基于一类新的半离散的小波变换的信号重构公式

在基小波ψ(t)为带限的条件下,基于半离散的小波变换(Wψf)(na,a)(n∈Z,a∈R)信号f(t)的重构公式被给出.     

广义典型流形上连续小波变换的性质

研究广义典型流形M上小波变换的性质,根据广义典型流形M的结构特征与广义典型流形M上连续小波变换的定义,讨论了广义典型流形M上的连续小波变换的重构公式,线性性质,伸缩平移性等,讨论了广义典型流形M上小波变换的性质.最后,给出了连续小波ψ的卷积公式.     

小波变换及其应用(续二)

(五 )离散小波变换正交小波基上面我们介绍了连续小波变换 ,但在实际问题及数值计算中更重要的是其离散形式 (在作具体数值计算时 ,连续小波的参数 a,b必然要离散化 )。对确定的小波母函数ψ( t) ,取定 a0 >1 ,b0 >0　　令ψmn( t) =am20 ψ( am0 t-nb0 ) ,　 m,n∈ Z ( 5.1 )这里 Z表示全体整数所构成的集合 ,我们称 ψmn( t)为离散小波。对于函数 f( t) ,相应的离散小波变换为 :Cf( m,n) =∫∞-∞f ( t)ψmn( t) dt,m,n∈ Z ( 5.2 )　　我们知道对连续小波 ,由 Wf( a,b) ,a,b∈ ( -∞ ,∞ ) ,a≠ 0可唯一确定函数 f ( t) (反演公式( 3 .…     

周期函数和非平稳周期函数的小波变换

探讨了三角函数、周期函数以及一类非平稳周期函数小波变换的一些性质,发现周期函数的小波能谱的峰高和峰宽均正比于信号的周期.提出了一个新的只利用与信号周期有关的一个尺度小波变换系数的重构公式,它可准确地重构三角函数,对一般周期函数的重构结果优于其Fourier级数中的任何一项,对一类均值和振幅变化的非平稳周期函数的重构结果与信号非常吻合.     

用再生核表示小波变换

本文研究了调制高斯函数的小波变换．利用再生核函数的特殊技巧，得到了该小波变换的等距恒等式和像空间的结构，同时给出了该小波变换的采样定理．使得小波变换能用再生核函数表示．这为一般的小波变换的像空间的研究提供了理论基础．     

连续切波变换反演公式的级数表示

我们主要研究连续切波变换反演公式的级数表示.首先引入两类由切波变换反演公式定义的无穷级数和有限级数,并研究了由Kittipoom等人介绍的切波生成空间,得到这个切波生成空间的一些重要性质.其次利用这些结果显示:对于这个切波生成空间,当采样密度趋于无穷时由我们定义的无穷级数按L~2-范数收敛于重构函数;对于可允许函数空间,当采样密度趋于无穷时由我们定义的有限级数按L~2-范数收敛于重构函数.     

广义小波框架

通过将分数阶Fourier变换(Fr FT)与传统小波框架(WF)相结合,本文引入了一类新的框架,即广义小波框架(GWF).首先介绍了分数阶Fourier变换、级数以及相关结论;然后通过分析母小波的Fourier变换,得到了L2(R)中广义小波框架的若干个充分条件和必要条件;最后在参数选择为a=2,b=1的特殊情形下,给出了L2(R)中紧广义小波框架的一个充要条件.     

一类二元半正交小波包的构造

本文给出伸缩矩阵行列式为2的一类二元半正交小波包的构造算法.该小波包是以频域给出的,随着用于小波包分裂的滤波器选取的不同会得到L2(R2)中形态各异的Riesz基,这样使得L2(R2)中小波基的选择更灵活.     

小波变换的一类新的反演公式

给出关于带限函数的小波变换的一类新的反演公式，这个公式具有比熟知的结果更清晰的表达式，并且含有可以自由选择的因子。     

紧支撑多重向量值正交小波包的性质

给出紧支撑多重向量值正交小波包的定义及构造方法.运用矩阵理论与积分变换,研究了多重向量值正交小波包的性质,得到三个正交性公式.进而,得到空间L2(R,Cr)的一个新的规范正交基.     

Uncertainty Principles for the Continuous Dunkl Gabor Transform and the Dunkl Continuous Wavelet Transform

In this paper we consider the Dunkl operators T _j , j = 1, . . . , d, on and the harmonic analysis associated with these operators. We define a continuous Dunkl Gabor transform, involving the Dunkl translation operator, by proceeding as mentioned in [20] by C.Wojciech and G. Gigante. We prove a Plancherel formula, an inversion formula and a weak uncertainty principle for it. Then, we show that the portion of the continuous Dunkl Gabor transform lying outside some set of finite measure cannot be arbitrarily too small. Similarly, using the basic theory for the Dunkl continuous wavelet transform introduced by K. Trimèche in [18], an analogous of this result for the Dunkl continuous wavelet transform is given. Finally, an analogous of Heisenberg’s inequality for a continuous Dunkl Gabor transform (resp. Dunkl continuous wavelet transform) is proved.      

Wavelet transform associated with linear canonical Hankel transform

The main goal of this paper is to study about the continuous as well as discrete wavelet transform in terms of linear canonical Hankel transform (LCH‐transform) and discuss some of its basic properties. Parseval's relation and reconstruction formula of continuous linear canonical Hankel wavelet transform (CLCH‐wavelet transform) is obtained. Moreover, semidiscrete and discrete LCH‐wavelet transform are also discussed.     

连续小波变换及小波框架算子的一些性质

以泛函分析的观点来考察连续小波变换及小波框架算子，得到了它们的一些性质，并给出了严格证明，弥补了有关献中的不足。     

Some Localization Properties of the Lp Continuous Wavelet Transform

We shall prove some simultaneous localization or concentration inequalities for the continuous wavelet transform. We will also show that simultaneous localization in the scale-time(space) is impossible, in the sense that the scale sections of the support of wavelet transform of a nonnull L^p-function can not have finite Lebesgue measure. Finally, some properties of the support of continuous wavelet transform of band-limited functions are studied.     

Continuous and discrete Bessel wavelet transforms

Using the theory of Hankel convolution, continuous and discrete Bessel wavelet transforms are defined. Certain boundedness results and inversion formula for the continuous Bessel wavelet transform are obtained. Important properties of the discrete Bessel wavelet transform are given.     

Wavelet transforms for integrable Boehmians

By the application of continuous wavelet transforms (or windowed Fourier transform) and employing Burzyk's conjecture, the wavelet transform for integrable Boehmians is obtained. Inversion formula is also established.     

再生核与小波变换

在再生核基本理论的基础上,介绍了再生核在小波变换中的作用,并且根据连续小波变换像空间是再生核Hilbert空间这一基本事实,借助再生核理论的特殊技巧,建立了Littlewood-Paley和Haar小波变换像空间的再生核函数与已知再生核空间的再生核的关系,为小波变换像空间的进一步研究提供理论基础.     

Gauss小波变换像空间的描述

在连续小波变换像空间是再生核Hilbert空间的基础上,针对经常用于边界检测并且使用效果非常好的Gauss小波,给出了其小波变换像空间的再生核具体表达式.并且当固定尺度因子和固定平移因子时,利用再生核空间理论,对Gauss小波变换像空间做了具体描述,分别给出了Gauss小波变换像空间中的等距恒等式和反演公式,这为进一步研究一般的小波变换像空间提供了理论基础.     

Frames for the Continuous Spherical Wavelet Transform

In this paper we will construct frames for the continuous spherical wavelet transform. (© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Homogeneous Approximation Property for Multivariate Continuous Wavelet Transforms

The homogeneous approximation property (HAP) for the continuous wavelet transform is useful in practice because it means that the measure of the building area involved in a reconstruction of a function up to some error is essentially invariant under timescale shifts. For the univariate case, it was shown that the pointwise HAP holds if and only if the Fourier transforms of both wavelets and the function to be reconstructed are compactly supported on ??{0}. In this paper, we study the HAP for multivariate wavelet transforms. We show that similar results hold for this case. However, the above condition is only sufficient but not necessary if the dimension of the variable is greater than 1, which is different from the univariate case. We also get a convergence result on the inverse of wavelet transforms, which improves similar results by Daubechies and Holschneider and Tchamitchain.