Abstract: Preserving edge structures is a challenge to image interpolation algorithms that reconstruct a high-resolution image from a low-resolution counterpart. We propose a new edge-guided nonlinear interpolation technique through directional filtering and data fusion. For a pixel to be interpolated, two observation sets are defined in two orthogonal directions, and each set produces an estimate of the pixel value. These directional estimates, modeled as different noisy measurements of the missing pixel are fused by the linear minimum mean square-error estimation (LMMSE) technique into a more robust estimate, using the statistics of the two observation sets. We also present a simplified version of the LMMSE-based interpolation algorithm to reduce computational cost without sacrificing much the interpolation performance. Experiments show that the new interpolation techniques can preserve edge sharpness and reduce ringing artifacts

Abstract: Interpolation of a spatially correlated random process is used in many scientific areas. The best unbiased linear predictor, often called a kriging predictor in geostatistical science, requires the solution of a (possibly large) linear system based on the covariance matrix of the observations. In this article, we show that tapering the correct covariance matrix with an appropriate compactly supported positive definite function reduces the computational burden significantly and still leads to an asymptotically optimal mean squared error. The effect of tapering is to create a sparse approximate linear system that can then be solved using sparse matrix algorithms. Monte Carlo simulations support the theoretical results. An application to a large climatological precipitation dataset is presented as a concrete and practical illustration.

Abstract: Super-resolution algorithms reconstruct a high-resolution image from a set of low-resolution images of a scene. Precise alignment of the input images is an essential part of such algorithms. If the low-resolution images are undersampled and have aliasing artifacts, the performance of standard registration algorithms decreases. We propose a frequency domain technique to precisely register a set of aliased images, based on their low-frequency, aliasing-free part. A high-resolution image is then reconstructed using cubic interpolation. Our algorithm is compared to other algorithms in simulations and practical experiments using real aliased images. Both show very good visual results and prove the attractivity of our approach in the case of aliased input images. A possible application is to digital cameras where a set of rapidly acquired images can be used to recover a higher-resolution final image.

Abstract: Fluorescence excitation-emission matrix (EEM) measurements are useful in fields such as food science, analytical chemistry, biochemistry and environmental science. EEMs contain information which can be modeled using the parallel factor analysis (PARAFAC) model but the data analysis is often complicated due to both Rayleigh and Raman scattering. There are several established ways to deal with scattering effects. However, all of these methods have associated problems. This paper develops a new method for handling scattering using interpolation in the areas affected by first- and second-order Rayleigh and Raman scatter in such a way that the interfering signal is, at best, removed. The suggested method is fast and requires no additional input other than specifying the scattering region. The results of the proposed method were compared with those obtained from common alternative approaches used for preprocessing fluorescence data before analysis with PARAFAC and were shown to be equally good for various types of EEM data. The main advantage of the interpolation method is in its lack of additional metaparameters, its algorithmic speed and subsequent speed-up of PARAFAC modeling. It also allows for using EEM data in software not able to handle missing data. Copyright © 2007 John Wiley & Sons, Ltd.

Abstract: This study presents a method for estimating daily rainfall on a 0.05° latitude/longitude grid covering all of New Zealand for the period 1960–2004 using a second order derivative trivariate thin plate smoothing spline spatial interpolation model. Use of a hand-drawn (and subsequently digitised) mean annual rainfall surface as an independent variable in the interpolation is shown to reduce the interpolation error compared with using an elevation surface. This result is confirmed when long-term average annual rainfall data, derived from the daily interpolations, are validated using long-term river flow data. Copyright © 2006 Royal Meteorological Society

Abstract: We present an interpolation-based planning and replanning algorithm for generating low-cost paths through uniform and nonuniform resolution grids. Most grid-based path planners use discrete state transitions that artificially constrain an agent's motion to a small set of possible headings (e.g., 0, π/4, π/2, etc.). As a result, even “optimal” grid-based planners produce unnatural, suboptimal paths. Our approach uses linear interpolation during planning to calculate accurate path cost estimates for arbitrary positions within each grid cell and produce paths with a range of continuous headings. Consequently, it is particularly well suited to planning low-cost trajectories for mobile robots. In this paper, we introduce a version of the algorithm for uniform resolution grids and a version for nonuniform resolution grids. Together, these approaches address two of the most significant shortcomings of grid-based path planning: the quality of the paths produced and the memory and computational requirements of planning over grids. We demonstrate our approaches on a number of example planning problems, compare them to related algorithms, and present several implementations on real robotic systems.

Abstract: Seismic surveys generally have irregular areas where data cannot be acquired. These data should often be interpolated. A projection onto convex sets (POCS) algorithm using Fourier transforms allows interpolation of irregularly populated grids of seismic data with a simple iterative method that produces high-quality results. The original 2D image restoration method, the Gerchberg-Saxton algorithm, is extended easily to higher dimensions, and the 3D version of the process used here produces much better interpolations than typical 2D methods. The only parameter that makes a substantial difference in the results is the number of iterations used, and this number can be overestimated without degrading the quality of the results. This simplicity is a significant advantage because it relieves the user of extensive parameter testing. Although the cost of the algorithm is several times the cost of typical 2D methods, the method is easily parallelized and still completely practical.

Abstract: Introduction Remit of This Book Local Models and Methods What Is Local? Spatial Dependence Spatial Scale Stationarity Spatial Data Models Data Sets Used for Illustrative Purposes A Note on Notation Overview Local Modeling Approaches to Local Adaptation Stratification or Segmentation of Spatial Data Moving Window/Kernel Methods Locally Varying Model Parameters Transforming and Detrending Spatial Data Overview Grid Data Exploring Spatial Variation in Single Variables Global Univariate Statistics Local Univariate Statistics Analysis of Grid Data Moving Windows for Grid Analysis Wavelets Segmentation Analysis of Digital Elevation Models Overview Spatial Relations Spatial Autocorrelation: Global Measures Spatial Autocorrelation: Local Measures Global Regression Local Regression Regression and Spatial Data Spatial Autoregressive Models Multilevel Modeling Allowing for Local Variation in Model Parameters Moving Window Regression (MWR) Geographically Weighted Regression (GWR) Spatially Weighted Classification Overview Spatial Prediction 1: Deterministic Methods Point Interpolation Global Methods Local Methods Areal Interpolation General Approaches: Overlay Local Models and Local Data Limitations: Point and Areal Interpolation Overview Spatial Prediction 2: Geostatistics Random Function Models Stationarity Global Models Exploring Spatial Variation Kriging Equivalence of Splines and Kriging Conditional Simulation The Change of Support Problem Other Approaches Local Approaches: Nonstationary Models Nonstationary Mean Nonstationary Models for Prediction Nonstationary Variogram Variograms in Texture Analysis Summary Point Patterns Point Patterns Visual Examination of Point Patterns Density and Distance Methods Statistical Tests of Point Patterns Global Methods Distance Methods Other Issues Local Methods Density Methods Accounting for the Population at Risk The Local K Function Point Patterns and Detection of Clusters Overview Summary: Local Models for Spatial Analysis Review Key Issues Software Future Developments Summary References Index

Abstract: 1D PROBLEMS 1D Model Elliptic Problem A Two-Point Boundary Value Problem Algebraic Structure of the Variational Formulation Equivalence with a Minimization Problem Sobolev Space H1(0, l) Well Posedness of the Variational BVP Examples from Mechanics and Physics The Case with "Pure Neumann" BCs Exercises Galerkin Method Finite Dimensional Approximation of the VBVP Elementary Convergence Analysis Comments Exercises 1D hp Finite Element Method 1D hp Discretization Assembling Element Matrices into Global Matrices Computing the Element Matrices Accounting for the Dirichlet BC Summary Assignment 1: A Dry Run Exercises 1D hp Code Setting up the 1D hp Code Fundamentals Graphics Element Routine Assignment 2: Writing Your Own Processor Exercises Mesh Refinements in 1D The h-Extension Operator. Constrained Approximation Coefficients Projection-Based Interpolation in 1D Supporting Mesh Refinements Data-Structure-Supporting Routines Programming Bells and Whistles Interpolation Error Estimates Convergence Assignment 3: Studying Convergence Definition of a Finite Element Exercises Automatic hp Adaptivity in 1D The hp Algorithm Supporting the Optimal Mesh Selection Exponential Convergence. Comparing with h Adaptivity Discussion of the hp Algorithm Algebraic Complexity and Reliability of the Algorithm Exercises Wave Propagation Problems Convergence Analysis for Noncoercive Problems Wave Propagation Problems Asymptotic Optimality of the Galerkin Method Dispersion Error Analysis Exercises 2D ELLIPTIC PROBLEMS 2D Elliptic Boundary-Value Problem Classical Formulation Variational (Weak) Formulation Algebraic Structure of the Variational Formulation Equivalence with a Minimization Problem Examples from Mechanics and Physics Exercises Sobolev Spaces Sobolev Space H1(O) Sobolev Spaces of an Arbitrary Order Density and Embedding Theorems Trace Theorem Well Posedness of the Variational BVP Exercises 2D hp Finite Element Method on Regular Meshes Quadrilateral Master Element Triangular Master Element Parametric Element Finite Element Space. Construction of Basis Functions Calculation of Element Matrices Modified Element. Imposing Dirichlet Boundary Conditions Postprocessing- Local Access to Element d.o.f Projection-Based Interpolation Exercises 2D hp Code Getting Started Data Structure in FORTRAN 90 Fundamentals The Element Routine Modified Element. Imposing Dirichlet Boundary Conditions Assignment 4: Assembly of Global Matrices The Case with "Pure Neumann" Boundary Conditions Geometric Modeling and Mesh Generation Manifold Representation Construction of Compatible Parametrizations Implicit Parametrization of a Rectangle Input File Preparation Initial Mesh Generation The hp Finite Element Method on h-Refined Meshes Introduction. The h Refinements 1-Irregular Mesh Refinement Algorithm Data Structure in Fortran 90 (Continued) Constrained Approximation for C0 Discretizations Reconstructing Element Nodal Connectivities Determining Neighbors for Midedge Nodes Additional Comments Automatic hp Adaptivity in 2D The Main Idea The 2D hp Algorithm Example: L-Shape Domain Problem Example: 2D "Shock" Problem Additional Remarks Examples of Applications A "Battery Problem" Linear Elasticity An Axisymmetric Maxwell Problem Exercises Exterior Boundary-Value Problems Variational Formulation. Infinite Element Discretization Selection of IE Radial Shape Functions Implementation Calculation of Echo Area Numerical Experiments Comments Exercises 2D MAXWELL PROBLEMS 2D Maxwell Equations Introduction to Maxwell's Equation Variational Formulation Exercises Edge Elements and the de Rham Diagram Exact Sequences Projection-Based Interpolation De Rham Diagram Shape Functions Exercises 2D Maxwell Code Directories. Data Structure The Element Routine Constrained Approximation. Modified Element Setting up a Maxwell Problem Exercises hp Adaptivity for Maxwell Equations Projection-Based Interpolation Revisited The hp Mesh Optimization Algorithm Example: The Screen Problem Exterior Maxwell Boundary-Value Problems Variational Formulation Infinite Element Discretization in 3D Infinite Element Discretization in 2D Stability Implementation Numerical Experiments Exercises A Quick Summary and Outlook Appendix Bibliography Index

Abstract: The particle level set method has proven successful for the simulation of two separate regions (such as water and air, or fuel and products). In this paper, we propose a novel approach to extend this method to the simulation of as many regions as desired. The various regions can be liquids (or gases) of any type with differing viscosities, densities, viscoelastic properties, etc. We also propose techniques for simulating interactions between materials, whether it be simple surface tension forces or more complex chemical reactions with one material converting to another or two materials combining to form a third. We use a separate particle level set method for each region, and propose a novel projection algorithm that decodes the resulting vector of level set values providing a "dictionary" that translates between them and the standard single-valued level set representation. An additional difficulty occurs since discretization stencils (for interpolation, tracing semi-Lagrangian rays, etc.) cross region boundaries naively combining non-smooth or even discontinuous data. This has recently been addressed via ghost values, e.g. for fire or bubbles. We instead propose a new paradigm that allows one to incorporate physical jump conditions in data "on the fly," which is significantly more efficient for multiple regions especially at triple points or near boundaries with solids.

Abstract: Kernels are valuable tools in various elds of Numerical Analysis, including approximation, interpolation, meshless methods for solving partial dieren tial equations, neural networks, and Machine Learning. This contribution explains why and how kernels are applied in these disciplines. It uncovers the links between them, as far as they are related to kernel techniques. It addresses non-expert readers and focuses on practical guidelines for using kernels in applications.

Abstract: This paper investigates the sensitivity of hedonic models of house prices to the spatial interpolation of measures of air quality. We consider three aspects of this question: the interpolation technique used, the inclusion of air quality as a continuous vs discrete variable in the model, and the estimation method. Using a sample of 115,732 individual house sales for 1999 in the South Coast Air Quality Management District of Southern California, we compare Thiessen polygons, inverse distance weighting, Kriging and splines to carry out spatial interpolation of point measures of ozone obtained at 27 air quality monitoring stations to the locations of the houses. We take a spatial econometric perspective and employ both maximum-likelihood and general method of moments techniques in the estimation of the hedonic. A high degree of residual spatial autocorrelation warrants the inclusion of a spatially lagged dependent variable in the regression model. We find significant differences across interpolators...

Abstract: A high-precision CMOS time-to-digital converter IC has been designed. Time interval measurement is based on a counter and two-level interpolation realized with stabilized delay lines. Reference recycling in the delay line improves the integral nonlinearity of the interpolator and enables the use of a low frequency reference clock. Multi-level interpolation reduces the number of delay elements and registers and lowers the power consumption. The load capacitor scaled parallel structure in the delay line permits very high resolution. An INL look-up table reduces the effect of the remaining nonlinearity. The digitizer measures time intervals from 0 to 204 /spl mu/s with 8.1 ps rms single-shot precision. The resolution of 12.2 ps from a 5-MHz external reference clock is divided by means of only 20 delay elements.

Abstract: The goal of this paper is to illustrate how the techniques of locally analytic p-adic representation theory (as developed in [28, 29, 30, 31] and [13, 14, 17]; see also [16] for a short summary of some of these results) may be applied to study arithmetic properties of automorphic representations. More specifically, we consider the problem of p-adically interpolating the systems of eigenvalues attached to automorphic Hecke eigenforms (as well as the corresponding Galois representations, in situations where these appear in the étale cohomology of Shimura varieties). We can summarize our approach to the problem as follows: rather than attempting to directly interpolate the systems of eigenvalues attached to eigenforms, we instead attempt to interpolate the automorphic representations that these eigenforms give rise to. To be more precise, we fix a connected reductive linear algebraic group G defined over a number field F , and a finite prime p of F . We let Fp denote the completion of F at p, let E be a finite extension of Fp over which the group G splits, let A denote the ring of adèles of F , and let Af denote the ring of finite adèles of F . The representations that we construct are admissible locally analytic representations of the group G(Af ) on certain locally convex topological E-vector spaces. These representations are typically not irreducible; rather, they contain as closed subrepresentations many locally algebraic representations of G(Af ) which are closely related to automorphic representations of G(A) of cohomological type. (It is for this reason that we regard the representations that we construct as forming an “interpolation” of those automorphic representations.) Once we have our locally analytic representations of G(Af ) in hand, we may apply to them the Jacquet module functors of [14]. In this way we obtain p-adic analytic families of systems of Hecke eigenvalues, which (under a suitable hypothesis, for which see the statement of Theorem 0.7 below) p-adically interpolate (in the

Abstract: This paper surveys a wide selection of the interpolation algorithms that are in use in financial markets for construction of curves such as forward curves, basis curves, and most importantly, yield curves. In the case of yield curves the issue of bootstrapping is reviewed and how the interpolation algorithm should be intimately connected to the bootstrap itself is discussed. The criterion for inclusion in this survey is that the method has been implemented by a software vendor (or indeed an inhouse developer) as a viable option for yield curve interpolation. As will be seen, many of these methods suffer from problems: they posit unreasonable expections, or are not even necessarily arbitrage free. Moreover, many methods lead one to derive hedging strategies that are not intuitively reasonable. In the last sections, two new interpolation methods (the monotone convex method and the minimal method) are introduced, which it is believed overcome many of the problems highlighted with the other methods discussed ...

Abstract: The analysis and improvement of an immersed boundary method (IBM) for simulating turbulent flows over complex geometries are presented. Direct forcing is employed. It consists in interpolating boundary conditions from the solid body to the Cartesian mesh on which the computation is performed. Lagrange and least squares high-order interpolations are considered. The direct forcing IBM is implemented in an incompressible finite volume Navier–Stokes solver for direct numerical simulations (DNS) and large eddy simulations (LES) on staggered grids. An algorithm to identify the body and construct the interpolation schemes for arbitrarily complex geometries consisting of triangular elements is presented. A matrix stability analysis of both interpolation schemes demonstrates the superiority of least squares interpolation over Lagrange interpolation in terms of stability. Preservation of time and space accuracy of the original solver is proven with the laminar two-dimensional Taylor–Couette flow. Finally, practicability of the method for simulating complex flows is demonstrated with the computation of the fully turbulent three-dimensional flow in an air-conditioning exhaust pipe. Copyright © 2006 John Wiley & Sons, Ltd.

Abstract: Thin plate smoothing splines are widely used to spatially interpolate surface climate, however, their application to large data sets is limited by computational efficiency. Standard analytic calculation of thin plate smoothing splines requires O(n3) operations, where n is the number of data points, making routine computation infeasible for data sets with more than around 2000 data points. An O(N) iterative procedure for calculating finite element approximations to bivariate minimum generalised cross validation (GCV) thin plate smoothing splines operations was developed, where N is the number of grid points. The key contribution of the method lies in the incorporation of an automatic procedure for optimising smoothness to minimise GCV. The minimum GCV criterion is commonly used to optimise thin plate smoothing spline fits to climate data. The method discretises the bivariate thin plate smoothing spline equations using hierarchical biquadratic B-splines, and uses a nested grid multigrid procedure to solve the system. To optimise smoothness, a double iteration is incorporated, whereby the estimate of the spline solution and the estimate of the optimal smoothing parameter are updated simultaneously. When the method was tested on temperature data from the African and Australian continents, accurate approximations to analytic solutions were obtained.

Abstract: [1] After decades of research on continental tectonics, there is still no consensus on the mode of deformation of continents or on the forces that drive their deformation In Asia the debate opposes edge-driven block models, requiring a strong lithosphere with strain localized on faults, to buoyancy-driven continuous models, requiring a viscous lithosphere with pervasive strain Discriminating between these models requires continent-wide estimates of lithospheric strain rates Previous efforts have relied on the resampling of heterogeneous geodetic and Quaternary faulting data sets using interpolation techniques We present a new velocity field based on the rigorous combination of geodetic solutions with relatively homogeneous station spacing, avoiding technique-dependent biases inherent to interpolation methods We find (1) unresolvable strain rates (<3 × 109/yr) over a large part of Asia, with current motions well-described by block or microplate rotations, and (2) internal strain, possibly continuous, limited to high-elevation areas

Abstract: A meshless numerical model is developed for analyzing transient heat conduction in non-homogeneous functionally graded materials (FGM), which has a continuously functionally graded thermal conductivity parameter First, the analog equation method is used to transform the original non-homogeneous problem into an equivalent homogeneous one at any given time so that a simpler fundamental solution can be employed to take the place of the one related to the original problem Next, the approximate particular and homogeneous solutions are constructed using radial basis functions and virtual boundary collocation method, respectively Finally, by enforcing satisfaction of the governing equation and boundary conditions at collocation points of the original problem, in which the time domain is discretized using the finite difference method, a linear algebraic system is obtained from which the unknown fictitious sources and interpolation coefficients can be determined Further, the temperature at any point can be easily computed using the results of fictitious sources and interpolation coefficients The accuracy of the proposed method is assessed through two numerical examples

Abstract: A new set of H(curl)-conforming hierarchical basis functions for tetrahedral meshes is presented. Contrary to previous bases, this one is designed such that higher order basis functions vanish when they are projected onto a lower order finite-element space using the interpolation operator defined by Nedelec. Consequently, to increase the polynomial order and improve the accuracy of the interpolated field, only additional degrees of freedom (DOFs) of higher order are added, whereas the original DOFs (the coefficients for the basis functions) remain unchanged. This makes this basis very well suited for use with efficient multilevel solvers and goal-oriented hierarchical error estimators, which is demonstrated through numerical examples

Abstract: We present a simple method for compression and management of very large molecular dynamics trajectories. The approach is based on the projection of the Cartesian snapshots collected along the trajectory into an orthogonal space defined by the eigenvectors obtained by diagonalization of the covariance matrix. The transformation is mathematically exact when the number of eigenvectors equals 3N-6 (N being the number of atoms), and in practice very accurate even when the number of eigenvectors is much smaller, permitting a dramatic reduction in the size of trajectory files. In addition, we have examined the ability of the method, when combined with interpolation, to recover dense samplings (snapshots collected at a high frequency) from more sparse (lower frequency) data as a method for further data compression. Finally, we have investigated the possibility of using the approach when extrapolating the behavior of the system to times longer than the original simulation period. Overall our results suggest that the method is an attractive alternative to current approaches for including dynamic information in static structure files such as those deposited in the Protein Data Bank.

Abstract: A first interpolation processing apparatus that engages in processing on image data which are provided in a calorimetric system constituted of first˜nth (n≧2) color components and include color information corresponding to a single color component provided at each pixel to determine an interpolation value equivalent to color information corresponding to the first color component for a pixel at which the first color component is missing, includes: an interpolation value calculation section that uses color information at pixels located in a local area containing an interpolation target pixel to undergo interpolation processing to calculate an interpolation value including, at least (1) local average information of the first color component with regard to the interpolation target pixel and (2) local curvature information corresponding to at least two color components with regard to the interpolation target pixel.

Abstract: The Components of Time Series.- The Cholette-Dagum Regression-Based Benchmarking Method - The Additive Model.- Covariance Matrices for Benchmarking and Reconciliation Methods.- The Cholette-Dagum Regression-Based Benchmarking Method - The Multiplicative Model.- The Denton Method and its Variants.- Temporal Distribution, Interpolation and Extrapolation.- Signal Extraction and Benchmarking.- Calendarization.- A Unified Regression-Based Framework for Signal Extraction, Benchmarking and Interpolation.- Reconciliation and Balancing Systems of Time Series.- Reconciling One-Way Classified Systems of Time Series.- Reconciling the Marginal Totals of Two-Way Classified Systems of Series.- Reconciling Two-Way Classifed Systems of Series.