Inequalities for Wavelet Frames

One of the fundamental problems in the study of wavelet frames is to find conditions on the wavelet function and the dilation and translation parameters so that the corresponding wavelet system forms a frame. In this article, we obtain some inequalities for a discrete wavelet system to be a frame. Our result improves known ones by Chui, Shi, and Chen.     

Sufficient conditions and stability of wavelet superframes

Superframes have been introduced and developed recently. In this article, we give some sufficient conditions for a super wavelet system to be a superframe with explicit frame bounds. We also study the stability of wavelet superframes and give explicit stability bounds.     

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.     

Construction of periodic wavelet frames with dilation matrix

An important tool for the construction of periodic wavelet frame with the help of extension principles was presented in the Fourier domain by Zhang and Saito [Appl. Comput. Harmon. Anal., 2008, 125： 68-186]. We extend their results to the dilation matrix cases in two aspects. We first show that the periodization of any wavelet frame constructed by the unitary extension principle formulated by Ron and Shen is still a periodic wavelet frame under weaker conditions than that given by Zhang and Saito, and then prove that the periodization of those generated by the mixed extension principle is also a periodic wavelet frame if the scaling functions have compact supports.     

Constructing pairs of dual bandlimited framelets with desired time localization

For sufficiently small translation parameters, we prove that any bandlimited function ψ, for which the dilations of its Fourier transform form a partition of unity, generates a wavelet frame with a dual frame also having the wavelet structure. This dual frame is generated by a finite linear combination of dilations of ψ with explicitly given coefficients. The result allows a simple construction procedure for pairs of dual wavelet frames whose generators have compact support in the Fourier domain and desired time localization. The construction is based on characterizing equations for dual wavelet frames and relies on a technical condition. We exhibit a general class of function satisfying this condition; in particular, we construct piecewise polynomial functions satisfying the condition.      

小波紧框架的显式构造

该文研究对应于3带尺度函数的小波紧框架,这个小波紧框架是由V_1中的l个函数ψ^1, ψ^2, ψ^n 构成.给出这l个函数构成小波紧框架的充分条件.由此给出由3 带尺度函数构造出一个小波紧框架的显式公式.特别的,如果给定尺度函数的符号是有理函数,则可以构造出符号为有理函数的小波紧框架.最后还给出类似于小波的小波紧框架的分解与重构算法.      

小波紧框架的构造

小波框架理论是小波分析的重要内容之一.本文对于4-带尺度函数,由V1中的l个函数ψ1,ψ2,…,ψl构造小波紧框架.首先给出这个l个函数构成小波紧框架的充分条件.由此给出由4-带尺度函数构造出一个小波紧框架的公式.最后还给出类似于小波的小波紧框架的分解与重构算法.     

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.

     

Minimally Supported Frequency Composite Dilation Parseval Frame Wavelets

A composite dilation Parseval frame wavelet is a collection of functions generating a Parseval frame for L ²(ℝ ⁿ ) 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.      

Directional Haar wavelet frames on triangles

Traditional wavelets are not very effective in dealing with images that contain orientated discontinuities (edges). To achieve a more efficient representation one has to use basis elements with much higher directional sensitivity. In recent years several approaches like curvelets and shearlets have been studied providing essentially optimal approximation properties for images that are piecewise smooth and have discontinuities along C²-curves. While curvelets and shearlets have compact support in frequency domain, we construct directional wavelet frames generated by functions with compact support in time domain. Our Haar wavelet constructions can be seen as special composite dilation wavelets, being based on a generalized multiresolution analysis (MRA) associated with a dilation matrix and a finite collection of ‘shear’ matrices. The complete system of constructed wavelet functions forms a Parseval frame. Based on this MRA structure we provide an efficient filter bank algorithm. The freedom obtained by the redundancy of the applied Haar functions will be used for an efficient sparse representation of piecewise constant images as well as for image denoising.     

Density of irregular wavelet frames

We show that if an irregular multi-generated wavelet system forms a frame, then both the time parameters and the logarithms of scale parameters have finite upper Beurling densities, or equivalently, both are relatively uniformly discrete. Moreover, if generating functions are admissible, then the logarithms of scale parameters possess a positive lower Beurling density. However, the lower Beurling density of the time parameters may be zero. Additionally, we prove that there are no frames generated by dilations of a finite number of admissible functions.

     

ON NECESSARY AND SUFFICIENT CONDITIONS FOR DUALS OF WAVELET FRAMES

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Pairs of Dual Wavelet Frames from Any Two Refinable Functions

Starting from any two compactly supported refinable functions in L₂(R) with dilation factor d,we show that it is always possible to construct 2d wavelet functions with compact support such that they generate a pair of dual d-wavelet frames in L₂(R). Moreover, the number of vanishing moments of each of these wavelet frames is equal to the approximation order of the dual MRA; this is the highest possible. In particular, when we consider symmetric refinable functions, the constructed dual wavelets are also symmetric or antisymmetric. As a consequence, for any compactly supported refinable function in L₂(R), it is possible to construct, explicitly and easily, wavelets that are finite linear combinations of translates (d · – k), and that generate a wavelet frame with an arbitrarily preassigned number of vanishing moments.We illustrate the general theory by examples of such pairs of dual wavelet frames derived from B-spline functions.     

ON THE STABILITY OF FUSION FRAMES （FRAMES OF SUBSPACES）

A frame is an orthonormal basis-like collection of vectors in a Hilbert space, but need not be a basis or orthonormal. A fusion frame (frame of subspaces) is a frame-like collection of subspaces in a Hilbert space, thereby constructing a frame for the whole space by joining sequences of frames for subspaces. Moreover the notion of fusion frames provide a framework for applications and providing efficient and robust information processing algorithms.In this paper we study the conditions under which removing an element from a fusion frame, again we obtain another fusion frame. We give another proof of [5, Corollary 3.3(iii)] with extra information about the bounds.     

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.     

Irregular Gabor frames and their stability

In this paper we give sufficient conditions for irregular Gabor systems to be frames. We show that for a large class of window functions, every relatively uniformly discrete sequence in with sufficiently high density will generate a Gabor frame. Explicit frame bounds are given. We also study the stability of irregular Gabor frames and show that every Gabor frame with arbitrary time-frequency parameters is stable if the window function is nice enough. Explicit stability bounds are given.

     

Homogeneous Approximation Property for Wavelet Frames with Matrix Dilations,Ⅱ

The homogeneous approximation property (HAP) states that the number of building blocks involved in a reconstruction of a function up to some error is essentially invariant under time-scale shifts. In this paper, we show that every wavelet frame with nice wavelet function and arbitrary expansive dilation matrix possesses the HAP. Our results improve some known ones.     

Super-Wavelets and Decomposable Wavelet Frames

A wavelet frame is called decomposable whenever it is equivalent to a superwavelet frame of length greater than one. Decomposable wavelet frames are closely related to some problems on super-wavelets. In this article we first obtain some necessary or sufficient conditions for decomposable Parseval wavelet frames. As an application of these conditions, we prove that for each n > 1 there exists a Parseval wavelet frame which is m-decomposable for any 1 < m ≤ n, but not k-decomposable for any k > n. Moreover, there exists a super-wavelet whose components are non-decomposable. Similarly we also prove that for each n > 1, there exists a Parseval wavelet frame that can be extended to a super-wavelet of length m for any 1 < m ≤ n, but can not be extended to any super-wavelet of length k with k > n. The connection between decomposable Parseval wavelet frames and super-wavelets is investigated, and some necessary or sufficient conditions for extendable Parseval wavelet frames are given.