Wavelets with Frame Multiresolution Analysis (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.

Parseval frame wavelets with -dilations

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.

Frames, modular functions for shift-invariant subspaces and FMRA wavelet frames

Proceedings of the American Mathematical Society, 2004

We introduce the concept of the modular function for a shiftinvariant subspace that can be represented by normalized tight frame generators for the shift-invariant subspace and prove that it is independent of the selections of the frame generators for the subspace. We shall apply it to study the connections between the dimension functions of wavelet frames for any expansive integer matrix A and the multiplicity functions for general multiresolution analysis (GMRA). Given a frame mutiresolution analysis (FMRA), we show that the standard construction formula for orthonormal multiresolution analysis wavelets does not yield wavelet frames unless the underlying FMRA is an MRA. A modified explicit construction formula for FMRA wavelet frames is given in terms of the frame scaling functions and the low-pass filters.

Parseval frame wavelets with E n ( 2 ) -dilations

Applied and Computational Harmonic …, 2005

Orthogonal wavelet frames and vector-valued wavelet transforms

Applied and Computational Harmonic Analysis, 2007

Motivated by the notion of orthogonal frames, we describe sufficient conditions for the construction of orthogonal MRA wavelet frames in L 2 (R) from a suitable scaling function. These constructions naturally lead to filter banks in 2 (Z) with similar orthogonality relations and, through these filter banks, the orthogonal wavelet frames give rise to a vector-valued discrete wavelet transform (VDWT). The novelty of these constructions lies in their potential for use with vector-valued data, where the VDWT seeks to exploit correlation between channels. Extensions to higher dimensions are natural and the constructions corresponding to the bidimensional case are presented along with preliminary results of numerical experiments in which the VDWT is applied to color image data.

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

Parseval frame wavelets with E ( 2 ) n-dilations ✩

2005

We study Parseval frame wavelets in L2(Rn) with matrix dilations of the form(Df )(x) = √2f (Ax), whereA is an arbitrary expanding n × n matrix with integer coefficients, such that |detA| = 2. We show that each A-MRA admits either Parseval frame wavelets, or Parseval frame bi-wavelets. The minimal number of generators seval frame associated with an A-MRA (i.e. 1 or 2) is determined in terms of a scaling function. All Parseval fra (bi)wavelets associated with A-MRA’s are described. We then introduce new classes of filter induced wavele bi-wavelets. It is proved that these new classes strictly contain the classes of all A-MRA Parseval frame wavelet and bi-wavelets, respectively. Finally, we demonstrate a method of constructing all filter induced Parseva (bi)wavelets from generalized low-pass filters.  2005 Elsevier Inc. All rights reserved.

An algebraic perspective on multivariate tight wavelet frames. II

In this paper we employ recent results from real algebraic geometry and theory of moment problems to make the first step towards resolving the question of existence of multivariate tight wavelet frames whose generators have at least one vanishing moment. The so-called Unitary Extension Principle from [30] and the results in [28] allow us to reformulate the question of existence of tight wavelet frame in terms of the existence of the sum of squares decomposition of a single trigonometric polynomial with real coefficients. Our main result confirms the existence of such decompositions in the two-dimensional case. We also give sufficient conditions for existence of tight wavelet frames in the dimension d ≥ 3 and illustrate our results with several examples.

An Algebraic Perspective on Multivariate Tight Wavelet Frames

Constructive Approximation, 2013

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],

Wavelets with Frame Multiresolution Analysis (original) (raw)

Related papers