pinecone/core-2020-05-10-deduplication huggingface dataset:
High Resolution Pursuit for Feature Extraction

AbstractRecently, adaptive approximation techniques have become popular for obtaining parsimonious representations of large classes of signals. These methods include method of frames, matching pursuit, and, most recently, basis pursuit. In this work, high resolution pursuit (HRP) is developed as an alternative to existing function approximation techniques. Existing techniques do not always efficiently yield representations which are sparse and physically interpretable. HRP is an enhanced version of the matching pursuit algorithm and overcomes the shortcomings of the traditional matching pursuit algorithm by emphasizing local fit over global fit at each stage. Further, the HRP algorithm has the same order of complexity as matching pursuit. In this paper, the HRP algorithm is developed and demonstrated on 1D functions. Convergence properties of HRP are also examined. HRP is also suitable for extracting features which may then be used in recognition
The Wavelet Element Method Part I. Construction and Analysis

AbstractThe Wavelet Element Method (WEM) combines biorthogonal wavelet systems with the philosophy of Spectral Element Methods in order to obtain a biorthogonal wavelet system on fairly general bounded domains in some Rn. The domain of interest is split into subdomains which are mapped to a simple reference domain, heren-dimensional cubes. Thus, one has to construct appropriate biorthogonal wavelets on the reference domain such that mapping them to each subdomain and matching along the interfaces leads to a wavelet system on the domain. In this paper we use adapted biorthogonal wavelet systems on the interval in such a way that tensor products of these functions can be used for the construction of wavelet bases on the reference domain. We describe the matching procedure in any dimensionnin order to impose continuity and prove that it leads to a construction of a biorthogonal wavelet system on the domain. These wavelet systems characterize Sobolev spaces measuring both piecewise and global regularity. The construction is detailed for a bivariate example and an application to the numerical solution of second order partial differential equations is given
Traveling Wavelets for the Simulation of a Hele–Shaw Flow with Surface Tension

AbstractThe singularity dynamics of two immiscible fluids in a Hele-Shaw cell is studied using the traveling wavelet method. The singularities of the conformal mapping of the flow are accurately approximated by the solutions of a system of ordinary differential equations for the position of couples of zeros and poles. This model also describes secondary tip-splitting instabilities and remains valid for values of the surface tension and time intervals where previous perturbative calculations do not apply
Error Estimates and Reproduction of Polynomials for Biorthogonal Local Trigonometric Bases

AbstractA general approach for biorthogonal local trigonometric bases in the two-overlapping setting was given by Chui and Shi. In this paper, we give error estimates for the approximation with such basis functions. In particular, it is shown that for a partition of the real axis into small intervals one obtains better approximation order if polynomials are reproduced locally. Furthermore, smooth trigonometric bases are constructed, which reproduce constants resp. linear functions by only one resp. a small number of basis functions for each interval
Biorthogonal Spline Wavelets on the Interval—Stability and Moment Conditions

AbstractThis paper is concerned with the construction of biorthogonal multiresolution analyses on [0, 1] such that the corresponding wavelets realize any desired order of moment conditions throughout the interval. Our starting point is the family of biorthogonal pairs consisting of cardinal B-splines and compactly supported dual generators on R developed by Cohen, Daubechies, and Feauveau. In contrast to previous investigations we preserve the full degree of polynomial reproduction also for the dual multiresolution and prove in general that the corresponding modifications of dual generators near the end points of the interval still permit the biorthogonalization of the resulting bases. The subsequent construction of compactly supported biorthogonal wavelets is based on the concept of stable completions. As a first step we derive an initial decomposition of the spline spaces where the complement spaces between two successive levels are spanned by compactly supported splines which form uniformly stable bases on each level. As a second step these initial complements are then projected into the desired complements spanned by compactly supported biorthogonal wavelets. Since all generators and wavelets on the primal and the dual sides have finitely supported masks, the corresponding decomposition and reconstruction algorithms are simple and efficient. The desired number of vanishing moments is implied by the polynomial exactness of the dual multiresolution. Again due to the polynomial exactness the primal and dual spaces satisfy corresponding Jackson estimates. In addition, Bernstein inequalities can be shown to hold for a range of Sobolev norms depending on the regularity of the primal and dual wavelets. Then it follows from general principles that the wavelets form Riesz bases forL2([0, 1]) and that weighted sequence norms for the coefficients of such wavelet expansions characterize Sobolev spaces and their duals on [0, 1] within a range depending on the parameters in the Jackson and Bernstein estimates. We conclude by addressing several issues concerning numerical implementation. In particular, we test the quantitative stability properties of corresponding multiscale transformations, indicate strategies for improving them, and present some numerical experiments
Harmonic Analysis of Neural Networks

AbstractIt is known that superpositions of ridge functions (single hidden-layer feedforward neural networks) may give good approximations to certain kinds of multivariate functions. It remains unclear, however, how to effectively obtain such approximations. In this paper, we use ideas from harmonic analysis to attack this question. We introduce a special admissibility condition for neural activation functions. The new condition is not satisfied by the sigmoid activation in current use by the neural networks community; instead, our condition requires that the neural activation function be oscillatory. Using an admissible neuron we construct linear transforms which represent quite general functionsfas a superposition of ridge functions. We develop&#x02022;•a continuous transform which satisfies a Parseval-like relation;&#x02022;•a discrete transform which satisfies frame bounds.Both transforms representfin a stable and effective way. The discrete transform is more challenging to construct and involves an interesting new discretization of time–frequency–direction space in order to obtain frame bounds for functions inL2(A) whereAis a compact set of Rn. Ideas underlying these representations are related to Littlewood–Paley theory, wavelet analysis, and group representation theory
Approximation Error for Quasi-Interpolators and (Multi-)Wavelet Expansions

AbstractWe investigate the approximation properties of general polynomial preserving operators that approximate a function into some scaled subspace of L2via an appropriate sequence of inner products. In particular, we consider integer shift-invariant approximations such as those provided by splines and wavelets, as well as finite elements and multi-wavelets which use multiple generators. We estimate the approximation error as a function of the scale parameterTwhen the function to approximate is sufficiently regular. We then present a generalized sampling theorem, a result that is rich enough to provide tight bounds as well as asymptotic expansions of the approximation error as a function of the sampling stepT. Another more theoretical consequence is the proof of a conjecture by Strang and Fix, which states the equivalence between the order of a multi-wavelet space and the order of a particular subspace generated by a single function. Finally, we consider refinable generating functions and use the two-scale relation to obtain explicit formulae for the coefficients of the asymptotic development of the error. The leading constants are easily computable and can be the basis for the comparison of the approximation power of wavelet and multi-wavelet expansions of a given orderL
A Multiresolution Method for Numerical Reduction and Homogenization of Nonlinear ODEs

AbstractThe multiresolution analysis (MRA) strategy for the reduction of a nonlinear differential equation is a procedure for constructing an equation directly for the coarse scale component of the solution. The MRA homogenization process is a method for building a simpler equation whose solution has the same coarse behavior as the solution to a more complex equation. We present two multiresolution reduction methods for nonlinear differential equations: a numerical procedure and an analytic method. We also discuss one possible appproach to the homogenization method
Experimental Evidence for Anomalous Scale Dependent Cascading Process in Turbulent Velocity Statistics

AbstractWe use a wavelet-based deconvolution method to extract some multiplicative cascading process from experimental turbulent velocity signals. We show that at the highest accessible Reynolds numbers, the experimental data do not allow us to discriminate between various phenomenological cascade models recently proposed to account for intermittency and their log-normal approximations. We further report evidence that velocity fluctuations are not scale invariant but possess more complex self-similarity properties that are likely to depend on the Reynolds number. We comment on the possible asymptotic validity of the multifractal description
Approximate Wavelets and the Approximation of Pseudodifferential Operators

AbstractThis paper studiesapproximate multiresolution analysisfor spaces generated by smooth functions providing high-order semi-analytic cubature formulas for multidimensional integral operators of mathematical physics. Since these functions satisfy refinement equations with any prescribed accuracy, methods from wavelet theory can be applied. We obtain an approximate decomposition of the finest scale space into almost orthogonal wavelet spaces. For the example of the Gaussian function we study some properties of the analytic prewavelets and describe the projection operators onto the wavelet spaces. The multivariate wavelets retain the property of the scaling function to provide efficient analytic expressions for the action of important integral operators, which leads to sparse and semi-analytic representations of these operators
Directional Wavelets Revisited: Cauchy Wavelets and Symmetry Detection in Patterns

AbstractThe analysis of oriented features in images requires two-dimensional directional wavelets. Among these, we study in detail the class of Cauchy wavelets, which are strictly supported in a (narrow) convex cone in spatial frequency space. They have excellent angular selectivity, as shown by a standard calibration test, and they have minimal uncertainty. In addition, we present a new application of directional wavelets, namely a technique for determining the symmetries of a given pattern with respect to rotations and dilation
From Local Cosine Bases to Global Harmonics

AbstractIn this paper, the reproduction of trigonometric polynomials with two-overlapping local cosine bases is investigated. This study is motivated by the need to represent most effectively a Fourier series in the form of a localized cosine series for the purpose of local analysis, thus providing a vehicle for the transition from classical harmonic analysis to analysis by Wilson-type wavelets. It is shown that there is one and only one class, which is a one-parameter family, of window functions that allows pointwise reproduction of all global harmonics, where the parameter is the order of smoothness of the window functions. It turns out that this class of window functions is also optimal in the sense that all global harmonics are reproduced by using a minimal number of the local trigonometric basis functions
Tomography, Approximate Reconstruction, and Continuous Wavelet Transforms

AbstractIt has been recognized for some time now that certain high-frequency information concerning planar densities f in a neighborhood of a point can be recovered from data which consist of averages of f over lines that are relatively close to that point. The wavelet transform of f is a classical tool for analyzing local frequency content. In this article we introduce continuous wavelet transforms which are particularly well suited to producing high-resolution local reconstructions from local data of the type described above. We also show how such transforms can be realized numerically via simple modifications of well-established convolution backprojection-type algorithms.As part of our development we review the concepts of “local tomography” and “pseudolocal tomography” introduced by several authors and indicate that, in effect, these notions basically involve the computation of a wavelet transform. The results in this paper are based on the observation that Radon's classical inversion formula is a summability formula with an integrable convolution-type summability kernel
Wavelet Filtering for Filtered Backprojection in Computed Tomography

AbstractIn this paper, we present a characterization of MRA biorthogonal wavelet filters with full frequency supports. Based on this characterization, it is established that wavelet ramp filters are biorthogonal wavelets if the original wavelets are sufficiently regular. An efficient subband coding algorithm is developed for wavelet filtering in filtered backprojection, which is the most popular method in computed tomography (CT). Computer simulation suggests that this wavelet filtering process is a useful tool for improving image quality and reducing computational time in local CT reconstruction
Locally Supported Wavelets on Manifolds with Applications to the 2D Sphere

AbstractIn this paper we present a construction principle for locally supported wavelets on manifolds once a multiresolution analysis is given. The wavelets provide a stable (or unconditional) basis for a scale of Sobolev spaces Hs, 0 ≤ s ≤ s̄. We examine a fast wavelet transform with almost optimal complexity. For the two-dimensional sphere we construct a multiresolution analysis generated by continuous splines that are bilinear with respect to some special spherical grid. In our approach the poles are not exceptional points concerning the approximation power or the stability of the wavelet basis. Finally we present some numerical applications to singularity detection and the analysis of observed atmospheric data
A Contribution to Wavelet-Based Subgrid Modeling

AbstractA systematic technique for the derivation of subgrid scale models in the numerical solution of partial differential equations is described. The technique is based on Haar wavelet projections of the discrete operator followed by a sparse approximation. As numerical testing suggests, the resulting numerical method will accurately represent subgrid scale phenomena on a coarse grid. Applications to numerical homogenization and wave propagation in materials with subgrid inhomogeneities are presented
Compactly Supported Orthogonal Symmetric Scaling Functions

AbstractDaubechies (1988, Comm. Pure Appl. Math.41, 909–996) showed that, except for the Haar function, there exist no compactly supported orthogonal symmetric scaling functions for the dilation q = 2. Nevertheless, such scaling functions do exist for dilations q>2 (as evidenced by Chui and Lian's construction (1995, Appl. Comput. Harmon. Anal.2, 68–84) for q = 3); these functions are the main object of this paper. We construct new symmetric scaling functions and introduce the “Batman” family of continuous symmetric scaling functions with very small supports. We establish the exact smoothness of the “Batman” scaling functions using the joint spectral radius technique
Compactly Supported Wavelets Based on Almost Interpolating and Nearly Linear Phase Filters (Coiflets)

AbstractNew compactly supported wavelets for which both the scaling and wavelet functions have a high number of vanishing moments are presented. Such wavelets are a generalization of the so-called coiflets and they are useful in applications where interpolation and linear phase are of importance. The new approach is to parameterize coiflets by the first moment of the scaling function. By allowing noninteger values for this parameter, the interpolation and linear phase properties of coiflets are optimized. Besides giving a new definition for coiflets, a new system for the filter coefficients is introduced. This system has a minimal set of defining equations and can be solved with algebraic or numerical methods. Examples are given of the various types of coiflets that can be obtained from such systems. The corresponding filter coefficients are listed and their properties are illustrated
Wavelets on the 2-Sphere: A Group-Theoretical Approach

AbstractWe present a purely group-theoretical derivation of the continuous wavelet transform (CWT) on the 2-sphere S2, based on the construction of general coherent states associated to square integrable group representations. The parameter space X of our CWT is the product of SO(3) for motions and R+* for dilations on S2, which are embedded into the Lorentz group SO0(3, 1) via the Iwasawa decomposition, so that X ≃ SO0(3, 1) M Y S O L N, where N ≃ C. We select an appropriate unitary representation of SO0(3, 1) acting in the space L2(S2, d μ) of finite energy signals on S2. This representation is square integrable over X; thus it yields immediately the wavelets on S2 and the associated CWT. We find a necessary condition for the admissibility of a wavelet, in the form of a zero mean condition. Finally, the Euclidean limit of this CWT on S2 is obtained by redoing the construction on a sphere of radius R and performing a group contraction for R → ∞. Then the parameter space goes into the similitude group of R2 and one recovers exactly the CWT on the plane, including the usual zero mean necessary condition for admissibility
A Direct Interpolation Method for Irregular Sampling

AbstractIn this paper, band-limited functions are reconstructed from their values taken at a sequence of irregularly spaced sample points. We use a modified Lagrange formula, which is attributed to Boas and Bernstein. The formula used in this paper differs from the classical Boas–Bernstein formula in the following way. Instead of using infinite canonical products with respect to the whole sequence of sample points, we use canonical products with respect to sequences of sample points which are irregularly spaced only on finite intervals. Estimates for the truncation error of this reconstruction method are given
Riesz Bases and Multiresolution Analyses

AbstractRecently we found a family of nearly orthonormal affine Riesz bases of compact support and arbitrary degrees of smoothness, obtained by perturbing the one-dimensional Haar mother wavelet using B-splines. The mother wavelets thus obtained are symmetric and given in closed form, features which are generally lacking in the orthogonal case. We also showed that for an important subfamily the wavelet coefficients can be calculated in O(n) steps, just as for orthogonal wavelets. It was conjectured by Aldroubi, and independently by the author, that these bases cannot be obtained by a multiresolution analysis. Here we prove this conjecture. The work is divided into four sections. The first section is introductory. The main feature of the second is simple necessary and sufficient conditions for an affine Riesz basis to be generated by a multiresolution analysis, valid for a large class of mother wavelets. In the third section we apply the results of the second section to several examples. In the last section we show that our bases cannot be obtained by a multiresolution analysis
Symmetric–Antisymmetric Orthonormal Multiwavelets and Related Scalar Wavelets

AbstractFor compactly supported symmetric–antisymmetric orthonormal multiwavelet systems with multiplicity 2, we first show that any length-2Nmultiwavelet system can be constructed from a length-(2N+1) multiwavelet system and vice versa. Then we present two explicit formulations for the construction of multiwavelet functions directly from their associated multiscaling functions. This is followed by the relationship between these multiscaling functions and the scaling functions of related orthonormal scalar wavelets. Finally, we present two methods for constructing families of symmetric–antisymmetric orthonormal multiwavelet systems via the construction of the related scalar wavelets
Criteria for Hierarchical Bases in Sobolev Spaces

AbstractSeveral approaches to solving elliptic problems numerically are based on hierarchical Riesz bases in Sobolev spaces. We are interested in determining the exact range of Sobolev exponents for which a system of compactly supported functions derived from a multiresolution analysis forms such a Riesz basis. This involves determining the smoothness of the dual system. The elements of the dual system typically consist of noncompactly supported functions, whose smoothness can be treated by extending the results of 7, 9, and 22. We show how to determine the exact range of Sobolev exponents in the multivariate case, both theoretically and numerically, from spectral properties of transfer operators. This technique is applied to several bases deriving from linear finite elements which have been proposed in the literature. For 29hierarchical basis, we find that it forms a Riesz basis in Hs(Rd) for −0.990236…<s<3/2
Fast Algorithm for Best Anisotropic Walsh Bases and Relatives

AbstractWe introduce a new family of Nlog Nbest basis search algorithms for functions of more than one variable. These algorithms search the collection of anisotropic wavelet packet and cosine packet bases and output a minimum entropy basis for a given function. These algorithms are constructed after treating the model problem of computing best Walsh packet bases. Several intermediate algorithms for conducting mixed isotropic/anisotropic best basis searches in the function's various coordinate directions are also presented
Wavelets of Wilson Type with Arbitrary Shapes

AbstractMotivated by the Gaussian bases of Coifman and Meyer and the need of bases with arbitrary shapes which may have to be different at different locations, we derive complete characterizations of window functions and their duals for localization of all appropriate sines and cosines that give rise to biorthogonal Schauder bases, Riesz bases, and frames. In addition, when the window functions are simply integer translates of a single window function, we give an explicit formulation of its dual that generates the biorthogonal basis, regardless of the shape and support of the window function. Besides the Coifman–Meyer Gaussian bases, several other examples of wavelets of Wilson type are given
A Construction of Biorthogonal Functions to B-Splines with Multiple Knots

AbstractWe present a construction of a refinable compactly supported vector of functions which is biorthogonal to the vector of B-splines of a given degree with multiple knots at the integers with prescribed multiplicity. The construction is based on Hermite interpolatory subdivision schemes, and on the relation between B-splines and divided differences. The biorthogonal vector of functions is shown to be refinable, with a mask related to that of the Hermite scheme. For simplicity of presentation the special (scalar) case, corresponding to B-splines with simple knots, is treated separately