Wavelets with composite dilations and their MRA properties (original) (raw)

Scaling sets and orthonormal wavelets with dilations induced by expanding matrices

2000

The paper studies orthonormal wavelets in L2(Rn) with dilations induced by expanding matrices with integer coe-cients of arbitrary deter- minant. We provide a method of construction of all scaling sets and, hence, of all orthonormal MSF wavelets with the additional property that the core space of the underlying multiresolution structure is singly generated. Several examples on the real line and

On multiresolution analysis (MRA) wavelets in ℝ n

The Journal of Fourier Analysis and Applications, 2000

We prove that for any expansive n x n integral matrix A with l det A [ = 2, there exist A-dilation minimally supported frequency (MSF) wavelets that are associated with a multiresolution analysis (MRA). The condition I det A I = 2 was known to be necessary, and we prove that it is sufficient. A wavelet set is the support set of the Fourier transform of an MSF wavelet. We give some concrete examples of MRA wavelet sets in the plane. The same technique of proof is also applied to yield an existence result for A-dilation MRA subspace wavelets. An orthonormal wavelet for a dilation factor a > 0 in R is a single function ~p E L2(R) with the property that {a~r n, lEZ} is an orthonormal basis for L2(R). The proof of the existence of wavelets for any dilation factor a > 1 can be found in [6]. Similarly, one can consider wavelets in R n. If A is a real expansive matrix (equivalently, all the eigenvalues of A are required to have absolute value greater than 1), an A-dilation wavelet is a single function ~p E L2(R n) (product Lebesgue measure) with the property that {[detAl~r~(Amt-k) : m EZ, k EZ n} is an orthonormal basis for L2(Rn). In the article [7], Dai, Larson and Speegle proved the existence of wavelets for any expansive dilation matrix A. This was surprising since prior to this, several researchers had suspected that single function wavelets did not exist for A = 21 in the case n > 1. The method used in [7] was the construction of special wavelets of the form 1 ff~-I (~XE) (*) Math Subject Classifications. 42C15, 46E15.

A New Construction of Multiwavelets with Composite Dilations

Journal of Advances in Mathematics, 2014

‎Consider an affine system mathcalAAB(Psi)\mathcal{A}_{AB} (\Psi)mathcalAAB(Psi) with composite dilations Da‎,‎DbD_{a}‎, ‎D_{b}Da‎,‎Db‎, ‎in which ainA‎,‎binB‎,‎A‎,‎BsubseteqGLn(mathbbR)a \in A‎, ‎b \in B‎, ‎\ A‎, ‎B \subseteq GL_{n} (\mathbb{R})ainA‎,‎binB‎,‎A‎,‎BsubseteqGLn(mathbbR) and PsiinL2(mathbbRn)\Psi \in L^{2} (\mathbb{R}^n)PsiinL2(mathbbRn)‎. ‎It can be made an orthonormal ABABAB-multiwavelet Psi\PsiPsi or a parsval frame ABABAB-wavelet Psi\PsiPsi‎, ‎by choosing appropriate sets AAA and BBB‎. ‎In this paper‎, ‎we constructe an orthonormal ABABAB-multiwavelet that arises from ABABAB-multiresolution analysis‎ . ‎Our construction is useful since the group BBB is shear group‎. ‎More generally‎, ‎we give a parsval frame ABABAB-wavelet‎.

The Theory of Wavelets with Composite Dilations

Applied and Numerical Harmonic Analysis

A wavelet with composite dilations is a function generating an orthonormal basis or a Parseval frame for L 2 (R n) 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 shear matrices (all eigenvalues are one) while the members of A are matrices expanding or contracting on a proper subspace of R n. These wavelets are of interest in applications because of their tendency to produce "long, narrow" window functions well suited to edge detection. In this paper, we discuss the remarkable extent to which the theory of wavelets with composite dilations parallels the theory of classical wavelets, and present several examples of such systems.

Dilation Matrices for Nonseparable Bidimensional Wavelets

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.

An algebraic approach to discrete dilations. Application to discrete wavelet transforms

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.

Wavelets on irregular grids with arbitrary dilation matrices and frame atoms for

Applied and Computational Harmonic Analysis, 2004

In this article, we develop a general method for constructing wavelets {| det A j | 1/2 ψ(A j x − x j,k ) : j ∈ J, k ∈ K} on irregular lattices of the form X = {x j,k ∈ R d : j ∈ J, k ∈ K}, and with an arbitrary countable family of invertible d × d matrices {A j ∈ GL d (R) : j ∈ J} that do not necessarily have a group structure. This wavelet construction is a particular case of general atomic frame decompositions of L 2 (R d ) developed in this article, that allow other time frequency decompositions such as non-harmonic Gabor frames with non-uniform covering of the Euclidean space R d . Possible applications include image and video compression, speech coding, image and digital data transmission, image analysis, estimations and detection, and seismology.

Wavelets on irregular grids with arbitrary dilation matrices and frame atoms for L 2 ( R d )

Appl Comput Harmonic Anal, 2004

In this article, we develop a general method for constructing wavelets {| det A j | 1/2 ψ(A j x − x j,k ) : j ∈ J, k ∈ K} on irregular lattices of the form X = {x j,k ∈ R d : j ∈ J, k ∈ K}, and with an arbitrary countable family of invertible d × d matrices {A j ∈ GL d (R) : j ∈ J} that do not necessarily have a group structure. This wavelet construction is a particular case of general atomic frame decompositions of L 2 (R d ) developed in this article, that allow other time frequency decompositions such as non-harmonic Gabor frames with non-uniform covering of the Euclidean space R d . Possible applications include image and video compression, speech coding, image and digital data transmission, image analysis, estimations and detection, and seismology.

Wavelets with composite dilations

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 ...