Abstract: Image interpolation techniques often are required in medical imaging for image generation (e.g., discrete back projection for inverse Radon transform) and processing such as compression or resampling. Since the ideal interpolation function spatially is unlimited, several interpolation kernels of finite size have been introduced. This paper compares 1) truncated and windowed sine; 2) nearest neighbor; 3) linear; 4) quadratic; 5) cubic B-spline; 6) cubic; g) Lagrange; and 7) Gaussian interpolation and approximation techniques with kernel sizes from 1/spl times/1 up to 8/spl times/8. The comparison is done by: 1) spatial and Fourier analyses; 2) computational complexity as well as runtime evaluations; and 3) qualitative and quantitative interpolation error determinations for particular interpolation tasks which were taken from common situations in medical image processing. For local and Fourier analyses, a standardized notation is introduced and fundamental properties of interpolators are derived. Successful methods should be direct current (DC)-constant and interpolators rather than DC-inconstant or approximators. Each method's parameters are tuned with respect to those properties. This results in three novel kernels, which are introduced in this paper and proven to be within the best choices for medical image interpolation: the 6/spl times/6 Blackman-Harris windowed sinc interpolator, and the C2-continuous cubic kernels with N=6 and N=8 supporting points. For quantitative error evaluations, a set of 50 direct digital X-rays was used. They have been selected arbitrarily from clinical routine. In general, large kernel sizes were found to be superior to small interpolation masks. Except for truncated sine interpolators, all kernels with N=6 or larger sizes perform significantly better than N=2 or N=3 point methods (p/spl Lt/0.005). However, the differences within the group of large-sized kernels were not significant. Summarizing the results, the cubic 6/spl times/6 interpolator with continuous second derivatives, as defined in (24), can be recommended for most common interpolation tasks. It appears to be the fastest six-point kernel to implement computationally. It provides eminent local and Fourier properties, is easy to implement, and has only small errors. The same characteristics apply to B-spline interpolation, but the 6/spl times/6 cubic avoids the intrinsic border effects produced by the B-spline technique. However, the goal of this study was not to determine an overall best method, but to present a comprehensive catalogue of methods in a uniform terminology, to define general properties and requirements of local techniques, and to enable the reader to select that method which is optimal for his specific application in medical imaging.

Abstract: The accuracy of many schemes for interpolating scattered data with radial basis functions depends on a shape parameter c of the radial basis function. In this paper we study the effect of c on the quality of fit of the multiquadric, inverse multiquadric and Gaussian interpolants. We show, numerically, that the value of the optimal c (the value of c that minimizes the interpolation error) depends on the number and distribution of data points, on the data vector, and on the precision of the computation. We present an algorithm for selecting a good value for c that implicitly takes all the above considerations into account. The algorithm selects c by minimizing a cost function that imitates the error between the radial interpolant and the (unknown) function from which the data vector was sampled. The cost function is defined by taking some norm of the error vector E = (E 1, ... , EN)T where E k = Ek = fk - Sk xk) and S k is the interpolant to a reduced data set obtained by removing the point x k and the corresponding data value f k from the original data set. The cost function can be defined for any radial basis function and any dimension. We present the results of many numerical experiments involving interpolation of two dimensional data sets by the multiquadric, inverse multiquadric and Gaussian interpolants and we show that our algorithm consistently produces good values for the parameter c.

Abstract: Coupling atmosphere, ocean, sea ice, and land surface models requires a means for remapping fields between grids in an accurate and conservative manner. A method is described here for computing interpolation weights for first- and second-order conservative remappings. The method is completely general and can be used for any grid on a sphere.

Abstract: This paper presents a technique for adapting existing motion of a human-like character to have the desired features that are specified by a set of constraints This problem can be typically formulated as a spacetime constraint problem Our approach combines a hierarchical curve fitting technique with a new inverse kinematics solver Using the kinematics solver, we can adjust the configuration of an articulated figure to meet the constraints in each frame Through the fitting technique, the motion displacement of every joint at each constrained frame is interpolated and thus smoothly propagated to frames We are able to adaptively add motion details to satisfy the constraints within a specified tolerance by adopting a multilevel Bspline representation which also provides a speedup for the interpolation The performance of our system is further enhanced by the new inverse kinematics solver We present a closed-form solution to compute the joint angles of a limb linkage This analytical method greatly reduces the burden of a numerical optimization to find the solutions for full degrees of freedom of a human-like articulated figure We demonstrate that the technique can be used for retargetting a motion to compensate for geometric variations caused by both characters and environments Furthermore, we can also use this technique for directly manipulating a motion clip through a graphical interface CR Categories: I37 [Computer Graphics]: Threedimensional Graphics—Animation; G12 [Numerical Analysis]: Approximation—Spline and piecewise polynomial approximation

Abstract: Spatiotemporal applications, such as fleet management and air traffic control, involving continuously moving objects are increasingly at the focus of research efforts. The representation of the continuously changing positions of the objects is fundamentally important in these applications. This paper reports on on-going research in the representation of the positions of moving-point objects. More specifically, object positions are sampled using the Global Positioning System, and interpolation is applied to determine positions in-between the samples. Special attention is given in the representation to the quantification of the position uncertainty introduced by the sampling technique and the interpolation. In addition, the paper considers the use for query processing of the proposed representation in conjunction with indexing. It is demonstrated how queries involving uncertainty may be answered using the standard filter-and-refine approach known from spatial query processing.

Abstract: A factorial, computational experiment was conducted to compare the spatial interpolation accuracy of ordinary and universal kriging and two types of inverse squared-distance weighting. The experiment considered, in addition to these four interpolation methods, the effects of four data and sampling characteristics: surface type, sampling pattern, noise level, and strength of small-scale spatial correlation. Interpolation accuracy was measured by the natural logarithm of the mean squared interpolation error. Main effects of all five factors, all two-factor interactions, and several three-factor interactions were highly statistically significant. Among numerous findings, the most striking was that the two kriging methods were substantially superior to the inverse distance weighting methods over all levels of surface type, sampling pattern, noise, and correlation.

Abstract: Traditionally, shape transformation using implicit functions is performed in two distinct steps: 1) creating two implicit functions, and 2) interpolating between these two functions. We present a new shape transformation method that combines these two tasks into a single step. We create a transformation between two N-dimensional objects by casting this as a scattered data interpolation problem in N + 1 dimensions. For the case of 2D shapes, we place all of our data constraints within two planes, one for each shape. These planes are placed parallel to one another in 3D. Zero-valued constraints specify the locations of shape boundaries and positive-valued constraints are placed along the normal direction in towards the center of the shape. We then invoke a variational interpolation technique (the 3D generalization of thin-plate interpolation), and this yields a single implicit function in 3D. Intermediate shapes are simply the zero-valued contours of 2D slices through this 3D function. Shape transformation between 3D shapes can be performed similarly by solving a 4D interpolation problem. To our knowledge, ours is the first shape transformation method to unify the tasks of implicit function creation and interpolation. The transformations produced by this method appear smooth and natural, even between objects of differing topologies. If desired, one or more additional shapes may be introduced that influence the intermediate shapes in a sequence. Our method can also reconstruct surfaces from multiple slices that are not restricted to being parallel to one another.

Abstract: Data collection of MRI which is sampled nonuniformly in k-space is often interpolated onto a Cartesian grid for fast reconstruction. The collected data must be properly weighted before interpolation, for accurate reconstruction. We propose a criterion for choosing the weighting function necessary to compensate for nonuniform sampling density. A numerical iterative method to find a weighting function that meets that criterion is also given. This method uses only the coordinates of the sampled data; unlike previous methods, it does not require knowledge of the trajectories and can easily handle trajectories that "cross" in k-space. Moreover, the method can handle sampling patterns that are undersampled in some regions of k-space and does not require a post-gridding density correction. Weighting functions for various data collection strategies are shown. Synthesized and collected in vivo data also illustrate aspects of this method.

Abstract: This paper presents a new processing algorithm for spotlight SAR data processing. The algorithm performs the range cell migration correction for non-chirped raw data without interpolation by using a novel frequency scaling operation. The azimuth processing is based on a spectral analysis approach which is made highly accurate by azimuth scaling. In almost all processing stages, a subaperture approach is introduced for efficient azimuth processing. In this paper, the complete derivation of the algorithm is presented. A very useful formulation for non-chirped SAR signals in the range Doppler domain is also proposed where the residual video phase is expressed by a chirp convolution. The algorithm performance is shown by several simulations. A spotlight image, which has been extracted from stripmap raw data of the experimental SAR system of DLR, shows the validity of the frequency scaling algorithm.

Abstract: A finite element implementation is reported of the Fleck–Hutchinson phenomenological strain gradient theory. This theory fits within the Toupin–Mindlin framework and deals with first-order strain gradients and the associated work-conjugate higher-order stresses. In conventional displacement-based approaches, the interpolation of displacement requires C1-continuity in order to ensure convergence of the finite element procedure for higher-order theories. Mixed-type finite elements are developed herein for the Fleck–Hutchinson theory; these elements use standard C0-continuous shape functions and can achieve the same convergence as C1 elements. These C0 elements use displacements and displacement gradients as nodal degrees of freedom. Kinematic constraints between displacement gradients are enforced via the Lagrange multiplier method. The elements developed all pass a patch test. The resulting finite element scheme is used to solve some representative linear elastic boundary value problems and the comparative accuracy of various types of element is evaluated. Copyright © 1999 John Wiley & Sons, Ltd.

Abstract: For certain medical applications resampling of data is required. In magnetic resonance tomography (MRT) or computer tomography (CT), e.g., data may be sampled on nonrectilinear grids in the Fourier domain. For the image reconstruction a convolution-interpolation algorithm, often called gridding, can be applied for resampling of the data onto a rectilinear grid. Resampling of data from a rectilinear onto a nonrectilinear grid are needed, e.g., if projections of a given rectilinear data set are to be obtained. In this paper the authors introduce the application of the convolution interpolation for resampling of data from one arbitrary grid onto another. The basic algorithm can be split into two steps. First, the data are resampled from the arbitrary input grid onto a rectilinear grid and second, the rectilinear data is resampled onto the arbitrary output grid. Furthermore, the authors like to introduce a new technique to derive the sampling density function needed for the first step of their algorithm. For fast, sampling-pattern-independent determination of the sampling density function the Voronoi diagram of the sample distribution is calculated. The volume of the Voronoi cell around each sample is used as a measure for the sampling density. It is shown that the introduced resampling technique allows fast resampling of data between arbitrary grids. Furthermore, it is shown that the suggested approach to derive the sampling density function is suitable even for arbitrary sampling patterns. Examples are given in which the proposed technique has been applied for the reconstruction of data acquired along spiral, radial, and arbitrary trajectories and for the fast calculation of projections of a given rectilinearly sampled image.

Abstract: This paper proposes a new method of dual-Doppler radar analysis based on a variational approach. In it, a cost function, defined as the distance between the analysis and the observations at the data points, is minimized through a limited memory, quasi-Newton conjugate gradient algorithm with the mass continuity equation imposed as a weak constraint. The analysis is performed in Cartesian space. Compared with traditional methods, the variational method offers much more flexibility in its use of observational data and various constraints. Using the radar data directly at observation locations avoids an interpolation step, which is often a source of error, especially in the presence of data voids. In addition, using the mass continuity equation as a weak instead of strong constraint avoids the error accumulation and the subsequent somewhat arbitrary adjustment associated with the explicit vertical integration of the continuity equation. The current method is tested on both model-simulated and observed datasets of supercell storms. It is shown that the circulation inside and around the storms, including the strong updraft and associated downdraft, is well analyzed in both cases. Furthermore, the authors found that the analysis is not very sensitive to the specification of boundary conditions and to data contamination. The method also has the potential for retrieving, with reasonable accuracy, the wind in regions of single-Doppler radar coverage.

Abstract: This paper presents a test system for conducting on-line tests in a real time and a series of real-time on-line tests conducted to verify the effectiveness of the system. The proposed system is characterized by (1) use of a Digital Signal Processor (DSP) now readily available, (2) adoption of the C language to ensure easy programming, and (3) separation of response analysis and displacement signal generation to apply the system for tests with complex structures. To create displacement signals successively without being interrupted by the computation of equations of motion, extrapolation and interpolation procedures using present and past target displacements are developed. Base-isolated building models were chosen for the real-time on-line test. The effectiveness of the extrapolation and interpolation procedures was demonstrated through a series of real-time on-line tests applied to the models treated as SDOF structures. A five-storey base-isolated building model (treated as a six DOF structure) was tested for various ground motions, and it was verified that the system is able to simulate earthquake responses involving large displacements and large velocities. The number of DOFs that can be handled in the proposed system was investigated, and it was found that the system is capable of performing the test with reasonable accuracy for up to 10 DOF structures with a range of response frequency not greater than 3·0 Hz, or 12 DOF structures with a range of response frequency not greater than 2·0 Hz. Copyright © 1999 John Wiley & Sons, Ltd.

Abstract: 1. Applications of sampling theory to combintorial analysis, Stirling numbers, special functions and the Riemann zeta function 2. Sampling theory and the arithmetic Fourier transform 3. Derivative sampling - a paradigm example of multi-channel methods 4. Computational methods in linear prediction for band-limited signals based on past samples 5. Interpolation and sampling theories, and linear ordinary boundary value problems 6. Sampling by generalized kernels 7. Sampling theory and wavelets 8. Approximation by translates of a radial function 9. Almost sure sampling restoration of band-limited stochastic signals 10. Abstract harmonic analysis and the sampling theorem

Abstract: Solving large radial basis function (RBF) interpolation problems with non‐customised methods is computationally expensive and the matrices that occur are typically badly conditioned. For example, using the usual direct methods to fit an RBF with N centres requires O(N 2) storage and O(N 3) flops. Thus such an approach is not viable for large problems with N ≥10,000.

Abstract: In this paper, the conventional moving least squares interpolation scheme is generalized, to incorporate the information concerning the derivative of the field variable into the interpolation scheme. By using this generalized moving least squares interpolation, along with the MLPG (Meshless Local Petrov–Galerkin) paradigm, a new numerical approach is proposed to deal with 4th order problems of thin beams. Through numerical examples, convergence tests are performed; and problems of thin beams under various loading and boundary conditions are analyzed by the proposed method, and the numerical results are compared with analytical solutions.

Abstract: The authors propose a method of interpolating linear time-invariant controllers with observer state feedback structure in order to generate a continuously varying family of controllers that stabilizes a family of linear plants. Gain scheduling is a motivation for this work, and the interpolation method yields guidelines for the design of gain scheduled controllers. The method is illustrated with the design of a missile autopilot using loop-shaping H-infinity controllers.

Abstract: We present a high-resolution digital elevation model (DEM) of the Antarctic. It was created in a geographic information system (GIS) environment by integrating the best available topographic data from a variety of sources. Extensive GIS-based error detection and correction operations ensured that our DEM is free of gross errors. The carefully designed interpolation algorithms for different types of source data and incorporation of surface morphologic information preserved and enhanced the fine surface structures present in the source data. The effective control of adverse edge effects and the use of the Hermite blending weight function in data merging minimized the discontinuities between different types of data, leading to a seamless and topographically consistent DEM throughout the Antarctic. This new DEM provides exceptional topographical details and represents a substantial improvement in horizontal resolution and vertical accuracy over the earlier, continental-scale renditions, particularly in mountainous and coastal regions. It has a horizontal resolution of 200 m over the rugged mountains, 400 m in the coastal regions, and approximately 5 km in the interior. The vertical accuracy of the DEM is estimated at about 100–130 m over the rugged mountainous area, better than 2 m for the ice shelves, better than 15 m for the interior ice sheet, and about 35 m for the steeper ice sheet perimeter. The Antarctic DEM can be obtained from the authors.

Abstract: The paper describes a method for the spatial interpolation of the site-specific LARS-WG stochastic weather generator to produce 'realistic' daily weather data for the gaps between observed sites. One of the uses of LARS-WG has been site-scale climate change impact assessments. However, such assessments are often applied across regions and so there is a need for an interpolation method to provide input daily weather at many sites or grid-boxes where observed weather data is not available. The interpolation method devised combines the local interpolation of the weather generator parame- ters from observed sites near the unobserved location with the use of globally interpolated monthly mean statistics for a large number of sites. Thin plate smoothing splines with elevation as an indepen- dent variable were used for the global interpolation of mean monthly rainfall and temperature. The data sets used allow daily weather to be generated for any location in Great Britain and the methodol- ogy was tested at 3 locations with different local characteristics. The interpolation method showed a good performance at the 3 sites when compared to the observed data, the main differences occurring when the spline method was unable to reproduce closely the observed mean values. The limitations of the interpolation method, its applicability to other regions and its potential use in climate change and other studies are also discussed.

Abstract: In this paper, the operational 3D variational data assimilation system (3D‐var) of the Canadian Meteorological Centre (CMC) is described and its performance is compared to that of the previously operational statistical interpolation analysis. Deliberately configuring the 3D‐var to be as close as possible to the statistical interpolation system permits an evaluation of the impact of data selection on both the analysis and the resulting forecasts. The current implementation of the 3D‐var is incremental in the horizontal and the vertical since the analysis increments are constructed at a lower horizontal resolution on prescribed pressure levels. They are subsequently interpolated vertically to the σ levels of the model. The results show that although there could be significant differences in the single analysis increments, the impact on the resulting forecasts is neutral. The 3D‐var implements a multivariate covariance model implicitly through changes of variables. It is shown that the implicit cova...

Abstract: A previously presented method for modeling Kolmogorov phase fluctuations over a finite aperture is both formalized and improved on. The method relies on forming an initial low-resolution Kolmogorov phase screen from direct factorization of a covariance. The resolution of the screen is then increased by a randomized interpolation to produce a Kolmogorov phase screen of the desired size. The computational requirement is asymptotically proportional to the number of points in the phase screen.

Abstract: A method to perform seismic trace interpolation known as the Spitz method handles spatially aliased events. The Spitz method uses the unit‐step prediction filter to estimate data spaced at Δx/2. The missing data are obtained by solving a complex linear system of equations whose unknowns are the coefficients at the interpolated location. We attack this problem by introducing a half‐step prediction filter that makes trace interpolation significantly more efficient and easier for implementation. A complex half‐step prediction filter at frequency f/2 is computed in the least‐squares sense to predict odd data components from even ones. At the frequency f, the prediction operator is shrunk and convolved with the input data spaced at Δx to predict data at Δx/2 directly. Instead of solving two systems of linear equations, as proposed by Spitz, only a system for the half‐step prediction filter has to be solved. Numerical examples using a marine seismic common‐midpoint (CMP) gather and a poststack seismic section w...

Abstract: We present an approach for the reconstruction and approximation of 3D CAD models from an unorganized collection of points. Applications include rapid reverse engineering of existing objects for use in a virtual prototyping environment, including computer aided design and manufacturing. Our reconstruction approach is flexible enough to permit interpolation of both smooth surfaces and sharp features, while placing few restrictions on the geometry or topology of the object. Our algorithm is based on alpha-shapes to compute an initial triangle mesh approximating the surface of the object. A mesh reduction technique is applied to the dense triangle mesh to build a simplified approximation, while retaining important topological and geometric characteristics of the model. The reduced mesh is interpolated with piecewise algebraic surface patches which approximate the original points. The process is fully automatic, and the reconstruction is guaranteed to be homeomorphic and error bounded with respect to the original model when certain sampling requirements are satisfied. The resulting model is suitable for typical CAD modeling and analysis applications.

Abstract: Interpolating values of climate variables from measurement stations to large areas is important in a variety of disciplines. Each of the 38 climate observation stations in the Israel area represents an average area of 725 km 2 . Therefore it is important to minimize the extent of interpolation errors by using a suitable interpolation method. In this study we compared the performance of 2 local interpolation methods, Spline and Inverse Distance Weighting (IDW), with the performance of multiple regression models. These interpolation methods were applied to 4 temperature variables: mean daily temperature of the coldest month (January), mean daily temperature of the warmest month (August), the lowest mean monthly minimum temperature (January) and the highest mean monthly maximum temperature (June). Spline and IDW models with a range of parameter settings were applied to eleva- tion detrended temperature data. The multiple regression models were based on geographic longitude, latitude and elevation and included terms of first and second order. Two methods of variable selection (Stepwise, Forced Entry) were used to construct 2 regression models for each temperature variable. Accuracy was assessed by a one-left-out cross validation test. Mean daily temperature variables proved more predictable than mean monthly extreme temperature variables. Mean daily temperature vari- ables were predicted more accurately by using a regression model, whereas mean monthly extreme temperature variables were somewhat better predicted by a local interpolation method. The Spline interpolator predicted more accurately than IDW for the 2 summer temperature variables, while IDW performed better for the winter temperature variables. Combining multiple regression and local inter- polation methods improved prediction accuracy by about 5% for the extreme temperature variables but did not effect the prediction of mean daily temperatures. Errors in the estimation increased with the use of local interpolation methods in areas where neighboring data were not 'local enough' to show micro-climatic influences. Where the data supported strong short-range climatic factors (such as the cooling effect of the Mediterranean Sea on the shoreline in summer), local methods were more effec- tive than regression models, which became complicated and tended to over extrapolate. These findings suggest that in some instances simple overall regression models can be as effective as sophisticated local interpolation methods, especially when dealing with mean climatic fields.

Abstract: Neural network simulations often spend a large proportion of their time computing exponential functions. Since the exponentiation routines of typical math libraries are rather slow, their replacement with a fast approximation can greatly reduce the overall computation time. This paper describes how exponentiation can be approximated by manipulating the components of a standard (IEEE-754) floating-point representation. This models the exponential function as well as a lookup table with linear interpolation, but is significantly faster and more compact.

Abstract: The value of a target variable is predicted by obtaining historical values for the target variable at each of several time points and obtaining previously predicted values and currently predicted values for each of several predictor variables, the predictor variables being different from the target variable. Values are assigned to parameters of a forecasting model to obtain the best fit of the previously predicted values for the predictor variables to the historical values for the target variable. Finally, a value of the target variable is predicted from the currently predicted values for at least a subset of the predictor variables using the forecasting model and the values assigned to the parameters of the forecasting model.