Two-directional Wavelet Research Articles

The Mallat Algorithm of Biorthogonal Two-direction Bivariate Wavelets and Application in Communication Engineering

In this paper we investigate the Mallat algorithms of biorthogonal bivariate two-direction wavelets. We introduce the concepts of biorthogonal bivariate two-direction refinement function and biorthogonal two-direction bivariate wavelet. The Mallat algorithms of biorthogonal two-direction bivariate wavelets was investigated. Moreover, a necessary and sufficient condition of complete reconstruction with decomposition is deduced. This algorithm is important for the decomposition and reconstruction of discrete signal with finite energy.

Construction of two-direction tight wavelet frames

We investigate the construction of two-direction tight wavelet frames. First, a sufficient condition for a two-direction refinable function generating two-direction tight wavelet frames is derived. Second, a simple constructive method of two-direction tight wavelet frames is given. Third, based on the obtained two-direction tight wavelet frames, one can construct a symmetric multiwavelet frame easily. Finally, some examples are given to illustrate the results.

The Feature of Symmetric Frames and Two-Directional Vector Wavelets and Applications in Material Science

Material science is an interdisciplinary field applying the properties of matter to various areas of science and engineering. In this work, we study construction and properties of orthogonal two-direction vector-valued wavelet with poly-scale. Firstly, the concepts concerning two-direct-ional vector-valued waelets and wavelet wraps with multi-scale are provided. Secondly, we prop ose a construction algorim for compactly supported orthogonal two-directional vector-valued wave lets. Lastly, properties of a sort of orthogonal two-directional vector-valued wavelet wraps are char acterized by virtue of the matrix theory and the time-frequency analysis method.

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 Study of Compactly Supported Orthogonal Two-Direction Vector-Valued Wavelet Packet

The concept of two-direction vector-valued wavelet packets is proposed. An approach for constructing orthogonal two-direction vector-valued wavelet packets is developed and their properties are discussed by means of time-frequency analysis method, algebra theory and functional analysis method. Orthogonality formulas concerning these two-direction vector-valued wavelet packets are established.

A class of multiwavelets and projected frames from two-direction wavelets

This article aims at studying two-direction refinable functions and two-direction wavelets in the setting ℝs, s > 1. We give a sufficient condition for a two-direction refinable function belonging to L2(ℝs). Then, two theorems are given for constructing biorthogonal (orthogonal) two-direction refinable functions in L2(ℝs) and their biorthogonal (orthogonal) two-direction wavelets, respectively. From the constructed biorthogonal (orthogonal) two-direction wavelets, symmetric biorthogonal (orthogonal) multiwaveles in L2(ℝs can be obtained easily. Applying the projection method to biorthogonal (orthogonal) two-direction wavelets in L2(ℝs, we can get dual (tight) two-direction wavelet frames in L2(ℝm, where. m ≥ s From the projected dual (tight) two-direction wavelet frames in L2(ℝm, symmetric dual (tight) frames in L2(ℝm can be obtained easily. In the end, an example is given to illustrate theoretical results.

The Characterization of Two-Direction Vector-Valued Wavelets and its Applications in Material Engineering

Material science is an interdisciplinary field applying the properties of matter to various areas of science and engineering. In this work, we study construction and properties of orthogonal two-direction vector-valued wavelet with poly-scale. Firstly, the concepts concerning two-direct-ional vector-valued waelets and wavelet wraps with multi-scale are provided. Secondly, we prop ose a construction algorim for compactly supported orthogonal two-directional vector-valued wave lets. Lastly, properties of a sort of orthogonal two-directional vector-valued wavelet wraps are char acterized by virtue of the matrix theory and the time-frequency analysis method.

Construction of Compactly Supported Orthonormal Symmetric and Antisymmetric Two-Direction Refinable Functions and its Corresponding Wavelet with Scale M

In this paper, the concepts and related conclusions of two-direction multiresolution analysis and two-direction refinable functions are introduced; two-direction wavelet with scale 2 is researched and is extended to orthonormal symmetric and antisymmetric two-direction wavelet with scale M and relevant properties are obtained. A condition that stable solution of two-direction refinable functions can create for MRA is given; finally construction of two-direction wavelet by two-direction compactly orthonormal refinable functions is discussed.

Refinement type equations: sources and results

It has been proved recently that the two-direction refinement equation of the form \[ f(x)=\sum_{n\in\mathbb Z}c_{n,1}f(kx-n)+\sum_{n\in\mathbb Z}c_{n,-1}f(-kx-n) \] can be used in wavelet theory for constructing two-direction wavelets, biorthogonal wavel

The Traits of Nontensor Five-Variant Wavelet Packs and Applications in Management Science and Materials Science

Materials science is an interdisciplinary field applying the properties of matter to various areas of science and engineering. In this article, the notion of orthogonal nonseparable five-variant wavelet packages is presented. A novel approach for constructing them is presented by iteration method and functional analysis method. A feasible approach for constructing two-directional orthogonal wavelet packs is developed. The orthogonality property of five-variant wavelet packs is discussed. Three orthogonality formulas concerning these wavelet packs are estabished. A constructive method for affine frames of is proposed. The sufficient condition for the existence of a class of affine pseudoframes with filter banks is obtained by virtue of a generalized multiresolution analysis.

Designment Algorithm on Orthogonal Two-Directional Vector-Valued Wavelets with Three-Scale Dilation Factor

In this work, the notion of orthogonal two-directional shortly supported wavelets with poly-scale is developed. A new method for designing two-directional orthogonal wavelet wraps is proposed and their properties is investigated by means of time-frequency analysis methodand, operator theory. The existence of shortly supported orthogonal two-directional vector-valued wavelets associated with a pair of orthogonal compactly supported vector-valued scaling functions is investigated. A new method for designing a sort of orthogonal shortly supported two-directional vector-valued wavelet functions is presented by using multiresolution analysis and matrix theory.

A study on orthogonal two-direction vector-valued wavelets and two-direction wavelet packets

Abstract In this paper, the notion of two-direction vector-valued multiresolution analysis and the two-direction orthogonal vector-valued wavelets are introduced. The definition for two-direction orthogonal vector-valued wavelet packets is proposed. An algorithm for constructing a class of two-direction orthogonal vector-valued compactly supported wavelets corresponding to the two-direction orthogonal vector-valued compactly supported scaling functions is proposed by virtue of matrix theory and time–frequency analysis method. The properties of the two-direction vector-valued wavelet packets are investigated. At last, the direct decomposition relation for space L 2 ( R ) r is presented.

The Excellent Traits of a Class of Orthogonal Quarternary Wavelet Wraps with Short Support

The frame theory has been one of powerful tools for researching into wavelets. In this article, the notion of orthogonal nonseparable quarternary wavelet wraps, which is the generalizati- -on of orthogonal univariate wavelet wraps, is presented. A novel approach for constructing them is presented by iteration method and functional analysis method. A liable approach for constructing two-directional orthogonal wavelet wraps is developed. The orthogonality property of quarternary wavelet wraps is discussed. Three orthogonality formulas concerning these wavelet wraps are estabished. A constructive method for affine frames of L2(R4) is proposed. The sufficient condition for the existence of a class of affine pseudoframes with filter banks is obtained by virtue of a generalized multiresolution analysis. The pyramid decomposition scheme is established based on such a generalized multiresolution structure.

The Nice Characters of Biorthogonal Four-Element Wavelet Packets with Two-Scale

In this article, the notion of orthogonal nonseparable four-dimensional wavelet packets, which is the generalization of orthogonal univariate wavelet packets, is introduced. A new approach for constructing them is presented by iteration method. A novel approach for constructing two-directional biorthogonal wavelet packets is developed. The biorthogonality property of four-dimensional wavelet packets is discussed. Three biorthogonality formulas concerning these wavelet packets are estabished. A constructive method for affine frames of is proposed.

Constructive Algorithm to Two-Directional Biorthogonal Shortly Supported Wavelets with Poly-Scale Dilation Factor

In this work, the notion of biorthogonal two-directional shortly supported wavelets with poly-scale is developed. A new method for designing two-directional biorthogonal wavelet packets is proposed and their properties is investigated by means of time-frequency analysis methodand, operator theory. The existence of shortly supported biorthogonal vector-valued wavelets associated with a pair of biorthogonal compactly supported vector-valued scaling functions is investigated. A new method for designing a class of biorthogonal shortly supported vector-valued wavelet functions is presented by using multiresolution analysis and matrix theory.

A CLASS OF ORTHOGONAL TWO-DIRECTION REFINABLE FUNCTIONS AND TWO-DIRECTION WAVELETS

In this paper, an algorithm for constructing a class of orthogonal two-direction refinable functions and the corresponding orthogonal two-direction wavelets is obtained. In addition, the relation of both two-direction refinable functions and multiwavelets is discussed. We discover that Chui–Lian's orthogonal symmetric/antisymmetric multiscaling functions and the corresponding multiwavelets can be recovered by using an orthogonal two-direction refinable function and the corresponding two-direction wavelets, respectively. Finally, some construction examples are given.

Construction of biorthogonal two-direction refinable function and two-direction wavelet with dilation factor m

An algorithm for constructing a pair of biorthogonal two-direction refinable function and the corresponding biorthogonal two-direction wavelet is presented. In particular, the algorithm can also be used to construct two-direction orthogonal refinable function. Finally, we give an example illustrating how to use our method to construct biorthogonal two-direction scaling function with dilation factor 3.

Two-direction refinable functions and two-direction wavelets with high approximation order and regularity

The concept of two-direction refinable functions and two-direction wavelets is introduced. We investigate the existence of distributional(or L 2-stable) solutions of the two-direction refinement equation: $$\phi (x) = \sum\limits_k {p_k^ + } \phi (mx - k) + \sum\limits_k {p_k^ - } \phi (k - mx)$$ , where m ⩾ 2 is an integer. Based on the positive mask {p k + } and negative mask {p k − }, the conditions that guarantee the above equation has compactly distributional solutions or L 2-stable solutions are established. Furthermore, the condition that the L 2-stable solution of the above equation can generate a two-direction MRA is given. The support interval of ϕ(x) is discussed amply. The definition of orthogonal two-direction refinable function and orthogonal two-direction wavelets is presented, and the orthogonality criteria for two-direction refinable functions are established. An algorithm for constructing orthogonal two-direction refinable functions and their two-direction wavelets is presented. Another construction algorithm for two-direction L 2-refinable functions, which have nonnegative symbol masks and possess high approximation order and regularity, is presented. Finally, two construction examples are given.

Two-direction refinable functions and two-direction wavelets with dilation factor m

Motivated by cardinal B-spline scaling functions, we introduce the concept of two-direction refinable functions and two-direction wavelets with dilation factor m. We investigate the following two-direction refinement equation: ϕ ( x ) = ∑ k p k + ϕ ( mx - k ) + ∑ k p k - ϕ ( k - mx ) , where m ⩾ 2 is an integer. Here, two sequences { p k + } k and { p k - } k are called the positive mask and negative mask respectively. Based on the positive mask and negative mask, the conditions that the above equation has compactly distributional solutions are established. The conditions that the above equation has L 2-stable solutions are also presented. The support of ϕ( x) is discussed amply. An algorithm for constructing orthogonal two-direction refinable functions and their two-direction wavelets is presented. Based on orthogonal two-direction scaling functions and two-direction wavelets, orthogonal and symmetric multiscaling functions and their multiwavelets can be constructed easily. In the end, a construction example is given.

Biorthogonal two-direction refinable function and two-direction wavelet

The concept of two-direction refinable function and two-direction wavelet is generalized to the biorthogonal setting. The definition of biorthogonal two-direction refinable function is introduced, and the biorthogonality criteria for two-direction refinable function is established. We present an algorithm for constructing a pair of biorthogonal two-direction refinable function and the corresponding biorthogonal two-direction wavelet. Finally, we give two examples illustrating how to use our method to construct a pair of biorthogonal refinable function and its corresponding biorthogonal two-direction wavelet pair.