Showing papers on "Interpolation published in 2003"

Poisson image editing

Abstract: Using generic interpolation machinery based on solving Poisson equations, a variety of novel tools are introduced for seamless editing of image regions. The first set of tools permits the seamless importation of both opaque and transparent source image regions into a destination region. The second set is based on similar mathematical ideas and allows the user to modify the appearance of the image seamlessly, within a selected region. These changes can be arranged to affect the texture, the illumination, and the color of objects lying in the region, or to make tileable a rectangular selection.

Radial Basis Functions: Theory and Implementations

Abstract: Preface 1. Introduction 2. Summary of methods and applications 3. General methods for approximation and interpolation 4. Radial basis function approximation on infinite grids 5. Radial basis functions on scattered data 6. Radial basis functions with compact support 7. Implementations 8. Least squares methods 9. Wavelet methods with radial basis functions 10. Further results and open problems Appendix Bibliography Index.

Radial Basis Functions

Abstract: From the Publisher: "In many areas of mathematics, science and engineering, from computer graphics to inverse methods to signal processing it is necessary to estimate parameters, usually multidimensional, by approximation and interpolation. Radial basis functions are a modern and powerful tool which work well in very general circumstances, and so are becoming of widespread use, as the limitations of other methods, such as least squares, polynomial interpolation or wavelet-based, become apparent." This is the first book devoted to the subject and the author's aim is to give a thorough treatment from both the theoretical and practical implementation viewpoints. For example, he emphasises the many positive features of radial basis functions such as the unique solvability of the interpolation problem, the computation of interpolants, their smoothness and convergence, and provides a careful classification of the radial basis functions into types that have different convergence. A comprehensive bibliography rounds off what will prove a very valuable work.

Nonuniform fast Fourier transforms using min-max interpolation

Abstract: The fast Fourier transform (FFT) is used widely in signal processing for efficient computation of the FT of finite-length signals over a set of uniformly spaced frequency locations. However, in many applications, one requires nonuniform sampling in the frequency domain, i.e., a nonuniform FT. Several papers have described fast approximations for the nonuniform FT based on interpolating an oversampled FFT. This paper presents an interpolation method for the nonuniform FT that is optimal in the min-max sense of minimizing the worst-case approximation error over all signals of unit norm. The proposed method easily generalizes to multidimensional signals. Numerical results show that the min-max approach provides substantially lower approximation errors than conventional interpolation methods. The min-max criterion is also useful for optimizing the parameters of interpolation kernels such as the Kaiser-Bessel function.

Poisson image editing

Abstract: Using generic interpolation machinery based on solving Poisson equations, a variety of novel tools are introduced for seamless editing of image regions. The first set of tools permits the seamless ...

Numerical Analysis of Wavelet Methods

Abstract: Introduction. Notations. 1. Basic examples. 1.1 Introduction. 1.2 The Haar system. 1.3 The Schauder hierarchical basis. 1.4 Multivariate constructions. 1.5 Adaptive approximation. 1.6 Multilevel preconditioning. 1.7 Conclusions. 1.8 Historical notes. 2. Multiresolution approximation. 2.1 Introduction. 2.2 Multiresolution analysis. 2.3 Refinable functions. 2.4 Subdivision schemes. 2.5 Computing with refinable functions. 2.6 Wavelets and multiscale algorithms. 2.7 Smoothness analysis. 2.8 Polynomial exactness. 2.9 Duality, orthonormality and interpolation. 2.10 Interpolatory and orthonormal wavelets. 2.11 Wavelets and splines. 2.12 Bounded domains and boundary conditions. 2.13 Point values, cell averages, finite elements. 2.14 Conclusions. 2.15 Historical notes. 3. Approximation and smoothness. 3.1 Introduction. 3.2 Function spaces. 3.3 Direct estimates. 3.4 Inverse estimates. 3.5 Interpolation and approximation spaces. 3.6 Characterization of smoothness classes. 3.7 Lp-unstable approximation and 0 1. 3.8 Negative smoothness and Lp-spaces. 3.9 Bounded domains. 3.10 Boundary conditions. 3.11 Multilevel preconditioning. 3.12 Conclusions. 3.13 Historical notes. 4. Adaptivity. 4.1 Introduction. 4.2 Nonlinear approximation in Besov spaces. 4.3 Nonlinear wavelet approximation in Lp. 4.4 Adaptive finite element approximation. 4.5 Other types of nonlinear approximations. 4.6 Adaptive approximation of operators. 4.7 Nonlinear approximation and PDE's. 4.8 Adaptive multiscale processing. 4.9 Adaptive space refinement. 4.10 Conclusions. 4.11 Historical notes. References. Index.

Efficient Implementation of Marching Cubes' Cases with Topological Guarantees

Abstract: Marching Cubes methods first offered visual access to experimental and theoretical volumetric data. The implementation of this method usually relies on a small look-up table; many enhancements and optimizations of Marching Cubes still use it. However, this look-up table can lead to cracks and inconsistent topology. This paper introduces a full implementation of Chernyaev's technique to ensure a topologically correct result, i.e., a manifold mesh, for any input data. It completes the original paper for the ambiguity resolution and for the feasibility of the implementation. Moreover, the cube interpolation provided here can be used in a wider range of methods. The source code is available online.

Comparative analysis of interpolation methods in the middle Ebro Valley (Spain): application to annual precipitation and temperature

Abstract: This paper analyzes the validity of various precipitation and temperature maps obtained by means of diverse interpolation methods. The study was carried out in an area where geographic differences and spatial climatic diversity are significant (the middle Ebro Valley in the northeast of Spain). Two variables, annual precipitation and temperature, and several interpolation methods were used in the climate mapping: global interpolators (trend surfaces and regression models), local inter- polators (Thiessen polygons, inverse distance weighting, splines), geostatistical methods (simple kriging, ordinary kriging, block kriging, directional kriging, universal kriging and co-kriging) and mixed methods (combined global, local and geostatistical methods). The validity of the maps was checked through independent test weather stations (30% of the original stations). Different statistical accuracy measurements determined the quality of the models. The results show that some interpola- tion methods are very similar. Nevertheless, in the case of precipitation maps, we obtained the best results using geostatistical methods and a regression model formed by 4 geographic and topographic variables. The best results for temperature mapping were obtained using the regression-based method. The accuracy measurements obtained by the different interpolation methods change signif- icantly depending on the climatic variable mapped. The validity of interpolation methods in the creation of climatic maps, useful for agricultural and hydrologic management, is discussed.

Effective color interpolation in CCD color filter arrays using signal correlation

Abstract: We propose an effective color filter array (CFA) interpolation method for digital still cameras (DSCs) using a simple image model that correlates the R,G,B channels. In this model, we define the constants K/sub R/ as green minus red and K/sub B/ as green minus blue. For real-world images, the contrasts of K/sub R/ and K/sub B/ are quite flat over a small region and this property is suitable for interpolation. The main contribution of this paper is that we propose a low-complexity interpolation method to improve the image quality. We show that the frequency response of the proposed method is better than the conventional methods. Simulation results also verify that the proposed method obtain superior image quality on typical images. The luminance channel of the proposed method outperforms by 6.34-dB peak SNR the bilinear method, and the chrominance channels have a 7.69-dB peak signal-to-noise ratio improvement on average. Furthermore, the complexity of the proposed method is comparable to conventional bilinear interpolation. It requires only add and shift operations to implement.

Estimating differential quantities using polynomial fitting of osculating jets

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.

Flexible automatic motion blending with registration curves

Abstract: Many motion editing algorithms, including transitioning and multitarget interpolation, can be represented as instances of a more general operation called motion blending. We introduce a novel data structure called a registration curve that expands the class of motions that can be successfully blended without manual input. Registration curves achieve this by automatically determining relationships involving the timing, local coordinate frame, and constraints of the input motions. We show how registration curves improve upon existing automatic blending methods and demonstrate their use in common blending operations.

Influence of Spatial Structure on Accuracy of Interpolation Methods

Abstract: Effectiveness of precision agriculture depends on accurate and efficient mapping of soil properties. Among the factors that most affect soil property mapping are the number of soil samples, the distance between sampling locations, and the choice of interpolation procedures. The objective of this study is to evaluate the effect of data variability and the strength of spatial correlation in the data on the performance of (i) grid soil sampling of different sampling density and (ii) two interpolation procedures, ordinary point kriging and optimal inverse distance weighting (IDW). Soil properties with coefficients of variation (CV) ranging from 12 to 67% were sampled in a 20-ha field using a regular grid with a 30-m distance between grid points. Data sets with different spatial structures were simulated based on the soil sample data using a simulated annealing procedure. The strength of simulated spatial structures ranged from weak with nugget to sill (N/S) ratio of 0.6 to strong (N/S ratio of 0.1). The results indicated that regardless of CV values, soil properties with a strong spatial structure were mapped more accurately than those that had weak spatial structure. Kriging with known variogram parameters performed significantly better than the IDW for most of the studied cases (P < 0.01). However, when variogram parameters were determined from sample variograms, kriging was as accurate as the IDW only for sufficiently large data sets, but was less precise when a reliable sample variogram could not be obtained from the data.

Kriging for interpolation in random simulation

Abstract: Whenever simulation requires much computer time, interpolation is needed. Simulationists use different interpolation techniques (eg linear regression), but this paper focuses on Kriging. This technique was originally developed in geostatistics by DG Krige, and has recently been widely applied in deterministic simulation. This paper, however, focuses on random or stochastic simulation. Essentially, Kriging gives more weight to ‘neighbouring’ observations. There are several types of Kriging; this paper discusses—besides Ordinary Kriging—a novel type, which ‘detrends’ data through the use of linear regression. Results are presented for two examples of input/output behaviour of the underlying random simulation model: Ordinary and Detrended Kriging give quite acceptable predictions; traditional linear regression gives the worst results.

Moving kriging interpolation and element‐free Galerkin method

Abstract: A new formulation of the element-free Galerkin (EFG) method is presented in this paper. EFG has been extensively popularized in the literature in recent years due to its flexibility and high convergence rate in solving boundary value problems. However, accurate imposition of essential boundary conditions in the EFG method often presents difficulties because the Kronecker delta property, which is satisfied by finite element shape functions, does not necessarily hold for the EFG shape function. The proposed new formulation of EFG eliminates this shortcoming through the moving kriging (MK) interpolation. Two major properties of the MK interpolation: the Kronecker delta property (Φ I (S J ) = δ I J ) and the consistency property (Σ n I Φ I (x) = I and Σ n I )Φ I (x)x l i = x i ) are proved. Some preliminary numerical results arc given.

An automatic modeling of human bodies from sizing parameters

Abstract: In this paper, we present an automatic, runtime modeler for modeling realistic, animatable human bodies. A user can generate a new model or modify an existing one simply by inputting a number of sizing parameters.We approach the problem by forming deformation functions that are devoted to the generation of appropriate shape and proportion of the body geometry by taking the parameters as input. Starting from a number of 3D scanned data of human body models as examples, we derive these functions by using radial basis interpolation. A prerequisite of such formulation is to have correspondence among example models in the database. We obtain the correspondence by fitting a template onto each scanned data. Throughout the paper, body geometry is considered to have two distinct entities, namely rigid and elastic component of the deformation. The rigid deformation is represented by the corresponding joint parameters, which will determine the linear approximation of the physique. The elastic deformation is essentially vertex displacements, which, when added to the rigid deformation, depicts the detail shape of the body.Having these interpolators formulated, the runtime modeling can be reduced to the function evaluation and application of the evaluated results to the template model. We demonstrate our method by applying different parameters to generate a wide range of different body models.

Sensitivity of Precipitation Forecast Skill Scores to Bilinear Interpolation and a Simple Nearest-Neighbor Average Method on High-Resolution Verification Grids

Abstract: Grid transformations are common postprocessing procedures used in numerical weather prediction to transfer a forecast field from one grid to another. This paper investigates the statistical effects of two different interpolation techniques on widely used precipitation skill scores like the equitable threat score and the Hanssen–Kuipers score. The QUADRICS Bologna Limited Area Model (QBOLAM), which is a parallel version of the Bologna Limited Area Model (BOLAM) described by Buzzi et al., is used, and it is verified on grids of about 10 km (grid-box size). The precipitation analysis is obtained by means of a Barnes objective analysis scheme. The rain gauge data are from the Piedmont and Liguria regions, in northwestern Italy. The data cover 243 days, from 1 October 2000 to 31 May 2001. The interpolation methods considered are bilinear interpolation and a simple nearest-neighbor averaging method, also known as remapping or budget interpolation, which maintains total precipitation to a desired degree...

A multi-scale approach to 3D scattered data interpolation with compactly supported basis functions

Abstract: We propose a hierarchical approach to 3D scattered data interpolation with compactly supported basis functions. Our numerical experiments suggest that the approach integrates the best aspects of scattered data fitting with locally and globally supported basis functions. Employing locally supported functions leads to an efficient computational procedure, while a coarse-to-fine hierarchy makes our method insensitive to the density of scattered data and allows us to restore large parts of missed data. Given a point cloud distributed along a surface, we first use spatial down sampling to construct a coarse-to-fine hierarchy of point sets. Then we interpolate the sets starting from the coarsest level. We interpolate a point set of the hierarchy, as an offsetting of the interpolating function computed at the previous level. An original point set and its coarse-to-fine hierarchy of interpolated sets is presented. According to our numerical experiments, the method is essentially faster than the state-of-the-art scattered data approximation with globally supported RBFs (Carr et al., 2001) and much simpler to implement.

On marching cubes

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.

Performance of mutual information similarity measure for registration of multitemporal remote sensing images

Abstract: Accurate registration of multitemporal remote sensing images is essential for various change detection applications. Mutual information has recently been used as a similarity measure for registration of medical images because of its generality and high accuracy. Its application in remote sensing is relatively new. There are a number of algorithms for the estimation of joint histograms to compute mutual information, but they may suffer from interpolation-induced artifacts under certain conditions. In this paper, we investigate the use of a new joint histogram estimation algorithm called generalized partial volume estimation (GPVE) for computing mutual information to register multitemporal remote sensing images. The experimental results show that higher order GPVE algorithms have the ability to significantly reduce interpolation-induced artifacts. In addition, mutual-information-based image registration performed using the GPVE algorithm produces better registration consistency than the other two popular similarity measures, namely, mean squared difference (MSD) and normalized cross correlation (NCC), used for the registration of multitemporal remote sensing images.

Seismic trace interpolation in the Fourier transform domain

Abstract: A data adaptive interpolation method is designed and applied in the Fourier transform domain (f‐k or f‐kx‐ky for spatially aliased data. The method makes use of fast Fourier transforms and their cyclic properties, thereby offering a significant cost advantage over other techniques that interpolate aliased data.The algorithm designs and applies interpolation operators in the f‐k (or f‐kx‐ky domain to fill zero traces inserted in the data in the t‐x (or t‐x‐y) domain at locations where interpolated traces are needed. The interpolation operator is designed by manipulating the lower frequency components of the stretched transforms of the original data. This operator is derived assuming that it is the same operator that fills periodically zeroed traces of the original data but at the lower frequencies, and corresponds to the f‐k (or f‐kx‐ky domain version of the well‐known f‐x (or f‐x‐y) domain trace interpolators.The method is applicable to 2D and 3D data recorded sparsely in a horizontal plane. The most comm...

Extraction of red edge optical parameters from Hyperion data for estimation of forest leaf area index

Abstract: A correlation analysis was conducted between forest leaf area index (LAI) and two red edge parameters: red edge position (REP) and red well position (RWP), extracted from reflectance image retrieved from Hyperion data. Field spectrometer data and LAI measurements were collected within two days after the Earth Observing One satellite passed over the study site in the Patagonia region of Argentina. The two red edge parameters were extracted with four approaches: four-point interpolation, polynomial fitting, Lagrangian technique, and inverted-Gaussian (IG) modeling. Experimental results indicate that the four-point approach is the most practical and suitable method for extracting the two red edge parameters from Hyperion data because only four bands and a simple interpolation computation are needed. The polynomial fitting approach is a direct method and has its practical value if hyperspectral data are available. However, it requires more computation time. The Lagrangian method is applicable only if the first derivative spectra are available; thus, it is not suitable to multispectral remote sensing. The IG approach needs further testing and refinement for Hyperion data.

Mutual information-based CT-MR brain image registration using generalized partial volume joint histogram estimation

Abstract: Mutual information (MI)-based image registration has been found to be quite effective in many medical imaging applications. To determine the MI between two images, the joint histogram of the two images is required. In the literature, linear interpolation and partial volume interpolation (PVI) are often used while estimating the joint histogram for registration purposes. It has been shown that joint histogram estimation through these two interpolation methods may introduce artifacts in the MI registration function that hamper the optimization process and influence the registration accuracy. In this paper, we present a new joint histogram estimation scheme called generalized partial volume estimation (GPVE). It turns out that the PVI method is a special case of the GPVE procedure. We have implemented our algorithm on the clinically obtained brain computed tomography and magnetic resonance image data furnished by Vanderbilt University. Our experimental results show that, by properly choosing the kernel functions, the GPVE algorithm significantly reduces the interpolation-induced artifacts and, in cases that the artifacts clearly affect registration accuracy, the registration accuracy is improved.

Mutual information based image registration for remote sensing data

Abstract: Registration is a fundamental operation in image processing to align images taken at different times, from different sensors or from different viewing angles. Automatic image registration procedures are gaining importance to efficiently register the large volumes of remote sensing data available these days. In this Letter, we investigate an automated mutual information-based registration technique for remote sensing data. The performance of a number of interpolation algorithms to compute mutual information for registration of multi-sensor and multi-resolution Landsat TM, Radarsat SAR and IRS PAN images is evaluated.

A parametric interpolator with confined chord errors, acceleration and deceleration for NC machining

Abstract: Parametric interpolation has many advantages over linear interpolation in machining curves. Real time parametric interpolation research so far has addressed achieving a uniform feed rate, confined chord errors and jerk limited trajectory planning. However, simultaneous consideration of confined chord errors that respect the acceleration and deceleration capabilities of the machine has not been attempted. In this paper, the offline detection of feed rate sensitive corners is proposed. The velocity profile in these zones is planned so that chord errors are satisfied while simultaneously accommodating the machine's acceleration and deceleration limits. Outside the zone of the feed rate sensitive corners, the feed rate is planned using the Taylor approximation. Simulation results indicate that the offline detection of feed rate sensitive corners improves parametric interpolation. For real time interpolation, the parametric curve information can be augmented with the detected feed rate sensitive corners that are stored in 2×2 matrices.

Radial basis function interpolation: numerical and analytical developments

Abstract: The Radial Basis Function (RBF) method is one of the primary tools for interpolating multidimensional scattered data. The methods' ability to handle arbitrarily scattered data, to easily generalize to several space dimensions, and to provide spectral accuracy have made it particularly popular in several different types of applications. Some of the more recent of these applications include cartography, neural networks, medical imaging, and the numerical solution of partial differential equations (PDEs). In this thesis we study three issues with the RBF method that have received very little attention in the literature. First, we focus on the behavior of RBF interpolants near boundaries. Like most interpolation methods, a common feature of the RBF method is how relatively inaccurate the interpolants are near boundaries. Such boundary induced errors can severely limit the utility of the RBF method for numerically solving certain PDEs. With that as motivation, we investigate the behavior of RBF interpolants near boundaries and propose the first practical techniques for ameliorating the errors there. We next focus on some numerical developments for the RBF method based on infinitely smooth RBFs. Most infinitely smooth RBFs feature a free “shape” parameter e such that, as the magnitude of e decreases, the RBFs become increasingly flat. While small values of e typically result in more accurate interpolants, the direct method of computing the interpolants suffers from severe numerical ill-conditioning as e → 0. Until recently, this ill-conditioning has severely limited the range of e that could be considered in the RBF method. We present a novel numerical approach that largely overcomes the numerical ill-conditioning and allows for the stable computation of RBF interpolants for all values of e, including the limiting e = 0 case. This new method provides the first tool for the numerical exploration of RBF interpolants as e → 0. The third focus of the thesis is on the behavior of RBF interpolants as e → 0. In most cases the interpolants converge to a finite degree multivariate polynomial interpolant as e → 0. However, in rare situations the interpolants may diverge. We investigate this phenomenon in great detail both numerically and analytically, and link it directly to the failure of a condition known as “polynomial unisolvency”. We also find that the Gaussian RBF is inherently different from the other standard infinitely smooth RBFs in that it appears to result in an interpolant that never diverges as e → 0. We conclude with a brief overview of two future research opportunities related to the topics of the thesis. The first involves using RBF interpolants to generate scattered-node finite difference formulas. The second involves using RBF interpolants to generate linear multistep methods for solving ordinary differential equations.

Block boundary artifact reduction for block-based image compression

Abstract: Block boundary artifact reduction in decompressed digital images is accomplished by filtering the intensity of pixels in the vicinity of the block boundary. The filter utilizes filter coefficients selected from tables on the basis of the distribution of a scalar quantity describing the pixels neighboring the pixel to which the intensity adjustment is to be applied. The intensities of pixels at the boundary and one pixel removed from the boundary are adjusted. If interpolation pixels in each of neighboring blocks are of relatively constant intensity, the intensities of additional pixels, more remote from the boundary, are adjusted. Filter coefficients can be selected from different arrays for pixels on the boundary or removed from the boundary or if the pixel intensity adjustment is based on the intensities of pixels in a horizontal row or vertical column of interpolation pixels.