Abstract

Iterative algorithms with feedback are among the most powerful and versatile optimization methods for phase retrieval. Among these, the hybrid input–output algorithm has demonstrated practical solutions to giga-element nonlinear phase retrieval problems, escaping local minima and producing images at resolutions beyond the capabilities of lens-based optical methods. Here the input–output iteration is improved by a lower-dimensional subspace saddle-point optimization.

© 2007 Optical Society of America

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2007 (2)

S. Marchesini, "A unified evaluation of iterative projection algorithms for phase retrieval," Rev. Sci. Instrum. 78, 011301 1-10 (2007).
[CrossRef]

V. Elser, I. Rankenburg, and P. Thibault, "Searching with iterated maps," Proc. Natl. Acad. Sci. U.S.A. 104, 418-423 (2007).
[CrossRef] [PubMed]

2006 (5)

V. Elser and I. Rankenburg, "Deconstructing the energy landscape: Constraint-based algorithms for folding heteropolymers," Phys. Rev. E 73, 026702 1-8 (2006).
[CrossRef]

J. Miao, C-C. Chen, C. Song, Y. Nishino, Y. Kohmura, T. Ishikawa, D. Ramunno-Johnson, T-K. Lee, and S. H. Risbud, "Three-dimensional GaN-Ga2O3 core shell structure revealed by x-ray diffraction microscopy," Phys. Rev. Lett. 97, 215503-1-4 (2006).
[CrossRef]

M. A. Pfeifer, G. J. Williams, I. A. Vartanyants, R. Harder, and I. K. Robinson, "Three-dimensional mapping of a deformation field inside a nanocrystal," Nature (London) 442, 63-67 (2006).
[CrossRef]

H. N. Chapman, A. Barty, S. Marchesini, A. Noy, C. Cui, M. R. Howells, R. Rosen, H. He, J. C. H. Spence, U. Weierstall, T. Beetz, C. Jacobsen, and D. Shapiro, "High-resolution ab initio three-dimensional X-ray diffraction microscopy," J. Opt. Soc. Am. A 23, 1179-1200 (2006).
[CrossRef]

H. N. Chapman, A. Barty, M. J. Bogan, S. Boutet, M. Frank, S. P. Hau-Riege, S. Marchesini, B. W. Woods, S. Bajt, W. H. Benner, R. A. London, E. Plönjes, M. Kuhlmann, R. Treusch, S. Düterer, T. Tschentscher, J. R. Schneider, E. Spiller, T. Möller, C. Bostedt, M. Hoener, D. A. Shapiro, K. O. Hodgson, D. Van Der Spoel, F. Burmeister, M. Bergh, C. Caleman, Gösta Huldt, M. M. Seibert, F. R. N. C. Maia, R. W. Lee, A. Szöke, N. Timneanu, and J. Hajdu, "Femtosecond diffractive imaging with a soft-X-ray free-electron laser," Nat. Phys. 2, 789-862 (2006).
[CrossRef]

2005 (3)

D. Shapiro, P. Thibault, T. Beetz, V. Elser, M. Howells, C. Jacobsen, J. Kirz, E. Lima, H. Miao, A. Neiman, and D. Sayre, "Biological imaging by soft x-ray diffraction microscopy," Proc. Natl. Acad. Sci. U.S.A. 102, 1543-1546 (2005).
[CrossRef]

D. R. Luke, "Relaxed averaged alternating reflections for diffraction imaging," Inverse Probl. 21, 37-50 (2005).
[CrossRef]

G. Oszlányi and A. Süto, "Ab initio structure solution by charge flipping. II. Use of weak reflections," Acta Crystallogr. A61, 147-152 (2005).

2003 (3)

H. H. Bauschke, P. L. Combettes, and D. R. Luke, "Hybrid projection-reflection method for phase retrieval," J. Opt. Soc. Am. A 20, 1025-1034 (2003).
[CrossRef]

V. Elser, "Phase retrieval by iterated projections," J. Opt. Soc. Am. A 20, 40-55 (2003).
[CrossRef]

S. Marchesini, H. He, H. N. Chapman, S. P. Hau-Riege, A. Noy, M. R. Howells, U. Weierstall, and J. C. H. Spence, "X-ray image reconstruction from a diffraction pattern alone," Phys. Rev. B 68, 140101(R) (2003).
[CrossRef]

2002 (2)

1999 (2)

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 (London) 400, 342-344 (1999).
[CrossRef]

T. Isernia, G. Leone, R. Pierri, and F. Soldovieri, "Role of support information and zero locations in phase retrieval by a quadratic approach," J. Opt. Soc. Am. A 16, 1845-1856 (1999).
[CrossRef]

1996 (1)

1986 (1)

1984 (2)

M. J. D. Powell, "Nonconvex minimization calculations and the conjugate gradient method," Lect. Notes Math. 1066, 122-141 (1984).
[CrossRef]

A. Levi and H. Stark, "Image restoration by the method of generalized projections with application to restoration from magnitude," J. Opt. Soc. Am. A 1, 932-943 (1984).
[CrossRef]

1982 (2)

J. R. Fienup, "Phase retrieval algorithms: a comparison," Appl. Opt. 21, 2758-2769 (1982).
[CrossRef] [PubMed]

D. C. Youla and H. Webb, "Image restoration by the method of projections onto convex sets Part I," IEEE Trans. Med. Imaging TMI-1, 81-94 (1982).
[CrossRef]

1978 (1)

1972 (1)

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

Artyukhin, A. B.

A. Barty, S. Marchesini, H. N. Chapman, C. Cui, M. R. Howells, D. A. Shapiro, A. M. Minor, J. C. H. Spence,U. Weierstall, J. Ilavsky, A. Noy, S. P. Hau-Riege,A. B. Artyukhin, T. Baumann, T. Willey, J. Stolken, T. van Buuren, and J. H. Kinney, "Three-dimensional coherent X-ray diffraction imaging of a ceramic nanofoam: determination of structural deformation mechanisms," http://arxiv.org/abs/0708.4035.

Bajt, S.

H. N. Chapman, A. Barty, M. J. Bogan, S. Boutet, M. Frank, S. P. Hau-Riege, S. Marchesini, B. W. Woods, S. Bajt, W. H. Benner, R. A. London, E. Plönjes, M. Kuhlmann, R. Treusch, S. Düterer, T. Tschentscher, J. R. Schneider, E. Spiller, T. Möller, C. Bostedt, M. Hoener, D. A. Shapiro, K. O. Hodgson, D. Van Der Spoel, F. Burmeister, M. Bergh, C. Caleman, Gösta Huldt, M. M. Seibert, F. R. N. C. Maia, R. W. Lee, A. Szöke, N. Timneanu, and J. Hajdu, "Femtosecond diffractive imaging with a soft-X-ray free-electron laser," Nat. Phys. 2, 789-862 (2006).
[CrossRef]

Barty, A.

H. N. Chapman, A. Barty, M. J. Bogan, S. Boutet, M. Frank, S. P. Hau-Riege, S. Marchesini, B. W. Woods, S. Bajt, W. H. Benner, R. A. London, E. Plönjes, M. Kuhlmann, R. Treusch, S. Düterer, T. Tschentscher, J. R. Schneider, E. Spiller, T. Möller, C. Bostedt, M. Hoener, D. A. Shapiro, K. O. Hodgson, D. Van Der Spoel, F. Burmeister, M. Bergh, C. Caleman, Gösta Huldt, M. M. Seibert, F. R. N. C. Maia, R. W. Lee, A. Szöke, N. Timneanu, and J. Hajdu, "Femtosecond diffractive imaging with a soft-X-ray free-electron laser," Nat. Phys. 2, 789-862 (2006).
[CrossRef]

H. N. Chapman, A. Barty, S. Marchesini, A. Noy, C. Cui, M. R. Howells, R. Rosen, H. He, J. C. H. Spence, U. Weierstall, T. Beetz, C. Jacobsen, and D. Shapiro, "High-resolution ab initio three-dimensional X-ray diffraction microscopy," J. Opt. Soc. Am. A 23, 1179-1200 (2006).
[CrossRef]

A. Barty, S. Marchesini, H. N. Chapman, C. Cui, M. R. Howells, D. A. Shapiro, A. M. Minor, J. C. H. Spence,U. Weierstall, J. Ilavsky, A. Noy, S. P. Hau-Riege,A. B. Artyukhin, T. Baumann, T. Willey, J. Stolken, T. van Buuren, and J. H. Kinney, "Three-dimensional coherent X-ray diffraction imaging of a ceramic nanofoam: determination of structural deformation mechanisms," http://arxiv.org/abs/0708.4035.

S. Marchesini, H. N. Chapman, A. Barty, C. Cui, M. R. Howells, J. C. H. Spence, U. Weierstall, and A. M. Minor, "Phase aberrations in diffraction microscopy," in Proceedings of the 8th International Conference on X-ray Microscopy, IPAP Conference Series 7, 380-382 (2006).

Baumann, T.

A. Barty, S. Marchesini, H. N. Chapman, C. Cui, M. R. Howells, D. A. Shapiro, A. M. Minor, J. C. H. Spence,U. Weierstall, J. Ilavsky, A. Noy, S. P. Hau-Riege,A. B. Artyukhin, T. Baumann, T. Willey, J. Stolken, T. van Buuren, and J. H. Kinney, "Three-dimensional coherent X-ray diffraction imaging of a ceramic nanofoam: determination of structural deformation mechanisms," http://arxiv.org/abs/0708.4035.

Bauschke, H. H.

Beetz, T.

H. N. Chapman, A. Barty, S. Marchesini, A. Noy, C. Cui, M. R. Howells, R. Rosen, H. He, J. C. H. Spence, U. Weierstall, T. Beetz, C. Jacobsen, and D. Shapiro, "High-resolution ab initio three-dimensional X-ray diffraction microscopy," J. Opt. Soc. Am. A 23, 1179-1200 (2006).
[CrossRef]

D. Shapiro, P. Thibault, T. Beetz, V. Elser, M. Howells, C. Jacobsen, J. Kirz, E. Lima, H. Miao, A. Neiman, and D. Sayre, "Biological imaging by soft x-ray diffraction microscopy," Proc. Natl. Acad. Sci. U.S.A. 102, 1543-1546 (2005).
[CrossRef]

Benner, W. H.

H. N. Chapman, A. Barty, M. J. Bogan, S. Boutet, M. Frank, S. P. Hau-Riege, S. Marchesini, B. W. Woods, S. Bajt, W. H. Benner, R. A. London, E. Plönjes, M. Kuhlmann, R. Treusch, S. Düterer, T. Tschentscher, J. R. Schneider, E. Spiller, T. Möller, C. Bostedt, M. Hoener, D. A. Shapiro, K. O. Hodgson, D. Van Der Spoel, F. Burmeister, M. Bergh, C. Caleman, Gösta Huldt, M. M. Seibert, F. R. N. C. Maia, R. W. Lee, A. Szöke, N. Timneanu, and J. Hajdu, "Femtosecond diffractive imaging with a soft-X-ray free-electron laser," Nat. Phys. 2, 789-862 (2006).
[CrossRef]

Bergh, M.

H. N. Chapman, A. Barty, M. J. Bogan, S. Boutet, M. Frank, S. P. Hau-Riege, S. Marchesini, B. W. Woods, S. Bajt, W. H. Benner, R. A. London, E. Plönjes, M. Kuhlmann, R. Treusch, S. Düterer, T. Tschentscher, J. R. Schneider, E. Spiller, T. Möller, C. Bostedt, M. Hoener, D. A. Shapiro, K. O. Hodgson, D. Van Der Spoel, F. Burmeister, M. Bergh, C. Caleman, Gösta Huldt, M. M. Seibert, F. R. N. C. Maia, R. W. Lee, A. Szöke, N. Timneanu, and J. Hajdu, "Femtosecond diffractive imaging with a soft-X-ray free-electron laser," Nat. Phys. 2, 789-862 (2006).
[CrossRef]

Bogan, M. J.

H. N. Chapman, A. Barty, M. J. Bogan, S. Boutet, M. Frank, S. P. Hau-Riege, S. Marchesini, B. W. Woods, S. Bajt, W. H. Benner, R. A. London, E. Plönjes, M. Kuhlmann, R. Treusch, S. Düterer, T. Tschentscher, J. R. Schneider, E. Spiller, T. Möller, C. Bostedt, M. Hoener, D. A. Shapiro, K. O. Hodgson, D. Van Der Spoel, F. Burmeister, M. Bergh, C. Caleman, Gösta Huldt, M. M. Seibert, F. R. N. C. Maia, R. W. Lee, A. Szöke, N. Timneanu, and J. Hajdu, "Femtosecond diffractive imaging with a soft-X-ray free-electron laser," Nat. Phys. 2, 789-862 (2006).
[CrossRef]

Bostedt, C.

H. N. Chapman, A. Barty, M. J. Bogan, S. Boutet, M. Frank, S. P. Hau-Riege, S. Marchesini, B. W. Woods, S. Bajt, W. H. Benner, R. A. London, E. Plönjes, M. Kuhlmann, R. Treusch, S. Düterer, T. Tschentscher, J. R. Schneider, E. Spiller, T. Möller, C. Bostedt, M. Hoener, D. A. Shapiro, K. O. Hodgson, D. Van Der Spoel, F. Burmeister, M. Bergh, C. Caleman, Gösta Huldt, M. M. Seibert, F. R. N. C. Maia, R. W. Lee, A. Szöke, N. Timneanu, and J. Hajdu, "Femtosecond diffractive imaging with a soft-X-ray free-electron laser," Nat. Phys. 2, 789-862 (2006).
[CrossRef]

Boutet, S.

H. N. Chapman, A. Barty, M. J. Bogan, S. Boutet, M. Frank, S. P. Hau-Riege, S. Marchesini, B. W. Woods, S. Bajt, W. H. Benner, R. A. London, E. Plönjes, M. Kuhlmann, R. Treusch, S. Düterer, T. Tschentscher, J. R. Schneider, E. Spiller, T. Möller, C. Bostedt, M. Hoener, D. A. Shapiro, K. O. Hodgson, D. Van Der Spoel, F. Burmeister, M. Bergh, C. Caleman, Gösta Huldt, M. M. Seibert, F. R. N. C. Maia, R. W. Lee, A. Szöke, N. Timneanu, and J. Hajdu, "Femtosecond diffractive imaging with a soft-X-ray free-electron laser," Nat. Phys. 2, 789-862 (2006).
[CrossRef]

Burke, J. V.

D. R. Luke, J. V. Burke, and R. G. Lyon, "Optical wavefront reconstruction: theory and numerical methods," SIAM Rev. 44, 169-224 (2002).
[CrossRef]

Burmeister, F.

H. N. Chapman, A. Barty, M. J. Bogan, S. Boutet, M. Frank, S. P. Hau-Riege, S. Marchesini, B. W. Woods, S. Bajt, W. H. Benner, R. A. London, E. Plönjes, M. Kuhlmann, R. Treusch, S. Düterer, T. Tschentscher, J. R. Schneider, E. Spiller, T. Möller, C. Bostedt, M. Hoener, D. A. Shapiro, K. O. Hodgson, D. Van Der Spoel, F. Burmeister, M. Bergh, C. Caleman, Gösta Huldt, M. M. Seibert, F. R. N. C. Maia, R. W. Lee, A. Szöke, N. Timneanu, and J. Hajdu, "Femtosecond diffractive imaging with a soft-X-ray free-electron laser," Nat. Phys. 2, 789-862 (2006).
[CrossRef]

Caleman, C.

H. N. Chapman, A. Barty, M. J. Bogan, S. Boutet, M. Frank, S. P. Hau-Riege, S. Marchesini, B. W. Woods, S. Bajt, W. H. Benner, R. A. London, E. Plönjes, M. Kuhlmann, R. Treusch, S. Düterer, T. Tschentscher, J. R. Schneider, E. Spiller, T. Möller, C. Bostedt, M. Hoener, D. A. Shapiro, K. O. Hodgson, D. Van Der Spoel, F. Burmeister, M. Bergh, C. Caleman, Gösta Huldt, M. M. Seibert, F. R. N. C. Maia, R. W. Lee, A. Szöke, N. Timneanu, and J. Hajdu, "Femtosecond diffractive imaging with a soft-X-ray free-electron laser," Nat. Phys. 2, 789-862 (2006).
[CrossRef]

Chapman, H. N.

H. N. Chapman, A. Barty, M. J. Bogan, S. Boutet, M. Frank, S. P. Hau-Riege, S. Marchesini, B. W. Woods, S. Bajt, W. H. Benner, R. A. London, E. Plönjes, M. Kuhlmann, R. Treusch, S. Düterer, T. Tschentscher, J. R. Schneider, E. Spiller, T. Möller, C. Bostedt, M. Hoener, D. A. Shapiro, K. O. Hodgson, D. Van Der Spoel, F. Burmeister, M. Bergh, C. Caleman, Gösta Huldt, M. M. Seibert, F. R. N. C. Maia, R. W. Lee, A. Szöke, N. Timneanu, and J. Hajdu, "Femtosecond diffractive imaging with a soft-X-ray free-electron laser," Nat. Phys. 2, 789-862 (2006).
[CrossRef]

H. N. Chapman, A. Barty, S. Marchesini, A. Noy, C. Cui, M. R. Howells, R. Rosen, H. He, J. C. H. Spence, U. Weierstall, T. Beetz, C. Jacobsen, and D. Shapiro, "High-resolution ab initio three-dimensional X-ray diffraction microscopy," J. Opt. Soc. Am. A 23, 1179-1200 (2006).
[CrossRef]

S. Marchesini, H. He, H. N. Chapman, S. P. Hau-Riege, A. Noy, M. R. Howells, U. Weierstall, and J. C. H. Spence, "X-ray image reconstruction from a diffraction pattern alone," Phys. Rev. B 68, 140101(R) (2003).
[CrossRef]

S. Marchesini, H. N. Chapman, A. Barty, C. Cui, M. R. Howells, J. C. H. Spence, U. Weierstall, and A. M. Minor, "Phase aberrations in diffraction microscopy," in Proceedings of the 8th International Conference on X-ray Microscopy, IPAP Conference Series 7, 380-382 (2006).

A. Barty, S. Marchesini, H. N. Chapman, C. Cui, M. R. Howells, D. A. Shapiro, A. M. Minor, J. C. H. Spence,U. Weierstall, J. Ilavsky, A. Noy, S. P. Hau-Riege,A. B. Artyukhin, T. Baumann, T. Willey, J. Stolken, T. van Buuren, and J. H. Kinney, "Three-dimensional coherent X-ray diffraction imaging of a ceramic nanofoam: determination of structural deformation mechanisms," http://arxiv.org/abs/0708.4035.

Charalambous, P.

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 (London) 400, 342-344 (1999).
[CrossRef]

Chen, C-C.

J. Miao, C-C. Chen, C. Song, Y. Nishino, Y. Kohmura, T. Ishikawa, D. Ramunno-Johnson, T-K. Lee, and S. H. Risbud, "Three-dimensional GaN-Ga2O3 core shell structure revealed by x-ray diffraction microscopy," Phys. Rev. Lett. 97, 215503-1-4 (2006).
[CrossRef]

Chen, P-T.

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Cui, C.

H. N. Chapman, A. Barty, S. Marchesini, A. Noy, C. Cui, M. R. Howells, R. Rosen, H. He, J. C. H. Spence, U. Weierstall, T. Beetz, C. Jacobsen, and D. Shapiro, "High-resolution ab initio three-dimensional X-ray diffraction microscopy," J. Opt. Soc. Am. A 23, 1179-1200 (2006).
[CrossRef]

S. Marchesini, H. N. Chapman, A. Barty, C. Cui, M. R. Howells, J. C. H. Spence, U. Weierstall, and A. M. Minor, "Phase aberrations in diffraction microscopy," in Proceedings of the 8th International Conference on X-ray Microscopy, IPAP Conference Series 7, 380-382 (2006).

A. Barty, S. Marchesini, H. N. Chapman, C. Cui, M. R. Howells, D. A. Shapiro, A. M. Minor, J. C. H. Spence,U. Weierstall, J. Ilavsky, A. Noy, S. P. Hau-Riege,A. B. Artyukhin, T. Baumann, T. Willey, J. Stolken, T. van Buuren, and J. H. Kinney, "Three-dimensional coherent X-ray diffraction imaging of a ceramic nanofoam: determination of structural deformation mechanisms," http://arxiv.org/abs/0708.4035.

Düterer, S.

H. N. Chapman, A. Barty, M. J. Bogan, S. Boutet, M. Frank, S. P. Hau-Riege, S. Marchesini, B. W. Woods, S. Bajt, W. H. Benner, R. A. London, E. Plönjes, M. Kuhlmann, R. Treusch, S. Düterer, T. Tschentscher, J. R. Schneider, E. Spiller, T. Möller, C. Bostedt, M. Hoener, D. A. Shapiro, K. O. Hodgson, D. Van Der Spoel, F. Burmeister, M. Bergh, C. Caleman, Gösta Huldt, M. M. Seibert, F. R. N. C. Maia, R. W. Lee, A. Szöke, N. Timneanu, and J. Hajdu, "Femtosecond diffractive imaging with a soft-X-ray free-electron laser," Nat. Phys. 2, 789-862 (2006).
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D. Shapiro, P. Thibault, T. Beetz, V. Elser, M. Howells, C. Jacobsen, J. Kirz, E. Lima, H. Miao, A. Neiman, and D. Sayre, "Biological imaging by soft x-ray diffraction microscopy," Proc. Natl. Acad. Sci. U.S.A. 102, 1543-1546 (2005).
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V. Elser, "Phase retrieval by iterated projections," J. Opt. Soc. Am. A 20, 40-55 (2003).
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R. Fletcher, Practical Methods of Optimization (Wiley, 2000).

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H. N. Chapman, A. Barty, M. J. Bogan, S. Boutet, M. Frank, S. P. Hau-Riege, S. Marchesini, B. W. Woods, S. Bajt, W. H. Benner, R. A. London, E. Plönjes, M. Kuhlmann, R. Treusch, S. Düterer, T. Tschentscher, J. R. Schneider, E. Spiller, T. Möller, C. Bostedt, M. Hoener, D. A. Shapiro, K. O. Hodgson, D. Van Der Spoel, F. Burmeister, M. Bergh, C. Caleman, Gösta Huldt, M. M. Seibert, F. R. N. C. Maia, R. W. Lee, A. Szöke, N. Timneanu, and J. Hajdu, "Femtosecond diffractive imaging with a soft-X-ray free-electron laser," Nat. Phys. 2, 789-862 (2006).
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R. Gerchberg and W. Saxton, "A practical algorithm for the determination of phase from image and diffraction plane pictures," Optik (Stuttgart) 35, 237-246 (1972).

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H. N. Chapman, A. Barty, M. J. Bogan, S. Boutet, M. Frank, S. P. Hau-Riege, S. Marchesini, B. W. Woods, S. Bajt, W. H. Benner, R. A. London, E. Plönjes, M. Kuhlmann, R. Treusch, S. Düterer, T. Tschentscher, J. R. Schneider, E. Spiller, T. Möller, C. Bostedt, M. Hoener, D. A. Shapiro, K. O. Hodgson, D. Van Der Spoel, F. Burmeister, M. Bergh, C. Caleman, Gösta Huldt, M. M. Seibert, F. R. N. C. Maia, R. W. Lee, A. Szöke, N. Timneanu, and J. Hajdu, "Femtosecond diffractive imaging with a soft-X-ray free-electron laser," Nat. Phys. 2, 789-862 (2006).
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M. A. Pfeifer, G. J. Williams, I. A. Vartanyants, R. Harder, and I. K. Robinson, "Three-dimensional mapping of a deformation field inside a nanocrystal," Nature (London) 442, 63-67 (2006).
[CrossRef]

Hau-Riege, S. P.

H. N. Chapman, A. Barty, M. J. Bogan, S. Boutet, M. Frank, S. P. Hau-Riege, S. Marchesini, B. W. Woods, S. Bajt, W. H. Benner, R. A. London, E. Plönjes, M. Kuhlmann, R. Treusch, S. Düterer, T. Tschentscher, J. R. Schneider, E. Spiller, T. Möller, C. Bostedt, M. Hoener, D. A. Shapiro, K. O. Hodgson, D. Van Der Spoel, F. Burmeister, M. Bergh, C. Caleman, Gösta Huldt, M. M. Seibert, F. R. N. C. Maia, R. W. Lee, A. Szöke, N. Timneanu, and J. Hajdu, "Femtosecond diffractive imaging with a soft-X-ray free-electron laser," Nat. Phys. 2, 789-862 (2006).
[CrossRef]

S. Marchesini, H. He, H. N. Chapman, S. P. Hau-Riege, A. Noy, M. R. Howells, U. Weierstall, and J. C. H. Spence, "X-ray image reconstruction from a diffraction pattern alone," Phys. Rev. B 68, 140101(R) (2003).
[CrossRef]

A. Barty, S. Marchesini, H. N. Chapman, C. Cui, M. R. Howells, D. A. Shapiro, A. M. Minor, J. C. H. Spence,U. Weierstall, J. Ilavsky, A. Noy, S. P. Hau-Riege,A. B. Artyukhin, T. Baumann, T. Willey, J. Stolken, T. van Buuren, and J. H. Kinney, "Three-dimensional coherent X-ray diffraction imaging of a ceramic nanofoam: determination of structural deformation mechanisms," http://arxiv.org/abs/0708.4035.

He, H.

H. N. Chapman, A. Barty, S. Marchesini, A. Noy, C. Cui, M. R. Howells, R. Rosen, H. He, J. C. H. Spence, U. Weierstall, T. Beetz, C. Jacobsen, and D. Shapiro, "High-resolution ab initio three-dimensional X-ray diffraction microscopy," J. Opt. Soc. Am. A 23, 1179-1200 (2006).
[CrossRef]

S. Marchesini, H. He, H. N. Chapman, S. P. Hau-Riege, A. Noy, M. R. Howells, U. Weierstall, and J. C. H. Spence, "X-ray image reconstruction from a diffraction pattern alone," Phys. Rev. B 68, 140101(R) (2003).
[CrossRef]

Hestenes, M. R.

M. R. Hestenes, Conjugate Direction Methods in Optimization (Springer-Verlag, 1980).

Hodgson, K. O.

H. N. Chapman, A. Barty, M. J. Bogan, S. Boutet, M. Frank, S. P. Hau-Riege, S. Marchesini, B. W. Woods, S. Bajt, W. H. Benner, R. A. London, E. Plönjes, M. Kuhlmann, R. Treusch, S. Düterer, T. Tschentscher, J. R. Schneider, E. Spiller, T. Möller, C. Bostedt, M. Hoener, D. A. Shapiro, K. O. Hodgson, D. Van Der Spoel, F. Burmeister, M. Bergh, C. Caleman, Gösta Huldt, M. M. Seibert, F. R. N. C. Maia, R. W. Lee, A. Szöke, N. Timneanu, and J. Hajdu, "Femtosecond diffractive imaging with a soft-X-ray free-electron laser," Nat. Phys. 2, 789-862 (2006).
[CrossRef]

Hoener, M.

H. N. Chapman, A. Barty, M. J. Bogan, S. Boutet, M. Frank, S. P. Hau-Riege, S. Marchesini, B. W. Woods, S. Bajt, W. H. Benner, R. A. London, E. Plönjes, M. Kuhlmann, R. Treusch, S. Düterer, T. Tschentscher, J. R. Schneider, E. Spiller, T. Möller, C. Bostedt, M. Hoener, D. A. Shapiro, K. O. Hodgson, D. Van Der Spoel, F. Burmeister, M. Bergh, C. Caleman, Gösta Huldt, M. M. Seibert, F. R. N. C. Maia, R. W. Lee, A. Szöke, N. Timneanu, and J. Hajdu, "Femtosecond diffractive imaging with a soft-X-ray free-electron laser," Nat. Phys. 2, 789-862 (2006).
[CrossRef]

Howells, M.

D. Shapiro, P. Thibault, T. Beetz, V. Elser, M. Howells, C. Jacobsen, J. Kirz, E. Lima, H. Miao, A. Neiman, and D. Sayre, "Biological imaging by soft x-ray diffraction microscopy," Proc. Natl. Acad. Sci. U.S.A. 102, 1543-1546 (2005).
[CrossRef]

Howells, M. R.

H. N. Chapman, A. Barty, S. Marchesini, A. Noy, C. Cui, M. R. Howells, R. Rosen, H. He, J. C. H. Spence, U. Weierstall, T. Beetz, C. Jacobsen, and D. Shapiro, "High-resolution ab initio three-dimensional X-ray diffraction microscopy," J. Opt. Soc. Am. A 23, 1179-1200 (2006).
[CrossRef]

S. Marchesini, H. He, H. N. Chapman, S. P. Hau-Riege, A. Noy, M. R. Howells, U. Weierstall, and J. C. H. Spence, "X-ray image reconstruction from a diffraction pattern alone," Phys. Rev. B 68, 140101(R) (2003).
[CrossRef]

S. Marchesini, H. N. Chapman, A. Barty, C. Cui, M. R. Howells, J. C. H. Spence, U. Weierstall, and A. M. Minor, "Phase aberrations in diffraction microscopy," in Proceedings of the 8th International Conference on X-ray Microscopy, IPAP Conference Series 7, 380-382 (2006).

A. Barty, S. Marchesini, H. N. Chapman, C. Cui, M. R. Howells, D. A. Shapiro, A. M. Minor, J. C. H. Spence,U. Weierstall, J. Ilavsky, A. Noy, S. P. Hau-Riege,A. B. Artyukhin, T. Baumann, T. Willey, J. Stolken, T. van Buuren, and J. H. Kinney, "Three-dimensional coherent X-ray diffraction imaging of a ceramic nanofoam: determination of structural deformation mechanisms," http://arxiv.org/abs/0708.4035.

Huldt, Gösta

H. N. Chapman, A. Barty, M. J. Bogan, S. Boutet, M. Frank, S. P. Hau-Riege, S. Marchesini, B. W. Woods, S. Bajt, W. H. Benner, R. A. London, E. Plönjes, M. Kuhlmann, R. Treusch, S. Düterer, T. Tschentscher, J. R. Schneider, E. Spiller, T. Möller, C. Bostedt, M. Hoener, D. A. Shapiro, K. O. Hodgson, D. Van Der Spoel, F. Burmeister, M. Bergh, C. Caleman, Gösta Huldt, M. M. Seibert, F. R. N. C. Maia, R. W. Lee, A. Szöke, N. Timneanu, and J. Hajdu, "Femtosecond diffractive imaging with a soft-X-ray free-electron laser," Nat. Phys. 2, 789-862 (2006).
[CrossRef]

Ilavsky, J.

A. Barty, S. Marchesini, H. N. Chapman, C. Cui, M. R. Howells, D. A. Shapiro, A. M. Minor, J. C. H. Spence,U. Weierstall, J. Ilavsky, A. Noy, S. P. Hau-Riege,A. B. Artyukhin, T. Baumann, T. Willey, J. Stolken, T. van Buuren, and J. H. Kinney, "Three-dimensional coherent X-ray diffraction imaging of a ceramic nanofoam: determination of structural deformation mechanisms," http://arxiv.org/abs/0708.4035.

Isernia, T.

Ishikawa, T.

J. Miao, C-C. Chen, C. Song, Y. Nishino, Y. Kohmura, T. Ishikawa, D. Ramunno-Johnson, T-K. Lee, and S. H. Risbud, "Three-dimensional GaN-Ga2O3 core shell structure revealed by x-ray diffraction microscopy," Phys. Rev. Lett. 97, 215503-1-4 (2006).
[CrossRef]

Jacobsen, C.

H. N. Chapman, A. Barty, S. Marchesini, A. Noy, C. Cui, M. R. Howells, R. Rosen, H. He, J. C. H. Spence, U. Weierstall, T. Beetz, C. Jacobsen, and D. Shapiro, "High-resolution ab initio three-dimensional X-ray diffraction microscopy," J. Opt. Soc. Am. A 23, 1179-1200 (2006).
[CrossRef]

D. Shapiro, P. Thibault, T. Beetz, V. Elser, M. Howells, C. Jacobsen, J. Kirz, E. Lima, H. Miao, A. Neiman, and D. Sayre, "Biological imaging by soft x-ray diffraction microscopy," Proc. Natl. Acad. Sci. U.S.A. 102, 1543-1546 (2005).
[CrossRef]

Kinney, J. H.

A. Barty, S. Marchesini, H. N. Chapman, C. Cui, M. R. Howells, D. A. Shapiro, A. M. Minor, J. C. H. Spence,U. Weierstall, J. Ilavsky, A. Noy, S. P. Hau-Riege,A. B. Artyukhin, T. Baumann, T. Willey, J. Stolken, T. van Buuren, and J. H. Kinney, "Three-dimensional coherent X-ray diffraction imaging of a ceramic nanofoam: determination of structural deformation mechanisms," http://arxiv.org/abs/0708.4035.

Kirz, J.

D. Shapiro, P. Thibault, T. Beetz, V. Elser, M. Howells, C. Jacobsen, J. Kirz, E. Lima, H. Miao, A. Neiman, and D. Sayre, "Biological imaging by soft x-ray diffraction microscopy," Proc. Natl. Acad. Sci. U.S.A. 102, 1543-1546 (2005).
[CrossRef]

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 (London) 400, 342-344 (1999).
[CrossRef]

Kohmura, Y.

J. Miao, C-C. Chen, C. Song, Y. Nishino, Y. Kohmura, T. Ishikawa, D. Ramunno-Johnson, T-K. Lee, and S. H. Risbud, "Three-dimensional GaN-Ga2O3 core shell structure revealed by x-ray diffraction microscopy," Phys. Rev. Lett. 97, 215503-1-4 (2006).
[CrossRef]

Kuhlmann, M.

H. N. Chapman, A. Barty, M. J. Bogan, S. Boutet, M. Frank, S. P. Hau-Riege, S. Marchesini, B. W. Woods, S. Bajt, W. H. Benner, R. A. London, E. Plönjes, M. Kuhlmann, R. Treusch, S. Düterer, T. Tschentscher, J. R. Schneider, E. Spiller, T. Möller, C. Bostedt, M. Hoener, D. A. Shapiro, K. O. Hodgson, D. Van Der Spoel, F. Burmeister, M. Bergh, C. Caleman, Gösta Huldt, M. M. Seibert, F. R. N. C. Maia, R. W. Lee, A. Szöke, N. Timneanu, and J. Hajdu, "Femtosecond diffractive imaging with a soft-X-ray free-electron laser," Nat. Phys. 2, 789-862 (2006).
[CrossRef]

Lee, R. W.

H. N. Chapman, A. Barty, M. J. Bogan, S. Boutet, M. Frank, S. P. Hau-Riege, S. Marchesini, B. W. Woods, S. Bajt, W. H. Benner, R. A. London, E. Plönjes, M. Kuhlmann, R. Treusch, S. Düterer, T. Tschentscher, J. R. Schneider, E. Spiller, T. Möller, C. Bostedt, M. Hoener, D. A. Shapiro, K. O. Hodgson, D. Van Der Spoel, F. Burmeister, M. Bergh, C. Caleman, Gösta Huldt, M. M. Seibert, F. R. N. C. Maia, R. W. Lee, A. Szöke, N. Timneanu, and J. Hajdu, "Femtosecond diffractive imaging with a soft-X-ray free-electron laser," Nat. Phys. 2, 789-862 (2006).
[CrossRef]

Lee, T-K.

J. Miao, C-C. Chen, C. Song, Y. Nishino, Y. Kohmura, T. Ishikawa, D. Ramunno-Johnson, T-K. Lee, and S. H. Risbud, "Three-dimensional GaN-Ga2O3 core shell structure revealed by x-ray diffraction microscopy," Phys. Rev. Lett. 97, 215503-1-4 (2006).
[CrossRef]

Leone, G.

Levi, A.

Liao, C-W.

Lima, E.

D. Shapiro, P. Thibault, T. Beetz, V. Elser, M. Howells, C. Jacobsen, J. Kirz, E. Lima, H. Miao, A. Neiman, and D. Sayre, "Biological imaging by soft x-ray diffraction microscopy," Proc. Natl. Acad. Sci. U.S.A. 102, 1543-1546 (2005).
[CrossRef]

London, R. A.

H. N. Chapman, A. Barty, M. J. Bogan, S. Boutet, M. Frank, S. P. Hau-Riege, S. Marchesini, B. W. Woods, S. Bajt, W. H. Benner, R. A. London, E. Plönjes, M. Kuhlmann, R. Treusch, S. Düterer, T. Tschentscher, J. R. Schneider, E. Spiller, T. Möller, C. Bostedt, M. Hoener, D. A. Shapiro, K. O. Hodgson, D. Van Der Spoel, F. Burmeister, M. Bergh, C. Caleman, Gösta Huldt, M. M. Seibert, F. R. N. C. Maia, R. W. Lee, A. Szöke, N. Timneanu, and J. Hajdu, "Femtosecond diffractive imaging with a soft-X-ray free-electron laser," Nat. Phys. 2, 789-862 (2006).
[CrossRef]

Luke, D. R.

Lyon, R. G.

D. R. Luke, J. V. Burke, and R. G. Lyon, "Optical wavefront reconstruction: theory and numerical methods," SIAM Rev. 44, 169-224 (2002).
[CrossRef]

Maia, F. R. N. C.

H. N. Chapman, A. Barty, M. J. Bogan, S. Boutet, M. Frank, S. P. Hau-Riege, S. Marchesini, B. W. Woods, S. Bajt, W. H. Benner, R. A. London, E. Plönjes, M. Kuhlmann, R. Treusch, S. Düterer, T. Tschentscher, J. R. Schneider, E. Spiller, T. Möller, C. Bostedt, M. Hoener, D. A. Shapiro, K. O. Hodgson, D. Van Der Spoel, F. Burmeister, M. Bergh, C. Caleman, Gösta Huldt, M. M. Seibert, F. R. N. C. Maia, R. W. Lee, A. Szöke, N. Timneanu, and J. Hajdu, "Femtosecond diffractive imaging with a soft-X-ray free-electron laser," Nat. Phys. 2, 789-862 (2006).
[CrossRef]

Marchesini, S.

S. Marchesini, "A unified evaluation of iterative projection algorithms for phase retrieval," Rev. Sci. Instrum. 78, 011301 1-10 (2007).
[CrossRef]

H. N. Chapman, A. Barty, M. J. Bogan, S. Boutet, M. Frank, S. P. Hau-Riege, S. Marchesini, B. W. Woods, S. Bajt, W. H. Benner, R. A. London, E. Plönjes, M. Kuhlmann, R. Treusch, S. Düterer, T. Tschentscher, J. R. Schneider, E. Spiller, T. Möller, C. Bostedt, M. Hoener, D. A. Shapiro, K. O. Hodgson, D. Van Der Spoel, F. Burmeister, M. Bergh, C. Caleman, Gösta Huldt, M. M. Seibert, F. R. N. C. Maia, R. W. Lee, A. Szöke, N. Timneanu, and J. Hajdu, "Femtosecond diffractive imaging with a soft-X-ray free-electron laser," Nat. Phys. 2, 789-862 (2006).
[CrossRef]

H. N. Chapman, A. Barty, S. Marchesini, A. Noy, C. Cui, M. R. Howells, R. Rosen, H. He, J. C. H. Spence, U. Weierstall, T. Beetz, C. Jacobsen, and D. Shapiro, "High-resolution ab initio three-dimensional X-ray diffraction microscopy," J. Opt. Soc. Am. A 23, 1179-1200 (2006).
[CrossRef]

S. Marchesini, H. He, H. N. Chapman, S. P. Hau-Riege, A. Noy, M. R. Howells, U. Weierstall, and J. C. H. Spence, "X-ray image reconstruction from a diffraction pattern alone," Phys. Rev. B 68, 140101(R) (2003).
[CrossRef]

S. Marchesini, H. N. Chapman, A. Barty, C. Cui, M. R. Howells, J. C. H. Spence, U. Weierstall, and A. M. Minor, "Phase aberrations in diffraction microscopy," in Proceedings of the 8th International Conference on X-ray Microscopy, IPAP Conference Series 7, 380-382 (2006).

A. Barty, S. Marchesini, H. N. Chapman, C. Cui, M. R. Howells, D. A. Shapiro, A. M. Minor, J. C. H. Spence,U. Weierstall, J. Ilavsky, A. Noy, S. P. Hau-Riege,A. B. Artyukhin, T. Baumann, T. Willey, J. Stolken, T. van Buuren, and J. H. Kinney, "Three-dimensional coherent X-ray diffraction imaging of a ceramic nanofoam: determination of structural deformation mechanisms," http://arxiv.org/abs/0708.4035.

Miao, H.

D. Shapiro, P. Thibault, T. Beetz, V. Elser, M. Howells, C. Jacobsen, J. Kirz, E. Lima, H. Miao, A. Neiman, and D. Sayre, "Biological imaging by soft x-ray diffraction microscopy," Proc. Natl. Acad. Sci. U.S.A. 102, 1543-1546 (2005).
[CrossRef]

Miao, J.

J. Miao, C-C. Chen, C. Song, Y. Nishino, Y. Kohmura, T. Ishikawa, D. Ramunno-Johnson, T-K. Lee, and S. H. Risbud, "Three-dimensional GaN-Ga2O3 core shell structure revealed by x-ray diffraction microscopy," Phys. Rev. Lett. 97, 215503-1-4 (2006).
[CrossRef]

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 (London) 400, 342-344 (1999).
[CrossRef]

Minor, A. M.

A. Barty, S. Marchesini, H. N. Chapman, C. Cui, M. R. Howells, D. A. Shapiro, A. M. Minor, J. C. H. Spence,U. Weierstall, J. Ilavsky, A. Noy, S. P. Hau-Riege,A. B. Artyukhin, T. Baumann, T. Willey, J. Stolken, T. van Buuren, and J. H. Kinney, "Three-dimensional coherent X-ray diffraction imaging of a ceramic nanofoam: determination of structural deformation mechanisms," http://arxiv.org/abs/0708.4035.

S. Marchesini, H. N. Chapman, A. Barty, C. Cui, M. R. Howells, J. C. H. Spence, U. Weierstall, and A. M. Minor, "Phase aberrations in diffraction microscopy," in Proceedings of the 8th International Conference on X-ray Microscopy, IPAP Conference Series 7, 380-382 (2006).

Möller, T.

H. N. Chapman, A. Barty, M. J. Bogan, S. Boutet, M. Frank, S. P. Hau-Riege, S. Marchesini, B. W. Woods, S. Bajt, W. H. Benner, R. A. London, E. Plönjes, M. Kuhlmann, R. Treusch, S. Düterer, T. Tschentscher, J. R. Schneider, E. Spiller, T. Möller, C. Bostedt, M. Hoener, D. A. Shapiro, K. O. Hodgson, D. Van Der Spoel, F. Burmeister, M. Bergh, C. Caleman, Gösta Huldt, M. M. Seibert, F. R. N. C. Maia, R. W. Lee, A. Szöke, N. Timneanu, and J. Hajdu, "Femtosecond diffractive imaging with a soft-X-ray free-electron laser," Nat. Phys. 2, 789-862 (2006).
[CrossRef]

Neiman, A.

D. Shapiro, P. Thibault, T. Beetz, V. Elser, M. Howells, C. Jacobsen, J. Kirz, E. Lima, H. Miao, A. Neiman, and D. Sayre, "Biological imaging by soft x-ray diffraction microscopy," Proc. Natl. Acad. Sci. U.S.A. 102, 1543-1546 (2005).
[CrossRef]

Nishino, Y.

J. Miao, C-C. Chen, C. Song, Y. Nishino, Y. Kohmura, T. Ishikawa, D. Ramunno-Johnson, T-K. Lee, and S. H. Risbud, "Three-dimensional GaN-Ga2O3 core shell structure revealed by x-ray diffraction microscopy," Phys. Rev. Lett. 97, 215503-1-4 (2006).
[CrossRef]

Nocedal, J.

J. Nocedal and S. J. Wright, Numerical Optimization (Springer Verlag, 2006).

Noy, A.

H. N. Chapman, A. Barty, S. Marchesini, A. Noy, C. Cui, M. R. Howells, R. Rosen, H. He, J. C. H. Spence, U. Weierstall, T. Beetz, C. Jacobsen, and D. Shapiro, "High-resolution ab initio three-dimensional X-ray diffraction microscopy," J. Opt. Soc. Am. A 23, 1179-1200 (2006).
[CrossRef]

S. Marchesini, H. He, H. N. Chapman, S. P. Hau-Riege, A. Noy, M. R. Howells, U. Weierstall, and J. C. H. Spence, "X-ray image reconstruction from a diffraction pattern alone," Phys. Rev. B 68, 140101(R) (2003).
[CrossRef]

A. Barty, S. Marchesini, H. N. Chapman, C. Cui, M. R. Howells, D. A. Shapiro, A. M. Minor, J. C. H. Spence,U. Weierstall, J. Ilavsky, A. Noy, S. P. Hau-Riege,A. B. Artyukhin, T. Baumann, T. Willey, J. Stolken, T. van Buuren, and J. H. Kinney, "Three-dimensional coherent X-ray diffraction imaging of a ceramic nanofoam: determination of structural deformation mechanisms," http://arxiv.org/abs/0708.4035.

Oszlányi, G.

G. Oszlányi and A. Süto, "Ab initio structure solution by charge flipping. II. Use of weak reflections," Acta Crystallogr. A61, 147-152 (2005).

Pfeifer, M. A.

M. A. Pfeifer, G. J. Williams, I. A. Vartanyants, R. Harder, and I. K. Robinson, "Three-dimensional mapping of a deformation field inside a nanocrystal," Nature (London) 442, 63-67 (2006).
[CrossRef]

Pierri, R.

Plönjes, E.

H. N. Chapman, A. Barty, M. J. Bogan, S. Boutet, M. Frank, S. P. Hau-Riege, S. Marchesini, B. W. Woods, S. Bajt, W. H. Benner, R. A. London, E. Plönjes, M. Kuhlmann, R. Treusch, S. Düterer, T. Tschentscher, J. R. Schneider, E. Spiller, T. Möller, C. Bostedt, M. Hoener, D. A. Shapiro, K. O. Hodgson, D. Van Der Spoel, F. Burmeister, M. Bergh, C. Caleman, Gösta Huldt, M. M. Seibert, F. R. N. C. Maia, R. W. Lee, A. Szöke, N. Timneanu, and J. Hajdu, "Femtosecond diffractive imaging with a soft-X-ray free-electron laser," Nat. Phys. 2, 789-862 (2006).
[CrossRef]

Polak, E.

E. Polak, Computational Methods in Optimization (Academic Press, 1971).

Pommet, D. A.

Powell, M. J. D.

M. J. D. Powell, "Nonconvex minimization calculations and the conjugate gradient method," Lect. Notes Math. 1066, 122-141 (1984).
[CrossRef]

Press, W. H.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes in C (Cambridge U. Press, 1992).

Ramunno-Johnson, D.

J. Miao, C-C. Chen, C. Song, Y. Nishino, Y. Kohmura, T. Ishikawa, D. Ramunno-Johnson, T-K. Lee, and S. H. Risbud, "Three-dimensional GaN-Ga2O3 core shell structure revealed by x-ray diffraction microscopy," Phys. Rev. Lett. 97, 215503-1-4 (2006).
[CrossRef]

Rankenburg, I.

V. Elser, I. Rankenburg, and P. Thibault, "Searching with iterated maps," Proc. Natl. Acad. Sci. U.S.A. 104, 418-423 (2007).
[CrossRef] [PubMed]

V. Elser and I. Rankenburg, "Deconstructing the energy landscape: Constraint-based algorithms for folding heteropolymers," Phys. Rev. E 73, 026702 1-8 (2006).
[CrossRef]

Risbud, S. H.

J. Miao, C-C. Chen, C. Song, Y. Nishino, Y. Kohmura, T. Ishikawa, D. Ramunno-Johnson, T-K. Lee, and S. H. Risbud, "Three-dimensional GaN-Ga2O3 core shell structure revealed by x-ray diffraction microscopy," Phys. Rev. Lett. 97, 215503-1-4 (2006).
[CrossRef]

Robinson, I. K.

M. A. Pfeifer, G. J. Williams, I. A. Vartanyants, R. Harder, and I. K. Robinson, "Three-dimensional mapping of a deformation field inside a nanocrystal," Nature (London) 442, 63-67 (2006).
[CrossRef]

Rosen, R.

Saxton, W.

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

Sayre, D.

D. Shapiro, P. Thibault, T. Beetz, V. Elser, M. Howells, C. Jacobsen, J. Kirz, E. Lima, H. Miao, A. Neiman, and D. Sayre, "Biological imaging by soft x-ray diffraction microscopy," Proc. Natl. Acad. Sci. U.S.A. 102, 1543-1546 (2005).
[CrossRef]

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 (London) 400, 342-344 (1999).
[CrossRef]

Schneider, J. R.

H. N. Chapman, A. Barty, M. J. Bogan, S. Boutet, M. Frank, S. P. Hau-Riege, S. Marchesini, B. W. Woods, S. Bajt, W. H. Benner, R. A. London, E. Plönjes, M. Kuhlmann, R. Treusch, S. Düterer, T. Tschentscher, J. R. Schneider, E. Spiller, T. Möller, C. Bostedt, M. Hoener, D. A. Shapiro, K. O. Hodgson, D. Van Der Spoel, F. Burmeister, M. Bergh, C. Caleman, Gösta Huldt, M. M. Seibert, F. R. N. C. Maia, R. W. Lee, A. Szöke, N. Timneanu, and J. Hajdu, "Femtosecond diffractive imaging with a soft-X-ray free-electron laser," Nat. Phys. 2, 789-862 (2006).
[CrossRef]

Seibert, M. M.

H. N. Chapman, A. Barty, M. J. Bogan, S. Boutet, M. Frank, S. P. Hau-Riege, S. Marchesini, B. W. Woods, S. Bajt, W. H. Benner, R. A. London, E. Plönjes, M. Kuhlmann, R. Treusch, S. Düterer, T. Tschentscher, J. R. Schneider, E. Spiller, T. Möller, C. Bostedt, M. Hoener, D. A. Shapiro, K. O. Hodgson, D. Van Der Spoel, F. Burmeister, M. Bergh, C. Caleman, Gösta Huldt, M. M. Seibert, F. R. N. C. Maia, R. W. Lee, A. Szöke, N. Timneanu, and J. Hajdu, "Femtosecond diffractive imaging with a soft-X-ray free-electron laser," Nat. Phys. 2, 789-862 (2006).
[CrossRef]

Shapiro, D.

H. N. Chapman, A. Barty, S. Marchesini, A. Noy, C. Cui, M. R. Howells, R. Rosen, H. He, J. C. H. Spence, U. Weierstall, T. Beetz, C. Jacobsen, and D. Shapiro, "High-resolution ab initio three-dimensional X-ray diffraction microscopy," J. Opt. Soc. Am. A 23, 1179-1200 (2006).
[CrossRef]

D. Shapiro, P. Thibault, T. Beetz, V. Elser, M. Howells, C. Jacobsen, J. Kirz, E. Lima, H. Miao, A. Neiman, and D. Sayre, "Biological imaging by soft x-ray diffraction microscopy," Proc. Natl. Acad. Sci. U.S.A. 102, 1543-1546 (2005).
[CrossRef]

Shapiro, D. A.

H. N. Chapman, A. Barty, M. J. Bogan, S. Boutet, M. Frank, S. P. Hau-Riege, S. Marchesini, B. W. Woods, S. Bajt, W. H. Benner, R. A. London, E. Plönjes, M. Kuhlmann, R. Treusch, S. Düterer, T. Tschentscher, J. R. Schneider, E. Spiller, T. Möller, C. Bostedt, M. Hoener, D. A. Shapiro, K. O. Hodgson, D. Van Der Spoel, F. Burmeister, M. Bergh, C. Caleman, Gösta Huldt, M. M. Seibert, F. R. N. C. Maia, R. W. Lee, A. Szöke, N. Timneanu, and J. Hajdu, "Femtosecond diffractive imaging with a soft-X-ray free-electron laser," Nat. Phys. 2, 789-862 (2006).
[CrossRef]

A. Barty, S. Marchesini, H. N. Chapman, C. Cui, M. R. Howells, D. A. Shapiro, A. M. Minor, J. C. H. Spence,U. Weierstall, J. Ilavsky, A. Noy, S. P. Hau-Riege,A. B. Artyukhin, T. Baumann, T. Willey, J. Stolken, T. van Buuren, and J. H. Kinney, "Three-dimensional coherent X-ray diffraction imaging of a ceramic nanofoam: determination of structural deformation mechanisms," http://arxiv.org/abs/0708.4035.

Soldovieri, F.

Song, C.

J. Miao, C-C. Chen, C. Song, Y. Nishino, Y. Kohmura, T. Ishikawa, D. Ramunno-Johnson, T-K. Lee, and S. H. Risbud, "Three-dimensional GaN-Ga2O3 core shell structure revealed by x-ray diffraction microscopy," Phys. Rev. Lett. 97, 215503-1-4 (2006).
[CrossRef]

Spence, J. C. H.

H. N. Chapman, A. Barty, S. Marchesini, A. Noy, C. Cui, M. R. Howells, R. Rosen, H. He, J. C. H. Spence, U. Weierstall, T. Beetz, C. Jacobsen, and D. Shapiro, "High-resolution ab initio three-dimensional X-ray diffraction microscopy," J. Opt. Soc. Am. A 23, 1179-1200 (2006).
[CrossRef]

S. Marchesini, H. He, H. N. Chapman, S. P. Hau-Riege, A. Noy, M. R. Howells, U. Weierstall, and J. C. H. Spence, "X-ray image reconstruction from a diffraction pattern alone," Phys. Rev. B 68, 140101(R) (2003).
[CrossRef]

S. Marchesini, H. N. Chapman, A. Barty, C. Cui, M. R. Howells, J. C. H. Spence, U. Weierstall, and A. M. Minor, "Phase aberrations in diffraction microscopy," in Proceedings of the 8th International Conference on X-ray Microscopy, IPAP Conference Series 7, 380-382 (2006).

A. Barty, S. Marchesini, H. N. Chapman, C. Cui, M. R. Howells, D. A. Shapiro, A. M. Minor, J. C. H. Spence,U. Weierstall, J. Ilavsky, A. Noy, S. P. Hau-Riege,A. B. Artyukhin, T. Baumann, T. Willey, J. Stolken, T. van Buuren, and J. H. Kinney, "Three-dimensional coherent X-ray diffraction imaging of a ceramic nanofoam: determination of structural deformation mechanisms," http://arxiv.org/abs/0708.4035.

Spiller, E.

H. N. Chapman, A. Barty, M. J. Bogan, S. Boutet, M. Frank, S. P. Hau-Riege, S. Marchesini, B. W. Woods, S. Bajt, W. H. Benner, R. A. London, E. Plönjes, M. Kuhlmann, R. Treusch, S. Düterer, T. Tschentscher, J. R. Schneider, E. Spiller, T. Möller, C. Bostedt, M. Hoener, D. A. Shapiro, K. O. Hodgson, D. Van Der Spoel, F. Burmeister, M. Bergh, C. Caleman, Gösta Huldt, M. M. Seibert, F. R. N. C. Maia, R. W. Lee, A. Szöke, N. Timneanu, and J. Hajdu, "Femtosecond diffractive imaging with a soft-X-ray free-electron laser," Nat. Phys. 2, 789-862 (2006).
[CrossRef]

Stark, H.

Stolken, J.

A. Barty, S. Marchesini, H. N. Chapman, C. Cui, M. R. Howells, D. A. Shapiro, A. M. Minor, J. C. H. Spence,U. Weierstall, J. Ilavsky, A. Noy, S. P. Hau-Riege,A. B. Artyukhin, T. Baumann, T. Willey, J. Stolken, T. van Buuren, and J. H. Kinney, "Three-dimensional coherent X-ray diffraction imaging of a ceramic nanofoam: determination of structural deformation mechanisms," http://arxiv.org/abs/0708.4035.

Süto, A.

G. Oszlányi and A. Süto, "Ab initio structure solution by charge flipping. II. Use of weak reflections," Acta Crystallogr. A61, 147-152 (2005).

Szöke, A.

H. N. Chapman, A. Barty, M. J. Bogan, S. Boutet, M. Frank, S. P. Hau-Riege, S. Marchesini, B. W. Woods, S. Bajt, W. H. Benner, R. A. London, E. Plönjes, M. Kuhlmann, R. Treusch, S. Düterer, T. Tschentscher, J. R. Schneider, E. Spiller, T. Möller, C. Bostedt, M. Hoener, D. A. Shapiro, K. O. Hodgson, D. Van Der Spoel, F. Burmeister, M. Bergh, C. Caleman, Gösta Huldt, M. M. Seibert, F. R. N. C. Maia, R. W. Lee, A. Szöke, N. Timneanu, and J. Hajdu, "Femtosecond diffractive imaging with a soft-X-ray free-electron laser," Nat. Phys. 2, 789-862 (2006).
[CrossRef]

Teukolsky, S. A.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes in C (Cambridge U. Press, 1992).

Thibault, P.

V. Elser, I. Rankenburg, and P. Thibault, "Searching with iterated maps," Proc. Natl. Acad. Sci. U.S.A. 104, 418-423 (2007).
[CrossRef] [PubMed]

D. Shapiro, P. Thibault, T. Beetz, V. Elser, M. Howells, C. Jacobsen, J. Kirz, E. Lima, H. Miao, A. Neiman, and D. Sayre, "Biological imaging by soft x-ray diffraction microscopy," Proc. Natl. Acad. Sci. U.S.A. 102, 1543-1546 (2005).
[CrossRef]

Timneanu, N.

H. N. Chapman, A. Barty, M. J. Bogan, S. Boutet, M. Frank, S. P. Hau-Riege, S. Marchesini, B. W. Woods, S. Bajt, W. H. Benner, R. A. London, E. Plönjes, M. Kuhlmann, R. Treusch, S. Düterer, T. Tschentscher, J. R. Schneider, E. Spiller, T. Möller, C. Bostedt, M. Hoener, D. A. Shapiro, K. O. Hodgson, D. Van Der Spoel, F. Burmeister, M. Bergh, C. Caleman, Gösta Huldt, M. M. Seibert, F. R. N. C. Maia, R. W. Lee, A. Szöke, N. Timneanu, and J. Hajdu, "Femtosecond diffractive imaging with a soft-X-ray free-electron laser," Nat. Phys. 2, 789-862 (2006).
[CrossRef]

Treusch, R.

H. N. Chapman, A. Barty, M. J. Bogan, S. Boutet, M. Frank, S. P. Hau-Riege, S. Marchesini, B. W. Woods, S. Bajt, W. H. Benner, R. A. London, E. Plönjes, M. Kuhlmann, R. Treusch, S. Düterer, T. Tschentscher, J. R. Schneider, E. Spiller, T. Möller, C. Bostedt, M. Hoener, D. A. Shapiro, K. O. Hodgson, D. Van Der Spoel, F. Burmeister, M. Bergh, C. Caleman, Gösta Huldt, M. M. Seibert, F. R. N. C. Maia, R. W. Lee, A. Szöke, N. Timneanu, and J. Hajdu, "Femtosecond diffractive imaging with a soft-X-ray free-electron laser," Nat. Phys. 2, 789-862 (2006).
[CrossRef]

Tschentscher, T.

H. N. Chapman, A. Barty, M. J. Bogan, S. Boutet, M. Frank, S. P. Hau-Riege, S. Marchesini, B. W. Woods, S. Bajt, W. H. Benner, R. A. London, E. Plönjes, M. Kuhlmann, R. Treusch, S. Düterer, T. Tschentscher, J. R. Schneider, E. Spiller, T. Möller, C. Bostedt, M. Hoener, D. A. Shapiro, K. O. Hodgson, D. Van Der Spoel, F. Burmeister, M. Bergh, C. Caleman, Gösta Huldt, M. M. Seibert, F. R. N. C. Maia, R. W. Lee, A. Szöke, N. Timneanu, and J. Hajdu, "Femtosecond diffractive imaging with a soft-X-ray free-electron laser," Nat. Phys. 2, 789-862 (2006).
[CrossRef]

van Buuren, T.

A. Barty, S. Marchesini, H. N. Chapman, C. Cui, M. R. Howells, D. A. Shapiro, A. M. Minor, J. C. H. Spence,U. Weierstall, J. Ilavsky, A. Noy, S. P. Hau-Riege,A. B. Artyukhin, T. Baumann, T. Willey, J. Stolken, T. van Buuren, and J. H. Kinney, "Three-dimensional coherent X-ray diffraction imaging of a ceramic nanofoam: determination of structural deformation mechanisms," http://arxiv.org/abs/0708.4035.

Van Der Spoel, D.

H. N. Chapman, A. Barty, M. J. Bogan, S. Boutet, M. Frank, S. P. Hau-Riege, S. Marchesini, B. W. Woods, S. Bajt, W. H. Benner, R. A. London, E. Plönjes, M. Kuhlmann, R. Treusch, S. Düterer, T. Tschentscher, J. R. Schneider, E. Spiller, T. Möller, C. Bostedt, M. Hoener, D. A. Shapiro, K. O. Hodgson, D. Van Der Spoel, F. Burmeister, M. Bergh, C. Caleman, Gösta Huldt, M. M. Seibert, F. R. N. C. Maia, R. W. Lee, A. Szöke, N. Timneanu, and J. Hajdu, "Femtosecond diffractive imaging with a soft-X-ray free-electron laser," Nat. Phys. 2, 789-862 (2006).
[CrossRef]

Vartanyants, I. A.

M. A. Pfeifer, G. J. Williams, I. A. Vartanyants, R. Harder, and I. K. Robinson, "Three-dimensional mapping of a deformation field inside a nanocrystal," Nature (London) 442, 63-67 (2006).
[CrossRef]

Vetterling, W. T.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes in C (Cambridge U. Press, 1992).

Wackerman, C. C.

Webb, H.

D. C. Youla and H. Webb, "Image restoration by the method of projections onto convex sets Part I," IEEE Trans. Med. Imaging TMI-1, 81-94 (1982).
[CrossRef]

Weierstall, U.

H. N. Chapman, A. Barty, S. Marchesini, A. Noy, C. Cui, M. R. Howells, R. Rosen, H. He, J. C. H. Spence, U. Weierstall, T. Beetz, C. Jacobsen, and D. Shapiro, "High-resolution ab initio three-dimensional X-ray diffraction microscopy," J. Opt. Soc. Am. A 23, 1179-1200 (2006).
[CrossRef]

S. Marchesini, H. He, H. N. Chapman, S. P. Hau-Riege, A. Noy, M. R. Howells, U. Weierstall, and J. C. H. Spence, "X-ray image reconstruction from a diffraction pattern alone," Phys. Rev. B 68, 140101(R) (2003).
[CrossRef]

S. Marchesini, H. N. Chapman, A. Barty, C. Cui, M. R. Howells, J. C. H. Spence, U. Weierstall, and A. M. Minor, "Phase aberrations in diffraction microscopy," in Proceedings of the 8th International Conference on X-ray Microscopy, IPAP Conference Series 7, 380-382 (2006).

A. Barty, S. Marchesini, H. N. Chapman, C. Cui, M. R. Howells, D. A. Shapiro, A. M. Minor, J. C. H. Spence,U. Weierstall, J. Ilavsky, A. Noy, S. P. Hau-Riege,A. B. Artyukhin, T. Baumann, T. Willey, J. Stolken, T. van Buuren, and J. H. Kinney, "Three-dimensional coherent X-ray diffraction imaging of a ceramic nanofoam: determination of structural deformation mechanisms," http://arxiv.org/abs/0708.4035.

Willey, T.

A. Barty, S. Marchesini, H. N. Chapman, C. Cui, M. R. Howells, D. A. Shapiro, A. M. Minor, J. C. H. Spence,U. Weierstall, J. Ilavsky, A. Noy, S. P. Hau-Riege,A. B. Artyukhin, T. Baumann, T. Willey, J. Stolken, T. van Buuren, and J. H. Kinney, "Three-dimensional coherent X-ray diffraction imaging of a ceramic nanofoam: determination of structural deformation mechanisms," http://arxiv.org/abs/0708.4035.

Williams, G. J.

M. A. Pfeifer, G. J. Williams, I. A. Vartanyants, R. Harder, and I. K. Robinson, "Three-dimensional mapping of a deformation field inside a nanocrystal," Nature (London) 442, 63-67 (2006).
[CrossRef]

Woods, B. W.

H. N. Chapman, A. Barty, M. J. Bogan, S. Boutet, M. Frank, S. P. Hau-Riege, S. Marchesini, B. W. Woods, S. Bajt, W. H. Benner, R. A. London, E. Plönjes, M. Kuhlmann, R. Treusch, S. Düterer, T. Tschentscher, J. R. Schneider, E. Spiller, T. Möller, C. Bostedt, M. Hoener, D. A. Shapiro, K. O. Hodgson, D. Van Der Spoel, F. Burmeister, M. Bergh, C. Caleman, Gösta Huldt, M. M. Seibert, F. R. N. C. Maia, R. W. Lee, A. Szöke, N. Timneanu, and J. Hajdu, "Femtosecond diffractive imaging with a soft-X-ray free-electron laser," Nat. Phys. 2, 789-862 (2006).
[CrossRef]

Wright, S. J.

J. Nocedal and S. J. Wright, Numerical Optimization (Springer Verlag, 2006).

Youla, D. C.

D. C. Youla and H. Webb, "Image restoration by the method of projections onto convex sets Part I," IEEE Trans. Med. Imaging TMI-1, 81-94 (1982).
[CrossRef]

Acta Crystallogr. (1)

G. Oszlányi and A. Süto, "Ab initio structure solution by charge flipping. II. Use of weak reflections," Acta Crystallogr. A61, 147-152 (2005).

Appl. Opt. (1)

IEEE Trans. Med. Imaging (1)

D. C. Youla and H. Webb, "Image restoration by the method of projections onto convex sets Part I," IEEE Trans. Med. Imaging TMI-1, 81-94 (1982).
[CrossRef]

Inverse Probl. (1)

D. R. Luke, "Relaxed averaged alternating reflections for diffraction imaging," Inverse Probl. 21, 37-50 (2005).
[CrossRef]

J. Opt. Soc. Am. A (8)

Lect. Notes Math. (1)

M. J. D. Powell, "Nonconvex minimization calculations and the conjugate gradient method," Lect. Notes Math. 1066, 122-141 (1984).
[CrossRef]

Nat. Phys. (1)

H. N. Chapman, A. Barty, M. J. Bogan, S. Boutet, M. Frank, S. P. Hau-Riege, S. Marchesini, B. W. Woods, S. Bajt, W. H. Benner, R. A. London, E. Plönjes, M. Kuhlmann, R. Treusch, S. Düterer, T. Tschentscher, J. R. Schneider, E. Spiller, T. Möller, C. Bostedt, M. Hoener, D. A. Shapiro, K. O. Hodgson, D. Van Der Spoel, F. Burmeister, M. Bergh, C. Caleman, Gösta Huldt, M. M. Seibert, F. R. N. C. Maia, R. W. Lee, A. Szöke, N. Timneanu, and J. Hajdu, "Femtosecond diffractive imaging with a soft-X-ray free-electron laser," Nat. Phys. 2, 789-862 (2006).
[CrossRef]

Nature (London) (2)

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 (London) 400, 342-344 (1999).
[CrossRef]

M. A. Pfeifer, G. J. Williams, I. A. Vartanyants, R. Harder, and I. K. Robinson, "Three-dimensional mapping of a deformation field inside a nanocrystal," Nature (London) 442, 63-67 (2006).
[CrossRef]

Opt. Lett. (1)

Optik (Stuttgart) (1)

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

Phys. Rev. B (1)

S. Marchesini, H. He, H. N. Chapman, S. P. Hau-Riege, A. Noy, M. R. Howells, U. Weierstall, and J. C. H. Spence, "X-ray image reconstruction from a diffraction pattern alone," Phys. Rev. B 68, 140101(R) (2003).
[CrossRef]

Phys. Rev. E (1)

V. Elser and I. Rankenburg, "Deconstructing the energy landscape: Constraint-based algorithms for folding heteropolymers," Phys. Rev. E 73, 026702 1-8 (2006).
[CrossRef]

Phys. Rev. Lett. (1)

J. Miao, C-C. Chen, C. Song, Y. Nishino, Y. Kohmura, T. Ishikawa, D. Ramunno-Johnson, T-K. Lee, and S. H. Risbud, "Three-dimensional GaN-Ga2O3 core shell structure revealed by x-ray diffraction microscopy," Phys. Rev. Lett. 97, 215503-1-4 (2006).
[CrossRef]

Proc. Natl. Acad. Sci. U.S.A. (2)

D. Shapiro, P. Thibault, T. Beetz, V. Elser, M. Howells, C. Jacobsen, J. Kirz, E. Lima, H. Miao, A. Neiman, and D. Sayre, "Biological imaging by soft x-ray diffraction microscopy," Proc. Natl. Acad. Sci. U.S.A. 102, 1543-1546 (2005).
[CrossRef]

V. Elser, I. Rankenburg, and P. Thibault, "Searching with iterated maps," Proc. Natl. Acad. Sci. U.S.A. 104, 418-423 (2007).
[CrossRef] [PubMed]

Rev. Sci. Instrum. (1)

S. Marchesini, "A unified evaluation of iterative projection algorithms for phase retrieval," Rev. Sci. Instrum. 78, 011301 1-10 (2007).
[CrossRef]

SIAM Rev. (1)

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Figures (4)

Fig. 1
Fig. 1

Algorithms seek the intersection between two sets (data and constraints represented by two lines) using different types of fixed points. ER projects back and forth between the two sets, moving toward the minimum of ε m within the constraint (horizontal line). HIO seeks the saddle point ( min s max s ̱ ) of ( L ) , represented in the background in a pseudocolor map. L is the difference of the square distances ( ε 2 ) between the current point and the two sets. In order to reach the saddle point, HIO spirals toward the solution by inverting the component of the gradient ( L in black) parallel to the constraint (horizontal direction) following a descent–ascent direction ( [ P s + P s ̱ ] L in light blue) toward the solution. Marker symbols are plotted every five iterations. See also [19] for a comparison with other algorithms.

Fig. 2
Fig. 2

Pseudocolor and contour maps of the lower-dimensional function ψ ( α , β ) are depicted in the background. This ψ was computed from one iteration of the test described in Section 6. The saddle, typical of this test, is wide in the horizontal direction and narrow in the vertical, showing the importance of the relaxation parameter β ¯ < 1 to reduce (precondition) the vertical move. However, the condition number increases near the saddle point as it becomes even narrower in the vertical direction. Often the dependence of ψ with respect to the fitting parameter β (vertical axis) resembled a V centered near the solution, and successive iterations using just a preconditioner jumped up and down near the saddle point. Iterations using the SR1 quasi-Newton update (starting from τ ( 0 ) = 0 , H ( 0 ) = H ̂ HIO ) shown here converge rapidly to the solution.

Fig. 3
Fig. 3

Test figure used for benchmarking (total size: 256 2 , object size: 128 2 pixels). The support was slightly larger than the object: 129 2 pixels.

Fig. 4
Fig. 4

Successful phase retrievals, starting from random phases, as a function of iterations. When the normalized error metric ε ¯ m 2 falls below a threshold ( 10 4 ) it is considered a successful reconstruction. The plot represents the cumulative sum of the number of successful reconstructions versus the number of iterations required. Values for 50% (dotted line) and 100% success are listed in Table 1. Comparison with other algorithms is shown in [19].

Tables (1)

Tables Icon

Table 1 Benchmark of Various Algorithms a Summarizing the Results in Fig. 4

Equations (66)

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I ( k ) = ρ ̃ ( k ) 2 ,
ρ ( r ) = 0 , if r S .
P s ρ ( r ) = { ρ ( r ) if r S , 0 otherwise .
P ̃ m ρ ̃ ( k ) = P ̃ m ρ ̃ ( k ) e i φ ( k ) = I ( k ) e i φ ( k ) .
P m = F 1 P ̃ m F .
ρ 2 = ρ ρ = r ρ ( r ) 2 = k ρ ̃ ( k ) 2 .
ε s ( ρ ) = P s ρ ρ ,
ε m ( ρ ) = P m ρ ρ ,
ε m 2 ( ρ ) = 2 [ P m I ] ρ ,
ε s 2 ( ρ ) = 2 [ P s I ] ρ .
min ρ ε m 2 ( ρ ) ,
subject to P s ̱ ρ = 0 .
min ρ s ε m 2 ( ρ s ) .
ρ ( n + 1 ) = ρ ( n ) + Δ ρ ( n ) ,
Δ ρ ( n ) = 1 2 s ε m 2 ( ρ ( n ) ) ,
= P s [ I P m ] ρ ( n ) ,
ρ ( n + 1 ) = P s P m ρ ( n ) ;
min δ ε m 2 ( ρ + δ Δ ρ ) .
δ ε m 2 ( ρ + δ Δ ρ s ) = Δ ρ s 2 P s [ I P m ] ( ρ + δ Δ ρ s ) r ,
0 = Δ ρ s [ I P m ] ( ρ + δ Δ ρ s ) r ,
P ̃ s = F P s F 1 ,
min ρ s max ρ s ̱ L ( ρ s + ρ s ̱ ) ,
Δ ρ ( n ) = { P s + β ¯ P s ̱ } 1 2 L ( ρ ( n ) ) .
L ( ρ ) = 2 [ P s P m ] ρ ,
Δ ρ ( n ) = { P s [ P m I ] β ¯ P s ̱ P m } ρ ( n ) ,
ρ ( n + 1 ) = ρ ( n ) + Δ ρ ( n ) .
ρ ( n + 1 ) = [ P s P m + P s ̱ [ I β ¯ P m ] ] ρ ( n ) = { P m ρ ( n ) ( r ) if r S . ( I β ¯ P m ) ρ ( n ) ( r ) otherwise
Δ ρ [ P s + β ¯ P s ̱ ] L ( ρ + δ Δ ρ ) r = 0 .
min α max β ψ ( α , β ) ,
ψ ( α , β ) = L ( ρ + α Δ ρ s + β Δ ρ s ̱ ) ,
Δ ρ s = 1 2 s L ( ρ ) , Δ ρ s ̱ = 1 2 s ̱ L ( ρ ) ,
ψ α = Δ ρ s L ( ρ + α Δ ρ s + β Δ ρ s ̱ ) r = 0 ,
ψ β = Δ ρ s ̱ L ( ρ + α Δ ρ s + β Δ ρ s ̱ ) r = 0 .
τ ψ ( τ ) = Δ ρ ρ L ( ρ + τ T Δ ρ r = 0 ,
τ = ( α β ) , Δ ρ = ( Δ ρ s Δ ρ s ̱ ) , ρ = ( s s ̱ ) .
τ ψ ( 0 ) = Δ ρ s s L ( ρ ) Δ ρ s ̱ s ̱ L ( ρ ) , = 2 ( Δ ρ s 2 + Δ ρ s ̱ 2 ) = 1 2 ( s L ( ρ ) 2 + s ̱ L ( ρ ) 2 ) .
min τ Φ ( τ ) , Φ = 1 2 τ ψ ( τ ) 2 ,
H τ = [ α α α β β α β β ] ψ , { H α , α 0 H β , β 0 .
Δ τ = H 1 τ ψ .
H ̂ SDA = 2 [ Δ ρ s 2 0 0 Δ ρ s ̱ 2 ] .
H ̂ HIO = [ 1 0 0 1 β ¯ ] H ̂ SDA .
Δ τ H ̂ 1 τ ψ ( τ + δ Δ τ ) r = 0 .
Λ τ ( n + 1 ) = Δ τ ( n + 1 ) + γ ( n ) Λ τ ( n ) ,
γ ( n ) = Δ τ ( n + 1 ) H ̂ ( 1 ) ( ψ ( τ ( n + 1 ) ) ψ ( τ ( n ) ) ) Δ τ ( n ) H ̂ 1 Δ ψ ( τ ( n ) ) .
y = τ ψ ( τ + Δ τ ) τ ψ ( τ ) ,
Δ H 1 = Δ τ H 1 y 2 ( Δ τ H 1 y ) T y ,
H 1 H 1 + Δ H 1 .
min α ( n , n + 1 ) max β ( n , n + 1 ) L ( ρ + ( τ ( n + 1 ) ) T Δ ρ ( n + 1 ) + ( τ ( n ) ) T Δ ρ ( n ) ) .
{ min ρ s ε m 2 ( ρ ) = min ρ s [ I P m ] ρ 2 min ρ s ̱ ε s 2 ( ρ ) ε m 2 ( ρ ) = min ρ s ̱ 2 P m ρ ρ r + c .
P ̃ m ρ ̃ ρ ̃ = I ρ ̃ s + ρ ̃ s ̱ .
L ( ρ ̃ s + ρ ̃ s ̱ ) = [ I P ̃ m ] ( ρ ̃ s + ρ ̃ s ̱ ) 2 ρ ̃ s ̱ 2 = ρ ̃ s 2 2 I ρ ̃ s + ρ ̃ s ̱ + I
L = ( P s L P s ̱ L ) = 2 ( P s [ I P m ] ( ρ s + ρ s ̱ ) P s ̱ P m ( ρ s + ρ s ̱ ) ) .
ψ ( α , β ) = [ I P ̃ m ] ( ρ ̃ + α Δ ρ ̃ s + β Δ ρ ̃ s ̱ ) 2 ρ ̃ s ̱ + β Δ ρ ̃ s ̱ 2 = k ρ ̃ + α Δ ρ ̃ s + β Δ ρ ̃ s ̱ I 2 ρ ̃ s ̱ + β Δ ρ ̃ s ̱ 2 = k I + ρ ̃ s + α Δ ρ ̃ s 2 2 I ρ ̃ + α Δ ρ ̃ s + β Δ ρ ̃ s ̱ .
t ψ = Δ ρ ̃ L ( ρ ̃ τ ) r = 2 Δ ρ ̃ [ P ̃ s P ̃ m ] ρ ̃ τ r ,
ψ α = 2 Δ ρ ̃ s [ I P ̃ m ] ρ ̃ τ r ,
ψ β = 2 Δ ρ ̃ s ̱ [ P ̃ m ] ρ ̃ τ r .
x x = x x ,
α x + α Δ x = R ( Δ x ( x + α Δ x ) ) x + α Δ x ,
α x + α Δ x x + α Δ x = Δ x x + α Δ x x + α Δ x x + α Δ x × R ( Δ x ( x + α Δ x ) ) x + α Δ x 2 ,
α x + α Δ x 2 = 2 R ( Δ x ( x + α Δ x ) ) ,
2 α 2 x + α Δ x = Δ x 2 x + α Δ x R ( Δ x ( x + α Δ x ) ) 2 x + α Δ x 3 ,
2 α 2 x + α Δ x 2 = Δ x 2 ,
2 x + α Δ x + β Δ y α β = R ( Δ x Δ y ) x + α Δ x + β Δ y R ( Δ x ( x + α Δ x + β Δ y ) ) x + α Δ x + β Δ y × R ( Δ y ( x + α Δ x + β Δ y ) ) x + α Δ x + β Δ y 2 ,
2 ψ β 2 = 2 Δ ρ ̃ s ̱ P ̃ m ρ ̃ τ β = 2 Δ ρ ̃ s ̱ 2 I ρ ̃ τ + R ( Δ ρ ̃ s ̱ ρ ̃ τ ) 2 I ρ ̃ τ 3 = 2 Δ ρ ̃ s ̱ 2 I ρ ̃ τ + I 2 ρ ̃ τ R ( Δ ρ ̃ s ̱ ρ ̃ τ ) ( Δ ρ ̃ s ̱ ρ ̃ τ + Δ ρ ̃ s ̱ ρ ̃ τ ) ρ ̃ τ 2 = 2 Δ ρ ̃ s ̱ I 2 ρ ̃ τ ( 1 ρ ̃ τ 2 Δ ρ ̃ s ̱ 2 Δ ρ ̃ s ̱ 2 ρ ̃ τ 2 ) Δ ρ ̃ s ̱ r ,
2 ψ α 2 = 2 Δ ρ ̃ s [ I P ̃ m ] ρ ̃ τ α r = 2 Δ ρ ̃ s 2 Δ ρ ̃ s 2 I ρ ̃ τ + R ( Δ ρ ̃ s ρ ̃ τ ) 2 I ρ ̃ τ 3 = 2 Δ ρ ̃ s [ 1 I 2 ρ ̃ τ ( 1 ρ ̃ τ 2 Δ ρ ̃ s 2 Δ ρ ̃ s 2 ρ ̃ τ 2 ) ] Δ ρ ̃ s r .
2 ψ β α = 2 Δ ρ ̃ s [ I P ̃ m ] ρ ̃ τ β = 2 Δ ρ ̃ s I 2 ρ ̃ τ ( 1 ρ ̃ τ 2 Δ ρ ̃ s ̱ 2 Δ ρ ̃ s ̱ 2 ρ ̃ τ 2 ) Δ ρ ̃ s ̱ r = 2 Δ ρ ̃ s ̱ I 2 ρ ̃ τ ( 1 ρ ̃ τ 2 Δ ρ ̃ s 2 Δ ρ ̃ s 2 ρ ̃ τ 2 ) Δ ρ ̃ s r .

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