Invited article: a [corrected] unified evaluation of iterative projection algorithms for phase retrieval.

Iterative projection algorithms are successfully being used as a substitute of lenses to recombine, numerically rather than optically, light scattered by illuminated objects. Images obtained computationally allow aberration-free diffraction-limited imaging and the possibility of using radiation for which no lenses exist. The challenge of this imaging technique is transferred from the lenses to the algorithms. We evaluate these new computational "instruments" developed for the phase-retrieval problem, and discuss acceleration strategies.

[1]  G. G. Stokes.  "J." , 1890, The New Yale Book of Quotations.

[2]  D. Sayre.  Some implications of a theorem due to Shannon , 1952 .

[3]  H. Jagodzinski.  Vector space and its application in crystal-structure investigation , 1959 .

[4]  C. M. Reeves,et al.  Function minimization by conjugate gradients , 1964, Comput. J..

[5]  J R Fienup,et al.  Reconstruction of an object from the modulus of its Fourier transform. , 1978, Optics letters.

[6]  D. Sayre.  Prospects for long-wavelength X-ray microscopy and diffraction , 1980 .

[7]  James R. Fienup,et al.  Iterative Method Applied To Image Reconstruction And To Computer-Generated Holograms , 1980 .

[8]  Magnus R. Hestenes,et al.  Conjugate Direction Methods in Optimization , 1980 .

[9]  J R Fienup,et al.  Phase retrieval algorithms: a comparison. , 1982, Applied optics.

[10]  J. Solem,et al.  Microholography of Living Organisms , 1982, Science.

[11]  G. Newsam,et al.  Necessary conditions for a unique solution to two‐dimensional phase recovery , 1984 .

[12]  M. Powell.  Nonconvex minimization calculations and the conjugate gradient method , 1984 .

[13]  Aharon Levi,et al.  Image restoration by the method of generalized projections with application to restoration from magnitude , 1984 .

[14]  J. Walkup,et al.  Statistical optics , 1986, IEEE Journal of Quantum Electronics.

[15]  James R. Fienup,et al.  Phase-retrieval stagnation problems and solutions , 1986 .

[16]  Henry Stark,et al.  Image recovery: Theory and application , 1987 .

[17]  R G Paxman,et al.  Phase retrieval from experimental far-field speckle data. , 1988, Optics letters.

[18]  Rick P. Millane,et al.  Phase retrieval in crystallography and optics , 1990 .

[19]  J. H. Seldin,et al.  Hubble Space Telescope characterized by using phase-retrieval algorithms. , 1993, Applied optics.

[20]  S. Forte,et al.  To be published in the proceedings , 1995 .

[21]  J. Abrahams,et al.  Methods used in the structure determination of bovine mitochondrial F1 ATPase. , 1996, Acta crystallographica. Section D, Biological crystallography.

[22]  R. Millane.  Multidimensional phase problems , 1996 .

[23]  M. Fiddy,et al.  Blind deconvolution and phase retrieval from point zeros , 1996 .

[24]  J. Miao,et al.  Phase retrieval from the magnitude of the Fourier transforms of nonperiodic objects , 1998 .

[25]  J. Miao,et al.  Extending the methodology of X-ray crystallography to allow imaging of micrometre-sized non-crystalline specimens , 1999, Nature.

[26]  T. Isernia,et al.  Role of support information and zero locations in phase retrieval by a quadratic approach , 1999 .

[27]  J. Hajdu,et al.  Potential for biomolecular imaging with femtosecond X-ray pulses , 2000, Nature.

[28]  Andrew G. Glen,et al.  APPL , 2001 .

[29]  J. Miao,et al.  Imaging whole Escherichia coli bacteria by using single-particle x-ray diffraction , 2002, Proceedings of the National Academy of Sciences of the United States of America.

[30]  Heinz H. Bauschke,et al.  Phase retrieval, error reduction algorithm, and Fienup variants: a view from convex optimization. , 2002, Journal of the Optical Society of America. A, Optics, image science, and vision.

[31]  J. Miao,et al.  High resolution 3D x-ray diffraction microscopy. , 2002, Physical review letters.

[32]  James V. Burke,et al.  Optical Wavefront Reconstruction: Theory and Numerical Methods , 2002, SIAM Rev..

[33]  M. Howells,et al.  Phase recovery and lensless imaging by iterative methods in optical, X-ray and electron diffraction , 2002, Philosophical Transactions of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences.

[34]  Malcolm R. Howells,et al.  Off-axis zone plate monochromator for high-power undulator radiation , 2002, SPIE Optics + Photonics.

[35]  James V. Burke,et al.  Variational Analysis Applied to the Problem of Optical Phase Retrieval , 2003, SIAM J. Control. Optim..

[36]  S. Marchesini,et al.  X-ray image reconstruction from a diffraction pattern alone , 2003, physics/0306174.

[37]  Veit Elser.  Phase retrieval by iterated projections. , 2003, Journal of the Optical Society of America. A, Optics, image science, and vision.

[38]  I. Robinson,et al.  Three-dimensional imaging of microstructure in Au nanocrystals. , 2003, Physical review letters.

[39]  J. Zuo,et al.  Atomic Resolution Imaging of a Carbon Nanotube from Diffraction Intensities , 2003, Science.

[40]  Heinz H. Bauschke,et al.  Hybrid projection-reflection method for phase retrieval. , 2003, Journal of the Optical Society of America. A, Optics, image science, and vision.

[41]  J. Spence,et al.  Single molecule diffraction. , 2004, Physical review letters.

[42]  S. Marchesini,et al.  SPEDEN: reconstructing single particles from their diffraction patterns. , 2004, Acta crystallographica. Section A, Foundations of crystallography.

[43]  G. Oszlányi,et al.  Ab initio structure solution by charge flipping. , 2003, Acta crystallographica. Section A, Foundations of crystallography.

[44]  L J Allen,et al.  Retrieval of a complex-valued object from its diffraction pattern. , 2004, Physical review letters.

[45]  J. Spence,et al.  Application of a modified Oszlányi and Süto ab initio charge-flipping algorithm to experimental data. , 2004, Acta crystallographica. Section A, Foundations of crystallography.

[46]  M. Howells,et al.  Coherence and sampling requirements for diffractive imaging. , 2004, Ultramicroscopy.

[47]  J. Kirz,et al.  Apparatus for X-ray diffraction microscopy and tomography of cryo specimens , 2005 .

[48]  D. R. Luke.  Relaxed Averaged Alternating Reflections for Diffraction Imaging , 2004, math/0405208.

[49]  J. Kirz,et al.  Biological imaging by soft x-ray diffraction microscopy , 2005, Proceedings of the National Academy of Sciences of the United States of America.

[50]  J. Spence,et al.  Diffractive electron imaging of nanoparticles on a substrate , 2005, Nature materials.

[51]  D. Sayre,et al.  Electronic Reprint Foundations of Crystallography Reconstruction of a Yeast Cell from X-ray Diffraction Data Foundations of Crystallography Reconstruction of a Yeast Cell from X-ray Diffraction Data , 2022 .

[52]  Garth J. Williams,et al.  Three-dimensional mapping of a deformation field inside a nanocrystal , 2006, Nature.

[53]  W. H. Benner,et al.  Femtosecond diffractive imaging with a soft-X-ray free-electron laser , 2006, physics/0610044.

[54]  S. Marchesini,et al.  High-resolution ab initio three-dimensional x-ray diffraction microscopy. , 2005, Journal of the Optical Society of America. A, Optics, image science, and vision.

[55]  J. Spence,et al.  Ab initio phasing of X-ray powder diffraction patterns by charge flipping , 2006, Nature materials.