Showing papers on "Multivariate interpolation published in 2003"
23 Jun 2003
TL;DR: This paper addresses the point-wise estimation of differential properties of a smooth manifold S--a curve in the plane or a surface in 3D--assuming a point cloud sampled over S is provided, and is among the first ones providing accurate estimates for differential quantities of order three and more.
Abstract: This paper addresses the pointwise estimation of differential properties of a smooth manifold S---a curve in the plane or a surface in 3D--- assuming a point cloud sampled over S is provided. The method consists of fitting the local representation of the manifold using a jet, by either interpolating or approximating. A jet is a truncated Taylor expansion, and the incentive for using jets is that they encode all local geometric quantities---such as normal or curvatures.On the way to using jets, the question of estimating differential properties is recasted into the more general framework of multivariate interpolation/approximation, a well-studied problem in numerical analysis. On a theoretical perspective, we prove several convergence results when the samples get denser. For curves and surfaces, these results involve asymptotic estimates with convergence rates depending upon the degree of the jet used. For the particular case of curves, an error bound is also derived. To the best of our knowledge, these results are among the first ones providing accurate estimates for differential quantities of order three and more. On the algorithmic side, we solve the interpolation/approximation problem using Vandermonde systems. Experimental results for surface of R3 are reported. These experiments illustrate the asymptotic convergence results, but also the robustness of the methods on general Computer Graphics models.
313 citations
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TL;DR: The algorithm described here will produce a triangular mesh surface approximation to an isosurface which preserves the same connectivity/separation of vertices as given by the isOSurface of trilinear interpolation.
Abstract: A characterization and classification of the isosurfaces of trilinear functions is presented. Based upon these results, a new algorithm for computing a triangular mesh approximation to isosurfaces for data given on a 3D rectilinear grid is presented. The original marching cubes algorithm is based upon linear interpolation along edges of the voxels. The asymptotic decider method is based upon bilinear interpolation on faces of the voxels. The algorithm of this paper carries this theme forward to using trilinear interpolation on the interior of voxels. The algorithm described here will produce a triangular mesh surface approximation to an isosurface which preserves the same connectivity/separation of vertices as given by the isosurface of trilinear interpolation.
236 citations
TL;DR: In this article, a 1-km spacing was used for the gridding of snow telemetry (SNOTEL) measurements for the years 1993, 1998, and 1999, which on average represented higher than average, average, and lower than average snow years.
Abstract: [1] Inverse weighted distance and regression nonexact techniques were evaluated for interpolating methods snow water equivalent (SWE) across the entire Colorado River Basin of the western United States. A 1-km spacing was used for the gridding of snow telemetry (SNOTEL) measurements for the years 1993, 1998, and 1999, which on average, represented higher than average, average, and lower than average snow years. Because of the terrain effects, the regression techniques (hypsometric elevation and multivariate physiographic parameter) were found to be superior to the weighted distance approaches (inverse distance weighting squared, and optimal power inverse distance weighting). A regression detrended inverse weighted distance method was developed for the hypsometric and multivariate techniques in order to preserve the point SNOTEL data. On the basis of root mean square error analysis and estimates of SWE volumes in different elevation zones for the entire basin and for subbasins the elevation detrended method with a point by point regression was found to be the most appropriate technique. Various search radii and anisotropies of the search ellipse were tested with the hypsometric method, producing only small difference in the root mean square error and SWE volumes.
202 citations
TL;DR: In this paper, the authors proposed a spatial probabilistic modeling of slope failure using a combined Geographic Information System (GIS), infinite-slope stability model and Monte Carlo simulation approach and applied in the landslide-prone area of Sasebo city, southern Japan.
183 citations
TL;DR: A link between classical osculatory interpolation and modern convolution-based interpolation is established and it is shown that two well-known cubic convolution schemes are formally equivalent to two osculation interpolation schemes proposed in the actuarial literature about a century ago.
Abstract: We establish a link between classical osculatory interpolation and modern convolution-based interpolation and use it to show that two well-known cubic convolution schemes are formally equivalent to two osculatory interpolation schemes proposed in the actuarial literature about a century ago. We also discuss computational differences and give examples of other cubic interpolation schemes not previously studied in signal and image processing.
138 citations
Patent•
9 Jul 2003TL;DR: In this paper, a method and system for coding a video sequence based on motion compensated prediction (642), wherein an interpolation filter (640) is used to generate predicted pixel values for picture blocks in the video sequence.
Abstract: A method and system for coding a video sequence based on motion compensated prediction (642), wherein an interpolation filter (640) is used to generate predicted pixel values for picture blocks in the video sequence. The interpolation filter for use in conjunction with a multi-picture type is shorter or having fewer coefficients than the interpolation filter for use in conjunction with a single-picture type. As such, the complexity of the interpolation filter for the multi-picture type can be reduced. Furthermore, the interpolation filter may be changed based on the characteristics of the block, the size and/or the shape of the block.
124 citations
TL;DR: The radial point interpolation method (RPIM) based on local supported radial basis function (RBF) and the Galerkin weak form has been developed and successfully applied to many engineering problems as discussed by the authors.
Abstract: The radial point interpolation method (RPIM) based on local supported radial basis function (RBF) and the Galerkin weak form has been developed and successfully applied to many engineering problems. Recently, a new meshfree method was proposed based on the universal moving Kriging interpolation. This paper studies the difference between the meshfree shape functions created based on the point interpolation and the Kriging interpolation. It is found that both the two methods yield the same shape function as long as the same radial basis function or semivariogram is adopted for interpolation. Although the two methods lead to the same shape function, the theorem in Kriging formulation may provide an alternative theoretical support for the RPIM. Some common semivariograms used in Kriging may also be incorporated in the RPIM. In addition, in order to satisfy the conformability requirements, a penalty technique is introduced in this paper to form a conforming Kriging, which can pass the standard patch test exactly.
109 citations
TL;DR: In this paper, a general formulation for developing reproducing kernel (RK) interpolation is presented based on the coupling of a primitive function and an enrichment function, which introduces discrete Kronecker delta properties, while the enrichment function constitutes reproducing conditions.
Abstract: A general formulation for developing reproducing kernel (RK) interpolation is presented. This is based on the coupling of a primitive function and an enrichment function. The primitive function introduces discrete Kronecker delta properties, while the enrichment function constitutes reproducing conditions. A necessary condition for obtaining a RK interpolation function is an orthogonality condition between the vector of enrichment functions and the vector of shifted monomial functions at the discrete points. A normalized kernel function with relative small support is employed as the primitive function. This approach does not employ a finite element shape function and therefore the interpolation function can be arbitrarily smooth. To maintain the convergence properties of the original RK approximation, a mixed interpolation is introduced. A rigorous error analysis is provided for the proposed method. Optimal order error estimates are shown for the meshfree interpolation in any Sobolev norms. Optimal order convergence is maintained when the proposed method is employed to solve one-dimensional boundary value problems. Numerical experiments are done demonstrating the theoretical error estimates. The performance of the method is illustrated in several sample problems. Copyright © 2003 John Wiley & Sons, Ltd.
102 citations
TL;DR: A new deinterlacing algorithm, which uses directional interpolation and motion compensation, and experimental results demonstrate that the proposed method provides better performances than conventional deInterlacing algorithms.
Abstract: In this paper, we propose a new deinterlacing algorithm, which uses directional interpolation and motion compensation. In the proposed method, intrafield interpolation is first performed in the direction that shows the highest correlation. Second, motion estimation is performed between two fields of the same parity. The motion vector is further refined in half-pixel accuracy. In the conventional motion compensated methods, a prefilter such as line averaging, is applied to interpolate missing lines prior to motion estimation between opposite parity fields. The proposed method does not require this prefilter since block matching is performed between the same parity fields. Finally, we apply a test and use either directional interpolation or motion compensated interpolation depending on the test result. Experimental results demonstrate that the proposed method provides better performances than conventional deinterlacing algorithms.
94 citations
Patent•
23 Jul 2003TL;DR: In this article, a motion compensation unit implementing motion-compensated temporal interpolation for each of estimated motion vectors with reference to a previous field and a next field, which are respectively ahead of and behind a current field to be interpolated, producing interpolation values of a pixel, and outputting a selected value from the interpolations as a first interpolation value.
Abstract: A deinterlacing apparatus and method thereof include a motion compensation unit implementing motion-compensated temporal interpolation for each of estimated motion vectors with reference to a previous field and a next field, which are respectively ahead of and behind a current field to be interpolated, producing interpolation values of a pixel to be interpolated, and outputting a selected value from the interpolation values as a first interpolation value. Further, a spatial interpolation unit producing a second interpolation value of the pixel to be interpolated using values of pixels around the pixel to be interpolated, and an output unit mixing the first and second interpolation.
77 citations
TL;DR: The hypothesis presented here, is that higher-order interpolation techniques will always be more accurate than the popular bilinear algorithm, and this hypothesis is evaluated through an assessment of the accuracy with which DEMs can be interpolated to higher spatial resolutions.
Abstract: The fundamental aim of a digital elevation model (DEM) is to represent a surface accurately, such that elevations can be estimated for any given location. It is, therefore, necessary to have efficient and precise algorithms for the computation of surface elevations between given points. The hypothesis presented here, is that higher-order interpolation techniques will always be more accurate than the likes of the popular bilinear algorithm. This hypothesis will be evaluated through an assessment of the accuracy with which DEMs can be interpolated to higher spatial resolutions. A variety of interpolation techniques are assessed, ranging from the one-term level plane to the 36-term biquintic polynomial. In general, techniques that take account of the local terrain neighbourhood are more consistent and accurate, reducing the rms. error by up to 20% of the bilinear interpolant.
7 May 2003
TL;DR: In this article, a new deinterlacing algorithm utilizing directional interpolation and motion compensation is proposed, where intra-field interpolation is first performed in the spatial direction that shows the highest correlation, and motion estimation is performed between two fields of the same parity.
Abstract: In this paper, we propose a new deinterlacing algorithm utilizing directional interpolation and motion compensation. In particular, intrafield interpolation is first performed in the spatial direction that shows the highest correlation. Then, motion estimation is performed between two fields of the same parity. In order to interpolate missing lines prior to motion estimation between opposite parity fields, the proposed method does not require a prefilter such as line averaging, since block matching is performed between the same parity fields. Finally, we apply a test and use either directional interpolation or motion compensated interpolation depending on the test result. Experimental results are promising.
TL;DR: In this paper, the spatial distribution of the Canadian Drought code (DC) is analyzed in the region of Andaluc{@a (south Spain) following two procedures.
Abstract: Traditionally, the estimation of fire danger is performed from meteorological danger indices that are computed for single locations, where the weather stations are located. Frequently, these locations are far from forested areas, and there is a need to spatially interpolate danger variables. Methods for spatial interpolation are always prone to error, especially for those variables that show a greater spatial variability (wind, mainly). Satellite images may be considered a good alternative for interpolation of danger values, since they perform a spatially exhaustive observation of the territory. This paper analyses the spatial distribution of the Canadian Drought Code (DC), part of the Canadian Forest Fire Weather Index System (CFFWIS), in the region of Andaluc{@a (south Spain) following two procedures. First, maps of DC values were obtained from spatial interpolation of a network of 30 weather stations using the squared inverse distance algorithm. These results were compared with interpolation based on l...
TL;DR: Two flexible and computationally efficient algorithms for boundary effects free and adaptive discrete sinc interpolation are presented: frame-wise (global) sine interpolation in the discrete cosine transform (DCT) domain and local adaptive sinc extrapolation in the DCT domain of a sliding window.
Abstract: The problem of digital signal and image resampling with discrete sinc interpolation is addressed. Discrete sinc interpolation is theoretically the best one among the digital convolution-based signal resampling methods because it does not distort the signal as defined by its samples and is completely reversible. However, sinc interpolation is frequently not considered in applications because it suffers from boundary effects, tends to produce signal oscillations at the image edges, and has relatively high computational complexity when irregular signal resampling is required. A solution that enables the elimination of these limitations of the discrete sinc interpolation is suggested. Two flexible and computationally efficient algorithms for boundary effects free and adaptive discrete sinc interpolation are presented: frame-wise (global) sinc interpolation in the discrete cosine transform (DCT) domain and local adaptive sinc interpolation in the DCT domain of a sliding window. The latter offers options not available with other interpolation methods: interpolation with simultaneous signal restoration/enhancement and adaptive interpolation with super resolution.
Patent•
25 Jul 2003TL;DR: In this article, the authors propose an apparatus for and a method of deinterlacing an interlaced image signal, in which an interpolation value is calculated based on pixels in previous and next fields corresponding to the pixel to be interpolated.
Abstract: An apparatus for and a method of deinterlacing an interlaced image signal. A weight value is calculated after detecting degree of a motion between a pixel of a previous field and a pixel of a next field relative to a pixel of the current field to be interpolated. An inter-field interpolation value is calculated based on pixels in previous and next fields corresponding to the pixel to be interpolated. An intra-field interpolation value is calculated based adjacent pixels in the same field as the pixel to be interpolated. A final interpolation value is calculated based on the on the inter-field interpolation value, the intra-field interpolation value and the weight value.
6 Apr 2003
TL;DR: This paper addresses the problem of data independent robust array interpolation over large angular sectors and proposes a new interpolation approach which enjoys both simple design procedure and fast implementation and offers reliable DOA estimation for a wide range of different scenarios.
Abstract: We address the problem of data independent robust array interpolation over large angular sectors. Previous interpolation methods apply the root-MUSIC principle to interpolation data of a predefined virtual ULA manifold. These methods either suffer from severely biased direction-of-arrival estimates due to interpolation errors or rely on data dependent interpolation matrix design. In this paper a new interpolation approach is proposed. Instead of transforming the original array geometry to the rather restrictive ULA structure, here interpolation is performed with the objective to create a virtual array manifold which is a shifted version of the real array manifold. This artificial shift-invariance can be exploited by the well-known ESPRIT algorithm. A joint design of virtual array geometry and interpolation matrix yields additional degrees of freedom which reduce interpolation errors and allow us to increase the interpolation sector. The new algorithm enjoys both simple design procedure and fast implementation and offers reliable DOA estimation for a wide range of different scenarios.
TL;DR: A drastic simplification of the minimization process is demonstrated, showing that for an arbitrary number of stages K, the solution of the K-1-dimensional problem can be reduced to one dimension, giving an enormous saving in computation.
Abstract: The optimization problem for the design of multistage decimators and interpolators is considered. The corresponding objective function for sample rate increase or decrease is based upon the number of multiplies and adds per second. The structure of the multidimensional gradient equations for the decimation or interpolation ratios is investigated. A drastic simplification of the minimization process is demonstrated. Even for an arbitrary number of stages K, the solution of the K-1-dimensional problem can be reduced to one dimension, giving an enormous saving in computation. The highly applicable cases of K=3 and K=4 are treated explicitly.
TL;DR: This paper proposes to enlarge an image using a particular class of nonlinear techniques via an adaptive interpolatory procedure that aims to avoid blurred edges by not allowing the averaging of data from both sides of an edge.
18 Mar 2003
TL;DR: Results show that the adaptive weighting of pixels in interpolation gives better results than that obtained using traditional interpolation methods only or by using the warped distance technique.
Abstract: In this paper, an adaptive warped distance method is suggested for image interpolation. This method depends on modifying the warped distance technique for image interpolation taking into consideration the level of activity in local regions of the image. This is performed by weighting the pixels used in the interpolation process with different adaptive weights. The adaptation can be extended to different traditional interpolation techniques such as bilinear, bicubic and cubic spline techniques as well as to the warped distance technique. Results show that the adaptive weighting of pixels in interpolation gives better results than that obtained using traditional interpolation methods only or by using the warped distance technique.
TL;DR: It is shown that the interpolation positive operators of a wide class satisfy also the approximation property, giving the conditions under which a sequence of operators of the considered class converges to a continuous function in a convex compact set in Rm (m∈N).
Abstract: The purpose of this paper is to show that the interpolation positive operators of a wide class satisfy also the approximation property. Such a situation of simultaneous interpolation and approximation may be very desirable, but is rather unusual. Our attention is focused on the convergence problem, giving the conditions under which a sequence of operators of the considered class converges to a continuous function in a convex compact set in Rm (m∈N). It must be recalled that many of these operators are very interesting in applications and that suitable algorithms can be devised for parallel, multistage and iterative computation.
TL;DR: In this article, a new method of estimating total water levels relative to a datum is proposed, called tidal constituent and residual interpolation (TCARI), which takes values at the tide gauges and spatially interpolates them throughout the region.
TL;DR: This paper characterize MPH curves in R2,1 by the roots of the hodographs of their complexified spine curves, and presents two schemes for this interpolation problem: one is a subdivision scheme using direct C1 interpolation and the other is a two step scheme using a new concept, C1/2 interpolation.
25 May 2003
TL;DR: This paper derives the relation between various polynomial-based interpolation filters (Farrow structure and its modifications) and polyphase FIR model filters and observes possible applications of these relations, such as filter design, implementation complexity reduction, and response distortion analysis.
Abstract: If sampling rate conversion (SRC) is performed between arbitrary sampling rates, then the SRC factor can be a ratio of two very large integers or even an irrational number. An efficient way to reduce the implementation complexity of a SRC system in those cases is to use polynomial-based interpolation filters that mimic digitally the hybrid analogue/digital system. In practice, the sampling rate conversion is approximated with a rational factor. In this case, the hybrid analogue/digital model used to represent the SRC process may be represented by an equivalent discrete-time model. The discrete-time modeling of the rational SRC has been used earlier for the zeroth order interpolation. This paper extends this idea to arbitrary polynomial-based interpolation. Furthermore, this paper derives the relation between various polynomial-based interpolation filters (Farrow structure and its modifications) and polyphase FIR model filters. This paper observes possible applications of these relations, such as filter design, implementation complexity reduction, and response distortion analysis.
TL;DR: In this article, a non-parametric kernel density regression (KDR) was used for the analysis of heavy metal contaminated soil and its treatment using the Mollifier interpolation, which was capable of including additional independent variables (beyond the spatial dimensions x and y) in the spatial interpolation.
Abstract: To obtain data on heavy metal contaminated soil requires laborious and time-consuming data sampling and analysis. Not only has the contamination to be measured, but also additional data characterizing the soil and the boundary conditions of the site, such as pH, land use, and soil fertility. For an integrative approach, combining the analysis of spatial distribution, and of factors influencing the contamination, and its treatment, the Mollifier interpolation was used, which is a non-parametric kernel density regression. The Mollifier was capable of including additional independent variables (beyond the spatial dimensions x and y) in the spatial interpolation and hence explored the combined influence of spatial and other variables, such as land use, on the heavy metal distribution. The Mollifier could also represent the interdependence between different heavy metal concentrations and additional site characteristics. Although the uncertainty measure supplied by the Mollifier at first seems somewhat unusual, it is a valuable feature and supplements the geostatistical uncertainty assessment.
TL;DR: A new interpolation kernel for SAR interferometric registration is presented and discussed in comparison with five classical ones, based on the Knab sampling window that has been successfully employed in various electromagnetic problems and which has been proposed as capable of best preserving the stochastic information.
Abstract: Estimation of digital elevation maps from electromagnetic signals received at an interferometric synthetic aperture radar (SAR) requires significant processing. In this framework, a key step is the interferometric registration, which can be mathematically framed as an interpolation procedure. A new interpolation kernel for SAR interferometric registration is presented and discussed in comparison with five classical ones. It is based on the Knab sampling window that has been successfully employed in various electromagnetic problems and which has been proposed as capable of best preserving the stochastic information. The numerical experiments show the superiority of the new interpolation kernel.
TL;DR: In this paper, a combination of downward continuation with reference parameters, explicit correction filter application to account for laterally varying anisotropy, and interpolation between reference wavefields to ensure correct vertical propagation velocity is proposed.
Abstract: We describe an approach to performing wavefield extrapolation in VTI media in the case when the Thomsen anisotropy parameters δ and and vertical propagation velocity vary laterally. Our method is a combination of downward continuation with reference parameters, explicit correction filter application to account for laterally varying anisotropy, and interpolation between reference wavefields to ensure correct vertical propagation velocity. We address the questions of explicit filter design, spatial interpolation between reference wavefields, and operator splitting in 3-D. Both synthetic and real data examples are used to illustrate the performance of our algorithm.
Patent•
25 Nov 2003
TL;DR: In this article, a low angle tilt condition is used to reduce the computational complexity of high angle spatial interpolation in a display processor integrated circuit, where the pipelining is employed to write parts of segment buffers at the same time that other parts are being read to perform the interpolation process.
Abstract: A display processor integrated circuit includes a display processor portion and an on-chip programmable logic portion. The programmable logic portion can be configured to implement custom video and/or image enhancement functions. The display processor portion performs block-based motion detection. If no motion is detected for a given block of pixels, then interline gaps in the block are filled using temporal interpolation. If motion is detected, then interline gaps are filled using spatial interpolation. To maintain accuracy without unduly increasing computational complexity, a less complex high angle spatial interpolation method is employed where a low angle tilt condition is not detected. A more computationally intensive low angle spatial interpolation method can therefore be employed in low angle tilt conditions. Integrated circuit cost is reduced by employing pipelining to write parts of segment buffers at the same time that other parts are being read to perform the interpolation process.
TL;DR: In this paper, the spatial variability of soil variables is a critical component of modeling, estimation, prediction and risk assessment in soil science, and spatial variability must be taken into account for optimal spatial interpolation (e.g., kriging) and risk assessing (i.e., evaluating the probabilistic risk of evaluating the soil variables).
Abstract: The spatial variability of soil variables is a critical component of modeling, estimation, prediction and risk assessment in soil science. On one hand, spatial variability must be taken into account for optimal spatial interpolation (e.g., kriging) and risk assessment (e.g., evaluating the probabili
24 Nov 2003
TL;DR: This paper first considers only optimization of the interpolation filter, then considers the joint optimization over both the decimation and interpolation filters, using the variable projection method, and demonstrates a significant improvement over other approaches.
Abstract: Block coders are among the most common compression tools available for still images and video sequences. Their low computational complexity along with their good performance make them a popular choice for compression of natural images. Yet, at low bit-rates, block coders introduce visually annoying artifacts into the image. One approach that alleviates this problem is to downsample the image, apply the coding algorithm, and interpolate back to the original resolution. In this paper, we consider the use of optimal decimation and interpolation filters in this scheme. We first consider only optimization of the interpolation filter, by formulating the problem as least-squares minimization. We then consider the joint optimization over both the decimation and the interpolation filters, using the variable projection method. The experimental results presented clearly exhibit a significant improvement over other approaches.