Abstract: A computational theory of visible-surface representations is developed. The visible-surface reconstruction process that computes these quantitative representations unifies formal solutions to the key problems of: (1) integrating multiscale constraints on surface depth and orientation from multiple-visual sources; (2) interpolating dense, piecewise-smooth surfaces from these constraints; (3) detecting surface depth and orientation discontinuities to apply boundary conditions on interpolation; and (4) structuring large-scale, distributed-surface representations to achieve computational efficiency. Visible-surface reconstruction is an inverse problem. A well-posed variational formulation results from the use of a controlled-continuity surface model. Discontinuity detection amounts to the identification of this generic model's distributed parameters from the data. Finite-element shape primitives yield a local discretization of the variational principle. The result is an efficient algorithm for visible-surface reconstruction. >

Abstract: This paper presents a method of constructing a smooth function of two or more variables that interpolates data values at arbitrarily distributed points. Shepard's method for fitting a surface to data values at scattered points in the plane has the advantages of a small storage requirement and an easy generalization to more than two independent variables, but suffers from low accuracy and a high computational cost relative to some alternative methods. Localizations of this method have reasonably low computational costs, but remain relatively inaccurate. We describe a modified Shepard's method that, without sacrificing the advantages, has accuracy comparable to other local methods. Computational efficiency is also improved by using a cell method for nearest-neighbor searching. Test results for two and three independent variables are presented.

Abstract: A study of different cubic interpolation kernels in the frequency domain is presented that reveals novel aspects of both cubic spline and cubic convolution interpolation. The kernel used in cubic convolution is of finite support and depends on a parameter to be chosen at will. At the Nyquist frequency, the spectrum attains a value that is independent of this parameter. Exactly the same value is found at the Nyquist frequency in the cubic spline interpolation. If a strictly positive interpolation kernel is of importance in applications, cubic convolution with the parameter value zero is recommended. >

Abstract: A technique for registration of images with geometric distortions is described. This technique uses two surface splines to represent the X-component and the Y-component of a mapping function. A mapping function is described in such a way that it would map corresponding control points in the image exactly on top of each other and map other points in the image by interpolation using information and local geometric distortion between the images. >

Abstract: A short characteristic method based on parabolic approximation of the source function is developed and applied to the solution of the two-dimensional radiative transfer problem on Cartesian meshes. The method is significantly faster for the evaluation of multidimensional radiation fields than those currently in use. Convergence as a functional of the grid resolution is discussed and linear and parabolic upwind interpolation are compared.

Abstract: Polynomial interpolation is known to be ill-conditioned if the interpolating points are not chosen in special ways; classical rational interpolation can give better results, but does not work in all cases and the corresponding functions can show poles in the interval of interpolation. We present here rational functions which guarantee well-conditioned interpolation on a real interval or a circle and cannot have any poles there. They can be evaluated at least as efficiently as the corresponding interpolation polynomials and the accuracy of their approximation to a given function often compares favorably with that of spline interpolants.

Abstract: A criterion for the positivity of a cubic polynomial on a given interval is derived. By means of this result a necessary and sufficient condition is given under which cubicC 1-spline interpolants are nonnegative. Further, since such interpolants are not uniquely determined, for selecting one of them the geometric curvature is minimized. The arising optimization problem is solved numerically via dualization.

Abstract: A polynomial displacement basis for the three-node plate bending element (Zienkiewicz-triangle) is developed from a relaxed C1-continuity requirement called the interpolation test. The test provides a general convergence criterion for non-conforming shape functions and a practical guideline to select a proper displacement basis. The resulting simple displacement type element passes the patch test. Several reduced numerical integration schemes are discussed and numerical testing provides a comparison with the standard element formulated by Zienkiewicz.

Abstract: An 8-bit 100-MHz full-Nyquist analog-to-digital (A/D) converter using a folding and interpolation architecture is presented. In a folding system a multiple use of comparator stages is implemented. A reduction in the number of comparators, equal to the number of times the signal is folded, is obtained. However, every quantization level requires a folding stage, thus no reduction in input circuitry is found. Interpolation between the outputs of the folding stages generates additional folding signals without the need for input stages. A reduction in input circuitry equal to the number of interpolations is obtained. The converter is implemented in an oxide-isolated bipolar process, requiring 800 mW from a single 5.2-V supply. A high-level model describing distortion caused by timing errors is presented. Considering clock timing accuracies needed to obtain the speed requirement, this distortion is thought to be the main speed limitation. >

Abstract: A method is presented for finding a threshold surface which involves the ideas used in other methods but attempts to overcome some of their disadvantages. The method uses the gradient map of the image to point at well-defined portions of object boundaries in it. Both the location and gray levels at these boundary points make them good choices for local thresholds. These point values are then interpolated, yielding the threshold surface. A method for fitting a surface to this set of points, which are scattered in a manner unknown in advance, becomes necessary. Several possible approaches are discussed, and the implementation of one of them is described in detail. Two versions of the C.K. Chow and T. Kaneko algorithm (1972) and the present algorithm are applied to a few images, and the latter is shown to give better results, matching human performance quite well. >

Abstract: Motion Interpolation, which arises in many situations such as Keyframe Animation, is the synthesis of a sequence of images portraying continuous motion by interpolating between a set of keyframes. If the keyframes are specified by parameters of moving objects at several instants of time, (e.g., position, orientation, velocity) then the goal is to find their values at the intermediate instants of time. Previous approaches to this problem have been to construct these intermediate, or in-between, frames by interpolating each of the motion parameters independently. This often produces unnatural motion since the physics of the problem is not considered and each parameter is obtained independently. Our approach models the motion of objects and their environment by differential equations obtained from classical mechanics. In order to satisfy the constraints imposed by the keyframes we apply external control. We show how smooth and natural looking interpolations can be obtained by minimizing a combination of the control energy and the roughness of the trajectory of the objects in 3D-space. A general formulation is presented which allows several trade-offs between various parameters that control motion. Although optimal parameter values resulting in the best subjectively looking motion are not yet known, our simulations have produced smooth and natural motion that is subjectively better than that produced by other interpolation methods, such as the cubic splines.

Abstract: A new analytical technique is proposed to interpolate data between the lowest model level and the earth's surface in numerical weather prediction systems. The data can be interpolated to any height and especially to the one of routine measurements. The method can therefore be applied either to the verification of forecasts or to the determination of “observation minus guess” increments in data assimilation. The procedure assumes that a Monin-Obukhov-type flux calculation has been previously performed when integrating the model. DOI: 10.1111/j.1600-0870.1988.tb00352.x

Abstract: We describe a new algorithm for scattered data interpolation. The method is similar to that of Algorithm 660 but achieves cubic precision and C2 continuity at very little additional cost. An accompanying article presents test results that show the method to be among the most accurate available.

Abstract: A rapid nonrecursive prestack Kirchhoff migration is implemented (for 2-D or 2.5-D media) by computing the Green’s functions (both traveltimes and amplitudes) in variable velocity media with the paraxial ray method. Since the paraxial ray method allows the Green’s functions to be determined at points which do not lie on the ray, two‐point ray tracing is not required. The Green’s functions between a source or receiver location and a dense grid of thousands of image points can be estimated to a desired accuracy by shooting a sufficiently dense fan of rays. For a given grid of image points, the paraxial ray method reduces computation time by one order of magnitude compared with interpolation schemes. The method is illustrated using synthetic data generated by acoustic ray tracing. Application to VSP data collected in a borehole adjacent to a reef in Michigan produces an image that clearly shows the location of the reef.

Abstract: The Shankland-Koelling-Wood Fourier interpolation scheme is altered by a more physical choice of roughness function. Examples are presented demonstrating the improvement in the resulting band-structure plots; a more critical application is to the computation of the transport properties of crystalline solids which require second derivatives of the electronic energy bands.

Abstract: A fresh approach is taken to the embarrassingly difficult problem of adequately modeling simple pure advection. An explicit conservative control-volume formation makes use of a universal limiter for transient interpolation modeling of the advective transport equations. This ULTIMATE conservative difference scheme is applied to unsteady, one-dimensional scalar pure advection at constant velocity, using three critical test profiles: an isolated sine-squared wave, a discontinuous step, and a semi-ellipse. The goal, of course, is to devise a single robust scheme which achieves sharp monotonic resolution of the step without corrupting the other profiles. The semi-ellipse is particularly challenging because of its combination of sudden and gradual changes in gradient. The ULTIMATE strategy can be applied to explicit conservation schemes of any order of accuracy. Second-order schemes are unsatisfactory, showing steepening and clipping typical of currently popular so-called high resolution shock-capturing of TVD schemes. The ULTIMATE third-order upwind scheme is highly satisfactory for most flows of practical importance. Higher order methods give predictably better step resolution, although even-order schemes generate a (monotonic) waviness in the difficult semi-ellipse simulation. Little is to be gained above ULTIMATE fifth-order upwinding which gives results close to the ultimate for which one might hope.

Abstract: Bearing estimation is a fundamental array processing task for which attractive algorithms exist when the array is linear and uniformly-sampled. Since circumstances often require the use of an irregular two-dimensional array, the problem of interpolating a synthetic linear, uniformly sampled array from the real array is considered. Accurate interpolation is achieved by interpolating several synthetic arrays, each of which represents the real array over a limited sector of bearing angles. The utility of the method is demonstrated through a design example and simulation using a circular array and an eigenvector-based bearing estimator for linear, uniformly-sampled arrays. >

Abstract: A motion compensated interpolator for digital television images comprises a three-dimensional variable separable finite impulse response interpolation filter operating in the horizontal, vertical and temporal domains.

Abstract: A sub-Nyguist sampling encoder and decoder has a plurality of interpolators provided in an encoder and corresponding plural interpolators provided in a decoder. Information indicating which interpolator can be used to minimize the interpolation error is superimposed on a value of a non-thinned-out pixel and transmitted from the encoder to the decoder, and the decoder receiving the information is permitted to always perform optimum interpolation. The interpolation information is superimposed on the least significant bit of the value of the non-thinned-out pixel, with the view of effecting the superimposition without increasing the amount of transmission data and without degrading quality of the video signal.

Abstract: This paper presents an introduction for the nonspecialist to the use of geostatistics to estimate and map contaminant concentrations and estimation errors in a groundwater plume from a set of measured contaminant concentrations. The paper begins with a brief review of the essential elements of geostatistical theory. The remainder of the paper describes the four steps of a geostatistical analysis with special emphasis on the interpolation technique known as point kriging. This procedure can be used to obtain the best (i.e., minimum estimation error), linear (i.e., the estimated concentration at an unmeasured point is given by a linear combination of nearby measured concentrations), unbiased estimate, and the estimation error for any point within the plume. These point values can then be mapped (i.e., contours of equal values of expected contaminant concentration and estimation error can be drawn), e.g., to display the extent and severity of groundwater contamination at a site.

Abstract: A method for the spatial analysis of EEG and EP data, based on the spherical harmonic Fourier expansion (SHE) of scalp potential measurements, is described. This model provides efficient and accurate formulas for: (1) the computation of the surface Laplacian and (2) the interpolation of electrical potentials, current source densities, test statistics and other derived variables. Physiologically based simulation experiments show that the SHE method gives better estimates of the surface Laplacian than the commonly used finite difference method. Cross-validation studies for the objective comparison of different interpolation methods demonstrate the superiority of the SHE over the commonly used methods based on the weighted (inverse distance) average of the nearest three and four neighbor values.

Abstract: The author addresses the problem of reconstructing a bandlimited signal from a finite number of its unevenly spaced sampled data. A Fourier analysis of the available unevenly spaced sampled data is presented. The result is utilized to develop an interpolation scheme from the available data. Conditions for accurate reconstruction are examined. Algorithms to implement the reconstruction scheme are discussed. The method's application to one-dimensional and two-dimensional reconstruction problems is shown. >

Abstract: New flat-top windows by which per forming the measurement of the parameters that characterise periodic signals are presented. It is shown how, by applying these windows in the frequency domain beyond the FFT, the negative effects of both the spectral leakage and the harmonic interference are reduced without employing interpolation algorithms. The high accuracy of the results is confirmed by an error analysis. These windows are very useful in the determination of the harmonic content of the inverter output PWM waveforms supplying electrical motors, in order to qualify the entire drive.

Abstract: The authors present two methods for very-high-precision estimation of straight-line parameters from the Hough transform and compare them with the standard method of taking the absolute peak in the Hough array and with least-squares fitting using both extensive simulation and a number of tests with real target images. Both methods use preprocessing and interpolation in the Hough array, and are based on compensating for effects that cause a spreading of the peak in Hough space. By interpolation, the authors achieve accuracy better than the accumulator cell size. A complete set of simulations show that the two methods produce similar results, which are much better than taking the absolute peak in Hough space. They also compare well with least-square fitting, which was considered optimal in the case of zero mean noise. Results of experiments with real images are reported, confirming that the Hough transform can yield very accurate results, almost as good as least-squares fitting for zero mean noise. >