Theory Of Multi-band Wavelet And Its Application

Wavelet analysis is a new offset of mathematics which developing very fast in late 1980s.The appearance of wavelet has brought extensive effect in science research fields.The theory and application of wavelet analysis have obtained plentiful fruits in last two decates.Wavelet analysis,samilary as the traditional analysis,deploy and research an arbitrary function by base functions.But,compare to other bases,wavelet bases have some advantages.For example,wavelet bases have localize character,mathematical microtelescope feature and adaptive feature.Multiplicity of property is the main reason that wavelet analysis has been applied in many fields. people are possibility to tailor wavelet bases to a given application field.Therefore, the construction of wavelet bases is a very important task in the wavelet academic research.The main contents of this thesis are about the construction and application of multi-band wavelet and multiwavelet bases.The outline of this thesis is as follows:In chapter one,the history of wavelet development,the motivation and the primary results relating to the thesis are introduced.In chaper two,we summarize some methods to construct multi-band scaling functions firstly.A method to construct the compacted orthogonal symmtric multiband scaling function is given in details.Under the condition of given regular order, we consider the construction procedure for orthogonal symmetric scaling function with the shortest support.In chaper three,the theory on constructing multi-band wavelet frames is summarized at firstly.utilizing the UEP regulation,we consider in detail the method to construct the filter banks of symmetric and orthogonal multi-band wavelet.By analyzing the polyphase matrix of multi-band wavelet,we research a class of symmetric orthogonal filter banks of multi-band wavelet which are expressed by angle parameter.via choosing angle parameter,one can obtain multi-band wavelet bases with different property.In chaper four,we put forward the concept of optimal multi-band Haar wavelets and give a general method to construct the filter banks of optimal Haar wavelets. Optimal Haar wavelet transform is superior to the discrete cosine transform(DCT) which have been shown experimentally in image and video compression.In chaper five,we define a new operation between wavelet matrix.Applying the operation to orthonomal wavelet matrix,we obtain a method to construct orthononal wavelet matrix with especial structure such as pair-symmetric and beautiful structure.In chaper six,we firstly summarize some theory of multi-wavelet and method to construct orthogonal multi-wavelet.at last,in order to heighten p.p.o and vanishing moment,we research an algorithm for dimension extension of orthogonal symmetric multi-wavelets.In chaper seven,summarize and prospect,and we put forward some issue relating to the thesis which deserved to rescarch.The main innovations of this thsis are as follows:1.A general way to construct symmetric orthogonal multi-band scaling function is provided.A scheme for constructing scaling sequence with the shortest length and a given regularly order is obtained.2.The factorization of polyphase matrix of a class of orthogonal symmetric multi-band wavelet filter banks has been considered,method to construct orthogonal symmetric multi-band wavelet filter banks has been obtained.A class of orthogonal symmetric filter banks are provided by angle parameters.3.The concept of optimal multi-bank Haar wavelets was put forward,and a general method of constructing the optimal multi-bank Haar wavelets is provided.The optimal multi-bank Haar wavelets perform better than DCT in image compression has been shown experimentally.4.Appling wavelet matrix operation,we obtain a method to construct a class of orthononal wavelet matrix with especial structure such as pair-synunetric and beautiful structure wavelet matrix.In order to heighten vanishing moment,we put forward an algorithm for dimension extension of orthogonal symmetric multi-wavelets.