Year: 2015
Analysis in Theory and Applications, Vol. 31 (2015), Iss. 3 : pp. 221–235
Abstract
The single 2 dilation orthogonal wavelet multipliers in one dimensional case and single $A$-dilation (where $A$ is any expansive matrix with integer entries and $|detA|=2$) wavelet multipliers in high dimensional case were completely characterized by the Wutam Consortium (1998) and Z. Y. Li, et al. (2010). But there exist no more results on orthogonal multivariate wavelet matrix multipliers corresponding integer expansive dilation matrix with the absolute value of determinant not 2 in $L^2(\mathbb{R}^2)$. In this paper, we choose $$2I_2=\left(\begin{array}{cc}2 & 0\\0 & 2\end{array}\right)$$ as the dilation matrix and consider the $2I_2$-dilation orthogonal multivariate wavelet$\Psi=\{\psi_1,\psi_2,\psi_3\}$, (which is called a dyadic bivariate wavelet) multipliers. We call the $3\times 3$ matrix-valued function $A(s)=[f_{i,j}(s)]_{3\times 3}$, where $f_{i,j}$ are measurable functions, a dyadic bivariate matrix Fourier wavelet multiplier if the inverse Fourier transform of $A(s)(\widehat{\psi_{1}}(s),\widehat{\psi_{2}}(s),\widehat{\psi_{3}}(s))^{\top}=(\widehat{g_1}(s),\widehat{g_2}(s),\widehat{g_3}(s))^{\top}$ is a dyadic bivariate wavelet whenever $(\psi_{1},\psi_{2},\psi_{3})$ is any dyadic bivariate wavelet. We give some conditions for dyadic matrix bivariate wavelet multipliers. The results extended that of Z. Y. Li and X. L. Shi (2011). As an application, we construct some useful dyadic bivariate wavelets by using dyadic Fourier matrix wavelet multipliers and use them to image denoising.
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Journal Article Details
Publisher Name: Global Science Press
Language: English
DOI: https://doi.org/10.4208/ata.2015.v31.n3.1
Analysis in Theory and Applications, Vol. 31 (2015), Iss. 3 : pp. 221–235
Published online: 2015-01
AMS Subject Headings: Global Science Press
Copyright: COPYRIGHT: © Global Science Press
Pages: 15
Keywords: Multi-wavelets Fourier multipliers image denoising.