Abstract: Several results are obtained appertaining to the reconstruction of a two- dimensional image from a finite number of projections. Several schemes are considered for interpolating between the given data. When a trigonometrical Fourier series is used for angular interpolation then one finds, firstly, a consistency condition whereby a posteriori estimates can be made of the errors in the given data, and secondly, a basic image which contains only that information common to all physically permissible interpolation schemes. This basic image is necessarily free of misleading artefacts but it is computationally slow. Several computationally rapid interpolation schemes (based on the fast Fourier transform algorithm) are found to give good quality images, provided the given number of projections is sufficient to resolve the major details of the true image. A computational example is presented showing that a contrived image can be accurately reconstructed from a single projection.

Abstract: : This report describes a method for using bidirectional reflectance information previously reported in the Eleventh Supplement to the Target Signature Analysis Center: Data Compilation and further validates the bidirectional reflectance model originated and extended under recent contracts. It includes bidirectional reflectance model parameters for a variety of paints. Parameters were extracted from measurement data reported in the Eleventh Supplement. Reduced reflectance data are also provided; these data may be used with the computer model or, optionally, in an interpolation procedure for estimating reflectances without the aid of a computer. The computer model makes it possible to calculate bidirectional reflectance data from a very small amount of measured data. Accuracy demonstrated in the Model Validation section indicates that the model is very effective, although improvement can still be obtained at large receiver zenith angles. The interpolation procedure also shows excellent agreement with measurement. (Author)

Abstract: A numerical technique for mathematically modeling the vibrational response of a complex structure immersed in an infinite acoustic medium is presented. The elastic response of the structure is modeled using the finite element method, and the acoustic radiation loading on the structure is modeled by approximating the surface Helmholtz integral equation formulation of the acoustic radiation problem. Arbitrary (and distinct) nodal point distributions and interpolation functions can be used in the finite element and acoustic radiation models. A technique defined in terms of these nodes and interpolation functions is presented for combining the results of these models into a combined equation of motion for the acoustically damped structure. The application of this technique to sonar transducers is discussed, including the modeling of piezoelectric material. The problem of obtaining reliable piezoelectric material parameters is discussed. Mathematical models are given for a piezoelectric sphere, a piezoelectric...

Abstract: A method is described for defining complex two- and threedimensional objects with a minimal amount of data. The method employs a ''''preprocessor'''' which receives the coordinate values of an ordered set of points on the object and calculates one or more state vectors which concisely define the object. Two numerical control systems are described in which these state vectors are applied to a ''''mapping'''' interpolator that controls the servomechanisms of a machine tool. The mapping interpolator controls machine tool motion to generate a three-dimensional surface which not only passes through each of the data points used to describe the object, but which also assumes a smooth shape of minimum strain energy.

Abstract: In [3] Boneva, Kendall and Stefanov (B.K.S.) have effectively rediscovered the essential features of what I like to call cardinal cubic spline interpolation. Moreover, and this is an important point, the data are not the usual function values that are to be interpolated, but rather approximations of the derivative (i.e. the unknown density function) in the form of a histogram. This (pershaps only apparent) difference is bridged by the ingenious area-matching condition.

Abstract: A method for dividing a polynomial of degree (2n-l) by a precomputed nth degree polynomial in 0(n log n) arithmetic operations is given. This is used to prove that the evaluation of an nth degree polynomial at n+1 arbitrary points can be done in 0(n log^ n) arithmetic operations, and consequently, its dual problem, interpolation of an nth degree polynomial from 2 n+1 arbitrary points can be performed in 0(n log n) arithmetic operations. The best previously known algorithms required 0(n log^ n) arithmetic operations .

Abstract: An objective cross-section analysis scheme based upon Hermite Polynomial interpolation on surfaces of constant potential temperature is shown to define frontal-scale horizontal and vertical gradients of potential temperature and geostrophic wind using synoptic upper air sounding observations as input data. The objective routine was also found applicable to analysis of the thermal properties of oceanic frontal zones from bathythermograph sounding data. Calculations of potential vorticity, vertical wind shear, and Richardson number suggest the usefulness of the objective scheme for real-time specification of regions of stratospheric-tropospheric mass exchange and probable locations of jet-stream, frontal-zone-related, clear-air turbulence.

Abstract: An alternate proof is presented of Kershaw’s result that the $L_\infty $-norm of the error in natural spline interpolation to a function $f \in C^4 [a,b]$ is $O(h^4 )$ in a closed subinterval which is asymptotic to $[a,b]$ as $h \to 0$. The case $f \in C^m [a,b],m = 2$ or 3, is also considered. These univariate results are then used to investigate natural bicubic spline interpolation over a rectangle $\mathcal{R}$. For $f \in C^m [\mathcal{R}]$, the $L_\infty $-norm of the error is shown to be $O(h^{m - 2} )$ throughout $\mathcal{R}$ and $O(h^m )$ in a closed subrectangle which is asymptotic to $\mathcal{R}$ as $h \to 0$. Arbitrary sequences of partitions are considered throughout.

Abstract: Publisher Summary This chapter demonstrates the similarity of the finite-difference energy method and the finite-element method by the application of both methods to problems involving shells of revolution. Curved finite elements are introduced into the BOSOR3 computer program and rates of convergence and computer times are established for stress, buckling, and vibration analyses of an elastic hemisphere. In both the finite-element and finite-difference energy methods, the unknowns of the problem are certain generalized displacement components located at discrete nodes in the domain. Integration can then be performed analytically or numerically. In the finite-difference calculations the “element” properties corresponding to the centroid of each element were provided as input. No interpolation is required because the energy is evaluated at only one point within each element. The eigenvalues and modes are determined by the inverse power iteration method with spectral shifts.

Abstract: A new, fast and economical automated procedure for implementing the traditional V-g method of flutter solution is described. The procedure requires as input the generalized aerodynamic forces for a range of reduced frequencies obtained from an aerodynamic program. These aerodynamic forces are interpolated with respect to reduced frequency using a newly developed, partially tabulated cubic spline that is both fast in execution and economical in storage. The flutter solution is then obtained using an eigenvalue routine that has been developed to take advantage of the parametric nature of the V-g type of solution. Furthermore, the routine takes care of the fundamental and troublesome problem of properly sorting the output eigenvalues. By solving the root-sorting problem, the interpolation for flutter crossings and automatic plotting are accomplished efficiently. The computational techniques used in this new program are described and some sample results are given.

Abstract: Recently, Bramble and Schatz have proposed a projection method for approximating the solution of Dirichlet's problem. Error estimates are derived by the authors using arguments based on certain interpolation theorems for linear operators on Hilbert spaces. It is shown here that simpler and shorter methods can be used to obtain these error estimates. 1. Introduction. The purpose of this note is to give a simplified proof of the results obtained by Bramble and Schatz in (1), where they proposed a method of least squares for obtaining approximations to solutions of Dirichlet's problem. To obtain error estimates in (1), Bramble and Schatz employed certain interpola- tion theorems for linear operators on Hilbert space, and, in particular, used these theorems in an iterative argument to get the results. Subsequently, an observation of Thomee (7) showed that the iterative argument mentioned above could be avoided with the use of appropriate trace theorems and a reformulation of the approximability assumptions on the subspaces used. This observation resulted in a slight simplification of the methods used in (1). Here, we present a new technique for obtaining the results of (1) which is much simpler and shorter than the prior ones. An entirely different approach is used which in essence involves a basic a priori estimate, and an argument based on duality. This new technique also yields a slight extention of the results of (1). In particular, the estimate (5.3) of Theorem 5.1 is new.

Abstract: Pocklington's integral equation for a straight, slim wire element is solved for the distribution of wire current, using the Bubnov-Galerkin projective method. Interpolation polynomials are employed as approximators, and special weighted Gaussian quadratures are used to avoid difficulties in the near-singular integrals encountered. It is shown that Kirchhoff's current law, applied wherever several such wire segments are electrically connected, leads to a modification of the approximating matrix equation by elementary row and column operations. Computer programs based on this method are used to analyse several different types of wire antenna, and the results are compared with experimental data. Agreement is generally good, and computing times pleasingly low, indicating that this method is likely to be generally preferable to point-matching methods that lead to much larger matrix sizes.

Abstract: Spline surfaces are interpolated for top of the Dundee Limestone of the central Michigan Basin, USA. The requirement of gridded data render spline functions inappropriate tools for representing many types of geological mapped data. Comparisons are drawn with maps for the same Michigan data based on trend surfaces and spatial filtering.

Abstract: A multi-axis machine tool or the like employing controls for an open loop stepping motor system of two or more axes in which variable frequency feedrate clock pulses are generated to ultimately control the speed of the various stepping motors. The frequency of the feedrate clock pulses is modified in accordance with programmed information; additional information defines the tool path which can either be straight line or circular arc segments and characteristically the feedrate clock pulse generation means increases pulse frequency at the beginning of a segment in such manner as to increase motor speed in accordance with system inertial limitations from a speed below to a speed above the slewing rate of the stepping motors. Thereafter upon sensing a predetermined distance from the end of the segment, pulse frequency is reduced to reduce the stepping motor speed below the slewing rate. The feedrate clock pulse chain is used by the interpolation logic to trigger two consecutive calculations, the time lapse between the clock pulse and the second calculation being so short compared to the time lapse between consecutive clock pulses, that the action initiated by the results of the calculations can be considered for all intents and purposes to be coincident with the clock pulse. The first calculation always generates a major axis interpolation pulse, during the second calculation an error accumulation register is tested to determine whether a minor axis interpolation pulse should be generated to minimize the error from the programmed path. Thus the major axis interpolation pulse chain corresponds directly to the frequency of the feedrate clock chain, the minor axis interpolation pulse chain may be irregular with respect to the chain of feedrate clock pulses. A predetermined member of interpolation pulses are then used to generate each of the major and minor axis step pulses so that any irregularity in minor axis pulse frequency is eliminated and resolution is improved.

Abstract: The interpolation and extrapolation techniques of evaluating data required for zone integration are briefly described and discussed. In particular, the characteristic errors incurred by each approach are considered.

Abstract: A comparison of the interpolation of index of refraction data for Czochralski sapphire, cyclohexane, and polystyrene dissolved in cyclohexane using a three-term Sellmeier equation, the Lorentz-Lorenz equation with six terms, third and fifth order polynomials, and a cubic-spline technique indicates that the cubic spline method is extremely valuable for simple interpolation. Not only were the magnitudes of the rms and average absolute residuals the smallest, but the fits showed no systematic errors.

Abstract: An electronic watch having a liquid crystal display and presenting time information in a substantially conventional format of hour and minute hands. A set of 12 hour and minute hands in a nested relationship are provided as thin film transparent conductor patterns in the liquid crystl display and are electronically actuated to provide visible indication of time to the nearest 5 minutes. Additional liquid crystal conductor patterns are provided as interpolation elements to indicate each minute. A control logic system is provided for sequentially activating and setting the hands to a particular indication and for additionally providing calendar data employing the existing pattern of hands.

Abstract: Publisher Summary This chapter discusses finite elements with harmonic interpolation functions. The basis of all finite element solutions of partial differential equations must be the satisfaction of three types of condition for best fit, namely, in the element interior on the interfaces, and on the external boundary of the entire region considered. It is pointed out that these conditions can be satisfied exactly, or in the mean, and that they are independent. Convergence of a solution may be obtained either by using the same type of element and increasing the number of elements toward infinity, while the size of each element tends to zero, or by keeping the division into elements unchanged and letting the number of degrees of freedom in each individual element increase toward infinity. Many engineering problems involve the solution of Laplace's equation. For this, if polynomial solutions are selected as interpolation functions in the elements, the condition in the interior is satisfied exactly. The energy integral method can no longer be used, but an alternative functional is proposed that consists solely of surface integrals. The method offers a possibility of satisfying both natural and essential boundary conditions for both the unknown function and its derivative. The chapter discusses the possibilities of solving biharmonic problems and also a nonlinear system as in the elasto-plastic problem.