紧支半正交小波的构造

§ 1　IntroductionIt is well known that wavelets with dilation factor two can be constructed from amultiresolution analysis and lots of their applications have been found.It is also knownthat wavelets with general dilation factor M may be constructed from the multiresolutionanalysis with dilation factor M≥ 2 [1— 3] .Wavelets are closely related to M-channel filterbanks[4,5] ,so some important applications such as in audio coding and communication aredeveloped.Semi-orthogonal wavelets const…     

构造三角样条小波的一种新方法

首先给出了三角样条函数及其性质,然后在此基础上给出了一种构造三角样条小波的新方法.该方法简单易行,而且构造出的小波具有许多良好的性质,如对称性(具有线性相位或广义线性相位)、良好的时频局部特性、短支集及半正交性等,这些对信号处理是非常重要的.     

Designing Multiresolution Analysis-type Wavelets and Their Fast Algorithms

Often, the Dyadic Wavelet Transform is performed and implemented with the Daubechies wavelets, the Battle-Lemarie wavelets, or the splines wavelets, whereas in continuous-time wavelet decomposition a much larger variety of mother wavelets is used. Maintaining the dyadic time-frequency sampling and the recursive pyramidal computational structure, we present various methods for constructing wavelets ψ_wanted, with some desired shape and properties and which are associated with semi-orthogonal multiresolution analyses. We explain in detail how to design any desired wavelet, starting from any given multiresolution analysis. We also explicitly derive the formulae of the filter bank structure that implements the designed wavelet. We illustrate these wavelet design techniques with examples that we have programmed with Matlab routines.     

具有特殊伸缩矩阵的Parseval框架小波集的结构

揭示具有特殊伸缩矩阵的Parseval框架小波集的丰富结构.借助于平移不变空间和维数函数,研究了具有特殊伸缩矩阵M的Parseval框架小波(M-PFW)、半正交M-PFW和MRA M-PFW的各种性质,探讨了M-PFW集合的各种子类,给出了这些子类的构造性算例.     

有限元多尺度小波

利用有限元插值和多尺度分析理论构造出了有限元多尺度小波.这些小波函数集许多优良性质于一身,如固定的短支集、高阶的消失矩、半正交性及正则性等.     

The characterization of a class of multivariate MRA and semi-orthogonal parseval frame wavelets

Let A be a d × d expansive matrix with ∣detA∣ = 2. This paper addresses Parseval frame wavelets (PFWs) in the setting of reducing subspaces of L²(R^d). We prove that all semi-orthogonal PFWs (semi-orthogonal MRA PFWs) are precisely the ones with their dimension functions being non-negative integer-valued (0 or 1). We also characterize all MRA PFWs. Some examples are provided.     

Compactly supported wavelet bases for Sobolev spaces

In this paper we investigate compactly supported wavelet bases for Sobolev spaces. Starting with a pair of compactly supported refinable functions φ and in satisfying a very mild condition, we provide a general principle for constructing a wavelet ψ such that the wavelets ψ_jk:=2^j/2ψ(2^j·−k) ( ) form a Riesz basis for . If, in addition, φ lies in the Sobolev space , then the derivatives 2^j/2ψ^(m)(2^j·−k) ( ) also form a Riesz basis for . Consequently, is a stable wavelet basis for the Sobolev space . The pair of φ and are not required to be biorthogonal or semi-orthogonal. In particular, φ and can be a pair of B-splines. The added flexibility on φ and allows us to construct wavelets with relatively small supports.     

Tight Frame Wavelets, their Dimension Functions, MRA Tight Frame Wavelets and Connectivity Properties

This is a continuation of our study of generalized low pass filters and MRA frame wavelets. In this first study we concentrated on the construction of such functions. Here we are particularly interested in the role played by the dimension function. In particular we characterize all semi-orthogonal Tight Frame Wavelets (TFW) by showing that they correspond precisely to those for which the dimension function is non-negative integer-valued. We also show that a TFW arises from our MRA construction if and only if the dimension of a particular linear space is either zero or one. We present many examples. In addition we obtain a result concerning the connectivity of TFW's that are MSF tight frame wavelets.     

L~2(R~n)上的半正交多小波框架

本文研究L2(Rn)上伸缩矩阵A满足|detA|1的半正交多小波框架.本文得到半正交和严格半正交框架的一系列性质及刻画.本文证明半正交Parseval多小波框架与广义多分辨分析(GMRA)Parseval多小波框架是等价的.特别地,本文利用最小频率支撑(MSF)多小波框架和小波集,构造若干半正交多小波框架的例子.     

The Poisson problem in a domain with a cut

With the help of harmonic wavelets, we study the behavior of solutions to the Poisson problem in an elliptic ring when the interior bound shrinks to a segment. It is demonstrated that only partial derivatives of a solution have unbounded singularities near the ends of this segment.     

Directional wavelets on n-dimensional spheres

Directional Poisson wavelets, being directional derivatives of Poisson kernel, are introduced on n-dimensional spheres. It is shown that, slightly modified and together with another wavelet family, they are an admissible wavelet pair according to the definition derived from the theory of approximate identities. We investigate some of the properties of directional Poisson wavelets, such as recursive formulae for their Fourier coefficients or explicit representations as functions of spherical variables (for some of the wavelets). We derive also an explicit formula for their Euclidean limits.     

A Local Parameterization of Orthogonal and Semi-Orthogonal Matrices with Applications

This article describes a local parameterization of orthogonal and semi-orthogonal matrices. The parameterization leads to a unified approach for obtaining the asymptotic joint distributions of estimators of singular-values and -vectors, and of eigen-values and -vectors. The singular- or eigen-values can have arbitrary multiplicities. The approach is illustrated on principal components analyzes, canonical correlation analysis, inter-battery factory analysis, and reduced-rank regression.     

On the detection of singularities of a periodic function

We discuss the problem of detecting the location of discontinuities of derivatives of a periodic function, given either finitely many Fourier coefficients of the function, or the samples of the function at uniform or scattered data points. Using the general theory, we develop a class of trigonometric polynomial frames suitable for this purpose. Our methods also help us to analyze the capabilities of periodic spline wavelets, trigonometric polynomial wavelets, and some of the classical summability methods in the theory of Fourier series. This revised version was published online in June 2006 with corrections to the Cover Date.     

A numerical assessment of parabolic partial differential equations using Haar and Legendre wavelets

In this paper we present two new numerically stable methods based on Haar and Legendre wavelets for one- and two-dimensional parabolic partial differential equations (PPDEs). This work is the extension of the earlier work ,  and  from one- and two-dimensional boundary-value problems to one- and two- dimensional PPDEs. Two generic numerical algorithms are derived in two phases. In the first stage a numerical algorithm is derived by using Haar wavelets and then in the second stage Haar wavelets are replaced by Legendre wavelets in quest for better accuracy. In the proposed methods the time derivative is approximated by first order forward difference operator and space derivatives are approximated using Haar (Legendre) wavelets. Improved accuracy is obtained in the form of wavelets decomposition. The solution in this process is first obtained on a coarse grid and then refined towards higher accuracy in the high resolution space. Accuracy wise performance of the Legendre wavelets collocation method (LWCM) is better than the Haar wavelets collocation method (HWCM) for problems having smooth initial data or having no shock phenomena in the solution space. If sharp transitions exists in the solution space or if there is a discontinuity between initial and boundary conditions, LWCM loses its accuracy in such cases, whereas HWCM produces a stable solution in such cases as well. Contrary to the existing methods, the accuracy of both HWCM and LWCM do not degrade in case of Neumann’s boundary conditions. A distinctive feature of the proposed methods is its simple applicability for a variety of boundary conditions. Performances of both HWCM and LWCM are compared with the most recent methods reported in the literature. Numerical tests affirm better accuracy of the proposed methods for a range of benchmark problems.     

On divergence-free wavelets

This paper is concerned with the construction of compactly supported divergence-free vector wavelets. Our construction is based on a large class of refinable functions which generate multivariate multiresolution analyses which includes, in particular, the non tensor product case.For this purpose, we develop a certain relationship between partial derivatives of refinable functions and wavelets with modifications of the coefficients in their refinement equation. In addition, we demonstrate that the wavelets we construct form a Riesz-basis for the space of divergence-free vector fields.Work supported by the Deutsche Forschungsgemeinschaft in the Graduiertenkolleg Analyse und Konstruktion in der Mathematik at the RWTH Aachen.     

Continuous and Discrete Wavelets for Acceleration Kinematics: A Motion Estimation Based on Lie Groups and Variational Principles

A group-theoretic framework is presented for acceleration transformations. The main purpose is to show the existence of families of spatio-temporal continuous wavelets, frames, and discrete wavelets related to these transformations. The main application of interest is the analysis of motion in space–time signals. The construction of this framework starts with the enumeration of Lie algebras as building blocks that provide all the observable kinematics that comply with the properties of the space–time under analysis. These classes of accelerated kinematics generalize the kinematics defined in the Galilei group. Exponentiation from Lie algebras defines locally compact exponential groups. Unitary, irreducible, and square-integrable group representations are thereafter derived in the function spaces and the signals to be analyzed, leading to the existence of continuous and discrete wavelets, frames all indexed with higher orders of temporal derivatives of the translational motion. Group representations and wavelets are tools that perform the local optimum estimation of pieces of trajectory. The adjunction of a variational principle of optimality is further necessary for building a global trajectory and for performing tracking. The Euler–Lagrange equation provides the motion equation of the moving system and the Noether's theorem derives the related constants of motion. Dynamic programming implements the algorithms for tracking and constructing the global trajectory. Finally, tight frames and bases enable signal decompositions along the trajectory of interest.     

Directional Wavelet Bases Constructions with Dyadic Quincunx Subsampling

We construct directional wavelet systems that will enable building efficient signal representation schemes with good direction selectivity. In particular, we focus on wavelet bases with dyadic quincunx subsampling. In our previous work (Yin, in: Proceedings of the 2015 international conference on sampling theory and applications (SampTA), 2015), we show that the supports of orthonormal wavelets in our framework are discontinuous in the frequency domain, yet this irregularity constraint can be avoided in frames, even with redundancy factor <2. In this paper, we focus on the extension of orthonormal wavelets to biorthogonal wavelets and show that the same obstruction of regularity as in orthonormal schemes exists in biorthogonal schemes. In addition, we provide a numerical algorithm for biorthogonal wavelets construction where the dual wavelets can be optimized, though at the cost of deteriorating the primal wavelets due to the intrinsic irregularity of biorthogonal schemes.     

对称-反对称复合伸缩多小波的构造

基于已知的复合伸缩小波,本文给出一个构造对称反对称复合伸缩多小波的简单方法.这个方法可增加复合伸缩小波的数量,而且构造的多小波保持了原来小波的大部分性质.     

Wavelets with differential relation

Divergence-free wavelets are successfully applied to numerical solutions of Navier-Stokes equation and to analysis of incompressible flows. They closely depend on a pair of one-dimensional wavelets with some differential relations. In this paper, we point out some restrictions of those wavelets and study scaling functions with the differential relation; Wavelets and their duals are discussed; In addition to the differential relation, we are particularly interested in a class of examples with the interpolatory property; It turns out there is a connection between our examples and Micchelli’s work.     

Semi-orthogonal decomposability of the derived category of a curve

We show that the bounded derived category of coherent sheaves on a smooth projective curve except the projective line admits no non-trivial semi-orthogonal decompositions.