan, and I. Schlichting, “Femtosecond free-electron laser x-ray diffraction data sets for algorithm development,” Opt. Express 20, 4149–4158 (2012).
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A. V. Martin, F. Wang, N. D. Loh, T. Ekeberg, F. R. N. C. Maia, M. Hantke, G. van der Schot, C. Y. Hampton, R. G. Sierra, A. Aquila, S. Bajt, M. Barthelmess, C. Bostedt, J. D. Bozek, N. Coppola, S. W. Epp, B. Erk, H. Fleckenstein, L. Foucar, M. Frank, H. Graafsma, L. Gumprecht, A. Hartmann, R. Hartmann, G. Hauser, H. Hirsemann, P. Holl, S. Kassemeyer, N. Kimmel, M. Liang, L. Lomb, S. Marchesini, K. Nass, E. Pedersoli, C. Reich, D. Rolles, B. Rudek, A. Rudenko, J. Schulz, R. L. Shoeman, H. Soltau, D. Starodub, J. Steinbrener, F. Stellato, L. Strüder, J. Ullrich, G. Weidenspointner, T. A. White, C. B. Wunderer, A. Barty, I. Schlichting, M. J. Bogan, and H. N. Chapman, “Noise-robust coherent diffractive imaging with a single diffraction pattern,” Opt. Express 20, 16650–16661 (2012).
[Crossref]

Barty, A.

A. V. Martin, F. Wang, N. D. Loh, T. Ekeberg, F. R. N. C. Maia, M. Hantke, G. van der Schot, C. Y. Hampton, R. G. Sierra, A. Aquila, S. Bajt, M. Barthelmess, C. Bostedt, J. D. Bozek, N. Coppola, S. W. Epp, B. Erk, H. Fleckenstein, L. Foucar, M. Frank, H. Graafsma, L. Gumprecht, A. Hartmann, R. Hartmann, G. Hauser, H. Hirsemann, P. Holl, S. Kassemeyer, N. Kimmel, M. Liang, L. Lomb, S. Marchesini, K. Nass, E. Pedersoli, C. Reich, D. Rolles, B. Rudek, A. Rudenko, J. Schulz, R. L. Shoeman, H. Soltau, D. Starodub, J. Steinbrener, F. Stellato, L. Strüder, J. Ullrich, G. Weidenspointner, T. A. White, C. B. Wunderer, A. Barty, I. Schlichting, M. J. Bogan, and H. N. Chapman, “Noise-robust coherent diffractive imaging with a single diffraction pattern,” Opt. Express 20, 16650–16661 (2012).
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S. Kassemeyer, J. Steinbrener, L. Lomb, E. Hartmann, A. Aquila, A. Barty, A. V. Martin, C. Y. Hampton, S. Bajt, M. Barthelmess, T. R. Barends, C. Bostedt, M. Bott, J. D. Bozek, N. Coppola, M. Cryle, D. P. DePonte, R. B. Doak, S. W. Epp, B. Erk, H. Fleckenstein, L. Foucar, H. Graafsma, L. Gumprecht, A. Hartmann, R. Hartmann, G. Hauser, H. Hirsemann, A. Hömke, P. Holl, O. Jönsson, N. Kimmel, F. Krasniqi, M. Liang, F. R. Maia, S. Marchesini, K. Nass, C. Reich, D. Rolles, B. Rudek, A. Rudenko, C. Schmidt, J. Schulz, R. L. Shoeman, R. G. Sierra, H. Soltau, J. C. H. Spence, D. Starodub, F. Stellato, S. Stern, G. Stier, M. Svenda, G. Weidenspointner, U. Weierstall, T. A. White, C. Wunderer, M. Frank, H. N. Chapman, J. Ullrich, L. Strüder, M. J. Bogan, and I. Schlichting, “Femtosecond free-electron laser x-ray diffraction data sets for algorithm development,” Opt. Express 20, 4149–4158 (2012).
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Bauschke, H.

H. Bauschke, P. Combettes, and D. Luke, “Hybrid projection-reflection method for phase retrieval,” J. Opt. Soc. Am. A. 20, 1025–1034 (2003).
[Crossref]

H. Bauschke, P. Combettes, and D. Luke, “Phase retrieval, error reduction algorithm, and fienup variants: a view from convex optimization,” J. Opt. Soc. Am. A. 19, 1334–1345 (2002).
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Bogan, M. J.

A. V. Martin, F. Wang, N. D. Loh, T. Ekeberg, F. R. N. C. Maia, M. Hantke, G. van der Schot, C. Y. Hampton, R. G. Sierra, A. Aquila, S. Bajt, M. Barthelmess, C. Bostedt, J. D. Bozek, N. Coppola, S. W. Epp, B. Erk, H. Fleckenstein, L. Foucar, M. Frank, H. Graafsma, L. Gumprecht, A. Hartmann, R. Hartmann, G. Hauser, H. Hirsemann, P. Holl, S. Kassemeyer, N. Kimmel, M. Liang, L. Lomb, S. Marchesini, K. Nass, E. Pedersoli, C. Reich, D. Rolles, B. Rudek, A. Rudenko, J. Schulz, R. L. Shoeman, H. Soltau, D. Starodub, J. Steinbrener, F. Stellato, L. Strüder, J. Ullrich, G. Weidenspointner, T. A. White, C. B. Wunderer, A. Barty, I. Schlichting, M. J. Bogan, and H. N. Chapman, “Noise-robust coherent diffractive imaging with a single diffraction pattern,” Opt. Express 20, 16650–16661 (2012).
[Crossref]

S. Kassemeyer, J. Steinbrener, L. Lomb, E. Hartmann, A. Aquila, A. Barty, A. V. Martin, C. Y. Hampton, S. Bajt, M. Barthelmess, T. R. Barends, C. Bostedt, M. Bott, J. D. Bozek, N. Coppola, M. Cryle, D. P. DePonte, R. B. Doak, S. W. Epp, B. Erk, H. Fleckenstein, L. Foucar, H. Graafsma, L. Gumprecht, A. Hartmann, R. Hartmann, G. Hauser, H. Hirsemann, A. Hömke, P. Holl, O. Jönsson, N. Kimmel, F. Krasniqi, M. Liang, F. R. Maia, S. Marchesini, K. Nass, C. Reich, D. Rolles, B. Rudek, A. Rudenko, C. Schmidt, J. Schulz, R. L. Shoeman, R. G. Sierra, H. Soltau, J. C. H. Spence, D. Starodub, F. Stellato, S. Stern, G. Stier, M. Svenda, G. Weidenspointner, U. Weierstall, T. A. White, C. Wunderer, M. Frank, H. N. Chapman, J. Ullrich, L. Strüder, M. J. Bogan, and I. Schlichting, “Femtosecond free-electron laser x-ray diffraction data sets for algorithm development,” Opt. Express 20, 4149–4158 (2012).
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Bostedt, C.

S. Kassemeyer, J. Steinbrener, L. Lomb, E. Hartmann, A. Aquila, A. Barty, A. V. Martin, C. Y. Hampton, S. Bajt, M. Barthelmess, T. R. Barends, C. Bostedt, M. Bott, J. D. Bozek, N. Coppola, M. Cryle, D. P. DePonte, R. B. Doak, S. W. Epp, B. Erk, H. Fleckenstein, L. Foucar, H. Graafsma, L. Gumprecht, A. Hartmann, R. Hartmann, G. Hauser, H. Hirsemann, A. Hömke, P. Holl, O. Jönsson, N. Kimmel, F. Krasniqi, M. Liang, F. R. Maia, S. Marchesini, K. Nass, C. Reich, D. Rolles, B. Rudek, A. Rudenko, C. Schmidt, J. Schulz, R. L. Shoeman, R. G. Sierra, H. Soltau, J. C. H. Spence, D. Starodub, F. Stellato, S. Stern, G. Stier, M. Svenda, G. Weidenspointner, U. Weierstall, T. A. White, C. Wunderer, M. Frank, H. N. Chapman, J. Ullrich, L. Strüder, M. J. Bogan, and I. Schlichting, “Femtosecond free-electron laser x-ray diffraction data sets for algorithm development,” Opt. Express 20, 4149–4158 (2012).
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S. Kassemeyer, J. Steinbrener, L. Lomb, E. Hartmann, A. Aquila, A. Barty, A. V. Martin, C. Y. Hampton, S. Bajt, M. Barthelmess, T. R. Barends, C. Bostedt, M. Bott, J. D. Bozek, N. Coppola, M. Cryle, D. P. DePonte, R. B. Doak, S. W. Epp, B. Erk, H. Fleckenstein, L. Foucar, H. Graafsma, L. Gumprecht, A. Hartmann, R. Hartmann, G. Hauser, H. Hirsemann, A. Hömke, P. Holl, O. Jönsson, N. Kimmel, F. Krasniqi, M. Liang, F. R. Maia, S. Marchesini, K. Nass, C. Reich, D. Rolles, B. Rudek, A. Rudenko, C. Schmidt, J. Schulz, R. L. Shoeman, R. G. Sierra, H. Soltau, J. C. H. Spence, D. Starodub, F. Stellato, S. Stern, G. Stier, M. Svenda, G. Weidenspointner, U. Weierstall, T. A. White, C. Wunderer, M. Frank, H. N. Chapman, J. Ullrich, L. Strüder, M. J. Bogan, and I. Schlichting, “Femtosecond free-electron laser x-ray diffraction data sets for algorithm development,” Opt. Express 20, 4149–4158 (2012).
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H. Bauschke, P. Combettes, and D. Luke, “Hybrid projection-reflection method for phase retrieval,” J. Opt. Soc. Am. A. 20, 1025–1034 (2003).
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A. V. Martin, F. Wang, N. D. Loh, T. Ekeberg, F. R. N. C. Maia, M. Hantke, G. van der Schot, C. Y. Hampton, R. G. Sierra, A. Aquila, S. Bajt, M. Barthelmess, C. Bostedt, J. D. Bozek, N. Coppola, S. W. Epp, B. Erk, H. Fleckenstein, L. Foucar, M. Frank, H. Graafsma, L. Gumprecht, A. Hartmann, R. Hartmann, G. Hauser, H. Hirsemann, P. Holl, S. Kassemeyer, N. Kimmel, M. Liang, L. Lomb, S. Marchesini, K. Nass, E. Pedersoli, C. Reich, D. Rolles, B. Rudek, A. Rudenko, J. Schulz, R. L. Shoeman, H. Soltau, D. Starodub, J. Steinbrener, F. Stellato, L. Strüder, J. Ullrich, G. Weidenspointner, T. A. White, C. B. Wunderer, A. Barty, I. Schlichting, M. J. Bogan, and H. N. Chapman, “Noise-robust coherent diffractive imaging with a single diffraction pattern,” Opt. Express 20, 16650–16661 (2012).
[Crossref]

Hartmann, E.

Hartmann, R.

S. Kassemeyer, J. Steinbrener, L. Lomb, E. Hartmann, A. Aquila, A. Barty, A. V. Martin, C. Y. Hampton, S. Bajt, M. Barthelmess, T. R. Barends, C. Bostedt, M. Bott, J. D. Bozek, N. Coppola, M. Cryle, D. P. DePonte, R. B. Doak, S. W. Epp, B. Erk, H. Fleckenstein, L. Foucar, H. Graafsma, L. Gumprecht, A. Hartmann, R. Hartmann, G. Hauser, H. Hirsemann, A. Hömke, P. Holl, O. Jönsson, N. Kimmel, F. Krasniqi, M. Liang, F. R. Maia, S. Marchesini, K. Nass, C. Reich, D. Rolles, B. Rudek, A. Rudenko, C. Schmidt, J. Schulz, R. L. Shoeman, R. G. Sierra, H. Soltau, J. C. H. Spence, D. Starodub, F. Stellato, S. Stern, G. Stier, M. Svenda, G. Weidenspointner, U. Weierstall, T. A. White, C. Wunderer, M. Frank, H. N. Chapman, J. Ullrich, L. Strüder, M. J. Bogan, and I. Schlichting, “Femtosecond free-electron laser x-ray diffraction data sets for algorithm development,” Opt. Express 20, 4149–4158 (2012).
[Crossref] [PubMed]

A. V. Martin, F. Wang, N. D. Loh, T. Ekeberg, F. R. N. C. Maia, M. Hantke, G. van der Schot, C. Y. Hampton, R. G. Sierra, A. Aquila, S. Bajt, M. Barthelmess, C. Bostedt, J. D. Bozek, N. Coppola, S. W. Epp, B. Erk, H. Fleckenstein, L. Foucar, M. Frank, H. Graafsma, L. Gumprecht, A. Hartmann, R. Hartmann, G. Hauser, H. Hirsemann, P. Holl, S. Kassemeyer, N. Kimmel, M. Liang, L. Lomb, S. Marchesini, K. Nass, E. Pedersoli, C. Reich, D. Rolles, B. Rudek, A. Rudenko, J. Schulz, R. L. Shoeman, H. Soltau, D. Starodub, J. Steinbrener, F. Stellato, L. Strüder, J. Ullrich, G. Weidenspointner, T. A. White, C. B. Wunderer, A. Barty, I. Schlichting, M. J. Bogan, and H. N. Chapman, “Noise-robust coherent diffractive imaging with a single diffraction pattern,” Opt. Express 20, 16650–16661 (2012).
[Crossref]

Hatsui, T.

C. Song, K. Tono, J. Park, T. Ebisu, S. Kim, H. Shimada, S. Kim, M. Gallagher-Jones, D. Nam, T. Sato, T. Togashi, K. Ogawa, Y. Joti, T. Kameshima, S. Ono, T. Hatsui, S. Iwata, M. Yabashi, and T. Ishikawa, “Multiple application X-ray imaging chamber for single-shot diffraction experiments with femtosecond X-ray laser pulses,” J. Appl. Crystallogr. 47, 188–197 (2014).
[Crossref]

Hauser, G.

A. V. Martin, F. Wang, N. D. Loh, T. Ekeberg, F. R. N. C. Maia, M. Hantke, G. van der Schot, C. Y. Hampton, R. G. Sierra, A. Aquila, S. Bajt, M. Barthelmess, C. Bostedt, J. D. Bozek, N. Coppola, S. W. Epp, B. Erk, H. Fleckenstein, L. Foucar, M. Frank, H. Graafsma, L. Gumprecht, A. Hartmann, R. Hartmann, G. Hauser, H. Hirsemann, P. Holl, S. Kassemeyer, N. Kimmel, M. Liang, L. Lomb, S. Marchesini, K. Nass, E. Pedersoli, C. Reich, D. Rolles, B. Rudek, A. Rudenko, J. Schulz, R. L. Shoeman, H. Soltau, D. Starodub, J. Steinbrener, F. Stellato, L. Strüder, J. Ullrich, G. Weidenspointner, T. A. White, C. B. Wunderer, A. Barty, I. Schlichting, M. J. Bogan, and H. N. Chapman, “Noise-robust coherent diffractive imaging with a single diffraction pattern,” Opt. Express 20, 16650–16661 (2012).
[Crossref]

S. Kassemeyer, J. Steinbrener, L. Lomb, E. Hartmann, A. Aquila, A. Barty, A. V. Martin, C. Y. Hampton, S. Bajt, M. Barthelmess, T. R. Barends, C. Bostedt, M. Bott, J. D. Bozek, N. Coppola, M. Cryle, D. P. DePonte, R. B. Doak, S. W. Epp, B. Erk, H. Fleckenstein, L. Foucar, H. Graafsma, L. Gumprecht, A. Hartmann, R. Hartmann, G. Hauser, H. Hirsemann, A. Hömke, P. Holl, O. Jönsson, N. Kimmel, F. Krasniqi, M. Liang, F. R. Maia, S. Marchesini, K. Nass, C. Reich, D. Rolles, B. Rudek, A. Rudenko, C. Schmidt, J. Schulz, R. L. Shoeman, R. G. Sierra, H. Soltau, J. C. H. Spence, D. Starodub, F. Stellato, S. Stern, G. Stier, M. Svenda, G. Weidenspointner, U. Weierstall, T. A. White, C. Wunderer, M. Frank, H. N. Chapman, J. Ullrich, L. Strüder, M. J. Bogan, and I. Schlichting, “Femtosecond free-electron laser x-ray diffraction data sets for algorithm development,” Opt. Express 20, 4149–4158 (2012).
[Crossref] [PubMed]

Hirsemann, H.

S. Kassemeyer, J. Steinbrener, L. Lomb, E. Hartmann, A. Aquila, A. Barty, A. V. Martin, C. Y. Hampton, S. Bajt, M. Barthelmess, T. R. Barends, C. Bostedt, M. Bott, J. D. Bozek, N. Coppola, M. Cryle, D. P. DePonte, R. B. Doak, S. W. Epp, B. Erk, H. Fleckenstein, L. Foucar, H. Graafsma, L. Gumprecht, A. Hartmann, R. Hartmann, G. Hauser, H. Hirsemann, A. Hömke, P. Holl, O. Jönsson, N. Kimmel, F. Krasniqi, M. Liang, F. R. Maia, S. Marchesini, K. Nass, C. Reich, D. Rolles, B. Rudek, A. Rudenko, C. Schmidt, J. Schulz, R. L. Shoeman, R. G. Sierra, H. Soltau, J. C. H. Spence, D. Starodub, F. Stellato, S. Stern, G. Stier, M. Svenda, G. Weidenspointner, U. Weierstall, T. A. White, C. Wunderer, M. Frank, H. N. Chapman, J. Ullrich, L. Strüder, M. J. Bogan, and I. Schlichting, “Femtosecond free-electron laser x-ray diffraction data sets for algorithm development,” Opt. Express 20, 4149–4158 (2012).
[Crossref] [PubMed]

A. V. Martin, F. Wang, N. D. Loh, T. Ekeberg, F. R. N. C. Maia, M. Hantke, G. van der Schot, C. Y. Hampton, R. G. Sierra, A. Aquila, S. Bajt, M. Barthelmess, C. Bostedt, J. D. Bozek, N. Coppola, S. W. Epp, B. Erk, H. Fleckenstein, L. Foucar, M. Frank, H. Graafsma, L. Gumprecht, A. Hartmann, R. Hartmann, G. Hauser, H. Hirsemann, P. Holl, S. Kassemeyer, N. Kimmel, M. Liang, L. Lomb, S. Marchesini, K. Nass, E. Pedersoli, C. Reich, D. Rolles, B. Rudek, A. Rudenko, J. Schulz, R. L. Shoeman, H. Soltau, D. Starodub, J. Steinbrener, F. Stellato, L. Strüder, J. Ullrich, G. Weidenspointner, T. A. White, C. B. Wunderer, A. Barty, I. Schlichting, M. J. Bogan, and H. N. Chapman, “Noise-robust coherent diffractive imaging with a single diffraction pattern,” Opt. Express 20, 16650–16661 (2012).
[Crossref]

Hodgson, K.

J. Miao, K. Hodgson, T. Ishikawa, C. Larabell, M. Legros, and Y. Nishino, “Imaging whole escherichia coli bacteria by using single-particle x-ray diffraction,” Proc. Natl. Acad. Sci. USA 100, 110–112 (2003).
[Crossref] [PubMed]

Holl, P.

A. V. Martin, F. Wang, N. D. Loh, T. Ekeberg, F. R. N. C. Maia, M. Hantke, G. van der Schot, C. Y. Hampton, R. G. Sierra, A. Aquila, S. Bajt, M. Barthelmess, C. Bostedt, J. D. Bozek, N. Coppola, S. W. Epp, B. Erk, H. Fleckenstein, L. Foucar, M. Frank, H. Graafsma, L. Gumprecht, A. Hartmann, R. Hartmann, G. Hauser, H. Hirsemann, P. Holl, S. Kassemeyer, N. Kimmel, M. Liang, L. Lomb, S. Marchesini, K. Nass, E. Pedersoli, C. Reich, D. Rolles, B. Rudek, A. Rudenko, J. Schulz, R. L. Shoeman, H. Soltau, D. Starodub, J. Steinbrener, F. Stellato, L. Strüder, J. Ullrich, G. Weidenspointner, T. A. White, C. B. Wunderer, A. Barty, I. Schlichting, M. J. Bogan, and H. N. Chapman, “Noise-robust coherent diffractive imaging with a single diffraction pattern,” Opt. Express 20, 16650–16661 (2012).
[Crossref]

S. Kassemeyer, J. Steinbrener, L. Lomb, E. Hartmann, A. Aquila, A. Barty, A. V. Martin, C. Y. Hampton, S. Bajt, M. Barthelmess, T. R. Barends, C. Bostedt, M. Bott, J. D. Bozek, N. Coppola, M. Cryle, D. P. DePonte, R. B. Doak, S. W. Epp, B. Erk, H. Fleckenstein, L. Foucar, H. Graafsma, L. Gumprecht, A. Hartmann, R. Hartmann, G. Hauser, H. Hirsemann, A. Hömke, P. Holl, O. Jönsson, N. Kimmel, F. Krasniqi, M. Liang, F. R. Maia, S. Marchesini, K. Nass, C. Reich, D. Rolles, B. Rudek, A. Rudenko, C. Schmidt, J. Schulz, R. L. Shoeman, R. G. Sierra, H. Soltau, J. C. H. Spence, D. Starodub, F. Stellato, S. Stern, G. Stier, M. Svenda, G. Weidenspointner, U. Weierstall, T. A. White, C. Wunderer, M. Frank, H. N. Chapman, J. Ullrich, L. Strüder, M. J. Bogan, and I. Schlichting, “Femtosecond free-electron laser x-ray diffraction data sets for algorithm development,” Opt. Express 20, 4149–4158 (2012).
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Hömke, A.

Ishikawa, T.

J. Miao, T. Ishikawa, I. Robinson, and M. Murnane, “Beyond crystallography: Diffractive imaging using coherent x-ray light sources,” Science 348, 530–535 (2015).
[Crossref] [PubMed]

C. Song, K. Tono, J. Park, T. Ebisu, S. Kim, H. Shimada, S. Kim, M. Gallagher-Jones, D. Nam, T. Sato, T. Togashi, K. Ogawa, Y. Joti, T. Kameshima, S. Ono, T. Hatsui, S. Iwata, M. Yabashi, and T. Ishikawa, “Multiple application X-ray imaging chamber for single-shot diffraction experiments with femtosecond X-ray laser pulses,” J. Appl. Crystallogr. 47, 188–197 (2014).
[Crossref]

H. Jiang, C. Song, C.-C. Chen, R. Xu, K. S. Raines, B. P. Fahimian, C.-H. Lu, T.-K. Lee, A. Nakashima, J. Urano, T. Ishikawa, F. Tamanoi, and J. Miao, “Quantitative 3d imaging of whole, unstained cells by using x-ray diffraction microscopy,” Proc. Natl. Acad. Sci. USA 107, 11234–11239 (2010).
[Crossref] [PubMed]

J. Miao, Y. Nishino, Y. Kohmura, B. Johnson, C. Song, S. H. Risbud, and T. Ishikawa, “Quantitative image reconstruction of gan quantum dots from oversampled diffraction intensities alone,” Phys. Rev. Lett. 95, 085503 (2005).
[Crossref] [PubMed]

J. Miao, K. Hodgson, T. Ishikawa, C. Larabell, M. Legros, and Y. Nishino, “Imaging whole escherichia coli bacteria by using single-particle x-ray diffraction,” Proc. Natl. Acad. Sci. USA 100, 110–112 (2003).
[Crossref] [PubMed]

Iwata, S.

C. Song, K. Tono, J. Park, T. Ebisu, S. Kim, H. Shimada, S. Kim, M. Gallagher-Jones, D. Nam, T. Sato, T. Togashi, K. Ogawa, Y. Joti, T. Kameshima, S. Ono, T. Hatsui, S. Iwata, M. Yabashi, and T. Ishikawa, “Multiple application X-ray imaging chamber for single-shot diffraction experiments with femtosecond X-ray laser pulses,” J. Appl. Crystallogr. 47, 188–197 (2014).
[Crossref]

Jiang, H.

H. Jiang, C. Song, C.-C. Chen, R. Xu, K. S. Raines, B. P. Fahimian, C.-H. Lu, T.-K. Lee, A. Nakashima, J. Urano, T. Ishikawa, F. Tamanoi, and J. Miao, “Quantitative 3d imaging of whole, unstained cells by using x-ray diffraction microscopy,” Proc. Natl. Acad. Sci. USA 107, 11234–11239 (2010).
[Crossref] [PubMed]

K. Raines, S. Salha, R. Sandberg, H. Jiang, J. Rodriguez, B. Fahimian, H. Kapteyn, J. Du, and J. Miao, “Threedimensional structure determination from a single view,” Nature 463, 214–217 (2010).
[Crossref]

Johnson, B.

J. Miao, Y. Nishino, Y. Kohmura, B. Johnson, C. Song, S. H. Risbud, and T. Ishikawa, “Quantitative image reconstruction of gan quantum dots from oversampled diffraction intensities alone,” Phys. Rev. Lett. 95, 085503 (2005).
[Crossref] [PubMed]

Jönsson, O.

Joti, Y.

C. Song, K. Tono, J. Park, T. Ebisu, S. Kim, H. Shimada, S. Kim, M. Gallagher-Jones, D. Nam, T. Sato, T. Togashi, K. Ogawa, Y. Joti, T. Kameshima, S. Ono, T. Hatsui, S. Iwata, M. Yabashi, and T. Ishikawa, “Multiple application X-ray imaging chamber for single-shot diffraction experiments with femtosecond X-ray laser pulses,” J. Appl. Crystallogr. 47, 188–197 (2014).
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Kameshima, T.

C. Song, K. Tono, J. Park, T. Ebisu, S. Kim, H. Shimada, S. Kim, M. Gallagher-Jones, D. Nam, T. Sato, T. Togashi, K. Ogawa, Y. Joti, T. Kameshima, S. Ono, T. Hatsui, S. Iwata, M. Yabashi, and T. Ishikawa, “Multiple application X-ray imaging chamber for single-shot diffraction experiments with femtosecond X-ray laser pulses,” J. Appl. Crystallogr. 47, 188–197 (2014).
[Crossref]

Kapteyn, H.

K. Raines, S. Salha, R. Sandberg, H. Jiang, J. Rodriguez, B. Fahimian, H. Kapteyn, J. Du, and J. Miao, “Threedimensional structure determination from a single view,” Nature 463, 214–217 (2010).
[Crossref]

Kassemeyer, S.

S. Kassemeyer, J. Steinbrener, L. Lomb, E. Hartmann, A. Aquila, A. Barty, A. V. Martin, C. Y. Hampton, S. Bajt, M. Barthelmess, T. R. Barends, C. Bostedt, M. Bott, J. D. Bozek, N. Coppola, M. Cryle, D. P. DePonte, R. B. Doak, S. W. Epp, B. Erk, H. Fleckenstein, L. Foucar, H. Graafsma, L. Gumprecht, A. Hartmann, R. Hartmann, G. Hauser, H. Hirsemann, A. Hömke, P. Holl, O. Jönsson, N. Kimmel, F. Krasniqi, M. Liang, F. R. Maia, S. Marchesini, K. Nass, C. Reich, D. Rolles, B. Rudek, A. Rudenko, C. Schmidt, J. Schulz, R. L. Shoeman, R. G. Sierra, H. Soltau, J. C. H. Spence, D. Starodub, F. Stellato, S. Stern, G. Stier, M. Svenda, G. Weidenspointner, U. Weierstall, T. A. White, C. Wunderer, M. Frank, H. N. Chapman, J. Ullrich, L. Strüder, M. J. Bogan, and I. Schlichting, “Femtosecond free-electron laser x-ray diffraction data sets for algorithm development,” Opt. Express 20, 4149–4158 (2012).
[Crossref] [PubMed]

A. V. Martin, F. Wang, N. D. Loh, T. Ekeberg, F. R. N. C. Maia, M. Hantke, G. van der Schot, C. Y. Hampton, R. G. Sierra, A. Aquila, S. Bajt, M. Barthelmess, C. Bostedt, J. D. Bozek, N. Coppola, S. W. Epp, B. Erk, H. Fleckenstein, L. Foucar, M. Frank, H. Graafsma, L. Gumprecht, A. Hartmann, R. Hartmann, G. Hauser, H. Hirsemann, P. Holl, S. Kassemeyer, N. Kimmel, M. Liang, L. Lomb, S. Marchesini, K. Nass, E. Pedersoli, C. Reich, D. Rolles, B. Rudek, A. Rudenko, J. Schulz, R. L. Shoeman, H. Soltau, D. Starodub, J. Steinbrener, F. Stellato, L. Strüder, J. Ullrich, G. Weidenspointner, T. A. White, C. B. Wunderer, A. Barty, I. Schlichting, M. J. Bogan, and H. N. Chapman, “Noise-robust coherent diffractive imaging with a single diffraction pattern,” Opt. Express 20, 16650–16661 (2012).
[Crossref]

Kim, S.

C. Song, K. Tono, J. Park, T. Ebisu, S. Kim, H. Shimada, S. Kim, M. Gallagher-Jones, D. Nam, T. Sato, T. Togashi, K. Ogawa, Y. Joti, T. Kameshima, S. Ono, T. Hatsui, S. Iwata, M. Yabashi, and T. Ishikawa, “Multiple application X-ray imaging chamber for single-shot diffraction experiments with femtosecond X-ray laser pulses,” J. Appl. Crystallogr. 47, 188–197 (2014).
[Crossref]

C. Song, K. Tono, J. Park, T. Ebisu, S. Kim, H. Shimada, S. Kim, M. Gallagher-Jones, D. Nam, T. Sato, T. Togashi, K. Ogawa, Y. Joti, T. Kameshima, S. Ono, T. Hatsui, S. Iwata, M. Yabashi, and T. Ishikawa, “Multiple application X-ray imaging chamber for single-shot diffraction experiments with femtosecond X-ray laser pulses,” J. Appl. Crystallogr. 47, 188–197 (2014).
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Kimmel, N.

A. V. Martin, F. Wang, N. D. Loh, T. Ekeberg, F. R. N. C. Maia, M. Hantke, G. van der Schot, C. Y. Hampton, R. G. Sierra, A. Aquila, S. Bajt, M. Barthelmess, C. Bostedt, J. D. Bozek, N. Coppola, S. W. Epp, B. Erk, H. Fleckenstein, L. Foucar, M. Frank, H. Graafsma, L. Gumprecht, A. Hartmann, R. Hartmann, G. Hauser, H. Hirsemann, P. Holl, S. Kassemeyer, N. Kimmel, M. Liang, L. Lomb, S. Marchesini, K. Nass, E. Pedersoli, C. Reich, D. Rolles, B. Rudek, A. Rudenko, J. Schulz, R. L. Shoeman, H. Soltau, D. Starodub, J. Steinbrener, F. Stellato, L. Strüder, J. Ullrich, G. Weidenspointner, T. A. White, C. B. Wunderer, A. Barty, I. Schlichting, M. J. Bogan, and H. N. Chapman, “Noise-robust coherent diffractive imaging with a single diffraction pattern,” Opt. Express 20, 16650–16661 (2012).
[Crossref]

S. Kassemeyer, J. Steinbrener, L. Lomb, E. Hartmann, A. Aquila, A. Barty, A. V. Martin, C. Y. Hampton, S. Bajt, M. Barthelmess, T. R. Barends, C. Bostedt, M. Bott, J. D. Bozek, N. Coppola, M. Cryle, D. P. DePonte, R. B. Doak, S. W. Epp, B. Erk, H. Fleckenstein, L. Foucar, H. Graafsma, L. Gumprecht, A. Hartmann, R. Hartmann, G. Hauser, H. Hirsemann, A. Hömke, P. Holl, O. Jönsson, N. Kimmel, F. Krasniqi, M. Liang, F. R. Maia, S. Marchesini, K. Nass, C. Reich, D. Rolles, B. Rudek, A. Rudenko, C. Schmidt, J. Schulz, R. L. Shoeman, R. G. Sierra, H. Soltau, J. C. H. Spence, D. Starodub, F. Stellato, S. Stern, G. Stier, M. Svenda, G. Weidenspointner, U. Weierstall, T. A. White, C. Wunderer, M. Frank, H. N. Chapman, J. Ullrich, L. Strüder, M. J. Bogan, and I. Schlichting, “Femtosecond free-electron laser x-ray diffraction data sets for algorithm development,” Opt. Express 20, 4149–4158 (2012).
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Kirz, J.

J. Miao, P. Charalambous, J. Kirz, and D. Sayre, “Extending the methodology of x-ray crystallography to allow imaging of micrometre-sized non-crystalline specimens,” Nature 400, 342–344 (1999).
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Kohmura, Y.

J. Miao, Y. Nishino, Y. Kohmura, B. Johnson, C. Song, S. H. Risbud, and T. Ishikawa, “Quantitative image reconstruction of gan quantum dots from oversampled diffraction intensities alone,” Phys. Rev. Lett. 95, 085503 (2005).
[Crossref] [PubMed]

Krasniqi, F.

Larabell, C.

J. Miao, K. Hodgson, T. Ishikawa, C. Larabell, M. Legros, and Y. Nishino, “Imaging whole escherichia coli bacteria by using single-particle x-ray diffraction,” Proc. Natl. Acad. Sci. USA 100, 110–112 (2003).
[Crossref] [PubMed]

Lee, T.

C. Chen, J. Miao, C. Wang, and T. Lee, “Application of optimization technique to noncrystalline x-ray diffraction microscopy: Guided hybrid input-output method,” Phys. Rev. B 76, 064113 (2007).
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Lee, T.-K.

H. Jiang, C. Song, C.-C. Chen, R. Xu, K. S. Raines, B. P. Fahimian, C.-H. Lu, T.-K. Lee, A. Nakashima, J. Urano, T. Ishikawa, F. Tamanoi, and J. Miao, “Quantitative 3d imaging of whole, unstained cells by using x-ray diffraction microscopy,” Proc. Natl. Acad. Sci. USA 107, 11234–11239 (2010).
[Crossref] [PubMed]

Legros, M.

J. Miao, K. Hodgson, T. Ishikawa, C. Larabell, M. Legros, and Y. Nishino, “Imaging whole escherichia coli bacteria by using single-particle x-ray diffraction,” Proc. Natl. Acad. Sci. USA 100, 110–112 (2003).
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Li, X.

E. Candès, X. Li, and M. Soltanolkotabi, “Phase retrieval via wirtinger flow: Theory and algorithms,” IEEE Trans. Info. Theory 61, 1985–2007 (2015).
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S. Kassemeyer, J. Steinbrener, L. Lomb, E. Hartmann, A. Aquila, A. Barty, A. V. Martin, C. Y. Hampton, S. Bajt, M. Barthelmess, T. R. Barends, C. Bostedt, M. Bott, J. D. Bozek, N. Coppola, M. Cryle, D. P. DePonte, R. B. Doak, S. W. Epp, B. Erk, H. Fleckenstein, L. Foucar, H. Graafsma, L. Gumprecht, A. Hartmann, R. Hartmann, G. Hauser, H. Hirsemann, A. Hömke, P. Holl, O. Jönsson, N. Kimmel, F. Krasniqi, M. Liang, F. R. Maia, S. Marchesini, K. Nass, C. Reich, D. Rolles, B. Rudek, A. Rudenko, C. Schmidt, J. Schulz, R. L. Shoeman, R. G. Sierra, H. Soltau, J. C. H. Spence, D. Starodub, F. Stellato, S. Stern, G. Stier, M. Svenda, G. Weidenspointner, U. Weierstall, T. A. White, C. Wunderer, M. Frank, H. N. Chapman, J. Ullrich, L. Strüder, M. J. Bogan, and I. Schlichting, “Femtosecond free-electron laser x-ray diffraction data sets for algorithm development,” Opt. Express 20, 4149–4158 (2012).
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A. V. Martin, F. Wang, N. D. Loh, T. Ekeberg, F. R. N. C. Maia, M. Hantke, G. van der Schot, C. Y. Hampton, R. G. Sierra, A. Aquila, S. Bajt, M. Barthelmess, C. Bostedt, J. D. Bozek, N. Coppola, S. W. Epp, B. Erk, H. Fleckenstein, L. Foucar, M. Frank, H. Graafsma, L. Gumprecht, A. Hartmann, R. Hartmann, G. Hauser, H. Hirsemann, P. Holl, S. Kassemeyer, N. Kimmel, M. Liang, L. Lomb, S. Marchesini, K. Nass, E. Pedersoli, C. Reich, D. Rolles, B. Rudek, A. Rudenko, J. Schulz, R. L. Shoeman, H. Soltau, D. Starodub, J. Steinbrener, F. Stellato, L. Strüder, J. Ullrich, G. Weidenspointner, T. A. White, C. B. Wunderer, A. Barty, I. Schlichting, M. J. Bogan, and H. N. Chapman, “Noise-robust coherent diffractive imaging with a single diffraction pattern,” Opt. Express 20, 16650–16661 (2012).
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Song, C.

C. Song, K. Tono, J. Park, T. Ebisu, S. Kim, H. Shimada, S. Kim, M. Gallagher-Jones, D. Nam, T. Sato, T. Togashi, K. Ogawa, Y. Joti, T. Kameshima, S. Ono, T. Hatsui, S. Iwata, M. Yabashi, and T. Ishikawa, “Multiple application X-ray imaging chamber for single-shot diffraction experiments with femtosecond X-ray laser pulses,” J. Appl. Crystallogr. 47, 188–197 (2014).
[Crossref]

H. Jiang, C. Song, C.-C. Chen, R. Xu, K. S. Raines, B. P. Fahimian, C.-H. Lu, T.-K. Lee, A. Nakashima, J. Urano, T. Ishikawa, F. Tamanoi, and J. Miao, “Quantitative 3d imaging of whole, unstained cells by using x-ray diffraction microscopy,” Proc. Natl. Acad. Sci. USA 107, 11234–11239 (2010).
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J. Miao, Y. Nishino, Y. Kohmura, B. Johnson, C. Song, S. H. Risbud, and T. Ishikawa, “Quantitative image reconstruction of gan quantum dots from oversampled diffraction intensities alone,” Phys. Rev. Lett. 95, 085503 (2005).
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Spence, J. C. H.

Starodub, D.

S. Kassemeyer, J. Steinbrener, L. Lomb, E. Hartmann, A. Aquila, A. Barty, A. V. Martin, C. Y. Hampton, S. Bajt, M. Barthelmess, T. R. Barends, C. Bostedt, M. Bott, J. D. Bozek, N. Coppola, M. Cryle, D. P. DePonte, R. B. Doak, S. W. Epp, B. Erk, H. Fleckenstein, L. Foucar, H. Graafsma, L. Gumprecht, A. Hartmann, R. Hartmann, G. Hauser, H. Hirsemann, A. Hömke, P. Holl, O. Jönsson, N. Kimmel, F. Krasniqi, M. Liang, F. R. Maia, S. Marchesini, K. Nass, C. Reich, D. Rolles, B. Rudek, A. Rudenko, C. Schmidt, J. Schulz, R. L. Shoeman, R. G. Sierra, H. Soltau, J. C. H. Spence, D. Starodub, F. Stellato, S. Stern, G. Stier, M. Svenda, G. Weidenspointner, U. Weierstall, T. A. White, C. Wunderer, M. Frank, H. N. Chapman, J. Ullrich, L. Strüder, M. J. Bogan, and I. Schlichting, “Femtosecond free-electron laser x-ray diffraction data sets for algorithm development,” Opt. Express 20, 4149–4158 (2012).
[Crossref] [PubMed]

A. V. Martin, F. Wang, N. D. Loh, T. Ekeberg, F. R. N. C. Maia, M. Hantke, G. van der Schot, C. Y. Hampton, R. G. Sierra, A. Aquila, S. Bajt, M. Barthelmess, C. Bostedt, J. D. Bozek, N. Coppola, S. W. Epp, B. Erk, H. Fleckenstein, L. Foucar, M. Frank, H. Graafsma, L. Gumprecht, A. Hartmann, R. Hartmann, G. Hauser, H. Hirsemann, P. Holl, S. Kassemeyer, N. Kimmel, M. Liang, L. Lomb, S. Marchesini, K. Nass, E. Pedersoli, C. Reich, D. Rolles, B. Rudek, A. Rudenko, J. Schulz, R. L. Shoeman, H. Soltau, D. Starodub, J. Steinbrener, F. Stellato, L. Strüder, J. Ullrich, G. Weidenspointner, T. A. White, C. B. Wunderer, A. Barty, I. Schlichting, M. J. Bogan, and H. N. Chapman, “Noise-robust coherent diffractive imaging with a single diffraction pattern,” Opt. Express 20, 16650–16661 (2012).
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Steinbrener, J.

A. V. Martin, F. Wang, N. D. Loh, T. Ekeberg, F. R. N. C. Maia, M. Hantke, G. van der Schot, C. Y. Hampton, R. G. Sierra, A. Aquila, S. Bajt, M. Barthelmess, C. Bostedt, J. D. Bozek, N. Coppola, S. W. Epp, B. Erk, H. Fleckenstein, L. Foucar, M. Frank, H. Graafsma, L. Gumprecht, A. Hartmann, R. Hartmann, G. Hauser, H. Hirsemann, P. Holl, S. Kassemeyer, N. Kimmel, M. Liang, L. Lomb, S. Marchesini, K. Nass, E. Pedersoli, C. Reich, D. Rolles, B. Rudek, A. Rudenko, J. Schulz, R. L. Shoeman, H. Soltau, D. Starodub, J. Steinbrener, F. Stellato, L. Strüder, J. Ullrich, G. Weidenspointner, T. A. White, C. B. Wunderer, A. Barty, I. Schlichting, M. J. Bogan, and H. N. Chapman, “Noise-robust coherent diffractive imaging with a single diffraction pattern,” Opt. Express 20, 16650–16661 (2012).
[Crossref]

S. Kassemeyer, J. Steinbrener, L. Lomb, E. Hartmann, A. Aquila, A. Barty, A. V. Martin, C. Y. Hampton, S. Bajt, M. Barthelmess, T. R. Barends, C. Bostedt, M. Bott, J. D. Bozek, N. Coppola, M. Cryle, D. P. DePonte, R. B. Doak, S. W. Epp, B. Erk, H. Fleckenstein, L. Foucar, H. Graafsma, L. Gumprecht, A. Hartmann, R. Hartmann, G. Hauser, H. Hirsemann, A. Hömke, P. Holl, O. Jönsson, N. Kimmel, F. Krasniqi, M. Liang, F. R. Maia, S. Marchesini, K. Nass, C. Reich, D. Rolles, B. Rudek, A. Rudenko, C. Schmidt, J. Schulz, R. L. Shoeman, R. G. Sierra, H. Soltau, J. C. H. Spence, D. Starodub, F. Stellato, S. Stern, G. Stier, M. Svenda, G. Weidenspointner, U. Weierstall, T. A. White, C. Wunderer, M. Frank, H. N. Chapman, J. Ullrich, L. Strüder, M. J. Bogan, and I. Schlichting, “Femtosecond free-electron laser x-ray diffraction data sets for algorithm development,” Opt. Express 20, 4149–4158 (2012).
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Stellato, F.

S. Kassemeyer, J. Steinbrener, L. Lomb, E. Hartmann, A. Aquila, A. Barty, A. V. Martin, C. Y. Hampton, S. Bajt, M. Barthelmess, T. R. Barends, C. Bostedt, M. Bott, J. D. Bozek, N. Coppola, M. Cryle, D. P. DePonte, R. B. Doak, S. W. Epp, B. Erk, H. Fleckenstein, L. Foucar, H. Graafsma, L. Gumprecht, A. Hartmann, R. Hartmann, G. Hauser, H. Hirsemann, A. Hömke, P. Holl, O. Jönsson, N. Kimmel, F. Krasniqi, M. Liang, F. R. Maia, S. Marchesini, K. Nass, C. Reich, D. Rolles, B. Rudek, A. Rudenko, C. Schmidt, J. Schulz, R. L. Shoeman, R. G. Sierra, H. Soltau, J. C. H. Spence, D. Starodub, F. Stellato, S. Stern, G. Stier, M. Svenda, G. Weidenspointner, U. Weierstall, T. A. White, C. Wunderer, M. Frank, H. N. Chapman, J. Ullrich, L. Strüder, M. J. Bogan, and I. Schlichting, “Femtosecond free-electron laser x-ray diffraction data sets for algorithm development,” Opt. Express 20, 4149–4158 (2012).
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A. V. Martin, F. Wang, N. D. Loh, T. Ekeberg, F. R. N. C. Maia, M. Hantke, G. van der Schot, C. Y. Hampton, R. G. Sierra, A. Aquila, S. Bajt, M. Barthelmess, C. Bostedt, J. D. Bozek, N. Coppola, S. W. Epp, B. Erk, H. Fleckenstein, L. Foucar, M. Frank, H. Graafsma, L. Gumprecht, A. Hartmann, R. Hartmann, G. Hauser, H. Hirsemann, P. Holl, S. Kassemeyer, N. Kimmel, M. Liang, L. Lomb, S. Marchesini, K. Nass, E. Pedersoli, C. Reich, D. Rolles, B. Rudek, A. Rudenko, J. Schulz, R. L. Shoeman, H. Soltau, D. Starodub, J. Steinbrener, F. Stellato, L. Strüder, J. Ullrich, G. Weidenspointner, T. A. White, C. B. Wunderer, A. Barty, I. Schlichting, M. J. Bogan, and H. N. Chapman, “Noise-robust coherent diffractive imaging with a single diffraction pattern,” Opt. Express 20, 16650–16661 (2012).
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Stern, S.

Stier, G.

Strohmer, T.

E. Candès, T. Strohmer, and V. Voroninski, “Phaselift: Exact and stable signal recovery from magnitude measurements via convex programming,” Comm. Pure Appl. Math. 66, 1241–1274 (2013).
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E. Candès, Y. Eldar, T. Strohmer, and V. Voroninski, “Phase retrieval via matrix completion,” SIAM J. Imaging Sci. 6, 199–225 (2013).
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Strüder, L.

A. V. Martin, F. Wang, N. D. Loh, T. Ekeberg, F. R. N. C. Maia, M. Hantke, G. van der Schot, C. Y. Hampton, R. G. Sierra, A. Aquila, S. Bajt, M. Barthelmess, C. Bostedt, J. D. Bozek, N. Coppola, S. W. Epp, B. Erk, H. Fleckenstein, L. Foucar, M. Frank, H. Graafsma, L. Gumprecht, A. Hartmann, R. Hartmann, G. Hauser, H. Hirsemann, P. Holl, S. Kassemeyer, N. Kimmel, M. Liang, L. Lomb, S. Marchesini, K. Nass, E. Pedersoli, C. Reich, D. Rolles, B. Rudek, A. Rudenko, J. Schulz, R. L. Shoeman, H. Soltau, D. Starodub, J. Steinbrener, F. Stellato, L. Strüder, J. Ullrich, G. Weidenspointner, T. A. White, C. B. Wunderer, A. Barty, I. Schlichting, M. J. Bogan, and H. N. Chapman, “Noise-robust coherent diffractive imaging with a single diffraction pattern,” Opt. Express 20, 16650–16661 (2012).
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S. Kassemeyer, J. Steinbrener, L. Lomb, E. Hartmann, A. Aquila, A. Barty, A. V. Martin, C. Y. Hampton, S. Bajt, M. Barthelmess, T. R. Barends, C. Bostedt, M. Bott, J. D. Bozek, N. Coppola, M. Cryle, D. P. DePonte, R. B. Doak, S. W. Epp, B. Erk, H. Fleckenstein, L. Foucar, H. Graafsma, L. Gumprecht, A. Hartmann, R. Hartmann, G. Hauser, H. Hirsemann, A. Hömke, P. Holl, O. Jönsson, N. Kimmel, F. Krasniqi, M. Liang, F. R. Maia, S. Marchesini, K. Nass, C. Reich, D. Rolles, B. Rudek, A. Rudenko, C. Schmidt, J. Schulz, R. L. Shoeman, R. G. Sierra, H. Soltau, J. C. H. Spence, D. Starodub, F. Stellato, S. Stern, G. Stier, M. Svenda, G. Weidenspointner, U. Weierstall, T. A. White, C. Wunderer, M. Frank, H. N. Chapman, J. Ullrich, L. Strüder, M. J. Bogan, and I. Schlichting, “Femtosecond free-electron laser x-ray diffraction data sets for algorithm development,” Opt. Express 20, 4149–4158 (2012).
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Svenda, M.

Tamanoi, F.

H. Jiang, C. Song, C.-C. Chen, R. Xu, K. S. Raines, B. P. Fahimian, C.-H. Lu, T.-K. Lee, A. Nakashima, J. Urano, T. Ishikawa, F. Tamanoi, and J. Miao, “Quantitative 3d imaging of whole, unstained cells by using x-ray diffraction microscopy,” Proc. Natl. Acad. Sci. USA 107, 11234–11239 (2010).
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Togashi, T.

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Ullrich, J.

A. V. Martin, F. Wang, N. D. Loh, T. Ekeberg, F. R. N. C. Maia, M. Hantke, G. van der Schot, C. Y. Hampton, R. G. Sierra, A. Aquila, S. Bajt, M. Barthelmess, C. Bostedt, J. D. Bozek, N. Coppola, S. W. Epp, B. Erk, H. Fleckenstein, L. Foucar, M. Frank, H. Graafsma, L. Gumprecht, A. Hartmann, R. Hartmann, G. Hauser, H. Hirsemann, P. Holl, S. Kassemeyer, N. Kimmel, M. Liang, L. Lomb, S. Marchesini, K. Nass, E. Pedersoli, C. Reich, D. Rolles, B. Rudek, A. Rudenko, J. Schulz, R. L. Shoeman, H. Soltau, D. Starodub, J. Steinbrener, F. Stellato, L. Strüder, J. Ullrich, G. Weidenspointner, T. A. White, C. B. Wunderer, A. Barty, I. Schlichting, M. J. Bogan, and H. N. Chapman, “Noise-robust coherent diffractive imaging with a single diffraction pattern,” Opt. Express 20, 16650–16661 (2012).
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Urano, J.

H. Jiang, C. Song, C.-C. Chen, R. Xu, K. S. Raines, B. P. Fahimian, C.-H. Lu, T.-K. Lee, A. Nakashima, J. Urano, T. Ishikawa, F. Tamanoi, and J. Miao, “Quantitative 3d imaging of whole, unstained cells by using x-ray diffraction microscopy,” Proc. Natl. Acad. Sci. USA 107, 11234–11239 (2010).
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van der Schot, G.

Voroninski, V.

E. Candès, Y. Eldar, T. Strohmer, and V. Voroninski, “Phase retrieval via matrix completion,” SIAM J. Imaging Sci. 6, 199–225 (2013).
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E. Candès, T. Strohmer, and V. Voroninski, “Phaselift: Exact and stable signal recovery from magnitude measurements via convex programming,” Comm. Pure Appl. Math. 66, 1241–1274 (2013).
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C. Chen, J. Miao, C. Wang, and T. Lee, “Application of optimization technique to noncrystalline x-ray diffraction microscopy: Guided hybrid input-output method,” Phys. Rev. B 76, 064113 (2007).
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Yabashi, M.

C. Song, K. Tono, J. Park, T. Ebisu, S. Kim, H. Shimada, S. Kim, M. Gallagher-Jones, D. Nam, T. Sato, T. Togashi, K. Ogawa, Y. Joti, T. Kameshima, S. Ono, T. Hatsui, S. Iwata, M. Yabashi, and T. Ishikawa, “Multiple application X-ray imaging chamber for single-shot diffraction experiments with femtosecond X-ray laser pulses,” J. Appl. Crystallogr. 47, 188–197 (2014).
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Figures (8)

Fig. 1:
Fig. 1: Flowchart of GPS algorithm for stage 1. For simplicity, we denote ϵ = σ σ + 1 in Eq. (14). The input z0 is initialized with random phases and magnitudes are equal the measurements, while y0 is a zero vector. The proj S * and the convolution ∗ follows Eq. (19) and Eq. (20) respectively. zk and yk with the smallest RF will be fed as an input into the next stage where a Gaussian kernel G γ with larger γ is applied.
Fig. 2:
Fig. 2: First row: vesicle model with log-scaled diffraction pattern (left), zoom-in image (center) and residual (right). Second row: HIO: RF = 12.87%, Rreal = 21.14%. Third row: OSS: RF = 6.08%, Rreal = 3.59%. Fourth row: GPS-R RF = 5.90%, Rreal = 2.85%. Fifth row: GPS-F RF = 5.89% and Rreal = 0.7%. Third column: residual.
Fig. 3:
Fig. 3: Histogram (first row), and convergence (second row) of RF and Rreal on Vesicle data using HIO, OSS, GPS. GPS consistently produces smaller RF and Rreal than HIO or OSS. Moreover, GPS converges fastest with the fewest oscillations.
Fig. 4:
Fig. 4: The (scaled) means and variances of the best five of 500 independent reconstructions by HIO, OSS, GPS-F and GPS-R
Fig. 5:
Fig. 5: S. pombe yeast spore log-scaled diffraction pattern, size 500 × 500 (top). Means (first row) and variance (second row) are computed from the best 5 of 500 independent reconstructions. HIO: RF = 15.697% ± 0.526%, OSS: RF = 9.775% ± 0.202%, and GPS: RF = 8.672% ± 0.025%.
Fig. 6:
Fig. 6: Top: histogram of RF in 500 independent runs (top). Bottom: the convergence curves of a singe construction of RF on S. pombe yeast spore by GPS-RF, OSS, and HIO.
Fig. 7:
Fig. 7: Diffraction pattern of nanorice1 and nanorice2 253 × 253 (first column) and reconstructions using OSS: RF = 18.23%, 16.32% and GPS-F: RF = 17.40%, 15.83% respectively. GPS obtains less noise on the boundary and lower relative error RF.
Fig. 8:
Fig. 8: Histogram (first row) and convergence (second row) of OSS and GPS-F on nanorice1 (first column) and nanorice2 (second column). The results of HIO are omitted due to lack of convergence.

Tables (2)

Tables Icon

Algorithm 1 GPS-R: GPS with smoothing in real space.

Tables Icon

Algorithm 2 GPS-F: GPS with smoothing in Fourier space.

Equations (41)

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find u n , such that u S T .
min u n f ( u ) + g ( u ) ,
min u , z n f ( u ) + g ( z ) subject to z = u .
g σ ( z ) = 1 2 σ | z | b 2 .
f G ( u ) : = inf v { f ( v ) + 1 2 v u G 1 2 } = inf v S 1 2 v u G 1 2
min u , z n f G ( u ) + g σ ( z ) subject to z = u .
( u , z ; y ) = f G ( u ) + g σ ( z ) + Re y , * z u ,
y f G ( u ) , y g σ ( z ) * z = u f G * ( y ) , y g σ ( z ) ,
min z max y g σ ( z ) f G * ( y ) + Re z , y ,
f G * ( y ) = f * ( y ) + 1 2 y G 2 ,
f * ( y ) = sup v S Re y , v = { 0 if Re ( y ) 0 on S , otherwise .
min z max y S * 1 2 σ | z | b 2 + 1 2 y G 2 + Re z , y .
{ z k + 1 = prox t g σ ( z k t y k ) y k + 1 = prox s f G * ( y k + s 1 ( 2 z k + 1 z k ) )
prox t g σ ( z ) = arg min v n 1 2 σ | v | b 2 + 1 2 t v z 2 = b exp ( i arg ( z ) ) + ( σ / t ) z 1 + σ / t ,
prox s f G * ( y ) = arg min v n f G * ( v ) + 1 2 s v y 2 = arg min v n f * ( v ) + 1 2 v G 2 + 2 2 s v y 2 = arg min v S * 1 2 v G 2 + 2 2 s v y 2 .
f G * ( y ) = f * ( y ) + γ 2 D y 2 ,
prox s f G * ( y ) ( I + s γ D D ) 1 proj S * ( y ) .
y k + 1 = G s γ proj S * ( y k + s 1 ( 2 z k + 1 z k ) ) ,
proj S * ( y ) i = { Re ( y i ) + i Im ( y i ) if i S , y i otherwise .
G γ u = Re { 1 ( W γ u ) } .
R F ( u k ) = i | | u k | i b i | i b i .
f G ( u ) = { i | u proj S ( u ) | i 2 2 γ r i 2 if u i 0 at the center , otherwise ,
| d d ξ u ^ ( ξ ) | 2 d ξ = | x u ( x ) | 2 d x ,
D u 2 = r u 2 ,
f G * ( y ) = f * ( y ) + γ 2 r y 2 = f * ( y ) + γ 2 D y 2 .
prox s f G * ( y ) = 1 1 + s γ r 2 proj S * ( y ) exp ( s γ r 2 ) proj S * ( y ) .
y k + 1 = W s γ 4 π 2 proj S * ( y k + s 1 ( 2 z k + 1 z k ) ) .
z i k + 1 = { ( prox t g σ ( z k t y k ) ) i if b i is known , ( z k t y k ) i otherwise .
b i = Poisson ( | u o | i 2 flux u o 2 ) + N ( 0 , σ ) ,
R n o i s e = i | b i | u o | i | i | u o | i ,
R r e a l = i | u i k u i o | i u i o .
h G ( u ) : = inf v n { h ( v ) + 1 2 v u G 1 2 } .
h t ( u ) : = inf v n { h ( v ) + 1 2 t v u 2 } .
h t ( u ) : = 1 2 t inf v X v u 2 = 1 2 t u proj X ( u ) 2
g σ ( z ) : = 1 2 σ inf | v | = b v z 2 .
g σ ( z ) = 1 2 σ b exp ( i arg ( z ) ) z 2 = 1 2 σ ( b | z | ) exp ( i arg ( z ) ) 2 = 1 2 σ | z | b 2
h ( u ) : = h ˜ ( R e ( u ) , Im ( u ) ) = h ˜ ( u ˜ ) ,
Re u , y = Re ( u ) , Re ( y ) + Im ( u ) , Im ( y ) = u ˜ , y ˜ .
h * ( y ) : = s u p u n { Re y , u h ( u ) } .
f G * ( y ) = f * ( y ) + ( 1 2 G 1 2 ) * ( y ) ,
f G ( u ) = f G * * ( u ) = ( f G * ( y ) ) * = sup y n { Re u , y f G * ( y ) } = sup y n { Re u , y f * ( y ) 1 2 y G 2 } = sup y S * { Re u , y 1 2 y G 2 } .

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