The Orthogonality of Two-scale Three-dimensional Eight-direction Wavelet

Jing Zhang,Xiaoying Yang,Gang Wang,Chuanyan Hou

doi:10.11648/j.ajam.20210904.15

Jing Zhang, Xiaoying Yang + Show 2 more

Open Access

https://doi.org/10.11648/j.ajam.20210904.15

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Abstract

The construction of wavelets is a key problem in wavelet analysis. In the background of the one-dimensional double wavelet theory and the one dimensional biorthogonal bidirectional wavelet construction theory, this paper extends the one-dimensional bidirectional wavelet to the two-scale three-dimensional eight-direction biorthogonal wavelet. By using the method of tensor products to construct higher dimensional wavelets, the two-scale three-dimensional eight-direction multi-resolution analysis, two-scale three-dimensional eight-direction scale function and wavelet function are obtained. In addition, the conditions satisfied of the orthogonal and biorthogonal properties of the two-scale three-dimensional eight-direction wavelet are studied.

Highlights

The core problem of wavelet theory is to study of the structure and properties of wavelet
In order to make up for the deficiency of single wavelet, people put forward multi-wavelet theory, such as Gabor wavelet [1]
Professor Yang proposed the concepts of bidirectional subdivision equation and bidirectional wavelet, and discussed the orthogonality, approximation order and regularity of bidirectional subdivision function [13,14]

Summary

Introduction

The core problem of wavelet theory is to study of the structure and properties of wavelet. Yang Shouzhi et al gave the method of the constructing compactly supported orthonormal multiscale functions [4,5,6,7,8], and J Lian et al studied the orthogonal multiwavelet problem [9,10,11,12]. Based on the construction algorithm of a-scale three-dimensional eight-direction plus fine wave scale function [15] and biorthogonal a-scale three-dimensional eight-direction scale function and wavelet function [16], this paper extended the one-dimensional bidirectional wavelet to the two-scale three-dimensional eight-direction biorthogonal wavelet by tensor product. The two-scale three-dimensional eight-direction multi-resolution analysis is given, the construction of two-scale three-dimensional eight-direction scale function and wavelet function is studied, and the two-scale three-dimensional eight-directionl wavelet under orthogonal and biorthogonal conditions is obtained.

Two-Scale Three-dimensional Eight-direction Multi-resolution Analysis

Conclusion