Equations for the propagation of phase and irradiance are derived, and a Green’s function solution for the phase in terms of irradiance and perimeter phase values is given A measurement scheme is discussed, and the results of a numerical simulation are given Both circular and slit pupils are considered An appendix discusses the local validity of the parabolic-wave equation based on the factorized Helmholtz equation approach to the Rayleigh–Sommerfeld and Fresnel diffraction theories Expressions for the diffracted-wave field in the near-field region are given

Deterministic phase retrieval: a Green’s function solution

Suppose we wish to recover a signal x in C^n from m intensity measurements of the form | |^2, i = 1, 2,..., m; that is, from data in which phase information is missing. We prove that if the vectors z_i are sampled independently and uniformly at random on the unit sphere, then the signal x can be recovered exactly (up to a global phase factor) by solving a convenient semidefinite program---a trace-norm minimization problem; this holds with large probability provided that m is on the order of n log n, and without any assumption about the signal whatsoever. This novel result demonstrates that in some instances, the combinatorial phase retrieval problem can be solved by convex programming techniques. Finally, we also prove that our methodology is robust vis a vis additive noise.

PhaseLift: Exact and Stable Signal Recovery from Magnitude Measurements via Convex Programming

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

X-ray sources are developing rapidly and their coherent output is growing correspondingly. The increased coherent flux from modern X-ray sources is being matched with an associated development in experimental methods. This article reviews the literature describing the ideas that utilize the increased brilliance from modern X-ray sources. It explores how ideas in coherent X-ray science are leading to developments in other areas, and vice versa. The article describes measurements of coherence properties and uses this discussion as a base from which to describe partially coherent diffraction and X-ray phase-contrast imaging, with applications in materials science, engineering and medicine. Coherent diffraction imaging methods are reviewed along with associated experiments in materials science. Proposals for experiments to be performed with the new X-ray free-electron lasers are briefly discussed. The literature on X-ray photon-correlation spectroscopy is described and the features it has in common with other coherent X-ray methods are identified. Many of the ideas used in the coherent X-ray literature have their origins in the optical and electron communities and these connections are explored. A review of the areas in which ideas from coherent X-ray methods are contributing to methods for the neutron, electron and optical communities is presented.

Coherent methods in the X-ray sciences

There exists a vast literature on convergence rates results for Tikhonov regularized minimizers. We are concerned with the solution of nonlinear ill-posed operator equations. The first convergence rates results for such problems were developed by Engl, Kunisch and Neubauer in 1989. While these results apply for operator equations formulated in Hilbert spaces, the results of Burger and Osher from 2004, more generally, apply to operators formulated in Banach spaces. Recently, Resmerita and co-workers presented a modification of the convergence rates result of Burger and Osher which turns out to be a complete generalization of the rates result of Engl and co-workers. In all these papers relatively strong regularity assumptions are made. However, it has been observed numerically that violations of the smoothness assumptions of the operator do not necessarily affect the convergence rate negatively. We take this observation and weaken the smoothness assumptions on the operator and prove a novel convergence rate result. The most significant difference in this result from the previous ones is that the source condition is formulated as a variational inequality and not as an equation as previously. As examples, we present a phase retrieval problem and a specific inverse option pricing problem, both previously studied in the literature. For the inverse finance problem, the new approach allows us to bridge the gap to a singular case, where the operator smoothness degenerates just when the degree of ill-posedness is minimal.

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A convergence rates result for Tikhonov regularization in Banach spaces with non-smooth operators

A method for solving the phase retrieval problem is proposed. The method consists of measuring the modulus of the Fourier transform of the unknown function and the modulus of the Fourier transform of the product of the unknown function and an exponential. From these two measurements, the location of the complex zeros of the analytic continuation of the Fourier transform of the unknown function may easily and quickly be deduced and the unknown function constructed.

J.G. Walker

Papers

The Phase Retrieval Problem