Abstract: From Shepard's (1968) local-search method, algorithms are developed for contouring on spherical surfaces and in Cartesian two-space. These algorithms are used to investigate errors on small-scale climate maps caused by the common practice of interpolating—from irregularly-spaced data points to regular-lattice nodes—and contouring in Cartesian two-space. Using mean annual air temperatures drawn from 100 irregularly-spaced weather stations, the annual air-temperature field over the western half of the northern hemisphere is estimated both on the sphere (assumed to be correct) and in Cartesian two-space. When these fields are mapped and compared, error magnitudes as large as 5° to 10° C appear in the air-temperature field approximated in Cartesian two-space.

Abstract: Sampling is a fundamental operation in all image communication systems A time-varying image, which is a function of three independent variables, must be sampled in at least two dimensions for transmission over a one-dimensional analog communication channel, and in three dimensions for digital processing and transmission At the receiver, the sampled image must be interpolated to reconstruct a continuous function of space and time In imagery destined for human viewing, the visual system forms an integral part of the reconstruction process This paper presents an overview of the theory of sampling and reconstruction of multidimensional signals The concept of sampling structures based on lattices is introduced The important problem of conversion between different sampling structures is also treated This theory is then applied to the sampling of time-varying imagery, including the role of the camera and display apertures, and the human visual system Finally, a class of nonlinear interpolation algorithms which adapt to the motion in the scene is presented

Abstract: The most common technique for the control of structures is modal control. In modal control, the differential equations in terms of actual coordinates are replaced by a set of ordinary differential equations in terms of the modal coordinates known as modal equations. In designing feedback controls in conjunction with the modal equations, one must know the modal states for the modes targeted for control. The sensors measure actual states, however. The modal states can be estimated by means of a Luenberger observer or modal filters. The modal filters produce estimates of the modal states from distributed measurements of the states. If distributed measurements are not available, then they can be reconstructed from measurements at discrete points via interpolation. This paper examines various questions associated with the implementation of modal filters, such as the effect of choice of interpolation functions and sensors locations, as well as of measurement errors, on the state estimation process. The method is demonstrated by means of two numerical examples.

Abstract: An explicit representation of a $C^1 $ piecewise rational cubic function is developed which can be used to solve the problem of shape preserving interpolation. It is shown that the interpolation method can be applied to convex and/or monotonic sets of data and an error analysis of the interpolant is given. The scheme includes, as a special case, the monotonic rational quadratic interpolant considered by the authors in [1] and [5]. However, the requirement of convexity necessitates the generalization to the rational cubic form employed here.

Abstract: At how many points is it possible to prescribe values to a polynomial of given degree, together with derivatives up to order, say, t-1? We solve this problem in case of polynomials in two variables for t=2 and t=3 and in case of polynomials in three variables for t=2. Proofs develop in the frame of modern projective geometry.

Abstract: Parametric keyframing is a popular animation technique where values for parameters which control the position, orientation, size, and shape of modeled objects are determined at key times, then interpolated for smooth animation. Typically the parameter values defined by the keyframes are interpolated by spline techniques with the result that the parameter change kinetics are implicitly defined by the given keyframe times and data points. Existing interpolation systems for animation are examined and found to lack certain desirable features such as continuity of acceleration or convenient kinetic control. The requirements of interpolation for animation are analyzed in order to determine the characteristics of a satisfactory system. A new interpolation system is developed and implemented which incorporates second-derivative continuity (continuity of acceleration), local control, convenient kinetic control, and joining and phrasing of successive motions. Phrasing control includes the ability to parametrically control the degree and extent of smooth motion flow between separately defined motions.

Abstract: In the local basis-function approach, a reconstruction is represented as a linear expansion of basis functions, which are arranged on a rectangular grid and possess a local region of support. The basis functions considered here are positive and may overlap. It is found that basis functions based on cubic B-splines offer significant improvements in the calculational accuracy that can be achieved with iterative tomographic reconstruction algorithms. By employing repetitive basis functions, the computational effort involved in these algorithms can be minimized through the use of tabulated values for the line or strip integrals over a single-basis function. The local nature of the basis functions reduces the difficulties associated with applying local constraints on reconstruction values, such as upper and lower limits. Since a reconstruction is specified everywhere by a set of coefficients, display of a coarsely represented image does not require an arbitrary choice of an interpolation function.

Abstract: Fluid mechanics is understood in a descriptive way through experimental observation, mathematical analysis, and numerical simulation. When the understanding is required for flows with complex internal structure and with complicated regional boundaries, insight is gained primarily from experiments and simulations. Numerical simulations are motivated by the prospect of economically obtaining a detailed flow-field description. In each instance, a governing system of flow equations is analytically formulated over the region and is solved in an approximate form on a computer. Various numerical methods have been devised for such ap­ proximations, and most depend upon some representation of the flow-field region by means of finitely many points and the interconnections between them. When each regional boundary is given by a sequence of connected points, the efficiency and accuracy of the various methods are enhanced, since boundary conditions can be applied without interpolation. Further enhancement comes when the connectivity pattern is regular. The most regular and thus preferable patterns are those that result from coordinate transformations. With the application of transformations, regions with topological complexity are consistently treated even when points are in motion. During the course of simulation, motion can be advantageously used to adaptively resolve the significantly varying solution quantities. From a geometric viewpoint, the quantities determine a surface over the physical region. The resolution of the surface as it evolves then determines the adaptive movement.

Abstract: $C^1 $ monotone quadratic splines are analyzed. This motivates an algorithm for interpolating monotone data, given on a rectangular grid, with a $C^1 $ monotone quadratic spline surface. Error estimates, an operation count and numerical examples are given.

Abstract: Wider use of hologram interferometry for quantitative measure-ments has been delayed by the fact that interpolation between the fringe maxima and minima to obtain the optical path difference at a particular point in the field is laborious and inaccurate. A solution to this problem is quasi-hetero-dyne interferometry, which permits rapid and accurate measurements simultaneously at a number of points distributed over the interference pattern. In this technique a television camera is used in conjunction with digital electronics to measure and store the irradiance values at points on a rectangular sampling grid covering the real-time interference fringes. The phase difference between the interfering wavefronts at each point is then calculated from the irradiance values obtained from successive scans of the camera made while the phase of one of the wavefronts is shifted either continuously or in steps. A practical system is described with which values of the optical path difference for 10,000 data points can be obtained with an accuracy of ± A/200 in less than 10 s. The application of quasi-heterodyne hologram interferometry to the measurement of vector displacements and to holographic contouring is discussed.

Abstract: This paper deals with methods for obtaining near-optimum step sizes for finite difference approximations to first derivatives with particular application to sensitivity analysis. A technique denoted the finite difference (FD) algorithm, previously described in the literature and applicable to one derivative at a time, is extended to the calculation of several simultaneously. Both the original and extended FD algorithms are applied to sensitivity analysis for a data-fitting problem in which derivatives of the coefficients of an interpolation polynomial are calculated with respect to uncertainties in the data. The methods are also applied to sensitivity analysis of the structural response of a finite-element-modeled swept wing. In a previous study, this sensitivity analysis of the swept wing required a time-consuming trial-and-error effort to obtain a suitable step size, but it proved to be a routine application for the extended FD algorithm herein.

Abstract: In this note we discuss the H?-sensitivity minimitization problem for linear time-invariant delay systems. While the unweighted case reduces to simple Nevanlinna-Pick interpolation, the weighted case turns out to be much more complicated and demands certain functional - analytic techniques for its solution.

Abstract: This paper presents finite impulse response switched-capacitor (SC) decimator and interpolator circuits based on nonrecursive polyphase structures, which are particularly suitable for narrow-band SC bandpass filter systems. The circuits are attractive for integration and their good performance is demonstrated using practical discrete component models.

Abstract: A study is conducted of the stability of mesh refinement in space and time for several different interface equations and finite-difference approximations. First, a root condition which implies stability for the initial-boundary value problem for this type of interface is derived. From the root condition, the stability of several interface equations is proved, using the maximum principle. In some cases, the final verification steps can be done analytically; in other cases, a simple computer program has been written to check the condition for values of a parameter along the boundary of the unit circle. Using this method, stability for Lax-Wendroff with all the interface conditions considered, and for Leapfrog with interpolation interface conditions when the fine and coarse grids overlap is proved.

Abstract: Recently, an iterative reconstruction-reprojection (IRR) algorithm has been suggested for application to limited view computed tomography (CT). In the IRR, the interpolation operation is performed in the object space during backprojection-reprojection. The errors associated with the interpolation degrade the reconstructed image and may cause divergence unless a large number of rays is used. In this paper, we propose a projection space iterative reconstruction-reprojection (PSIRR) algorithm based on backprojection-reprojection in the projection space. In the process of the backprojection-reprojection, iteration is performed with a single equation instead of multiple equations and interpolation is eliminated. Computer simulation results are presented, and image quality of the PSIRR algorithm shows a substantial improvement compared to the original IRR algorithm. In addition, the new algorithm was applied to ultrasonic attenuation CT using a sponge phantom.

Abstract: An adaptive delay-estimation (ADE) algorithm is proposed for the continuous tracking of time-delay. The method uses an adaptive delay line which is interpolated by a first-order filter. Two delay-line interpolating filters are considered, each having a single coefficient which is estimated in real time. The first implements linear interpolation, and the second interpolates using a first-order allpass filter. Since the ADE algorithm is derived from recursive Gauss-Newton optimization, it can be viewed as a recursive maximum likelihood (RML) algorithm for time-delay estimation.

Abstract: Two incremental linear interpolation algorithms are derived and analyzed for speed and accuracy. The first is a version of a “simple” digital differential analyzer (DDA) employing fixed-point arithmetic, whereas the second is a new algorithm that uses only integral arithmetic and is a generalization of Bresenham's line-drawing algorithm. The new algorithm is shown to achieve perfect accuracy and, depending on the underlying processor, may be faster than the fixed-point algorithm.

Abstract: We give a sufficient condition for a sequence of points in the unit ball of ℂ n to be an interpolating sequence for H ∞ . This result generalizes Carleson's interpolation theorem in one complex variable, but an example shows that the condition is not necessary when n ≥1.

Abstract: Bloch Functions: The Basic Theory.- A Survey of Some Results on Subnormal Operators.- Optimization, Engineering, and a More General Corona Theorem.- Minimal Factorization, Linear Systems and Integral Operators.- Ha-Plitz Operators: A Survey of Some Recent Results.- Stochastic Processes, Infinitesimal Generators and Function Theory.- Paracommutators and Minimal Spaces.- Decomposition Theorems for Bergman Spaces and their Applications.- Operator-Theoretic Aspects of the Nevanlinna-Pick Interpolation Problem.- Cyclic Vectors in Banach Spaces of Analytic Functions.- Interpolation by Analytic Matrix Functions.

Abstract: Etude des proprietes supplementaires d'un operateur borne sur une famille parametree d'espaces de Banach. Etablissement d'estimations du second ordre dans la theorie de l'interpolation reelle