Abstract: Under the horizontally layered velocity assumption, migration is defined by a set of independent ordinary differential equations in the wavenumber‐frequency domain. The wave components are extrapolated downward by rotating their phases. This paper shows that one can generalize the concepts of the phase‐shift method to media having lateral velocity variations. The wave extrapolation procedure consists of two steps. In the first step, the wave field is extrapolated by the phase‐shift method using l laterally uniform velocity fields. The intermediate result is l reference wave fields. In the second step, the actual wave field is computed by interpolation from the reference wave fields. The phase shift plus interpolation (PSPI) method is unconditionally stable and lends itself conveniently to migration of three‐dimensional data. The performance of the methods is demonstrated on synthetic examples. The PSPI migration results are then compared with those obtained from a finite‐difference method.

Abstract: A new approach to implement computationally efficient finite impulse response (FIR) digital filters is presented. The filter structure is a cascade of two sections. The first section generates a sparse set of impulse response samples and the other section generates the remaining samples by using interpolation. The method can be used to implement most practical FIR filters with significant savings in the number of arithmetic operations. Typically 1/2 to 1/8 of the number of multipliers and adders of conventional FIR filters are required in the implementation. The saving is achieved both in the linear phase and the non-linear phase cases. In addition, the new implementation gives smaller coefficient sensitivities and better roundoff noise properties than conventional implementations.

Abstract: 1, Introduction. An assumption implicit in the usual scheme for self-calibration of radio interferometer data is one of i»oplanaii»m: that oyer each element of the array, all wavefronts arriving from different parts of the sky to which the interferometer pairs are sensitive are subject to identical tropospheric/ionospheric path delays. Approximate validity of the isoplanatism assumption is a necessary condition for the success of selfcalibration. This memorandum is an outline of a means by which the self-calibration algorithm might be modified in order to deal with the anisoplanatic case. Anisoplanatism is a severe problem with a low-frequency array, such as the one which has been proposed by R. A. Perley and W. C. Erickson [8] for construction at the VLA site. This is because, of the extreme magnitude of ionospheric effects at long wavelengths, and the large field of view of such an instrument. An initial attempt at a scheme for self-calibration of low-frequency array data is outlined in Perley and Erickson's proposal; and the need for a generalization of the self-calibration algorithm is reiterated in [2] and [4]. In § 2 is described a method of incorporating an interpolation formula in the selfcalibration solution for antenna phases. The idea is to express the phase corruption seen by a given array element, in an arbitrary direction, as a linear combination (i.e., as an interpolation) of the phase corruptions {/r}£Li toward the centers of some small number m of \"isoplanatic patches\". Setting m — 5 to 20, or so—with the patches judiciously centered—might suffice in a typical instance. When the source model used for self-calibration is given by a set of CLEAN point source components, it is easy to modify the solution scheme so as to yield the /{. Choice of an appropriate interpolation formula is discussed in § 3. Having obtained from the ^self-calibration solution algorithm a set of n spacevariant antenna phases, one for each antenna, the next problem is finding a way to make use of this information in mapping. The usual mapping/deconvolution schemes, such as Fourier inversion combined with CLEAN or with the maximum entropy deconvolution algorithm, are not designed to cope with space-variant effects. A means of utilizing the space-variant antenna phases in a modified, mosaicing version of the usual map/CLEAN combination is outlined in § 4. A drawback of the method described in § 2 is the increase (by a factor « m) over the usual number of solution parameters, or degrees of freedom, in the self-calibration solution algorithm. Because of this increase, a better source model, higher signal-tonoise ratio (S/N), or a larger number of antenna elements, (or a combination of all three) becomes desirable. By incorporating assumptions of spatial and temporal correlation of the antenna phases^ one may try to hold this larger number of degrees of freedom in check; this idea is pursued in §5 5-6. Perley and Erickson argue that for the proposed low-frequency array, which is designed to operate at 75 and 150 MHz, simple and accurate source models often will be available (perhaps from 327 MHz observations). And their data suggest that the spatial extent and the velocities of the ionospheric irregularities responsible for the severest phase fluctuations at 75 MHz are such that the techniques of §§ 5-6 would be useful. They report that during this summer they have found the typical case in 327 MHz VLA observations to be one of

Abstract: Time domain and frequency domain concepts which aid in the design and efficient implementation of a digital beamformer have been described at various times in the literature. The numerous beam-former structures that result are discussed with an emphasis on hardware requirements and spectral areas of application. Time domain procedures which include delay-sum, partial-sum, interpolation and shifted-sideband beamforming, and frequency domain techniques which include the application of discrete Fourier transforms and phase shift beam-forming are considered. Hardware considerations are primarily in the areas of analog-to-digital conversion, data storage, and computational throughput requirements.

Abstract: This paper discusses methods and software for C/sup 1/ interpolation at arbitrarily distributed data points in the plane. The primary results presented here are derivative-estimation procedures which lead to interpolatory surfaces constituting very accurate approximations for a variety of test functions.

Abstract: This paper presents an algorithm for azimuth correlation of synthetic aperture radar (SAR) data with extraordinarily large-range migration which cannot be accommodated by the existing frequency domain interpolation approach used in current Seasat SAR processing. We first provide a mathematical model for the SAR range-correlated point-target response both in the spatial and frequency domains. A simple and efficient processing algorithm derived from the exact two-dimensional correlation algorithm is given. This algorithm enables azimuth correlation by two cascaded one-dimensional correlation steps. The first step is to transform the range-correlated SAR data into the frequency domain in azimuth, followed by a newly developed range convolution filter to correct the range-dispersed spectrum of the range-correlated point-target response (RCPTR). The second step is the frequency-domain range migration correction approach for the azimuth compression. Simulation results provided here indicate that this processing algorithm yields a satisfactory compressed impulse response for SAR data with large-range migration whereas previous methods were subject to a broadening of the impulse response along range.

Abstract: This paper studies a method of solving the matrix Nevanlinna-Pick, Caratheodory, and Hamburger problems by using the theory of analytic J-expanding matrix-valued functions (J-theory). The solution of each problem is represented as a linear fractional transformation whose coefficient matrix is an entity of J-theory.

Abstract: A signal processing method and apparatus for providing interpolated values between sampled values in a sampled image signal, is characterized by providing a plurality of different interpolation routines for producing interpolated signal values for appropriately completing a respective plurality of known features, detecting which of the geometrical image features is represented by a neighborhood of sample values and applying the interpolation routine appropriate for completing the detected feature, to produce the interpolated signal value. The signal processing method and apparatus has the advantage that reconstruction errors in the reproduced image are reduced. In one mode of the invention for processing images to be viewed by humans, reconstruction errors are forced to occur in areas of the image composed of natural textures where the errors are not readily visible to the human observer. The appearance of the reconstructed image is thereby improved.

Abstract: The problem treated is that of constructing a C ~ interpolant of data values associated with arbitrarily distributed nodes on the surface of a sphere. A local interpolation method that has proved very successful for fitting data on the plane consists of generating a triangulation of the nodes, estimating gradients at the nodes, and constructing a triangle-based mterpolant of the data and gradient estamates Methods and software that extend thas solution procedure to the surface of the sphere are described, and test results are presented. The method is shown to be quite efficient and accurate for data taken from a variety of test functions.

Abstract: The vast majority of numerical investigations of transport phenomena use an Eulerian formulation for the convenience that the computational grids are fixed in space. An Eulerian-Lagrangian method (ELM) of solution for the convection-dispersion equation is discussed and analyzed. The ELM uses the Lagrangian concept in an Eulerian computational grid system. The values of the dependent variable off the grid are calculated by interpolation. When a linear interpolation is used, the method is a slight improvement over the upwind difference method. At this level of approximation both the ELM and the upwind difference method suffer from large numerical dispersion. However, if second-order Lagrangian polynomials are used in the interpolation, the ELM is proven to be free of artificial numerical dispersion for the convection-dispersion equation. The concept of the ELM is extended for treatment of anisotropic dispersion in natural coordinates. In this approach the anisotropic properties of dispersion can be conveniently related to the properties of the flow field. Several numerical examples are given to further substantiate the results of the present analysis.

Abstract: The performance of PN spread-spectrum communication systems in the presence of narrow-band interference is studied when linear interpolation filters are employed for the estimation and subsequent suppression of the interference. Closed-form analytical expressions for the system's performance are established for a broad class of interference processes. The advantages of linear interpolation filters over predictive filters with identical number of taps are examined analytically and some unexpected results are obtained. The analytical results are illustrated by examples.

Abstract: An algorithm is described for constructing a smooth computable function, f, defined over the surface of a sphere and interpolating a set of n data values, u sub i, associated with n locations, P sub i, on the surface of the sphere. The interpolation function, f, will be continuous and have continuous first partial derivatives. The locations, p sub i, are not required to lie on any type of regular grid.

Abstract: Triangle based interpolation is introduced by an outline of two classical planar interpolation methods, viz. linear triangular facets and proximal polygons. These are shown to have opposite local bias. By applying cross products of triangles to obtain local gradients, a method designated “slant-top proximal polygon interpolation” is introduced that is intermediate between linear facets and polygonal interpolation in its local bias. This surface is not continuous, but, by extending and weighting the gradient planes, a C1 surface can be obtained. The gradients also allow a roughness index to be calculated for each data point in the set. This index is used to control the shape of a blending function that provides a weighted combination of the gradient planes and linear interpolation. This results in a curvilinear, C1,interpolation of the data set that is bounded by the linear interpolation and the weighted gradient planes and is tangent to the slant-top interpolation at the data points. These procedures may be applied to data with two, three, or four independent variables.

Abstract: : This paper presents some experimental results that indicate the plausibility of using non-convex variational principles to reconstruct piecewise smooth surfaces from sparse and noisy data. This method uses prior generic knowledge about the geometry of the discontinuities to prevent the blurring of the boundaries between continuous subregions. The author includes examples of the application of this approach to the reconstruction of dynthetic surfaces, and to the interpolation of disparity data from the stereo processing of real images. (Author)

Abstract: A new method of instrument calibration and data analysis is presented for single-etalon interferometric measurements of winds, temperatures, and emission line intensities. The technique has been developed for the multichannel Fabry-Perot interferometer on the Dynamics Explorer spacecraft. A numerical representation of the instrumental transfer function is used based on a truncated Fourier series with empirically determined coefficients. The numerical form is compared with the conventional analytic form. The Fourier coefficients describing the instrument function are generated at the wavelength of a stable He–Ne laser and are translated to other wavelengths using an interpolation technique for both phase and power. A quasi-linear least-squares fitting process involving matrices provides for a rapid and accurate data reduction.

Abstract: This algorithm is a 1966 American National Standard FORTRAN implementation of the methods discussed in [1] and [2]. The software consists of a set of triangulation modules (which have application in problems other than interpolation) and a set of interpolation modules that require the triangulation routines. The triangulation software constructs a Thiessen triangulatiofi of a set of N nodes (x,, y,), i = 1 . . . . . N, arbitrarily distributed in the x-y plane. Only 7N storage locations are required to represent the triangulation and the software provides the capability of updating the data structure with the addition of a new node. Given data values z, associated with the nodes, the interpolation problem is to construct a C 1 function F such that

Abstract: Zero-crossing or feature-point based stereo algorithms can, by definition, determine explicit depth information only at particular points in the image. To compute a complete surface description, this sparse depth map must be interpolated. A computational theory of this interpolation or reconstruction process, based on a surface consistency constraint, has previously been proposed, implemented, and tested. In order to provide stronger boundary conditions for the interpolation process, other visual cues to surface shape are examined in this paper. In particular, it is shown that in theory, shading information from the two views can be used to determine the orientation of the surface normal along the feature-point contours, provided the photometric properties of the surface material are known. This computation can be performed by using a simple modification of existing photometric stereo algorithms. It is further shown that these photometric properties need not be known a priori, but can be computed directly from image irfadiance information for a particular class of surface materials. The numerical stability of the resulting equations is also examined.

Abstract: The representation and approximation of three- and four-dimensional surfaces is accomplished by means of local, piecewise defined, smooth interpolation methods. In order to interpolate to arbitrarily located data, the schemes are defined on geometric domains of triangles or tetrahedra, respectively.