Abstract: At the National Meteorological Center (NMC), a new analysis system was implemented into the operational Global Data Assimilation System on 25 June 1991. This analysis system is referred to as Spectral Statistical Interpolation (SSI) because the spectral coefficients used in the NMC spectral model are analyzed directly using the same basic equations as statistical (optimum) interpolation. The major differences between the SSI analysis system and the conventional optimum interpolation (OI) analysis system previously used operationally at NMC are: –The analysis variables are closely related to the coefficients of the NMC spectral model. –Temperature observations are used, not heights as in the previous procedure. As a result, aircraft temperatures are being used for the first time at NMC. –Nonstandard observations, such as satellite estimates of total precipitable water and ocean-surface wind speeds, can be easily included. –No data selection is necessary. All observations are used simultaneously. –...

Abstract: CT scanning in spiral geometry is achieved by continuously transporting the patient through the gantry in synchrony with continuous data acquisition over a multitude of 360-deg scans. Data for reconstruction of images in planar geometry are estimated from the spiral data by interpolation. The influence of spiral scanning on image quality is investigated. Most of the standard physical performance parameters, e.g., spatial resolution, image uniformity, and contrast, are not affected; results differ for pixel noise and slice sensitivity profiles. For linear interpolation, pixel noise is expected to be reduced by a factor of 0.82; reduction factors of 0.81 to 0.83 were measured. Slice sensitivity profiles are changed as a function of table feed d, measured in millimeters per 360-deg scan; they are smoothed as the original profile is convolved with the object motion function. The motion function is derived for linear interpolation that constitutes a triangle with a base line width of 2d and a maximal height equal to 1/d. Calculations of both the full width at half-maximum and the shape of the profiles were in good agreement with experimental results. The effect of the widened profiles, in particular of their extended tail ends, on image quality is demonstrated in phantom measurements.

Abstract: The multivariate scattered data interpolation problem is introduced and the reasons for the difficulty of the problem compared to the one dimensional case are discussed. Basic ideas for interpolation (or approximation) of scattered data are introduced. Various types of data sets and some strategies for dealing with some of them are given. Readily available algorithms for the solution of the problem are discussed and suitability for various types of data, along with discussion of situations where they have been useful is given. Some related ideas are briefly mentioned. Throughout there are bountiful references to the existing literature.

Abstract: The usual Bramble-Hilbert theory is extended for proving more refined estimates of the interpolation error For a large class of finite elements, it is shown that one can derive benefit from the presence of small and even large angles of the elements For bilinear shape functions on rectangular grids it is proved that interpolation and finite element approximation error coincide As an example, we consider the finite element approximation for problems on domains containing edges

Abstract: The frequency-dependent resistance and inductance of uniform transmission lines are calculated with a hybrid technique that combines a cross-section coupled circuit method with a surface integral equation approach. The coupled circuit approach is most applicable for low-frequency calculations, while the integral equation approach is best for high frequencies. The low-frequency method consists in subdividing the cross section of each conductor into triangular filaments, each with an assumed uniform current distribution. The high-frequency method expresses the resistance and inductance of each conductor in terms of the current normal to the surface. An interpolation between the results of these two methods gives very good results over the entire frequency range, even when few basis functions are used. Results for a variety of configurations are shown and are compared with experimental data and other numerical techniques. >

Abstract: We consider system identification in H∞ in the framework proposed by Helmicki, Jacobson and Nett. An algorithm using the Jackson polynomials is proposed that achieves an exponential convergence rate for exponentially stable systems. It is shown that this, and similar identification algorithms, can be successfully combined with a model reduction procedure to produce low-order models. Connections with the Nevanlinna-Pick interpolation problem are explored, and an algorithm is given in which the identified model interpolates the given noisy data. Some numerical results are provided for illustration. Finally, the case of unbounded random noise is discussed and it is shown that one can still obtain convergence with probability 1 under natural assumptions.

Abstract: This paper reports on a research project concerned with the areal interpolation problem — the problem of comparing different data sets when they have been made available for different zonal systems. Our approach is based on using additional information to guide the interpolation process. This paper emphasizes recent work applying the method to Poisson and binomially distributed data. There is also discussion of how the method can best be implemented in a geographic information system.

Abstract: An optimal sampling interpolation algorithm which allows the accurate recovery of plane-rectangular near-field samples from the knowledge of the plane-polar ones is developed. This enables the standard near-field-far-field (NF-FF) transformation, which takes full advantage of the fast Fourier transform (FFT) algorithm, to be applied to plane-polar scanning. The maximum allowable sample spacing is also rigorously derived, and it is shown that it can be significantly greater than lambda /2 as the measurement place moves away from the source. This allows a remarkable reduction of both measurement time and memory storage requirements. The sampling approach is compared with that based on the bivariate Lagrange interpolation (BLI) method. The sampling reconstruction agrees with the exact results significantly better than the BLI, in spite of the significantly lower number of required measurements. >

Abstract: Cauchy's technique for interpolating a rational function from samples of frequency responses and/or their derivatives is investigated. This technique can be used to speed up the numerical computations of parameters, including input impedance and RCS of any linear time-invariant electromagnetic system. This technique is utilized to find the far field of a slit conducting cylinder (TM incidence) over a bandwidth utilizing the information about the current and its derivatives at a few sample points. The numerical results are presented are in good agreement with exact computational data. This technique is a true interpolation/extrapolation technique as it provides the same defective result as the original electric field integral equation at a frequency which corresponds to the internal resonance of the closed structure. >

Abstract: Experience with the handling of spatial data on a computer led to the identification of a variety of “awkward” problems, including interpolation, error estimation and dynamic polygon building and e...

Abstract: This paper presents the motivation for and construction of coordinate transformations that generate optimally efficient meshes for linear interpolation. The coordinate transformations are derived from a result in differential geometry characterizing a “flat” space. The optimality results are demonstrated for some numerical examples. Adaptive meshes produced by PLTMG [R. E. Bank, PLTMG: A Software Package for Solving Elliptic Partial Differential Equations, Society for Industrial and Applied Mathematics, Philadelphia, PA, 1990] are included for comparison. The paper concludes that coordinate transformation is a promising strategy for investigation into more complex optimal meshing problems in finite element analysis.

Abstract: The method of “wave superposition” is based on the idea that an acoustic radiator can be approximately represented by the sum of the fields due to a finite number of interior point sources. The accuracy of this representation depends upon how well the velocity boundary condition on the surface of the body is approximated. The ultimate objective of this study, then, is to provide some guidelines for improving the accuracy of the surface velocity reconstruction and, consequently, the accuracy of the superposition solutions. In general, this is dependent upon the particular surface velocity distribution to be reconstructed, as well as other formulation factors such as the acoustic wave number, the number and locations of the surface nodes, and the number and locations of the point sources. Velocity interpolation functions are introduced as a means of quantifying the dependence of reconstruction errors on the acoustic wave number and the placement of the surface nodes and point sources. Numerical experiments on cylindrical radiators with different velocity distributions are performed to further illustrate how the solution accuracy depends on the surface velocity boundary conditions as well as the other formulation factors.

Abstract: An imaging system apparatus and operating method thereof utilize a volumetric resampling technique with interpolation to determine sample point values on user-defined paths traced through a volumetric image comprising a group of predefined data values, which are themselves samples of a continuous volumetric object. Data values to be displayed are determined from the sample point values using user-selected interpolation and visualization operating modes.

Abstract: The goal of this research is to develop interpolation techniques which preserve or enhance the local structure critical to image quality. Preliminary results are presented which exploit either the properties of vision or the properties of the image in order to achieve the goals. Directional image interpolation is considered which is based on a local analysis of the spatial image structure. The extension of techniques for the design of linear filters based on properties of human perception reported previously to enhance the perceived quality of interpolated images is considered. >

Abstract: An efficient method is proposed for performing the grid interpolations required at each advective time step of a multilevel, limited-area semi-Lagrangian model. The distinctive feature of the method is that it is composed of a cascade of one-dimensional interpolations of entire fields of data through a sequence of intermediate grids. These intermediate grids are formed as hybrid combinations of the standard model grid coordinates, which, together with the Lagrangian coordinates, are delineated by the origins of the trajectories that characterize the semi-Lagrangian method. When applied at Nth-order accuracy, the cascade method requires only O(N) operations compared with O(N3) for the conventional three-dimensional interpolation methods, making the adoption of high-order schemes attractive. The technique was tested on a large number (100) of 48-h forecasts and was found to be as accurate as the conventional interpolation procedures based on point-by-point Cartesian products of one-dimensional inte...

Abstract: Fractional Brownian motion is a surface model through which the topographic surface may be described. This model provides a simple method for estimating the fractal dimension of a digital elevation model. By computing the fractal dimension value at different directions, interpolation artifacts may be revealed, contributing to digital elevation model quality assessment.

Abstract: Accurate estimation of the parameters of a curve present in a grey-level image is required in various machine-vision and computer-vision problems. Quadratic curves are more common than other curve types in these fields. The accuracy of the estimated parameters depends not only on the global interpolation technique used, but, as well, on compensation of major sources of error. In this paper, first, as a preliminary step in accurate parameter estimation of quadratic curves, a sequential distortion-compensation procedure is formulated. This procedure addresses the major distortion factors involved in the transformation of a curve from the object space to the image space. Subsequently, as a means for accurate estimation of the coordinates of edge points, a new subpixel edge detector based on the principle of the sample-moment-preserving transform (SMPT) is developed. A circular-arc geometry is assumed for the boundary inside the detection area. The new arc-edge detector is designed as a cascade process using a linear-edge detector and a look-up table. Its performance is compared with that of a linear subpixel edge detector. Then, as a part of the main theme of the paper, the estimation of the five basic parameters of an elliptical shape based on its edge-point data is addressed. To achieve the desired degree of accuracy, a new error function is introduced and as the basis for a comparative study, an objective and independent measure for “goodness” of fit is derived. The proposed new error function and two other error functions previously developed are applied to six different situations. The comparative performance of these error functions is discussed. Finally, as the basis for evaluation of the total process, a 3D location estimation problem is considered. The objective is to accurately estimate the orientation and position in 3D of a set of circular features. The experimental results obtained are significant in two separate ways: in general, they show the validity of the overall process introduced here in the accurate estimation of 3D location; in particular, they demonstrate the effectiveness of the sub-pixel edge detector and the global interpolation technique, both developed here.

Abstract: In this paper we consider the dynamic interpolation problem for control systems in which certain dynamic variables of state trajectories are forced to pass through specific points by suitable choices of controls. This problem can be viewed as an extension of the spline problem. Following Noakes, Heinzinger and Paden [16], we give a derivation of suitable interpolating cubic splines on a Riemannian manifold extending the variational approach in Milnor [15]. For the special case of compact Lie groups, the relation with optimal control problems and singular Riemannian Geometry is spelled out in detail.

Abstract: A finite impulse response (FIR) filter is presented which can synthesize any fractional sample delay by nonlinear interpolation. This circuit can be used as a digital interpolator capable of compensating for the delay time between the echo canceller output and the receiver input for the V.32 full-duplex modem or the U-transceiver in digital subscriber loop (DSL). Analytic closed-forms for the tap weights of such an FIR filter and numerical verifications are presented. >

Abstract: The authors present an active triangulation-based range finding system composed of an independent laser system generating a sheet of light projected on the object to be measured, which is placed on a linear or a rotary table driven by a personal computer. This computer includes a video digitizer board to which two cameras, looking at the scene from both sides of the sheet of light, are connected. Besides its low cost, this system has several advantages over similar systems. First of all, two cameras are used to limit the occlusion problem, and a method is proposed to integrate range data obtained from these cameras into a single range image. The calibration of each camera is very simple, provides subpixel accuracy, and is performed only once as the laser or the camera does not move. The data acquisition uses an interpolation technique that produces very accurate depth measurements. The system also provides intensity data in registration with the range data. The application of all these techniques is illustrated by showing numerous examples of the range and intensity data acquisition from various complex objects. >

Abstract: The nonlinear dynamics of air flow during speech production may often result in some small or large degree of turbulence. The author quantifies the geometry of speech turbulence, as reflected in the fragmentation of the time signal, by using fractal models. He describes an efficient algorithm for estimating the short-time fractal dimension of speech segmentation and sound classification. He also develops a method for fractal speech interpolation which can be used to synthesize controlled amounts of turbulence in speech or to increase its sampling rate by preserving not its bandwidth (as is classically done) but rather its fractal dimension. >

Abstract: An analytic method of digital interpolator optimization is proposed. Performance approaching or equaling that of full polyphase filter designs is achieved through the generation of orthogonal filters via singular value decomposition (SVD). This approach preserves the magnitude, group delay, and composite filter response and can eliminate the traditional restriction to ratio of integer interpolation and decimation factors. >

Abstract: An adaptive local mesh refinement strategy for two-phase Stefan problems is discussed in light of its efficiency and computational complexity. Three local parameters are used to equidistribute interpolation errors in maximum norm for temperature and a fourth one, in the event of mushy regions, to equidistribute $L^1 $-interpolation errors for enthalpy within the mush. If certain quality mesh tests fail, then the current mesh is discarded and a new one completely regenerated by an efficient mesh generator, which in turn is briefly described. A typical triangulation is strongly graded to become very fine near computed interfaces and coarse away from them. Consecutive meshes are not compatible. The use of quadtree data structures is discussed as a means to reach a nearly optimal computational complexity in various tasks to be performed, mainly in generating a mesh and interpolating. Various implementation details are given so as to derive the computational complexity of each relevant subroutine. The approximation of both solutions and interfaces is drastically improved. The proposed method is robust in that it can handle the formation of cusps and mushy regions as well as the spontaneous appearance of phases. It is also superior in terms of computing time or a desired accuracy. Several numerical experiments illustrate these facts and provide quantitative information about each task complexity.

Abstract: We investigate how compact operators behave under J and K interpolation methods for N spaces and two parameters. First we study those methods: relationship with those already existing in the literature, estimates for the norms of interpolated operators, examples, characterization as Aronszajn-Gagliardo functors,.... We also describe the relationship between Sparr and Fernandez methods and we derive sharp estimates for the norms of interpolated operators in Fernandez' case. Then we investigate the behaviour of compact operators. We begin with the case when one of the N-tuples reduces to a single Banach space, and later we treat the general case by means of the approach developed in [8].

Abstract: The links between adaptive layered networks, functional interpolation and dynamical systems are considered and applied to the nonlinear predictive analysis of time series. The ability of networks to produce interpolation surfaces to generators of data (i.e. differential equations, iterative maps) is used to analyse a variety of time series. If network may be trained to approximate a static) generator of data, the network may be iterated on its own output to produce a time series with the same characteristics as the training waveform. However, since iterated networks are one example of nonlinear dynamical systems, this raises problems of sensitive dependence upon initial conditions leading ultimately to deterministic chaos. An introduction to the relevant concepts is presented and illustrations are provided from simple chaotic maps, nonlinear differential equations, and stock-market prediction. The latter example is included to illustrate the problems which often occur in real-world data due to noise, undersampling, high dimensionality and insufficient data.