Abstract: This paper explores a Reproducing Kernel Particle Method (RKPM) which incorporates several attractive features. The emphasis is away from classical mesh generated elements in favour of a mesh free system which only requires a set of nodes or particles in space. Using a Gaussian function or a cubic spline function, flexible window functions are implemented to provide refinement in the solution process. It also creates the ability to analyse a specific frequency range in dynamic problems reducing the computer time required. This advantage is achieved through an increase in the critical time step when the frequency range is low and a large window is used. The stability of the window function as well as the critical time step formula are investigated to provide insight into RKPMs. The predictions of the theories are confirmed through numerical experiments by performing reconstructions of given functions and solving elastic and elastic–plastic one-dimensional (1-D) bar problems for both small and large deformation as well as three 2-D large deformation non-linear elastic problems. Numerical and theoretical results show the proposed reproducing kernel interpolation functions satisfy the consistency conditions and the critical time step prediction; furthermore, the RKPM provides better stability than Smooth Particle Hydrodynamics (SPH) methods. In contrast with what has been reported in SPH literature, we do not find any tensile instability with RKPMs.

Abstract: Techniques used in a previous study of the objective identification and tracking of meteorological features in model data are extended to the unit sphere. An alternative feature detection scheme is described based on cubic interpolation for the sphere and local maximization. The extension of the tracking technique, used in the previous study, to the unit sphere is described. An example of the application of these techniques to a global relative vorticity field from a model integration are presented and discussed.

Abstract: This paper describes the method for modeling human figure locomotions with emotions. Fourier expansions of experimental data of actual human behaviors serve as a basis from which the method can interpolate or extrapolate the human locomotions. This means, for instance, that transition from a walk to a run is smoothly and realistically performed by the method. Moreover an individual's character or mood, appearing during the human behaviors, is also extracted by the method. For example, the method gets "briskness" from the experimental data for a "normal" walk and a "brisk" walk. Then the "brisk" run is generated by the method, using another Fourier expansion of the measured data of running. The superposition of these human behaviors is shown as an efficient technique for generating rich variations of human locomotions. In addition, step-length, speed, and hip position during the locomotions are also modeled, and then interactively controlled to get a desired animation. Abstract

Abstract: Two “smart” interpolation procedures are presented and assessed with respect to their ability to estimate annual-average air temperatures at unsampled points in space from available station averages. Smart approaches examined here improve upon commonly used procedures in that they incorporate spatially high-resolution digital elevation information, an average environmental lapse rate, and/or another higher-resolution longer-term average temperature field. Two other straightforward or commonly used interpolation methods also are presented and evaluated as benchmarks to which the smart interpolators can be compared. Interpolation from a spatially high-resolution, long-term-average air temperature climatology serves as a first approximation, while “traditional” interpolation (from a single realization of annual average air temperature on a single station network) is the other benchmark. Traditional interpolation continues to be the most commonly used interpolation approach within many of the atmosph...

Abstract: SUMMARY An approach is presented for interpolating a property of the Earth (for example temperature or seismic velocity) specified at a series of ‘reference’ points with arbitrary distribution in two or three dimensions. The method makes use of some powerful algorithms from the field of computational geometry to efficiently partition the medium into ‘Delaunay’ triangles (in 2-D) or tetrahedra (in 3-D) constructed around the irregularly spaced reference points. The field can then be smoothly interpolated anywhere in the medium using a method known as natural-neighbour interpolation. This method has the following useful properties: (1) the original function values are recovered exactly at the reference points; (2) the interpolation is entirely local (every point is only influenced by its natural-neighbour nodes); and (3) the derivatives of the interpolated function are continuous everywhere except at the reference points. In addition, the ability to handle highly irregular distributions of nodes means that large variations in the scale-lengths of the interpolated function can be represented easily. These properties make the procedure ideally suited for ‘gridding’ of irregularly spaced geophysical data, or as the basis of parametrization in inverse problems such as seismic tomography. We have extended the theory to produce expressions for the derivatives of the interpolated function. These may be calculated efficiently by modifying an existing algorithm which calculates the interpolated function using only local information. Full details of the theory and numerical algorithms are given. The new theory for function and derivative interpolation has applications to a range of geophysical interpolation and parametrization problems. In addition, it shows much promise when used as the basis of a finite-element procedure for numerical solution of partial differential equations.

Abstract: The hurricane model initialization scheme developed at GFDL was modified to improve the representation of the environmental fields in the initial condition. The filter domain defining the extent of the tropical cyclone in the global analysis is determined from the distribution of the low-level disturbance winds. The shape of the domain is generally not circular in order to minimize the removal of important nonhurricane features near the storm region. An optimum interpolation technique is used to determine the environmental fields within the filter domain. Outside of the domain, the environmental fields are identical to the original global analysis. The generation process of the realistic and model-compatible vortex has also undergone some minor modifications so that reasonable vortices are produced for various data conditions. The upgraded hurricane prediction system was tested for a number of cases and compared against the previous version and yielded an overall improvement in the forecasts of s...

Abstract: An algorithm is presented to solve the elastic-wave equation by replacing the partial differentials with finite differences. It enables wave propagation to be simulated in three dimensions through generally anisotropic and heterogeneous models. The space derivatives are calculated using discrete convolution sums, while the time derivatives are replaced by a truncated Taylor expansion. A centered finite difference scheme in cartesian coordinates is used for the space derivatives leading to staggered grids. The use of finite difference approximations to the partial derivatives results in a frequency-dependent error in the group and phase velocities of waves. For anisotropic media, the use of staggered grids implies that some of the elements of the stress and strain tensors must be interpolated to calculate the Hook sum. This interpolation induces an additional error in the wave properties. The overall error depends on the precision of the derivative and interpolation operators, the anisotropic symmetry system, its orientation and the degree of anisotropy. The dispersion relation for the homogeneous case was derived for the proposed scheme. Since we use a general description of convolution sums to describe the finite difference operators, the numerical wave properties can be calculated for any space operator and an arbitrary homogeneous elastic model. In particular, phase and group velocities of the three wave types can be determined in any direction. We demonstrate that waves can be modeled accurately even through models with strong anisotropy when the operators are properly designed.

Abstract: We consider the dynamic interpolation problem for nonlinear control systems modeled by second-order differential equations whose configuration space is a Riemannian manifoldM. In this problem we are given an ordered set of points inM and would like to generate a trajectory of the system through the application of suitable control functions, so that the resulting trajectory in configuration space interpolates the given set of points. We also impose smoothness constraints on the trajectory and typically ask that the trajectory be also optimal with respect to some physically interesting cost function. Here we are interested in the situation where the trajectory is twice continuously differentiable and the Lagrangian in the optimization problem is given by the norm squared acceleration along the trajectory. The special cases whereM is a connected and compact Lie group or a homogeneous symmetric space are studied in more detail.

Abstract: Electronic cameras using a single CCD detector acquire scene color by subsampling in three, color planes and subsequently interpolating the information to reconstruct three, full-resolution color planes. The nature and size of the interpolation errors are a function of the algorithm used. When interpolation errors are propagated through the rest of the imaging chain, it becomes evident that synergistic effects among image processing operations must be considered when selecting and tuning an interpolation algorithm. This presentation demonstrates and comments on these image processing interactions.

Abstract: In this paper a solution is provided to the problem of obtaining a high resolution image from several low resolution images that have been subsampled and displaced by different amounts of sub-pixel shifts. In its most general form, this problem can be broken up into three sub-problems: registration, restoration, and interpolation. Previous work has either solved all three sub-problems independently, or more recently, solved either the first two steps (registration and restoration) or the last two steps together. However, none of the existing methods solve all three sub-problems simultaneously. This paper poses the low resolution to high resolution problem as a maximum likelihood (ML) problem which is solved by the expectation-maximization (EM) algorithm. By exploiting the structure of the matrices involved, the problem ran be solved in the discrete frequency domain. The ML problem is then the estimation of the sub-pixel shifts, the noise variances of each image, the power spectra of the high resolution image, and the high resolution image itself. Experimental results are shown which demonstrate the effectiveness of this approach.

Abstract: In this paper we formalize the observation that filtering and interpolation induce complementary, or "dual," decompositions of the space of positive real rational functions of degree less than or equal to n. From this basic result about the geometry of the space of positive real functions, we are able to deduce two complementary sets of conclusions about positive rational extensions of a given partial covariance sequence. On the one hand, by viewing a certain fast filtering algorithm as a nonlinear dynamical system defined on this space, we are able to develop estimates on the asymptotic behavior of the Schur parameters (1918) of positive rational extensions. On the other hand we are also able to provide a characterization of all positive rational extensions of a given partial covariance sequence. Indeed, motivated by its application to signal processing, speech processing, and stochastic realization theory, this characterization is in terms of a complete parameterization using familiar objects from systems theory and proves a conjecture made by Georgiou (1983, 1987). Our basic result, however, also enables us to analyze the robustness of this parameterization with respect to variations in the problem data. The methodology employed is a combination of complex analysis, geometry, linear systems, and nonlinear dynamics. >

Abstract: A method and apparatus are disclosed for interpolating pixels to obtain subpels for use by a video decompression processor. A prediction area is defined from which subpels are necessary to decompress a portion of a video image. Instead of reading all of the pixels from the prediction area and then processing them together to perform the necessary interpolation, portions of the pixel data are read and simultaneously averaged using in-place computation in order to reduce hardware requirements. Rounding of subpixel results is achieved using the carry input of conventional adders to add a binary "1" to the averaged pixels, which are subsequently truncated to provide the interpolated subpels.

Abstract: We introduce an acquisition method, "block regional interpolation scheme for k-space" (BRISK), to reduce the acquisition time for cardiac imaging. The method exploits the high degree of correlation that exists between time-resolved cardiac images. For representative k-space data sets, Fourier analysis was applied along the cardiac phase dimension to reveal that different regions of k-space can be effectively sampled at different rates. A reduced sampling strategy was implemented, and unsampled points were generated by Fourier interpolation. Time savings of up to 75% are quite feasible and 25% BRISK scans compare well with 100% scans. Simulations and acquisitions using a normal volunteer and patients are presented.

Abstract: We present a new dual‐level approach to representing potential energy surfaces in which a very small number of high‐level electronic structure calculations are combined with a lower‐level global surface, e.g., one defined implicitly by neglect‐of‐diatomic‐differential‐overlap calculations with specific reaction parameters, to generate the potential at any geometry where it may be needed. We interpolate the potential energy surface with a small number of accurate data points (the higher level) that are placed along the reaction path by using information on the global shape of the potential from less accurate calculations (the lower level). We confirm the findings of Ischtwan and Collins on the usefulness of single‐level schemes including Hessians, and we delineate the regime of usefulness of single‐level schemes based on gradients or even single‐point energies. Furthermore we find that dual‐level interpolation can offer cost savings over single‐level schemes, and dual‐level methods employing Hessians, gradients, or even only simple energy evaluations can yield reasonable potential energy surfaces with relatively low cost, with the potentials being more accurate along the reaction path. For all methods considered in this paper the accuracy of the interpolation for our test cases is lower when the potentials at points significantly removed from the reaction path are predicted from data that lie entirely on the reaction path.

Abstract: Preface. A Class of Interpolating Positive Linear Operators: Theoretical and Computational Aspects G. Allasia. Quasi-Interpolation E.W. Cheney. Approximation and Interpolation on Spheres E.W. Cheney. Exploring Covariance, Consistency and Convergence in Pade Approximation Theory A. Cuyt. Dijkstra's Cyclic Projections Algorithm: the Rate of Convergence F. Deutsch. Interpolation from a Convex Subset of Hilbert Space: a Survey of Some Recent Results F. Deutsch. The Angle between Subspaces of a Hilbert Space F. Deutsch. Neville Elimination and Approximation Theory M. Gasca, J.M. Pena. Approximation with Weights, the Chebyshev Measure and the Equilibrium Measure M.V. Golitschek. A One-Parameter Class of B-Splines Z.F. Kocak, G.M. Phillips. Interpolation on the Triangle and Simplex S.L. Lee, G.M. Phillips. Knot Removal for Scattered Data A. le Mehaute. Error Estimates for Approximation by Radial Basic Functions W. Light, H. Wayne. Wavelets on the Interval E. Quak, N. Weyrich. Best Approximations and Fixed Point Theorems S.P. Singh, B. Watson. How to Approximate the Inverse Operator J. Appell. On Some Averages of Trigonometric Interpolating Operators M. Campiti, G. Metafune, D. Pallara. On the Zeros Locations of K > 2 Consecutive Orthogonal Polynomials and of their Derivatives F. Costabile, R. Luceri. Can Irregular Subdivisions Preserve Convexity? S. de Marchi, M.M. Cecchi. On Functions Approximation by Shepart-Type Operators -- a Survey B. della Vecchia, G. Mastroianni. Wavelet Representation of the Potential Integral Equations M. Dorobantu. Lyapunov Theorem in Approximation Theory C. Franchetti. On the Order Monotonicity of the Metric Projection Operator G. Isac. Pointwise Estimates forMultivariate Interpolation Using Conditionally Positive Definite Functions J. Levesley. Experiments with a Wavelet Based Image Denoising Method M. Malfait, D. Roose. Proximity Maps: Some Continuity Results G. Marino, P. Pietramala. Non-Smooth Wavelets: Graphing Functions Unbounded in Every Interval D.L. Ragozin, A. Bruce, H.-Y. Gao. On the Possible Wavelet Packets Orthonormal Bases S. Saliani. A Case Study in Multivariate Lagrange Interpolation T. Sauer, Y. Xu. Trigonometric Wavelets for Time-Frequency-Analysis K. Selig. Interpolating Subspaces in Rn B. Shekhtman. Multivariate Periodic Interpolating Wavelets F. Sprengel. Finite Element Multiwavelets V. Strela, G. Strang. Polynomial Wavelets on [-1,1] M. Tasche. On the Solution of Discretely Given Fredholm Integral Equations over Lines E. Venturino. De-Noising Using Wavelets and Cross Validation N. Weyrich, G.T. Warhola. On the Construction of Two Dimensional Spatial Varying FIR Filter Banks with Perfect Reconstruction X. Xia, B.W. Suter. Recursions for Tchebycheff B-Splines and their Jumps Y. Xu. Quasi-Interpolation on Compact Domains J. Levesley, M. Roach. Index.

Abstract: Magnetic resonance (MR) tagging has shown great potential for noninvasive measurement of the motion of a beating heart. In MR tagged images, the heart appears with a spatially encoded pattern that moves with the tissue. The position of the tag pattern in each frame of the image sequence can be used to obtain a measurement of the 3-D displacement field of the myocardium. The measurements are sparse, however, and interpolation is required to reconstruct a dense displacement field from which measures of local contractile performance such as strain can be computed. Here, the authors propose a method for estimating a dense displacement field from sparse displacement measurements. Their approach is based on a multidimensional stochastic model for the smoothness and divergence of the displacement field and the Fisher estimation framework. The main feature of this method is that both the displacement field model and the resulting estimate equation are defined only on the irregular domain of the myocardium. The authors' methods are validated on both simulated and in vivo heart data.

Abstract: A recently developed multiresolution estimation framework offers the possibility of highly efficient statistical analysis, interpolation, and smoothing of extremely large data sets in a multiscale fashion. This framework enjoys a number of advantages not shared by other statistically-based methods. In particular, the algorithms resulting from this framework have complexity that scales only linearly with problem size, yielding constant complexity load per grid point independent of problem size. Furthermore these algorithms directly provide interpolated estimates at multiple resolutions, accompanying error variance statistics of use in assessing resolutionlaccuracy tradeoffs and in detecting statistically significant anomalies, and maximum likelihood estimates of parameters such as spectral power law coefficients. Moreover, the efficiency of these algorithms is completely insensitive to irregularities in the sampling or spatial distribution of measurements and to heterogeneities in measurement errors or model parameters. For these reasons this approach has the potential of being an effective tool in a variety of remote sensing problems. In this paper, we demonstrate a realization of this potential by applying the multiresolution framework to a problem of considerable current interest-the interpolation and statistical analysis of ocean surface data from the TOPEXPOSEIDON altimeter

Abstract: In this paper we describe two error-recovery approaches for MPEG encoded video over ATM networks. The first approach aims at reconstructing each lost pixel by spatial interpolation from the nearest undamaged pixels. The second approach recovers lost macroblocks by minimizing intersample variations within each block and across its boundaries. Moreover, a new technique for packing ATM cells with compressed data is also proposed.

Abstract: A numerical method based on the method of fundamental solutions, thin plate spine interpolation and monotone iteration is devised to find the minimal solution of the steady-state blow-up problem. The method of fundamental solutions requires neither domain nor boundary discretization and results in high accuracy and efficiency. For illustration, critical values of the Frank-Kamenetskii parameter are given for different geometrical boundaries in the two-dimensional case.

Abstract: The paper introduces a method for the sinc interpolation of discrete periodic signals. The convolution of the sinc kernel with the infinite sequence of a periodic function is rewritten as a finite summation. The method is equivalent to trigonometrical interpolation by Fourier series expansion. >

Abstract: In this paper we evaluate the use of higher order derivatives in the construction of an interpolated potential energy surface for the OH+H2→H2O+H reaction. The surface construction involves interpolating between local Taylor expansions about a set of known data points. We examine the use of first, second, third, and fourth order Taylor expansions in the interpolation scheme. The convergence of the various interpolated surfaces is evaluated in terms of the probability of reaction. We conclude that first order Taylor expansions (and by implication zeroth order expansions) are not suitable for constructing potential energy surfaces for reactive systems. We also conclude that it is inefficient to use fourth order derivatives. The factors differentiating between second and third order Taylor expansions are less clear. Although third order surfaces require substantially fewer data points to converge than second order surfaces, this faster convergence does not offset the large cost incurred in calculating numeri...

Abstract: Presents a new approach to rendering arbitrary views of real-world 3D objects of complex shapes. We propose to represent an object by a sparse set of corresponding 2D views, and to construct any other view as a combination of these reference views. We show that this combination can be linear, assuming proximity of the views, and we suggest how the visibility of constructed points can be determined. Our approach makes it possible to avoid difficult 3D reconstruction, assuming only rendering is required. Moreover, almost no calibration of views is needed. We present preliminary results on real objects, indicating that the approach is feasible. >

Abstract: 1. The prediction and mapping of climate in areas between climate stations is of increasing importance in ecology. 2. Four categories of model, simple interpolation, thin plate splines, multiple linear regression and mixed spline-regression, were tested for their ability to predict the spatial distribution of temperature on the British mainland. The models were tested by external cross-verification. 3. The British distribution of mean daily temperature was predicted with the greatest accuracy by using a mixed model: a thin plate spline fitted to the surface of the country, after correction of the data by a selection from 16 independent topographical variables (such as altitude, distance from the sea, slope and topographic roughness), chosen by multiple regression from a digital terrain model (DTM) of the country. 4. The next most accurate method was a pure multiple regression model using the DTM. Both regression and thin plate spline models based on a few variables (latitude, longitude and altitude) only were comparatively unsatisfactory, but some rather simple methods of surface interpolation (such as bilinear interpolation after correction to sea level) gave moderately satisfactory results. Differences between the methods seemed to be dependent largely on their ability to model the erect of the sea on land temperatures. 5. Prediction of temperature by the best methods was greater than 95% accurate in all months of the year, as shown by the correlation between the predicted and actual values. The predicted temperatures were calculated at real altitudes, not subject to sea-level correction. 6. A minimum of just over 30 temperature recording stations would generate a satisfactory surface, provided the stations were well spaced. 7. Maps of mean daily temperature, using the best overall methods are provided; further important variables, such as continentality and length of growing season, were also mapped. Many of these are believed to be the first detailed representations at real altitude. 8. The interpolated monthly temperature surfaces are available on disk

Abstract: Through the use of the dimensional splitting “cascade” method of grid-to-grid interpolation, it is shown that consistently high-order-accurate semi-Lagrangian integration of a three-dimensional hydrostatic primitive equations model can be carried out using forward (downstream) trajectories instead of the backward (upstream) trajectory computations that are more commonly employed in semi-Lagrangian models. Apart from the efficiency resulting directly from the adoption of the cascade method, improved computational performance is achieved partly by the selective implicit treatment of only the deepest vertical gravity modes and partly by obviating the need to iterate the estimation of each trajectory's location. Perhaps the main distinction of our present semi-Lagrangian method is its inherent exact conservation of mass and passive tracers. This is achieved by adopting a simple variant of the cascade interpolation that incorporates mass (and tracer) conservation directly and at only a very modest add...