Matrix-valued Wavelet Research Articles

Matrix-valued nonstationary frames associated with the Weyl–Heisenberg group and the extended affine group

We study nonstationary frames of matrix-valued Gabor systems and wavelet systems in the matrix-valued function space [Formula: see text]. First, we show that a diagonal matrix-valued window function constitutes a frame for [Formula: see text] whenever each diagonal entry constitutes a frame for the space [Formula: see text]. This is not true for arbitrary nonzero matrix-valued function. Using this, we prove the existence of nonstationary matrix-valued Gabor frames associated with the Weyl–Heisenberg group in terms of density of real numbers. We give a representation of the frame operator of matrix-valued nonstationary Gabor system. A necessary condition with explicit frame bounds for nonstationary matrix-valued Gabor frames associated with the Weyl–Heisenberg group is given. We discuss matrix-valued frame preserving maps in terms of adjointablity, with respect to the matrix-valued inner product, of bounded linear operators acting on [Formula: see text]. It is shown that the image of a matrix-valued Gabor frame under bounded, linear and invertible operator on [Formula: see text] may not be a frame for [Formula: see text]. In this direction, we give sufficient conditions on bounded linear operators which can preserve frame conditions. Finally, we give necessary and sufficient condition for the existence of nonstationary matrix-valued wavelet frames associated with the extended affine group.

A Matrix-Valued Inner Product for Matrix-Valued Signals and Matrix-Valued Lattices

A matrix-valued inner product was proposed before to construct orthonormal matrix-valued wavelets for matrix-valued signals. It introduces a weaker orthogonality for matrix-valued signals than the orthogonality of all components in a matrix that is commonly used in orthogonal multiwavelet constructions. With the weaker orthogonality, it is easier to construct orthonormal matrix-valued wavelets. In this paper, we re-study the matrix-valued inner product more from the inner product viewpoint that is more fundamental and propose a new but equivalent norm for matrix-valued signals. We show that although it is not scalar-valued, it maintains most of the scalarvalued inner product properties. We introduce a new linear independence concept for matrix-valued signals and present some related properties. We then present the Gram-Schmidt orthonormalization procedure for a set of linearly independent matrix-valued signals. Finally we define matrix-valued lattices, where the newly introduced Gram-Schmidt orthogonalization might be applied.

Time-varying spectral matrix estimation via intrinsic wavelet regression for surfaces of Hermitian positive definite matrices

Intrinsic wavelet transforms and denoising methods are introduced for the purpose of time-varying Fourier spectral matrix estimation. A non-degenerate time-varying spectral matrix constitutes a surface of Hermitian positive definite matrices across time and frequency and any spectral matrix estimator ideally adheres to these geometric constraints. Spectral matrix estimation of a locally stationary time series by means of linear or nonlinear wavelet shrinkage naturally respects positive definiteness at each time-frequency point, without any postprocessing. Moreover, the spectral matrix estimator enjoys equivariance in the sense that it does not nontrivially depend on the chosen basis or coordinate system of the multivariate time series. The algorithmic construction is based on a second-generation average-interpolating wavelet transform in the space of Hermitian positive definite matrices equipped with an affine-invariant metric. The wavelet coefficient decay and linear wavelet thresholding convergence rates of intrinsically smooth surfaces of Hermitian positive definite matrices are derived. Furthermore, practical nonlinear thresholding based on the trace of the matrix-valued wavelet coefficients is investigated. Finally, the time-varying spectral matrix of a nonstationary multivariate electroencephalography (EEG) time series recorded during an epileptic brain seizure is estimated.

Existence of Matrix-Valued Multiresolution Analysis-Based Matrix-Valued Tight Wavelet Frames

ABSTRACTIn this article, we introduce and study the matrix-valued tight wavelet frames for analyzing matrix-valued signal based on matrix-valued multiresolution analysis (MMRA). We put our emphasis on the existence of the MMRA-based matrix-valued tight wavelet frames by establishing the correspondence with their the unitary extension principle (UEP). Here in particular we introduce the square brackets product and the quasi-interpolatory operator, which makes the certificating process for UEP become relatively simple. Some interesting byproducts, such as features on the quasi-interpolatory operator 𝒫n in the matrix-valued function space case, are the critical foundation for our main work.

The Research of Orthogonal Symmetric Matrix-Valued Wavelets Packets with Multiscale and Applications

Material science is an interdisciplinary field applying the properties of matter to various areas of science and engineering. In this work, we introduce orthogonal matrix-valued wavelets with poly-scale, which are wavelets for vector fields, based on the notion of full rank subdivision operators. It is demonstrated that, like in the scalar and multiwavelet case, the existence of an orthogonal matrix-valued scaling function guarantees the existence of orthogonal matrix-valued wavelet functions. Secondly, we propose a construction algorim for compactly supported orthog onal two-directional matrix-valued wavelet packets. Lastly, A method for constructing a class of compactly supported orthogonal matrix-valued wavelets is presented by means of matrix theory.

The Characteristics Concerning a Class of Orthogonal Multivariate Matrix-Valued Scaling Functions and Frame Packets

Mechanical engineering is a discipline of engineering that applies the principles of physics and materials science for analysis, design, manufacturing, and maintenance of mechanical systems. In this work, the notion of matrix-valued multiresolution analysis of space is introduced. A method for constructing orthogonal matrix-valued ternary wavelet packs is developed and their properties are discussed by means of time-frequency analysis method, matrix theory and functional analysis method. Three orthogonality formulas concerning these wavelet packets are provided. Finally, new orthonormal wavelet pack bases of space are obtained by constructing a series of subspaces of orthogonal matrix-valued wavelet packets.

Matrix-Valued and Quaternion Wavelets

Wavelet transforms using matrix-valued wavelets (MVWs) can process the components of vector-valued signals jointly. We construct some novel families of non-trivial orthogonal n×n MVWs for n=2 and 4 having several vanishing moments. Some useful uniqueness and non-existence results for filters with certain lengths and numbers of vanishing moments are proved. The matrix-based method for n=4 is used for the construction of a non-trivial symmetric quaternion wavelet with compact support. This is an important addition to the literature where existing quaternion wavelet designs suffer from some critical problems.

The Noval Properties and Construction of Multi-scale Matrix-valued Bivariate Wavelet wraps

Abstract In this paper, we introduce matrix-valued multi-resolution structure and matrix-valued bivariate wavelet wraps. A constructive method of semi-orthogonal matrix-valued bivari-ate wavelet wraps is presented. Their properties have been characterized by using time-frequency analysis method, unitary extension principle and operator theory. The direct decom-position relation is obtained.

Constructions of Vector‐Valued Filters and Vector‐Valued Wavelets

Let a = (a1, a2, …, am)∈ℂm be an m‐dimensional vector. Then, it can be identified with an m × m circulant matrix. By using the theory of matrix‐valued wavelet analysis (Walden and Serroukh, 2002), we discuss the vector‐valued multiresolution analysis. Also, we derive several different designs of finite length of vector‐valued filters. The corresponding scaling functions and wavelet functions are given. Specially, we deal with the construction of filters on symmetric matrix‐valued functions space.

On Existence of Matrix-Valued Wavelets

Wavelet analysis has been a powerful tool for exploring and solving many complicated problems in natural science and engineering computation. In this paper, we investigate the existence of matrix-valued wavelet associated with a matrix-valued multireslution analysis. By using operator polar decomposition, we provide a new proof for the existence of matrix-valued wavelets. We prove that, like in the scalar case, every matrix-valued multiresolution analysis guarantees the existence of an orthogonal matrix-valued wavelet.

An Algorithm and Characteristics Concerning a Sort of Orthogonal Ternary Matrix-Valued Frame Wavelet Packs

In this work, the notion of matrix-valued multiresolution analysis of space L2(Rn, Cs×s) is introduced. A method for constructing orthogonal matrix-valued ternary wavelet packs is developed and their properties are discussed by means of time-frequency analysis method, matrix theory and functional analysis method. Three orthogonality formulas concerning these wavelet packets are provided. Finally, new orthonormal wavelet pack bases of space L2(Rn, Cs×s) are obtained by constructing a series of subspaces of orthogonal matrix-valued wavelet packets.

Optimal Interpolatory Wavelets Transform for Multiresolution Triangular Meshes

In recent years, several matrix-valued subdivisions have been proposed for triangular meshes. The matrix-valued subdivisions can simulate and extend the traditional scalar-valued subdivision, such as loop and 3 subdivision. In this paper, we study how to construct the matrix-valued subdivision wavelets, and propose the novel biorthogonal wavelet based on matrix-valued subdivisions on multiresolution triangular meshes. The new wavelets transform not only inherits the advantages of subdivision, but also offers more resolutions of complex models. Based on the matrix-valued wavelets proposed, we further optimize the wavelets transform for interactive modeling and visualization applications, and develop the efficient interpolatory loop subdivision wavelets transform. The transform simplifies the computation, and reduces the memory usage of matrix-valued wavelets transform. Our experiments showed that the novel wavelets transform is sufficiently stable, and performs well for noise reduction and fitting quality especially for multiresolution semi-regular triangular meshes.

The Orthogonality Character of Matrix Univariate Wavelet Packets Concerning an Integer Factor

The notion of matrix-valued multiresolution analysis. A procedure for designing orthogonal matrix-valued univariate wavelet packets is presented and their orthogonality properties are discussed by means of time-frequency analysis method, matrix theory and functional analysis method. Three orthogonality formulas concerning these wavelet packets are obtained. Finally, one new orthonormal basis of are obtained by constructing a series of subspaces of orthogonal matrix-valued wavelet packets.

A Study of Matrix-Valued Trivariate Wavelet Packets Associated with an Orthogonal Matrix-Valued Trivariate Scaling Function

Wavelet analysis has become a developing branch of mathematics for over twenty years. In this paper, the notion of matrix-valued multiresolution analysis of space is introduced. A method for constructing biorthogonal matrix–valued trivariate wavelet packets is developed and their properties are discussed by means of time-frequency analysis method, matrix theory and functional analysis method. Three biorthogonality formulas concerning these wavelet packets are provided. Finally, new Riesz bases of space is obtained by constructing a series of subspaces of biorthogonal matrix-valued wavelet packets.

Existence and design of biorthogonal matrix-valued wavelets

In this paper, we study biorthogonal matrix-valued wavelets for analyzing matrix-valued signals based on matrix multiresolution analysis. Firstly, sufficient conditions for the existence of biorthogonal matrix-valued scaling function are established in terms of two-scale matrix symbols. Then we focus on the construction of biorthogonal matrix-valued wavelet. Two designs based on factorizations of biorthogonal two-scale matrix symbol are presented. In particular, explicit constructing formulations for biorthogonal matrix-valued wavelets are given. With these formulations, highpass filters { G k } and { G ̃ k } of biorthogonal matrix-valued wavelets can be given explicitly by lowpass filters { H k } and { H ̃ k } of their corresponding biorthogonal scaling functions. Finally, according to our designs, two examples of two-scale matrix filters are given.

Construction and properties of orthogonal matrix-valued wavelets and wavelet packets

In this paper, we introduce matrix-valued multiresolution analysis and orthogonal matrix-valued wavelets with arbitrary integer dilation factor m. A necessary and sufficient condition on the existence of orthogonal matrix-valued wavelets is derived by virtue of paraunitary vector filter bank theory. An algorithm for constructing compactly supported m- scale orthogonal matrix-valued wavelets is presented. The notion of orthogonal matrix-valued wavelet packets is proposed. Their properties are investigated by means of time–frequency method, operator theory and matrix theory. In particular, it is shown how to construct various orthonormal bases of space L 2( R, C r× r ) from these wavelet packets, and the orthogonal decomposition relation is also given.

The properties of orthogonal matrix-valued wavelet packets in higher dimensions

In the paper matrix-valued multiresolution analysis and matrix-valued wavelet packets of spaceL 2(R n ,C s x s) are introduced. A procedure for constructing a class of matrix-valued wavelet packets in higher dimensions is proposed. The properties for the matrix-valued multivariate wavelet packets are investigated by using integral transform, algebra theory and operator theory. Finally, a new orthonormal basis ofL 2(R n ,C s x s) is derived from the orthogonal multivariate matrix-valued wavelet packets.

WAVELET ANALYSIS OF QUATERNION-VALUED TIME-SERIES

In this paper we introduce quaternion-valued multi-resolution analysis. Applying the theory of matrix-valued wavelet analysis, we give the construction of scaling functions and wavelets by identifying the quaternion-valued function with the complex duplex matrix-valued function.

A Matrix-Valued Wavelet KL-Like Expansion for Wide-Sense Stationary Random Processes

Matrix-valued wavelet series expansions for wide-sense stationary processes are studied in this paper. The expansion coefficients a are uncorrelated matrix random process, which is a property similar to that of a matrix Karhunen-Loe/spl grave/ve (MKL) expansion. Unlike the MKL expansion, however, the matrix wavelet expansion does not require the solution of the eigen equation. This expansion also has advantages over the Fourier series, which is often used as an approximation to the MKL expansion in that it completely eliminates correlation. The basis functions of this expansion can be obtained easily from wavelets of the Matrix-valued Lemarie/spl acute/-Meyer type and the power-spectral density of the process.