The Fourier synthesis technique of image formation has been in use in radio astronomy since the 1950's. Interferometric measurements provide values of the complex Fourier transform of a brightness distribution at a finite set of spatial frequencies, and it is required to reconstruct the brightness distribution. Originally, the reconstruction consisted simply of computing the inverse transform with the values at all unmeasured spatial frequencies set to zero. With good spatial frequency coverage, this often gives a satisfactory result. However, observations are often made with limited coverage. Also, the increasing sensitivity of radio telescopes has led to a demand for larger dynamic range in the images. There has therefore been an effort to develop improved imaging techniques; these are reviewed here.

A simple iterative algorithm for reconstructing the images of compact radio sources from very-long-baseline interferometry (VLBI) measurements of visibility amplitudes and "closure" phases is described. The method makes efficient use of redundant sampling of spatial frequencies by making a global solution of phase "closure" equations for a complete set of observetions.

Radio interferometer observations are, essentially, samples of V, the Fourier transform of the radio brightness, I, on a patch of sky. The dominant errors usually are ascribable to individual elements of the array. Most damaging, often, are phase errors due to atmospheric refraction, but amplitude errors also may be serious. The sampling distribution improves as an observing run progresses since, as the earth rotates, any pair of elements samples V at different spatial frequencies lying along a curve. Adaptive calibration can be achieved as follows: Assumptions on the analytic properties of I lead to a measure of the consistency of the observations, from a given instant, with the total set of data; given an appropriate error model, an overdetermined system of equations in the unknown errors results; correcting the data accordingly, an improved estimate of I is derived; and the procedure may be repeated. This encompasses schemes proposed by Readhead and Wilkinson, and by Cotton, for compensation of phase errors in very-long-baseline interferometry (VLBI); similar methods are proposed by Muller and Buffington for optical telescopes. Results in analysis of Very Large Array (VLA) data are extremely encouraging. The computational expense is a few times that of standard methods.

In aperture synthesis measurements in Radioastronomy, the measurement set is frequently three-dimensional (in space) and therefore cannot be simply represented in two-dimensions. As a result, the point spread function for the instrument varies as a function of direction. This has been considered in recent publications in relation to digital data processing and some of the features of the three-dimensional problem, and methods for implementing the computation using one or two-dimensional transforms will be presented.

One of the most significant developments in radio astronomy has been the recent discovery of over 50 different molecules in the interstellar medium. These observations have changed our picture of the distribution of mass in the galaxy, altered our understanding of the process of star formation, and has also opened up a new and lively field of interstellar chemistry. This achievement was made possible not only by the development of sensitive heterodyne receivers (front-end))in the centimeter and millimeter range, but also by the construction of sensitive RF spectrometers (back-end) which enabled the spectral lines of molecules to be detected and identified. Traditionally spectrometers have been constructed as banks of discrete adjacently tuned RF filters or as digital auto-correlators. However, a new technique combining acoustic bending of a collimated coherent light beam by a Bragg cell followed by detection by a sensitive array of photodetectors (thus forming a RF acousto-optic spectrometer (AOS) promises to have distinct advantages over older spectrometer technology. An AOS has wide bandwidth, large number of channels, high resolution, and is compact, light weight, and energy efficient. These factors become very important as heterodyne receivers are developed into the submillimeter, far infrared, and 10 micronspectral ranges and as more observations are performed from remote, airborne, or spaceborne platforms. We give a short description and report of existing AOS backends in Australia and Japan but will concentrate on our recent construction of a proto-type AOS at Goddard Space Flight Center. The GSFC AOS uses a discrete bulk acoustic wave Itek Bragg Cell, 5 mW Helium-Neon laser, and a 1024 element Reticon CCPD array. The analog signals from the photodiode array are digitized, added and stored in a very high-speed custom built multiplexer board which allows us to perform synchronous detection of weak signals. The experiment is controlled and the data is displayed and stored with an LSI-11 microcomputer system with dual floppy disks. We will report the performance of the GSFC AOS obtained from our initial tests. We also will give a description of an integrated SAW Bragg cell which will miniaturize a complete AOS system into a 1x3 inch package.

Optical Fourier transform processing of radiotelescope visibility function data is reviewed. Emphasis is placed on the aspects of the processor design that arise from the unusual characteristics of the data. The complex visibility function is available only over a partially filled aperture consisting of a set of elliptical paths. It is recorded at the input to the optical processor on a carrier frequency as a real nonnegative transmittance. Since a bipolar sky brightness function output is to be computed, the output of the optical Fourier transform channel is mixed with a reference beam and the difference is taken of two successive measurements differing in reference phase by 180°. Experimental results demonstrating processor concepts are shown and a processor system design approach is described.

It is shown that an optical Fourier transform processor can convert VLA telescope visibility data at its input to a high accuracy ski-brightness map at its output. A particularly attractive Fourier transform processor that minimized the number of lens elements and hence potential background scattering sources is one for which the input is placed after the Fourier transform lens. This lens in its simplest form illuminates the input signal with a converging spherical wave. Such a system does not produce a perfect Fourier transform, but rather produces an aberrated one. The aberration is effectively analyzed as an error term within a Fourier transform integral. Error reduction techniques for the output map are described and numerical results from digital simulations are presented for the VLA A-array aperture.

An optical Fourier transform processor converts visibility data at its input to a sky brightness function at its output. Scattered light and spurious terms resulting from the method of encoding the complex visibility function as a real-valued and nonnegative transmittance result in errors in the processor output. In this paper, we discuss the degree to which each of these undesired terms degrades the signal-to-noise ratio of the output for different encoding methods and classes of images. We suggest alternative methods of encoding the input that greatly reduce the errors. Two particularly powerful methods of minimizing noise are "complementary weighting" and "bias equalization". Two data dependent parameters are found to be most important: (1) the ratio of the mean-squared visibility magnitude to the square of its maximum, and (2) the ratio of the squared maximum of the brightness function to its mean squared value.

Computer experiments were performed to determine the effects of photon noise on reconstructing images from stellar speckle interferometer data. Estimates of the modulus of the Fourier transform of an extended space object were computed using the method of Labeyrie on degraded images simulated to include the effects of atmospheric turbulence and photon noise (Poisson statistics). Several Fourier modulus estimates with different signal-to-noise ratios were computed by assuming different brightness levels (photons per pixel) for the object in each case. Images were reconstructed from the Fourier modulus estimates using an iterative method based on the nonnegativity of the object. It was found that the rms error of a reconstructed image is roughly equal to the square root of the rms error of the Fourier modulus estimate.

In this paper we develop an algorithm by which, in general, a 2-dimensional wavefront can be reconstructed uniquely, up to conjugation, from intensity measurements in a single plane. Our technique consists of a detailed mathematical analysis employing Fourier theoretic concepts and differential equations. The algorithm is intended not as a practical algorithm, since efficient iterative algorithms have been developed which incorporate known prior information concerning the wavefront, but rather as a means of addressing the uniqueness question.

Two phase retrieval methods are tested with two-dimensional simulated noisy speckle image data: a simple phase unwrapping algorithm and a Knox-Thompson algorithm. These are used in combination with the blind deconvolution and the Welter-Worden autocorrelation seeing correction methods. The final images are retouched with the iterative Fienup method.

Given a truncated portion of a bandlimited image with known x and y bandwidths, the extrapolation problem is to determine the signal over all x and y. An iterative extrapolation algorithm, recently proposed by Gerchberg, requires only the repeated operations of Fourier transformation and truncation. This paper presents a coherent optical processor which implements Gerchberg's iterative extrapolation algorithm in two dimensions. Iteration is performed by simple passive feedback. Experimental results are presented to illustrate the effectiveness of the processor.

This paper describes an approach to the problem of reconstructing a star which has been blurred by atmospheric turbulence and further corrupted by sensor noise. The Knox-Thompson speckle-imaging technique is used to obtain initial estimates of the amplitude and phase spectra of the object. When the noise level is high, these estimates are quite inaccurate at the higher spatial frequencies, and the associated reconstruction is very poor. We attempted to alleviate the problem by throwing away the noisy high-frequency terms and then using a spectral extrapolation algorithm to reconstruct them from the more accurate low frequencies. Spectral extrapolation was done using both the Gerchberg algorithm and a new algorithm described in the paper. These two algorithms and others are discussed in terms of a convenient eigenfunction expansion, and some experimental results are presented. Although these particular results favor the new algorithm, general conclusions must be drawn with caution.

Recently, a set of conditions has been developed under which a sequence is uniquely specified by the phase or samples of the phase of its Fourier transform. These conditions are distinctly different from the minimum or maximum phase requirement and are applicable to both one-dimensional and multi-dimensional sequences. Under the specified conditions, several numerical algorithms have been developed to reconstruct a sequence from its phase. In this paper, we review the recent theoretical results pertaining to the phase-only reconstruction problem and we discuss in detail two iterative numerical algorithms for performing the reconstruction.

In a coherent imaging system, the complex amplitude and phase of the images is necessary to perform any linear processing or wavefront reconstruction. Using the image intensities measured in two defocused planes, it is possible to reconstruct the lost phase by iterative techniques. A merit function is defined as the sum over the second image plane of the squared differences between the known modulus in that plane and the modulus calculated by defocusing the image in the first plane with a phase estimate. This merit function is used to evaluate the convergence properties of two types of iterative schemes:the Misell (Gerchberg-Saxton) algorithm, and a gradient searching steepest descent method. The convergence is studied as a function of the amount of defocus between the images, complexity of the image, and noise present in the measured intensities. Variations and improvements of the methods are discussed. For example, for small amounts of defocus, the Misell algorithm has difficulty converging; the application of alternating constraints in Fourier space may help convergence. For the steepest descent and other gradient related methods, convergence of the phase depends on the value of the modulus at that point. Results of computer experiments using simulated images and pupil distributions are shown.

The principles of limited-angle reconstruction of space-limited objects using the concepts of "allowed cone" and "missing cone" in Fourier space are discussed. The distortion of a point source resulting from setting the Fourier components in the missing cone to zero has been calculated mathematically, and its bearing on the convergence of an iteration scheme involving Fourier transforms has been analysed in detail. It was found that the convergence rate is fairly insensitive to the position of the point source within the boundary of the object, apart from an edge effect which tends to enhance some parts of the boundary in reconstructing the object. Another iteration scheme involving Radon transforms was introduced and compared to the Fourier transform method in such areas as root mean square error, stability with respect to noise, and computer reconstruction time.

High angular resolution is achievable with a large-aperture instrument even if the aperture is distorted, provided that adaptive signal processing compensates for the distortion. The radio camera is an instrument designed for this purpose. Its algorithm is discussed and experimental results are given of a 3 cm wavelength demonstration system.

This paper has two purposes. The first is to report experimental results of an application of computer-reconstructed, radio-wave holography, for imaging the spatial structure of the ionosphere, and to briefly discuss their implications. The second is to point out that most imaging applications are really inverse scattering problems, and that holographic reconstruction is usually a poor solution to such problems. The correct solution to inverse scattering problems is presented, and the criteria which must be satisfied for holographic reconstruction to be an adequate approximation to this solution are discussed.

A new approach to real-time, long-wavelength holography is described. The method utilizes one wavelength for formation and another for reconstruction. The reconstruction is done with a millimeter wave lens illuminated by an antenna array operating at the shorter wave-lengths. The amplitude distribution from the array is a replica of the sampled microwave hologram. PIN-diode modulators control the reconstruction array. Experimental verification is given.

In tomography, a cross section of an object is imaged with as little interference as possible from structures not in the cross section. In reconstructive tomography, the image is computed from elementary measured data. Both X rays and nuclear emissions from injected radioactive isotopes are being successfully used in clinical applications at the present time. However, these modalities suffer the disadvantage of being invasive. Fortunately, other forms of energy can also be used in reconstructive tomography, including microwaves and ultrasound. As opposed to X rays, micro and acoustic waves can be reflected, and in this paper we explore methods to obtain images from such reflected waves. Our methods include pulse-echo reconstructive tomography and tomographic extensions of CW Doppler processing. The use of these forms of energy, as opposed to X rays, also permits the possibility of coherent signal generation and processing. In this paper, our emphasis is on Doppler processing. Analysis is presented as well as exprimental verification of the analysis. It is both theoretically and experimentally shown that high-quality reconstruction of object points can be obtained. Two equal points separated by a quarter wavelength may be resolved in the case of the coherent Doppler tomography.

This is a review article about radars based upon the hologram matrix principle. The principles proposed and the experimental results obtained by various authors are summarized. The hologram matrix radars may be classified into three broad categories: HISS radar, Multifrequency Hologram Matrix Radar, and Step Frequency Radar. The advantages and disadvantages of each method are presented.

Whereas in the optical region commonly catan hoeognaphy is used to describe the process of wavefront reconstruction, it is shown that polarization effects can no longer be neglected in the microwave and mm-wave regions and an extension to vegan hotogitaphy is required. Based on a succint review of previous studies on considering polarization properties in holography, it is apparent that a complete vector holographic treatment has yet to be formulated. Therefore, various basic properties of the object scattering matrices and their transformation invariants are introduced together with some pertinent descriptions of coherent versus non/incoherent scatter effects from rough surfaces which then allow a complete formulation of vector holography. Finally it is shown that a further generalization of vector holography incorporating recording of magnetic (energy) intensities in addition to the electric intensity, may enable us to recover simultaneously the shape and the averaged surface impedance of a scatterer relative to some reference surface impedance.

The concepts of reference wave slowness (reciprocal of velocity) and an associated free reference space Green's fuction slowness spectrum are introduced. A modified Kirchhoff surface integral, containing only the imaginary part of this free reference space Green's function slowness spectrum, is formulated, yielding an integral equation for the unknown fields and sources in the interior of a closed surface on which the (remotely sensed) fields are known. A we analytic closed form solution of this integral equation is obtained.

The problem of 3-D visualization of biomedical objects attracts increasing attention of medical specialists. Although tomographic techniques are advanced methods of volume representation, they are hard to apply in X-ray and ultrasound diagnostics. The reasons are well discussed in the literature. The computerized transverse axial scanning tomography opened a new field in X-ray diagnostics, and it was thought that the same principle could be applied also to ultrasonic diagnostics, even in two different ways. In the first case, ultrasound transmission tomograms of attenuation (UTTAR), while in the second case, ultrasonic transmission tomograms of velocity reconstruction (UTTVR) are formed. However, as long as X rays are used, diffraction and scattering problems are less severe than when ultrasonic waves are the information carriers. These image degradation effects issuing from the interaction of the ultrasonic beam with the tissue could be compensated for without the need of a computer if the reconstruction is performed optically and suitable filters and lenses illuminated by coherent light are used.

The reconstruction of a two-dimensional sectional image from one-dimensional projections can be achieved in parallel by a deconvolution, if the projections are arranged in properly chosen coordinates. In most cases the deconvolution result is still to be coordinate-transformed. The reconstruction theory is briefly discussed and illustrated by digital computer simulations. A coherent optical realization of the deconvolution is described and results are shown. Problems due to spatial truncation are discussed.

A method is described for image reconstruction which processes the projection data in polar rather than rectangular coordinates, and does not require backprojection. It is based on the decomposition of the object and its shadow (set of projections) into "circular harmonics," or radial modulators of angular Fourier components. An inverse filter is derived which enables the radial modulators of the object to be reconstructed from those of the shadow. Experimental results from digital and coherent optical implementations are given. A real time optical reconstruction processor is proposed.

Several new polar sampling theorems are discussed and applied to the Problem of reconstructing images from their samples or projections. An application of a Polar sampling theorem to computer-aided tomography (CAT) is illustrated and compared with standard CAT reconstructions. The advantages and disadvantages of our technique versus convolution-back projection are given.

Expressions for the mean and variance of the images produced by Fourier Multiaperture Emission Tomography are derived. Only quantum noise (counting statistics) is considered. The signal-to-noise ratio achieved by the system for a thin disc-shaped object is computed from these expresions. An example is used to demonstrate some significant features of this single photon emission tomogrphic imaging system.

The distribution of noise in coded aperture images is known to depend in a complex manner upon the encoding technique, the decoding technique and upon the object distribution. We have examined the S/N characteristics of a classs of, planar, pseudorandom, time-modulated coded apertures in order to optimize the aperture design for a defined object distribution. Relative standard deviation (RSD) in the reconstructed image is studied both theoretically and by computer simulation. Results are shown for uniform, planar source distributions of varying size as a function of mean code plate transmission and aperture hole spacing. In each case, effects of solid angle and finite geometry are taken into account. For simplicity, image reconstruction is accomplished by backprojection. For a source size equal to 20% of a full field flood, a code of 12% mean transmission gives a near optimum S/N. The RSD with this code for an on-axis image element is equal to .33 of that for a single scanning pinhole covering an identical field of view. Even for a 100% field flood an optimum code exists which has a mean transmission of nearly 4%. The RSD in this case is smaller compared to the scanning pinhole by a factor of .85.

A hybrid digital-optical processor for reconstructing a 2-dimensional scalar field from its 1-dimensional projections will be presented. This processor represents a further development in a continuing effort to incorporate and utilize the power of optical data processing in x-ray computed tomography. The processor to be described operates on a data set recorded on x-ray film. The reconstruction of the field from its projection data is obtained via the filtered back-projection algorithm. Specifically, the time consuming operations of filtering and back-projection are done optically. Final addition and subtraction operations are done in digital hardware, with the final image residing in a digital memory. Sophisticated image enhancement operations are then possible with computer back-up. The interface between the optical and digital hardware is simple video. This particular processor has the very attractive feature that full digital resolution can in principle be obtained over any desired area of the reconstructed image so that the final system should be limited by the x-ray detector, in this case film, which inherently has a much greater resolution capability than the discrete detector arrays found in conventional CT scanners.

A Coded Aperture Imaging System (CAIS) has been developed at Sandia National Laboratories to image the motion of nuclear fuel rods undergoing tests simulating accident conditions within a liquid metal fast breeder reactor. The tests require that the motion of the test fuel be monitored while it is immersed in a liquid sodium coolant precluding the use of normal optical means of imaging. However, using the fission gamma rays emitted by the fuel itself and coded aperture techniques, images with 1.5 mm radial and 5 mm axial resolution have been attained. Using an electro-optical detection system coupled to a high speed motion picture camera a time resolution of one millisecond can be achieved. This paper will discuss the application of coded aperture imaging to the problem, including the design of the one-dimensional Fresnel zone plate apertures used and the special problems arising from the reactor environment and use of high energy gamma ray photons to form the coded image. Also to be discussed will be the reconstruction techniques employed and the effect of various noise sources on system performance. Finally, some experimental results obtained using the system will be presented.

The methods of computerized tomography (CT), developed for medical X-ray applications, can be adapted for use in studying plasma X-ray emissivity distributions in tokamaks and other magnetic confinement devices. Current generation CT scanners reconstruct maps of X-ray absorptivity on body cross sections by processing transmission data from a number of fan-shaped beams of X-rays. Analogous fan beam emission data can be obtained from confined plasmas by collimating emitted soft X-rays with a "pin hole" or slit and detecting them with a linear array of solid state detectors. Data from a number of such one-dimensional views of the plasma can be used to reconstruct a two-dimensional "photograph" of the absolute X-ray emission in cross section. No a priori assumptions about the nature of the emissivity distribution are necessary. In this paper we demonstrate the feasibility of the technique by reconstructing test patterns with data simulated for a number of different types of detector arrangements. We also use the technique with real data to reconstruct a rotating emissivity feature on a cross section of Massachusetts Institute of Technology's Alcator A tokamak.

The complex spectrogram of a signal φ(t) is defined by ∫φ(t)g*(t-to)exp[-iwot]dt ; it is, in fact, the Fourier transform of the product of the signal and the complex conjugated and shifted version of a so-called window function g(t). From the complex spectrogram the signal can be reconstructed uniquely. It is shown that the complex spectrogram is completely determined by its values on the points (to=mT, wo=nΩ), where ΩT= 2tt and m and n take all integer values. The lattice of points (mT,nΩ) is exactly the lattice suggested by Gabor as early as 1946; it arose in connection with Gabor's suggestion to expand a signal into a discrete set of Gaussian elementary signals. Such an expansion is a special case of the general expansion ∑∑mn amng(t-mT)exp[inΩt] of a signal into a discrete set of shifted and modulated window functions. It is shown that this expansion exists. Furthermore, a set of functions is constructed, which is bi-orthonormal to the set of shifted and modulated window functions. With the help of this bi-orthonormal set of functions, the expansion coefficients amn can be determined easily.

A coherent optical correlator using a diode injection laser source spatially modulated with a liquid crystal light valve was demonstrated for cross correlation and tracking of vehicles in scenes and of areas in aerial scenes. The light valve was optically addressed with a TV monitor allowing imagery from a TV camera or video tapes to be used for input to the correlator. The long wavelength and low coherence make filter recording with laser diodes difficult if not impossible. Filters were therefore recorded with a helium-neon laser for use with the correlator. Correlations obtained with these filters had similar line widths and signal-to-noise ratios when used with helium-neon or diode laser sources. The small size and low power consumption of laser diodes permit the miniaturization of a coherent optical correlator. Two 3 compact correlators using available components were designed. One, a cylindrical package, required 300 cm volume and another more novel arrangement resulted in a 60 cm3 package.

A number of acousto-optical spectrometers (AOS) have been constructed for radio astronomy in recent years. Although such spectrometers are simple and relatively cheap, great care is needed in their construction if the stability necessary for radio astronomy is to be achieved. An experimental AOS, with a bandwidth of 100 MHz, has been built at Caltech and used for a study of the various sources of drifts. The results of this study are presented. The experimental AOS can already operate with negligible drift at a chopping rate of 0.02 Hz. Improvements planned for the next version should permit operation at a chopping rate of 0.01 Hz.

In synthetic-aperture radar (SAR), the radar signal, recorded as an interferogram, exhibits a defect known as range curvature. The two dimensions of the interferogram are coupled and cannot be processed separately, leading to image degradation. Two variations of a method of correcting for range curvature are presented. They both involve the insertion of a cylindrical lens or lenses near the region of the Fourier-transform plane of a standard tilted-plane SAR processor. Variable tilt of the lens(es) controls the amount of correction. In the presence of pointing errors of the SAR antenna, an extra cross-coupling factor is introduced. It is corrected by a simple rotation of the range-curvature corrector.

This paper is the brief summary of a series of papers published in China, in Acta Physica Sinica. The main question discussed here is the relation between the holographic optical system, the pattern transformation, and the pattern recognition. We have analyzed the following problems: (1) What kind of pattern transformation can be realized by the holographic optical system. (2) How to design a hologaphic optical system to realize an arbitrarily given unitary transformation by the usual optimization methods. (3) How to realize a projection operator by the holographic system in a practical way. The possibility of optical pattern recognition also has been discussed.