High Balanced Biorthogonal Multiwavelets with Symmetry

Youfa Li,Gengrong Zhang,Yanfeng Shen,Shouzhi Yang

doi:10.1155/2014/154269

Youfa Li, Gengrong Zhang + Show 2 more

Open Access

https://doi.org/10.1155/2014/154269

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Abstract

Balanced multiwavelet transform can process the vector-valued data sparsely while preserving a polynomial signal. Yang et al. (2006) constructed balanced multiwavelets from the existing nonbalanced ones. It will be proved, however, in this paper that if the nonbalanced multiwavelets have antisymmetric component, it is impossible for the balanced multiwavelets by the method mentioned above to have symmetry. In this paper, we give an algorithm for constructing a pair of biorthogonal symmetric refinable function vectors from any orthogonal refinable function vector, which has symmetric and antisymmetric components. Then, a general scheme is given for high balanced biorthogonal multiwavelets with symmetry from the constructed pair of biorthogonal refinable function vectors. Moreover, we discuss the approximation orders of the biorthogonal symmetric refinable function vectors. An example is given to illustrate our results.

Highlights

Multiwavelets have been studied extensively in the literature; for example, to mention only a few here, see [1,2,3,4,5,6,7] and the references therein
Let us introduce the relationship between the mask symbol of a refinable function vector and its symmetry
Note. (I) The algorithm for constructing Û(ξ) in Lemma 2 can be seen in the proof of [2, Theorem 2.5]. (II) There exist a number of orthogonal multiwavelets in the literature, whose mask symbols take the form of (15); just see [12, 13] for some examples

Summary

Introduction

Multiwavelets have been studied extensively in the literature; for example, to mention only a few here, see [1,2,3,4,5,6,7] and the references therein. We will construct pairs of biorthogonal symmetric multiwavelets with high balanced orders from any orthogonal refinable function vector, which simultaneously has symmetric and antisymmetric components. Let us introduce the relationship between the mask symbol of a refinable function vector and its symmetry. Φr)T be an orthogonal d-refinable function vector We say that it has κ + 1 balanced orders if. Remark 1 (a balanced refinable function vector does not has antisymmetric component). Since many famous refinable function vectors satisfy (13), for example, CL multiwavelets [12], we naturally face the following problem. How can we construct symmetric multiwavelets with high balanced orders from such ones that have antisymmetric component?. Note. (I) The algorithm for constructing Û(ξ) in Lemma 2 can be seen in the proof of [2, Theorem 2.5]. (II) There exist a number of orthogonal multiwavelets in the literature, whose mask symbols take the form of (15); just see [12, 13] for some examples

Main Results

High Balanced Biorthogonal Multiwavelets from CL Multiwavelets

Future work

Conflict of Interests