Iterative algorithms for phase retrieval from intensity data are compared to gradient search methods. Both the problem of phase retrieval from two intensity measurements (in electron microscopy or wave front sensing) and the problem of phase retrieval from a single intensity measurement plus a non-negativity constraint (in astronomy) are considered, with emphasis on the latter. It is shown that both the error-reduction algorithm for the problem of a single intensity measurement and the Gerchberg-Saxton algorithm for the problem of two intensity measurements converge. The error-reduction algorithm is also shown to be closely related to the steepest-descent method. Other algorithms, including the input-output algorithm and the conjugate-gradient method, are shown to converge in practice much faster than the error-reduction algorithm. Examples are shown.

Phase retrieval algorithms: a comparison.

Publisher Summary This chapter discusses the effects of atmospheric turbulence in optical astronomy, summarizes the present state of the theory, reviews the experimental checks that have been made, and discusses the implications in the domain of astronomical observations. Diffraction limited resolution has been obtained with large telescopes up to magnitude 13 by means of speckle interferometry. A Michelson interferometer is already working in the visible on a 20 m baseline up to magnitude 4, and many projects are nearing completion. However, such technological progress can be useful only if accurate quantitative measurements prove to be feasible through atmospheric turbulence. In addition, much work remains to be done before achieving a full understanding of atmospheric effects. The relationship between isoplanicity or speckle boiling and the structure of the atmosphere is still not clear.

V The Effects of Atmospheric Turbulence in Optical Astronomy

i»?The problem of phase retrieval, i.e., the recovery of a function given the magnitude of its Fourier transform, arises in various fields of science and engineering, including electron microscopy, crystallography, astronomy, and optical imaging. Exploring phase retrieval in optical settings, specifically when the light originates from a laser, is natural since optical detection devices [e.g., charge-coupled device (CCD) cameras, photosensitive films, and the human eye] cannot measure the phase of a light wave. This is because, generally, optical measurement devices that rely on converting photons to electrons (current) do not allow for direct recording of the phase: the electromagnetic field oscillates at rates of ~1015âHz, which no electronic measurement device can follow. Indeed, optical measurement/detection systems measure the photon flux, which is proportional to the magnitude squared of the field, not the phase. Consequently, measuring the phase of optical waves (electromagnetic fields oscillating at 1015âHz and higher) involves additional complexity, typically by requiring interference with another known field, in the process of holography.

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Phase Retrieval with Application to Optical Imaging: A contemporary overview

Phase problems occur in many scientific disciplines, particularly those involving remote sensing using a wave field. Although there has been much interest in phase retrieval in optics and in imaging in general over the past decade, phase retrieval has a much longer history in x-ray crystallography, and a variety of powerful and practical techniques have been developed. The nature of crystallography means that crystallographic phase problems are distinct from those in other imaging contexts, but there are a number of commonalities. Here the principles of phase retrieval in crystallography are outlined and are compared and contrasted with phase retrieval in general imaging. Uniqueness results are discussed, but the emphasis is on phase-retrieval algorithms and areas in which results in one discipline have, and may, contribute to the other.

http://xrm.phys.northwestern.edu/research/pdf_papers/1990/millane_josaa_1990.pdf

Phase retrieval in crystallography and optics

It is suggested that, given the magnitude of Fourier transforms sampled at the Bragg density, the phase problem is underdetermined by a factor of 2 for 1D, 2D, and 3D objects. It is therefore unnecessary to oversample the magnitude of Fourier transforms by 2× in each dimension (i.e., oversampling by 4× for 2D and 8× for 3D) in retrieving the phase of 2D and 3D objects. Our computer phasing experiments accurately retrieved the phase from the magnitude of the Fourier transforms of 2D and 3D complex-valued objects by using positivity constraints on the imaginary part of the objects and loose supports, with the oversampling factor much less than 4 for 2D and 8 for 3D objects. Under the same conditions we also obtained reasonably good reconstructions of 2D and 3D complex-valued objects from the magnitude of their Fourier transforms with added noise and a central stop.

/pdf/phase-retrieval-from-the-magnitude-of-the-fourier-transforms-1erncptehx.pdf

Phase retrieval from the magnitude of the Fourier transforms of nonperiodic objects

On the ambiguity of the image reconstruction problem

A method suggested earlier for the restoration of speckle images from the average phase of the spectrum is described and investigated in detail. Much attention is concentrated on the reconstruction ambiguity that arises because of inexact knowledge of the spectrum phase, uncertainty about the position and the size of the object, and the probable presence of symmetric factors in the spectrum. Several algorithms that permit unambiguous object image reconstruction (to within arbitrary symmetric factors) have been developed. Results of model calculations for simple one- and two-dimensional images are given.

Speckle interferometry image reconstruction from the Fourier transform phase.

The present method of image restoration from the phase of its spectrum requires the solution of sets of linear equations whose number is equal to that of the image elements. For real two-dimensional images, this leads to sets involving about 103 equations or more. A method is suggested in which the initial stage consists of reconstructing the real and imaginary components of discrete Fourier transforms of image sections (or scans), rather than the image proper. The multi-dimensional problems is thus reduced to several one-dimensional ones, and the reconstruction procedure is substantially simplified.

An Improved Method for Reconstructing Two- and Multidimensional Images from the Phase of Their Fourier Spectrum

A hybrid device comprising two identical hybrid transformers is studied both theoretically and experimentally. The circuit is shown to possess several properties unknown even for the most advanced prior configurations : viz. identity of all inputs over a wide frequency range with respect to matching and isolation ; a theoretically unbounded frequency range of isolation ; a possibility of obtaining arbitrary power division ratios from nought to infinity with the turns ratio varying over a rather restricted range. A number of broadband hybrid devices with varied parameters have been developed, as well as circuit elements on the basis thereof : viz. directional couplers ; phase shifters, phase changing (including quadrature) hybrids; beam forming matrices and directional filters.

L. G. Sodin

Papers

On the ambiguity of the image reconstruction problem

Speckle interferometry image reconstruction from the Fourier transform phase.

An Improved Method for Reconstructing Two- and Multidimensional Images from the Phase of Their Fourier Spectrum

A broadband hybrid device