Volume 31, Issue 3
Dyadic Bivariate Fourier Multipliers for Multi-Wavelets in L2(R2)

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 onedimensional case and single A-dilation (where A is any expansivematrix with integer entries and |detA|=2) wavelet multipliers inhigh dimensional case were completely characterized by the WutamConsortium (1998) and Z. Y. Li, et al. (2010). But there exist no moreresults on orthogonal multivariate wavelet matrix multiplierscorresponding integer expansive dilation matrix with the absolutevalue of determinant not 2 in L2(R2). In this paper, wechoose $$2I_2=\left(\begin{array}{cc}2 & 0\\0 & 2\end{array}\right)$$ as the dilation matrix and consider the2I2-dilation orthogonal multivariate wavelet$\Psi=\{\psi_1,\psi_2,\psi_3\}$, (which is called a dyadic bivariatewavelet) multipliers. We call the $3\times 3$ matrix-valued function$A(s)=[f_{i,j}(s)]_{3\times 3}$, where $f_{i,j}$ are measurablefunctions, a dyadic bivariate matrix Fourier wavelet multiplier ifthe 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. Wegive some conditions for dyadic matrix bivariate waveletmultipliers. The results extended that of Z. Y. Li and X. L. Shi (2011). As an application, we construct some useful dyadicbivariate wavelets by using dyadic Fourier matrix waveletmultipliers and use them to image denoising.

• Keywords

Multi-wavelets Fourier multipliers image denoising

42C15 46C05 47B10

@Article{ATA-31-221, author = {Z. Li and X. Xu}, title = {Dyadic Bivariate Fourier Multipliers for Multi-Wavelets in L2(R2)}, journal = {Analysis in Theory and Applications}, year = {2017}, volume = {31}, number = {3}, pages = {221--235}, abstract = {The single 2 dilation orthogonal wavelet multipliers in onedimensional case and single A-dilation (where A is any expansivematrix with integer entries and |detA|=2) wavelet multipliers inhigh dimensional case were completely characterized by the WutamConsortium (1998) and Z. Y. Li, et al. (2010). But there exist no moreresults on orthogonal multivariate wavelet matrix multiplierscorresponding integer expansive dilation matrix with the absolutevalue of determinant not 2 in L2(R2). In this paper, wechoose $$2I_2=\left(\begin{array}{cc}2 & 0\\0 & 2\end{array}\right)$$ as the dilation matrix and consider the2I2-dilation orthogonal multivariate wavelet$\Psi=\{\psi_1,\psi_2,\psi_3\}$, (which is called a dyadic bivariatewavelet) multipliers. We call the $3\times 3$ matrix-valued function$A(s)=[f_{i,j}(s)]_{3\times 3}$, where $f_{i,j}$ are measurablefunctions, a dyadic bivariate matrix Fourier wavelet multiplier ifthe 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. Wegive some conditions for dyadic matrix bivariate waveletmultipliers. The results extended that of Z. Y. Li and X. L. Shi (2011). As an application, we construct some useful dyadicbivariate wavelets by using dyadic Fourier matrix waveletmultipliers 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} }