pinecone/core-2020-05-10-deduplication huggingface dataset:
Validity of WH-Frame Bound Conditions Depends on Lattice Parameters

AbstractIn the study of Weyl–Heisenberg frames the assumption of having a finite frame upper bound appears recurrently. In this article it is shown that it actually depends critically on the time–frequency lattice used. Indeed, for any irrational α>0 we can construct a smooth g∈ L2(R) such that for any two rationals a >0 and b >0 the collection (gna, mb)n, m∈ Zof time–frequency translates of ghas a finite frame upper bound, while for any β>0 and any rational c> 0 the collection (gncα, mβ)n, m∈ Zhas no such bound. It follows from a theorem of I. Daubechies, as well as from the general atomic theory developed by Feichtinger and Gröchenig, that for any nonzero g∈ L2(R) which is sufficiently well behaved, there exist ac>0, bc>0 such that (gn a, m b)n, m∈ Zis a frame whenever 0 < a < ac, 0 < b < bc. We present two examples of a nonzero g∈ L2(R), bounded and supported by (0, 1), for which such numbers ac, bcdo not exist. In the first one of these examples, the frame bound equals 0 for all a >0, b >0, b <1. In the second example, the frame lower bound equals 0 for all aof the form l· 3− kwith l, k∈ N and all b, 0 < b <1, while the frame lower bound is at least 1 for all aof the form (2 m)− 1with m∈ N and all b, 0 < b <1
The Wavelet Element Method Part II. Realization and Additional Features in 2D and 3D

AbstractThe Wavelet Element Method (WEM) provides a construction of multiresolution systems and biorthogonal wavelets on fairly general domains. These are split into subdomains that are mapped to a single reference hypercube. Tensor products of scaling functions and wavelets defined on the unit interval are used on the reference domain. By introducing appropriate matching conditions across the interelement boundaries, a globally continuous biorthogonal wavelet basis on the general domain is obtained. This construction does not uniquely define the basis functions but rather leaves some freedom for fulfilling additional features. In this paper we detail the general construction principle of the WEM to the 1D, 2D, and 3D cases. We address additional features such as symmetry, vanishing moments, and minimal support of the wavelet functions in each particular dimension. The construction is illustrated by using biorthogonal spline wavelets on the interval
A Characterization of Affine Dual Frames in L2(Rn)

AbstractWe give a characterization of all (quasi)affine frames in L2(Rn) which have a (quasi)affine dual in terms of the two simple equations in the Fourier transform domain. In particular, if the dual frame is the same as the original system, i.e., it is a tight frame, we obtain the well-known characterization of wavelets. Although these equations have already been proven under some special conditions we show that these characterizations are valid without any decay assumptions on the generators of the affine system
An Algebraic Structure of Orthogonal Wavelet Space

AbstractIn this paper we study the algebraic structure of the space of compactly supported orthonormal wavelets over real numbers. Based on the parameterization of wavelet space, one can define a parameter mapping from the wavelet space of rank 2 (or 2-band, scale factor of 2) and genus gto the (g−1) dimensional real torus (the products of unit circles). By the uniqueness and exactness of factorization, this mapping is well defined and one-to-one. Thus we can equip the rank 2 orthogonal wavelet space with an algebraic structure of the torus. Because of the degenerate phenomenon of the paraunitary matrix, the parameterization map is not onto. However, there exists an onto mapping from the torus to the closure of the wavelet space. And with such mapping, a more complete parameterization is obtained. By utilizing the factorization theory, we present a fast implementation of discrete wavelet transform (DWT). In general, the computational complexity of a rank morthogonal DWT is O(m2g). In this paper we start with a given scaling filter and construct additional (m−1) wavelet filters so that the DWT can be implemented in O(mg). With a fixed scaling filter, the approximation order, the orthogonality, and the smoothness remain unchanged; thus our fast DWT implementation is quite general
Compactly Supported Tight Frames Associated with Refinable Functions

AbstractIt is well known that in applied and computational mathematics, cardinal B-splines play an important role in geometric modeling (in computer-aided geometric design), statistical data representation (or modeling), solution of differential equations (in numerical analysis), and so forth. More recently, in the development of wavelet analysis, cardinal B-splines also serve as a canonical example of scaling functions that generate multiresolution analyses of L2(−∞,∞). However, although cardinal B-splines have compact support, their corresponding orthonormal wavelets (of Battle and Lemarie) have infinite duration. To preserve such properties as self-duality while requiring compact support, the notion of tight frames is probably the only replacement of that of orthonormal wavelets. In this paper, we study compactly supported tight frames Ψ={ψ1,…,ψN} for L2(−∞,∞) that correspond to some refinable functions with compact support, give a precise existence criterion of Ψ in terms of an inequality condition on the Laurent polynomial symbols of the refinable functions, show that this condition is not always satisfied (implying the nonexistence of tight frames via the matrix extension approach), and give a constructive proof that when Ψ does exist, two functions with compact support are sufficient to constitute Ψ, while three guarantee symmetry/anti-symmetry, when the given refinable function is symmetric
Fast Functional Magnetic Resonance Imaging via Prolate Wavelets

AbstractOur objective is to improve the time-resolution of functional magnetic resonance imaging by sampling only a small fraction of the Fourier transform of the spin density and using a prolate wavelet (P-wavelet) filter to approximately obtain not the usual susceptibility map, but instead the integral, I(t), of the difference in susceptibility between task and pretask, over a prespecified region of interest in the brain at successive time-points, t. This space–time trade-off thus allows us to obtain, at high time-resolution, the total activity, I(t), in a specified region, B, of the brain which processes the specific stimulus or task to learn or verify where the brain function takes place. We find that for a typical region, B, of the brain, say describing the hippocampus, believed to be involved in memory, consisting of say 100 points in a 64×64 brain image space, that our optimal choice of Fourier sampling region, A, has a=400 points, which then gives a 10-fold speed-up compared to the usual method of sampling, since the usual sampling needs 642 points which exceeds 10 times 400. Of course we get this speed-up at the price of spatial resolution. Even faster sampling of the integral of the susceptibility difference ought to be possible for a set B of 100 points in this level of pixelization. Once A is fixed, the mathematical problem of choosing the optimal P-wavelet filter can be viewed as a natural generalization to 2 and 3 dimensions of the theory of prolate spheroidal wave functions, due to Landau, Pollak, and Slepian. The first prolate spheroidal wave function for fixed A, B is that function φ which is maximally concentrated on B and whose Fourier transform vanishes except on A. The problem of choosing the sampling region A, for fixed size a=∣A∣, to maximize the concentration on the given B remains open and is probably NP-complete, i.e., probably involves exponential search. However, we give a very simple heuristic for choosing a set A, which seems to give excellent results in the examples we have studied, and we give the optimal P-wavelet filter, φ=φB, for the choice of A, based on our heuristic.The heuristic is to take as it A the scaled polar set of B, i.e., A=k: maxx,y∈B(k,x−y)<c, (1) where c is chosen so that the size of A is a, as desired to achieve a given time-resolution for I(t). We give evidence to support our claim that A gives good results
Efficient Computation of Oscillatory Integrals via Adaptive Multiscale Local Fourier Bases

AbstractThe integral ∫0Leiνφ(s,t)f(s)dswith a highly oscillatory kernel (large ν, ν is up to 2000) is considered. This integral is accurately evaluated with an improved trapezoidal rule and effectively transcribed using local Fourier basis and adaptive multiscale local Fourier basis. The representation of the oscillatory kernel in these bases is sparse. The coefficients after the application of local Fourier transform are smoothed. Sometimes this enables us to obtain further compression with wavelets
Zeros of a Mask Function and the Orthogonality of the Related Scaling Function

AbstractSuppose that m(ξ) is a trigonometric polynomial with period 1 satisfying m(0)=1 and ∣m(ξ)∣2+∣m(ξ+ 12 )∣2=1 for all ξ in R. Let ϕ̂(ξ)=∏∞j=1m(2−jξ), ϕ(x)=∫+∞−∞ϕ̂(ξ)e2πixξdξ. The orthogonality of ϕ(x), i.e., ∫+∞−∞ϕ(x−m)ϕ(x−n) dx=δm,n, is related to the zeros of m(ξ). In 1995, A. Cohen and R. D. Ryan, “Wavelets and Multiscale Signal Processing,” Chapman & Hall, proved that if m(ξ) has no zeros in [− 16 , 16 ], then ϕ(x) is an orthogonal function. In (X. Zhou and W. Su, Appl. Comput. Harmon. Anal. 8, 197–202 (2000)) we proved that if m(ξ) has no zeros in [− 110 , 110 ] and ∣m( 16 )∣−∣m(− 16 )∣>0, then ϕ(x) is also an orthogonal function. A natural question, then, is whether this procedure can be extended to arbitrarily small intervals, i.e., whether for any Δ∈(0, 12 ) there exists a finite set ZΔ such that the orthogonality of ϕ(x) is ensured by the requirement that ∣m(ξ)∣>0 for ∣ξ∣≤Δ and for some ξ∈ZΔ. In this paper, we show that this is true if and only if Δ exceeds a strictly positive Δ0 which we derive explicitly
Spectral Approximation of the Free-Space Heat Kernel

AbstractMany problems in applied mathematics, physics, and engineering require the solution of the heat equation in unbounded domains. Integral equation methods are particularly appropriate in this setting for several reasons: they are unconditionally stable, they are insensitive to the complexity of the geometry, and they do not require the artificial truncation of the computational domain as do finite difference and finite element techniques. Methods of this type, however, have not become widespread due to the high cost of evaluating heat potentials. When mpoints are used in the discretization of the initial data, Mpoints are used in the discretization of the boundary, and Ntime steps are computed, an amount of work of the order O(N2M2+NMm) has traditionally been required. In this paper, we present an algorithm which requires an amount of work of the order O(NMlog M+mlog m) and which is based on the evolution of the continuousspectrum of the solution. The method generalizes an earlier technique developed by Greengard and Strain (1990, Comm. Pure Appl. Math.43, 949) for evaluating layer potentials in bounded domains
Translation Invariance of Orthogonal Multiresolution Analyses of L2(R)

AbstractTwo numerical characteristics for estimation of translation invariance of a multiresolution analysis (MRA) of L2(R) are proposed. These two characteristics are defined either through the scaling function or through the scaling symbol of the MRA. Some simple conditions under which both translation invariance characteristics have similar numerical values were found. Translation invariance of Daubechies' compactly supported scaling function, defined by 2N-length low-pass filter, is of order O(N−1/2). We have designed some compactly supported scaling functions having a translation invariance O(N−1) and proved that this order of decay is optimal. For any MRA of L2(R) a Heisenberg translation uncertainty principle is proved
A Construction of Orthogonal Compactly Supported Multiwavelets on R2

AbstractThis paper will provide the general construction of the continuous, orthogonal, compactly supported multiwavelets associated with a class of continuous, orthogonal, compactly supported scaling functions that contain piecewise linears on a uniform triangulation of R2. This class of scaling functions is a generalization of a set of scaling functions first constructed by Donovan, Geronimo, and Hardin. A specific set of scaling functions and associated multiwavelets with symmetry properties will be constructed
Almost Everywhere Behavior of General Wavelet Shrinkage Operators

AbstractWavelet shrinkage estimators are obtained by applying a shrinkage rule on the empirical wavelet coefficients. Such simple estimators are now well explored and widely used in wavelet-based nonparametrics. Results of Tao (1996, Appl. Comput. Harmon. Anal.3, 384–387) demonstrated that hard and soft thresholding shrinkage estimators absolutely converge almost everywhere to the original function when the threshold value goes to zero. Such natural and intuitive behavior of threshold estimators is expected, yet this result does not translate to the Fourier expansions. In this paper we show that almost everywhere convergence of shrinkage estimators holds for a range of shrinkage rules, not necessarily thresholding, subject to some mild technical conditions. Comments about the norm convergence are provided as well
Smooth Multiwavelet Duals of Alpert Bases by Moment-Interpolating Refinement

AbstractUsing refinement subdivision techniques, we construct smooth multiwavelet bases for L2(R) and L2([0,1]) which are in an appropriate sense dual to Alpert orthonormal multiwavelets. Our new multiwavelets allow one to easily give smooth reconstructions of a function purely from knowledge of its local moments. At the heart of our construction is the concept of moment-interpolating (MI) refinement schemes, which interpolate sequences from coarse scales to finer scales while preserving the underlying local moments on dyadic intervals. We show that MI schemes have smooth refinement limits. Our proof technique exhibits an intimate intertwining relation between MI schemes and Hermite schemes. This intertwining relation is then used to infer knowledge about moment-interpolating schemes from knowledge about Hermite schemes. Our MI multiwavelets make Riesz bases for L2 and unconditional bases of a variety of smoothness spaces, so they can efficiently represent smooth functions. We here derive an algorithm which rapidly develops a piecewise polynomial fit to data by recursive dyadic partitioning and then rapidly produces a smooth reconstruction with matching local moments on pieces of the partition. This avoids the blocking effect suffered by piecewise polynomial fitting
Orthonormal Wavelets and Tight Frames with Arbitrary Real Dilations

AbstractThe objective of this paper is to establish a complete characterization of tight frames, and particularly of orthonormal wavelets, for an arbitrary dilation factor a>1, that are generated by a family of finitely many functions in L2:=L2(R). This is a generalization of the fundamental work of G. Weiss and his colleagues who considered only integer dilations. As an application, we give an example of tight frames generated by one single L2 function for an arbitrary dilation a>1 that possess “good” time-frequency localization. As another application, we also show that there does not exist an orthonormal wavelet with good time-frequency localization when the dilation factor a>1 is irrational such that aj remains irrational for any positive integer j. This answers a question in Daubechies' Ten Lectures book for almost all irrational dilation factors. Other applications include a generalization of the notion of s-elementary wavelets of Dai and Larson to s-elementary wavelet families with arbitrary dilation factors a>1. Generalization to dual frames is also discussed in this paper
Walsh-Type Wavelet Packet Expansions

AbstractWe consider a family of basic nonstationary wavelet packets generated using the Haar filters except for a finite number of scales where we allow the use of arbitrary filters. Such a system, which we call a system of Walsh-type wavelet packets, can be considered as a smooth generalization of the Walsh functions. We show that the basic Walsh-type wavelet packets share a number of metric properties with the Walsh system. We prove that the system constitutes a Schauder basis for Lp(R), 1<p<∞, and we construct an explicit function in L1(R) for which the expansion fails. Then we prove that expansions of Lp(R)-functions, 1<p<∞, in the Walsh-type wavelet packets converge pointwise a.e. Finally, we prove that the analogous results are true for periodic Walsh-type wavelet packets in Lp[0,1)
Construction of Biorthogonal Multiwavelets Using the Lifting Scheme

AbstractThe lifting scheme has been found to be a flexible method for constructing scalar wavelets with desirable properties. Here it is extended to the construction of multiwavelets. It is shown that any set of compactly supported biorthogonal multiwavelets can be obtained from the Lazy matrix filters with a finite number of lifting steps. As an illustration of the general theory, compactly supported biorthogonal multiwavelets with optimum time–frequency resolution are constructed. In addition, experimental results of applying these multiwavelets to image compression are presented
Algorithms for Fast Convolutions on Motion Groups

AbstractIn this paper we apply techniques from noncommutative harmonic analysis to the development of fast algorithms for the computation of convolution integrals on motion groups. In particular, we focus on the group of rigid-body motions in 3-space, which is denoted here as SE(3). The general theory of irreducible unitary representations (IURs) of the 3D motion group is described briefly. Using IURs in operator form, we write the Fourier transform of functions on the motion group as an integral over the product space SE(3)×S2. The integral form of the Fourier transform matrix elements allows us to apply fast Fourier transform (FFT) methods developed previously for R3, S2, and SO(3) to speed up considerably the computation of convolutions of functions on SE(3). Such convolutions have been shown to play an important role in a number of engineering disciplines. An algorithm for the fast computation of the Fourier transform is given and its complexity is discussed. The Fourier transform for the 3D “discrete motion group” (semi-direct product of the icosahedral group with the translation group) is also developed and the computational complexity is discussed
On Frame Wavelets Associated with Frame Multiresolution Analysis

AbstractWe present some necessary and sufficient conditions for a frame multiresolution analysis (FMRA) to admit a frame wavelet whose dyadic dilations and integer translates generate a frame for L2(R) and propose a construction of a wavelet, if it exists, which reduces to the classical orthonormal wavelet in the case of orthonormal multiresolution analysis. We also show that there always exists a frame for the detail space W0 of a frame MRA consisting of the integer translates of two functions and give an explicit construction of them
M-Band Burt–Adelson Biorthogonal Wavelets

AbstractFor every integer M>2 we introduce a new family of biorthogonal MRAs with dilation factor M, generated by symmetric scaling functions with small support. This construction generalizes Burt–Adelson biorthogonal 2-band wavelets. For M∈{3,4} we are able to find simple explicit expressions for two different families of wavelets associated with these MRAs: one with better localization and the other with interesting symmetry–antisymmetry properties. We study the regularity of our scaling functions by determining their Sobolev exponent, for every value of the parameter and every M. We also study the critical exponent when M=3
A Construction of C1-Wavelets on the Two-Dimensional Sphere

AbstractIn this paper a construction of C1-wavelets on the two-dimensional sphere is presented. First, we focus on the construction of a multiresolution analysis leading to C1-functions on S2. We show refinability of the constructed tensor product generators. Second, for the wavelet construction we employ a factorization of the refinement matrices which leads to refinement matrices characterizing complement spaces. With this method we achieve an initial stable completion. A desired stable completion can be gained by lifting the initial stable completion. The result is a biorthogonal wavelet basis leading to C1-functions on the sphere
Information Cost Functions

AbstractA best orthogonal basis for a vector is selected from a library to minimize a cost function of the expansion coefficients. How it depends on the cost function and under what conditions it provides the fastest nonlinear approximation are still open questions which we partially answer in this paper. Squared expansion coefficients may be considered a discrete probability density function, or pdf. We apply some inequalities for pdfs to obtain three positive results and two counterexamples. We use the notion of subexponentiality, derived from the classical proof of an entropy inequality, to derive a number of curious inequalities relating different information costs of a single pdf. We then generalize slightly the classical result that one pdf majorizes another if it is cheaper with respect to a large-enough set of information cost functions. Finally, we present inequalities that bracket any information cost for a pdf between two functions of norms of the pdf, plus a counterexample showing that our result has a certain optimality. Another counterexample shows that, unfortunately, the set of norm-type pdfs is not large enough to imply majorization. We conclude that all information cost functions are weakly comparable to norms, but this is not quite enough to guarantee in general that the cheapest-norm pdf majorizes
Smooth Wavelet Tight Frames with Zero Moments

AbstractThis paper considers the design of wavelet tight frames based on iterated oversampled filter banks. The greater design freedom available makes possible the construction of wavelets with a high degree of smoothness, in comparison with orthonormal wavelet bases. In particular, this paper takes up the design of systems that are analogous to Daubechies orthonormal wavelets—that is, the design of minimal length wavelet filters satisfying certain polynomial properties, but now in the oversampled case. Gröbner bases are used to obtain the solutions to the nonlinear design equations. Following the dual-tree DWT of Kingsbury, one goal is to achieve near shift invariance while keeping the redundancy factor bounded by 2, instead of allowing it to grow as it does for the undecimated DWT (which is exactly shift invariant). Like the dual tree, the overcomplete DWT described in this paper is less shift-sensitive than an orthonormal wavelet basis. Like the examples of Chui and He, and Ron and Shen, the wavelets are much smoother than what is possible in the orthonormal case
Nonseparable, Compactly Supported Interpolating Refinable Functions with Arbitrary Smoothness

AbstractIn this paper we construct families of compactly supported nonseparable interpolating refinable functions with arbitrary smoothness (or regularity). The symbols for the newly constructed scaling functions are given by a simple formula related to the Bernstein polynomials. The emphasis of the paper is to show that under an easy-to-verify geometric condition these families satisfy Cohenrs condition, and they have arbitrarily high regularity. Furthermore, the constructed scaling functions satisfy, under the same geometrical condition, the Strang–Fix conditions of arbitrarily high order, which implies that corresponding interpolating schemes have arbitrarily high accuracy