Generalized multi-resolution analyses and a construction procedure for all wavelet sets in ℝ n (original) (raw)

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.

Irregular multiresolution analysis and associated wavelet

Arabian Journal of Mathematics, 2014

We introduce two generalizations, the first of which generalizes the concept of multiresolution analysis. We define the irregular generalized multiresolution analysis (IGMRA). This structure is defined taking translations on sets that are not necessarily regular lattices, for which certain density requirements are required, and without using dilations, also allows each subspace of IGMRA to be generated by outer frames of translations of different functions. The second generalization concerns the concept of association of wavelets to these new structures. We take frames of translations of a countable set of functions, which we called generalized wavelets, and define the concept of association of these generalized wavelets to those previously defined IGMRA. In the next stage, we prove two existence theorems. In the first theorem, we prove existence of IGMRA, and in the second existence of generalized wavelets associated with it. In the latter, we show that we are able to associate frames of translations with optimal localization properties, to IGMRA. In the last section of this paper, concrete examples of these structures are presented for L 2 (R) and for L 2 (R 2). Mathematics Subject Classification 42C40 • 42C30 1 Introduction From the classic concept of multiresolution analysis (MRA), introduced and further developed by Meyer [27,28], and Mallat [24,25], which provides a systematic way to construct orthonormal wavelet bases of L 2 (R), research in this area has been extended in various ways. These concepts are generalized to L 2 (R d) [14],

On the construction of multivariate (pre)wavelets

Constructive Approximation, 1993

A new approach for the construction of wavelets and prewavelets on R d from multiresolution is presented. The method uses only properties of shiftinvariant spaces and orthogonal projectors from L2(R d) onto these spaces, and requires neither decay nor stability of the scaling function. Furthermore, this approach allows a simple derivation of previous, as well as new, constructions of wavelets, and leads to a complete resolution of questions concerning the nature of the intersection and the union of a scale of spaces to be used in a multiresolution.

Multiresolution analysis using biorthogonal and interpolating wavelets

2004

This paper discusses a multiresolution analysis for the discretisation of Maxwell equations based on second-generation wavelets (biorthogonal and interpolating). When this type of wavelets is used the physical representation and the scaling representation are essentially the same. The scaling coefficients and the sampling values coincide. Since the coefficients are directly obtained from the physical representation it is possible to implement nonlinear operators (products, squares etc.) in an easy way. The implementation of the derivative is more complex and takes advantage of the reduction of the dimensionality given by multiresolution representation.

Wavelets with Frame Multiresolution Analysis

Journal of Fourier Analysis and Applications, 2003

A frame multiresolution (FMRA for short) orthogonal wavelet is a single-function orthogonal wavelet such that the associated scaling space V 0 admits a normalized tight frame (under translations). In this paper, we prove that for any expansive matrix A with integer entries, there exist A-dilation FMRA orthogonal wavelets. FMRA orthogonal wavelets for some other expansive matrix with non integer entries are also discussed.

Generalized Multiresolution Analyses with Given Multiplicity Functions

Journal of Fourier Analysis and Applications, 2009

Generalized multiresolution analyses are increasing sequences of subspaces of a Hilbert space H that fail to be multiresolution analyses in the sense of wavelet theory because the core subspace does not have an orthonormal basis generated by a fixed scaling function. Previous authors have studied a multiplicity function m which, loosely speaking, measures the failure of the GMRA to be an MRA. When the Hilbert space H is L 2 (R n ), the possible multiplicity functions have been characterized by Baggett and Merrill. Here we start with a function m satisfying a consistency condition which is known to be necessary, and build a GMRA in an abstract Hilbert space with multiplicity function m.

BLaC-wavelets: a multiresolution analysis with non-nested spaces

Proceedings of Seventh Annual IEEE Visualization '96, 1996

In the last ve y ears, there has been numerous applications of wavelets and m ultiresolution analysis in many elds of computer graphics as di erent as geometric modelling, volume visualization or illumination modelling. Classical multiresolution analysis is based on the k n o wledge of a nested set of functional spaces in which t he s u ccessive a p proximations of a given function converge to t hat f u nction, and can be e ciently computed. This paper rst proposes a theoretical framework which e n ables multiresolution analysis even if the f u nctional spaces are not nested, as long a s t hey still have t he property that t he s u ccessive a p proximations converge to t he g i v en function. Based on this concept we n ally introduce a new multiresolution analysis with exact reconstruction for large data sets de ned on uniform grids. We construct a one-parameter family of multiresolution analyses which is a blending o f H a a r and linear multiresolution.

The Wavelet Element Method Part II. Realization and Additional Features in 2D and 3D

Applied and Computational Harmonic Analysis, 2000

The Wavelet Element Method (WEM) provides a construction of multiresolution systems and biorthogonal wavelets on fairly general domains. These are split into subdomains that are mapped to a single reference hypercube. Tensor products of scaling functions and wavelets de ned on the unit interval are used on the reference domain. By introducing appropriate matching conditions across the interelement boundaries, a globally continuous biorthogonal wavalet basis on the general domain is obtained. This construction does not uniquely de ne the basis functions but rather leaves some freedom for ful lling additional features.