Abstract: This paper presents a chronological overview of the developments in interpolation theory, from the earliest times to the present date. It brings out the connections between the results obtained in different ages, thereby putting the techniques currently used in signal and image processing into historical perspective. A summary of the insights and recommendations that follow from relatively recent theoretical as well as experimental studies concludes the presentation.

Abstract: Numerous problems in electronic imaging systems in- volve the need to interpolate from irregularly spaced data. One ex- ample is the calibration of color input/output devices with respect to a common intermediate objective color space, such as XYZ or L*a*b*. In the present report we survey some of the most impor- tant methods of scattered data interpolation in two-dimensional and in three-dimensional spaces. We review both single-valued cases, where the underlying function has the form f:R 2 !R or f:R 3 !R, and multivalued cases, where the underlying function is f:R 2 !R 2 or f:R 3 !R 3 . The main methods we review include linear triangular (or tetrahedral) interpolation, cubic triangular (Clough-Tocher) interpo- lation, triangle based blending interpolation, inverse distance weighted methods, radial basis function methods, and natural neigh- bor interpolation methods. We also review one method of scattered data fitting, as an illustration to the basic differences between scat- tered data interpolation and scattered data fitting. © 2002 SPIE and

Abstract: Prerequisites and notation Introduction Kernels and function spaces Hardy spaces $P^2(\mu)$ Pick redux Qualitative properties of the solution of the Pick problem in $H^\infty(\mathbb{D})$ Characterizing kernels with the complete Pick property The universal Pick kernel Interpolating sequences Model theory I: Isometries The bidisk The extremal three point problem on $\mathbb{D}^2$ Collections of kernels Model theory II: Function spaces Localization Schur products Parrott's lemma Riesz interpolation The spectral theorem for normal $m$-tuples Bibliography Index.

Abstract: Two deficiencies in the original Noise algorithm are corrected: second order interpolation discontinuity and unoptimal gradient computation. With these defects corrected, Noise both looks better and runs faster. The latter change also makes it easier to define a uniform mathematical reference standard.

Abstract: When a mesh of simplicial elements (triangles or tetrahedra) is used to form a piecewise linear approximation of a function, the accuracy of the approximation depends on the sizes and shapes of the elements. In finite element methods, the conditioning of the stiffness matrices also depends on the sizes and shapes of the elements. This paper explains the mathematical connections between mesh geometry, interpolation errors, and stiffness matrix conditioning. These relationships are expressed by error bounds and element quality measures that determine the fitness of a triangle or tetrahedron for interpolation or for achieving low condition numbers. Unfortunately, the quality measures for these two purposes do not agree with each other; for instance, small angles are bad for matrix conditioning but not for interpolation. Several of the upper and lower bounds on interpolation errors and element stiffness matrix conditioning given here are tighter than those that have appeared in the literature before, so the quality measures are likely to be unusually precise indicators of element fitness.

Abstract: Many types of radial basis functions, such as multiquadrics, contain a free parameter In the limit where the basis functions become increasingly flat, the linear system to solve becomes highly ill-conditioned, and the expansion coefficients diverge Nevertheless, we find in this study that limiting interpolants often exist and take the form of polynomials In the 1-D case, we prove that with simple conditions on the basis function, the interpolants converge to the Lagrange interpolating polynomial Hence, differentiation of this limit is equivalent to the standard finite difference method We also summarize some preliminary observations regarding the limit in 2-D

Abstract: The survey is devoted to the modern state of the theory of interpolation of linear operators acting in Banach spaces. Principal attention is devoted to real and complex methods and applications of the theory of interpolation to analysis.

Abstract: Recently, modern manufacturing systems have been designed which can machine arbitrary parametric curves while greatly reducing data communication between CAD/CAM and CNC systems. However, a constant feedrate and chord accuracy between two interpolated points along parametric curves are generally difficult to achieve due to the non-uniform map between curves and parameters. A speed-controlled interpolation algorithm with an adaptive feedrate is proposed in this paper. Since the chord error in interpolation depends on the curve speed and the radius of curvature, the feedrate in the proposed algorithm is automatically adjusted so that a specified limit on the chord error is met. Both simulation and experimental results for non-uniform rational B-spline (NURBS) examples are provided to verify the feasibility and precision of the proposed interpolation algorithm.

Abstract: It is well known that a sparse hyperbolic Radon transform (RT) can be used to extend the aperture of aperture limited data, filter noise, and fill gaps. In the same manner, an elliptical RT can achieve similar results when applied to slant stack sections. A problem with these transformations is that they have a time-variant kernel that results in slow implementation. By defining a model space in terms of an irregularly sampled velocity space to minimize the number of unknowns during the inversion and using sparse matrices, however, the computation time can be kept low enough for practical application. We implement hyperbolic and elliptical time domain RTs by using inversion via weighted conjugate gradient methods with a sparseness constraint. The hyperbolic RT performs accurate interpolation in common-midpoint (CMP) gathers, while the elliptical RT attenuates sampling artifacts in slant stack sections obtained from CMP gathers with poor sampling and gaps.

Abstract: This paper describes the error reduction of frequency and amplitude estimates of the periodic signals with multipoint interpolated discrete Fourier transform (DFT). The bias removal and noise sensitivity properties of the interpolation algorithms are studied for rectangular and Hanning windows. The correction improves with increasing the number of the interpolation points of the DFT. The use of a suitable interpolation algorithm depends on the effective bits of the A/D conversion, on the position of the frequency component of the signal and on the mutual component interspacing along the frequency axis. Using different algorithms, we change adaptively the apparent window shape for the particular component.

Abstract: In this paper, we develop a set of data processing algorithms for generating textured facade meshes of cities from a series of vertical 2D surface scans and camera images, obtained by a laser scanner and digital camera while driving on public roads under normal traffic conditions. These processing steps are needed to cope with imperfections and non-idealities inherent in laser scanning systems such as occlusions and reflections from glass surfaces. The data is divided into easy-to-handle quasi-linear segments corresponding to approximately straight driving direction and sequential topological order of vertical laser scans; each segment is then transformed into a depth image. Dominant building structures are detected in the depth images, and points are classified into foreground and background layers. Large holes in the background layer, caused by occlusion from foreground layer objects, are filled in by planar or horizontal interpolation. The depth image is further processed by removing isolated points and filling remaining small holes. The foreground objects also leave holes in the texture of building facades, which are filled by horizontal and vertical interpolation in low frequency regions, or by a copy-paste method otherwise. We apply the above steps to a large set of data of downtown Berkeley with several million 3D points, in order to obtain texture-mapped 3D models.

Abstract: A new variational formulation for boundary node method (BNM) using a hybrid displacement functional is presented here. The formulation is expressed in terms of domain and boundary variables, and the domain variables are interpolated by classical fundamental solution; while the boundary variables are interpolated by moving least squares (MLS). The main idea is to retain the dimensionality advantages of the BNM, and get a truly meshless method, which does not require a ‘boundary element mesh’, either for the purpose of interpolation of the solution variables, or for the integration of the ‘energy’. All integrals can be easily evaluated over regular shaped domains (in general, semi-sphere in the 3-D problem) and their boundaries. Numerical examples presented in this paper for the solution of Laplace's equation in 2-D show that high rates of convergence with mesh refinement are achievable, and the computational results for unknown variables are most accurate. No further integrations are required to compute the unknown variables inside the domain as in the conventional BEM and BNM. Copyright © 2001 John Wiley & Sons, Ltd.

Abstract: A boundary point interpolation method (BPIM) is proposed for solving boundary value problems of solid mechanics In the BPIM, the boundary of a problem domain is represented by properly scattered nodes The boundary integral equation (BIE) for 2-D elastostatics has been discretized using point interpolants based only on a group of arbitrarily distributed boundary points In the present BPIM formulation, the shape functions constructed using polynomial basis function in a curvilinear coordinate possess Dirac delta function property The boundary conditions can be implemented with ease as in the conventional boundary element method (BEM) The BPIM for 2-D elastostatics has been coded in FORTRAN, and used to obtain numerical results for stress analysis of two-dimensional solids

Abstract: In a geostatistical analysis, spatial interpolation or smoothing of the observed values is often carried out by a procedure known as kriging. In its basic form, kriging involves the construction of a linear predictor for an unobserved value of the process, and the form of this linear predictor is chosen with reference to the covariance structure of the data as estimated by a data-analytic tool known as the variogram. Often, no explicit underlying stochastic model is declared. We adopt a model-based approach to this class of problems, by which we mean that we start with an explicit stochastic model and derive associated methods of parameter estimation, interpolation and smoothing by the application of general statistical principles. In particular, we use Bayesian methods of inference so as to make proper allowance for the uncertainty associated with estimating the unknown values of model parameters. To illustrate the model-based approach we analyse data on precipitation levels in Parana State, Brazil.

Abstract: Error estimates for scattered data interpolation by "shifts" of a positive definite function for target functions in the associated reproducing kernel Hilbert space (RKHS) have been known for a long time. However, apart from special cases where data is gridded, these estimates do not apply when the target functions generating the data are outside of the associated RKHS, and in fact no estimates were known for such target functions. In this paper, working with the n-sphere as the underlying manifold, we obtain Sobolev-type error estimates for interpolating functions $f\in C^{2k}(S^n)$ from "shifts" of a smoother positive definite function $\phi$ defined on Sn. Moreover, the estimates are close to the optimal approximation order. We also introduce a class of locally supported positive definite functions on Sn, functions based on Wendland's compactly supported radial basis functions (RBFs) [H. Wendland, Adv. Comput. Math., 4 (1995), pp. 389--396], which can be both explicitly and easily computed and also ana...

Abstract: Regularized Spline with Tension (RST) is an accurate, flexible and efficient method for multivariate interpolation of scattered data. This study evaluates its capabilities to interpolate daily and annual mean precipitation in regions with complex terrain. Tension, smoothing and anisotropy parameters are optimized using the cross-validation technique. In addition, smoothing and rescaling of the third variable (elevation) is used to minimize the predictive error. The approach is applied to data sets from Switzerland and Slovakia and interpolation accuracy is compared to the results obtained by several other methods, expert-drawn maps and measured runoff. The results demonstrate that RST performs as well or better than the methods tested in the literature. The incorporation of terrain improves the spatial model of precipitation in terms of its predictive error, spatial pattern and water balance.

Abstract: A new direction-oriented interpolation method is proposed and then applied to de-interlacing. The proposed method introduces the upper spatial direction vector. (USDV) and the lower spatial direction vector (LSDV) to obtain more accurate direction of the highest spatial correlation. Using the USDV and the LSDV, the proposed method reduces possibility of wrong decision for the highest-correlated spatial direction. Line-based directional interpolation is performed after the direction vector is found. Extensive simulations conducted for images and video sequences show the efficacy of the proposed method over the previous methods based on the ELA and over the line averaging method in terms of the objective and subjective image quality.

Abstract: This paper describes a novel Free-Viewpoint TV system based on Ray-Space representation. This system consists of a multi-view camera system for 3-D data capturing, a PC cluster with 16 PCs for data processing such as data compression and view interpolation, input device to specify a viewpoint, and a conventional 2-D display to show an arbitrary viewpoint image. To generate an arbitrary viewpoint image, the Ray-Space method is used. First, the multi-view image is converted to the Ray-Space data. Then, interpolation of the Ray-Space using adaptive filter is applied. Finally, an arbitrary view image is generated from the interpolated dense Ray-Space. This paper also describes various compression methods, such as model-based compression, arbitrary-shaped DCT(Discrete Cosine Transform), VQ(Vector Quantization), and subband coding. Finally, a demonstration of a full real-time system from capturing to display is explained.© (2002) COPYRIGHT SPIE--The International Society for Optical Engineering. Downloading of the abstract is permitted for personal use only.

Abstract: Seismic trace interpolation is implemented as a 2‐D (x, y) spatial prediction, performed separately on each frequency (f) slice. This so‐called f‐x‐y domain trace interpolation method is based on the relation that the linear prediction (LP) operator estimated at a given frequency may be used to predict data at a higher frequency but a smaller trace spacing. The relationship originally given for thef‐x domain trace interpolation is successfully extended to the f‐x‐y domain. The extension is achieved by masking the data samples selectively from the input frequency slice to design the LP operators. Two interpolation algorithms using the full‐step and the fractional‐step predictions, respectively, are developed. Both methods use an all‐azimuth prediction in the x‐y domain, but the fractional‐step prediction method is computationally more efficient. While the interpolation method can be applied to a common‐offset cube of 3‐D seismic, it can also be applied to 2‐D seismic traces for prestack data processing. Sy...

Abstract: The interpolation of first-order Hermite data by spatial Pythagorean-hodograph curves that exhibit closure under arbitrary 3-dimensional rotations is addressed. The hodographs of such curves correspond to certain combinations of four polynomials, given by Dietz et al. [4], that admit compact descriptions in terms of quaternions – an instance of the “PH representation map” proposed by Choi et al. [2]. The lowest-order PH curves that interpolate arbitrary first-order spatial Hermite data are quintics. It is shown that, with PH quintics, the quaternion representation yields a reduction of the Hermite interpolation problem to three “simple” quadratic equations in three quaternion unknowns. This system admits a closed-form solution, expressing all PH quintic interpolants to given spatial Hermite data as a two-parameter family. An integral shape measure is invoked to fix these two free parameters.

Abstract: A local radial point interpolation method (LRPIM) based on local residual formulation is presented and applied to solid mechanics in this paper. In LRPIM, the trial function is constructed by the radial point interpolation method (PIM) and establishes discrete equations through a local residual formulation, which can be carried out nodes by nodes. Therefore, element connectivity for trial function and background mesh for integration is not necessary. Radial PIM is used for interpolation so that singularity in polynomial PIM may be avoided. Essential boundary conditions can be imposed by a straightforward and effective manner due to its Delta properties. Moreover, the approximation quality of the radial PIM is evaluated by the surface fitting of given functions. Numerical performance for this LRPIM method is further studied through several numerical examples of solid mechanics.

Abstract: An active contour model for parametric curve and surface approximation is presented. The active curve or surface adapts to the model shape to be approximated in an optimization algorithm. The quasi-Newton optimization procedure in each iteration step minimizes a quadratic function which is built up with the help of local quadratic approximants of the squared distance function of the model shape and an internal energy which has a smoothing and regularization effect. The approach completely avoids the parametrization problem. We also show how to use a similar strategy for the solution of variational problems for curves on surfaces. Examples are the geodesic path connecting two points on a surface and interpolating or approximating spline curves on surfaces. Finally we indicate how the latter topic leads to the variational design of smooth motions which interpolate or approximate given positions.

Abstract: Using a far-field model, bistatic synthetic aperture radar (SAR) acquires Fourier data on a rather unusual, non-Cartesian grid in the Fourier domain. Previous image formation algorithms were mainly based on direct Fourier reconstruction to take advantage of the FFT, but the irregular coverage of the available Fourier domain data and the 2-D interpolation in the Fourier domain may adversely affect the accuracy of image reconstruction. Back-projection techniques avoid Fourier-domain interpolation, but ordinarily have huge computational cost. We present a fast back-projection algorithm for bistatic SAR imaging, motivated by a fast back-projection algorithm previously proposed for tomography. It has a reduced computational cost, on the same order as that of direct Fourier reconstruction. Furthermore, this approach can be used for near-field imaging. Simulation results verify the performance of this new algorithm.

Abstract: We study the relationship between power spectra of stationary stochastic inputs to a linear filter and the corresponding state covariances, and identify the structure of positive-semidefinite matrices that qualify as state covariances of the filter. This structure is best revealed by a rank condition pertaining to the solvability of a linear equation involving the state covariance and the system matrices. We then characterize all input power spectra consistent with any specific state covariance. The parametrization of input spectra is achieved through a relation to solutions of an analytic interpolation problem which is analogous, but not equivalent, to a matricial Nehari problem.