Construction of Multivariate Tight Framelet Packets Associated with Dilation Matrix

In this paper, we present a method for constructing multivariate tight framelet packets associated with an arbitrary dilation matrix using unitary extension principles.We also prove how to construct various tight frames for L2(Rd) by replac-ing some mother framelets.     

A-Parseval框架小波的特征刻画

研究与伸缩矩阵A相关的Parseval框架小波(A-PFW)的特征刻画,其中伸缩矩阵A满足A~3=2I_3且A的每一列元素之和均为偶数.首先,讨论了与两个特殊伸缩矩阵B,C相关的Parseval框架小波(B-PFW,C-PFW)之间的关系,并得到C-PFW分别与两类特殊伸缩矩阵D,■相关的Parseval框架小波(D-PFW,■-PFW)之间的等价关系.其次,探讨了伪尺度函数和源于多分辨分析的A-PFW(MRA A-PFW)的特征刻画.最后,借助于维数函数,给出了A-PFW是MRA A-PFW的一个充要条件.     

多尺度紧支撑向量值正交小波的构造

引入了多尺度向量值多分辨分析．利用矩阵理论,给出多尺度紧支撑向量值正交小波存在的必要条件及其构造方法．     

具有矩阵伸缩的双正交小波基

在这篇文章里,我们研究了伸缩为矩阵的双正交小波基的构造问题,在适当条件下,我们得到了Ｌ~２（Ｒ~ｎ）的小波框架或双正交小波基｛ｓｊ,ｋ｝和｛ｓｊ,ｋ｝,其中ｓｊｋ（ｘ）＝ｄｅｔＡｓ（Ａｊｘ－ｋ）,ｓｊ,ｋ（ｘ）＝ｄｅｔＡｊ２ｓ（Ａｊｘ－ｋ）（ｊ Ｚ． ｋ Ｚ~ｎ）及 Ａ是一伸缩矩阵．     

具有矩阵伸缩的尺度函数双正交特征

考虑了高维具有矩阵伸缩的尺度函数双正交性问题．给出了这种情形下尺度函数双正交性的三个特征刻划及尺度函数属于Ｌ～２（Ｒ～ｄ）的几个充分条件．     

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.      

On Multivariate Compactly Supported Bi-Frames

In this article, we construct compactly supported multivariate pairs of dual wavelet frames, shortly called bi-frames, for an arbitrary dilation matrix. Our construction is based on the mixed oblique extension principle, and it provides bi-frames with few wavelets. In the examples, we obtain optimal bi-frames, i.e., primal and dual wavelets are constructed from a single fundamental refinable function, whose mask size is minimal w.r.t. sum rule order and smoothness. Moreover, the wavelets reach the maximal approximation orderw.r.t. the underlying refinable function. For special dilation matrices, we derive very simple but optimal arbitrarily smooth bi-frames in arbitrary dimensions with only two primal and dual wavelets.     

Decompositions of trigonometric polynomials with applications to multivariate subdivision schemes

We study multivariate trigonometric polynomials satisfying the “sum-rule” conditions of a certain order. Based on the polyphase representation of these polynomials relative to a general dilation matrix, we develop a simple constructive method for a special type of decomposition of such polynomials. These decompositions are of interest in the analysis of convergence and smoothness of multivariate subdivision schemes associated with general dilation matrices. The approach presented in this paper leads directly to constructive algorithms, and is an alternative to the analysis of multivariate subdivision schemes in terms of the joint spectral radius of certain operators. Our convergence results apply to arbitrary dilation matrices, while the smoothness results are limited to two classes of dilation matrices.     

任意矩阵伸缩的正交小波包

1 引言 Coifman和Meyer引入L~2(R)中正交小波包,可以用张量积形式构造L~2(R~2)上的二维正交小波包;Chui和Li研究单变量非正交小波包和对偶小波包;Shen给出矩阵伸缩为2I时L~2(R~s)上非张量积小波包的构造算法;程正兴给出矩阵小波包的构     

Dyadic Bivariate Fourier Multipliers for Multi-Wavelets in L2(R2)

The single 2 dilation orthogonal wavelet multipliers in one dimensional case and single A-dilation(where A is any expansive matrix with integer entries and|det A|=2) wavelet multipliers in high dimensional case were completely characterized by the Wutam Consortium(1998) and Z. Y. Li, et al.(2010). But there exist no more results on orthogonal multivariate wavelet matrix multipliers corresponding integer expansive dilation matrix with the absolute value of determinant not 2 in L~2(R~2). In this paper, we choose 2I2=(_0~2 _2~0)as the dilation matrix and consider the 2 I2-dilation orthogonal multivariate waveletΨ = {ψ_1, ψ_2, ψ_3},(which is called a dyadic bivariate wavelet) multipliers. We call the3 × 3 matrix-valued function A(s) = [ f_(i, j)(s)]_(3×3), where fi, jare measurable functions, a dyadic bivariate matrix Fourier wavelet multiplier if the inverse Fourier transform of A(s)( ψ_1(s), ψ_2(s), ψ_3(s)) ~T=( g_1(s), g_2(s), g_3(s))~ T is a dyadic bivariate wavelet whenever(ψ_1, ψ_2, ψ_3) is any dyadic bivariate wavelet. We give some conditions for dyadic matrix bivariate wavelet multipliers. The results extended that of Z. Y. Li and X. L.Shi(2011). As an application, we construct some useful dyadic bivariate wavelets by using dyadic Fourier matrix wavelet multipliers and use them to image denoising.     

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.     

L2(Rs)中插值小波包的分裂技巧和精细分解

对具有任意伸缩矩阵A的插值加细函数，给出对应于L^2(R^s)中的小波包的一个构造方法．采样空间被直接分解来取代对加细函数的符号分解．按照这个方法构造的插值小波包能对基插值空间提供较为精细的分解，因而对自适应的插值给出较好的局部化．     

On a Transform Method for the Efficient Computation of Conditional V@R (and V@R) with Application to Loss Models with Jumps and Stochastic Volatility

In this paper we consider Fourier transform techniques to efficiently compute the Value-at-Risk and the Conditional Value-at-Risk of an arbitrary loss random variable, characterized by having a computable generalized characteristic function. We exploit the property of these risk measures of being the solution of an elementary optimization problem of convex type in one dimension. An application to univariate loss models driven by Lévy or stochastic volatility risk factors dynamic is finally reported.     

Regularity of Multivariate Refinable Functions

The regularity of a univariate compactly supported refinable function is known to be related to the spectral properties of an associated transfer operator. In the case of multivariate refinable functions with a general dilation matrix A , although factorization techniques, which are typically used in the univariate setting, are no longer applicable, we derive similar results that also depend on the spectral properties of A . September 30, 1996. Dates revised: December 1, 1996; February 14, 1997; August 1, 1997; November 11, 1997. Date accepted: November 14, 1997.     

Wavelets and Wavelet Packets Related to a Class of Dilation Matrices

The construction of wavelets generated from an orthogonal multiresolution analysis can be reduced to the unitary extension of a matrix, which is not easy in most cases. Jia and Micchelli gave a solution to the problem in the case where the dilation matrix is 21 and the dimension does not exceed 3. In this paper, by the method of unitary extension of a matrix, we obtain the construction of wavelets and wavelet oackets related to a class of dilation matrices.     

On the Toeplitz embedding of an arbitrary matrix

It has been shown by Delosme and Morf that an arbitrary block matrix can be embedded into a block Toeplitz matrix; the dimension of this embedding depends on the complexity of the matrix structure compared to the block Toeplitz structure. Due to the special form of the embedding matrix, the algebra of matrix polynomials relative to block Toeplitz matrices can be interpreted directly in terms of the original matrix and therefore can be extended to arbitrary matrices. In fact, these polynomials turn out to provide an appropriate framework for the recently proposed generalized Levinson algorithm solving the general matrix inversion problem.     

Asymptotics of matrix integrals and tensor invariants of compact Lie groups

In this paper we give an asymptotic formula for a matrix integral which plays a crucial role in the approach of Diaconis et al. to random matrix eigenvalues. The choice of parameter for the asymptotic analysis is motivated by an invariant-theoretic interpretation of this type of integral. For arbitrary regular irreducible representations of arbitrary connected semisimple compact Lie groups, we obtain an asymptotic formula for the trace of permutation operators on the space of tensor invariants, thus extending a result of Biane on the dimension of these spaces.

     

A very fast algorithm for matrix factorization

We present a very fast algorithm for general matrix factorization of a data matrix for use in the statistical analysis of high-dimensional data via latent factors. Such data are prevalent across many application areas and generate an ever-increasing demand for methods of dimension reduction in order to undertake the statistical analysis of interest. Our algorithm uses a gradient-based approach which can be used with an arbitrary loss function provided the latter is differentiable. The speed and effectiveness of our algorithm for dimension reduction is demonstrated in the context of supervised classification of some real high-dimensional data sets from the bioinformatics literature.     

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.

     

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.