Tight compactly supported wavelet frames of arbitrarily high smoothness

Based on Ron and Shen's new method for constructing tight wave-let frames, we show that one can construct, for any dilation matrix, and in any spatial dimension, tight wavelet frames generated by compactly supported functions with arbitrarily high smoothness.

     

Dual generators for weighted irregular wavelet frames and reconstruction error

Dual frames are very useful tools to reconstruct a function and have been explored in many different aspects. In this paper, we give the conditions under which a class of special weighted irregular wavelet frames could have a dual generator with a very explicit form. We also prove that when the irregular translations are changed in some admissible range, the reconstruction formula has a very small perturbation. An example is given to show the use of our main results.     

Oversampling and preservation of tightness in affine frames

The problem of how an oversampling of translations affects the bounds of an affine frame has been proposed by Chui and Shi. In particular, they proved that tightness is preserved if the oversampling factor is coprime with the dilation factor. In this paper we study, in the dyadic dilation case, oversampling of translation by factors which do not satisfy the above condition, and prove that tightness is preserved only in the case of affine frames generated by wavelets having frequency support with very particular properties.

     

Construction of k-Angle Tight Frames

Frames have become standard tools in signal processing due to their robustness against transmission errors and their resilience to noise. Equiangular tight frames (ETFs) are particularly useful and have been shown to be optimal for transmission under a certain number of erasures. Unfortunately, ETFs do not exist in many cases and are hard to construct when they do exist. However, it is known that an ETF of d + 1 vectors in a d dimensional space always exists. This article gives an explicit construction of ETFs of d + 1 vectors in a d dimensional space. This construction works for both real and complex cases and is simpler than existing methods. The absence of ETFs of arbitrary sizes in a given space leads to generalizations of ETFs. One way to do this to consider tight frames where the set of (acute) angles between pairs of vectors has k distinct values. This article presents a construction of tight frames such that for a given value of k, the angles between pairs of vectors take at most k distinct values. These tight frames can be related to regular graphs and association schemes.     

Tight frames and geometric properties of wavelet sets

A construction for providing single dyadic orthonormal wavelets in Euclidean space ℝ^d is given. It is called the general neighborhood mapping construction. The fact that one wavelet is sufficient to generate an orthonormal basis for L²(ℝ^d) is the critical issue. The validity of the construction is proved, and the construction is implemented computationally to provide a host of examples illustrating various geometrical properties of such wavelets in the spectral domain. Because of the inherent complexity of these single orthonormal wavelets, the method is applied to the construction of single dyadic tight frame wavelets, and these tight frame wavelets can be surprisingly simple in nature. The structure of the spectral domains of the wavelets arising from the general neighborhood mapping construction raises a basic geometrical question. There is also the question of whether or not the general neighborhood mapping construction gives rise to all single dyadic orthonormal wavelets. Results are proved giving partial answers to both of these questions. Dedicated to Charles A. Micchelli for his 60th birthday Mathematics subject classification (2000) 42C40. John J. Benedetto: Both authors gratefully acknowledge support from ONR Grant N000140210398. The first named author also gratefully acknowledges support from NSF DMS Grant 0139759.     

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.     

Generalized inverses and similarity to partial isometries

We obtain some results related to the problems of Badea and Mbekhta (2005) [1] concerning the similarity to partial isometries using the generalized inverses. Especially, we involve the Moore-Penrose inverses. Also a characterization for such a similarity is given in the terms of dilations similar to unitary operators, which leads to a new criterion for the similarity to an isometry and to a quasinormal partial isometry.     

Tight frames of compactly supported multivariate multi-wavelets

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 C¹ and C² piecewise cubic schemes are quite involved, we also include the corresponding Matlab code in the Appendix.     

Tight Frames and Their Symmetries

The aim of this paper is to investigate the symmetry properties of tight frames, with a view to constructing tight frames of orthogonal polynomials in several variables which share the symmetries of the weight function, and other similar applications. This is achieved by using representation theory to give methods for constructing tight frames as orbits of groups of unitary transformations acting on a given finite-dimensional Hilbert space. Along the way, we show that a tight frame is determined by its Gram matrix and discuss how the symmetries of a tight frame are related to its Gram matrix. We also give a complete classification of those tight frames which arise as orbits of an abelian group of symmetries.     

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.     

Finite Normalized Tight Frames

Frames are interesting because they provide decompositions in applications where bases could be a liability. Tight frames are valuable to ensure fast convergence of such decompositions. Normalized frames guarantee control of the frame elements. Finite frames avoid the subtle and omnipresent approximation problems associated with the truncation of infinite frames. In this paper the theory of finite normalized tight frames (FNTFs) is developed. The main theorem is the characterization of all FNTFs in terms of the minima of a potential energy function, which was designed to measure the total orthogonality of a Bessel sequence. Examples of FNTFs abound, e.g., in R ³ the vertices of the Platonic solids and of a soccer ball are FNTFs.     

Generation of finite tight frames by Householder transformations

Finite tight frames are widely used for many applications. An important problem is to construct finite frames with prescribed norm for each vector in the tight frame. In this paper we provide a fast and simple algorithm for such a purpose. Our algorithm employs the Householder transformations. For a finite tight frame consisting of m vectors in ?ⁿ or ?ⁿ only O(nm) operations are needed. In addition, we also study the following question: Given a set of vectors in ?ⁿ or ?ⁿ, how many additional vectors, possibly with constraints, does one need to add in order to obtain a tight frame?     

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.      

Optimal D-RIP bounds in compressed sensing

This paper establishes new bounds on the restricted isometry constants with coherent tight frames in compressed sensing. It is shown that if the sensing matrix A satisfies the D-RIP condition δk 1/3 or δ2k2~(1/2)/2, then all signals f with D*f are k-sparse can be recovered exactly via the constrained 1 minimization based on y = Af, where D*is the conjugate transpose of a tight frame D. These bounds are sharp when D is an identity matrix, see Cai and Zhang's work. These bounds are greatly improved comparing to the condition δk 0.307 or δ2k 0.4931. Besides, if δk 1/3 or δ2k2~(1/2)/2, the signals can also be stably reconstructed in the noisy cases.     

四元数矩阵的极分解及其GL偏序

本文给出了四元数矩阵的唯一极分解定理和两个四元数矩阵可同时极分解的两种刻画；进而引进了四元数矩阵的GL偏序的概念，它是重要的Lǒwner偏序的一般化，并得到这个新偏序的6种刻画．