Unique reconstruction of band-limited signals by a Mallat-Zhong wavelet transform algorithm

We show that uniqueness and existence for signal reconstruction from multiscale edges in the Mallat and Zhong algorithm become possible if we restrict our signals to Paley-Wiener space, band-limit our wavelets, and irregularly sample at the wavelet transform (absolute) maxima—the edges—while possibly including (enough) extra points at each level. We do this in a setting that closely resembles the numerical analysis setting of Mallat and Zhong and that seems to capture something of the essence of their (practical) reconstruction method. Our work builds on a uniqueness result for reconstructing an L² signal from irregular sampling of its wavelet transform of Grochenig and the related work of Benedetto, Heller, Mallat, and Zhong. We show that the rate of convergence for this reconstruction algorithm is geometric and computable in advance. Finally, we consider the effect on the rate of convergence of not sampling enough local maxima.     

RESEARCH ANNOUNCEMENTS Reconstruction of Band-limited Signals From Uniform Random Samples

The Shannon sampling theorem is one of the foundations for modern signal processing.Assume that a signal f（t）∈L²（R）.σ>0 is a constant.Signal f（t） is calledσ-band-limitedif |ω|>σ,F（ω） = f_-∞^∞f（t）e^-iω≧tdt = 0.The Shannon sampling theorem says that aσ-band-limitedf（t） can be reconstructed exactly by its all sampling points at the equal interval h≤π/σ.The reconstructed formula is     

由一元连续小波变换得到的二元连续小波变换及重构公式

讨论了L2(R2)空间中连续小波变换,分别得到由一元变换函数构造二元变换函数的二元小波变换及重构公式,得到重构公式在L2(R2)中范数收敛意义下成立的条件.     

离散积分小波变换的快速算法

本文讨论了积分小波变换的快速算法,通过尺度函数与小波间的二尺度关系,导出了一个实现积分小波变换的快速计算方法及相应滤波器的构造方法。     

Square-like Functions Generated by a Composite Wavelet Transform

The classical square functions play important role in Harmonic Analysis and have a very direct connection with L ₂-estimates and Littlewood-Paley theory. In this paper we introduce a new variant of square-like functions generated by some composite wavelet transform. We establish Calderón-type reproducing formula and then prove L ₂-boundedness of newly defined square-like functions.     

Genetic Algorithm Assisted Wavelet Noise Reduction Scheme for Chaotic Signals

We present a Genetic Algorithm-based wavelet denoising method which incorporates a Genetic Algorithm within a wavelet framework for threshold optimization. The new method not only intelligently adapts itself to different types of noise without any prior knowledge of the noise, but also balances the preservation of dynamics against the degree of noise reduction by optimizing the Signal-to-Noise Ratio and the Liu’s error factor. The presented method performs better than the state-of-the-art wavelet-based denoising methods when applied to chaotic signals.     

一种基于Eno-Haar小波变换二维遗传算法

A two-dimensional genetic algorithm of wavelet coefficient is presented by using the ENO wavelet transform and the decomposed characterization of the two-dimensional Haar wavelet.And simulated by the ENO interpolation the article shows the affectivity and the superiority of this algorithm.     

Reconstruction of Wideband Reflectivity Densities by Wavelet Transforms

This paper is concerned with reconstruction problems arising in the context of radar signal analysis. The goal in radar is to obtain information about objects by emitting certain signals and analyzing the reflected echoes. In this paper, we shall focus on the general wideband model for radar echoes and on the case of continuously distributed objects D (reflectivity density). In this case, the echo is given by an inverse wavelet transform of the density D where the role of the analyzing wavelet is played by the transmitted signal. However, the null space of an inverse wavelet transform is nontrivial, it is described by the corresponding reproducing kernel. Following the approach of Naparst [14] and Rebolla-Neira et al. [16], we suggest to treat this problem by transmitting not just one signal but a family of signals. Indeed, a reconstruction formula for one- and 2-dimensional reflectivity densities can be derived, provided that the set of outgoing signals forms an orthogonal basis or – more general – a frame. We also present some rigorous error estimates for these reconstruction formulas. The theoretical results are confirmed by some numerical examples.     

农民收入增长波动关系预测分析:基于小波变换

新农村建设背景下,保持农民收入快速、稳定增长对于构建社会主义和谐社会、不断缩小城乡差距具有重要的战略意义。本文运用小波变换(WT)对1976-2006年间农民收入的波动关系进行预测分析,实证检验结果显示:全国农民人均纯收入增长具有7年和40年左右的特征时间尺度,与农民人均纯收入变化存在着7年和40年两个主要周期振荡保持一致。两个特征时间尺度叠加,可以预见在未来的数年内,全国农民人均纯收入将呈现绝对值快速增长、增长率在波动中有所放缓的特征。最后,文章就如何保持农民收入稳定增长提出了点简短的对策建议。     

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。     

小波变换在膜蛋白跨膜螺旋区段预测中的应用

膜蛋白跨膜螺旋区段的预测是当前生物信息学研究的一个重要领域.提出了一种基于离散小波变换的预测膜蛋白跨膜螺旋区段数目和位置的方法.以PDB代码为1KQG的膜蛋白为例,描述了该方法对跨膜螺旋区段的数目和位置的预测分析过程.为了测试方法的实际效果,从最新的MPtopo数据库中随机抽取80条三维结构已知的膜蛋白序列作为测试集(含325个跨膜螺旋区),采用该方法对跨膜螺旋区段进行预测,其中308个跨膜螺旋区被准确预测到,预测准确率为96.3%.并且按照膜蛋白的功能和类型,把80条序列划分成13组,对每一组膜蛋白序列进行跨膜螺旋区段预测,达到了令人满意的结果.     

广义函数框架下的小波变换及其性质

本文将L^2空间的小波变换推广到广义函数空间上，建立了广义函数框架下的小波变换，证明了广义函数的小波变换及其有关性质，使小波变换这一信号分析的数学工具有了更大的应用范围．     

Reconstruction of Sampled Signals with Fractal Functions

This paper proposes a procedure to build a fractal model for real sampled signals like financial series, climatic data, bioelectric recordings, etc. The mapping constructed owns in general a rich geometric structure. In a first step, the method provides a truncate Chebyshev approximant which performs a low-pass filtering of the signal, displaying in this way the leading cycles of the phenomenon observed. In the second, the polynomial is transformed in a fractal object. The Lipschitz properties of the original signal guarantee a good approximation of the represented variable, whenever the sampling frequency is high enough.     

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。     

On the Windowed Fourier Transform and Wavelet Transform of Almost Periodic Functions

We present the windowed Fourier transform and wavelet transform as tools for analyzing persistent signals, such as bounded power signals and almost periodic functions. We establish the analogous Parseval-type identities. We consider discretized versions of these transforms and construct generalized frame decompositions. Finally, we bring out some relations with shift-invariant operators and linear systems.     

DOG小波变换像空间的描述

本文给出了DOG小波变换像空间的再生核函数的具体表达式及等距恒等式,并利用再生核函数的结构对DOG小波变换的像空间作出了具体的描述,使得对其像空间的形成有了更直观和更深刻的认识.这既为一般小波变换像空间的描述奠定了基础,又为该小波变换的实际运用提供理论依据.     

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.     

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.