| Frame theory is the combining development result of the operator theory, the nonlinear approximation and information theory. Wavelet frame provides the fresh blood for frame theory, has greatly promoted the research of frame theory and ap-plication, and makes the frame theory strong vitality, which plays more and more important role in many fields such as signal processing. At the same time, the con-cept of frame is also one of the basic concept of wavelet analysis, and is one of the main tools to study wavelet analysis. The frame has played a very important role in the development of the wavelet analysis. This combination of the wavelet analysis and frame theory makes remarkable progress in a number of research fields, and has broad application prospects.This article mainly presents the wavelet compact frame, biorthogonal wavelet frame and orthogonal multiwavelet frame, which is wavelet function with some special properties, construction theory and related properties analysis. The outline of this thesis is as following:In chapter1, we introduce the development course and main application of the wavelet frame, some basic concepts and basic results of wavelet frame, the purpose and the research contents of this thesis and the major results obtained at last.In chapter2, we summarize the basic concepts of multi-resolution analysis which is the unified platform of the wavelet frame construction, and summarize basic theory, main method of compact wavelet frame construction. We present orthogonal wavelet frame construction steps. Using the special properties of the matrix structure, we give the polyphase matrix of a class of even-band orthogonal symmetric wavelet frame filter angle parameters expression. and discuss a new method for constructing the odd-band orthogonal symmetric wavelet frame. Finally, from a class of vector Laurent polynomial algebraic structure, we obtain a method to construct biorthogonal wavelet frame, which contributes to biothogonal frame construction computer programming.In chapter3,we first summarize the basic concepts and related basic theory based on L2(R) multiple multi-resolution analysis,and the theory on constructing the compact multiwavelet frame which based on multi-resolution analysis principle, then we discuss constrction method of multi-scaling function with high approximation order from unit-wavelet scaling function. For improving the orthogonal symmetric wavelet frame vanishing moments, given a pair of orthogonal symmetric multiwavelet filter banks,we research an algorithm by extenting dimension of orthogonal symmetric multiwavelet to lift vanishing moment of wavelet functions.In chaper4, we pointed out the main results of this paper and some key questions related to look forward to further research.In the main innovation of this paper are as follows:1. According to a class of orthogonal symmetric multi-band wavelet frame filter polyphase matrix special decomposition,we get a new method to construct a class of orthogonal symmetric multi-wavelet frame by using the matrix with special prop-erties.This method is easy to parameterize filter banks solution corresponding to orthogonal symmetric wavelet,and we can select appropriate angle parameters to get appropriate wavelet frame according to the requirements of the application domain.2. Using a kind of unitary vector matrix expansion principle,we consider the constructor of odd-band wavelet compact framework.Taking an example of three-band wavelet,we can obtain an algorithm about unitary expasions of odd-band orthogonal symmetric summetric wavelet polyphase matrix.3. We introduce the concept of reciprocal vector polynomial.Using two reversible vector algebraic decomposition structure, we obtain the constructor of biorthogonal wavelet dual scale Sequence and algebraic decomposition of biorthogonal filter bank which corresponding to three kinds of algebraic structure.4. We put forward an algorithm for dimension extension of orthogonal summer-tric multi-wavelet to lift vanishing moment. |
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