Phase retrieval by iterated projections.

By Veit Elser. Journal of The Optical Society of America A-optics Image Science and Vision, 2001

712 citations recorded by the source corpus.

Several strategies in phase retrieval are unified by an iterative "difference map" constructed from a pair of elementary projections and three real parameters. For the standard application in optics, where the two projections implement Fourier modulus and object support constraints, respectively, the difference map reproduces the "hybrid" form of Fienup's input-output map when a particular choice is made for two of the parameters. The geometric construction of the difference map illuminates the distinction between its fixed points and the recovered object, as well as the mechanism whereby the form of stagnation encountered by alternating projection schemes is avoided. When support constraints are replaced by object histogram or atomicity constraints, the difference map lends itself to crystallographic phase retrieval. Numerical experiments with synthetic data suggest that structures with hundreds of atoms can be solved.

Subfields: Materials Chemistry, Radiation

View Paper: https://www.semanticscholar.org/paper/fdcb0b71c70e1488b68bbad43ae5faeb0144c774

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Citations in the corpus

On the use of deep learning for computational imaging (2020, methodology): “Classical iterative solutions to this problem are the Gerchberg-Saxton algorithm; 39 the input-output algorithm and its variants proposed by Fienup; 39–43 and the gradient descent 44,45 with its well-known variants such as steepest descent and conjugate gradient algorithms.”
SURVEY: SIXTY YEARS OF DOUGLAS–RACHFORD (2018, methodology): “he augmented the study 19 of DR for phase retrieval by replacing support constraints with object histogram and atomicity constraints for crystallographic phase retrieval [82, Section 5]. In 2001 when [82] was submitted, Elser was not yet aware of Lions’ and Mercier’s characterization of DR as the Do”
Relaxed averaged alternating reflections for diffraction imaging (2004, methodology): “The HPR algorithm can also be viewed a special case of the three-parameter difference map recently proposed by Elser [7].”
Solving Random Quadratic Systems of Equations Is Nearly as Easy as Solving Linear Systems (2015, methodology): “It is worth noting that apart from WF, various other nonconvex procedures have been proposed as well for phase retrieval, including the error reduction schemes dating back to Gerchberg-Saxton and Fienup [20,21], iterated projections [19], alternating minimization [32], generalized approximate messag”
Phase retrieval algorithms: a personal tour [Invited]. (2013, neutral): “Elser [84] developed the “difference map” algorithm, which also has HIO as a special case.”
Fixed-Point Algorithms for Inverse Problems in Science and Engineering (2011, methodology): “A very interesting development originates with Elser [42], who has recently very successfully applied the Douglas-Rachford algorithm to various continuous and discrete, nonconvex problems [43, 50].”
An improved ptychographical phase retrieval algorithm for diffractive imaging. (2009, background): “Further examples of common constraints are nonnegativity, atomicity and realness [5].”
Highly integrated optical phased arrays: photonic integrated circuits for optical beam shaping and beam steering (2017, methodology): “Gerchberg-Saxton algorithm [24, 25], phase-only beam shaping can be used to obtain arbitrary far-field images.”