Abstract: Pose space deformation generalizes and improves upon both shape interpolation and common skeleton-driven deformation techniques. This deformation approach proceeds from the observation that several types of deformation can be uniformly represented as mappings from a pose space, defined by either an underlying skeleton or a more abstract system of parameters, to displacements in the object local coordinate frames. Once this uniform representation is identified, previously disparate deformation types can be accomplished within a single unified approach. The advantages of this algorithm include improved expressive power and direct manipulation of the desired shapes yet the performance associated with traditional shape interpolation is achievable. Appropriate applications include animation of facial and body deformation for entertainment, telepresence, computer gaming, and other applications where direct sculpting of deformations is desired or where real-time synthesis of a deforming model is required.

Abstract: An electron-positron equation of state based on table interpolation of the Helmholtz free energy is developed and analyzed. The interpolation scheme guarantees perfect thermodynamic consistency, independent of the interpolating function. The choice of a biquintic Hermite polynomial as the interpolating function results in accurately reproducing the underlying Helmholtz free energy data in the table, and yields derivatives of the pressure, specific entropy, and specific internal energy which are smooth and continuous. The execution speed—evaluated across several different machine architectures, compiler options, and modes of operation—suggests that the Helmholtz equation of state routine is faster than any of the five equation of state routines surveyed by Timmes & Arnett. When an optimal balance of accuracy, thermodynamic consistency, and speed is desirable then the tabular Helmholtz equation of state is an excellent choice, particularly for multidimensional models of stellar phenomena.

Abstract: A software suite consisting of 17 MATLAB functions for solving differential equations by the spectral collocation (i.e., pseudospectral) method is presented. It includes functions for computing derivatives of arbitrary order corresponding to Chebyshev, Hermite, Laguerre, Fourier, and sinc interpolants. Auxiliary functions are included for incorporating boundary conditions, performing interpolation using barycentric formulas, and computing roots of orthogonal polynomials. It is demonstrated how to use the package for solving eigenvalue, boundary value, and initial value problems arising in the fields of special functions, quantum mechanics, nonlinear waves, and hydrodynamic stability.

Abstract: Based on the theory of approximation, this paper presents a unified analysis of interpolation and resampling techniques. An important issue is the choice of adequate basis functions. The authors show that, contrary to the common belief, those that perform best are not interpolating. By opposition to traditional interpolation, the authors call their use generalized interpolation; they involve a prefiltering step when correctly applied. The authors explain why the approximation order inherent in any basis function is important to limit interpolation artifacts. The decomposition theorem states that any basis function endowed with approximation order ran be expressed as the convolution of a B spline of the same order with another function that has none. This motivates the use of splines and spline-based functions as a tunable way to keep artifacts in check without any significant cost penalty. The authors discuss implementation and performance issues, and they provide experimental evidence to support their claims.

Abstract: This is a survey of the main results on multivariate polynomial interpolation in the last twenty-five years, a period of time when the subject experienced its most rapid development. The problem is considered from two different points of view: the construction of data points which allow unique interpolation for given interpolation spaces as well as the converse. In addition, one section is devoted to error formulas and another to connections with computer algebra. An extensive list of references is also included.

Abstract: We study polynomial interpolation on a d-dimensional cube, where d is large. We suggest to use the least solution at sparse grids with the extrema of the Chebyshev polynomials. The polynomial exactness of this method is almost optimal. Our error bounds show that the method is universal, i.e., almost optimal for many different function spaces. We report on numerical experiments for d = 10 using up to 652 065 interpolation points.

Abstract: We develop a new paradigm for thin-shell finite-element analysis based on the use of subdivision surfaces for (i) describing the geometry of the shell in its undeformed configuration, and (ii) generating smooth interpolated displacement fields possessing bounded energy within the strict framework of the Kirchhoff–Love theory of thin shells. The particular subdivision strategy adopted here is Loop's scheme, with extensions such as required to account for creases and displacement boundary conditions. The displacement fields obtained by subdivision are H2 and, consequently, have a finite Kirchhoff–Love energy. The resulting finite elements contain three nodes and element integrals are computed by a one-point quadrature. The displacement field of the shell is interpolated from nodal displacements only. In particular, no nodal rotations are used in the interpolation. The interpolation scheme induced by subdivision is non-local, i.e. the displacement field over one element depend on the nodal displacements of the element nodes and all nodes of immediately neighbouring elements. However, the use of subdivision surfaces ensures that all the local displacement fields thus constructed combine conformingly to define one single limit surface. Numerical tests, including the Belytschko et al. [10] obstacle course of benchmark problems, demonstrate the high accuracy and optimal convergence of the method.

Abstract: Recently, digital image correlation as a tool for surface defor- mation measurements has found widespread use and acceptance in the field of experimental mechanics. The method is known to reconstruct displacements with subpixel accuracy that depends on various factors such as image quality, noise, and the correlation algorithm chosen. How- ever, the systematic errors of the method have not been studied in detail. We address the systematic errors of the iterative spatial domain cross- correlation algorithm caused by gray-value interpolation. We investigate the position-dependent bias in a numerical study and show that it can lead to apparent strains of the order of 40% of the actual strain level. Furthermore, we present methods to reduce this bias to acceptable lev- els. © 2000 Society of Photo-Optical Instrumentation Engineers. (S0091-3286(00)00911-9)

Abstract: This chapter presents a survey of interpolation and resampling techniques in the context of exact, separable interpolation of regularly sampled data. In this context, the traditional view of interpolation is to represent an arbitrary continuous function as a discrete sum of weighted and shifted synthesis functions—in other words, a mixed convolution equation. An important issue is the choice of adequate synthesis functions that satisfy interpolation properties. Examples of finite-support ones are the square pulse (nearest-neighbor interpolation), the hat function (linear interpolation), the cubic Keys' function, and various truncated or windowed versions of the sinc function. On the other hand, splines provide examples of infinite-support interpolation functions that can be realized exactly at a finite, surprisingly small computational cost. We discuss implementation issues and illustrate the performance of each synthesis function. We also highlight several artifacts that may arise when performing interpolation, such as ringing, aliasing, blocking and blurring. We explain why the approximation order inherent in the synthesis function is important to limit these interpolation artifacts, which motivates the use of splines as a tunable way to keep them in check without any significant cost penalty.

Abstract: The book systematically develops nonlinear potential theory and the Sobolev space theory covers results concerning approximation, extension, and interpolation, Sobolev and Poincare inequalities, Ma ...

Abstract: The main topics of this technical report are quaternions, their mathematical properties, and how they can be used to rotate objects. We introduce quaternion mathematics and discuss why quaternions are a better choice for implementing rotation than the well-known matrix implementations. We then treat di erent methods for interpolation between series of rotations. During this treatment we give complete proofs for the correctness of the important interpolation methods Slerp and Squad . Inspired by our treatment of the di erent interpolation methods we develop our own interpolation method called Spring based on a set of objective constraints for an optimal interpolation curve. This results in a set of di erential equations, whose analytical solution meets these constraints. Unfortunately, the set of di erential equations cannot be solved analytically. As an alternative we propose a numerical solution for the di erential equations. The di erent interpolation methods are visualized and commented. Finally we provide a thorough comparison of the two most convincing methods (Spring and Squad). Thereby, this report provides a comprehensive treatment of quaternions, rotation with quaternions, and interpolation curves for series of rotations.

Abstract: Interactive direct volume rendering has yet been restricted to high-end graphics workstations and special-purpose hardware, due to the large amount of trilinear interpolations, that are necessary to obtain high image quality. Implementations that use the 2D-texture capabilities of standard PC hardware, usually render object-aligned slices in order to substitute trilinear by bilinear interpolation. However the resulting images often contain visual artifacts caused by the lack of spatial interpolation. In this paper we propose new rendering techniques that significantly improve both performance and image quality of the 2D-texture based approach. We will show how in ulti-texturing capabilitiesof modern consumer PC graphboards are exploited to enable in teractive high quality volume visualization on low-cost hardware. Furthermore we demonstrate how multi-stage rasterization hardware can be used to efficiently render shaded isosurfaces and to compute diffuse illumination for semi-transparent volume rendering at interactive frame rates.

Abstract: A combination interpolation and lowpass filtering for Bayer color-filtered arrays with the green high frequency used as an estimate for red and blue high frequency.

Abstract: We present an algorithm to reconstruct smooth surfaces of arbitrary topology from unorganised sample points and normals. The method uses natural neighbour interpolation, works in any dimension and allows to deal with non uniform samples. The reconstructed surface is a smooth manifold passing through all the sample points. This surface is implicitly represented as the zero-set of some pseudo-distance function. It can be meshed so as to satisfy a user-defined error bound. Experimental results are presented for surfaces in R^3.

Abstract: Traditional computed tomography (CT) reconstructions of total joint prostheses are limited by metal artifacts from corrupted projection data. Published metal artifact reduction methods are based on the assumption that severe attenuation of X-rays by prostheses renders corresponding portions of projection data unavailable, hence the "missing" data are either avoided (in iterative reconstruction) or interpolated (in filtered back-projection with data completion; typically, with filling data "gaps" via linear functions). Here, the authors propose a wavelet-based multiresolution analysis method for metal artifact reduction, in which information is extracted from corrupted projection data. The wavelet method improves image quality by a successive interpolation in the wavelet domain. Theoretical analysis and experimental results demonstrate that the metal artifacts due to both photon starving and beam hardening can be effectively suppressed using the authors' method. As compared to the filtered back-projection after linear interpolation, the wavelet-based reconstruction is significantly more accurate for depiction of anatomical structures, especially in the immediate neighborhood of the prostheses. This superior imaging precision is highly advantageous in geometric modeling for fitting hip prostheses.

Abstract: In this chapter we survey geometric techniques which have been used to measure the similarity or distance between shapes, as well as to approximate shapes, or interpolate between shapes. Shape is a modality which plays a key role in many disciplines, ranging from computer vision to molecular biology. We focus on algorithmic techniques based on computational geometry that have been developed for shape matching, simplification, and morphing.

Abstract: We introduce AMGe, an algebraic multigrid method for solving the discrete equations that arise in Ritz-type finite element methods for partial differential equations. Assuming access to the element stiffness matrices, we have that AMGe is based on the use of two local measures, which are derived from global measures that appear in existing multigrid theory. These new measures are used to determine local representations of algebraically "smooth" error components that provide the basis for constructing effective interpolation and, hence, the coarsening process for AMG. Here, we focus on the interpolation process; choice of the coarse "grids" based on these measures is the subject of current research. We develop a theoretical foundation for AMGe and present numerical results that demonstrate the efficacy of the method.

Abstract: Smoothed particle hydrodynamics is extended to a normalized, staggered particle formulation with boundary conditions. A companion set of interpolation points is introduced that carry the stress, velocity gradient, and other derived field variables. The method is stable, linearly consistent, and has an explicit treatment of boundary conditions. Also, a new method for finding neighbours is introduced which selects a minimal and robust set and is insensitive to anisotropy in the particle arrangement. Test problems show that these improvements lead to increased accuracy and stability. Published in 2000 by John Wiley & Sons, Ltd.

Abstract: This paper presents a novel edge orientation adaptive interpolation scheme for resolution enhancement of still images. In order to achieve ideal orientation adaptation, we propose to estimate the local covariance characteristics at low resolution but cleverly use them to direct the interpolation at high resolution based on the resolution invariant property of edge orientation. The orientation adaptive property guarantees the interpolation always go along the edge orientation but not across it. Our new interpolation scheme can generate images with dramatically higher visual quality than linear interpolation techniques while keeping the computational complexity still modest.

Abstract: The development of serially complete (no missing values) daily maximum–minimum temperatures and total precipitation time series over the western United States is documented. Several estimation techniques based on spatial objective analysis schemes are used to estimate daily values, with the &ldquost” estimate chosen as a missing value replacement. The development of a continuous and complete daily dataset will be useful in a variety of meteorological and hydrological research applications. The spatial interpolation schemes are evaluated separately by interpolation method and calendar month. Cross validation of the results indicates a distinct seasonality to the efficiency (error) of the estimates, although no systematic bias in the estimation procedures was found. The resulting number of serially complete daily time series for the western United States (all states west of the Mississippi River) includes 2034 maximum–minimum temperature stations and 2962 total daily precipitation locations.

Abstract: We build discrete-time compactly supported biorthogonal wavelets and perfect reconstruction filter banks for any lattice in any dimension with any number of primal and dual vanishing moments. The associated scaling functions are interpolating. Our construction relies on the lifting scheme and inherits all of its advantages: fast transform, in-place calculation, and integer-to-integer transforms. We show that two lifting steps suffice: predict and update. The predict step can be built using multivariate polynomial interpolation, while update is a multiple of the adjoint of predict. While we concentrate on the discrete-time case, some discussion of convergence and stability issues together with examples is given.

Abstract: This paper introduces a filterbank interpretation of various sampling strategies, which leads to efficient interpolation and reconstruction methods An identity, which is referred to as the interpolation identity, is developed and is used to obtain particularly efficient discrete-time systems for interpolation of generalized samples as well as a class of nonuniform samples, to uniform Nyquist samples, either for further processing in that form or for conversion to continuous time The interpolation identity also leads to new sampling strategies including an extension of Papoulis' (1977) generalized sampling expansion

Abstract: We present a fast and accurate tool for semiautomatic segmentation of volumetric medical images based on the live wire algorithm, shape-based interpolation and a new optimization method.

Abstract: A new methodology for sampling plan design has been developed to reduce the costs associated with long-term monitoring of sites with groundwater contamination. The method combines a fate-and-transport model, plume interpolation, and a genetic algorithm to identify cost-effective sampling plans that accurately quantify the total mass of dissolved contaminant. The plume interpolation methods considered were inverse-distance weighting, ordinary kriging, and a hybrid method that combines the two approaches. Application of the methodology to Hill Air Force Base indicated that sampling costs could be reduced by as much as 60% without significant loss in accuracy of the global mass estimates. Inverse-distance weighting was shown to be most effective as a screening tool for evaluating whether more comprehensive geostatistical modeling is warranted. The hybrid method was effective for implementing such a tiered approach, reducing computational time by more than 60% relative to kriging alone.

Abstract: This paper presents an interpolation variance as an alternative to the measure of the reliability of ordinary kriging estimates Contrary to the traditional kriging variance, the interpolation variance is data-values dependent, variogram dependent, and a measure of local accuracy Natural phenomena are not homogeneous; therefore, local variability as expressed through data values must be recognized for a correct assessment of uncertainty The interpolation variance is simply the weighted average of the squared differences between data values and the retained estimate Ordinary kriging or simple kriging variances are the expected values of interpolation variances; therefore, these traditional homoscedastic estimation variances cannot properly measure local data dispersion More precisely, the interpolation variance is an estimate of the local conditional variance, when the ordinary kriging weights are interpreted as conditional probabilities associated to the n neighboring data This interpretation is valid if, and only if, all ordinary kriging weights are positive or constrained to be such Extensive tests illustrate that the interpolation variance is a useful alternative to the traditional kriging variance

Abstract: In numerous applications, such as communications, audio and music technology, speech coding and synthesis, antenna and transducer arrays, and time delay estimation, not only the sampling frequency but the actual sampling instants are of crucial importance. Digital fractional delay (FD) filters provide a useful building block that can be used for fine-tuning the sampling instants, i.e., implement the required bandlimited interpolation. In this paper an overview of design techniques and applications is given.

Abstract: Initialization of land surface prognostic variables is a crucial issue for short- and medium-range forecasting as well as at seasonal timescales. In this study, two sequential soil moisture analysis schemes are tested, both based on the comparison between observed and predicted 2-m parameters: the nudging technique used operationally at the European Centre for Medium-Range Weather Forecasts (ECMWF) and the optimum interpolation technique proposed by J. F. Mahfouf and used operationally at Meteo-France. Both techniques compute the soil moisture increments as a linear function of analysis increments of 2-m parameters (specific humidity at ECMWF, temperature and relative humidity at Meteo-France). Following the preliminary study by Y. Hu et al., the optimum interpolation technique has been adapted to the four soil-level ECMWF land surface scheme. Both methods are tested in the ECMWF single column model, which has been run for 4 months in 1987 at a grid point close to the location of the First Intern...