Abstract: A general binomial mixture model is proposed for the species accumulation function based on presence-absence (incidence) of species in a sample of quadrats or other sampling units. The model covers interpolation between zero and the observed number of samples, as well as extrapolation beyond the observed sample set. For interpolation (sample- based rarefaction), easily calculated, closed-form expressions for both expected richness and its confidence limits are developed (using the method of moments) that completely eliminate the need for resampling methods and permit direct statistical comparison of richness between sample sets. An incidence-based form of the Coleman (random-placement) model is developed and compared with the moment-based interpolation method. For ex- trapolation beyond the empirical sample set (and simultaneously, as an alternative method of interpolation), a likelihood-based estimator with a bootstrap confidence interval is de- scribed that relies on a sequential, AIC-guided algorithm to fit the mixture model parameters. Both the moment-based and likelihood-based estimators are illustrated with data sets for temperate birds and tropical seeds, ants, and trees. The moment-based estimator is confi- dently recommended for interpolation (sample-based rarefaction). For extrapolation, the likelihood-based estimator performs well for doubling or tripling the number of empirical samples, but it is not reliable for estimating the richness asymptote. The sensitivity of individual-based and sample-based rarefaction to spatial (or temporal) patchiness is dis- cussed.

Abstract: Barycentric interpolation is a variant of Lagrange polynomial interpolation that is fast and stable. It deserves to be known as the standard method of polynomial interpolation.

Abstract: The nonequispaced Fourier transform arises in a variety of application areas, from medical imaging to radio astronomy to the numerical solution of partial differential equations. In a typical problem, one is given an irregular sampling of N data in the frequency domain and one is interested in reconstructing the corresponding function in the physical domain. When the sampling is uniform, the fast Fourier transform (FFT) allows this calculation to be computed in O(N log N ) operations rather than O(N 2 ) operations. Unfortunately, when the sampling is nonuniform, the FFT does not apply. Over the last few years, a number of algorithms have been developed to overcome this limitation and are often referred to as nonuniform FFTs (NUFFTs). These rely on a mixture of interpolation and the judicious use of the FFT on an oversampled grid (A. Dutt and V. Rokhlin, SIAM J. Sci. Comput., 14 (1993), pp. 1368-1383). In this paper, we observe that one of the standard interpolation or "gridding" schemes, based on Gaussians, can be accelerated by a significant factor without precomputation and storage of the interpolation weights. This is of particular value in two- and three- dimensional settings, saving either 10 d N in storage in d dimensions or a factor of about 5-10 in CPUtime (independent of dimension).

Abstract: The Material Point Method (MPM) discrete solution procedure for computational solid mechanics is generalized using a variational form and a Petrov- Galerkin discretization scheme, resulting in a family of methods named the Generalized Interpolation Material Point (GIMP) methods. The generalization permits iden- tification with aspects of other point or node based dis- crete solution techniques which do not use a body-fixed grid, i.e. the "meshless methods". Similarities are noted and some practical advantages relative to some of these methods are identified. Examples are used to demon- strate and explain numerical artifact noise which can be expected in MPM calculations. This noise results in non- physical local variations at the material points, where constitutive response is evaluated. It is shown to destroy the explicit solution in one case, and seriously degrade it in another. History dependent, inelastic constitutive laws can be expected to evolve erroneously and report inac- curate stress states because of noisy input. The noise is due to the lack of smoothness of the interpolation func- tions, and occurs due to material points crossing compu- tational grid boundaries. The next degree of smoothness available in the GIMP methods is shown to be capable of eliminating cell crossing noise. keyword: MPM, PIC, meshless methods, Petrov- Galerkin discretization.

Abstract: It is shown that in model-based geostatistics, not all parameters in the Matern class can be estimated consistently if data are observed in an increasing density in a fixed domain, regardless of the estimation methods used. Nevertheless, one quantity can be estimated consistently by the maximum likelihood method, and this quantity is more important to spatial interpolation. The results are established by using the properties of equivalence and orthogonality of probability measures. Some sufficient conditions are provided for both Gaussian and non-Gaussian equivalent measures, and necessary conditions are provided for Gaussian equivalent measures. Two simulation studies are presented that show that the fixed-domain asymptotic properties can explain some finite-sample behavior of both interpolation and estimation when the sample size is moderately large.

Abstract: We present results of 3D numerical simulations using a finite difference code featuring fixed mesh refinement (FMR), in which a subset of the computational domain is refined in space and time. We apply this code to a series of test cases including a robust stability test, a nonlinear gauge wave and an excised Schwarzschild black hole in an evolving gauge. We find that the mesh refinement results are comparable in accuracy, stability and convergence to unigrid simulations with the same effective resolution. At the same time, the use of FMR reduces the computational resources needed to obtain a given accuracy. Particular care must be taken at the interfaces between coarse and fine grids to avoid a loss of convergence at higher resolutions, and we introduce the use of 'buffer zones' as one resolution of this issue. We also introduce a new method for initial data generation, which enables higher order interpolation in time even from the initial time slice. This FMR system, 'Carpet', is a driver module in the freely available Cactus computational infrastructure, and is able to endow generic existing Cactus simulation modules ('thorns') with FMR with little or no extra effort.

Abstract: Smooth interpolation of unstructured surface data is usually achieved by joining local patches, where each patch is an approximation (usually parametric) defined on a local reference domain. A basic mesh-independent projection strategy for general surface interpolation is proposed here. The projection is based upon the ’Moving-Least-Squares’ (MLS) approach, and the resulting surface is C ∞ smooth. The projection involves a first stage of defining a local reference domain and a second stage of constructing an MLS approximation with respect to the reference domain. The approach is presented for the general problem of approximating a (d − 1)-dimensional manifold in ℝ d , d ≥ 2. The approach is applicable for interpolating or smoothing curve and surface data, as demonstrated here by some graphical examples.

Abstract: This paper introduces a new interpolation technique for demosaicing of color images produced by single-CCD digital cameras. We show that the proposed simple linear filter can lead to an improvement in PSNR of over 5.5 dB when compared to bilinear demosaicing, and about 0.7 dB improvement in R and B interpolation when compared to a recently introduced linear interpolator. The proposed filter also outperforms most nonlinear demosaicing algorithms, without the artifacts due to nonlinear processing, and a much reduced computational complexity.

Abstract: SUMMARY Particle Methods are those in which the problem is represented by a discrete number of particles. Each particle moves accordingly with its own mass and the external/internal forces applied to it. Particle Methods may be used for both, discrete and continuous problems. In this paper, a Particle Method is used to solve the continuous fluid mechanics equations. To evaluate the external applied forces on each particle, the incompressible Navier–Stokes equations using a Lagrangian formulation are solved at each time step. The interpolation functions are those used in the Meshless Finite Element Method and the time integration is introduced by an implicit fractional-step method. In this manner classical stabilization terms used in the momentum equations are unnecessary due to lack of convective terms in the Lagrangian formulation. Once the forces are evaluated, the particles move independently of the mesh. All the information is transmitted by the particles. Fluid–structure interaction problems including free-fluid-surfaces, breaking waves and fluid particle separation may be easily solved with this methodology. Copyright 2004 John Wiley & Sons, Ltd.

Abstract: We present a simple, original method to improve piecewise-linear interpolation with uniform knots: we shift the sampling knots by a fixed amount, while enforcing the interpolation property. We determine the theoretical optimal shift that maximizes the quality of our shifted linear interpolation. Surprisingly enough, this optimal value is nonzero and close to 1/5. We confirm our theoretical findings by performing several experiments: a cumulative rotation experiment and a zoom experiment. Both show a significant increase of the quality of the shifted method with respect to the standard one. We also observe that, in these results, we get a quality that is similar to that of the computationally more costly "high-quality" cubic convolution.

Abstract: In seismic data processing, we often need to interpolate and extrapolate data at missing spatial locations. The reconstruction problem can be posed as an inverse problem where, from inadequate and incomplete data, we attempt to reconstruct the seismic wavefield at locations where measurements were not acquired.We propose a wavefield reconstruction scheme for spatially band‐limited signals. The method entails solving an inverse problem where a wavenumber‐domain regularization term is included. The regularization term constrains the solution to be spatially band‐limited and imposes a prior spectral shape. The numerical algorithm is quite efficient since the method of conjugate gradients in conjunction with fast matrix–vector multiplications, implemented via the fast Fourier transform (FFT), is adopted. The algorithm can be used to perform multidimensional reconstruction in any spatial domain.

Abstract: The spatial prediction of point values from areal data of the same attribute is addressed within the general geostatistical framework of change of support; the term support refers to the domain informed by each datum or unknown value. It is demonstrated that the proposed geostatistical framework can explicitly and consistently account for the support differences between the available areal data and the sought-after point predictions. In particular, it is proved that appropriate modeling of all area-to-area and area-to-point covariances required by the geostatistical framework yields coherent (mass-preserving or pycnophylactic) predictions. In other words, the areal average (or areal total) of point predictions within any arbitrary area informed by an areal-average (or areal-total) datum is equal to that particular datum. In addition, the proposed geostatistical framework offers the unique advantage of providing a measure of the reliability (standard error) of each point prediction. It is also demonstrated that several existing approaches for area-to-point interpolation can be viewed within this geostatistical framework. More precisely, it is shown that (i) the choropleth map case corresponds to the geostatistical solution under the assumption of spatial independence at the point support level; (ii) several forms of kernel smoothing can be regarded as alternative (albeit sometimes incoherent) implementations of the geostatistical approach; and (iii) Tobler’s smooth pycnophylactic interpolation, on a quasi-infinite domain without non-negativity constraints, corresponds to the geostatistical solution when the semivariogram model adopted at the point support level is identified to the free-space Green’s functions (linear in 1-D or logarithmic in 2-D) of Poisson’s partial differential equation. In lieu of a formal case study, several 1-D examples are given to illustrate pertinent concepts.

Abstract: Time-delay estimation (TDE), which aims at measuring the relative time difference of arrival (TDOA) between different channels is a fundamental approach for identifying, localizing, and tracking radiating sources Recently, there has been a growing interest in the use of TDE based locator for applications such as automatic camera steering in a room conferencing environment where microphone sensors receive not only the direct-path signal, but also attenuated and delayed replicas of the source signal due to reflections from boundaries and objects in the room This multipath propagation effect introduces echoes and spectral distortions into the observation signal, termed as reverberation, which severely deteriorates a TDE algorithm in its performance This paper deals with the TDE problem with emphasis on combating reverberation using multiple microphone sensors The multichannel cross correlation coefficient (MCCC) is rederived here, in a new way, to connect it to the well-known linear interpolation technique Some interesting properties and bounds of the MCCC are discussed and a recursive algorithm is introduced so that the MCCC can be estimated and updated efficiently when new data snapshots are available We then apply the MCCC to the TDE problem The resulting new algorithm can be treated as a natural generalization of the generalized cross correlation (GCC) TDE method to the multichannel case It is shown that this new algorithm can take advantage of the redundancy provided by multiple microphone sensors to improve TDE against both reverberation and noise Experiments confirm that the relative time-delay estimation accuracy increases with the number of sensors

Abstract: Bilinear interpolation is often used to improve image quality after performing spatial transformation operations such as digital zooming or rotation. In the traditional case where the input coordinates appear in a raster-based fashion, the required pixel values can be obtained from the previous calculation, the frame buffer and a single line cache. This paper presents a novel approach to performing real-time bilinear interpolation that is useful in applications such as lens distortion correction where the input coordinates follow a curved path that spans multiple rows. To help retrieve the required pixels in a single clock cycle under imposed data bandwidth constraints a unique caching system has been devised. In the event that constraints make it impossible to obtain the four required pixel values, the approach performs a modified three-point interpolation. An example field programmable gate array implementation of the bilinear interpolation method used in conjunction with a lens distortion correction algorithm has been successfully completed.

Abstract: In this paper we study the Cauchy problem to the linear damped wave equation utt-Δu+2aut=0 in (0,∞)×Rn(n≥2). It has been asserted that the above equation has the diffusive structure as t→∞. We give the precise interpolation of the diffusive structure, which is shown by Lp-Lq estimates. We apply the above Lp-Lq estimates to the Cauchy problem for the semilinear damped wave equation utt-Δu+2aut=|u|σu in (0,∞)×Rn(2≤n≤5). If the power σ is larger than the critical exponent 2/n(Fujita critical exponent) and it satisfies σ≤2/(n-2) when n≥3, then the time global existence of small solution is proved, and the decay estimates of several norms of the solution are derived.

Abstract: Many simulation experiments require much computer time, so they necessitate interpolation for sensitivity analysis and optimization. The interpolating functions are 'metamodels' (or 'response surfaces') of the underlying simulation models. Classic methods combine low-order polynomial regression analysis with fractional factorial designs. Modern Kriging provides 'exact' interpolation, i.e., predicted output values at inputs already observed equal the simulated output values. Such interpolation is attractive in deterministic simulation, and is often applied in computer aided engineering. In discrete-event simulation, however, Kriging has just started. Methodologically, a Kriging metamodel covers the whole experimental area; i.e., it is global (not local). Kriging often gives better global predictions than regression analysis. Technically, Kriging gives more weight to 'neighboring' observations. To estimate the Kriging metamodel, space filling designs are used; for example, latin hypercube sampling (LHS). This paper also presents novel, customized (application driven) sequential designs based on cross-validation and bootstrapping.

Abstract: A CFD model is proposed for numerical simulations of extremely nonlinear free-surface flows such as wave impact phenomena and violent wave–body interactions. The constrained interpolation profile (CIP) method is adopted as the base scheme for the model. The wave–body interaction is treated as a multiphase problem, which has liquid (water), gas (air), and solid (wave-maker and floating body) phases. The flow is represented by one set of governing equations, which are solved numerically on a nonuniform, staggered Cartesian grid by a finite-difference method. The free surface as well as the body boundary are immersed in the computation domain and captured by different methods. In this article, the proposed numerical model is first described. Then to validate the accuracy and demonstrate the capability, several two-dimensional numerical simulations are presented, and compared with experiments and with computations by other numerical methods. The numerical results show that the present computation model is both robust and accurate for violent free-surface flows.

Abstract: A stabilized conforming nodal integration scheme is implemented in the natural neighbour method in conjunction with non-Sibsonian interpolation. In this approach, both the shape functions and the integration scheme are defined through use of first-order Voronoi diagrams. The method illustrates improved performance and significant advantages over previous natural neighbour formulations. The method also shows substantial promise for problems with large deformations and for the computation of higher-order gradients. Copyright © 2004 John Wiley & Sons, Ltd.

Abstract: Variational interpolation in curved geometries has many applications, so there has always been demand for geometrically meaningful and efficiently computable splines in manifolds. We extend the definition of the familiar cubic spline curves and splines in tension, and we show how to compute these on parametric surfaces, level sets, triangle meshes, and point samples of surfaces. This list is more comprehensive than it looks, because it includes variational motion design for animation, and allows the treatment of obstacles via barrier surfaces. All these instances of the general concept are handled by the same geometric optimization algorithm, which minimizes an energy of curves on surfaces of arbitrary dimension and codimension.

Abstract: The finite element formulation of geometrically exact rod models depends crucially on the interpolation of the rotation field from the nodes to the integration points where the internal forces and tangent stiffness are evaluated. Since the rotational group is a nonlinear space, standard (isoparametric) interpolation of these degrees of freedom does not guarantee the orthogonality of the interpolated field hence, more sophisticated interpolation strategies have to be devised. We review and classify the rotation interpolation techniques most commonly used in the context of nonlinear rod models and suggest new ones. All of them are compared and their advantages and disadvantages discussed. In particular, their effect on the frame invariance of the resulting discrete models is analyzed.

Abstract: A method and system for producing digital orthophotos from imagery acquired as full or sparse stereo. The orthophotos can be produced in a variety of map coordinate systems without the need to convert or recompute DEM or photogrammetric solution data. In one embodiment, a two dimensional, planimetric free-network solution, utilizing arbitrary datum definition constraints, is used to provide a transitory coordinate system that is used to facilitate the image measurement process. It is utilized as a preliminary step to refine apriori block layout information to facilitate point picking and to provide general quality control capabilities before undertaking a rigorous 3D photogrammetric adjustment. In place of a general map conversion transformation, an identity transformation can be used, so that map coordinates and world coordinates are identical. With this process, given DEM data and photogrammetric solution data in a particular coordinate system, the orthophoto image data can be produced in any map coordinate system. In one embodiment, all geometric coordinate transformations are performed prior to performing the image intensity interpolation operation. Thus, only one image intensity interpolation operation is performed, using the geometric coordinate data. In another embodiment, a network constraint is introduced to the block adjustment process that assumes an average vertical direction in order to support the process of self rectification.

Abstract: Quadratic models of objective functions are highly useful in many optimization algorithms. They are updated regularly to include new information about the objective function, such as the difference between two gradient vectors. We consider the case, however, when each model interpolates some function values, so an update is required when a new function value replaces an old one. We let the number of interpolation conditions, m say, be such that there is freedom in each new quadratic model that is taken up by minimizing the Frobenius norm of the second derivative matrix of the change to the model. This variational problem is expressed as the solution of an (m+n+1)×(m+n+1) system of linear equations, where n is the number of variables of the objective function. Further, the inverse of the matrix of the system provides the coefficients of quadratic Lagrange functions of the current interpolation problem. A method is presented for updating all these coefficients in ({m+n}2) operations, which allows the model to be updated too. An extension to the method is also described that suppresses the constant terms of the Lagrange functions. These techniques have a useful stability property that is investigated in some numerical experiments.

Abstract: Existing methods for data interpolation or backdating are either univariate or based on a very limited number of series, due to data and computing constraints that were binding until the recent past. Nowadays large datasets are readily available, and models with hundreds of parameters are fastly estimated. We model these large datasets with a factor model, and develop an interpolation method that exploits the estimated factors as an efficient summary of all the available information. The method is compared with existing standard approaches from a theoretical point of view, by means of Monte Carlo simulations, and also when applied to actual macroeconomic series. The results indicate that our method is more robust to model misspecification, although traditional multivariate methods also work well while univariate approaches are systematically outperformed. When interpolated series are subsequently used in econometric analyses, biases can emerge, depending on the type of interpolation but again be reduced with multivariate approaches, including factor-based ones.