Asymptotic decorrelation of between-Scale Wavelet coefficients

In recent years there has been much interest in the analysis of time series using a discrete wavelet transform (DWT) based upon a Daubechies wavelet filter. Part of this interest has been sparked by the fact that the DWT approximately decorrelates certain stochastic processes, including stationary fractionally differenced (FD) processes with long memory characteristics and certain nonstationary processes such as fractional Brownian motion. It is shown that, as the width of the wavelet filter used to form the DWT increases, the covariance between wavelet coefficients associated with different scales decreases to zero for a wide class of stochastic processes. These processes are Gaussian with a spectral density function (SDF) that is the product of the SDF for a (not necessarily stationary) FD process multiplied by any bounded function that can serve as an SDF on its own. We demonstrate that this asymptotic theory provides a reasonable approximation to the between-scale covariance properties of wavelet coefficients based upon filter widths in common use. Our main result is one important piece of an overall strategy for establishing asymptotic results for certain wavelet-based statistics.     

一类最佳数据压缩FIR滤波器与Daubechies 小波

提出了一种设计数据压缩FIR滤波器的新方法．该方法在不引入信息失真的前提下以最有效分解原信号为目的设计滤波器．由此设计的滤波器可以实现数据基于压缩目的的最优分解,能够最大可能地提取信号中的有用信息同时去除冗余信息．该分解过程与具有一定正则性的Daubechies小波的离散小波变换十分相似,该滤波器也和Daubechies小波分解滤波器完全相同,其原因是Daubechies在构造小波时引入的余项应为零．     

Embolic Doppler ultrasound signal detection using discrete wavelet transform

Asymptomatic circulating emboli can be detected by Doppler ultrasound. Embolic Doppler ultrasound signals are short duration transient like signals. The wavelet transform is an ideal method for analysis and detection of such signals by optimizing time-frequency resolution. We propose a detection system based on the discrete wavelet transform (DWT) and study some parameters, which might be useful for describing embolic signals (ES). We used a fast DWT algorithm based on the Daubechies eighth-order wavelet filters with eight scales. In order to evaluate feasibility of the DWT of ES, two independent data sets, each comprising of short segments containing an ES (N=100), artifact (N=100) or Doppler speckle (DS) (N=100), were used. After applying the DWT to the data, several parameters were evaluated. The threshold values used for both data sets were optimized using the first data set. While the DWT coefficients resulting from artifacts dominantly appear at the higher scales (five, six, seven, and eight), the DWT coefficients at the lower scales (one, two, three, and four) are mainly dominated by ES and DS. The DWT is able to filter out most of the artifacts inherently during the transform process. For the first data set, 98 out of 100 ES were detected as ES. For the second data set, 95 out of 100 ES were detected as ES when the same threshold values were used. The algorithm was also tested with a third data set comprising 202 normal ES; 198 signals were detected as ES.     

Multiple scale correlation of signals by shift-invariant discrete wavelet transform

Correlation of signals at multiple scales of observation is useful for multiresolution interpretation of image, data and target signature analysis. Multiresolution analysis is inherent in the discrete wavelet transform (DWT), but shift-variance of the coefficients of the transform in dyadic orthogonal and biorthogonal basis spaces is the problem associated with it. Shift-variance of the transform and absence of a direct transform domain relationship make correlation of signals by the DWT inconvenient at multiple scales. The circulant shift property of the DWT coefficients is used in a novel way to produce correlation of signals at multiple scales with the critically sampled DWT only. The algorithm is derived in both discrete time and z-domain for signal vectors of finite duration. The algorithm is independent of signal waveform and wavelet kernel and is applied particularly for multiple scale correlation of radar signals, namely linear frequency modulated (LFM) chirp signals.     

The discrete wavelet transform: wedding the a trous and Mallatalgorithms

Two separately motivated implementations of the wavelet transform are brought together. It is observed that these algorithms are both special cases of a single filter bank structure, the discrete wavelet transform, the behavior of which is governed by the choice of filters. In fact, the a trous algorithm is more properly viewed as a nonorthonormal multiresolution algorithm for which the discrete wavelet transform is exact. Moreover, it is shown that the commonly used Lagrange a trous filters are in one-to-one correspondence with the convolutional squares of the Daubechies filters for orthonormal wavelets of compact support. A systematic framework for the discrete wavelet transform is provided, and conditions are derived under which it computes the continuous wavelet transform exactly. Suitable filter constraints for finite energy and boundedness of the discrete transform are also derived. Relevant signal processing parameters are examined, and it is observed that orthonormality is balanced by restrictions on resolution     

Image denoising with anisotropic bivariate shrinkage

This paper presents a novel image denoising algorithm based on the modeling of wavelet coefficients with an anisotropic bivariate Laplacian distribution function. The anisotropic bivariate Laplacian model not only captures the child-parent dependency between wavelet coefficients, but also fits the anisotropic property of the variances of wavelet coefficients in different scales of natural images. With this statistical model, we derive a closed-form anisotropic bivariate shrinkage function in the framework of Bayesian denoising and a new image denoising approach with local marginal variance estimation based on this newly derived shrinkage function is proposed in the discrete wavelet transform (DWT) domain. The proposed anisotropic bivariate shrinkage approach is also extended to the dual-tree complex wavelet transform (DT-CWT) domain to further improve the performance of image denoising. To take full advantage of DT-CWT, a more accurate noise variance estimator is proposed and the way the anisotropic bivariate shrinkage function applied to the magnitudes of DT-CWT coefficients is presented. Experiments were carried out in both the DWT and the DT-CWT domain to validate the effectiveness of the proposed method. Using a representative set of standard test images corrupted by additive white Gaussian noise, the simulation results show that the proposed method provides promising results and is competitive with the best wavelet-based denoising results reported in the literature both in terms of peak signal-to-noise ratio (PSNR) and in visual quality.     

Wavelet methods for spike detection in mouse renal sympathetic nerve activity

Abnormal autonomic nerve traffic has been associated with a number of peripheral neuropathies and cardiovascular disorders prompting the development of genetically altered mice to study the genetic and molecular components of these diseases. Autonomic function in mice can be assessed by directly recording sympathetic nerve activity. However, murine sympathetic spikes are typically detected using a manually adjusted voltage threshold and no unsupervised detection methods have been developed for the mouse. Therefore, we tested the performance of several unsupervised spike detection algorithms on simulated murine renal sympathetic nerve recordings, including an automated amplitude discriminator and wavelet-based detection methods which used both the discrete wavelet transform (DWT) and the stationary wavelet transform (SWT) and several wavelet threshold rules. The parameters of the wavelet methods were optimized by comparing basal sympathetic activity to postmortem recordings and recordings made during pharmacological suppression and enhancement of sympathetic activity. In general, SWT methods were found to outperform amplitude discriminators and DWT methods with similar wavelet coefficient thresholding algorithms when presented with simulations with varied mean spike rates and signal-to-noise ratios. A SWT method which estimates the noise level using a "noise-only" wavelet scale and then selectively thresholds scales containing the physiologically important signal information was found to have the most robust spike detection. The proposed noise-level estimation method was also successfully validated during pharmacological interventions.     

Analysis of error in the fixed-point implementation of two-dimensional discrete wavelet transforms

Computational error due to the fixed-point implementation of two-dimensional (2-D) discrete wavelet transform (DWT) is analyzed. This analysis is based on the exact knowledge of the DWT analysis and synthesis filters and the word length of the original image. In the fixed-point implementation, it is crucial to understand and analyze effects of finite precision in filters coefficients as well as rounding of intermediate calculations for the purpose of storage and/or transmission. Analyses and formulations are presented for both convolution and lifting approaches and they are validated by Monte Carlo simulations. The specific example used throughout this work is the lossy wavelet transformation used in the JPEG2000 compression standard.     

Multiview image coding using depth layers and an optimized bit allocation

In this paper, we present a novel wavelet-based compression algorithm for multiview images. This method uses a layer-based representation, where the 3-D scene is approximated by a set of depth planes with their associated constant disparities. The layers are extracted from a collection of images captured at multiple viewpoints and transformed using the 3-D discrete wavelet transform (DWT). The DWT consists of the 1-D disparity compensated DWT across the viewpoints and the 2-D shape-adaptive DWT across the spatial dimensions. Finally, the wavelet coefficients are quantized and entropy coded along with the layer contours. To improve the rate-distortion performance of the entire coding method, we develop a bit allocation strategy for the distribution of the available bit budget between encoding the layer contours and the wavelet coefficients. The achieved performance of our proposed scheme outperforms the state-of-the-art codecs for several data sets of varying complexity.     

Efficient wavelet prefilters with optimal time-shifts

A wavelet prefilter maps sample values of an analyzed signal to the scaling function coefficient input of standard discrete wavelet transform (DWT) algorithms. The prefilter is the inverse of a certain postfilter convolution matrix consisting of integer sample values of a noninteger-shifted wavelet scaling function. For the prefilter and the DWT algorithms to have similar computational complexity, it is often necessary to use a "short enough" approximation of the prefilter. In addition to well-known quadrature formula and identity matrix prefilter approximations, we propose a Neumann series approximation, which is a band matrix truncation of the optimal prefilter, and derive simple formulas for the operator norm approximation error. This error shows a dramatic dependence on how the postfilter noninteger shift is chosen. We explain the meaning of this shift in practical applications, describe how to choose it, and plot optimally shifted prefilter approximation errors for 95 different Daubechies, Symlet, and B-spline wavelets. Whereas the truncated inverse is overall superior, the Neumann filters are by far the easiest ones to compute, and for some short support wavelets, they also give the smallest approximation error. For example, for Daubechies 1-5 wavelets, the simplest Neumann prefilter provide an approximation error reduction corresponding to 100-10 000 times oversampling in a nonprefiltered system.     

Bottom-up motion compensated prediction in wavelet domain for spatially scalable video coding

A new in-band motion compensation algorithm for wavelet-based video coding is proposed: the bottom-up prediction algorithm (BUP). This algorithm overcomes the periodic shift-invariance of the discrete wavelet transform (DWT) and is formalised into prediction rules using filtering operations. The combination of all prediction rules of the BUP algorithm defines a new transform: the bottom-up overcomplete DWT or BUP ODWT, which is shift-invariant. The envisaged application for the BUP algorithm is spatially scalable wavelet video coding.     

探地雷达回波信号小波压缩方法研究

介绍了多分辨分析和小波变换的基本理论以及常用小波变换压缩数据的三种方法：只保留近似信号；全部保留近似信号及细节信号中的较大值；保留近似信号及细节信号中的较大值。将紧支集小波（Daubechies)和正交三次B-样条小波压缩探地雷达数据进行了对比，计算表明正交三次B-样条小波变换方法效果较好，而在全部保留近似信号及只保留细节信号中数值较大的系数时，压缩比大而重建雷达数据与原始雷达数据间的均方差较小。     

The double-density dual-tree DWT

This paper introduces the double-density dual-tree discrete wavelet transform (DWT), which is a DWT that combines the double-density DWT and the dual-tree DWT, each of which has its own characteristics and advantages. The transform corresponds to a new family of dyadic wavelet tight frames based on two scaling functions and four distinct wavelets. One pair of the four wavelets are designed to be offset from the other pair of wavelets so that the integer translates of one wavelet pair fall midway between the integer translates of the other pair. Simultaneously, one pair of wavelets are designed to be approximate Hilbert transforms of the other pair of wavelets so that two complex (approximately analytic) wavelets can be formed. Therefore, they can be used to implement complex and directional wavelet transforms. The paper develops a design procedure to obtain finite impulse response (FIR) filters that satisfy the numerous constraints imposed. This design procedure employs a fractional-delay allpass filter, spectral factorization, and filterbank completion. The solutions have vanishing moments, compact support, a high degree of smoothness, and are nearly shift-invariant.     

Image coding based on wavelet transform and uniform scalar dead zone quantizer

In this paper, a new wavelet transform image coding algorithm is presented. The discrete wavelet transform (DWT) is applied to the original image. The DWT coefficients are firstly quantized with a uniform scalar dead zone quantizer. Then the quantized coefficients are decomposed into four symbol streams: a binary significance map symbol stream, a binary sign stream, a position of the most significant bit (PMSB) symbol stream and a residual bit stream. An adaptive arithmetic coder with different context models is employed for the entropy coding of these symbol streams. Experimental results show that the compression performance of the proposed coding algorithm is competitive to other wavelet-based image coding algorithms reported in the literature.     

An efficient coding algorithm for the compression of ECG signals using the wavelet transform

A wavelet-based electrocardiogram (ECG) data compression algorithm is proposed in this paper. The ECG signal is first preprocessed, the discrete wavelet transform (DWT) is then applied to the preprocessed signal. Preprocessing guarantees that the magnitudes of the wavelet coefficients be less than one, and reduces the reconstruction errors near both ends of the compressed signal. The DWT coefficients are divided into three groups, each group is thresholded using a threshold based on a desired energy packing efficiency. A binary significance map is then generated by scanning the wavelet decomposition coefficients and outputting a binary one if the scanned coefficient is significant, and a binary zero if it is insignificant. Compression is achieved by 1) using a variable length code based on run length encoding to compress the significance map and 2) using direct binary representation for representing the significant coefficients. The ability of the coding algorithm to compress ECG signals is investigated, the results were obtained by compressing and decompressing the test signals. The proposed algorithm is compared with direct-based and wavelet-based compression algorithms and showed superior performance. A compression ratio of 24:1 was achieved for MIT-BIH record 117 with a percent root mean square difference as low as 1.08%.     

METHODS OF RADAR DATA COMPRESSION AND TARGET IDENTIFICATION BASED ON BIORTHOGONAL FDWT

In this paper, by using the biorthogonal quadrature filters, the biorthogonal mul-tiresolution analysis of finite dimension space equipped with inner product and the fast discrete wavelet transform (FDWT) are constructed. The dual transform method is proposed and the radar data storage is reduced by it. The method of choosing the wavelet coefficients, and the methods of correlation and nearest neighbor classification in wavelet domain based on the compressed data, are presented. The experimental results of the classification, using the high resolution range returns from six kinds of aircrafts, show that the methods of transform, compression and recognition are efficient.     

在线脑机接口中脑电信号的特征提取与分类方法

在脑机接口研究中,针对运动想象脑电信号的特征抽取,提出了一种基于离散小波变换和AR模型的方法.利用Daubechies类小波函数对脑电信号进行3层分解,抽取小波变换系数的统计特征;利用Burg算法提取脑电信号6阶AR模型系数.将这两类特征进行组合后使用神经网络、支持向量机、马氏距离线性判别进行分类并比较分析.采用BCI...     

多级多维离散小波变换的快速提升计算

提升方法是计算离散小波变换的有效手段,它由一系列的提升步和拉伸变换组成.在计算多级和多维离散小波变换时,现有方法在每一次小波分解的过程中都做完整的提升步计算和拉伸变换计算.我们发现该方法存在运算过程的冗余,为此本文提出了一种称之为后拉伸变换的提升方法,基本思想是计算完所有的提升步后,再统一进行拉伸变换.它能减少离散小波变换的乘法运算量.例如,对图像与视频压缩中应用广泛的Daubechies 9/7小波,做一维5级分解时与现有方法相比,乘法运算减少20%,而二维5级分解时,乘法运算减少28%.     

The slantlet transform

The discrete wavelet transform (DWT) is usually carried out by filterbank iteration; however, for a fixed number of zero moments, this does not yield a discrete-time basis that is optimal with respect to time localization. This paper discusses the implementation and properties of an orthogonal DWT, with two zero moments and with improved time localization. The basis is not based on filterbank iteration; instead, different filters are used for each scale. For coarse scales, the support of the discrete-time basis functions approaches two thirds that of the corresponding functions obtained by filterbank iteration. This basis, which is a special case of a class of bases described by Alpert (1992, 1993), retains the octave-band characteristic and is piecewise linear (but discontinuous). Closed-form expressions for the filters are given, an efficient implementation of the transform is described, and improvement in a denoising example is shown. This basis, being piecewise linear, is reminiscent of the slant transform, to which it is compared     

Multitaper power spectrum estimation and thresholding: wavelet packets versus wavelets

It was suggested that spectrum estimation can be accomplished by applying wavelet denoising methodology to wavelet packet coefficients derived from the logarithm of a spectrum estimate. The particular algorithm we consider consists of computing the logarithm of the multitaper spectrum estimator, applying an orthonormal transform derived from a wavelet packet tree to the log multitaper spectrum ordinates, thresholding the empirical wavelet packet coefficients, and then inverting the transform. For a small number of tapers, suitable transforms/partitions for the logarithm of the multitaper spectrum estimator are derived using a method matched to statistical thresholding properties. The partitions thus derived starting from different stationary time series are all similar and easily derived, and any differences between the wavelet packet and discrete wavelet transform (DWT) approaches are minimal. For a larger number of tapers, where the chosen parameters satisfy the conditions of a proven theorem, the simple DWT again emerges as appropriate. Hence, using our approach to thresholding and the method of partitioning, we conclude that the DWT approach is a very adequate wavelet-based approach and that the use of wavelet packets is unnecessary.