Construction of multivariate compactly supported orthonormal wavelets

We propose a constructive method to find compactly supported orthonormal wavelets for any given compactly supported scaling function φ in the multivariate setting. For simplicity, we start with a standard dilation matrix 2I_2×2 in the bivariate setting and show how to construct compactly supported functions ψ₁,. . .,ψ_n with n>3 such that {2^kψ_j(2^kx−ℓ,2^ky−m), k,ℓ,m∈Z, j=1,. . .,n} is an orthonormal basis for L₂(ℝ²). Here, n is dependent on the size of the support of φ. With parallel processes in modern computer, it is possible to use these orthonormal wavelets for applications. Furthermore, the constructive method can be extended to construct compactly supported multi-wavelets for any given compactly supported orthonormal multi-scaling vector. Finally, we mention that the constructions can be generalized to the multivariate setting. Dedicated to Professor Charles A. Micchelli on the occasion of his 60th birthday Mathematics subject classifications (2000) 42C15, 42C30.     

Meyer型正交小波基的构造与性质

本文基于多分辨分析理论与A．W．W方法将Meyer正交小波的构造规范化,给出其设计方法,并证明此类Meyer型小波母函数ψ（x)及相应的尺度函数ψ(x)具有优良的性质,如速降性O（│x│^-N-1(│x│→∞）、N阶消失矩、线性相位、对称性、频谱有限性、并且双尺度序列（滤波器）hn=ψ（n/2）等,并给出N＝2时构造小波函的具体实例。     

二元紧支撑正交小波的构造

§ 1.　Introduction　　SinceDAUBECHIES [1 ]gavethewellknownconstructionofunivariatecompactlysup portedorthonormalwavelets,considerableattertionhasbeenspentonconstructingmultivariatecompactlysupportedorthonormalwavelets [2— 5etc.] .Althoughmanyspecialbivariatenon separablewaveletshavebeenconstructed ,itisstillanopenproblemhowtoconstructbivariatecompactlyorthonormalwaveletsforanygivencompactlysupportedscalingfunction .Thepur poseofthispaperistoconstructcompactlysupportedorthogonalwaveletass…     

Construction of multivariate biorthogonal wavelets with arbitrary vanishing moments

We present a concrete method to build discrete biorthogonal systems such that the wavelet filters have any number of vanishing moments. Several algorithms are proposed to construct multivariate biorthogonal wavelets with any general dilation matrix and arbitrary order of vanishing moments. Examples are provided to illustrate the general theory and the advantages of the algorithms. This revised version was published online in June 2006 with corrections to the Cover Date.     

α尺度r重正交共轭滤波器的构造

对于α尺度r重紧支撑正交多小波系统,给出了由长为L的α尺度r重正交共轭滤波器构造长为L+1的α尺度r重正交共轭滤波器的一般方法,也给出了由低阶矩阵滤波器构造高阶矩阵滤波器的方法.若给定的正交共轭滤波器满足完全重构条件,则利用算法构造新的滤波器也满足完全重构条件,算法还保持正交共轭滤波器对称性,这一点在信号处理方面具有很好的应用价值.     

具有特殊伸缩矩阵的三元不可分小波的构造

多元小波分析是分析和处理高维数字信号的有力工具.不可分多元小波被广泛地应用在模式识别、纹理分析和边缘检测等领域.本文给出了构造一类特殊伸缩矩阵的紧支撑三元不可分小波的算法,利用该算法得到的小波函数继承了来源于尺度函数和符号函数的对称性和消失矩性质,由于符号函数中的参数选取具有很大的自由度,因此可以根据不同的实际情况来动态地确定符号函数,从而为这类小波在信号处理方面的应用提供了便利.最后给出了相应的数值算例.     

具有特殊伸缩矩阵的三元不可分正交小波的构造

多元小波分析是分析和处理多维数字信号的有力工具.不可分多元小波被广泛地应用在模式识别、纹理分析和边缘检测等领域.给出了构造具有伸缩矩阵(101-1-110-10)的紧支撑三元不可分正交小波的算法,利用该算法得到的小波函数继承了来源于尺度函数和符号函数的对称性和消失矩性质,从而为这类小波在信号处理方面的应用提供了便利.最后给出了数值算例.     

A family of orthonormal wavelet bases with dilation factor 4

In this paper, we study a method for the construction of orthonormal wavelet bases with dilation factor 4. More precisely, for any integer M>0, we construct an orthonormal scaling filter mM(ξ) that generates a mother scaling function ?M, associated with the dilation factor 4. The computation of the different coefficients of ²|mM(ξ)| is done by the use of a simple iterative method. Also, this work shows how this construction method provides us with a whole family of compactly supported orthonormal wavelet bases with arbitrary high regularity. A first estimate of α(M), the asymptotic regularity of ?M is given by α(M)∼0.25M. Examples are provided to illustrate the results of this work.     

Construction of multiwavelets with high approximation order and symmetry

In this paper, based on existing symmetric multiwavelets, we give an explicit algorithm for constructing multiwavelets with high approximation order and symmetry. Concretely, suppose Φ(x) := (φ1(x), . . . , φr(x))T is a symmetric refinable function vectors with approximation order m. For an arbitrary nonnegative integer n, a new symmetric refinable function vector Φnew(x) := (φn1 ew(x), . . . , φrn ew(x))T with approximation order m + n can be constructed through the algorithm mentioned above. Additionally,...     

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．     

A topological structure on a set of continous functions with various domains

We construct a topological space of continuous functions which generalizes the previously studied space of functions defined on closed intervals. For the new space, metrizability properties are studied. The results can be applied in the topological theory of ordinary differential equations. Translated fromMatematicheskie Zametki, Vol. 66, No. 1, pp. 76–88, July, 1999.