In the choice of an estimator for the spectrum of a stationary time series from a finite sample of the process, the problems of bias control and consistency, or "smoothing," are dominant. In this paper we present a new method based on a "local" eigenexpansion to estimate the spectrum in terms of the solution of an integral equation. Computationally this method is equivalent to using the weishted average of a series of direct-spectrum estimates based on orthogonal data windows (discrete prolate spheroidal sequences) to treat both the bias and smoothing problems. Some of the attractive features of this estimate are: there are no arbitrary windows; it is a small sample theory; it is consistent; it provides an analysis-of-variance test for line components; and it has high resolution. We also show relations of this estimate to maximum-likelihood estimates, show that the estimation capacity of the estimate is high, and show applications to coherence and polyspectrum estimates.

Spectrum estimation and harmonic analysis

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.

Conditions to retrieve pulse intensity and chirp from autocorrelation signals are mathematically discussed. The interferometric SHG autocorrelation signal and the interferogram of fundamental light are shown to be a sufficient data set for the pulseform retrieval. This measurement scheme is experimentally demonstrated on laser diode pulses. The method utilizes Fourier analysis of the signals and iterative calculations, and is especially suited to the measurement of pulses having weak peak intensity. >

General method for ultrashort light pulse chirp measurement

A method is given for the evaluation, in transmission electron microscopy, of the amplitude and phase from the intensity distribution of an electron micrograph. The method requires a minimum of two micrographs taken under different defocus conditions. The iterative scheme requires only the relative defocus between micrographs, and the procedure is valid both in bright-field and dark-field microscopy for any specified coherence of the electron source. Assumptions on the scattering properties of the specimen, such as the weak-phase-weak-amplitude object, are not required. For a complete determination of the amplitude-phase distribution for electron transmission through the specimen, the electron micrograph must be corrected for the effect of lens aberrations and defocusing to give the electron wavefunction immediately after transmission; only in the case of a weak-phase object can this wavefunction be directly related to the projected potential distribution in the object. Inelastic electron scattering is explicitly omitted from the analysis presented.

Comment onA method for the solution of the phase problem in electron microscopy

This paper describes, analyzes, and implements an approach to far-field determination from phaseless measurements over two planar surfaces. A proper formulation of the problem is considered as a quadratic inverse one whose data is the square amplitude of the near field. A solution is introduced as the global minimum of an appropriate functional. Next, to perform such a minimization procedure, a finite dimensional representation of the field radiated by sources whose plane-wave spectrum becomes negligible outside a fixed angular domain is used. A detailed investigation of the properties of the mapping connecting the unknowns to the data makes it possible to analyze how to escape from the local minima possibly met in the course of the minimization procedure. To this end, the crucial role of the availability of phaseless data over two different sarfa,ces and of appropriate weights in the functional definition is emphasized, and a reliable iterative procedure converging on the solution, regardless of the starting point, is thus obtained. This property is confirmed by experimental results concerning near-zone data from a shaped reflector at 9 Ghz. It can be readily appreciated that when only the field intensity is detected the complexity and the cost of the equipment required for near-field techniques in antenna testing and diagnostics can be reduced to a very large extent.

Radiation pattern evaluation from near-field intensities on planes

The phase retrieval problem arises in applications of electromagnetic theory in which wave phase is apparently lost or impractical to measure and only intensity data are available. The mathematics of the problem provides unusual insights into the nature of electromagnetic fields. The theory is reviewed and illustrated. The basic issue of the phase retrieval problem, stated for a one-dimensional field, is that although a unique Fourier transform relation exists between the field F(x) in the Fraunhofer plane and the field u(x') in the object plane, the infinite fold phase ambiguity which appears as the result of the possibilities of conjugating the zeros of F(z), z = x + jy implies that additional information or processing of the object wave must be available to obtain the phase. Among the possible solutions which are described are reference beam addition, apodization and the use of multiple intensity distributions, permitting the use of iterative computational procedures in some applications.

L. Taylor

Papers

The phase retrieval problem