Wavelet footprints: theory, algorithms, and applications

Wavelet-based algorithms have been successful in different signal processing tasks. The wavelet transform is a powerful tool because it manages to represent both transient and stationary behaviors of a signal with few transform coefficients. Discontinuities often carry relevant signal information, and therefore, they represent a critical part to analyze. We study the dependency across scales of the wavelet coefficients generated by discontinuities. We start by showing that any piecewise smooth signal can be expressed as a sum of a piecewise polynomial signal and a uniformly smooth residual (Theorem 1). We then introduce the notion of footprints, which are scale space vectors that model discontinuities in piecewise polynomial signals exactly. We show that footprints form an overcomplete dictionary and develop efficient and robust algorithms to find the exact representation of a piecewise polynomial function in terms of footprints. This also leads to efficient approximation of piecewise smooth functions. Finally, we focus on applications and show that algorithms based on footprints outperform standard wavelet methods in different applications such as denoising, compression, and (nonblind) deconvolution. In the case of compression, we also prove that at high rates, footprint-based algorithms attain optimal performance (Theorem 3).     

Wavelet estimation

An important problem in seismic exploration is the estimation of and correction for the seismic wavelet. A seismic signal may be modeled as a convolutional model with the wavelet as one component. The wavelet propagated by the seismic energy source is complicated by transmission and recording filters. Some filters in the system can be deterministically defined while others are more conjectural. The estimation of the wavelet is useful in two major ways. Borehole measurements are used to model the surface seismograms. The wavelet used in the model needs to match that of the seismogram to correlate the two measurements. Conversely, the estimated wavelet can be used to design inverse filters which make the seismogram approach the borehole measures. Some well-known methods for estimation of the wavelet are based on assumptions about the wavelet or the earth reflectivity. Examples of the methods indicate success on some data even though each makes different assumptions. The methods serve to point out basic problems in reliably estimating the wavelet from the seismogram. Basic problems include noise, band-limiting, nonstationarity, uncertain theoretical models, assumption failure, and widely diverse geological sequences of the earth. Quality control or evaluation of the performance of an estimation algorithm is a nontrivial problem. The estimation of the wavelet from a seismic recording remains an area of challenging research and importance in exploration for hydrocarbons.     

基于小波变换/小波包变换的多载波调制技术

小波/小波包在通信系统中的应用是近年来一个新的研究领域,而基于小波/小波包变换的多载波调制技术是其中一大研究热点。由于小波/小波包基函数具有良好的正交性与时频局域性等特点,基于小波/小波包变换的多载波调制技术,能够有效地提高通信系统性能。本文介绍了当前几种主要的基于小波/小波包变换的多载波调制方案,分析了这些方案的性能特点及发展趋势,并与其他方案进行了比较。     

一种小波变换的小波系数均衡图像增强算法

介绍了几种经典的图像增强算法,依据经典直方图均衡化算法思想,提出了在小波变换下的小波系数均衡化图像增强算法,并给出了计算步骤。与经典直方图均衡化算法、同态滤波实验结果相比较,表明提出的算法可有效增强图像、抑制噪声。     

Scalable Interframe Wavelet Coding with Low Complex Spatial Wavelet Transform

In the decoding process associated with interframe wavelet coding, the inverse wavelet transform requires high computational complexity. However, as video technology starts to pervade all aspects of our lives, decoders are becoming required in various devices such as PDAs, notebooks, PCs, and set‐top boxes. Therefore, a decoder's complexity needs to be adapted to the processor's computational power, and consequently a low‐complexity codec is also required for scalable video coding. In this paper, we propose a method of controlling and lowering the complexity of the spatial wavelet transform while sustaining the same coding efficiency as that currently afforded. In addition, the proposed method may alleviate the ringing effect for slowly changing image sequences.     

小波分析和小波包在图像消噪中的应用

小波包理论是在小波基础上发展起来的时频分析理论,在图像消噪中取得了较好的应用。文中介绍了小波分析和小波包的基本理论,分析了基于小波和小波包进行图像消噪的方法,利用MATLAB7对相同舍噪图像进行了计算机仿真,并对仿真结果进行了分析。实验表明,图像消噪效果与阈值的选取密切相关,小波包消噪后的效果比小波更好。     

Wavelet harmonic balance

This letter introduces a new approach to steady state analysis of nonlinear microwave circuits under periodic excitation. The new method is similar to the well known technique of Harmonic Balance, but uses wavelets as basis functions instead of Fourier series. Use of wavelets allows significant increase in sparsity of the equation matrices and consequently decrease in CPU cost and storage requirements, while retaining accuracy and convergence of the traditional approach. The new method scales linearly with the size of the problem and is well suited for large scale simulations.     

关于小波空间的紧支集采样函数

余越、柯有安讨论了紧支集小波于空间内具紧支集的采样函数。本文证明了除Haar小波外没有任何紧支集正交小波空间具有紧支集采样函数。     

一种基于提升方案小波和小波块的图像水印算法

数字水印技术是多媒体信息安全领域的一个重要研究课题。文章提出一种基于提升方案小波和小波块的图象水印算法。首先，利用提升方案小波对宿主图像和水印图像进行分解，同时利用小波系数的树结构，将宿主图像的小波系数块按纹理密度分成两类，并根据嵌入位置所隶属的小波块的纹理强弱，嵌入不同强度的水印信息，这在一定程度上保证了所嵌入水印在不可见性前提下的极大鲁棒性。实验结果验证了所提出算法的有效性。     

Discrete Lattice Wavelet Transform

The discrete wavelet transform (DWT) has gained a wide acceptance in denoising and compression coding of images and signals. In this work we introduce a discrete lattice wavelet transform (DLWT). In the analysis part, the lattice structure contains two parallel transmission channels, which exchange information via two crossed lattice filters. For the synthesis part we show that the similar lattice structure yields a perfect reconstruction (PR) property. The PR condition can be used to design half-band filters, which effectively eliminate aliasing in decimated tree structured wavelet transform. The DLWT can be implemented directly to any of the existing DWT algorithms     

Wavelet applications in medicine

The author describes how, because of their compatibility with nonstationary random processes, wavelet transforms are particularly powerful when it comes to analyzing biomedical signals. In this paper, he presents a brief survey of the field which makes this fact clear. Applications discussed include detecting coronary artery disease, and heart sounds related to turbulent blood flow analysis     

Wavelet processing and optics

Fourier optics has offered a powerful global information analysis and processing tool to optical domain signals and images and firmly locked its respectable position in the history of optics. The evolution of wavelet theory in the past decade, which has created significant impacts on many areas of information science and engineering, has recently been frequently mentioned in the optics community. Whether or not this locality-oriented signal analysis and processing tool can play a comparably significant role in optics remains to be answered. The present paper is intended to summarize important developments and recent progress in areas where novel wavelet processing concepts are incorporated into optics research     

图像的小波矩

本文从一般特征矩的定义入手,导出极坐标系下图像的小波矩,进而给出对图像进行多分辨小波变换后,利用任一尺度下的逼近系数构造小波矩和基于小波模极大值边界图像构造小波矩两种方法.最后,经实验验证了本文构造图像小波矩方法的有效性.     

Adaptive Real-Time Wavelet Detection

Detection of wavelets in noisy signals is important for many biomedical as well as other applications. An adaptive wavelet detection system is described, based on the cross correlation method. The method is restricted to fnite-length, nonoverlapping wavelets. A random reference correlator (RRC) circuit is used to estimate the correlation between signal and template, eliminating the need for expensive hardware or time-consuming software correlation calculations.     

Generalized Daubechies Wavelet Families

We present a generalization of the orthonormal Daubechies wavelets and of their related biorthogonal flavors (Cohen-Daubechies-Feauveau, 9/7). Our fundamental constraint is that the scaling functions should reproduce a predefined set of exponential polynomials. This allows one to tune the corresponding wavelet transform to a specific class of signals, thereby ensuring good approximation and sparsity properties. The main difference with the classical construction of Daubechies is that the multiresolution spaces are derived from scale-dependent generating functions. However, from an algorithmic standpoint, Mallat's fast wavelet transform algorithm can still be applied; the only adaptation consists in using scale-dependent filter banks. Finite support ensures the same computational efficiency as in the classical case. We characterize the scaling and wavelet filters, construct them and show several examples of the associated functions. We prove that these functions are square-integrable and that they converge to their classical counterparts of the corresponding order.     

基于整数小波变换和改进嵌入零树编码的图像压缩

整数小波变换(IWT)具有输入输出都是整数、只需进行位内运算、便于硬件实现等优点,特别适于图像的无损压缩;而在有损压缩方面,其效果通常稍逊于传统的离散小波变换(DWT)。为了提高IWT图像有损压缩的性能,该文采用基于提升格式的IWT,结合基于形态膨胀运算的改进嵌入零树小波编码(EZW)方法。实验结果表明,在没有增加运算复杂度的情况下,此算法与传统的DWT相比,提高了峰值信噪比(PSNR),具有较好的有损压缩效果。     

小波变换图像压缩中最优小波基的选取方法

针对不同的小波基具有不同的时频特征,用不同的小波基分析同一个问题会产生不同的结果.故小波分析在应用中便存在一个最优小波基的选取问题.详述了选择小波基所依据的几个重要特征,并通过实例说明了在小波应用中要把握小波基的特征,根据应用需要,选择合适的小波基.     

具有抽样性质的双正交子波的逼近性能及其子波抽样的计算

本文主要讨论在具有抽样性质的双正交子波基下Mallat算法的逼近性能及其子波系数的计算。当尺度较小以及尺度较大时,我们依次得到了Mallat算法逼近误差的渐近公式和比较精确的定量估计。结果表明:在这样的子波基下,直接用均匀抽样点代替子波抽样点而无需进行预先滤波,其逼近速度可以达到K阶,这里K是综合尺度函数的阶。模拟实验也显示出它的优点。     

子波变换及其应用

子波 近几年来兴起的一种新的时频域信号分析工具,在信息分析、图像处理、量子物理、计算机视觉领域有着广阔的应用前景。介绍了子波函数、多分辨分析等子波变换的有关概念和理论,论述了它在信号自理图像处理语音处理等几个方面的具体应用。