Frame Multiresolution Analysis Research Articles

Multivariate FMRAs and FMRA frame wavelets for reducing subspaces of L2(Rd)

This article addresses frame multiresolution analyses (FMRAs) and FMRA frame wavelets in the setting of reducing subspaces of L2(Rd). For a general expansive matrix, we obtain a characterization and some conditions for a frame-scaling function to generate an FMRA, and we prove that an arbitrary reducing subspace must admit an FMRA. For an expansive matrix M with |det⁡M|=2, we establish a sufficient and necessary condition for FMRAs to admit a single FMRA frame wavelet, give an explicit construction of FMRA frame wavelets, and study the relation between s-frame wavelets and FMRA frame wavelets. These results are also new in the setting of L2(Rd).

Semiorthogonal Multiresolution Analysis Frames in Higher Dimensions

Benedetto and Li proposed the theory of frame multiresolution analysis (FMRA) in one dimension. This paper generalizes Benedetto and Li’s theory of FMRA to higher dimensions with arbitrary integral expansive matrix dilations, and gives two necessary and sufficient conditions to characterize (semiorthogonal) multiresolution analysis frames for L2(ℝn). One of the two conditions is put on the frame scaling function and the low-pass and high-pass filters only. Multiresolution analysis Parseval frames are also characterized. The theory is implemented to a bidimensional example with the nonseparable quincunx dilation \(\scriptsize\bigl(\begin{array}{c@{\ }c}1&-1\\1&1\end{array}\bigr)\) , along with its potential application in subband signal processing.

The spectrum sequences of periodic frame multiresolution analysis

The periodic multiresolution analysis (PMRA) and the periodic frame multiresolution analysis (PFMRA) provide a general recipe for the construction of periodic wavelets and periodic wavelet frames, respectively. This paper addresses PFMRAs by the introduction of the notion of spectrum sequence. In terms of spectrum sequences, the scaling function sequences generating a normalized PFMRA are characterized; a characterization of the spectrum sequences of PFMRAs is obtained, which provides a method to construct PFMRAs since its proof is constructive; a necessary and sufficient condition for a PFMRA to admit a single wavelet frame sequence is obtained; a necessary and sufficient condition for a PFMRA to be contained in a given PMRA is also obtained. What is more, it is proved that an arbitrary PFMRA must be contained in some PMRA. In the meanwhile, some examples are provided to illustrate the general theory.

Minimally Supported Frequency Composite Dilation Parseval Frame Wavelets

A composite dilation Parseval frame wavelet is a collection of functions generating a Parseval frame for L2(ℝn) under the actions of translations from a full rank lattice and dilations by products of elements of groups A and B. A minimally supported frequency composite dilation Parseval frame wavelet has generating functions whose Fourier transforms are characteristic functions of sets contained in a lattice tiling set. Constructive proofs are used to establish the existence of minimally supported frequency composite dilation Parseval frame wavelets in arbitrary dimension using any finite group B, any full rank lattice, and an expanding matrix generating the group A and normalizing the group B. Moreover, every such system is derived from a Parseval frame multiresolution analysis. Multiple examples are provided including examples that capture directional information.

Shannon multiresolution analysis on the Heisenberg group

We present a notion of frame multiresolution analysis on the Heisenberg group, abbreviated by FMRA, and study its properties. Using the irreducible representations of this group, we shall define a sinc-type function which is our starting point for obtaining the scaling function. Further, we shall give a concrete example of a wavelet FMRA on the Heisenberg group which is analogous to the Shannon MRA on R .

Reducing subspace frame multiresolution analysis and frame wavelets

Multiresolution analysis (<strong>MRA</strong>) and frame multiresolution analysis (<strong>FMRA</strong>) in $L^2(\mathbb R)$ play a significant role in the construction of wavelets and frame wavelets for $L^2(\mathbb R)$. In this paper, the notions of <strong>MRA</strong> and <strong>FMRA</strong> in a reducing subspace of $L^2(\mathbb R)$ are introduced, from which the construction of wavelets and frame wavelets for this subspace is obtained. Many examples are also provided to illustrate the general theory. In particular, most of them are about the space with its frequency lying in $[0,\infty)$, which is closely related to various Paley-Wiener Theorems.

A Class of Bidimensional FMRA Wavelet Frames

This paper addresses the construction of wavelet frame from a frame multiresolution analysis (FMRA) associated with a dilation matrix of determinant ±2. The dilation matrices of determinant ±2 can be classified as six classes according to integral similarity. In this paper, for four classes of them, the construction of wavelet frame from an FMRA is obtained, and, as examples, Shannon type wavelet frames are constructed, which have an independent value for their optimality in some sense.

The infimum cosine angle between two finitely generated shift-invariant spaces and its applications

An expression of the infimum cosine angle between two finite dimensional subspaces is given in terms of Gramians and is applied to find a useful formula for the infimum cosine angle between two finitely generated shift-invariant subspaces of L 2 ( R d ) in terms of the generators. We then present new equivalent conditions for the existence of the oblique projection between two finitely generated shift-invariant subspaces, thereby providing a constructive method to generate oblique dual frames. This result is a generalization of a result of Aldroubi and is closely related with the biorthogonality of two frame multiresolution analyses. Finally, we illustrate our results by examples.

On the spectrums of frame multiresolution analyses

We first give conditions for a univariate square integrable function to be a scaling function of a frame multiresolution analysis (FMRA) by generalizing the corresponding conditions for a scaling function of a multiresolution analysis (MRA). We also characterize the spectrum of the ‘central space’ of an FMRA, and then give a new condition for an FMRA to admit a single frame wavelet solely in terms of the spectrum of the central space of an FMRA. This improves the results previously obtained by Benedetto and Treiber and by some of the authors. Our methods and results are applied to the problem of the ‘containments’ of FMRAs in MRAs. We first prove that an FMRA is always contained in an MRA, and then we characterize those MRAs that contain ‘genuine’ FMRAs in terms of the unique low-pass filters of the MRAs and the spectrums of the central spaces of the FMRAs to be contained. This characterization shows, in particular, that if the low-pass filter of an MRA is almost everywhere zero-free, as is the case of the MRAs of Daubechies, then the MRA contains no FMRAs other than itself.

Frame multiresolution analysis and infinite trees in Banach spaces on locally compact abelian groups

We extend the concept of frame multiresolution analysis to a locally compact abelian group and use it to define certain weighted Banach spaces and the spaces of their antifunctionals. We define analysis and synthesis operators on these spaces and establish the continuity of their composition. Also, we prove a general result to characterize infinite trees in the above Banach spaces of antifunctionals. This paper paves the way for the study of corresponding problems associated with some other types of Banach spaces on locally compact abelian groups including modulation spaces.

Métastases et pseudo-tumeurs pulmonaires: Metastasis and pulmonary pseudo-tumors

We present some necessary and sufficient conditions for a frame multiresolution analysis (FMRA) to admit a frame wavelet whose dyadic dilations and integer translates generate a frame for L2(R) and propose a construction of a wavelet, if it exists, which reduces to the classical orthonormal wavelet in the case of orthonormal multiresolution analysis. We also show that there always exists a frame for the detail space W0 of a frame MRA consisting of the integer translates of two functions and give an explicit construction of them.

On the higher-dimensional wavelet frames

Every higher-dimensional wavelet frame is generated by dyadic dilations and integer translates of several mother functions. In this paper, associated with a frame multiresolution analysis { V m , ϕ} of L 2( R d ), many higher-dimensional wavelet frames are constructed with help of a splitting trick of the subspace V 1 and a very fine partition of R d . The least number of the mother functions is shown precisely in terms of the index of the FMRA. Finally, in order to explain our theory, an example is presented.

Local analysis of frame multiresolution analysis with a general dilation matrix

A multivariate semi-orthogonal frame multiresolution analysis with a general integer dilation matrix and multiple scaling functions is considered. We first derive the formulas of the lengths of the inital (central) shift-invariant space V0 and the next dilation space V1, and, using these formulas, we then address the problem of the number of the elements of a wavelet set, that is, the length of the shift-invariant space W0 := V1 ⊖ V0. Finally, we show that there does not exist a ‘genuine’ frame multiresolution analysis for which V0 and V1 are quasi-stable spaces satisfying the usual length condition.

Wavelets with Frame Multiresolution Analysis

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 article, 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.

Quasi-Biorthogonal Frame Multiresolution Analyses and Wavelets

We introduce the concepts of quasi-biorthogonal frame multiresolution analyses and quasi-biorthogonal frame wavelets which are natural generalizations of biorthogonal multiresolution analyses and biorthogonal wavelets, respectively. Necessary and sufficient conditions for quasi-biorthogonal frame multiresolution analyses to admit quasi-biorthogonal wavelet frames are given, and a non-trivial example of quasi-biorthogonal frame multiresolution analyses admitting quasi-biorthogonal frame wavelets is constructed. Finally, we characterize the pair of quasi-biorthogonal frame wavelets that is associated with quasi-biorthogonal frame multiresolution analyses.

Gramian analysis of multivariate frame multiresolution analyses

We perform a Gramian analysis of a frame multiresolution analysis to give a condition for it to admit a minimal wavelet set and to show that the frame bounds of the natural generator for the wavelet space of a degenerate frame multiresolution analysis shrink.

Frames and Single Wavelets for Unitary Groups

AbstractWe consider a unitary representation of a discrete countable abelian group on a separable Hilbert space which is associated to a cyclic generalized frame multiresolution analysis. We extend Robertson’s theorem to apply to frames generated by the action of the group. Within this setup we use Stone’s theorem and the theory of projection valued measures to analyze wandering frame collections. This yields a functional analytic method of constructing a wavelet from a generalized frame multiresolution analysis in terms of the frame scaling vectors. We then explicitly apply our results to the action of the integers given by translations on L2(ℝ).

Semi-orthogonal frame wavelets and frame multi-resolution analyses

We first characterise semi-orthogonal frame wavelets by generalising the characterisation of orthonormal wavelets. We then characterise those semi-orthogonal frame wavelets that are associated with frame multi-resolution analyses. This is a generalisation of a result of Wang and another result of Papadakis. Finally, we illustrate our results by an example.

On Frame Wavelets Associated with Frame Multiresolution Analysis

We present some necessary and sufficient conditions for a frame multiresolution analysis (FMRA) to admit a frame wavelet whose dyadic dilations and integer translates generate a frame for L 2( R ) and propose a construction of a wavelet, if it exists, which reduces to the classical orthonormal wavelet in the case of orthonormal multiresolution analysis. We also show that there always exists a frame for the detail space W 0 of a frame MRA consisting of the integer translates of two functions and give an explicit construction of them.

High-order wavelet packets and cumulant field analysis

In many applications it is necessary to characterize the statistical properties of the wavelet/wavelet packet coefficients of a stationary random signal. In particular, in a stationary non-Gaussian noise scenario it may be useful to determine the high-order statistics of the wavelet packet coefficients. In this work we prove that this task may be performed through multidimensional filter banks. In particular, we show how the cumulants of the M-band wavelet packet coefficients of a strictly stationary signal are derived from those of the signal and we provide scale-recursive decomposition and reconstruction formulae to compute these cumulants. High-order wavelet packets, associated with these multidimensional filter banks, are presented along with some of their properties. It is proved that under some conditions these high-order wavelet packets allow us to define frame multiresolution analyses. Finally, the asymptotic normality of the coefficients is studied by showing the geometric decay of their polyspectra/cumulants (of order greater than two) with respect to the resolution level.