Abstract: Although Bayesian analysis has been in use since Laplace, the Bayesian method of model-comparison has only recently been developed in depth. In this paper, the Bayesian approach to regularization and model-comparison is demonstrated by studying the inference problem of interpolating noisy data. The concepts and methods described are quite general and can be applied to many other data modeling problems. Regularizing constants are set by examining their posterior probability distribution. Alternative regularizers (priors) and alternative basis sets are objectively compared by evaluating the evidence for them. Occam's razor is automatically embodied by this process. The way in which Bayes infers the values of regularizing constants and noise levels has an elegant interpretation in terms of the effective number of parameters determined by the data set. This framework is due to Gull and Skilling.

Abstract: At the National Meteorological Center (NMC), a new analysis system is being extensively tested for possible use in the operational global data assimilation system. This analysis system is called the spectral statistical- interpolation (SSI) analysis system because the spectral coefficients used in the NMC spectral model are analyzed directly using the same basic equations as statistical (optimal) interpolation. Results from several months of parallel testing with the NMC spectral model have been very encouraging. Favorable features include smoother analysis increments, greatly reduced changes from initialization, and significant improvement of 1-5-day forecasts. Although the analysis is formulated as a variational problem, the objective function being minimized is formally the same one that forms the basis of all existing optimal interpolation schemes. This objective function is a combination of forecast and observation deviations from the desired analysis, weighted by the invent of the correspon...

Abstract: Fractional Brownian motion (FBM) offers a convenient modeling for nonstationary stochastic processes with long-term dependencies and 1/f-type spectral behavior over wide ranges of frequencies. Statistical self-similarity is an essential feature of FBM and makes natural the use of wavelets for both its analysis and its synthesis. A detailed second-order analysis is carried out for wavelet coefficients of FBM. It reveals a stationary structure at each scale and a power-law behavior of the coefficients' variance from which the fractal dimension of FBM can be estimated. Conditions for using orthonormal wavelet decompositions as approximate whitening filters are discussed, consequences of discretization are considered, and some connections between the wavelet point of view and previous approaches based on length measurements (analysis) or dyadic interpolation (synthesis) are briefly pointed out. >

Abstract: Finite element formulations based on stabilized bilinear and linear equal-order-interpolation velocity-pressure elements are presented for computation of steady and unsteady incompressible flows. The stabilization procedure involves a slightly modified Galerkin/least-squares formulation of the steady-state equations. The pressure field is interpolated by continuous functions for both the quadrilateral and triangular elements used. These elements are employed in conjunction with the one-step and multi-step time integration of the Navier-Stokes equations. The three test cases chosen for the performance evaluation of these formulations are the standing vortex problem, the lid-driven cavity flow at Reynolds number 400, and flow past a cylinder at Reynolds number 100.

Abstract: A new strategy based on the stabilized space-time finite element formulation is proposed for computations involving moving boundaries and interfaces. In the deforming-spatial-domain/space-time (DSD/ST) procedure the variational formulation of a problem is written over its space-time domain, and therefore the deformation of the spatial domain with respect to time is taken into account automatically. Because the space-time mesh is generated over the space-time domain of the problem, within each time step, the boundary (or interface) nodes move with the boundary (or interface). Whether the motion of the boundary is specified or not, the strategy is nearly the same. If the motion of the boundary is unknown, then the boundary nodes move as defined by the other unknowns at the boundary (such as the velocity or the displacement). At the end of each time step a new spatial mesh covers the new spatial domain. For computational feasibility, the finite element interpolation functions are chosen to be discontinuous in time, and the fully discretized equations are solved one space-time slab at a time.

Abstract: Focusing of SAR data requires a space-variant two-dimensional correlation. Different algorithms are compared with each other in terms of their focusing quality and their ability to handle the space-variance of the correlation kernel: the range-Doppler approach with and without secondary range compression, modified range-Doppler algorithms, and four versions of the wavenumber domain processor. The phase aberrations of the different algorithms are given in analytic form. Numerical examples are presented for Seasat and ERS-1. A novel systems theoretical derivation of the wavenumber domain algorithm is presented. >

Abstract: In such fields as geology and biology, a common problem is that of modelling complex surfaces that are defined by data of various types. Classical modelling techniques based on Bezier and spline interpolations can account for only some of these types of data. The paper proposes a different approach that is based on the discrete smooth interpolation method. In this approach, surfaces are modelled as 2D graphs whose node locations are determined for a wide variety of heterogeneous data.

Abstract: Convolution backprojection (CBP) image reconstruction has been proposed as a means of producing high-resolution synthetic-aperture radar (SAR) images by processing data directly in the polar recording format which is the conventional recording format for spotlight mode SAR. The CBP algorithm filters each projection as it is recorded and then backprojects the ensemble of filtered projections to create the final image in a pixel-by-pixel format. CBP reconstruction produces high-quality images by handling the recorded data directly in polar format. The CBP algorithm requires only 1-D interpolation along the filtered projections to determine the precise values that must be contributed to the backprojection summation from each projection. The algorithm is thus able to produce higher quality images by eliminating the inaccuracies of 2-D interpolation, as well as using all the data recorded in the spectral domain annular sector more effectively. The computational complexity of the CBP algorithm is O(N/sup 3/). >

Abstract: We give a complete description of sampling and interpolation in the Bargmann-Fock space, based on a density concept of Beurling. Roughly speaking, a discrete set is a set of sampling if and only if its density in every part of the plane is strictly larger than that of the von Neumann lattice, and similarly, a discrete set is a set of interpolation if and only if its density in every part of the plane is strictly smaller than that of the von Neumann lattice

Abstract: Various statistical procedures related to linear prediction and optimal filtering are developed for general, irregularly sampled, data sets. The data set may be a function of time, a spatial sample, or an unordered set. In the case of time series, the underlying process may be low-frequency divergent (weakly nonstationary)

Abstract: The problem of estimating the directions of arrival (DOAs) of signals, some of which may be perfectly correlated, is considered. A computationally efficient estimation algorithm which combines the ideas of spatial smoothing and array interpolation is derived. In one of its forms the proposed algorithm uses the root-MUSIC (root multiple signal classification) technique to compute the DOA estimates, thus avoiding the search procedure associated with the conventional MUSIC algorithm. The proposed technique can be applied to arbitrary array geometries and a general signal covariance matrix. The performance of the algorithm was evaluated by extensive simulations, and compared with the Cramer-Rao lower bound for the DOA estimates. >

Abstract: We describe an easily implemented and computationally feasible method for smoothly transitioning from one sampled volumetric model to another. This induces a transition between isosurfaces of the two models. The technique is based on interpolating smoothly between the Fourier transforms of the two volumetric models and then transforming the results back. A linear interpolation between the transformed datasets yields unsatisfactory results in some cases. We use a schedule for the interpolation in which the high frequencies of the first model are gradually removed, the low frequencies are interpolated to those of the second, and the high frequencies of the second model are gradually added in. Such scheduling yields more satisfactory results. We give several examples and comment briefly on preprocessing models to make the morphing smoother.

Abstract: The pair (6, P) of a point set 8 C R¿ and a polynomial space P on Rd is correct if the restriction map P —» E8 : p t-> P|Q is invertible, i.e., if there is, for any / defined on 0 , a unique p G P which matches /on 9. We discuss here a particular assignment 6 >-+ Fie , introduced by us previously, for which (8, lie) is always correct, and provide an algorithm for the construction of a basis for ne , which is related to Gauss elimination applied to the Vandermonde matrix (öa)öee a£Zd for 8. We also discuss some attractive properties of the above assignment and algorithmic details, and present some bivariate examples. We say that the pair (6, P) of a (finite) point set OcK1' and a (polynomial) space P of functions on Rd is correct if the restriction map

Abstract: In this paper we present methods to smoothly interpolate orientations, given N rotational key frames of an object along a trajectory. The methods allow the user to impose constraints on the rotational path, such as the angular velocity at the endpoints of the trajectory. We convert the rotations to quaternions, and then spline in that non-Euclidean space. Analogous to the mathematical foundations of flat-space spline curves, we minimize the net “tangential acceleration” of the quaternion path. We replace the flat-space quantities with curved-space quantities, and numerically solve the resulting equation with finite difference and optimization methods.

Abstract: An automatic method that can transform a sequence of tomographic image slices into an isotropic volume data set is described. In this method, correspondence is established between points in consecutive slices, and then this correspondence is used to estimate data between the slices by linear interpolation. The method takes advantage of the fact that consecutive slices have small geometric differences, and carries out the search in predicted small neighborhoods. Only points with high gradient magnitudes are used in the search process to increase the reliability of the correspondences. Mismatches that occur are detected and corrected using the continuity constraint in the correspondences. Experimental results showing the matching and interpolation of magnetic resonance slices and computed tomography slices are presented. >

Abstract: The implementation of the Ahmad-Cohen scheme based on a fourth-order Hermite integrator is described. With the fourth-order Hermite scheme, the force and the time derivative of the force are calculated analytically, and a third-order interpolation polynomial is constructed using two point in time. Compared with the standard scheme which is widely used, it allows a longer stepsize for the same accuracy, and the program is much simpler

Abstract: Multilayer perceptrons have recently been shown by M.A. Kramer and J.A. Leonard (1990, 1991) to give anomalous behavior in a diagnosis application. It is shown that this unsatisfactory behavior indicates a mismatch of application and technique, rather than any deficiency in the multilayer perceptron. It is further argued that this mismatch is indicative of the differences between applications that require interpolation and those which require extrapolation in addition. Simulation results to indicate a satisfactory performance of multilayer perceptrons in suitable applications are presented. >

Abstract: The underlying theory is statically and geometrically exact, and it naturally includes small strain and finite strain problems of thin as well as thick shells. This paper examines the development of computational procedures employing the Newton-Kantorovich linearization process and the Galerkin type discretization method, the treatment of finite rotations through an arbitrary parametrization of the rotation group, the interpolation procedure of SO(3)-valued functions underlying the construction of finite element basis.

Abstract: Two improvements on conventional ray bending are presented in this paper. First, gradient search methods are proposed to find the locations of successive points on a ray such that the travel time between its endpoints is minimal. Since only the integration of the travel time along the whole ray is involved, such methods are inherently more stable than methods that use first and second derivatives of the ray path to solve the raytracing equations. Second, it is shown that interpolation between successive points on a ray is generally necessary to obtain sufficient precision in the quadrature, but also advantageous in terms of efficiency when Beta-splines are used. Velocity discontinuities are easy to handle in the proposed bending algorithm. The target for the travel time precision is 1 part in 104, which can be considered necessary for many applications in seismic tomography. With the proposed method, this precision was reached for a constant gradient velocity model using the conjugate gradients algorithm on a five-point Beta-spline in single precision arithmetic. Several existing methods for parametrization and minimization failed to produce this target on this simple model even with computing times that are an order of magnitude larger. Calculations in a spherical Earth yield the required precision within feasible computation times. Finally, alternative search directions, which can be obtained from an approximate second order expansion of the travel time, accelerate the convergence in the first few iterations of the bending process.

Abstract: For Hermite-Birkhoff interpolation of scattered multidumensional data by radial basisfunction ,existence and characterization theorems and a variational principle are proved.Examples include (r)=r~b,Duchon’s thin-plate splines,Hardy’s multiquadrics,and inversemultiquadrics.

Abstract: Potential‐field geophysical data observed at scattered discrete points in three dimensions can be interpolated (gridded, for example, onto a level surface) by relating the point data to a continuous function of equivalent discrete point sources. The function used here is the inverse‐distance Newtonian potential. The sources, located beneath some of the data points at a depth proportional to distance to the nearest neighboring data point, are determined iteratively. Areas of no data are filled by minimum curvature. For two‐dimensional (2-D) data (all data points at the same elevation), grids calculated by minimum curvature and by equivalent sources are similar, but the equivalent‐source method can be tuned to reduce aliasing. Gravity data in an area of high topographic relief in southwest U.S.A. were gridded by minimum curvature (a 2-D algorithm) and also by equivalent sources (3-D). The minimum‐curvature grid shows strong correlation with topography, as expected, because variation in gravity effect due to...

Abstract: Part I. Curve Design: Properties of Minimal Energy Splines G. Brunnett Minimal Energy splines with Various End Constraints E. Jou and W. Han Interval Weighted Tau- splines D. Lasser and H. Hagen Curve and surface Interpolation using Quintic Weight Tau-splines D. Neuser Weighted splines Based on Piecewise Polynomial Weighted Functions K. Salkauskas Algorithms for Geometric spline Curves M. Eck On the Problem of Determining the distance Between parametric curves F. Fritsch and G. Nielson Part II. Non-Tensor Product Surfaces: A survey of scattered Data Fitting Using Triangular Interpolants T. De Rose Free-form surfaces from Partial Differential Equations M. I. G. Bloor and M. J. Wilson Modeling with Box spline surfaces M. Daehlen.

Abstract: Corrected expressions for parameter variances in linear (straight‐line) least‐squares fits given in an earlier paper [B. C. Reed, Am. J. Phys. 57, 642–646 (1989)] are presented and discussed. In addition, some simplifications to the solution given in that paper are presented, and some comments made as to the role of weighting.

Abstract: Part 1 Research and survey articles: on Bernstein-Durrineyer polynomials with Jacob weights, H.Berens and Y.Xu an overview of wavelets, C.Chui a note on weak inequalities in Orlicz and Lorentz spaces, G.A.Edgar and L.Sucheston bivariate Birkhoff interpolation - a survey, R.A.Lorentz linear approximations of functions with several restricted derivatives, Y.Makovoz some characterizations theorems for measures associated with orthogonal polynomials on the unit circle, K.Pan and E.B.Saff box splines, cardinal series, and wavelets, S.D.Reimenschneider and Z.Shen some aspects of the subspace structure of infinite dimensional banach space, H.Rosenthal fairness and monotone curvature, J.Roulier et al projections on 2-dimensional spaces, N.Tomczak-Jacgermann real versus complex best rational approximation, R.S.Varga and A.Ruttan.

Abstract: Spectral conditions are established for the possibility of extrapolating and interpolating stationary random sequences by a sufficiently large number of terms with any prescribed accuracy.