空间中二进二元小波矩阵乘子及其应用.docx

华北电力大学硕士学位论文摘 华北电力大学硕士学位论文 摘 要 本文主要讨论了小波乘子的充分条件以及去噪过程中分解层数自适应确定 的问题． 首先，给出了二进二元小波Fourier矩阵乘子的充分条件．通过对高维空间 小波以及紧框架的研究，证明了二进二元小波Fourier矩阵乘子需要满足的三个 条件，为构造新的小波提供了理论依据．同时，我们也给出了相关的实例进行去 噪研究，验证结论的可行性．主要结果见第四章定理4．1． 其次，给出了二进二元MRA小波Fourier矩阵乘子的充分条件．通过对小波 映射算子的应用、小波框架多分辨分析的存在性以及旋转不变空间的研究，证明 了在二进二元小波Fourier矩阵乘子的基础上，MRA小波需要满足的条件，为构 造新的MRA小波提供了理论依据．同时，我们也给出了相关的实例验证结论的可 行性．主要结果见第四章定理4．2． 最后讨论了基于能量分析和峭度分析的小波分解层数自适应选取的问题．在 此过程中，文章设计了一系列的自适应算法，并给出了实际数据，验证了算法的 可用性．主要结果见第六章6．3节． 关键词：二进二元小波；Fourier矩阵乘子；图像降噪；能量分析；峭度分析； 层数自适应 万方数据 华北电力大学硕士学位论文Abstract 华北电力大学硕士学位论文 Abstract 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 demenfional 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 risibility．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 proj ection 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 risibility．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 algorit