On The Construction And Algebraic Structure Of Compacted Orthogonal Wavelet Filter Banks

The theory of wavelet analysis has grown explosively in the last twenty years. But the theories and methods of wavelet are in developing and far from maturation, wavelet analysis and its application have great potentialities in many applied fields of nature science.In this paper we study the algebraic and geometric structure of the space of compactly supported orthogonal wavelet and bi-orthogonal wavelet. The main contributions are as follows:Firstly, we introduce development background of wavelet analysis, and its application fields.Secondly, we provide some knowledge and define of wavelet analysis which is very important in the next context, and introduce the construction methods to wavelet functions of Daubechies wavelet. In this part, we mainly give a method to construct multi-band orthogonal wavelets, we prove that any multi-band orthogonal wavelet poly-phase matrix (which consists of the scaling filters and wavelet filters) can be factored as the product of primitive para-unitary matrices, an invertible matrix and the canonical Haar matrix. For this theory, we give a method for constructing any multi-band wavelet filter banks.Thirdly, we study the bi-orthogonal wavelet filter bank construction. In this part, we define poly-phase matrix of bi-orthogonal filter bank. we study the algebraic structure and parametrization factorization of compactly supported biorthogonal wavelet filter banks. And give a program for constructing biorthogonal wavelets. For this program, we can construct any compactly supported biorthogonal wavelets. And the algebraic structure of this filter bank can be shown by the program. Fourthly, we well study some special property and factorization form of polyphase matrix of 4-band orthogonal symmetric wavelet. Based on the factorization, a program for constructing 4-band orthogonal symmetric orthogonal wavelet filter banks is presented. At last we dedicate on the construction of one special symmetric filter banks, and study the factorization of poly-phase matrix of filter banks.