An Equivalence Relation on Wavelets in Higher Dimensions

arXiv (Cornell University), 2016

This paper presents a full catalogue, up to conjugacy and subgroups of finite index, of all matrix groups H < GL(3, R) that give rise to a continuous wavelet transform with associated irreducible quasi-regular representation. For each group in this class, coorbit theory allows to consistently define spaces of sparse signals, and to construct atomic decompositions converging simultaneously in a whole range of these spaces. As an application of the classification, we investigate the existence of compactly supported admissible vectors and atoms for the groups.

The Journal of Fourier Analysis and Applications, 2000

We investigate the connections between continuous and discrete wavelet transforms on the basis of algebraic arguments. The discrete approach is formulated abstractly in terms of the action of a semidirect product A × Γ on ℓ 2 (Γ), with Γ a lattice and A an abelian semigroup acting on Γ. We show that several such actions may be considered, and investigate those which may be written as deformations of the canonical one. The corresponding deformed dilations (the pseudodilations) turn out to be characterized by compatibility relations of a cohomological nature. The connection with multiresolution wavelet analysis is based on families of pseudodilations of a different type.

Electron. Res. Announc. …, 2004

Abstract. A wavelet with composite dilations is a function generating an orthonormal basis or a Parseval frame for L2(Rn) under the action of lattice translations and dilations by products of elements drawn from non-commuting matrix sets A and B. Typically, the members of B are ...

Monatshefte für Mathematik, 2010

The wavelet subspaces of the space of square integrable functions on the affine group with respect to the left invariant Haar measure dν are studied using the techniques from [9] with respect to wavelets whose Fourier transforms are related to Laguerre polynomials. The orthogonal projections onto each of these wavelet subspaces are described and explicit forms of reproducing kernels are established. Isomorphisms between wavelet subspaces are given.

Applied and Computational Harmonic …, 2005

We study Parseval frame wavelets in L 2 (R n ) with matrix dilations of the form (Df )(x) = √ 2f (Ax), where A is an arbitrary expanding n × n matrix with integer coefficients, such that |det A| = 2. We show that each A-MRA admits either Parseval frame wavelets, or Parseval frame bi-wavelets. The minimal number of generators for a Parseval frame associated with an A-MRA (i.e. 1 or 2) is determined in terms of a scaling function. All Parseval frame (bi)wavelets associated with A-MRA's are described. We then introduce new classes of filter induced wavelets and bi-wavelets. It is proved that these new classes strictly contain the classes of all A-MRA Parseval frame wavelets and bi-wavelets, respectively. Finally, we demonstrate a method of constructing all filter induced Parseval frame (bi)wavelets from generalized low pass filters.

Comptes Rendus Mathematique, 2008

Let G be the "ax + b"-group with the left invariant Haar measure dν and ψ be a fixed real-valued admissible wavelet on L2(R). The complete decomposition of L2(G, dν) onto the space of wavelet transforms W ψ (L2(R)) is obtained after identifying the group G with the upper half-plane Π in C. To cite this article: O. Hutník, C. R. Acad. Sci. Paris, Ser. I 346 (2008).

Journal of Inequalities and Applications, 2016

The purpose of this paper is to study the decay properties of the discrete wavelet transform with n independent dilation parameters for functions f in L 2 (R n ) and a relationship with its continuity. The method we used to study the discrete wavelet transform of f with respect to a radially symmetric admissible function was through the fact of considering two parameters in Z n . We conclude that the continuity of f at x = 0 is determined by the existence of the limit of the discrete wavelet transform when each one of the independent dilation parameters tends to zero.

2006

For nonseparable bidimensional wavelet transforms, the choice of the dilation matrix is all–important, since it governs the downsampling and upsampling steps, determines the cosets that give the positions of the filters, and defines the elementary set that gives a tesselation of the plane. We introduce nonseparable bidimensional wavelets, and give formulae for the analysis and synthesis of images. We analyze several dilation matrices, and show how the wavelet transform operates visually. We also show some distorsions produced by some of these matrices. We show that the requirement of their eigenvalues being greater than 1 in absolute value is not enough to guarantee their suitability for image processing applications, and discuss other conditions.

Journal of Mathematics

In this paper, we define new real wavelets based on the Hartley kernel and Boas transforms. These wavelets have possible application in analyzing both the symmetries of an asymmetric real signal. We give various results to obtain their higher vanishing moments. Finally, we give a sufficient condition under which Hartley-Boas-Like wavelets associated with Riesz projector forms a convolution filter with transfer function vanishing for the positive frequencies.

Journal of Advances in Mathematics, 2014

‎Consider an affine system $\mathcal{A}_{AB} (\Psi)$ with composite dilations $D_{a}‎, ‎D_{b}$‎, ‎in which $a \in A‎, ‎b \in B‎, ‎\ A‎, ‎B \subseteq GL_{n} (\mathbb{R})$ and $\Psi \in L^{2} (\mathbb{R}^n)$‎. ‎It can be made an orthonormal $AB$-multiwavelet $\Psi$ or a parsval frame $AB$-wavelet $\Psi$‎, ‎by choosing appropriate sets $A$ and $B$‎. ‎In this paper‎, ‎we constructe an orthonormal $AB$-multiwavelet that arises from $AB$-multiresolution analysis‎ . ‎Our construction is useful since the group $B$ is shear group‎. ‎More generally‎, ‎we give a parsval frame $AB$-wavelet‎.