紧支撑正交对称和反对称小波的构造

１．引言 近年来,人们分别从数学和信号的观点对正交小波进行了广泛的研究．尤其是２尺度小波,它克服了短时 Ｆｏｕｒｉｅｒ变换的一些缺陷．目前最常用的 ２尺度小波是 Ｄａｕｂｅｃｈｉｅｓ 小波,但 ２尺度小波也存在一些问题：如 Ｄａｕｂｅｃｈｉｅｓ［２］已证明了除 Ｈａａｒ小波外不存在既正交又对称的紧支撑 ２尺度小波．因此人们提出了 ａ尺度小波理论［３］－［６］,文献［４］－［６］对 ４尺度小波迸行研究．本文的目的是研究４尺度因子时紧支撑正交对称和反对称小波的构造方法．并指出对同一紧支撑正交对称尺度函数而言,…     

Dimension Functions of Orthonormal Wavelets

The article is devoted to dimension functions of orthonormal wavelets on the real line with dyadic dilations. We describe properties of dimension functions and prove several characterization theorems. In addition, we provide a method of construction of dimension functions. Various new examples of dimension functions and orthonormal wavelets are included.     

Symmetric orthonormal scaling functions and wavelets with dilation factor 4

It is well known that in the univariate case, up to an integer shift and possible sign change, there is no dyadic compactly supported symmetric orthonormal scaling function except for the Haar function. In this paper we are concerned with the construction of symmetric orthonormal scaling functions with dilation factor d=4. Several examples of such orthonormal scaling functions are provided in this paper. In particular, two examples of C ¹ orthonormal scaling functions, which are symmetric about 0 and 1/6, respectively, are presented. We will then discuss how to construct symmetric wavelets from these scaling functions. We explicitly construct the corresponding orthonormal symmetric wavelets for all the examples given in this paper. This revised version was published online in June 2006 with corrections to the Cover Date.     

小波分析中一个逆问题

本文证明:如果来自多尺度分析(伸缩因子为矩阵)的小波是标准正交的,那么相对应的尺度函数也是标准正交的,其中函数f_s(x)∈L~2(R~n)(s=1,2,…,r,r是正整数)的标准正交性是指f_s(x)的整平移所构成的函数族为L~2(R~n)的标准正交系。结果表明,如果我们想从多尺度分析出发构造正交小波,那么该多尺度分析必须有正交尺度函数。     

A New Method of Constructions of Non-Tensor Product Wavelets

In this paper, we give a new method of constructions of non-tensor product wavelets. We start from the one-dimensional scaling functions to directly construct the two-dimensional non-tensor product wavelets. The wavelets constructed by us possess very simple, explicit representations and high regularity, and various symmetry (i.e., axial symmetry, central symmetry, and cyclic symmetry). Using this method, we construct various non-tensor product wavelets and show that there exists a sequence of non-tensor product wavelets with high regularity which tends to the tensor product Shannon wavelet in the L ²-norm.     

Compactly supported (bi)orthogonal wavelets generated by interpolatory refinable functions

This paper provides several constructions of compactly supported wavelets generated by interpolatory refinable functions. It was shown in [7] that there is no real compactly supported orthonormal symmetric dyadic refinable function, except the trivial case; and also shown in [10,18] that there is no compactly supported interpolatory orthonormal dyadic refinable function. Hence, for the dyadic dilation case, compactly supported wavelets generated by interpolatory refinable functions have to be biorthogonal wavelets. The key step to construct the biorthogonal wavelets is to construct a compactly supported dual function for a given interpolatory refinable function. We provide two explicit iterative constructions of such dual functions with desired regularity. When the dilation factors are larger than 3, we provide several examples of compactly supported interpolatory orthonormal symmetric refinable functions from a general method. This leads to several examples of orthogonal symmetric (anti‐symmetric) wavelets generated by interpolatory refinable functions. This revised version was published online in June 2006 with corrections to the Cover Date.     

Numerical solution of high-order differential equations by using periodized Shannon wavelets

In this paper, periodized Shannon wavelets are applied as basis functions in solution of the high-order ordinary differential equations and eigenvalue problem. The first periodized Shannon wavelets are defined. The second the connection coefficients of periodized Shannon wavelets are related by a simple variable transformation to the Cattani connection coefficients. Finally, collocation method is used for solving the high-order ordinary differential equations and eigenvalue problem. Some equations are solved in order to find out advantage of such choice of the basis functions.     

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.     

On progressive functions

Progressive functions at time t involve only the progressive functions at time before t and some nice compactly supported function at time t. We give sufficient conditions and explicit formulas to construct progressive functions with exponential decay and characterize the conditions on which the positive integer translates of a progressive function are orthonormal or a Riesz sequence. We provide explicit ways for construction of orthonormal progressive functions and for construction of the biorthogonal functions of nonorthogonal progressive functions. Such progressive functions can be used to construct wavelets with arbitrary smoothness on the half line if they are generated by a smooth refinable compactly supported function.     

High-Order Orthonormal Scaling Functions and Wavelets Give Poor Time-Frequency Localization

For a fairly general class of orthonormal scaling functions and wavelets with regularity exponents n, we prove that the areas of the time-frequency windows tend to infinity as n → ∞. This class includes those of Battle-Lemarie and Daubechies. In addition, if the scaling functions have at least asymptotic linear phase, then we prove that they converge to the "sinc" function and their corresponding orthonormal wavelets converge to the "difference" of two sinc functions.     

Wavelets for multichannel signals

In this paper, we introduce and investigate multichannel wavelets, which are wavelets for vector fields, based on the concept of full rank subdivision operators. We prove that, like in the scalar and multiwavelet case, the existence of a scaling function with orthogonal integer translates guarantees the existence of a wavelet function, also with orthonormal integer translates. In this context, however, scaling functions as well as wavelets turn out to be matrix-valued functions.     

Symmetric–Antisymmetric Orthonormal Multiwavelets and Related Scalar Wavelets

For 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.     

Band-limited Wavelets and Framelets in Low Dimensions

In this paper, we study the problem of constructing non-separable band-limited wavelet tight frames, Riesz wavelets and orthonormal wavelets in $\mathbb {R}^{2}$ and $\mathbb {R}^{3}$ . We first construct a class of non-separable band-limited refinable functions in low-dimensional Euclidean spaces by using univariate Meyer’s refinable functions along multiple directions defined by classical box-spline direction matrices. These non-separable band-limited definable functions are then used to construct non-separable band-limited wavelet tight frames via the unitary and oblique extension principles. However, these refinable functions cannot be used for constructing Riesz wavelets and orthonormal wavelets in low dimensions as they are not stable. Another construction scheme is then developed to construct stable refinable functions in low dimensions by using a special class of direction matrices. The resulting stable refinable functions allow us to construct a class of MRA-based non-separable band-limited Riesz wavelets and particularly band-limited orthonormal wavelets in low dimensions with small frequency support.     

a尺度正交小波的Mallat算法

本文研究了a尺度正交小波的Mallat算法,利用a重多分辨分析,得到了正交小波的分解与重构算法,给出了Haar小波的Mallat算法的矩阵表示,并简化了计算.     

与多通道正交小波滤波器相关的矩阵扩充的通解

本文研究了一类与多通道正交小波滤波器相关的矩阵方程。运用多相位分解的方法，获得了这类矩阵方程的通解。借助于该结果，可以从一组多通道正交小波能够产生许多组多通道正交小波。     

A Characterization of Generalized Frame MRAs Deriving Orthonormal Wavelets

In 2000, Papadakis announced that any orthonormal wavelet must be derived by a generalized frame MRA （GFMRA）. In this paper, we give a characterization of GFMRAs which can derive orthonormal wavelets, and show a general approach to the constructions of non-MRA wavelets. Finally we present two examples to illustrate the theory.     

Construction of orthonormal wavelets using Kampé de Fériet functions

One of the main aims of this paper is to bridge the gap between two branches of mathematics, special functions and wavelets. This is done by showing how special functions can be used to construct orthonormal wavelet bases in a multiresolution analysis setting. The construction uses hypergeometric functions of one and two variables and a generalization of the latter, known as Kampé de Fériet functions. The mother wavelets constructed by this process are entire functions given by rapidly converging power series that allow easy and fast numerical evaluation. Explicit representation of wavelets facilitates, among other things, the study of the analytic properties of wavelets.

     

Compactly supported orthonormal complex wavelets with dilation 4 and symmetry

In this paper, we provide a family of compactly supported orthonormal complex wavelets with dilation 4 such that their generating wavelet functions have symmetry and the shortest possible supports with respect to their increasing orders of vanishing moments.     

Characterizations of fundamental scaling functions and wavelets

The objective of this paper is to establish precise characterizations of scaling functions which are orthonormal or fundamental. A criterion for the corresponding wavelets is also given. Research supported by NSF Grant #DMS-89-01345 and ARO Contract DAAL 03-90-G-0091.     

Littlewood-Paley spline wavelets: a simple and efficient tool for signal and image processing in industrial applications

In this work, we present the Littlewood-Paley Spline Wavelets that are much more effective that the orthonormal discrete wavelets, although it cannot apply the multiresolution scheme. These wavelets carry the advantages of the splines functions what allows their implementation through simple numeric algorithms. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)