多小波、区间多小波和双向向量值小波的构造问题研究

发布时间：2018-02-11 02:35

本文关键词： 紧支撑正交多小波对称正定矩阵对角矩阵区间多小波伸缩因子正交性双正交多分辨分析双向向量值多重小波　出处：《新疆师范大学》2017年硕士论文　论文类型：学位论文

【摘要】：目前,小波分析是科学研究第一线的焦点。研究出带有诸多好的特性的小波是小波分析的中心课题,多小波由于满足这种需求而应运而生,现在多小波理论已经受到越来越多的关注,对于紧支撑正交多小波,从紧支撑正交性中使用多尺度函数来构造相应的多小波,到目前为止还没有平凡的构建方法,这使多小波和区间多小波运算相当复杂。因此,多小波和区间多小波的构造仍然是当前小波理论前沿研究的热点和难点问题,需要我们研究给出新的构造条件和方法。本文受杨守志教授思想影响,思考并研究了多小波、区间多小波和双向向量值小波的构造问题。本文具体有以下几个创新点:1.目前多小波的构造方法:仿酉矩阵的扩充,计算相当复杂,我们发现用现有文献中的方法构造多小波存在两个难点:第一,如何选择满足正定条件的系数矩阵;第二,在H~2=(aI-PiPiT)-1PiPiT中计算矩阵逆的过程,随着矩阵阶数的增大,其计算量也会增加,过程很复杂.当我们回忆对角矩阵的两个性质:(1)对角矩阵求逆时只需将对角线上的元素取倒数,(2)两个对角矩阵在相乘的时候可以交换次序,在第二章中,我们给出了新的构造条件和方法,并且成功的克服了以上两个难点.2.在第三章中,给出在[0,1]并且伸缩因子为a的区间多尺度函数以及区间多小波函数的定义,并给出其构造,同时得出相应的构造算例。3.在第四章中,给出一类双正交向量值双向多重小波(小波包)的定义,同时还深入研究了双向向量值双正交多重小波(小波包)的构造。
[Abstract]:At present, wavelet analysis is the focus of scientific research. Nowadays, more and more attention has been paid to the theory of multiwavelets. For compactly supported orthogonal multiwavelets, multiscale functions are used to construct the corresponding multiwavelets from compactly supported orthogonality, so far there is no ordinary method to construct them. Therefore, the construction of multiwavelets and interval multiwavelets is still a hot and difficult problem in the front research of wavelet theory. It is necessary for us to study and give new construction conditions and methods. This paper, under the influence of Professor Yang Shouzhi's thought, considers and studies multiwavelets. The construction of interval multiwavelets and bidirectional vector-valued wavelets. In this paper, there are some innovations as follows: 1. The present method of constructing multiwavelets: the expansion of para-unitary matrix, the computation is quite complex, We find that there are two difficulties in constructing multiwavelets by using the methods in existing literatures: first, how to select the coefficient matrix which satisfies the positive definite condition; secondly, the process of calculating the inverse of matrix in H ~ (2 +) ~ (2) I ~ (I) PiPiT ~ (-1) PiPiPiT, with the increase of matrix order, When we recall the two properties of the diagonal matrix: 1) when we recall the inverse of the diagonal matrix, we only need to take the reciprocal of the elements on the diagonal line, and when we multiply the two diagonal matrices, we can exchange the order of the two diagonal matrices. We give new construction conditions and methods, and successfully overcome the above two difficulties .2.In chapter 3, we give the definition of interval multiscale function and interval multiwavelet function with the expansion factor a, and give its construction. In Chapter 4th, the definition of a class of biorthogonal vector-valued biorthogonal biorthogonal biorthogonal multiplex wavelets (wavelet packets) is given, and the construction of biorthogonal vector-valued biorthogonal multiwavelets (wavelet packets) is also studied.
【学位授予单位】：新疆师范大学
【学位级别】：硕士
【学位授予年份】：2017
【分类号】：O174.2

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