相关论文: Groups, Wavelets, and Wavelet Sets
Wavelet and frames have become a widely used tool in mathematics, physics, and applied science during the last decade. In this article we discuss the construction of frames for $L^2(\R^n)$ using the action of closed subgroups $H\subset…
The continuous wavelet transform has become a widely used tool in applied science during the last decade. In this article we discuss some generalizations coming from actions of closed subgroups of $\mathrm{GL}(n,\mathbb{R})$ acting on…
Single wavelet sets, and thus single wavelets, are shown to exist for the actions of all crystallographic groups on $\mathbb R^2$ under all integer dilations. Examples of such sets satisfying the additional requirement that they are finite…
This review paper is intended to give a useful guide for those who want to apply discrete wavelets in their practice. The notion of wavelets and their use in practical computing and various applications are briefly described, but rigorous…
Analysis on the unit sphere $\mathbb{S}^{2}$ found many applications in seismology, weather prediction, astrophysics, signal analysis, crystallography, computer vision, computerized tomography, neuroscience, and statistics. In the last two…
The notion of wavelets is defined. It is briefly described {\it what} are wavelets, {\it how} to use them, {\it when} we do need them, {\it why} they are preferred and {\it where} they have been applied. Then one proceeds to the…
In recent years directional multiscale transformations like the curvelet- or shearlet transformation have gained considerable attention. The reason for this is that these transforms are - unlike more traditional transforms like wavelets -…
In \cite{AV99}, Antoine and Vandergheynst propose a group-theoretic approach to continuous wavelet frames on the sphere. The frame is constructed from a single so-called admissible function by applying the unitary operators associated to a…
The class of generalized shearlet dilation groups has recently been developed to allow the unified treatment of various shearlet groups and associated shearlet transforms that had previously been studied on a case-by-case basis. We consider…
Tight framelets on a smooth and compact Riemannian manifold $\mathcal{M}$ provide a tool of multiresolution analysis for data from geosciences, astrophysics, medical sciences, etc. This work investigates the construction, characterizations,…
Random Wavelet Series form a class of random processes with multifractal properties. We give three applications of this construction. First, we synthesize a random function having any given spectrum of singularities satisfying some…
Wavelet analysis and compression tools are reviewed and different applications to study MHD and plasma turbulence are presented. We introduce the continuous and the orthogonal wavelet transform and detail several statistical diagnostics…
In this article, we develop a general method for constructing wavelets {|det A_j|^{1/2} g(A_jx-x_{j,k}): j in J, k in K}, on irregular lattices of the form X={x_{j,k} in R^d: j in J, k in K}, and with an arbitrary countable family of…
We introduce a new concept of the so-called {\it composite wavelet transforms}. These transforms are generated by two components, namely, a kernel function and a wavelet function (or a measure). The composite wavelet transforms and the…
A recently developed new approach, called ``Empirical Wavelet Transform'', aims to build 1D adaptive wavelet frames accordingly to the analyzed signal. In this paper, we present several extensions of this approach to 2D signals (images). We…
The empirical wavelet transform is a data-driven time-scale representation consisting of an adaptive filter bank. Its robustness to data has made it the subject of intense developments and an increasing number of applications in the last…
Most of the examples of wavelet sets are for dilation sets which are groups. We find a necessary and sufficient condition under which subspace wavelet sets exist for dilation sets of the form $A B$, which are not necessarily groups. We…
We present a unified group-theoretical derivation of the Continuous Wavelet Transform (CWT) on the circle $\mathbb S^1$ and the real line $\mathbb{R}$, following the general formalism of Coherent States (CS) associated to unitary square…
The major goal of the paper is to prove that discrete frames of (directional) wavelets derived from an approximate identity exist. Additionally, a kind of energy conservation property is shown to hold in the case when a wavelet family is…
Wavelets, known to be useful in non-linear multi-scale processes and in multi-resolution analysis, are shown to have a q-deformed algebraic structure. The translation and dilation operators of the theory associate with any scaling equation…