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. The purpose of this paper 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 inputoutput 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. This work provides a theoretical framework to better understand and, potentially, improve existing phase recovery algorithms. 1 1

A computational model of auditory analysis is described that is inspired by psychoacoustical and neurophysiological findings in early and central stages of the auditory system. The model provides a unified multiresolution representation of the spectral and temporal features of sound likely critical in the perception of timbre. Several types of complex stimuli are used to demonstrate the spectrotemporal information extracted and represented by the model. Also outlined are several reconstruction algorithms to resynthesize the sound so as to evaluate the fidelity of the representation and contribution of different features and cues to the sound percept. Simplified versions of this model representations have already been used in a variety of applications, as in the assessment of speech intelligibility [Elhilali et al., 2003, Chi et al., 1999] and in explaining the perception of monaural phase sensitivity [Carlyon and Shamma, 2002]. 1 1.

Several strategies in phase retrieval are unified by an iterative “difference map” constructed from a pair of elementary projections and a single real parameter β. 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’s1 input-output map for β = 1. Other values of β are equally effective in retrieving phases but have no input-output counterparts. The geometric construction of the difference map illuminates the distinction between its fixed points and the recovered object, as well as the mechanism whereby stagnation 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. c○2008 Optical Society of America OCIS codes: 100.5070 1

this paper we show how such requirements can be handled by general-purpose procedures. The power of the algorithms lie in 1a2 their simplicity, and 1b2 the fact that the solutions are not confined to a predetermined structure, which leaves more flexibility to arrive at not only better solutions, but also at solutions that were not previously considered possible because of a, perhaps mistaken, a priori confinement of the solution. The purpose of the paper is thus twofold: 1a2 introduce, review, and enhance some new concepts in the design of optical PR systems 1for linear and nonlinear systems2, and 1b2 demonstrate the applicability of projection-based methods for the achievement of superior performance in the above PR systems under a wide and quite stringent range of requirements

The phase retrieval problem, fundamental in applied physics and engineering, asks to determine the phase of a complex-valued function from modulus data and additional a priori information. Recently, we identified two important methods for phase retrieval, namely Fienup’s Basic Input-Output (BIO) and Hybrid Input-Output (HIO) algorithms, with classical convex projection methods and suggested that further connections between convex optimization and phase retrieval should be explored. Following up on this work, we introduce a new projection-based method, termed the Hybrid Projection Reflection (HPR) algorithm, for solving phase retrieval problems featuring nonnegativity constraints in the object domain. Motivated by properties of the HPR algorithm for convex constraints, we recommend an error measure studied by Fienup more than twenty years ago. This error measure, which has received little attention in the literature, lends itself to an easily implementable stopping criterion. In numerical experiments, we found the HPR algorithm to be a competitive alternative to the HIO algorithm and the stopping criterion to be reliable and robust. 1 1

This paper proposes an associative memory neural network whose limiting state is the nearest point in a polyhedron from a given input. Two implementations of the proposed associative memory network are presented based on Dykstra's algorithm and a fixed point theorem for nonexpansive mappings. By these implementations, the set of all correctable errors by the network is characterized as a dual cone of the polyhedron at each pattern to be memorized, which leads to a simple amplifying technique to improve the error correction capability. It is shown by numerical examples that the proposed associative memory realizes much better error correction performance than the conventional one based on POCS at the expense of the increase of necessary number of iterations in the recalling stage.

this paper we address the following problem: What distribution of amplitude a(x) is needed in the transmittance function of a pure-amplitude hologram or spatial light modulator to generate a far-field light intensity I(u)? Here, x is the displacement vector in the plane of the hologram, and u is the displacement vector in the Fourier, or image plane. We examine both continuous and quantized amplitude holograms. With the increasing interest in diffractive optics,

this paper the authors attempt to demonstrate, with examples, the problems associated with convergence in iterative optical design by projections in a vector-space setting. In particular, we address the following: What is the difference between strong convergence, weak convergence, and summed-distance error (SDE) convergence, and when can the designer expect to see each? What are the hazards of imposing nonconvex constraints, especially when there are more than two such constraints? What modification in the projection algorithm will allow SDE convergence even when there are many nonconvex constraints ? What is the influence, if any, of the starting point in the iteration? And finally, what happens when constraints are inconsistent? Are reasonable solutions still possible?

The phase retrieval problem, fundamental in applied physics and engineering, asks to determine the phase of a complex-valued function from modulus data and additional a priori information. Recently, we identified two important methods for phase retrieval, namely Fienup's Basic Input-Output (BIO) and Hybrid Input-Output (HIO) algorithms, with classical convex projection methods and suggested that further connections between convex optimization and phase retrieval should be explored. Following up on this work, we introduce a new projection-based method, termed the Hybrid Projection Reflection (HPR) algorithm, for solving phase retrieval problems featuring nonnegativity constraints in the object domain. Motivated by properties of the HPR algorithm for convex constraints, we recommend an error measure studied by Fienup more than twenty years ago. This error measure, which has received little attention in the literature, lends itself to an easily implementable stopping criterion. In numerical experiments, we found the HPR algorithm to be a competitive alternative to the HIO algorithm and the stopping criterion to be reliable and robust.

The Schulz-Snyder iterative algorithm for phase retrieval attempts to recover a nonnegative function from its autocorrelation by minimizing the I-divergence between a measured autocorrelation and the autocorrelation of the estimated image. We illustrate that the Schulz-Snyder algorithm can become trapped in a local minimum of the I-divergence surface. To show that the estimates found are indeed local minima, sufficient conditions involving the gradient and Hessian matrix of the I-divergence are given. Then, we build a brief proof showing how an estimate that satisfies these conditions is a local minium. The conditions are used to perform numerical tests determining local minimality of estimates. Along with the tests, related numerical issues are examined,