The Theory of Multiresolution Analysis Frames and Applications to Filter Banks

The notion of a frame multiresolution analysis (FMRA) is formulated. An FMRA is a natural extension to affine frames of the classical notion of a multiresolution analysis (MRA). The associated theory of FMRAs is more complex than that of MRAs. A basic result of the theory is a characterization of frames of integer translates of a function φ in terms of the discontinuities and zero sets of a computable periodization of the Fourier transform of φ. There are subband coding filter banks associated with each FMRA. Mathematically, these filter banks can be used to construct new frames for finite energy signals. As with MRAs, the FMRA filter banks provide perfect reconstruction of all finite energy signals in any one of the successive approximation subspacesV_jdefining the FMRA. In contrast with MRAs, the perfect reconstruction filter bank associated with an FMRA can be narrow band. Because of this feature, in signal processing FMRA filter banks achieve quantization noise reduction simultaneously with reconstruction of a given narrow-band signal.     

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.     

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

We introduce the concept of the modular function for a shift-invariant 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 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.

     

p-Adic Multiresolution Analysis and Wavelet Frames

We study p-adic multiresolution analyses (MRAs). A complete characterization of test functions generating an MRA (scaling functions) is given. We prove that only 1-periodic test functions may be taken as orthogonal scaling functions and that all such scaling functions generate the Haar MRA. We also suggest a method for constructing sets of wavelet functions and prove that any set of wavelet functions generates a p-adic wavelet frame.     

Construction and reconstruction of tight framelets and wavelets via matrix mask functions

The paper develops construction procedures for tight framelets and wavelets using matrix mask functions in the setting of a generalized multiresolution analysis (GMRA). We show the existence of a scaling vector of a GMRA such that its first component exhausts the spectrum of the core space near the origin. The corresponding low-pass matrix mask has an especially advantageous form enabling an effective reconstruction procedure of the original scaling vector. We also prove a generalization of the Unitary Extension Principle for an infinite number of generators. This results in the construction scheme for tight framelets using low-pass and high-pass matrix masks generalizing the classical MRA constructions. We prove that our scheme is flexible enough to reconstruct all possible orthonormal wavelets. As an illustration we exhibit a pathwise connected class of non-MSF non-MRA wavelets sharing the same wavelet dimension function.     

Wavelets with Frame Multiresolution Analysis

A frame multiresolution (FMRA for short) orthogonalwavelet is a single-function orthogonal wavelet such that theassociated scaling space V0 admits a normalized tight frame(under translations). In this article, we prove that for anyexpansive matrix A with integer entries, there existA-dilation FMRA orthogonal wavelets. FMRA orthogonal waveletsfor some other expansive matrix with non integer entries are also discussed.     

Unitary mappings between multiresolution analyses of L2 (R) and a parametrization of low-pass filters

This article provides classes of unitary operators of L²(R) contained in the commutant of the Shift operator, such that for any pair of multiresolution analyses of L²(R) there exists a unitary operator in one of these classes, which maps all the scaling functions of the first multiresolution analysis to scaling functions of the other. We use these unitary operators to provide an interesting class of scaling functions. We show that the Dai-Larson unitary parametrization of orthonormal wavelets is not suitable for the study of scaling functions. These operators give an interesting relation between low-pass filters corresponding to scaling functions, which is implemented by a special class of unitary operators acting on L²([−π, π)), which we characterize. Using this characterization we recapture Daubechies' orthonormal wavelets bypassing the spectral factorization process. Acknowledgements and Notes. Partially supported by NSF Grant DMS-9157512, and Linear Analysis and Probability Workshop, Texas A&M University Dedicated to the memory of Professor Emeritus Vassilis Metaxas.     

Bi-frames with 4-fold axial symmetry for quadrilateral surface multiresolution processing

When bivariate filter banks and wavelets are used for surface multiresolution processing, it is required that the decomposition and reconstruction algorithms for regular vertices derived from them have high symmetry. This symmetry requirement makes it possible to design the corresponding multiresolution algorithms for extraordinary vertices. Recently lifting-scheme based biorthogonal bivariate wavelets with high symmetry have been constructed for surface multiresolution processing. If biorthogonal wavelets have certain smoothness, then the analysis or synthesis scaling function or both have big supports in general. In particular, when the synthesis low-pass filter is a commonly used scheme such as Loop’s scheme or Catmull-Clark’s scheme, the corresponding analysis low-pass filter has a big support and the corresponding analysis scaling function and wavelets have poor smoothness. Big supports of scaling functions, or in other words big templates of multiresolution algorithms, are undesirable for surface processing. On the other hand, a frame provides flexibility for the construction of “basis” systems. This paper concerns the construction of wavelet (or affine) bi-frames with high symmetry.In this paper we study the construction of wavelet bi-frames with 4-fold symmetry for quadrilateral surface multiresolution processing, with both the dyadic and refinements considered. The constructed bi-frames have 4 framelets (or frame generators) for the dyadic refinement, and 2 framelets for the refinement. Namely, with either the dyadic or refinement, a frame system constructed in this paper has only one more generator than a wavelet system. The constructed bi-frames have better smoothness and smaller supports than biorthogonal wavelets. Furthermore, all the frame algorithms considered in this paper are given by templates so that one can easily implement them.     

Existence theorem and minimal cardinality of UEP framelets and MEP bi-framelets

Based on multiresolution analysis (MRA) structures combined with the unitary extension principle (UEP), many frame wavelets were constructed, which are called UEP framelets. The aim of this letter is to derive general properties of UEP framelets based on the spectrum of the center space of the underlying MRA structures. We first give the existence theorem, that is, we give a necessary and sufficient condition that an MRA structure can derive UEP framelets. Second, we present a split trick that each mother function can be split into several functions such that the set consisting of these functions is still a UEP framelet. Third, we determine the minimal cardinality of UEP framelets. Finally, we directly construct UEP framelets with the minimal cardinality. Based on a pair of multiresolution analysis (MRA) structures, when their spectra intersect, we can always construct a pair of dual frame wavelets using mixed extension principle (MEP). This pair of dual frame wavelets is called a pair of MEP bi-framelets. We also give the split trick and find out the minimal cardinality of such MEP bi-framelets.     

L~2(R~n)中MRA Parseval框架小波和它们的乘子

刻画了L~2(R~n)中具有扩展矩阵伸缩的广义低通滤波器和多尺度分析Parseval框架小波(缩写为MRA PFW).首先,研究了伪逆的尺度函数、广义的低通滤波器和MRA PFW,给出它们的一些刻画.接着,我们给出与MRA PFW相联系的几类乘子的一些刻画.最后,给出了一个例子来证明的结论.     

矩阵值函数空间中尺度空间的稠密性

多分辨分析的概念在小波基构造中起着非常重要的作用,并经历了从经典多分辨分析到多重多分辨分析,再到矩阵值多分辨分析的研究历程.本文基于矩阵值多分辨分析,研究并给出了矩阵值函数空间中尺度空间稠密性的两个充要条件,并在此基础之上得到了稠密性的两个充分条件.     

Multiresolution Equivalence and Path Connectedness

An equivalence relation between multiresolution analyses was first introduced in 1996; an analogous definition for generalized multiresolution analyses was given in 2010. This article describes the relationship between the two notions and shows that both types of equivalence classes are path connected in an operator-theoretic sense. The GMRA paths are restricted to canonical GMRAs, and it is shown that whenever two MRAs in L ²(?) are equivalent, the GMRA path construction between their corresponding canonical GMRAs yields the natural analog of the MRA path. Examples are provided of GMRA paths that are distinct from MRA paths.     

Generalized low pass filters and MRA frame wavelets

A tight frame wavelet ψ is an L ²(ℝ) function such that {ψ jk(x)} = {2^j/2 ψ(2 ^j x −k), j, k ∈ ℤ},is a tight frame for L ² (ℝ).We introduce a class of “generalized low pass filters” that allows us to define (and construct) the subclass of MRA tight frame wavelets. This leads us to an associated class of “generalized scaling functions” that are not necessarily obtained from a multiresolution analysis. We study several properties of these classes of “generalized” wavelets, scaling functions and filters (such as their multipliers and their connectivity). We also compare our approach with those recently obtained by other authors.     

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.     

Dimension Functions of Rationally Dilated GMRAs and Wavelets

In this paper we study properties of generalized multiresolution analyses (GMRAs) and wavelets associated with rational dilations. We characterize the class of GMRAs associated with rationally dilated wavelets extending the result of Baggett, Medina, and Merrill. As a consequence, we introduce and derive the properties of the dimension function of rationally dilated wavelets. In particular, we show that any mildly regular wavelet must necessarily come from an MRA (possibly of higher multiplicity) extending Auscher’s result from the setting of integer dilations to that of rational dilations. We also characterize all 3 interval wavelet sets for all positive dilation factors. Finally, we give an example of a rationally dilated wavelet dimension function for which the conventional algorithm for constructing integer dilated wavelet sets fails.     

Basic properties of wavelets

A wavelet multiplier is a function whose product with the Fourier transform of a wavelet is the Fourier transform of a wavelet. We characterize the wavelet multipliers, as well as the scaling function multipliers and low pass filter multipliers. We then prove that if the set of all wavelet multipliers acts on the set of all MRA wavelets, the orbits are the sets of all MRA wavelets whose Fourier transforms have equal absolute values, and these are also equal to the sets, of all MRA wavelets with the corresponding scaling functions having the same absolute values of their Fourier transforms. As an application of these techniques, we prove that the set of MRA wavelets is arcwise connected in L²(R). Dedicated to Eugene Fabes The Wutam Consortium     

New Stable Biorthogonal Spline-Wavelets on the Interval

In this paper we present the construction of new stable biorthogonal spline-wavelet bases on the interval [0, 1] for arbitrary choice of spline-degree. As starting point, we choose the well-known family of compactly supported biorthogonal spline-wavelets presented by Cohen, Daubechies and Feauveau. Firstly, we construct biorthogonal MRAs (multiresolution analysis) on [0, 1]. The primal MRA consists of spline-spaces concerning equidistant, dyadic partitions of [0, 1], the so called Schoenberg-spline bases. Thus, the full degree of polynomial reproduction is preserved on the primal side. The construction, that we present for the boundary scaling functions on the dual side, guarantees the same for the dual side. In particular, the new boundary scaling functions on both, the primal and the dual side have staggered supports. Further, the MRA spaces satisfy certain Jackson- and Bernstein-inequalities, which lead by general principles to the result, that the associated wavelets are in fact L ₂([0, 1])-stable. The wavelets however are computed with aid of the method of stable completion. Due to the compact support of all occurring functions, the decomposition and reconstruction transforms can be implemented efficiently with sparse matrices. We also illustrate how bases with complementary or homogeneous boundary conditions can be easily derived from our construction.     

Generalized Multiresolution Analyses with Given Multiplicity Functions

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.