Factorizations and construction of linear phase paraunitary filter banks and higher multiplicity wavelets

It is known that paraunitary matrices can be factorized into shift products of orthogonal matrices or linear factors. When the number of rows of such a matrix (i.e. the number of channels of a paraunitary filter bank) is even, the symmetry constraints corresponding to the linear phase property of the filter bank can be expressed as restrictions on factors — except the very first one, all must be centrosymmetric. For odd numbers of rows the situation is more complicated. It turns out that paraunitary matrices comprising an even number of square blocks do not exist and quadratic centrosymmetric factors have to be used in the 0-shift product factorization. The centrosymmetric linear and quadratic factors can be easily obtained from partitions of centrosymmetric orthogonal matrices. Their parameterizations are also described.The characterizations of paraunitary matrices obtained from these factorizations are complete; the question of the number of free parameters is discussed. Furthermore, the proposed factorizations also allow us to derive lattice structures for linear phase paraunitary filter banks and, since the basic regularity conditions can be incorporated as a constraint on the first factor, they can be used also for the construction of symmetric higher multiplicity wavelets.and Cooperative Research Centre for Sensor Signal and Information ProcessingThe author is an Overseas Postgraduate Research Scholar supported by the Australian Government.     

基于Cayley变换的紧支撑二元正交小波滤波器组的构造

构造正交滤波器组,在多相域里就等价于构造仿酉矩阵,而仿酉矩阵的构造涉及到非线性方程组的求解.通过对Cayley变换的研究,把仿酉矩阵的构造转换为更易构造的仿斜厄米特矩阵,基于这种变换构造了二元紧支撑正交小波滤波器组,并给出了算例.     

基于仿酉矩阵的紧支撑二元正交小波滤波器组的构造

高维小波是处理多维信号的有力工具,张量积和栅格结构的小波有其自身的特点,但在实际应用中,我们仍需要构造小波滤波器来满足特定情形下的需要以提高滤波的效果,而构造正交滤波器,在多相域里就等价于构造仿酉阵,在本文中,我们通过对仿酉矩阵的研究,证明二元一次对称的仿酉阵一定能够块对角化,利用这种性质,给出了不可分离的二元正交小波滤波器组及线性相位小波滤波器的构造,并给出了相应的例子．     

中心对称仿酉矩阵的构造及二元具有线性相位正交小波滤波器组的参数化

中心对称仿酉矩阵(简记为CSPM)在线性相位的小波滤波器组的构造中起着重要的作用,本文给出偶数阶CSPM的表达式,矩阵中的元素为二元一次多项式.基于已给出的CSPM,给出具有线性相位的二元正交小波滤波器组的参数化,通过选取不同的参数可以得到的具有线性相位的正交小波滤波器组.最后给出算例.     

仿酉对称矩阵的构造及对称正交多小波滤波带的参数化

仿酉矩阵在小波、多小波、框架的构造中发挥了重要的作用.本文给出仿酉对称矩阵(简记为p.s.m.)的显式构造算法,其中仿酉对称矩阵是元素为对称或反对称多项式的仿酉矩阵.基于已构造的p.s.m.和已知的正交对称多小波(简记为o.s.m.),给出o.s.m.的参数化.恰当地选择一些参数,可得到具有一些优良性质的o.s.m.,例如Armlet.最后作这一个算例,构造出一类对称的Chui-Lian Armlet滤波带.     

An Algebraic Structure of Orthogonal Wavelet Space

In this paper we study the algebraic structure of the space of compactly supported orthonormal wavelets over real numbers. Based on the parameterization of wavelet space, one can define a parameter mapping from the wavelet space of rank 2 (or 2-band, scale factor of 2) and genus gto the (g−1) dimensional real torus (the products of unit circles). By the uniqueness and exactness of factorization, this mapping is well defined and one-to-one. Thus we can equip the rank 2 orthogonal wavelet space with an algebraic structure of the torus. Because of the degenerate phenomenon of the paraunitary matrix, the parameterization map is not onto. However, there exists an onto mapping from the torus to the closure of the wavelet space. And with such mapping, a more complete parameterization is obtained. By utilizing the factorization theory, we present a fast implementation of discrete wavelet transform (DWT). In general, the computational complexity of a rank morthogonal DWT is O(m²g). In this paper we start with a given scaling filter and construct additional (m−1) wavelet filters so that the DWT can be implemented in O(mg). With a fixed scaling filter, the approximation order, the orthogonality, and the smoothness remain unchanged; thus our fast DWT implementation is quite general.     

向量值正交小波的构造与向量值小波包的特征

The notion of vector-valued multiresolution analysis is introduced and the concept of orthogonal vector-valued wavelets with 3-scale is proposed.A necessary and sufficient condition on the existence of orthogonal vector-valued wavelets is given by means of paraunitary vector filter bank theory.An algorithm for constructing a class of compactly supported orthogonal vector-valued wavelets is presented.Their characteristics is discussed by virtue of operator theory,time-frequency method.Moreover,it is shown how to design various orthonormal bases of space L2(R,Cn) from these wavelet packets.     

LU分解与广义Schmidt正交化方法

利用对称内积的Schmidt正交化方法证明了各阶主子式不为零对称阵的LDLT分解.引入两个向量组关于弱内积广义正交的概念,并构造了将两组含相同个数向量的线性无关组化为广义正交组的广义Schmidt正交化方法.最后应用这一方法证明了各阶主子式不为零矩阵的LDU分解及一些相关的结果.     

一类向量值正交小波的设计与性质

引入整数因子伸缩的向量值正交小波与向量值小波包的概念.运用仿酉向量滤波器理论和矩阵理论,给出具有整数因子伸缩的向量值正交小波存在的充要条件.提供了紧支撑向量值正交的构建算法,给出了相应的构建算例.利用时频分析方法与算子理论,刻画了一类向量值正交小波包的性质,得到了整数伸缩的向量值小波包的正交公式.     

A study on compactly supported orthogonal vector-valued wavelets and wavelet packets

In this paper, vector-valued multiresolution analysis and orthogonal vector-valued wavelets are introduced. The definition for orthogonal vector-valued wavelet packets is proposed. A necessary and sufficient condition on the existence of orthogonal vector-valued wavelets is derived by means of paraunitary vector filter bank theory. An algorithm for constructing a class of compactly supported orthogonal vector-valued wavelets is presented. The properties of the vector-valued wavelet packets are investigated by using operator theory and algebra theory. In particular, it is shown how to construct various orthonormal bases of L²(R, C^s) from the orthogonal vector-valued wavelet packets.     

Orthogonal arrays constructed by generalized Kronecker product

In this paper, we propose a new general approach to construct asymmetrical orthogonal arrays, namely generalized Kronecker product. The operation is not usual Kronecker product in the theory of matrices, but it is interesting since the interaction of two columns of asymmetrical orthogonal arrays can be often written out by the generalized Kronecker product. As an application of the method, some new mixed-level orthogonal arrays of run sizes 72 and 96 are constructed.     

Smoothness Estimates for Soft-Threshold Denoising via Translation-Invariant Wavelet Transforms

In this paper, we study a generalization of the Donoho–Johnstone denoising model for the case of the translation-invariant wavelet transform. Instead of soft-thresholding coefficients of the classical orthogonal discrete wavelet transform, we study soft-thresholding of the coefficients of the translation-invariant discrete wavelet transform. This latter transform is not an orthogonal transformation. As a first step, we construct a level-dependent threshold to remove all the noise in the wavelet domain. Subsequently, we use the theory of interpolating wavelet transforms to characterize the smoothness of an estimated denoised function. Based on the fact that the inverse of the translation-invariant discrete transform includes averaging over all shifts, we use smoother autocorrelation functions in the representation of the estimated denoised function in place of Daubechies scaling functions.     

一类紧支撑矩阵值正交小波的构造

In this paper,we introduce matrix-valued multiresolution analysis and orthogonal matrix-valued wavelets.We obtain a necessary and sufficient condition on the existence of orthogonal matrix-valued wavelets by means of paraunitary vector filter bank theory.A method for constructing a class of compactly supported orthogonal matrix-valued wavelets is proposed by using multiresolution analysis method and matrix theory.     

Matrix orthogonal Laurent polynomials on the unit circle and Toda type integrable systems

Matrix orthogonal Laurent polynomials in the unit circle and the theory of Toda-like integrable systems are connected using the Gauss–Borel factorization of two, left and a right, Cantero–Morales–Velázquez block moment matrices, which are constructed using a quasi-definite matrix measure. A block Gauss–Borel factorization problem of these moment matrices leads to two sets of biorthogonal matrix orthogonal Laurent polynomials and matrix Szeg? polynomials, which can be expressed in terms of Schur complements of bordered truncations of the block moment matrix. The corresponding block extension of the Christoffel–Darboux theory is derived. Deformations of the quasi-definite matrix measure leading to integrable systems of Toda type are studied. The integrable theory is given in this matrix scenario; wave and adjoint wave functions, Lax and Zakharov–Shabat equations, bilinear equations and discrete flows — connected with Darboux transformations. We generalize the integrable flows of the Cafasso's matrix extension of the Toeplitz lattice for the Verblunsky coefficients of Szeg? polynomials. An analysis of the Miwa shifts allows for the finding of interesting connections between Christoffel–Darboux kernels and Miwa shifts of the matrix orthogonal Laurent polynomials.     

Factoring wavelet transforms into lifting steps

This article is essentially tutorial in nature. We show how any discrete wavelet transform or two band subband filtering with finite filters can be decomposed into a finite sequence of simple filtering steps, which we call lifting steps but that are also known as ladder structures. This decomposition corresponds to a factorization of the polyphase matrix of the wavelet or subband filters into elementary matrices. That such a factorization is possible is well-known to algebraists (and expressed by the formulaSL(n;R[z, z⁻¹])=E(n;R[z, z⁻¹])); it is also used in linear systems theory in the electrical engineering community. We present here a self-contained derivation, building the decomposition from basic principles such as the Euclidean algorithm, with a focus on applying it to wavelet filtering. This factorization provides an alternative for the lattice factorization, with the advantage that it can also be used in the biorthogonal, i.e., non-unitary case. Like the lattice factorization, the decomposition presented here asymptotically reduces the computational complexity of the transform by a factor two. It has other applications, such as the possibility of defining a wavelet-like transform that maps integers to integers. Research Tutorial Acknowledgements and Notes. Page 264.     

Parameterizations of Masks for Tight Affine Frames with Two Symmetric/Antisymmetric Generators

Parameterizations of FIR orthogonal systems are of fundamental importance to the design of filters with desired properties. By constructing paraunitary matrices, one can construct tight affine frames. In this paper we discuss parameterizations of paraunitary matrices which generate tight affine frames with two symmetric/antisymmetric generators (framelets). Based on the parameterizations, several symmetric/antisymmetric framelets are constructed.     

Fast Wavelet Transform for Toeplitz Matrices and Property Analysis

Fast wavelet transform algorithms for Toeplitz matrices are proposed in this paper. Distinctive from the well known discrete trigonometric transforms, such as the discrete cosine transform （DCT） and the discrete Fourier transform （DFT） for Toeplitz matrices, the new algorithms are achieved by compactly supported wavelet that preserve the character of a Toeplitz matrix after transform, which is quite useful in many applications involving a Toeplitz matrix. Results of numerical experiments show that the proposed method has good compression performance similar to using wavelet in the digital image coding. Since the proposed algorithms turn a dense Toeplitz matrix into a band-limited form, the arithmetic operations required by the new algorithms are O（N） that are reduced greatly compared with O（N log N） by the classical trigonometric transforms.     

基于反Krylov矩阵正交分解的半可分矩阵

在本文中，我们证明了对一个反Krylov矩阵作QR分解后，利用得到的正交矩阵可以将一个具有互异特征值的对称矩阵转化为一个半可分矩阵的形式，这个结果表明了反Krylov矩阵与半可分矩阵之间的联系．另外，我们还证明了这类对称半可分矩阵在QR达代下矩阵结构保持不变性．     

Multiscale Haar Orthonormal Matrices with the Corresponding Riesz Products and a Characterization of Cantor-Type Languages

We introduce a class of multiscale orthonormal matrices H(m) of order m×m, m = 2, 3,... . For m = 2 ^N, N = 1, 2,..., we get the well known Haar wavelet system. The term "multiscale" indicates that the construction of H(m) is achieved in different scales by an iteration process, determined through the prime integer factorization of m and by repetitive dilation and translation operations on matrices. The new Haar transforms allow us to detect the underlying ergodic structures on a class of Cantor-type sets or languages. We give a sufficient condition on finite data of lengthm, or step functions determined on the intervals [k/m, (k + 1)/m) , k = 0,...,m − 1 of [0, 1), to be written as a Riesz-type product in terms of the rows of H(m). This allows us to approximate in the weak-* topology continuous measures by Riesz-type products.     

Design of basic elements of multidimensional multirate systems. Multidimensional filters with fractional shift

The problem of designing filter banks for multidimensional multirate systems by using a lifting technique is considered. To solve it, we develop a design method for multidimensional digital filters with fractional shift. A symmetric structure is defined for τ = (1/2, 1/2) and a new structure is designed based on application of multidimensional Taylor series. Frequency and impulse responses are given for filters with fractional space shift and their L ₂-norm is found. Relevant wavelet functions are calculated and results of image compression by the designed filter banks are presented.