An effective wavelet matrix transform approach for efficientsolutions of electromagnetic integral equations

An effective numerical method based on wavelet matrix transforms for efficient solution of electromagnetic (EM) integral equations is proposed. Using the wavelet matrix transform produces highly sparse moment matrices which can be solved efficiently. A fast construction method for various orthonormal or nonorthonormal wavelet basis matrices is also given. It has been found that using nonsimilarity wavelet matrix transforms such as nonsimilarity nonorthonormal cardinal spline wavelet (NSNCSW) transform, one can obtain a much higher compression rate and much better accuracy of the approximate solutions than using similarity wavelet transforms such as Daubechies' (1992) orthonormal wavelet (DOW) transform. Numerical examples are given to show the validity and effectiveness of the method     

Construction of Hilbert Transform Pairs of Wavelet Bases and Gabor-Like Transforms

We propose a novel method for constructing Hilbert transform (HT) pairs of wavelet bases based on a fundamental approximation-theoretic characterization of scaling functions—the B-spline factorization theorem. In particular, starting from well-localized scaling functions, we construct HT pairs of biorthogonal wavelet bases of ${rm L}^2({BBR})$ by relating the corresponding wavelet filters via a discrete form of the continuous HT filter. As a concrete application of this methodology, we identify HT pairs of spline wavelets of a specific flavor, which are then combined to realize a family of complex wavelets that resemble the optimally-localized Gabor function for sufficiently large orders. Analytic wavelets, derived from the complexification of HT wavelet pairs, exhibit a one-sided spectrum. Based on the tensor-product of such analytic wavelets, and, in effect, by appropriately combining four separable biorthogonal wavelet bases of ${rm L}^2({BBR}^2)$, we then discuss a methodology for constructing 2-D directional-selective complex wavelets. In particular, analogous to the HT correspondence between the components of the 1-D counterpart, we relate the real and imaginary components of these complex wavelets using a multidimensional extension of the HT—the directional HT. Next, we construct a family of complex spline wavelets that resemble the directional Gabor functions proposed by Daugman. Finally, we present an efficient fast Fourier transform (FFT)-based filterbank algorithm for implementing the associated complex wavelet transform.      

On a choice of wavelet bases in the wavelet transform approach

The Daubechies orthogonal wavelet (DOW) is compared with the nonorthogonal cardinal spline wavelet (NCSW) in the wavelet transform approach and it is shown that the DOW is better than the NCSW in view of the computation cost. First, the computation cost required for the wavelet transform based on the DOW is less than that based on the NCSW because the DOW has smaller support provided the same number of vanishing moments of wavelets is used. Second, in contrast with the fact that the wavelet transform based on the DOW does not affect the condition number of the impedance matrix, that, based on the NCSW, has an effect to make it very large. As a result, even though the NCSW results in a sparser impedance matrix, it requires more computation cost for solving the resultant matrix equation in comparison with the DOW because the cost depends not only on the sparsity, but also on the condition number of the matrix.     

具有紧支采样函数的子波空间采样定理

从复制核Hilbert空间的观点出发，本文详细地讨论了子波空间采样定理，提出了子波空间推广的主尺度函数概念，证明了它是构造紧支子波空间采样函数的充要条件，从而得到具有紧支采样函数的子波空间采样定理。本文还详细地研究了推广的主正交尺度函数的性质，证明了紧支的推广主正交尺度函数所对应的子波函数仅有一阶消失矩，采样函数的紧支性和所对应的子波函数的光滑性是不可兼得的。     

基于方向正交小波基的声源定位

引入一种光滑性可调的对信号的方向性敏感的周期基插值小波,先对其进行正交化处理,构成二维空间的方向正交小波基,同时对相应的滤波器周期化,给出了快速方向小波变换的算法。对于方向性弱的声音信号可以选择阶数相对高的周期基插值小波来分析,而对于方向性强的信号只要阶数低的周期基插值小波就可以。用传声器阵列对声音信号进行采集,对采集的数据进行快速方向正交小波变换,通过角度和半径方向的图,得到两条比较清晰的半径线,它们的交点唯一确定声源的位置。     

Interpolating multiwavelet bases and the sampling theorem

This paper considers the classical sampling theorem in multiresolution spaces with scaling functions as interpolants. As discussed by Xia and Zhang (1993), for an orthogonal scaling function to support such a sampling theorem, the scaling function must be cardinal (interpolating). They also showed that the only orthogonal scaling function that is both cardinal and of compact support is the Haar function, which is not continuous. This paper addresses the same question, but in the multiwavelet context, where the situation is different. This paper presents the construction of compactly supported orthogonal multiscaling functions that are continuously differentiable and cardinal. The scaling functions thereby support a Shannon-like sampling theorem. Such wavelet bases are appealing because the initialization of the discrete wavelet transform (prefiltering) is the identity operator     

Fast implementation of the continuous wavelet transform withinteger scales

We describe a fast noniterative algorithm for the evaluation of continuous spline wavelet transforms at any integer scale m. In this approach, the input signal and the analyzing wavelet are both represented by polynomial splines. The algorithm uses a combination of moving sum and zero-padded filters, and its complexity per scale is O(N), where N is the signal length. The computation is exact, and the implementation is noniterative across scales. We also present examples of spline wavelets exhibiting properties that are desirable for either singularity detection (first and second derivative operators) or Gabor-like time-frequency signal analysis     

半导体激光器LIV测试中阈值电流的小波提取方法

针对传统半导体激光器阈值提取方法容易受噪声影响、存在重复性不好的问题,根据小波变换奇异性检测原理,提出一种采用小波变换的阈值电流提取法,其采用三阶样条函数的二倍膨胀作为光滑函数,取它的二阶导数为小波函数,并结合多尺度来对半导体激光器光功率-电流（L-I）信号中的奇异点和不规则突变点进行检测分析。实验结果表明,与传统的二段直线拟合法、一次微分法和二次微分法相比,小波提取法不受噪声影响,能够真实、准确地得到半导体激光器的阈值电流。     

Shift-orthogonal wavelet bases

Shift-orthogonal wavelets are a new type of multiresolution wavelet bases that are orthogonal with respect to translation (or shifts) within one level but not with respect to dilations across scales. We characterize these wavelets and investigate their main properties by considering two general construction methods. In the first approach, we start by specifying the analysis and synthesis function spaces and obtain the corresponding shift-orthogonal basis functions by suitable orthogonalization. In the second approach, we take the complementary view and start from the digital filterbank. We present several illustrative examples, including a hybrid version of the Battle-Lemarie (1987, 1988) spline wavelets. We also provide filterbank formulas for the fast wavelet algorithm. A shift-orthogonal wavelet transform is closely related to an orthogonal transform that uses the same primary scaling function; both transforms have essentially the same approximation properties. One experimentally confirmed benefit of relaxing the interscale orthogonality requirement is that we can design wavelets that decay faster than their orthogonal counterpart     

基于小波变换的扩频系统PN码捕获研究

由于小波分析具有多分辨率的特点,使得利用小波分析的工具在对突变信号的检测上提取信息方面有着很强的优势。利用小波的这一性质提出了一种小波捕获的方法。通过实际仿真证明B样条小波对有噪声污染的调制信号和白噪声信号的检测更优越;并基于Harr小波提出了一种步进快速捕获方法,通过理论分析,证明该方法可有效减少捕获时间。     

Generalized smoothing splines and the optimal discretization of the Wiener filter

We introduce an extended class of cardinal L/sup */L-splines, where L is a pseudo-differential operator satisfying some admissibility conditions. We show that the L/sup */L-spline signal interpolation problem is well posed and that its solution is the unique minimizer of the spline energy functional /spl par/Ls/spl par//sub L2//sup 2/, subject to the interpolation constraint. Next, we consider the corresponding regularized least squares estimation problem, which is more appropriate for dealing with noisy data. The criterion to be minimized is the sum of a quadratic data term, which forces the solution to be close to the input samples, and a "smoothness" term that privileges solutions with small spline energies. Here, too, we find that the optimal solution, among all possible functions, is a cardinal L/sup */L-spline. We show that this smoothing spline estimator has a stable representation in a B-spline-like basis and that its coefficients can be computed by digital filtering of the input signal. We describe an efficient recursive filtering algorithm that is applicable whenever the transfer function of L is rational (which corresponds to the case of exponential splines). We justify these algorithms statistically by establishing an equivalence between L/sup */L smoothing splines and the minimum mean square error (MMSE) estimation of a stationary signal corrupted by white Gaussian noise. In this model-based formulation, the optimum operator L is the whitening filter of the process, and the regularization parameter is proportional to the noise variance. Thus, the proposed formalism yields the optimal discretization of the classical Wiener filter, together with a fast recursive algorithm. It extends the standard Wiener solution by providing the optimal interpolation space. We also present a Bayesian interpretation of the algorithm.     

A local interpolatory cardinal spline method for the determinationof eigenstates in quantum-well structures with arbitrary potentialprofiles

A numerical method based on a variational approach and local interpolatory cardinal spline (LICS) functions is developed to compute the eigenstates of a quantum well with an arbitrary potential profile. The formulation of this method is straightforward, while the numerical solution is more accurate than results obtained by using other methods in the published literature     

Frequency domain volume rendering by the wavelet X-ray transform

We describe a wavelet based X-ray rendering method in the frequency domain with a smaller time complexity than wavelet splatting. Standard Fourier volume rendering is summarized and interpolation and accuracy issues are briefly discussed. We review the implementation of the fast wavelet transform in the frequency domain. The wavelet X-ray transform is derived, and the corresponding Fourier-wavelet volume rendering algorithm (FWVR) is introduced, FWVR uses Haar or B-spline wavelets and linear or cubic spline interpolation. Various combinations are tested and compared with wavelet splatting (WS). We use medical MR and CT scan data, as well as a 3-D analytical phantom to assess the accuracy, time complexity, and memory cost of both FWVR and WS. The differences between both methods are enumerated.     

Fully automated reduction of ocular artifacts in high-dimensional neural data

The reduction of artifacts in neural data is a key element in improving analysis of brain recordings and the development of effective brain-computer interfaces. This complex problem becomes even more difficult as the number of channels in the neural recording is increased. Here, new techniques based on wavelet thresholding and independent component analysis (ICA) are developed for use in high-dimensional neural data. The wavelet technique uses a discrete wavelet transform with a Haar basis function to localize artifacts in both time and frequency before removing them with thresholding. Wavelet decomposition level is automatically selected based on the smoothness of artifactual wavelet approximation coefficients. The ICA method separates the signal into independent components, detects artifactual components by measuring the offset between the mean and median of each component, and then removing the correct number of components based on the aforementioned offset and the power of the reconstructed signal. A quantitative method for evaluating these techniques is also presented. Through this evaluation, the novel adaptation of wavelet thresholding is shown to produce superior reduction of ocular artifacts when compared to regression, principal component analysis, and ICA.     

用二次B样条小波进行图像的自适应阈值边缘检测

根据边缘检测的评价标准 ,参照最佳边缘滤波器的设计要求 ,确定选择用于边缘检测的小波母函数的一般准则 ,并在此基础上构造出二次B样条小波 ,提出了基于小波变换的自适应阈值图像边缘检测的新方法。通过计算机仿真对该算法进行了验证 ,结果明显好于采用固定阈值的小波边缘检测。     

重构小波系数的分段三次样条播值新算法

利用小波变换模极大值原理对信号去噪之后,如何由保留下来的模极大值点恢复出满意的重构信号,是一个重要课题。本文首先分析模极大值与小波系数之间的内在关系,提出了模极大值实际上是小波系数在特定意义下的离散采样;然后给出了一种对模极大值进行预处理的方法,由此得到了一组新的伪模极大值序列;利用这组伪模极大值序列,提出了一种新的重构小波系数的分段三次样条播值（ＰＣＳＩ）新算法,该算法程序简单,易实现,克服了交替投影（ＡＰ）法计算量大、程序复杂等缺点;最后给出一个应用实例,实验结果表明,与经典的交替投影法相比,本文提出的ＰＣＳＩ算法可获得更高的重构信号信噪比增益和更小的相对均方误差,它是一种实际、有效的算法。     

Multiresolution Monogenic Signal Analysis Using the Riesz–Laplace Wavelet Transform

The monogenic signal is the natural 2D counterpart of the 1D analytic signal. We propose to transpose the concept to the wavelet domain by considering a complexified version of the Riesz transform which has the remarkable property of mapping a real-valued (primary) wavelet basis of L₂(R²) into a complex one. The Riesz operator is also steerable in the sense that it give access to the Hilbert transform of the signal along any orientation. Having set those foundations, we specify a primary polyharmonic spline wavelet basis of L₂(R²) that involves a single Mexican-hat-like mother wavelet (Laplacian of a B-spline). The important point is that our primary wavelets are quasi-isotropic: they behave like multiscale versions of the fractional Laplace operator from which they are derived, which ensures steerability. We propose to pair these real-valued basis functions with their complex Riesz counterparts to specify a multiresolution monogenic signal analysis. This yields a representation where each wavelet index is associated with a local orientation, an amplitude and a phase. We give a corresponding wavelet-domain method for estimating the underlying instantaneous frequency. We also provide a mechanism for improving the shift and rotation-invariance of the wavelet decomposition and show how to implement the transform efficiently using perfect-reconstruction filterbanks. We illustrate the specific feature-extraction capabilities of the representation and present novel examples of wavelet-domain processing; in particular, a robust, tensor-based analysis of directional image patterns, the demodulation of interferograms, and the reconstruction of digital holograms.     

Novel phase difference extraction method of FPP system based on DWT and OMP algorithm

In this paper, to reduce the non-linear phase error caused by gamma effect of the digital projector, a novel phase difference extraction method is proposed. First, through multi-layer discrete wavelet decomposition(DWT), the components of the original phase difference at different frequencies are extracted. Subsequently, the effective components are taken by the orthogonal matching pursuit(OMP) algorithm, and the phase difference is reconstructed. According to the upper and lower envelopes of the original phase difference extracted by cubic spline interpolation, the residual threshold of OMP is set. According to the results of three-dimensional(3-D) reconstruction experiment on a standard sphere, the proposed method is capable of reducing the errors in the original phase difference and achieving a better imaging result.     

一种基于小波变换的独立语音量分离方法

提出了一种基于小波变换的独立语音量分离方法,以改进卷积叠加语音分离方法.该方法在小波域进行独立元分析,把高阶去相关矩阵分解为小波子空间中一系列相应的低阶去相关矩阵,并在保证语音短时平稳性的基础上通过低阶处理跟踪时变的室内(存在室内回响)传输函数.实验结果表明,提出的方法改善了传统独立元分析在室内回响环境中独立语音量分离的效果.     

Analysis of pattern reversal visual evoked potentials (PRVEPs) byspline wavelets

The pattern-reversal visual evoked potentials (PRVEPs) collected from normal and demented subjects are investigated by applying the quadratic spline wavelet analysis. The data are decomposed into six octave frequency bands. For quantitative purposes, the wavelet coefficients in the residual waveform representing the delta-theta band activity (0-8 Hz) are explored to characterize the (N70-P100-N130) complex. Specifically, the coefficients corresponding to the location of N70, P100, and N130 peaks are investigated for their sign in order to test whether they represent a consistent (N70-P100-N130) complex in the averaged waveform. Waveforms with normal latency (N70-P100-N130) complex are observed to have positive second, negative third, and positive fourth coefficients in amplitude in their residual scale standing for the delta-theta (0-8 Hz) band activity. The method allows for the analysis of oscillatory-phase behaviour of the normal and pathological PRVEPs in their delta-theta band based on a few quantitative measures consistent with the time-frequency occurrence of the major components of the evoked potential