ORTHONORMAL MRA WAVELETS: SPECTRAL FORMULAS AND ALGORITHMS

2012 ◽  
Vol 10 (01) ◽  
pp. 1250008 ◽  
Author(s):  
F. GÓMEZ-CUBILLO ◽  
Z. SUCHANECKI ◽  
S. VILLULLAS
Keyword(s):  
Multiresolution Analysis ◽  
Matrix Representation ◽  
Infinite Product ◽  
Scaling Functions ◽  
Compactly Supported ◽  
Orthonormal Bases ◽  
Spectral Decompositions ◽  
Product Matrix ◽  
Orthonormal Wavelets

Spectral decompositions of translation and dilation operators are built in terms of suitable orthonormal bases of L2(ℝ), leading to spectral formulas for scaling functions and orthonormal wavelets associated with multiresolution analysis (MRA). The spectral formulas are useful to compute compactly supported scaling functions and wavelets. It is illustrated with a particular choice of the orthonormal bases, the so-called Haar bases, which yield a new algorithm related to the infinite product matrix representation of Daubechies and Lagarias.

A Study of a Two-Directional Vector Multivariate Wavelet Wraps and Applications in Engineering Materials

2014 ◽  
Vol 889-890 ◽  
pp. 1270-1274
Author(s):  
Jin Shun Feng ◽  
Qing Jiang Chen
Keyword(s):  
Multiresolution Analysis ◽  
Riesz Bases ◽  
Scaling Functions ◽  
Sufficient Condition ◽  
Analysis Method ◽  
Compactly Supported ◽  
Dilation Matrix ◽  
Engineering Materials ◽  
Vector Valued

The existence of compactly supported orthogonal two-directional vector-valued wavelets associated with a pair of orthogonal shortly supported vector-valued scaling functions is researched. We introduce a class of two-directional vector-valued four-dimensional wavelet wraps according to a dilation matrix, which are generalizations of univariate wavelet wraps. Three orthogonality formulas regarding the wavelet wraps are established. Finally, it is shown how to draw new Riesz bases of space from these wavelet wraps. The sufficient condition for the existence of four-dimensional wavelet wraps is established based on the multiresolution analysis method.

Existence and Designment of Filter Banks of Biorthogonal Vector-Valued Wavelets with Three-Scale Constant Factor

2010 ◽  
Vol 439-440 ◽  
pp. 938-943
Author(s):  
Hong Wei Gao ◽  
Li Ping Ding
Keyword(s):  
Multiresolution Analysis ◽  
Filter Banks ◽  
Matrix Theory ◽  
Constant Factor ◽  
New Method ◽  
Scaling Functions ◽  
Compactly Supported ◽  
Vector Valued

In this work, the notion of vector-valued multiresolution analysis and biort- ogonal vector–valued wavelets is introduced. The existence of compactly supported biorthogonal vector-valued wavelets associated with a pair of biorthogonal compactly supported vector-valued scaling functions is investigated. A new method for constructing a class of biorthogonal compactly supported vector-valued wavelet functions is presented by using multiresolution analysis and matrix theory.

Moments and Parametrizing Construction for 3-Band Biorthogonal Scaling Coefficients

2013 ◽  
Vol 284-287 ◽  
pp. 3189-3193
Author(s):  
Li Hong Cui ◽  
Yan Zhou ◽  
Bin Huang ◽  
Jian Jun Sun
Keyword(s):  
Symbolic Computation ◽  
Computer Algebra ◽  
Computation Method ◽  
Scaling Functions ◽  
Dual Scaling ◽  
Vanishing Moments ◽  
Compactly Supported ◽  
Orthonormal Wavelets ◽  
Discrete Moments ◽  
The Relationship

Regensburger and Scherzer described a symbolic computation method for moments and filter coefficients of scaling functions and obtained parametrizing compactly supported orthonormal wavelets. Following the idea, we are devoted to a study moments and parameterization construction for 3-band biorthogonal scaling coefficients with several vanishing moments. Firstly, we investigate the relations between filter lengths and symmetry. Then, we prove the relationship between dual continuous moments of 3-band biorthogonal scaling functions in theorem 2. This theorem reveals that the sum of continuous moments of dual scaling functions and is completely determined by the lower discrete moments. And we show the fact that the odd-indexed discrete moments are determined by the smaller even-indexed discrete moments. Finally, a family 3-band biorthogonal scaling coefficients with discrete moments as parameters are explicitly expressed based on computer algebra.

scholarly journals Wavelet transform of the dilation equation

1996 ◽  
Vol 37 (4) ◽  
pp. 474-489 ◽  
Author(s):  
Carlos A. Cabrelli ◽  
Ursula M. Molter
Keyword(s):  
Wavelet Transform ◽  
Multiresolution Analysis ◽  
Existence Of Solutions ◽  
Haar Wavelet ◽  
Scaling Functions ◽  
Infinite Products ◽  
Compactly Supported ◽  
Dilation Equation ◽  
Compactly Supported Solutions ◽  
Products Of Matrices

AbstractIn this article we study the dilation equation f(x) = ∑h ch f (2x − h) in ℒ2(R) using a wavelet approach. We see that the structure of Multiresolution Analysis adapts very well to the study of scaling functions. The equation is reduced to an equation in a subspace of ℒ2(R) of much lower resolution. This simpler equation is then “wavelet transformed” to obtain a discrete dilation equation. In particular we study the case of compactly supported solutions and we see that conditions for the existence of solutions are given by convergence of infinite products of matrices. These matrices are of the type obtained by Daubechies, and, when the analyzing wavelet is the Haar wavelet, they are exactly the same.

CONSTRUCTING MULTIPLE WAVELET PACKETS WITH FIFS

Fractals ◽  
2001 ◽  
Vol 09 (02) ◽  
pp. 165-169
Author(s):  
GANG CHEN ◽  
ZHIGANG FENG
Keyword(s):  
Multiresolution Analysis ◽  
Wavelet Packet ◽  
Wavelet Packets ◽  
Scaling Functions ◽  
Wavelet Packet Decomposition ◽  
Fractal Interpolation ◽  
Fractal Interpolation Functions ◽  
Interpolation Functions

By using fractal interpolation functions (FIF), a family of multiple wavelet packets is constructed in this paper. The first part of the paper deals with the equidistant fractal interpolation on interval [0, 1]; next, the proof that scaling functions ϕ1, ϕ2,…,ϕr constructed with FIF can generate a multiresolution analysis of L2(R) is shown; finally, the direct wavelet and wavelet packet decomposition in L2(R) are given.

scholarly journals Higher Rank Wavelets

2011 ◽  
Vol 63 (3) ◽  
pp. 689-720
Author(s):  
Sean Olphert ◽  
Stephen C. Power
Keyword(s):  
Orthonormal Basis ◽  
Fourier Transforms ◽  
Latin Square ◽  
Haar Wavelets ◽  
Scaling Functions ◽  
Wavelet Sets ◽  
Compactly Supported ◽  
Rank 2 ◽  
Higher Rank ◽  
Meyer Wavelet

Abstract A theory of higher rank multiresolution analysis is given in the setting of abelian multiscalings. This theory enables the construction, from a higher rank MRA, of finite wavelet sets whose multidilations have translates forming an orthonormal basis in L2(ℝd). While tensor products of uniscaled MRAs provide simple examples we construct many nonseparable higher rank wavelets. In particular we construct Latin square wavelets as rank 2 variants of Haar wavelets. Also we construct nonseparable scaling functions for rank 2 variants of Meyer wavelet scaling functions, and we construct the associated nonseparable wavelets with compactly supported Fourier transforms. On the other hand we show that compactly supported scaling functions for biscaled MRAs are necessarily separable.

Study of High Dimensional Orthogonal Vector-Valued Wavelet with Scale 4

2013 ◽  
Vol 790 ◽  
pp. 665-668
Author(s):  
Wei Qing Yang
Keyword(s):  
Multiresolution Analysis ◽  
High Dimensional ◽  
Orthogonal Vector ◽  
Compactly Supported ◽  
Definition Of ◽  
Vector Valued

In this paper, we introduce the definition of vector-valued multiresolution analysis with scale 4 and orthogonal vector-valued wavelet with scale 4 is gived. The properties of compactly supported orthogonal vector-valued wavelets with scale 4 are proved.

A Feasible Algorithm for Designing Biorthogonal Binary Finitely Supported Multiwavelet Functions

2011 ◽  
Vol 225-226 ◽  
pp. 1092-1095
Author(s):  
Bao Min Yu
Keyword(s):  
Wavelet Analysis ◽  
Multiresolution Analysis ◽  
Natural Science ◽  
Matrix Theory ◽  
Scaling Functions ◽  
Science And Engineering ◽  
Engineering Computation ◽  
Feasible Algorithm ◽  
Definition Of ◽  
Vector Valued

Wavelet analysis has been a powerful tool for exploring and solving many complicated problems in natural science and engineering computation. In this paper, the notion of vector-valued multiresolution analysis is introduced and the definition of the biorthogonal vector-valued bivariate wavelet functions is given. The existence of biorthogonal vector-valued binary wavelet functions associated with a pair of biorthogonal vector-valued finitely supported binary scaling functions is investigated. An algorithm for constructing a class of biorthogonal vector-valued finitely supported binary wavelet functions is presented by virtue of multiresolution analysis and matrix theory.

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