A Study On The Theorem Of Multi-wavelets And Its Construction Algorithm

In recent years, the theory of wavelets which is widely used in sig-nal processing, image processing and other aspects of applications develop rapidly. At the same time, the different wavelets have been constructed for requirements of practical application and the development of mathematics itself. The wavelet with 2-scale has been extensively studied from the beginning work by Haar. Although the 2-scale wavelet is researched perfectly, there still are some significant drawbacks, for example, Daubechies pointed out in 1988:for the 2-scale wavelet, in addition Haar wavelet outside, there is no any orthogonal, symmetric and compactly supported wavelets, and the corresponding proof is given. In order to overcome the short-comings of 2-scale wavelet, multi-wavelets, further more, vector-valued wavelets is extended to. Then, multiwavelets becomes the hot field of wavelets to research, and the vector-valued multiwavelets is to be the new hot spot. The theory of multi-scale wavelets and the algorithm of them are mainly studied in this thesis, and the basic natures of multi-scale wavelets are also investigated. The outline of this thesis is as follows:In the first chapter, the history of the development of wavelet theory and the motivation and the paimary results relating to the thesis are introduced.In the second chapter, an algorithm of constructing the muti-wavelets with the corresponding properties of a known compact support symmetric orthogonal muti-wavelets is proposed through selecting the elementary rotation matrix and the elementary reflection matrix. A class of balanced wavelets can also be constructed using this method. Finally.a specific example is proposed.In the third chapter, a method to construct the compactly supported biorthogo-nal multi-wavelets with scale 3 corresponding to the compactly supported biorthog-onal scaling functions is given in details through transfering the scaling functions with a long support with multiplicity r to be the ones with a short support with multiplicity ar. Only the orthogonal expansion of the matrix and solving equa-tions are required during the whole process. Finally, the natrues of the biorthogonal multiwavelet packets with scale 3 are investigated.In the fourth chapter, the method for parameterization of the orthogonal fil-ters with scale 4 is investigated by the way of singular value decomposition of the orthogonal matrix. The parametric filter form is obtained, and the wavelets are con-structed by this mehod. The wavelet filters are causal, paraunitary and symmetry. Finally, an examples is proposed.In the fifth chapter, the concepts of the two-direction MRA and two-direction wavelets are introduced. The natures of the orthogonal two-direction wavelets and the symmetry of the two-directin scaling functions are studied. Finally, an algorithm of constructing two-direction wavelets is proposed based on the compactly supported refinable two-direction functions.In the last chapter, summarizes and prospect, and we put forward some issue relating to the thesis which diserved to research.The main innovations of this thesis is as follows:1. An algorithm to construct compactly supported biorthogonal multi-wavelets corresponding to the compactly supported biorthogonal multi-scaling functions with scale 3 is studied through matrix orthogonal expansion and solving equations. The results researched by Douglas P. Hardin, Jeffrey A. Marasovich and others are spreaded to the multi-scale wavelets from the simple two-scale wavelet.2. The natures of biorthogonal wavelet packets are investigated through the wavelets with scale 3 and the corresponding decomposition relations are studied. The algorithm of the orthogonal filters with scale 4 is proposed through parameter-ization by matrix singular value decomposition and the wavelets obtained by this method are symmetric.3. The concept of orthogonal symmetric two-direction wavelets is proposed. the conditions to distinguish the natrue symmetry of the given two-direction scaling functions are given. An algorithm for constructing the orthogonal symmetric two-direction wavelets is given based on the compactly supported scaling functions by using two-direction MRA and the matrix theory.