Showing papers on "Interpolation published in 2011"
Book•
18 Nov 2011
TL;DR: In this paper, the authors define the Riesz-Thorin Theorem as a necessary and sufficient condition for interpolation spaces, and apply it to approximate spaces in the context of vector spaces.
Abstract: 1. Some Classical Theorems.- 1.1. The Riesz-Thorin Theorem.- 1.2. Applications of the Riesz-Thorin Theorem.- 1.3. The Marcinkiewicz Theorem.- 1.4. An Application of the Marcinkiewicz Theorem.- 1.5. Two Classical Approximation Results.- 1.6. Exercises.- 1.7. Notes and Comment.- 2. General Properties of Interpolation Spaces.- 2.1. Categories and Functors.- 2.2. Normed Vector Spaces.- 2.3. Couples of Spaces.- 2.4. Definition of Interpolation Spaces.- 2.5. The Aronszajn-Gagliardo Theorem.- 2.6. A Necessary Condition for Interpolation.- 2.7. A Duality Theorem.- 2.8. Exercises.- 2.9. Notes and Comment.- 3. The Real Interpolation Method.- 3.1. The K-Method.- 3.2. The J-Method.- 3.3. The Equivalence Theorem.- 3.4. Simple Properties of ??, q.- 3.5. The Reiteration Theorem.- 3.6. A Formula for the K-Functional.- 3.7. The Duality Theorem.- 3.8. A Compactness Theorem.- 3.9. An Extremal Property of the Real Method.- 3.10. Quasi-Normed Abelian Groups.- 3.11. The Real Interpolation Method for Quasi-Normed Abelian Groups.- 3.12. Some Other Equivalent Real Interpolation Methods.- 3.13. Exercises.- 3.14. Notes and Comment.- 4. The Complex Interpolation Method.- 4.1. Definition of the Complex Method.- 4.2. Simple Properties of ?[?].- 4.3. The Equivalence Theorem.- 4.4. Multilinear Interpolation.- 4.5. The Duality Theorem.- 4.6. The Reiteration Theorem.- 4.7. On the Connection with the Real Method.- 4.8. Exercises.- 4.9. Notes and Comment.- 5. Interpolation of Lp-Spaces.- 5.1. Interpolation of Lp-Spaces: the Complex Method.- 5.2. Interpolation of Lp-Spaces: the Real Method.- 5.3. Interpolation of Lorentz Spaces.- 5.4. Interpolation of Lp-Spaces with Change of Measure: p0 = p1.- 5.5. Interpolation of Lp-Spaces with Change of Measure: p0 ? p1.- 5.6. Interpolation of Lp-Spaces of Vector-Valued Sequences.- 5.7. Exercises.- 5.8. Notes and Comment.- 6. Interpolation of Sobolev and Besov Spaces.- 6.1. Fourier Multipliers.- 6.2. Definition of the Sobolev and Besov Spaces.- 6.3. The Homogeneous Sobolev and Besov Spaces.- 6.4. Interpolation of Sobolev and Besov Spaces.- 6.5. An Embedding Theorem.- 6.6. A Trace Theorem.- 6.7. Interpolation of Semi-Groups of Operators.- 6.8. Exercises.- 6.9. Notes and Comment.- 7. Applications to Approximation Theory.- 7.1. Approximation Spaces.- 7.2. Approximation of Functions.- 7.3. Approximation of Operators.- 7.4. Approximation by Difference Operators.- 7.5. Exercises.- 7.6. Notes and Comment.- References.- List of Symbols.
4,695 citations
Posted Content•
TL;DR: In this article, the authors deal with the fractional Sobolev spaces W^[s,p] and analyze the relations among some of their possible definitions and their role in the trace theory.
Abstract: This paper deals with the fractional Sobolev spaces W^[s,p]. We analyze the relations among some of their possible definitions and their role in the trace theory. We prove continuous and compact embeddings, investigating the problem of the extension domains and other regularity results. Most of the results we present here are probably well known to the experts, but we believe that our proofs are original and we do not make use of any interpolation techniques nor pass through the theory of Besov spaces. We also present some counterexamples in non-Lipschitz domains.
2,272 citations
TL;DR: Interpolation methods provided a high prediction accuracy of the mean concentration of soil heavy metals, however, the classic method based on percentages of polluted samples, gave a pollution area 23.54-41.92% larger than that estimated by interpolation methods.
444 citations
TL;DR: The quantitative and visual results are showing the superiority of the proposed technique over the conventional and state-of-art image resolution enhancement techniques.
Abstract: In this correspondence, the authors propose an image resolution enhancement technique based on interpolation of the high frequency subband images obtained by discrete wavelet transform (DWT) and the input image. The edges are enhanced by introducing an intermediate stage by using stationary wavelet transform (SWT). DWT is applied in order to decompose an input image into different subbands. Then the high frequency subbands as well as the input image are interpolated. The estimated high frequency subbands are being modified by using high frequency subband obtained through SWT. Then all these subbands are combined to generate a new high resolution image by using inverse DWT (IDWT). The quantitative and visual results are showing the superiority of the proposed technique over the conventional and state-of-art image resolution enhancement techniques.
410 citations
TL;DR: This paper presents an algorithm for the local implementation of Galerkin projection of discrete fields between meshes, which extends naturally to three dimensions and is very efficient.
363 citations
TL;DR: In this paper, the performance of two fundamentally different approaches to achieve sub-pixel precision of normalised cross-correlation when measuring surface displacements on mass movements from repeat optical images was evaluated.
362 citations
TL;DR: In this paper, a new algorithm is developed to improve the accuracy and efficiency of the material point method for problems involving extremely large tensile deformations and rotations, and a novel set of grid basis functions is proposed for efficiently calculating nodal force and consistent mass integrals on the grid.
Abstract: SUMMARY A new algorithm is developed to improve the accuracy and efficiency of the material point method for problems involving extremely large tensile deformations and rotations. In the proposed procedure, particle domains are convected with the material motion more accurately than in the generalized interpolation material point method. This feature is crucial to eliminate instability in extension, which is a common shortcoming of most particle methods. Also, a novel alternative set of grid basis functions is proposed for efficiently calculating nodal force and consistent mass integrals on the grid. Specifically, by taking advantage of initially parallelogram-shaped particle domains, and treating the deformation gradient as constant over the particle domain, the convected particle domain is a reshaped parallelogram in the deformed configuration. Accordingly, an alternative grid basis function over the particle domain is constructed by a standard 4-node finite element interpolation on the parallelogram. Effectiveness of the proposed modifications is demonstrated using several large deformation solid mechanics problems. Copyright 2011 John Wiley & Sons, Ltd.
357 citations
12 Dec 2011
TL;DR: The use of displacement interpolation is developed for this class of problem, which provides a generic method for interpolating between distributions or functions based on advection instead of blending.
Abstract: Interpolation between pairs of values, typically vectors, is a fundamental operation in many computer graphics applications. In some cases simple linear interpolation yields meaningful results without requiring domain knowledge. However, interpolation between pairs of distributions or pairs of functions often demands more care because features may exhibit translational motion between exemplars. This property is not captured by linear interpolation. This paper develops the use of displacement interpolation for this class of problem, which provides a generic method for interpolating between distributions or functions based on advection instead of blending. The functions can be non-uniformly sampled, high-dimensional, and defined on non-Euclidean manifolds, e.g., spheres and tori. Our method decomposes distributions or functions into sums of radial basis functions (RBFs). We solve a mass transport problem to pair the RBFs and apply partial transport to obtain the interpolated function. We describe practical methods for computing the RBF decomposition and solving the transport problem. We demonstrate the interpolation approach on synthetic examples, BRDFs, color distributions, environment maps, stipple patterns, and value functions.
348 citations
30 Oct 2011
TL;DR: In this paper, a simplified and faster implementation of Bradley's procedure is presented, and successful and unsuccessful attempts to improve it are discussed, as well as their experience with the algorithm suggests that it is stronger than interpolation on industrial problems.
Abstract: Last spring, in March 2010, Aaron Bradley published the first truly new bit-level symbolic model checking algorithm since Ken McMillan's interpolation based model checking procedure introduced in 2003. Our experience with the algorithm suggests that it is stronger than interpolation on industrial problems, and that it is an important algorithm to study further. In this paper, we present a simplified and faster implementation of Bradley's procedure, and discuss our successful and unsuccessful attempts to improve it.
295 citations
TL;DR: This paper presents bilinear and bicubic interpolation methods tailored for the division of focal plane polarization imaging sensor targeting a 1-Mega pixel polarization Imaging sensor operating in the visible spectrum.
Abstract: This paper presents bilinear and bicubic interpolation methods tailored for the division of focal plane polarization imaging sensor. The interpolation methods are targeted for a 1-Mega pixel polarization imaging sensor operating in the visible spectrum. The five interpolation methods considered in this paper are: bilinear, weighted bilinear, bicubic spline, an approximated bicubic spline and a bicubic interpolation method. The modulation transfer function analysis is applied to the different interpolation methods, and test images as well as numerical error analyses are also presented. Based on the comparison results, the full frame bicubic spline interpolation achieves the best performance for polarization images.
276 citations
TL;DR: This work develops two implementations of CBP for a one-dimensional translation-invariant source, one using a first-order Taylor approximation, and another using a form of trigonometric spline, and examines the tradeoff between sparsity and signal reconstruction accuracy in these methods.
Abstract: We consider the problem of decomposing a signal into a linear combination of features, each a continuously translated version of one of a small set of elementary features. Although these constituents are drawn from a continuous family, most current signal decomposition methods rely on a finite dictionary of discrete examples selected from this family (e.g., shifted copies of a set of basic waveforms), and apply sparse optimization methods to select and solve for the relevant coefficients. Here, we generate a dictionary that includes auxiliary interpolation functions that approximate translates of features via adjustment of their coefficients. We formulate a constrained convex optimization problem, in which the full set of dictionary coefficients represents a linear approximation of the signal, the auxiliary coefficients are constrained so as to only represent translated features, and sparsity is imposed on the primary coefficients using an L1 penalty. The basis pursuit denoising (BP) method may be seen as a special case, in which the auxiliary interpolation functions are omitted, and we thus refer to our methodology as continuous basis pursuit (CBP). We develop two implementations of CBP for a one-dimensional translation-invariant source, one using a first-order Taylor approximation, and another using a form of trigonometric spline. We examine the tradeoff between sparsity and signal reconstruction accuracy in these methods, demonstrating empirically that trigonometric CBP substantially outperforms Taylor CBP, which, in turn, offers substantial gains over ordinary BP. In addition, the CBP bases can generally achieve equally good or better approximations with much coarser sampling than BP, leading to a reduction in dictionary dimensionality.
TL;DR: This paper uses higher order piecewise interpolation polynomial to approximate the fractional integral and fractional derivatives, and uses the Simpson method to design a higher order algorithm for the fractionsal differential equations.
TL;DR: Different spatial interpolation algorithms have been evaluated to produce a reasonably good continuous dataset bridging the gaps in the historical series of precipitation records in Sicily and validation results indicate that the univariate methods, neglecting the information of elevation, are characterized by the largest errors.
TL;DR: In this article, a new type of interpolation function is introduced that has a zero slope at the equilibrium values of the nonconserved field variables representing the different phases and allows for a thermodynamically consistent interpolation of the free energies.
TL;DR: Two multi-material interpolation schemes as direct generalizations of the well-known SIMP and RAMP material interpolation scheme originally developed for isotropic mixtures of two isotropIC material phases are presented.
Abstract: This paper presents two multi-material interpolation schemes as direct generalizations of the well-known SIMP and RAMP material interpolation schemes originally developed for isotropic mixtures of two isotropic material phases. The new interpolation schemes provide generally applicable interpolation schemes between an arbitrary number of pre-defined materials with given (anisotropic) properties. The method relies on a large number of sparse linear constraints to enforce the selection of at most one material in each design subdomain. Topology and multi-material optimization is formulated within a unified parametrization.
TL;DR: A numerical scheme of computing quantities involving gradients of shape functions is introduced for the material point method, so that the quantities are continuous as material points move across cell boundaries, and is proved to satisfy mass and momentum conservations exactly.
TL;DR: In this article, a thorough analysis of the generalized delayed-signal cancellation (DSC) operator in both synchronous and stationary reference frames is first conducted, and the discretization error during digital implementation due to nonideal system sampling frequency and/or grid-frequency variation is quantified with the proposed concept of relative harmonic gain error.
Abstract: Phase-locked loop (PLL) is usually required to detect grid phase angle in grid-tied converters. Conventional PLL schemes have to compromise between steady-state accuracy and transient dynamics when grid voltage is polluted by unbalance and harmonics. To overcome this challenge, a generalized delayed-signal-cancellation (DSC) operator is proposed recently to form cascaded DSC (CDSC) operator to eliminate arbitrary harmonics. With the CDSC operator, the conditioned voltage can be used in PLL loop with very high bandwidth for fast tracking. However, for digital implementation, the CDSC operator may subject to delay-time error, which subsequently leads to residual distortions in the conditioned voltage. In this paper, a thorough analysis of the CDSC operator in both synchronous and stationary reference frames is first conducted. The discretization error during digital implementation due to nonideal system sampling frequency and/or grid-frequency variation is quantified with the proposed concept of relative harmonic gain error. An effective improvement method is then developed that is based on linear interpolation and is effective for all delay-based PLL schemes. Finally, experimental results are obtained to verify the harmonic elimination ability of CDSC in various scenarios and the effectiveness of the interpolation-based digital implementation scheme.
16 Jul 2011
TL;DR: The main results are model-theoretic characterizations of uniform interpolants and their existence in terms of bisimulations, tight complexity bounds for deciding the existence of Uniform interpolants, an approach to computing interpolants when they exist, and tight bounds on their size.
Abstract: We study uniform interpolation and forgetting in the description logic ALC. Our main results are model-theoretic characterizations of uniform interpolants and their existence in terms of bisimulations, tight complexity bounds for deciding the existence of uniform interpolants, an approach to computing interpolants when they exist, and tight bounds on their size. We use a mix of model-theoretic and automata-theoretic methods that, as a by-product, also provides characterizations of and decision procedures for conservative extensions.
Patent•
27 Sep 2011
TL;DR: In this paper, the authors present a method of operating a centralized healthcare management system that includes a data translation map database and a central interpolation server computer interconnected to a computer network.
Abstract: A method of operating a centralized healthcare management system that includes a data translation map database and a central interpolation server computer interconnected to a computer network The central server references a data translation map database for a desired translation map that enables the central interpolation server to translate data records from a source format to a destination format
TL;DR: This method is well suited for a topology optimization problem with a design domain containing higher-order elements or non-quadrilateral elements and has the ability to yield mesh-independent solutions if the radius of the influence domain is reasonably specified.
TL;DR: In this article, a new problem formulation with mass constraint is proposed, which is based on the common idea of using volume constraint instead of adopting the common concept of using a volume constraint.
Abstract: This work is focused on the topology optimization of lightweight structures consisting of multiphase materials. Instead of adopting the common idea of using volume constraint, a new problem formulation with mass constraint is proposed. Meanwhile, recursive multiphase materials interpolation (RMMI) and uniform multiphase materials interpolation (UMMI) schemes are discussed and compared based on numerical tests and theoretical analysis. It is indicated that the nonlinearity of the mass constraint introduced by RMMI brings numerical difficulties to attain the global optimum of the optimization problem. On the contrary, the UMMI-2 scheme makes it possible to formulate the mass constraint in a linear form with separable design variables. One such formulation favors very much the problem resolution by means of mathematical programming approaches, especially the convex programming methods. Moreover, numerical analysis indicates that fully uniform initial weighting is beneficial to seek the global optimum when UMMI-2 scheme is used. Besides, the relationship between the volume constraint and mass constraint is theoretically revealed. The filtering technique is adapted to avoid the checkerboard pattern related to the problem with multiphase materials. Numerical examples show that the UMMI-2 scheme with fully uniform initial weighting is reliable and efficient to deal with the structural topology optimization with multiphase materials and mass constraint. Meanwhile, the mass constraint formulation is evidently more significant than the volume constraint formulation. Copyright © 2011 John Wiley & Sons, Ltd.
TL;DR: A family of recursive interpolation schemes based on B-spline representation and its inverse gradient weighting version is employed to enhance the accuracy of DIC analysis.
Abstract: The interpolation algorithm plays an essential role in the digital image correlation (DIC) technique for shape, deformation, and motion measurements with subpixel accuracies. At the present, little effort has been made to improve the interpolation methods used in DIC. In this Letter, a family of recursive interpolation schemes based on B-spline representation and its inverse gradient weighting version is employed to enhance the accuracy of DIC analysis. Theories are introduced, and simulation results are presented to illustrate the effectiveness of the method as compared with the common bicubic interpolation.
TL;DR: In this article, a detailed analysis of natural frequencies of laminated composite plates using the mesh-free moving Kriging interpolation method is presented, and the convergence of the method on the natural frequency is also given.
Abstract: A detailed analysis of natural frequencies of laminated composite plates using the meshfree moving Kriging interpolation method is presented. The present formulation is based on the classical plate theory while the moving Kriging interpolation satisfying the delta property is employed to construct the shape functions. Since the advantage of the interpolation functions, the method is more convenient and no special techniques are needed in enforcing the essential boundary conditions. Numerical examples with different shapes of plates are presented and the achieved results are compared with reference solutions available in the literature. Several aspects of the model involving relevant parameters, fiber orientations, lay-up number, length-to-length, stiffness ratios, etc. affected on frequency are analyzed numerically in details. The convergence of the method on the natural frequency is also given. As a consequence, the applicability and the effectiveness of the present method for accurately computing natural frequencies of generally shaped laminates are demonstrated.
TL;DR: This paper discusses linear methods for interpolation, including nearest neighbor, bilinear, bicubic, splines, and sinc interpolation and focuses on separable interpolation.
Abstract: We discuss linear methods for interpolation, including nearest neighbor, bilinear, bicubic, splines, and sinc interpolation. We focus on separable interpolation, so most of what is said applies to one-dimensional interpolation as well as N-dimensional separable interpolation. Source Code The source code (ANSI C), its documentation, and the online demo are accessible at the IPOL web page of this article 1 .
TL;DR: This paper shows that very accurate solutions can be achieved using the optimal value of the constant shape parameter in PDE problems solved with a multiquadric based RBF finite difference (RBF-FD) method.
9 Dec 2011
TL;DR: This paper compares techniques to combine diverse parallel corpora for domain-specific phrase-based SMT system training and focuses on phrase table fill-up: a method that effectively exploits background knowledge to improve model coverage, while preserving the more reliable information coming from the in-domain corpus.
Abstract: This paper compares techniques to combine diverse parallel corpora for domain-specific phrase-based SMT system training We address a common scenario where little in-domain data is available for the task, but where large background models exist for the same language pair In particular, we focus on phrase table fill-up: a method that effectively exploits background knowledge to improve model coverage, while preserving the more reliable information coming from the in-domain corpus We present experiments on an emerging transcribed speech translation task – the TED talks While performing similarly in terms of BLEU and NIST scores to the popular log-linear and linear interpolation techniques, filled-up translation models are more compact and easy to tune by minimum error training
TL;DR: The results show that the proposed algorithms outperform the best known reduced-rank schemes, while requiring lower complexity.
Abstract: This work proposes a blind adaptive reduced-rank scheme and constrained constant-modulus (CCM) adaptive algorithms for interference suppression in wireless communications systems. The proposed scheme and algorithms are based on a two-stage processing framework that consists of a transformation matrix that performs dimensionality reduction followed by a reduced-rank estimator. The complex structure of the transformation matrix of existing methods motivates the development of a blind adaptive reduced-rank constrained (BARC) scheme along with a low-complexity reduced-rank decomposition. The proposed BARC scheme and a reduced-rank decomposition based on the concept of joint interpolation, switched decimation and reduced-rank estimation subject to a set of constraints are then detailed. The proposed set of constraints ensures that the multipath components of the channel are combined prior to dimensionality reduction. We develop low-complexity joint interpolation and decimation techniques, stochastic gradient, and recursive least squares reduced-rank estimation algorithms. A model-order selection algorithm for adjusting the length of the estimators is devised along with techniques for determining the required number of switching branches to attain a predefined performance. An analysis of the convergence properties and issues of the proposed optimization and algorithms is carried out, and the key features of the optimization problem are discussed. We consider the application of the proposed algorithms to interference suppression in DS-CDMA systems. The results show that the proposed algorithms outperform the best known reduced-rank schemes, while requiring lower complexity.
TL;DR: This paper has proposed and designed a specific parallel IDW interpolation algorithm, incorporating both single process, multiple data (SPMD) and master/slave (M/S) programming modes, which indicates that the parallel algorithm can greatly reduce processing time and maximize speed and performance.
TL;DR: In this article, hourly precipitation was spatially interpolated with the multivariate geostatistical method kriging with external drift (KED) using additional information from topography, rainfall data from the denser daily networks and weather radar data.
Abstract: . Hydrological modelling of floods relies on precipitation data with a high resolution in space and time. A reliable spatial representation of short time step rainfall is often difficult to achieve due to a low network density. In this study hourly precipitation was spatially interpolated with the multivariate geostatistical method kriging with external drift (KED) using additional information from topography, rainfall data from the denser daily networks and weather radar data. Investigations were carried out for several flood events in the time period between 2000 and 2005 caused by different meteorological conditions. The 125 km radius around the radar station Ummendorf in northern Germany covered the overall study region. One objective was to assess the effect of different approaches for estimation of semivariograms on the interpolation performance of short time step rainfall. Another objective was the refined application of the method kriging with external drift. Special attention was not only given to find the most relevant additional information, but also to combine the additional information in the best possible way. A multi-step interpolation procedure was applied to better consider sub-regions without rainfall. The impact of different semivariogram types on the interpolation performance was low. While it varied over the events, an averaged semivariogram was sufficient overall. Weather radar data were the most valuable additional information for KED for convective summer events. For interpolation of stratiform winter events using daily rainfall as additional information was sufficient. The application of the multi-step procedure significantly helped to improve the representation of fractional precipitation coverage.
TL;DR: A momentum-conserving sub-stepping technique is intro- duced into the fluid-particle coupling procedure, so that problems with a wide range of time scales can be solved without resorting to excessively small time steps in the CFD solver.
Abstract: A robust and efficient solver coupling computational fluid dynamics (CFD) with discrete element method (DEM) is developed to simulate particle-laden flows in various physical settings. An interpolation algorithm suitable for unstructured meshes is proposed to translate between mesh-based Eulerian fields and particle-based La- grangian quantities. The interpolation scheme reducesthemesh-dependence of the av- eraging and interpolation procedures. In addition, the fluid-particle interaction terms are treated semi-implicitly in this algorithm to improve stability and to maintain accu- racy. Finally, it is demonstrated that sub-stepping is desirable forfluid-particle systems with small Stokes numbers. A momentum-conserving sub-stepping technique is intro- duced into the fluid-particle coupling procedure, so that problems with a wide range of time scales can be solved without resorting to excessively small time steps in the CFD solver. Several numerical examples are presented to demonstrate the capabilities of the solver and the merits of the algorithm.