A feasible set approach to the crystallographic phase problem

Marks, Laurence D.; Sinkler, W.; Landree, Eric

doi:10.1107/s0108767398014408

Cited by 52 publications

(39 citation statements)

References 24 publications

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“…It is of fundamental importance in numerous areas of applied physics and engineering [1][2][3][4][5][6][7][8] and has been studied for over 40 years (see Refs. 9-11 and the references therein).…”

Section: Introductionmentioning

confidence: 99%

Phase retrieval, error reduction algorithm, and Fienup variants: a view from convex optimization

2002

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The phase retrieval problem is of paramount importance in various areas of applied physics and engineering. The state of the art for solving this problem in two dimensions relies heavily on the pioneering work of Gerchberg, Saxton, and Fienup. Despite the widespread use of the algorithms proposed by these three researchers, current mathematical theory cannot explain their remarkable success. Nevertheless, great insight can be gained into the behavior, the shortcomings, and the performance of these algorithms from their possible counterparts in convex optimization theory. An important step in this direction was made two decades ago when the error reduction algorithm was identified as a nonconvex alternating projection algorithm. Our purpose is to formulate the phase retrieval problem with mathematical care and to establish new connections between well-established numerical phase retrieval schemes and classical convex optimization methods. Specifically, it is shown that Fienup's basic input-output algorithm corresponds to Dykstra's algorithm and that Fienup's hybrid input-output algorithm can be viewed as an instance of the Douglas-Rachford algorithm. We provide a theoretical framework to better understand and, potentially, to improve existing phase recovery algorithms.

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Section: Introductionmentioning

confidence: 99%

Phase retrieval, error reduction algorithm, and Fienup variants: a view from convex optimization

2002

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“…Not unlike X-ray crystallography, the following methods are used to solve crystal structures from electron diffraction data: All of them assume that the data are at least pseudo-kinematic, and use techniques that are robust against systematic and random errors. The list is not exhaustive (see, for example, Marks, Sinkler and Landree, 1999), and new techniques are still being developed, but it covers more than 95% of published structures. We will examine each technique in turn.…”

Section: Solving Structures From Electron Diffraction Datamentioning

confidence: 98%

Nanocrystals: Solving Crystal Structures Using Electron Crystallography

Gilmore¹

2003

Crystallography Reviews

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Electron crystallography uses the electron microscope to generate diffraction patterns or high resolution images which can be used to solve crystal structures from nano-sized crystallites. The diffraction data collected are subject to systematic errors in intensity arising from n-beam dynamical scattering, secondary scattering, sample bending and radiation damage, but nonetheless the technique yields structural information when all other diffraction methods fail. Solving such structures can be routine especially from inorganic materials and intermetallic compounds, but, in general, the process can be difficult as can the validation of the proposed model. Direct methods, maximum entropy, Patterson techniques, model building and the use of image data can all be employed, and structures of high complexity can be solved and refined in favourable cases. Even low resolution protein data can be analysed in this way, and it is an important tool in surface crystallography.

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“…In pure mathematical terms (see Ref. [6] and references therein) there is no guarantee that they will always work (the problem is non-convex), and it is nearly impossible at present to predict when they will fail.…”

Section: Introductionmentioning

confidence: 99%