Tight wavelet frames for irregular multiresolution analysis

An important tool for the construction of tight wavelet frames is the Unitary Extension Principle first formulated in the Fourier-domain 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.     

具有整数值伸缩矩阵的三维紧框架小波的存在性

引进了三维紧框架小波的概念,它是由框架多分辨分析中子空间X_1中的若干个三维函数Γ~1(y),Γ~2(y),…,Γ~n(y)构成的.研究了对应于三维尺度函数的三维紧框架小波的存在性.运用时频分析方法、滤波器理论、算子理论,给出这n个三维函数生成小波紧框架的充分条件,得到了由一个尺度函数Ψ(y)构造三维紧框架小波的显式公式.     

Wavelet frames and shift-invariant subspaces of periodic functions

A general approach based on polyphase splines, with analysis in the frequency domain, is developed for studying wavelet frames of periodic functions of one or higher dimensions. Characterizations of frames for shift-invariant subspaces of periodic functions and results on the structure of these subspaces are obtained. Starting from any multiresolution analysis, a constructive proof is provided for the existence of a normalized tight wavelet frame. The construction gives the minimum number of wavelets required. As an illustration of the approach developed, the one-dimensional dyadic case is further discussed in detail, concluding with a concrete example of trigonometric polynomial wavelet frames.     

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.     

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.     

On the dimension function of orthonormal wavelets

We announce the following result: Every orthonormal wavelet of is associated with a multiresolution analysis such that for the subspace the integral translates of a countable at most family of functions is a tight frame.

     

The Matrix-Valued Riesz Lemma and Local Orthonormal Bases in Shift-Invariant Spaces

We use the matrix-valued Fejér–Riesz lemma for Laurent polynomials to characterize when a univariate shift-invariant space has a local orthonormal shift-invariant basis, and we apply the above characterization to study local dual frame generators, local orthonormal bases of wavelet spaces, and MRA-based affine frames. Also we provide a proof of the matrix-valued Fejér–Riesz lemma for Laurent polynomials.     

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.     

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.     

A unified characterization of reproducing systems generated by a finite family

This article presents a general result from the study of shift-invariant spaces that characterizes tight frame and dual frame generators for shift-invariant subspaces of L²(ℝⁿ). A number of applications of this general result are then obtained, among which are the characterization of tight frames and dual frames for Gabor and wavelet systems.     

单个生成元Walsh p-进制平移不变空间伸缩的交与并

p-进制MRA与GMRA是构造L~2(R_+)中小波框架的重要工具. L~2(R+)中嵌套子空间序列交集为{0},并集为L~2(R_+)是其构成p-进制MRA与GMRA的基本要求.本文研究单个生成元Walsh p-进制平移不变子空间伸缩的交与并,证明了:对任意单个生成元Walsh p-进制平移不变子空间,其p-进制伸缩的交是{0};若生成元分为Walsh p-细分函数,则其p-进制伸缩的并是L~2(R_+)中一个Walshp-进制约化子空间.特别地,其伸缩构成L~2(R_+)中p-进制GMRA当且仅当∪_(j∈z)p~j supp(■φ)=R+,其中■为定义在L~2(R_+)上的Walsh p-进制傅里叶变换.值得注意的是:形式上,我们的结果类似于通常L~2(R)的情形,然而其证明不是平凡的.这是因为定义在R_+上的p-进制加法"⊕"不同于定义在R上的通常加法"+".     

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.     

关于L2(Rd)中仿射子空间小波标架的一个注记

研究了L2(Rd)的有限生成仿射子空间中小波标架的构造.证明了任意有限生成仿射子空间都容许一个具有有限多个生成元的Parseval小波标架,并且得到了仿射子空间是约化子空间的一个充分条件.对其傅里叶变换是一个特征函数的单个函数生成的仿射子空间,得到了与小波标架构造相关的投影算子在傅里叶域上的明确表达式,同时也给出了一些例子.     

Compactly Supported Tight Affine Frames with Integer Dilations and Maximum Vanishing Moments

When a cardinal B-spline of order greater than 1 is used as the scaling function to generate a multiresolution approximation of L ²=L ²(R) with dilation integer factor M2, the standard matrix extension approach for constructing compactly supported tight frames has the limitation that at least one of the tight frame generators does not annihilate any polynomial except the constant. The notion of vanishing moment recovery (VMR) was introduced in our earlier work (and independently by Daubechies et al.) for dilation M=2 to increase the order of vanishing moments. This present paper extends the tight frame results in the above mentioned papers from dilation M=2 to arbitrary integer M2 for any compactly supported M-dilation scaling functions. It is shown, in particular, that M compactly supported tight frame generators suffice, but not M–1 in general. A complete characterization of the M-dilation polynomial symbol is derived for the existence of M–1 such frame generators. Linear spline examples are given for M=3,4 to demonstrate our constructive approach.     

Approximations for Gabor and wavelet frames

Let be a frame vector under the action of a collection of unitary operators . Motivated by the recent work of Frank, Paulsen and Tiballi and some application aspects of Gabor and wavelet frames, we consider the existence and uniqueness of the best approximation by normalized tight frame vectors. We prove that for any frame induced by a projective unitary representation for a countable discrete group, the best normalized tight frame (NTF) approximation exists and is unique. Therefore it applies to Gabor frames (including Gabor frames for subspaces) and frames induced by translation groups. Similar results hold for semi-orthogonal wavelet frames.

     

Using the refinement equation for the construction of pre-wavelets

A variety of methods have been proposed for the construction of wavelets. Among others, notable contributions have been made by Battle, Daubechies, Lemarié, Mallat, Meyer, and Stromberg. This effort has led to the attractive mathematical setting of multiresolution analysis as the most appropriate framework for wavelet construction. The full power of multiresolution analysis led Daubechies to the construction ofcompactly supported orthonormal wavelets with arbitrarily high smoothness. On the other hand, at first sight, it seems some of the other proposed methods are tied to special constructions using cardinal spline functions of Schoenberg. Specifically, we mention that Battle raises some doubt that his block spin method can produce only the Lemarié Ondelettes. A major point of this paper is to extend the idea of Battle to the generality of multiresolution analysis setup and address the easier job of constructingpre-wavelets from multiresolution.Research partially supported by DARPA and NSF Grant INT-87-12424     

Geometric aspects of frame representations of abelian groups

We consider frames arising from the action of a unitary representation of a discrete countable abelian group. We show that the range of the analysis operator can be determined by computing which characters appear in the representation. This allows one to compare the ranges of two such frames, which is useful for determining similarity and also for multiplexing schemes. Our results then partially extend to Bessel sequences arising from the action of the group. We apply the results to sampling on bandlimited functions and to wavelet and Weyl-Heisenberg frames. This yields a sufficient condition for two sampling transforms to have orthogonal ranges, and two analysis operators for wavelet and Weyl-Heisenberg frames to have orthogonal ranges. The sufficient condition is easy to compute in terms of the periodization of the Fourier transform of the frame generators.