Dyadic Bivariate Fourier Multipliers For Multi-Wavelets In L~2(R~2) And It’s Application

In this paper, we mainly discussed the sufficiency of the wavelet multipliers and the problem of the self-adaptive methods to choose the decomposition level.Firstly, we discussed some sufficiency conditions for the dyadic bivariate matrix Fourier wavelet multipliers. Through the study of the wavelets and the tight framelets in the high dementional space, we proved the three sufficiency conditions for the dyadic bivariate matrix Fourier wavelet multipliers. This provided us with a theory to construct some useful dyadic bivariate wavelets. At the same time, we indeed gave some examples of image denoising to prove its fisibility. The main result is theory 4.1.Secondly, we also discussed the suffifiency conditions for the dyadic bivariate matrix Fourier MRA wavelet multipliers. After the deep study for the applications of projection operators in wavelets, the excistence of MRA for framelets and the spectral function of shift-invariant spaces, we proved the sufficiency conditions for MRA wavelet multipliers based on the dyadic bivariate matrix Fourier wavelet multipliers. This provided us with a theory to construct some useful dyadic bivariate MRA wavelets. Surely, we also construct some examples to prove its fisibility. The main result is theory 4.2.Finally, we discussed the problem of self-adaptive methods to choose the decomposition level beased on the power analysis and the kurtosis analysis. In this process we designed some algorithms. After the running of the code, we got to know the usefulness of these algorithms. The main algorithms are in the section 6.3.