Wavelets with Frame Multiresolution Analysis

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.

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

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.

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.

Constructive Approximation, 2013

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.

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],

Journal of Mathematical Physics, 1996

This paper is concerned with the relations between discrete and continuous wavelet transforms on k-dimensional Euclidean space. We start with the construction of continuous wavelet transforms with the help of square-integrable representations of certain semidirect products, thereby generalizing results of Bernier and Taylor. We then turn to frames of L 2 (R k ) and to the question, when the functions occurring in a given frame are admissible for a given continuous wavelet transform. For certain frames we give a characterization which generalizes a result of Daubechies to higher dimensions.

Let A be a d × d real expansive matrix. We characterize the reducing subspaces of L 2 (R d ) for A-dilation and the regular translation operators acting on L 2 (R d ). We also characterize the Lebesgue measurable subsets E of R d such that the function defined by inverse Fourier transform of [1/(2π) d/2 ]χ E generates through the same A-dilation and the regular translation operators a normalized tight frame for a given reducing subspace. We prove that in each reducing subspace, the set of all such functions is non-empty and is also path connected in the regular L 2 (R d )-norm.

Applied and Computational Harmonic Analysis, 2008

An important tool for the construction of tight wavelet frames is the Unitary Extension Principle first formulated in the Fourierdomain by Ron and Shen. We show that the time-domain analogue of this principle provides a unified approach to the construction of tight frames based on many variations of multiresolution analyses, e.g., regular refinements of bounded L-shaped domains, refinements of subdivision surfaces around irregular vertices, and nonstationary subdivision. We consider the case of nonnegative refinement coefficients and develop a fully local construction method for tight frames. Especially, in the shift-invariant setting, our construction produces the same tight frame generators as the Unitary Extension Principle.

Journal of Computational and Applied Mathematics, 2010

This paper is devoted to the study and construction of compactly supported tight frames of multivariate multi-wavelets. In particular, a necessary condition for their existence is derived to provide some useful guide for constructing such MRA tight frames, by reducing the factorization task of the associated polyphase matrix-valued Laurent polynomial to that of certain scalar-valued non-negative ones. We illustrate our construction method with examples of both multivariate scalar-and vector-valued subdivision schemes. Since our constructions for C 1 and C 2 piecewise cubic schemes are quite involved, we also include the corresponding Matlab code in the Appendix.