Abstract: In the phase retrieval problem one seeks to recover an unknown function g(t) from the amplitude mod g(k) mod of its Fourier transform. Since phase and amplitude are, in general, independent of each other, it is necessary to make use of other kinds of information which implicitly or explicitly constrain the admissible solutions g(t). In this paper we survey a variety of results explaining circumstances under which g(t) may be uniquely recovered from mod g(k) mod and supplementary information. A number of explicit formulae for the phase are discussed. We pay particular attention to the phase retrieval problem as it arises in certain inverse-scattering applications.

Abstract: Recent developments in the application of zeros (of the analytically continued spectrum of a compact two-dimensional image) in solving deconvolution and phase retrieval problems are reviewed. New algorithms for use in the presence of noise are described and demonstrated. These include algorithms for deconvolution where the point-spread function is approximately known, for ensemble blind deconvolution (such as is required for ensembles of astronomical speckle images), and for phase retrieval (itself a special case of blind deconvolution). Many of the ideas embodied in the algorithms were foreshadowed by Bates et al. [ J. Opt. Soc. Am. A7, 468 ( 1990)]. Simulated images are employed in the examples shown, except for phase retrieval, where successful recovery of the phase error in the aperture of a radio telescope is demonstrated.

Abstract: We demonstrate a solution to the problem pertaining to the reconstruction of the profile of an interface separating two media of different refractive index, using near-field scattered intensity measurements. This is achieved by integration of the intensities of the scattered near fields over several angles of incidence, hence producing an effective incoherent source on the surface.

Abstract: A new method has been developed to remove noise from the deformation phase map obtained by a phase-shifting electronic speckle pattern interferometry. Unlike usual methods, it estimates almost noise-free phase values directly from the distributions of the intensity differences of four interference patterns by a least-squares fit. The fluctuations of uniform deformation phases are reduced to less than 0.05 rad with a 5 × 5 pixel fitting window. The so-called sawtooth phase jumps that are due to the use of arctangent functions are retained sharply in this method.

Abstract: A system and method for creating a three-dimensional image of an opaque object utilize frequency diverse coherent illumination in combination with an opacity constraint in performing three-dimensional phase retrieval and/or profile retrieval. An opaque object is one which exhibits only surface scattering and no volume scattering over volumes that extend beyond the desired range resolution. The system and method require only Fourier intensity information to create a three-dimensional image which avoids the difficulties associated with prior art systems and methods requiring phase information to produce similar images. Furthermore, the system and method of the present invention do not require imaging optics, precise alignment of optical components, or precise phase stability of the coherent illumination source.

Abstract: An analytical solution of the curvature-sensing equation is proposed. The approach developed is based on the construction of a set of special polynomials that allows a solution in terms of Zernike polynomials by direct integration. The calculation results show that it is possible to retrieve the Zernike coefficients of wave-front distortion independently of each other by use of a few points of integration. Because the method developed requires a small number of operations, it allows the wave-front distortion of interest to be obtained in real time, which is important for adaptive-optics applications.

Abstract: The maps of phase derivatives are extracted here by direct manipulation of phase-shifted interferograms. There are three main advantages: There is no need for prior phase evaluation or unwrapping procedures, and only a short processing time is needed. By digital integration of the derivatives the absolute phase map can also be retrieved without unwrapping procedures. A general description of the method is presented and discussed. For example, the proposed technique has been applied to the study of the deformation of a test object by the manipulation of four phase-shifted interferograms.

Abstract: Publisher Summary This chapter discusses the phase retrieval using the logarithmic Hilbert transform and the exponential filter method with their application to some related problems. A characteristic of the current phase retrieval method is that, the uniqueness of the solution by this method is ensured mathematically although it requires two (in 1D cases) or three (in 2D cases) Fourier intensity distributions. The theoretical background for the properties of entire functions is presented in the chapter. The chapter introduces the algorithm for applying the one-dimensional (1D) phase retrieval method to the two-dimensional (2D) case, and presents the simulated and experimental examples of reconstructing 2D objects. The application of the current method to the problems related to phase retrieval is discussed. The phase retrieval problem is only solvable if the complex amplitude function at the observation plane belongs to a particular class. This particular function is called “an entire function of exponential type.” The discussion in this chapter is also restricted to 1D case. The following two basic assumptions are set: (1) a scattering object is of finite extent and (2) a Fourier transform relationship exists between the object function and the scattered complex amplitude function in the diffraction region. According to these two assumptions, the phase retrieval problem on a Fourier-transforming optical system with a converging lens is discussed in the chapter.

Abstract: A phase retrieval algorithm and measured results applicable to the bipolar planar near-field technique in which the measurement planes have limited separation has been presented. An optimal sampling interpolation (OSI) algorithm operating on the measured squared amplitude bi-polar near-field data has also been presented. Phase retrieval results were generally found to be very good. The success of any such algorithm, however, is dependent on many factors including, for example, the proximity of the "initial guess" to the actual solution. The presented measured results indicate that the phase retrieval problem is a tractable problem in the bi-polar planar near-field measurement modality.

Abstract: The complex amplitude function in the waist of a one-dimensional Hermite–Gaussian model beam is measured by a noniterative method based on the Page distribution function in the two-dimensional spatial–angular phase space. The measurement procedure is realized by an extension of the knife-edge method. The model beam is generated by means of a phase plate in conjunction with a He–Ne laser.

Abstract: The authors demonstrate mathematically that the direction of an antenna beam (antenna pointing) can be determined by the centre-of-gravity of its beam intensity. The technique is validated using measurements on a 1.118 m 94 GHz Cassegrain reflector and is seen as pivotal in the future application of the phase retrieval technique for near-field/far-field prediction.

Abstract: A paraxial diffractive element is defined by a signal wave which is specified in a window of finite extent. The iterative Fourier transform algorithm is a well-known method to iteratively encode paraxial diffractive elements by making use of parameters of freedom. However, the algorithm suffers from a slow convergence if the parameters of freedom are limited. Thus, there is a demand for more efficient encoding strategies. Fienup considered to use descent methods for solving phase-retrieval problems which turned out to be very efficient. In this paper, we modify his theory that it can be applied for the design of diffractive phase elements. Computer simulations document that descent methods have a clearly better performance than the iterative Fourier transform algorithm.

Abstract: A spatial method of wave-front phase detection from an interferogram is presented. The method uses data-dependent systems methodology, an approach that extends and improves the way the stochastic autoregressive moving average models are obtained and interpreted. Its application to interference data addresses the fundamental problem of recovering the self-coherence function commonly used to retrieve the wave-front phase. The self-coherence function is efficiently computed by means of a complex autoregressive model and is used for surface reconstruction. The method is shown to be robust and suitable for surface testing. The correspondence of the data-dependent systems methodology and its physical meaning as related to the classical interferometry are presented. The theoretical development is illustrated by experimental implementation, with the results obtained from one- and two-dimensional interferometric fringe analysis of a computer hard disk.

Abstract: We discuss the method of time-resolved spectral phase measurement (TRSPM) as a technique for complete characterization of the electric field of a short optical pulse. Several phase retrieval algorithms, both deterministic and iterative, are compared.© (1995) COPYRIGHT SPIE--The International Society for Optical Engineering. Downloading of the abstract is permitted for personal use only.

Abstract: The discrete phase retrieval problem is to reconstruct a discrete-time signal whose support is known and compact from the magnitude of its discrete Fourier transform. We solve the 2-D discrete phase retrieval problem by partitioning it into a mostly-decoupled set of 1-D phase retrieval problems. The discrete and modulated Radon transforms are used to formulate two coupled 1-D problems, the solution to which then specifies solutions to the other decoupled 1-D problems. The latter may in turn be solved in parallel; however, using the solution to one problem as the input to a neighboring problem reduces the computation significantly for serial computers. Unlike other exact 2-D phase retrieval methods which rely on tracking zero curves of algebraic functions or equivalent operations, no continuous-function-based methods are used here. This makes the procedure more robust numerically.

Abstract: The Gerchberg-Saxton algorithm for iterative phase retrieval is applied in reconstructing an optical wave arriving from a coherently illuminated phase object. Computer simulations show that the algorithm converges to the correct solution when magnitude information for both image and Fourier domains is provided. Phase retrieval from measured intensities in an experimental optical system is compared with the simulations. Charge-coupled-device sensors, in which the output voltage is a nonlinear function of the light intensity, are used to perform measurements of the quasi magnitude information. Because of the accuracy of the measured data, phase retrieval for real simple objects is possible, and schlieren images can be generated on the computer screen.

Abstract: We consider the problem of making a minimum phase signal from an arbitrary one-dimensional signal by adding a point signal and its application to a two-dimensional phase retrieval problem. In particular, we show that a two-dimensional phase retrieval problem can be decomposed into several one-dimensional phase retrieval problems so that a M/spl times/N two-dimensional signal can be reconstructed from its Fourier transform magnitude by solving min.

Abstract: Phase retrieval holography has been used to determine the surface profile of the Swedish-ESO Sub-millimetre Telescope (SEST). The measurement was performed using a signal transmitted from the geosynchronous LES-8 satellite over a range of elevation angles. The results of the measurements have been used to reset the reflector thus reducing the surface errors from the initial 80 μm r.m.s. to 51 μm r.m.s. The improvement in the surface accuracy has been confirmed by telescope efficiency measurements at 232–250 GHz.

Abstract: The schlieren technique involves the manipulation of knife edges to limit the field of view in the image formation process. Practical implementation of this technique is not easy due to difficulties in positioning the knife edge in the optical system. A related problem concerns the reproducibility of an event to generate a series of schlieren images for different knife edge positions. A particularly successful method to overcome this problem is the use of the computer to generate such images from single pictures of the event. Computer generation of schlieren images involves the inverse Fourier transformation of the modified complex-valued diffraction pattern (magnitude and phase) of the event. Recording media in general respond only to light intensity and no difficulty is encountered in recording the intensity, and therefore the magnitude. The phase is either unobservable directly or cannot be determined anywhere nearly as accurately as the intensity. The Gerchberg and Saxton interative algorithm is used to recover the phase from records of intensity (magnitude) taken from the image and Fourier domains of the optical system. The knowledge of magnitude and phase in the Fourier domain (diffraction pattern) will enable us to modify it through a computer knife edge and generate the corresponding schlieren images.© (1995) COPYRIGHT SPIE--The International Society for Optical Engineering. Downloading of the abstract is permitted for personal use only.

Abstract: We demonstrate that the near field multiply scattered intensity registered by a detector, of light impinging on a surface, does not reproduce the structure of the surface. In other words, the images obtained in a near field optical microscope in conditions of multiple scattering of the light with the surface do not resemble the structure of the object. We present solutions to overcome this problem. One is a direct inverse scattering technique, which involves phase retrieval. Another procedure performs the reconstruction by adding near field intensities under several angles of incidence, and therefore overcomes the question of determining the phase.

Abstract: The paper presented deals with phase retrieval problem for image reconstruction from only the spectrum magnitude. Only two-dimensional spatially limited nonnegative objects, which are characterized by the analytical spectra, are considered assuming that the unique solution of the phase problem exists. It is proposed to use a nonlinear optimization approach, namely, the maximum entropy method (MEM) which has very good extrapolation features and a high noise stability. For solving the phase retrieval problem we introduce into the optimized entropy functional additional unknowns related to the real and imaginary parts of an object spectrum and represent the constraints, which are derived from measured spectrum magnitude data, as linear constraints, in order to reduce the optimization problem to the standard MEM. The whole computational algorithm is constructed as a combination of the standard MEM algorithm and an additional nonlinear constraint for the real and imaginary parts of the spectrum data which is realized during computational iterations. Images reconstructed by the proposed MEM approach may be, if necessary, further improved by Fienup's (1982) iterations. In this case the previous image is used as a starting point ensuring reliable convergence of Fienup's algorithm. Numerous simulation results demonstrate validity and high efficiency of the approach proposed.

Abstract: The determination of the surface quality of large reflector antennas by direct methods like the tape-theodolite method is not accurate enough for millimeter-wave operation. Indirect methods like holographic ones have been widely used. They are based on the Fourier transform (FT) relationship between the far-field pattern and the field distribution in the aperture, whose phase can be used to obtain a map of the axial deformations of the paraboloid by simple ray tracing. Measurement of the pattern phase requires a second antenna-receiver system and is difficult for high frequencies, so the possibility of recovering the aperture field from amplitude-only (or intensity) measurements of its FT (the pattern) has been studied and applied in radio telescope measurement. We present a discrete model for the aperture that enables us to approach this problem from an array processing point of view.

Abstract: An algorithm for image reconstruction from a series of short-exposure images with the use of the self-cross-correlation (self-cross spectrum in the Fourier domain) between an observation g(x, y) and a truncated sub-image of it, p(x, y), is described. The truncated subimages are chosen to be the brightest region in the observed images and of size comparable with that of the average point-spread function. We find that this self-cross spectrum retains the first-order approximation diffraction-limited phase information of the unknown object and does not rely on the integration of the second-order phase difference or the third-order phase closure estimate. For the case in which the components of the object corresponding to the truncated subimage p(x, y) have edges, a method to extract these edges and then use them to recover the phase of the object is presented. Results from computer-simulated data and from real data show the effectiveness of the method.

Abstract: Phase retrieval for reconstructing symmetric molecules from x-ray crystallographic data is investigated. The additional information contained in the symmetry should be sufficient to render the solution to such problems unique in many cases. This is supported by the application of iterative phase retrieval algorithms to simulated crystallographic data from a model of an icosahedral virus particle.© (1995) COPYRIGHT SPIE--The International Society for Optical Engineering. Downloading of the abstract is permitted for personal use only.

Abstract: The phase of the complex degree of temporal coherence of visible light is shown to be deducible from a three-beam interference experiment without any wavelength-scale accuracy on measuring a pathlength difference. Methods are discussed where the phase is obtained from the amplitude of rapid oscillations of the corresponding interferogram.

Abstract: We consider the reconstruction of a complex-valued object that is coherently illuminated and viewed through the same random-phase screen. The reconstruction is based on two intensity measurements: the intensity of the Fourier transform of the image and the intensity of the Fourier transform of the image when modulated with an exponential filter. The illumination beam has a Gaussian intensity profile of arbitrary width, and the phase screen is assumed to be described by a Gaussian random process of large variance and arbitrary correlation length. Computer-simulated examples of the reconstruction of a two-dimensional complex object demonstrate that the reconstruction is robust.

Abstract: Presents an iterative method, based on the use of a phase retrieval technique, for the accurate processing of SAR images starting from phase errors affected raw data.

Abstract: A simple zonal approach is proposed for estimating phase distribution on large grids. The estimation is based on phase differences that are precisely measured in two orthogonal directions by a lateral-shearing interferometer. It requires only O(N2) operations for reconstructing a phase distribution on an N × N grid. Computer simulation and experimental results are demonstrated to show the effectiveness of the new algorithm.