Volume 31, Issue 3
Dyadic Bivariate Fourier Multipliers for Multi-Wavelets in $L^2(\mathbb{R}^2)$

Anal. Theory Appl., 31 (2015), pp. 221-235.

Published online: 2017-07

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• 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.

• Keywords

Multi-wavelets, Fourier multipliers, image denoising.

42C15, 46C05, 47B10

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• RIS
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@Article{ATA-31-221, author = {}, title = {Dyadic Bivariate Fourier Multipliers for Multi-Wavelets in $L^2(\mathbb{R}^2)$}, journal = {Analysis in Theory and Applications}, year = {2017}, volume = {31}, number = {3}, pages = {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.

}, issn = {1573-8175}, doi = {https://doi.org/10.4208/ata.2015.v31.n3.1}, url = {http://global-sci.org/intro/article_detail/ata/4635.html} }
TY - JOUR T1 - Dyadic Bivariate Fourier Multipliers for Multi-Wavelets in $L^2(\mathbb{R}^2)$ JO - Analysis in Theory and Applications VL - 3 SP - 221 EP - 235 PY - 2017 DA - 2017/07 SN - 31 DO - http://doi.org/10.4208/ata.2015.v31.n3.1 UR - https://global-sci.org/intro/article_detail/ata/4635.html KW - Multi-wavelets, Fourier multipliers, image denoising. AB -

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.

Z. Li & X. Xu. (1970). Dyadic Bivariate Fourier Multipliers for Multi-Wavelets in $L^2(\mathbb{R}^2)$. Analysis in Theory and Applications. 31 (3). 221-235. doi:10.4208/ata.2015.v31.n3.1
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