MRA-框架的高通滤波器乘子矩阵

发布时间：2018-07-12 14:44

本文选题：MRA-框架 + 洛朗多项式　；参考：《陕西师范大学》2015年硕士论文

【摘要】：小波框架既能克服正交小波的不足,又增加了适当的冗余性,不但保持了除正交之外的所有小波的性质,例如很好的时频局部化特性和平移不变性。在实际应用中它可以把光滑性、紧支撑、对称性(或反对称性)等完美的结合在一起。对信号的重构较正交小波有更好的稳定性；而且框架比正交小波更易于设计。本文首先介绍了多分辨分析和小波框架；然后给出了由MRA出发构造的框架(MRA-框架),这样构造的框架具有极其类似于使用MRA构造的小波(MRA-小波)的分解和重构算法。这样的算法非常简单,仅仅是分层迭代(类似于Mallat算法)。还介绍了框架乘子,包括对应于单个生成元框架的框架乘子和对应于多个生成元框架的Fourier乘子矩阵,它们都可以从已经构造好的小波框架出发构造出不同于已经存在的小波框架。基于MRA-框架的这种优势和Fourier乘子矩阵的这种思想的基础之上,将Fourier乘子矩阵限制在MRA-框架的范围内,得到了高通滤波器乘子矩阵。高通滤波器乘子矩阵可以从已经构造好的MRA-框架出发构造出不同于已经存在的MRA-框架。并且,给出了一个高通滤波器乘子矩阵的充分条件。然后,给出了怎样使得具有不同对称类型的洛朗多项式的加法和乘法运算结果仍然是对称(或者反对称)的洛朗多项式条件。根据洛朗多项式的这个性质,得到了构造出具有特定的对称性的高通滤波器乘子矩阵,可以使得由它构造的MRA-框架均是对称(或者反对称),只要已经构造好的MRA-框架是对称(或者反对称)的,这就是算法1。然后,用算法1给出了几个例子,从已经构造好的具有两个(三个)高通滤波器的对称(或者反对称)MRA-框架出发,得到不同于它们的具有两个(三个)高通滤波器的对称(或者反对称)MRA-框架。最后,再用最基本的图像处理——图像去噪,说明了使用高通滤波器乘子矩阵构造出的MRA-框架在信号处理方面还是具有一定的使用价值。
[Abstract]:Wavelet frame can not only overcome the deficiency of orthogonal wavelet, but also increase proper redundancy. It not only preserves the properties of all wavelets except orthogonality, such as good time-frequency localization and translation invariance. In practical applications, it combines smoothness, compact support, symmetry (or antisymmetry) perfectly. The reconstruction of signal is more stable than orthogonal wavelet, and the frame is easier to design than orthogonal wavelet. In this paper, we first introduce the multi-resolution analysis and wavelet framework, and then give the frame (MRA-frame) constructed from MRA, which is very similar to the decomposition and reconstruction algorithm of MRA-constructed wavelet (MRA-wavelet). Such an algorithm is very simple and is simply a hierarchical iteration (similar to the Mallat algorithm). The frame multipliers, including the frame multipliers corresponding to a single generative meta-frame and the Fourier multiplier matrices corresponding to a plurality of generating meta-frames are also introduced. All of them can construct wavelet frames which are different from the existing ones. Based on the advantage of MRA-frame and the idea of Fourier multiplier matrix, the Fourier multiplier matrix is limited to the MRA-frame, and the high-pass filter multiplier matrix is obtained. The multiplier matrix of high pass filter can construct different MRA-frame from the MRA-frame which has already been constructed. Moreover, a sufficient condition for the multiplier matrix of high pass filter is given. Then, how to make the addition and multiplication results of Laurent polynomials of different symmetric types are still symmetric (or antisymmetric) conditions of Laurent polynomials are given. According to this property of Laurent polynomial, the multiplier matrix of high pass filter with special symmetry is constructed. We can make the MRA-frame constructed by it symmetric (or antisymmetric), so long as the constructed MRA-frame is symmetric (or anti-symmetric), this is the algorithm 1. Then, by using algorithm 1, several examples are given, starting from a symmetric (or antisymmetric) MRA-frame with two (three) high pass filters. A symmetric (or antisymmetric) MRA-frame with two (or three) high pass filters is obtained. Finally, by using the most basic image process-image denoising, it is shown that the MRA-frame constructed by using the multiplier matrix of high-pass filter is still valuable in signal processing.
【学位授予单位】：陕西师范大学
【学位级别】：硕士
【学位授予年份】：2015
【分类号】：TN713

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