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.     

Spectral Relations for Multidimensional Complex Improper Stationary and (Almost) Cyclostationary Processes

We study continuous-time multidimensional wide- sense stationary (WSS) and (almost) cyclostationary processes in the frequency domain. Under the assumption that the correlation function is uniformly continuous, we prove the existence of a unique sequence of spectral measures, which coincide with the restrictions to certain subdiagonals of the spectral measure in the strongly harmonizable case. Moreover, the off-diagonal measures are absolutely continuous with respect to the diagonal measure. As a consequence, for strongly harmonizable scalar improper almost cyclostationary processes, we obtain representation formulas for the components of the complementary spectral measure and the off-diagonal components of the spectral measure, in terms of the diagonal component of the spectral measure. We apply these results to analytic signals, which produces sufficient conditions for propriety for almost cyclostationary analytic signals.     

A binary wavelet decomposition of binary images

We construct a theory of binary wavelet decompositions of finite binary images. The new binary wavelet transform uses simple module-2 operations. It shares many of the important characteristics of the real wavelet transform. In particular, it yields an output similar to the thresholded output of a real wavelet transform operating on the underlying binary image. We begin by introducing a new binary field transform to use as an alternative to the discrete Fourier transform over GF(2). The corresponding concept of sequence spectra over GF(2) is defined. Using this transform, a theory of binary wavelets is developed in terms of two-band perfect reconstruction filter banks in GF(2). By generalizing the corresponding real field constraints of bandwidth, vanishing moments, and spectral content in the filters, we construct a perfect reconstruction wavelet decomposition. We also demonstrate the potential use of the binary wavelet decomposition in lossless image coding.     

A theory of polyspectra for nonstationary stochastic processes

Harmonizable processes constitute an important class of nonstationary stochastic processes. We present a theory of polyspectra (higher order moment spectra) for the harmonizable class. We define and discuss four basic quantities: the nth-order moment function, the nth-order time-frequency polyspectrum, the nth-order ambiguity function, and the nth-order frequency-frequency polyspectrum. The latter generalizes the conventional polyspectrum to nonstationary stochastic processes. These four functions are related to one another by Fourier transforms. We show that the frequency and time marginals of the time-frequency polyspectrum are the instantaneous nth-order moment and the conventional nth-order stationary polyspectrum, respectively. All quantities except the nth-order ambiguity function allow for insightful interpretations in terms of Hilbert space inner products. The inner product picture leads to two novel and very powerful definitions of polycoherence for a nonstationary stochastic process. The polycoherences are objective measures of stationarity to order n, which can be used to construct various statistical tests. Finally, we give some specific examples and apply the theory to linear time-varying systems, which are popular models for fading multipath communication channels.     

Kullback-Leibler approximation of spectral density functions

We introduce a Kullback-Leibler (1968) -type distance between spectral density functions of stationary stochastic processes and solve the problem of optimal approximation of a given spectral density /spl Psi/ by one that is consistent with prescribed second-order statistics. In general, such statistics are expressed as the state covariance of a linear filter driven by a stochastic process whose spectral density is sought. In this context, we show (i) that there is a unique spectral density /spl Phi/ which minimizes this Kullback-Leibler distance, (ii) that this optimal approximate is of the form /spl Psi//Q where the "correction term" Q is a rational spectral density function, and (iii) that the coefficients of Q can be obtained numerically by solving a suitable convex optimization problem. In the special case where /spl Psi/ = 1, the convex functional becomes quadratic and the solution is then specified by linear equations.     

短期电力负荷在小波包分解下的径向基神经网络预测方法

针对短期电力负荷预测问题,提出一种在小波包分解下的径向基神经网络预测方法。通过小波包分析,将电力载荷及其温度变量对称地分解为低频的近似系数和高频的细节系数。针对不同的小波系数,设计径向基神经网络作为预测器,并通过试错法确定网络合适的结构。网络的训练过程中,采用滑动窗口数据选择策略减少数据样本集,采用随机梯度法更新权值、中心位置和扩展参数。预测的小波系数用于重构出最终的电力载荷值。与前馈多层神经网络的对比数值,实验结果表明,新提出的方法具有较高的预测准确性。     

Estimation of the Number of Decomposition Levels for a Wavelet-Based Multiresolution Multisensor Image Fusion

The wavelet-based scheme for the fusion of multispectral (MS) and panchromatic (PAN) imagery has become quite popular due to its ability to preserve the spectral fidelity of the MS imagery while improving its spatial quality. This is important if the resultant imagery is used for automatic classification. Wavelet-based fusion results depend on the number of decomposition levels applied in the wavelet transform. Too few decomposition levels result in poor spatial quality fused images. On the other hand, too many levels reduce the spectral similarity between the original MS and the pan-sharpened images. If the shift-invariant wavelet transform is applied, each excessive decomposition level results in a large computational penalty. Thus, the choice of the number of decomposition levels is significant. In this paper, PAN and MS image pairs with different resolution ratios were fused using the shift-invariant wavelet transform, and the optimal decomposition levels were determined for each resolution ratio. In general, it can be said that the fusion of images with larger resolution ratios requires a higher number of decomposition levels. This paper provides the practitioner an understanding of the tradeoffs associated with the computational demand and the spatial and spectral quality of the wavelet-based fusion algorithm as a function of the number of decomposition levels     

一种新的二维非线性提升小波变换方法

根据图像的统计信息,本文构造了一种新的非线性算子即统计算子,提出了基于该算子的一种新的非线性提升小波分析方法.使图像经过该方法变换以后,在无量化失真的前提下,以较大概率取得零高频系数.本文将该方法与现存文献中所提出的非线性形态学小波等分析方法,进行了标准图像的测试分析,实验结果显示,利用本文所提出的基于统计算子的提升小波分析的方法所得到的高频子带的熵都低于其它几种非线性小波变换,取得了很好的分析结果.     

一种适于计算声场景分析的混叠语音基音检测方法

本文提出了一种在混叠语音信号中检测各自语音分量基音信息的方法.该方法采用小波变换作为基音检测模型中的滤波处理,并用广义自相关运算突出基音信息,用增强自相关累和消除冗余信息,并提出了用基音概率函数来预测并跟踪不同基音的变化以提高基音检测的准确性.本文提出的方法可应用于计算声场景分析中.实验结果表明,该方法对于混叠语音的基音检测是非常有效的.     

Fundamental tensor operations for large-scale data analysis using tensor network formats

We discuss extended definitions of linear and multilinear operations such as Kronecker, Hadamard, and contracted products, and establish links between them for tensor calculus. Then we introduce effective low-rank tensor approximation techniques including Candecomp/Parafac, Tucker, and tensor train (TT) decompositions with a number of mathematical and graphical representations. We also provide a brief review of mathematical properties of the TT decomposition as a low-rank approximation technique. With the aim of breaking the curse-of-dimensionality in large-scale numerical analysis, we describe basic operations on large-scale vectors, matrices, and high-order tensors represented by TT decomposition. The proposed representations can be used for describing numerical methods based on TT decomposition for solving large-scale optimization problems such as systems of linear equations and symmetric eigenvalue problems.     

Kernels and Multiple Windows for Estimation of the Wigner-Ville Spectrum of Gaussian Locally Stationary Processes

This paper treats estimation of the Wigner-Ville spectrum (WVS) of Gaussian continuous-time stochastic processes using Cohen's class of time-frequency representations of random signals. We study the minimum mean square error estimation kernel for locally stationary processes in Silverman's sense, and two modifications where we first allow chirp multiplication and then allow nonnegative linear combinations of covariances of the first kind. We also treat the equivalent multitaper estimation formulation and the associated problem of eigenvalue-eigenfunction decomposition of a certain Hermitian function. For a certain family of locally stationary processes which parametrizes the transition from stationarity to nonstationarity, the optimal windows are approximately dilated Hermite functions. We determine the optimal coefficients and the dilation factor for these functions as a function of the process family parameter     

Time-invariant orthonormal wavelet representations

A simple construction of an orthonormal basis starting with a so-called mother wavelet, together with an efficient implementation gained the wavelet decomposition easy acceptance and generated a great research interest in its applications. An orthonormal basis may not, however, always be a suitable representation of a signal, particularly when time (or space) invariance is a required property. The conventional way around this problem is to use a redundant decomposition. We address the time-invariance problem for orthonormal wavelet transforms and propose an extension to wavelet packet decompositions. We show that it,is possible to achieve time invariance and preserve the orthonormality. We subsequently propose an efficient approach to obtain such a decomposition. We demonstrate the importance of our method by considering some application examples in signal reconstruction and time delay estimation     

New perspectives and improvements on the symmetric extension filter bank for subband/wavelet image compression.

In subband/wavelet image coding, size-limited subband decompositions are ordinarily used to avoid increasing the number of samples that need to be coded. To reduce coding distortions that can occur at the borders, the symmetric extension filter bank is typically employed. This paper introduces some new perspectives and improvements to that decomposition. The symmetric extension filter bank is couched in the cyclic frequency domain, providing a framework that accommodates FIR and IIR filters in a natural way, all with perfect reconstruction. IIR filters with both rational and irrational transfer functions can be implemented and, in the context of symmetric extension, can accommodate IIRs that effectively have perfect stopband suppression. Enhancements to the filter bank at a tree-structured system level are also presented and include the application of spectral reversal correction and a transition band normalization approach to designing the constituent filters of the symmetric extension wavelet packet transform.     

A noise reduction technique of speech signal using ICA and spectral analysis

This paper discusses the new method on noise reduction exploiting the combined effects of wavelet decomposition, ICA and spectral analysis on noisy speech. The input noisy speech is wavelet decomposed into two signals. Wavelet entropy is computed based on the modified probability density function for the signal derived from the approximation coefficients during wavelet decomposition. By proper entropy comparison, the starting frame is detected. Between the two signals obtained from the wavelet decomposition, one is speech combined with noise and another one is noise alone. These two signals are analysed in independent component analysis (ICA) domain, in order to generate an enhanced speech. Zero-crossing rate is computed and used to discriminate between speech and noise. Then, spectral analysis is performed on the noise prior to starting frame and noisy speech. Elimination of noise frequencies in the noisy speech leads to noise reduced speech. Subjective analysis and experimental results show the considerable noise reduction capability of the proposed algorithm.     

Nonlinear multiresolution signal decomposition schemes. I.Morphological pyramids

Interest in multiresolution techniques for signal processing and analysis is increasing steadily. An important instance of such a technique is the so-called pyramid decomposition scheme. This paper presents a general theory for constructing linear as well as nonlinear pyramid decomposition schemes for signal analysis and synthesis. The proposed theory is based on the following ingredients: 1) the pyramid consists of a (finite or infinite) number of levels such that the information content decreases toward higher levels and 2) each step toward a higher level is implemented by an (information-reducing) analysis operator, whereas each step toward a lower level is implemented by an (information-preserving) synthesis operator. One basic assumption is necessary: synthesis followed by analysis yields the identity operator, meaning that no information is lost by these two consecutive steps. Several examples of pyramid decomposition schemes are shown to be instances of the proposed theory: a particular class of linear pyramids, morphological skeleton decompositions, the morphological Haar pyramid, median pyramids, etc. Furthermore, the paper makes a distinction between single-scale and multiscale decomposition schemes, i.e., schemes without or with sample reduction. Finally, the proposed theory provides the foundation of a general approach to constructing nonlinear wavelet decomposition schemes and filter banks.     

Efficient Block-Based Frequency Domain Wavelet Transform Implementations

Subband decompositions for image coding have been explored extensively over the last few decades. The condensed wavelet packet (CWP) transform is one such decomposition that was recently shown to have coding performance advantages over conventional decompositions. A special feature of the CWP is that its design and implementation are performed in the cyclic frequency domain. While performance gains have been reported, efficient implementations of the CWP (or more generally, efficient implementations of cyclic filter banks) have not yet been fully explored. In this paper, we present efficient block-based implementations of cyclic filter banks along with an analysis of the arithmetic complexity. Block-based cyclic filter bank implementations of the CWP coder are compared with conventional subband/wavelet image coders whose filter banks are implemented in the time domain. It is shown that block-based cyclic filter bank implementations can result in CWP coding systems that outperform the popular image coding systems both in terms of arithmetic complexity and coding performance.      

基于小波分析的PPL过程尺度指数估计

长记忆随机过程是一类重要的随机过程,可以将其建摸为完全幂规律（PPL）过程。鉴于PPL过程的尺度指数5能完全刻画其二阶统计特性,从而使得5的估计成为完成数学模型的关键。考虑到小波滤波器的近似带通特性以及平稳小波变换的性质,文章提出了一种基于小波分析的平稳FD过程分形指数估计的新方法。首先对过程进行平稳小波变换以获得各个尺度下的子过程,随后给出这些子过程方差的无偏估计,最后建立方差与尺度的函数关系,并在对数意义下对方差和尺度作线性回归,从而完成估计。计算机仿真表明该方法具有较高精度。     

Methods of calculation of nonstationary nonharmonic fields in guiding electrodynamic structures

Methods of calculation of nonstationary nonharmonic electromagnetic fields excited by electron streams in guiding electrodynamic structures (ESs) of microwave devices are considered. The methods can be applied in first- and second-level spectral models. The proposed discrete approximation is based on decompositions of the field in normal or partial modes of ESs. The continuous approximation proposed is based on decomposition of the field in regular modes of ESs.     

Multiresolution coding techniques for digital television: A review

Multiresolution decompositions for video coding are reviewed. Both nonrecursive and recursive coding schemes are considered. In nonrecursive schemes, it is shown that pyramid structures have certain advantages over subband or wavelet techniques, and a specific spatiotemporal pyramid coding of HDTV is discussed in some detail. It is shown that recursive, DPCM like schemes will incur a slight loss of optimality due to a restricted form of prediction if multiresolution decomposition with compatible decoding is required. Compatibility and transmission issues are also discussed. Multiresolution transmission for digital broadcast TV is introduced. This, when combined with multiresolution source coding, achieves spectrum efficiency, robustness and graceful degradation under channel impairments.Invited PaperWork supported in part by the National Science Foundation under grants ECD-88-11111, MIP-90-14189 and Bell Communications Research.Work supported by the National Science Foundation under grants ECD-88-11111. K.M. Uz is now with David Sarnoff Research Center in Princeton, NJ 08543.     

On the number of rectangular tilings.

Adaptive multiscale representations via quadtree splitting and two-dimensional (2-D) wavelet packets, which amount to space and frequency decompositions, respectively, are powerful concepts that have been widely used in applications. These schemes are direct extensions of their one-dimensional counterparts, in particular, by coupling of the two dimensions and restricting to only one possible further partition of each block into four subblocks. In this paper, we consider more flexible schemes that exploit more variations of multidimensional data structure. In the meantime, we restrict to tree-based decompositions that are amenable to fast algorithms and have low indexing cost. Examples of these decomposition schemes are anisotropic wavelet packets, dyadic rectangular tilings, separate dimension decompositions, and general rectangular tilings. We compute the numbers of possible decompositions for each of these schemes. We also give bounds for some of these numbers. These results show that the new rectangular tiling schemes lead to much larger sets of 2-D space and frequency decompositions than the commonly-used quadtree-based schemes, therefore bearing the potential to obtain better representation for a given image.