Quasi-exchange Formulas for Wavelet Transforms of Distributions

We investigate the wavelet transforms of tempered distributions in a way that closely links their Fourier transforms and wavelet transforms. Two exchange formulas of the convolution and the multiplication of wavelet transforms of tempered distributions are established. We call these formulas the quasi-exchange formulas for wavelet transforms of distributions, because of the resemblance between these formulas and the well-known exchange formula for Fourier transforms.     

Pseudo-exchange Formulas for Wavelet Transforms of Distributions

We study the wavelet transforms of tempered distributions via unitary irreducible representations, Fourier transforms, and the wavelet transforms of infinitely differentiable functions of rapid descent. We explore the general properties, and derive the pseudo-exchange formulas for wavelet transforms of tempered distributions. While the new formulas are rather different from the exchange formula for Fourier transforms of distributions, yet the pseudo-exchange formulas still exchange the products of convolution and the products of multiplication of tempered distributions.     

Numerical stability of biorthogonal wavelet transforms

Biorthogonal wavelets are essential tools for numerous practical applications. It is very important that wavelet transforms work numerically stable in floating point arithmetic. This paper presents new results on the worst-case analysis of roundoff errors occurring in floating point computation of periodic biorthogonal wavelet transforms, i.e. multilevel wavelet decompositions and reconstructions. Both of these wavelet algorithms can be realized by matrix–vector products with sparse structured matrices. It is shown that under certain conditions the wavelet algorithms can be remarkably stable. Numerous tests demonstrate the performance of the results.      

Frames and Coorbit Theory on Homogeneous Spaces with a Special Guidance on the Sphere

The topic of this article is a generalization of the theory of coorbit spaces and related frame constructions to Banach spaces of functions or distributions over domains and manifolds. As a special case one obtains modulation spaces and Gabor frames on spheres. Group theoretical considerations allow first to introduce generalized wavelet transforms. These are then used to define coorbit spaces on homogeneous spaces, which consist of functions having their generalized wavelet transform in some weighted L_p space. We also describe natural ways of discretizing those wavelet transforms, or equivalently to obtain atomic decompositions and Banach frames for the corresponding coorbit spaces. Based on these facts we treat aspects of nonlinear approximation and show how the new theory can be applied to the Gabor transform on spheres. For the S¹ we exhibit concrete examples of admissible Gabor atoms which are very closely related to uncertainty minimizing states.     

Detection of Singularities by Discrete Multiscale Directional Representations

One of the most remarkable properties of the continuous curvelet and shearlet transforms is their sensitivity to the directional regularity of functions and distributions. As a consequence of this property, these transforms can be used to characterize the geometry of edge singularities of functions and distributions by their asymptotic decay at fine scales. This ability is a major extension of the conventional continuous wavelet transform which can only describe pointwise regularity properties. However, while in the case of wavelets it is relatively easy to relate the asymptotic properties of the continuous transform to properties of discrete wavelet coefficients, this problem is surprisingly challenging in the case of discrete curvelets and shearlets where one wants to handle also the geometry of the singularity. No result for the discrete case was known so far. In this paper, we derive non-asymptotic estimates showing that discrete shearlet coefficients can detect, in a precise sense, the location and orientation of curvilinear edges. We discuss connections and implications of this result to sparse approximations and other applications.     

Wavelet transforms and edge detectors on digital images

The paper investigates the problem of locating edges on digital images. The process is completed through various mathematical transforms. The edges are found by implementing mathematical algorithms for convolutions that are adapted to locate the edges and highlights the technique on using wavelet transforms. We are especially addressing the problem of edge detection, because it has far reaching applications to all fields and especially within medicine. A patient can be diagnosed as having an aneurysm by studying an angiogram. An angiogram is the visual view of the blood vessels whereby the edges are highlighted through the implementation of edge detectors. This process can be completed through wavelet transforms and other convolution techniques.     

Discretizing continuous wavelet transforms using integrated wavelets

We present integrated wavelets as a method for discretizing the continuous wavelet transform. Using the language of group theory, the results are presented for wavelet transforms over semidirect product groups. We obtain tight wavelet frames for these wavelet transforms. Further integrated wavelets yield tight families of convolution operators independent of the choice of discretization of scale and orientation parameters. Thus these families can be adapted to specific problems. The method is more flexible than the well-known dyadic wavelet transform. We state an exact algorithm for implementing this transform. As an application the enhancement of digital mammograms is presented.     

Numerical Approximation of Probability Mass Functions via the Inverse Discrete Fourier Transform

First passage distributions of semi-Markov processes are of interest in fields such as reliability, survival analysis, and many others. Finding or computing first passage distributions is, in general, quite challenging. We take the approach of using characteristic functions (or Fourier transforms) and inverting them to numerically calculate the first passage distribution. Numerical inversion of characteristic functions can be unstable for a general probability measure. However, we show they can be quickly and accurately calculated using the inverse discrete Fourier transform for lattice distributions. Using the fast Fourier transform algorithm these computations can be extremely fast. In addition to the speed of this approach, we are able to prove a few useful bounds for the numerical inversion error of the characteristic functions. These error bounds rely on the existence of a first or second moment of the distribution, or on an eventual monotonicity condition. We demonstrate these techniques with two examples.     

Plancherel Inversion as Unified Approach to Wavelet Transforms and Wigner Functions

We demonstrate that the Plancherel transform for Type-I groups provides one with a natural, unified perspective for the generalized continuous wavelet transform, on the one hand, and for a class of Wigner functions, on the other. We first prove that a Plancherel inversion formula, well known for Bruhat functions on the group, holds for a much larger class of functions. This result allows us to view the wavelet transform as essentially the inverse Plancherel transform. The wavelet transform of a signal is an L²-function on an appropriately chosen group while the Wigner function is defined on a coadjoint orbit of the group and serves as an alternative characterization of the signal, which is often used in practical applications. The Plancherel transform maps L²-functions on a group unitarily to fields of Hilbert-Schmidt operators, indexed by unitary irreducible representations of the group. The wavelet transform can essentially be looked upon as a restricted inverse Plancherel transform, while Wigner functions are modified Fourier transforms of inverse Plancherel transforms, usually restricted to a subset of the unitary dual of the group. Some known results on both Wigner functions and wavelet transforms, appearing in the literature from very different perspectives, are naturally unified within our approach. Explicit computations on a number of groups illustrate the theory. Communicated by Gian Michele Graf submitted 05/06/01, accepted: 19/09/02     

Basic properties of wavelets

A wavelet multiplier is a function whose product with the Fourier transform of a wavelet is the Fourier transform of a wavelet. We characterize the wavelet multipliers, as well as the scaling function multipliers and low pass filter multipliers. We then prove that if the set of all wavelet multipliers acts on the set of all MRA wavelets, the orbits are the sets of all MRA wavelets whose Fourier transforms have equal absolute values, and these are also equal to the sets, of all MRA wavelets with the corresponding scaling functions having the same absolute values of their Fourier transforms. As an application of these techniques, we prove that the set of MRA wavelets is arcwise connected in L²(R). Dedicated to Eugene Fabes The Wutam Consortium     

Exploiting statistical properties of wavelet coefficient for face detection and recognition

Wavelet transforms (WT) are widely accepted as an essential tool for image processing and analysis. Image and video compression, image watermarking, content-base image retrieval, face recognition, texture analysis, and image feature extraction are all but few examples. It provides an alternative tool for short time analysis of quasi-stationary signals, such as speech and image signals, in contrast to the traditional short-time Fourier transform. The Discrete Wavelet Transform (DWT) is a special case of the WT, which provides a compact representation of a signal in the time and frequency domain. In particular, wavelet transforms are capable of representing smooth patterns as well anomalies (e.g. edges and sharp corners) in images. We are focusing here on using wavelet transforms statistical properties for facial feature detection, which allows us to extract the image facial feature/edges easily. Wavelet sub-bands segmentation method been developed and used to clean up the non-significant wavelet coefficients in wavelet sub-band (k) based on the (k-1) sub-band. Moreover, erosion which is considered as one of the fundamental operation in morphological image processing, been used to reduce the unwanted edges in certain directions. For face detection, face template profiles been built for both the face and the eyes for different wavelet sub-band levels to achieve better computational performance, these profiles used to match the extracted profiles from the wavelet domain of the input image using the Dynamic Time Warping technique DTW. The DTW smallest distance allows identifying the face and the eyes location. The performance of face features distances and ratio has been also tested for face verification purposes. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Clifford-hermite wavelets in euclidean space

Specific kernel functions for the continuous wavelet transform in higher dimension and new continuous wavelet transforms are presented within the framework of Clifford analysis. Their relationship with the heat equation and the newly introduced wavelet differential equation is established.     

多元小波变换及L^2（R^p）框架

提出了一般形式不可分离变量的多元小波变换,通常的小汉变换只是本的特例,进而相应地构造了多元小波框架以及多元函数小波框架展开式。     

Funk, Cosine, and Sine Transforms on Stiefel and Grassmann Manifolds

The Funk, cosine, and sine transforms on the unit sphere are indispensable tools in integral geometry and related harmonic analysis. The aim of this paper is to extend basic facts about these transforms to the more general context for Stiefel or Grassmann manifolds. The main topics are composition formulas, the Fourier functional relations for homogeneous distributions, analytic continuation, inversion formulas, and some applications.     

Inversion of Bessel Potentials with the Aid of Weighted Wavelet Transforms

A special type of weighted wavelet transforms is introduced and the relevant Calderón reproducing formula for functions f ∈ L^p(ℝⁿ ) is proved. By making use of these wavelet‐type transforms a new inversion formula of the classical Bessel potentials is obtained.     

过抽样M通道小波星图去噪方法研究

小波图像去噪已经成为目前图像去噪的主要方法之一,在分析了小波变换的基本理论和小波变换的多尺度分析基础上,根据多尺度小波变换的多分辨特性,提出了过抽样M通道小波变换去噪方法,并将此方法用于星图降噪处理中,收到良好的效果.     

基于方向多尺度变换和统计建模的纹理分类方法

纹理是图像分析和识别中经常使用的关键特征, 而小波变换则是图像纹理表示和分类中的常用工具. 然而, 基于小波变换的纹理分类方法常常忽略了小波低频子带信息, 并且无法提取图像纹理的块状奇异信息. 本文提出小波子带系数的局部能量直方图建模方法、轮廓波特征的Poisson 混合模型建模方法和基于轮廓波子带系数聚类的特征提取方法, 并将其应用于图像纹理分类上. 基于局部能量直方图的纹理分类方法解决了小波低频子带的建模难题, 基于Poisson 混合模型的纹理分类方法则首次将Poisson 混合模型用于轮廓子带特征的建模, 而基于轮廓波域聚类的纹理分类方法是一种快速的分类方法. 实验结果显示, 本文所提出的三类方法都超过了当前典型的纹理分类方法.     

Construction of Biorthogonal Discrete Wavelet Transforms Using Interpolatory Splines

We present a new family of biorthogonal wavelet and wavelet packet transforms for discrete periodic signals and a related library of biorthogonal periodic symmetric waveforms. The construction is based on the superconvergence property of the interpolatory polynomial splines of even degrees. The construction of the transforms is performed in a “lifting” manner that allows more efficient implementation and provides tools for custom design of the filters and wavelets. As is common in lifting schemes, the computations can be carried out “in place” and the inverse transform is performed in a reverse order. The difference with the conventional lifting scheme is that all the transforms are implemented in the frequency domain with the use of the fast Fourier transform. Our algorithm allows a stable construction of filters with many vanishing moments. The computational complexity of the algorithm is comparable with the complexity of the standard wavelet transform. Our scheme is based on interpolation and, as such, it involves only samples of signals and it does not require any use of quadrature formulas. In addition, these filters yield perfect frequency resolution.     

小波方法及其力学应用研究进展

小波理论在进行信号处理与函数逼近时体现出非常独特的时频局部性与多分辨分析能力,小波基函数则可兼具正交性、紧支性、低通滤波与插值性等优良的数学性质,这均使得小波分析理论在计算数学与计算力学领域具有很大的应用潜力,也进一步为这些领域的突破性发展带来了新的契机.自20世纪90年代以来,大量的研究已经证明,基于小波理论的数值方...     

The high frequency behaviour of continuous wavelet transforms

The high frequency behaviour of continuous wavelet transforms is characterized by the number of vanishing moments of the corresponding basic wavelets. As a consequence we give satisfying answers to the following questions of theoretical as well as practical interest:

What are the differences or similarities between transforms to different wavelets?

Why do wavelet transforms react so sensitively to abrupt signal changes ?