Abstract: A new method for solving the phase problem in two dimensions is presented. The solution is noniterative and is based on locating the zeros of one-dimensional strips of a single two-dimensional intensity distribution. A numerical example is included.

Abstract: Two extensions of our previously reported ‘crude phase estimation’ procedure are developed. They both serve to provide starting phases (for Fienup's iterative algorithms) which are ‘improved’ in the sense that rates of convergence of these algorithms are accelerated. The theory is illustrated by examples of phase retrieval from phaseless Fourier data. Images are reconstructed from Fourier magnitudes which were either computer-generated or measured in our optical laboratory, set up to simulate Labeyrie's speckle interferometry.

Abstract: A modification of a previously published two-dimensional phase-retrieval algorithm based on the sampling theorem [ Opt Commun47, 380 ( 1983)] is introduced With an appropriate choice of the intermediate sampling grid, a significant reduction in the number of computations is achieved, because an independent system of equations may be solved for each line of the Fourier transform of the object The results of computer simulations on real objects are given The algorithm converges rapidly

Abstract: Recent work on the Fourier-phase problem is shown to be relevant to the reconstruction of the relative phases of fields scattered to a large number of separated inaccurately surveyed locations. A proposed approach to phase retrieval is illustrated with a computational example. It is assumed that the waveforms of the scattered fields can be recorded at each location.

Abstract: The constraints laid on the phase of a Fourier transform by its intensity are reviewed in the contexts of well known phase problems. The considerable differences between phase problems involving one-dimensional and multi-dimensional images, and finite-sized (as arise in astronomy, for instance) and periodic (as occur in crystallography) images, are explained. The crucial importance, for uniqueness questions, of the concept of the image-form (and also its most compact manifestation) is emphasised, as is the almost always unique connection between the image-form of a positive multi-dimensional image and the intensity of its Fourier transform. The current status of phase recovery algorithms, as regards Fourier transforms of finite-sized images, is assessed. The necessity for composite algorithms, incorporating simple but powerful constructions, is pleaded and reinforced by computational examples illustrating our previously reported defogging routine and a new procedure called fringe magnification.

Abstract: The optical surface profile is derived from the intensity distribution of the light diffracted by the surface roughness using phase retrieval techniques. This method is applied to a surface of periodic roughness such as a diamond-turned metal surface. Computer simulations and experimental results make it clear that the method is useful for a surface whose root-mean-square roughness is more than ~30 A.

Abstract: The analytic properties of two-dimensional band-limited functions are discussed. In practice, only a limited number of intensity samples are available, and so we choose to model the spectrum as a finite degree polynomial. The set of reducible finite degree multi-variate polynomials is of measure zero and unique recovery from noise free Fourier magnitude is expected in almost all cases. We pursue a new algorithm based on finding the complex zeros of 1-D lines of the data set which requires only that the intensity is sampled at twice the Nyquist rate or greater. All solutions compatible with the Fourier magnitude samples are generated, including ambiguities should they exist. An exact solution to the phase retrieval problem, given a polynomial model, may be regarded as factorization. We discuss the relationship of this approach to factorization and iterative procedures and describe problems arising from data truncation and the presence of noise.

Abstract: The phase problem is examined as it occurs in stellar speckle interferometry, wherein one assumes that the image to be reconstructed is of finite spatial extent Algorithms for reconstructing the Fourier phase of an image are studied, as well as a detailed investigation of the two-dimensional uniqueness question undertaken. Criteria for unique phase retrieval are derived for two-dimensional discrete objects, with particular emphasis placed on the importance of support constraints. The support is shown to have a profound effect on the uniqueness properties of the function defined upon it. It is established that some, such as Eisenstein's support, ensure a single solution independent of the function defined on that region, excluding mild restrictions necessary to define the region. An alternate method of demonstrating the solution uniqueness associated with Eisenstein's support is presented. Moreover, this approach is generalized to produce a large number of supports which are not described by Eisenstein's criterion, but which nonetheless guarantee solution uniqueness. An algorithm is developed to test an arbitrary discrete support for membership in this special family. A related criterion is also derived which, when satisfied, ensures solution uniqueness when support information is explicitly incorporated into a reconstruction algorithm. A one-dimensional phase retrieval technique for stellar speckle interferometric imaging is presented, which is based on the identification and manipulation of complex zeros. The algorithm is applied to data produced in a laboratory simulation of stellar speckle. It is found that it is possible to recover one-dimensional images in this manner, however, the quality of the reconstruction is generally no better than that produced by the Knox-Thompson algorithm, and the roots method is substantially more difficult to implement. A brief survey of modulus-only reconstruction algorithms is undertaken, with an emphasis on Fienup's iterative approach. An implementation of Fienup's algorithm proves to be capable of reconstructing a limited number of objects from the Fourier modulus, but in general requires stronger constraints than nonnegativity to converge to the correct solution. The algorithm is capable of reconstructing rather complex images defined on Eisenstein's support, in addition to others which are defined on a sufficiently irregular region.

Abstract: A phase retrieval algorithm, developed and implemented by Perkin-Elmer, is used as a technique for in-orbit alignment of the optical system in NASA's Space Telescope. Reasonably accurate estimates of wavefront aberrations are obtained from measurements of the system point spread function (PSF). Further, the accuracy of the estimates increases as the magnitude of the aberrations decreases. Therefore, even if initial aberrations are large and result in crude estimates, a partial correction of the system can be made. A reapplication of the algorithm to the improved PSF will yield more accurate estimates of the uncorrected aberrations. In this fashion, a few iterations of the phase retrieval algorithm will allow perfect system correction from an aberrated state. The algorithm consists of initially defining an error function. The goal is to minimize the error function with respect to a set of parameters. Using Zernike polynomials to describe the phase in the pupil, we are able to write an analytic expression representing the PSF. An error function that can be minimized with respect to the coefficients of the Zernike polynomials is then calculated.

Abstract: Fienup's iterative phase retrieval algorithms are applied to an image (confined to a support having a concave perimeter) whose phase can be chosen either zero, or varying arbitrarily between 0 and 7, or quite arbitrary. Reconstructed images are confined (or, more precisely, we attempt to confine them) to either the support or the rectangle just enclosing it. Both positive reconstructed images are faithful (as expected) when positivity is enforced. Bipolar and complex images reconstructed within the support are recognizable, but are scarcely recognizable when reconstructed within the rectangle. The bipolar reconstructed images are superior to the complex ones. These results reinforce the general experience that a support-constraint tends by itself to be too weak to ensure faithful image reconstruction.

Abstract: The paper contains a discussion of the phase retrieval problem in two dimensions and proposes criteria to select those resolutions of the discrete ambiguity of the zero trajectories which are compatible with the analyticity in two variables of the scattered field.

Abstract: In this paper we deal with the problem of retrieving a finite-extent signal from the magnitude of its Fourier transform. We will present a brief review of the algebraic problem of the uniqueness of the solution for both discrete and continuous phase retrieval models. Several important issues which are yet unresolved will be pointed out and discussed. We will then consider the discrete phase retrieval problem as a special case of a more general problem which consists of recovering a real-valued signal x from the magnitude of the output of a linear distortion: |Hx|(j), j = 1, ..., n . An important result concerning the conditioning of this problem will be obtained for this general setting by means of algebraic-geometric techniques. In particular, the problems of the existence of a solution for phase retrieval, conditioning of the problem and stability of the (essentially) unique solution will be addressed.

Abstract: In this paper, we deal with the problem of retrieving a finite-extent function from the magnitude of its Fourier transform. This so-called phase retrieval problem will first be posed under its different underlying models. We will present a brief review of the main results known in this area for both discrete and continuous phase retrieval models giving special emphasis to the algebraic problem of the uniqueness of the solution. Several important issues which are yet unresolved will be pointed out and discussed. We will then consider the discrete phase retrieval problem as a special case of a more general problem which consists of recovering a real-valued sequence x from the magnitude of the output of a linear distortion: $| Hx | ( j ),\, j = 1, \cdots ,n$. A number of important results will be obtained for this general setting by means of algebraic-geometric techniques. In particular, the problems of the existence of a solution for phase retrieval, number of feasible solutions, stability of the (essential...