【作者】 邹庆云；

【导师】 王国秋；

【作者基本信息】 湖南师范大学 ， 基础数学， 2005， 硕士

【摘要】 王国秋提出了离散超小波变换,它扩大了小波的视野,特别是离散部分。本文以离散超小波变换为工具,提出了用于图像压缩的最优双正交小波基,通过分析实验数据,验证了其正确性。我们将最优条件拓展,通过变换矩阵的谱分析,揭示了它们的联系。 第一章介绍了小波发展史,多分辨分析,正交小波和双正交小波的性质。第二章提出了最优双正交小波基的定义,给出了最优条件。第三章根据最优条件构造出常见的双正交小波基并在第四章进行测试,以证明最优小波基的定义是合理的。第五章得出了当维数增加时,变换矩阵的谱是有界的。我们在理论上证明了双正交小波变换不可能无穷接近一个正交变换,并给出了双正交小波变换谱不可能达到的区间估计,给出了谱半径的一个下界!更多还原

【Abstract】 Discrete hyper-wavelet transforms which is put forward by Guoqiu Wang expand the scope of wavelet transforms, especially for its discrete part. In this paper, we put forward the definition for optimal biorthogonal wavelets base by discrete hyper-wavelet transforms and prove it is true by analyzing the data on image compression. We show the interaction between the optimal condition and the Biorthogonal wavelet filters spectrum.Chapter 1 introduce the wavelet development, Multi Resolution Analysis, the qualities of orthogonal wavelet filters and Biorthogonal wavelet filters. Chapter 2 put forward the definition for optimal biorthogonal wavelets base .Chapter 3 construct some different classes of optimal biorthogonal wavelets filter basing on optimal condition . We prove the definition for optimal biorthogonal wavelets base is reasonable in chapter 4. Chapter 5 show wavelet filters spectrum by discrete hyper-wavelet transforms is bouned . We find it is impossible for any biorthogonal wavelet filters spectrum to infinitely approach to a bounded interval . We find a interval estimation. We also prove that it is impossible for any biorthogonal wavelet transforms to infinitely approach to orthogonal wavelet transforms!更多还原