Abstract: The paper describes a fast algorithm for scattered data interpolation and approximation. Multilevel B-splines are introduced to compute a C/sup 2/ continuous surface through a set of irregularly spaced points. The algorithm makes use of a coarse to fine hierarchy of control lattices to generate a sequence of bicubic B-spline functions whose sum approaches the desired interpolation function. Large performance gains are realized by using B-spline refinement to reduce the sum of these functions into one equivalent B-spline function. Experimental results demonstrate that high fidelity reconstruction is possible from a selected set of sparse and irregular samples.

Abstract: This dissertation focuses on efficiently forming reduced-order models for large, linear dynamic systems. Projections onto unions of Krylov subspaces lead to a class of reduced-order models known as rational interpolants. The cornerstone of this dissertation is a collection of theory relating Krylov projection to rational interpolation. Based on this theoretical framework, three algorithms for model reduction are proposed. The first algorithm, dual rational Arnoldi, is a numerically reliable approach involving orthogonal projection matrices. The second, rational Lanczos, is an efficient generalization of existing Lanczos-based methods. The third, rational power Krylov, avoids orthogonalization and is suited for parallel or approximate computations. The performance of the three algorithms is compared via a combination of theory and examples. Independent of the precise algorithm, a host of supporting tools are also develop ed to form a complete model-reduction package. Techniques for choosing the matching frequencies, estimating the modeling error, insuring the model's stability, treating multiple-input multiple-output systems, implementing parallelism, and avoiding a need for exact factors of large matrix pencils are all examined to various degrees

Abstract: Low-order vector basis functions compatible with the Nedelec (1980) representations are widely used for electromagnetic field problems. Higher-order functions are receiving wider application, but their development is hampered by the complex procedures used to generate them and lack of a consistent notation for both elements and bases. In this paper, fully interpolatory higher order vector basis functions of the Nedelec type are defined in a unified and consistent manner for the most common element shapes. It is shown that these functions can be obtained as the product of zeroth-order Nedelec representations and interpolatory polynomials with specially arranged arrays of interpolation points. The completeness properties of the vector functions are discussed, and expressions for the vector functions of arbitrary polynomial order are presented. Sample numerical results confirm the faster convergence of the higher order functions.

Abstract: Radial basis functions are presented as a practical solution to the problem of interpolating incomplete surfaces derived from three-dimensional (3-D) medical graphics. The specific application considered is the design of cranial implants for the repair of defects, usually holes, in the skull. Radial basis functions impose few restrictions on the geometry of the interpolation centers and are suited to problems where the Interpolation centers do not form a regular grid. However, their high computational requirements have previously limited their use to problems where the number of interpolation centers is small (<300). Recently developed fast evaluation techniques have overcome these limitations and made radial basis interpolation a practical approach for larger data sets. In this paper radial basis functions are fitted to depth-maps of the skull's surface, obtained from X-ray computed tomography (CT) data using ray-tracing techniques. They are used to smoothly interpolate the surface of the skull across defect regions. The resulting mathematical description of the skull's surface can be evaluated at any desired resolution to be rendered on a graphics workstation or to generate instructions for operating a computer numerically controlled (CNC) mill.

Abstract: A modeling method that takes into account known points on a geological interface and plane orientation data such as stratification or foliation planes is described and tested. The orientations data do not necessarily belong to one of the interfaces but are assumed to sample the main anisotropy of a geological formation as in current geological situations. The problem is to determine the surfaces which pass through the known points on interfaces and which are compatible with the orientation data. The method is based on the interpolation of a scalar field defined in the space the gradient in which is orthogonal to the orientations, given that some points have the same but unknown scalar value (points of the same interface), and that scalar gradient is known on the other points (foliations). The modeled interfaces are represented as isovalues of the interpolated field. Preliminary two-dimensional tests carried-out with different covariance models demonstrate the validity of the method, which is easily transposable in three dimensions.

Abstract: A proof of the (propositional) Craig interpolation theorem for cut-free sequent calculus yields that a sequent with a cut-free proof (or with a proof with cut-formulas of restricted form; in particular, with only analytic cuts) with k inferences has an interpolant whose circuit-size is at most k. We give a new proof of the interpolation theorem based on a communication complexity approach which allows a similar estimate for a larger class of proofs. We derive from it several corollaries: (1) Feasible interpolation theorems for the following proof systems:(a) resolution(b) a subsystem of LK corresponding to the bounded arithmetic theory (α)(c) linear equational calculus(d) cutting planes.(2) New proofs of the exponential lower bounds (for new formulas)(a) for resolution ([15])(b) for the cutting planes proof system with coefficients written in unary ([4]).(3) An alternative proof of the independence result of [43] concerning the provability of circuit-size lower bounds in the bounded arithmetic theory (α).In the other direction we show that a depth 2 subsystem of LK does not admit feasible monotone interpolation theorem (the so called Lyndon theorem), and that a feasible monotone interpolation theorem for the depth 1 subsystem of LK would yield new exponential lower bounds for resolution proofs of the weak pigeonhole principle.

Abstract: A computationally efficient method for analyzing meteorological and oceanographic historical data sets has been developed. The method combines data reduction and least squares optimal estimation. The data reduction involves computing empirical orthogonal functions (EOFs) of the data based on their recent, high-quality portion and using a leading EOF subset as a basis for the analyzed solution and for fitting a first-order linear model of time transitions. We then formulate optimal estimation problems in terms of the EOF projection of the analyzed field to obtain reduced space analogues of the optimal smoother, the Kalman filter, and optimal interpolation techniques. All reduced space algorithms are far cheaper computationally than their full grid prototypes, while their solutions are not necessarily inferior since the sparsity and error in available data often make estimation of small-scale features meaningless. Where covariance patterns can be estimated from the available data, the analysis methods fill gaps, correct sampling errors, and produce spatially and temporally coherent analyzed data sets. As with classical least squares estimation, the reduced space versions also provide theoretical error estimates for analyzed values. The methods are demonstrated on Atlantic monthly sea surface temperature (SST) anomalies for 1856–1991 from the United Kingdom Meteorological Office historical sea surface temperature data set (version MOHSST5). Choice of a reduced space dimension of 30 is shown to be adequate. The analyses are tested by withholding a significant part of the data and prove to be robust and in agreement with their own error estimates; they are also consistent with a partially independent optimal interpolation (OI) analysis by Reynolds and Smith [1994] produced in the National Centers for Environmental Prediction (NCEP)(known as the NCEP OI analysis). A simple statistical model is used to depict the month-to-month SST evolution in the optimal smoother algorithm. Results are somewhat superior to both the Kalman filter, which relies less on the model, and the optimal interpolation, which does not use it at all. The method generalizes a few recent works on using a reduced space for data set analyses. Difficulties of methods which simply fit EOF patterns to observed data are pointed out, and the more complete analysis procedures developed here are suggested as a remedy.

Abstract: A simple new procedure called STRAIGHT (speech transformation and representation using adaptive interpolation of weighted spectrum) has been developed. STRAIGHT uses pitch-adaptive spectral analysis combined with a surface reconstruction method in the time-frequency region, and an excitation source design based on phase manipulation. It preserves the bilinear surface in the time-frequency region and allows for over 600% manipulation of such speech parameters as pitch, vocal tract length, and speaking rate, without further degradation due to the parameter manipulation.

Abstract: Fuzzy control is at present still the most important area of real applications for fuzzy theory. It is a generalized form of expert control using fuzzy sets in the definition of vague/linguistic predicates, modeling a system by If...then rules. In the classical approaches it is necessary that observations on the actual state of the system partly match (fire) one or several rules in the model (fired rules), and the conclusion is calculated by the evaluation of the degrees of matching and the fired rules. Interpolation helps reduce the complexity as it allows rule bases with gaps. Various interpolation approaches are shown. It is proposed that dense rule bases should be reduced so that only the minimal necessary number of rules remain still containing the essential information in the original base, and all other rules are replaced by the interpolation algorithm that however can recover them with a certain accuracy prescribed before reduction. The interpolation method used for demonstration is the Lagrange method supplying the best fitting minimal degree polynomial. The paper concentrates on the reduction technique that is rather independent from the style of the interpolation model, but cannot be given in the form of a tractable algorithm. An example is shown to illustrate possible results and difficulties with the method.

Abstract: Keywords: traitement des signaux ; techniques de : mesure ; mathematiques ; methodes : numeriques Reference Record created on 2005-11-18, modified on 2016-08-08

Abstract: In this paper, it is shown that Cauchy's method can be used effectively to interpolate/extrapolate narrow-band system responses. The given information can either be theoretical datapoints or measured experimental data over a band. For theoretical data extrapolation or interpolation, the sampled values of the function and, optionally, a few of its derivatives have been used to reconstruct the function. For measured data, only measured values of the parameter are used to create broadband information from limited data as derivative information is too noisy. Cauchy's method assumes that the parameter to be extrapolated/interpolated, as a function of frequency, is a ratio of two polynomials. The problem is to determine the order of the polynomials and the coefficients therein. The method of total least squares (TLS) has been used to solve the resulting matrix equation involving the coefficients of the polynomials. Typical numerical results have been presented to show that reliable interpolation/extrapolation can be done for various system responses.

Abstract: The loose coupling of computational fluid dynamics and computational structural dynamics solvers introduces some problems related to the information transfer between the codes. Some techniques developed to solve the problems of the load transfer and interface surface tracking are presented. The main criterion is to achieve conservation of total loads and total energy. The load projection scheme is based on Gaussian integration and fast interpolation algorithms for unstructured grids. The surface tracking algorithm, also based on interpolation, is important for many applications, including aeroelastic deformation of wings due to aerodynamic loads. The methodologies not only improve present fluid-structure interaction simulations, but also increase the range of their applicability. These techniques are of general character and can be used in other multidisciplinary applications as well.

Abstract: Compressed bitstreams are, in general, very sensitive to channel errors. For instance, a single bit error in a coded video bitstream may cause severe degradation on picture quality. When bit errors occur during transmission and cannot be corrected by an error correction scheme, error concealment is needed to conceal the corrupted image at the receiver. Error concealment algorithms attempt to repair damaged portions of the picture by exploiting both the spatial and the temporal redundancies in the received and reconstructed video signal. We discuss several temporal, spatial, and transform-domain error concealment techniques for MPEG coded pictures, and propose a new scheme based on directional interpolation. We also compare the performance of these techniques by computer simulation.

Abstract: We describe a new method for analyzing, classifying, and evaluating filters that can be applied to interpolation filters as well as to arbitrary derivative filters of any order. Our analysis is based on the Taylor series expansion of the convolution sum. Our analysis shows the need and derives the method for the normalization of derivative filter weights. Under certain minimal restrictions of the underlying function, we are able to compute tight absolute error bounds of the reconstruction process. We demonstrate the utilization of our methods to the analysis of the class of cubic BC-spline filters. As our technique is not restricted to interpolation filters, we are able to show that the Catmull-Rom spline filter and its derivative are the most accurate reconstruction and derivative filters, respectively, among the class of BC-spline filters. We also present a new derivative filter which features better spatial accuracy than any derivative BC-spline filter, and is optimal within our framework. We conclude by demonstrating the use of these optimal filters for accurate interpolation and gradient estimation in volume rendering.

Abstract: A suite of methods to interpolate a digital elevation model from a ground survey was evaluated with respect to precision and ability to maintain the shape of the original height data. This shape reliability was evaluated by comparing the spatial patterns of secondary terrain parameters derived from the interpolated elevation data. The best interpolation method for this study area was found to be a spline interpolation, which is somewhat contradictory to findings in the literature. The error and uncertainty found in the results for terrain analysis and modelling tools is important and sometimes distressingly high, even for some frequently used local or context operations on altitude. Positional operations, in which the output is determined more by the position in the topographic structure, seem to give more reliable results. Therefore, the results obtained by terrain analysis and spatial modelling need careful interpretation. © 1997 John Wiley & Sons, Ltd.

Abstract: Experimental error analysis of a digital angular stereoscopic PIV system is presented. The paper firstly describes an experimental rig which includes the design of a novel PIV test block for in situ calibration. This allowed the user to set up a static seeded flow volume which was translated in and out of plane to record PIV images using two megapixel CCD cameras positioned for angular stereoscopic viewing. PIV data were collected for a range of camera angles up to and for a range of flow displacements and processed by cross correlation into a set of two-dimensional calibration and flow displacement vectors. These 2D data were then processed into three-dimensional data by the use of geometric and bicubic spline interpolation algorithms and an error analysis performed on the predicted displacements. Results from this analysis have shown optimum system performance will be obtained by using camera angles of between 20 and and f numbers of f16 and higher. The results have also shown a theoretical prediction of system performance derived in previous work, which considers the ratio of out of plane to in plane errors, matches to within 8 and 18% of the experimental system performance.

Abstract: This paper presents a discrete technique specially designed for modeling the geometry and the properties of natural objects as those encountered in biology and geology. Contrary to classical Computer-Aided Design methods based on continuous (polynomial) functions, the proposed approach is based on a discretization of the objects close to the finite-element techniques used for solving differential equations. Each object is modeled as a set of interconnected nodes holding the geometry and the physical properties of the objects and the Discrete Smooth Interpolation method is used for fitting the geometry and the properties to complex data. Data are turned into linear constraints and some constraints related to typical information encountered in geology are presented.

Abstract: Introduction. 1. Operator Identities and Interpolation Problems. 2. Interpolation Problems in the Unite Circle. 3. Hermitian-Positive Functions of Several Variables. 4. De Branges Spaces of Entire Functions. 5. Degenerate Problems (Matrix Case). 6. Concrete Interpolation Problems. 7. Extremal Problems. 8. Spectral Problems for Canonical Systems of Difference Equations. 9. Integrable Nonlinear Equations (Discrete Case). 10. On Semi-Infinite Toda Chain. 11. Functions with an Operator Argument. Commentaries and Remarks. Bibliography. Index.

Abstract: This survey presents several techniques for solving variants of the following scattered data interpolation problem: given a finite set of N points in R3, find a surface that interpolates the given set of points. Problems of this variety arise in numerous areas of applications such as geometric modeling and scientific visualization. A large class of solutions exists for these problems and many excellent surveys exist as well.The focus of this survey is on presenting techniques that are relatively recent. Some discussion of two popular variants of the scattered data interpolation problem -- trivariate (or volumetric) case and surface-on-surface -- is also included.Solutions are classified into one of the five categories: piecewise polynomial or rational parametric solutions, algebraic solutions, radial basis function methods, Shepard's methods and subdivision surfaces. Discussion on parametric solutions includes global interpolation by a single polynomial, interpolants based on data dependent triangulations, piecewise linear solutions such as alpha-shapes, and interpolants on irregular mesh.Algebraic interpolants based on cubic A-patches are described. Interpolants based on radial basis functions include Hardy's multiquadrics, inverse multiquadrics and thin plate splines. Techniques for blending local solutions and natural neighbor interpolants are described as variations of Shepard's methods. Subdivision techniques include Catmull-Clark subdivision technique and its variants and extensions. A brief discussion on surface interrogation techniques and visualization techniques is also included.

Abstract: We present some refinements of a recently developed scheme for interpolating and iteratively improving molecular potential-energy surfaces (PES). By comparison with an analytic surface for the OH+H 2 →H 2 O+H reaction, we show that an accurate and smooth PES may be constructed using of the order of 100–200 calculations of the energy, energy gradient and second derivatives. The refinements rely, in part, on improved methods for determining the optimum locations for these calculations.

Abstract: A survey of effective finite element formulations for the analysis of shell structures is presented. First, the basic requirements for shell elements are discussed, in which it is emphasized that generality and reliability are most important items. A general displacement-based formulation is then briefly reviewed. This formulation is not effective, but it is used as a starting point for developing a general and effective approach using the mixed interpolation of the tensorial components. The formulation of various MITC elements (that is, elements based on Mixed Interpolation of Tensorial Components) are presented. Theoretical results (applicable to plate analysis) and various numerical results of analyses of plates and shells are summarized. These illustrate some current capabilities and the potential for further finite element developments.

Abstract: There is a wide range of applications for which surface interpolation or approximation from scattered data points in space is important. Dependent on the field of application and the related properties of the data, many algorithms were developed in the past. This contribution gives a survey of existing algorithms, and identifies basic methods common to independently developed solutions. We distinguish surface construction based on spatial subdivision, distance functions, warping, and incremental surface growing. The systematic analysis of existing approaches leads to several interesting open questions for further research.

Abstract: There are many signal processing tasks for which convolution-based continuous signal representations such as splines and wavelets provide an interesting and practical alternative to the more traditional sine-based methods. The coefficients of the corresponding signal approximations are typically obtained by direct sampling (interpolation or quasi-interpolation) or by using least squares techniques that apply a prefilter prior to sampling. We compare the performance of these approaches and provide quantitative error estimates that can be used for the appropriate selection of the sampling step h. Specifically, we review several results in approximation theory with a special emphasis on the Strang-Fix (1971) conditions, which relate the general O(h/sup L/) behavior of the error to the ability of the representation to reproduce polynomials of degree n=L-1. We use this theory to derive pointwise error estimates for the various algorithms and to obtain the asymptotic limit of the L/sub 2/-error as h tends to zero. We also propose a new improved L/sub 2/-error bound for the least squares case. In the process, we provide all the relevant bound constants for polynomial splines. Some of our results suggest the existence of an intermediate range of sampling steps where the least squares method is roughly equivalent to an interpolator with twice the order. We present experimental examples that illustrate the theory and confirm the adequacy of our various bound and limit determinations.