Locally supported, piecewise polynomial biorthogona wavelets on non-uniform meshes

In this paper, biorthogonal wavelets are constructed on non-uniform meshes. Both primal and dual wavelets are explicitly given locally supported, continuous piecewise polynomials. The wavelets generate Riesz bases for the Sobolev spaces H s for j s j < 3 2 . The wavelets at the primal side span standard Lagrange nite element spaces

Dunkl wavelets and applications to inversion of the Dunkl intertwining operator and its dual

We define and study Dunkl wavelets and the corresponding Dunkl wavelets transforms, and we prove for these transforms Plancherel and reconstruction formulas. We give as application the inversion of the Dunkl intertwining operator and its dual

Steerable filters generated with the hypercomplex dual-tree wavelet transform

The use of wavelets in the image processing domain is still in its infancy, and largely associated with image compression. With the advent of the dual-tree hypercomplex wavelet transform (DHWT) and its improved shift invariance and directional selectivity, applications in other areas of image processing are more conceivable. This paper discusses the problems and solutions in developing the DHWT and its inverse. It also offers a practical implementation of the algorithms involved. The aim of this work is to apply the DHWT in machine vision. Tentative work on a possible new way of feature extraction is presented. The paper shows that 2-D hypercomplex basis wavelets can be used to generate steerable filters which allow rotation as well as translation.</p

Efficient Adjoint Computation for Wavelet and Convolution Operators

First-order optimization algorithms, often preferred for large problems, require the gradient of the differentiable terms in the objective function. These gradients often involve linear operators and their adjoints, which must be applied rapidly. We consider two example problems and derive methods for quickly evaluating the required adjoint operator. The first example is an image deblurring problem, where we must compute efficiently the adjoint of multi-stage wavelet reconstruction. Our formulation of the adjoint works for a variety of boundary conditions, which allows the formulation to generalize to a larger class of problems. The second example is a blind channel estimation problem taken from the optimization literature where we must compute the adjoint of the convolution of two signals. In each example, we show how the adjoint operator can be applied efficiently while leveraging existing software.Comment: This manuscript is published in the IEEE Signal Processing Magazine, Volume 33, Issue 6, November 201

From Frazier-Jawerth characterizations of Besov spaces to Wavelets and Decomposition spaces

This article describes how the ideas promoted by the fundamental papers published by M. Frazier and B. Jawerth in the eighties have influenced subsequent developments related to the theory of atomic decompositions and Banach frames for function spaces such as the modulation spaces and Besov-Triebel-Lizorkin spaces. Both of these classes of spaces arise as special cases of two different, general constructions of function spaces: coorbit spaces and decomposition spaces. Coorbit spaces are defined by imposing certain decay conditions on the so-called voice transform of the function/distribution under consideration. As a concrete example, one might think of the wavelet transform, leading to the theory of Besov-Triebel-Lizorkin spaces. Decomposition spaces, on the other hand, are defined using certain decompositions in the Fourier domain. For Besov-Triebel-Lizorkin spaces, one uses a dyadic decomposition, while a uniform decomposition yields modulation spaces. Only recently, the second author has established a fruitful connection between modern variants of wavelet theory with respect to general dilation groups (which can be treated in the context of coorbit theory) and a particular family of decomposition spaces. In this way, optimal inclusion results and invariance properties for a variety of smoothness spaces can be established. We will present an outline of these connections and comment on the basic results arising in this context

The solution of multi-scale partial differential equations using wavelets

Wavelets are a powerful new mathematical tool which offers the possibility to treat in a natural way quantities characterized by several length scales. In this article we will show how wavelets can be used to solve partial differential equations which exhibit widely varying length scales and which are therefore hardly accessible by other numerical methods. As a benchmark calculation we solve Poisson's equation for a 3-dimensional Uranium dimer. The length scales of the charge distribution vary by 4 orders of magnitude in this case. Using lifted interpolating wavelets the number of iterations is independent of the maximal resolution and the computational effort therefore scales strictly linearly with respect to the size of the system