Some Research On Bi-directional Wavelet

This paper is devoted to some research about two-direction wavelets. Since the two-direction refinable function and two-direction wavelets were introduced by professor Yang Shouzhi a few years ago, people gradually had begun to do more discovery and study on this road. Compared with the traditional wavelets, two-direction wavelets are the more general cases and own more flexibility in practical application. So two-direction wavelets have become a hot object of research and attract much attention in the study of wavelet analysis.In this paper, firstly, orthogonal two-direction refinable function and orthogonal two-direction wavelets with dilation factor a are introduced. Then, using the the-ory of two-direction multi-resolution analysis, the decomposition and reconstruction Mallat algorithm of orthogonal two-direction wavelets with dilation a is obtained. Moreover, the corresponding matrix representation and the necessary and sufficient condition of complete reconstruction after decomposition are given. Secondly, the multivariate two-direction multi-wavelets refinement equation in which the dilation factor is a matrix is builded. Then, by using the unitary expansion principle and the polyphase decomposition of the corresponding matrix-valued Laurent polyno-mial, the construction algorithm and a sufficient condition about tight frames of the compactly supported multivariate two-direction multi-wavelets are deduced. The specific contents of this paper are devided into the following four chapters.Chapter one gives a brief introduction of the generation and development of wavelets analysis and a current research status about two-direction wavelets.Chapter two is about two-direction wavelets and multivariate two-direction multi-wavelets. Firstly, the definition of orthogonal two-direction refinable func-tion with dilation factor a is introduced from the two-direction refinement equation. Then, on this bases, two-direction multi-resolution analysis and orthogonal two-direction wavelets are defined. In addition, the definition of compactly supported two-direction refinable function vector is introduced by using two-direction refine-ment equation in which the dilation factor is a matrix. Then, this equation in the form of Fourier transform is deduced. Finally, the definition of multivariate two-direction multi-resolution analysis and the corresponding wavelets are given.Chapter three mainly investigates the Mallat algorithm of orthogonal two- direction wavelets with dilation fantor a. At the beginning, the theory of two-direction multi-resolution analysis and the given orthogonal two-direction refinable function or wavelets are recalled. Then, the decomposition and reconstruction al-gorithm of orthogonal two-direction wavelets with dilation a is obtained by the orthogonality of refinable function and wavelets. Besides, the corresponding matrix representation and the necessary and sufficient condition of complete reconstruction after decomposition are also given in the end. Which largely promotes the research about two-direction wavelets.Chapter four mainly studies the tight frames of the compactly supported mul-tivariate two-direction multi-wavelets. On the basis of the construction about tight frames of the compactly supported multivariate multi-wavelets, the construction algorithm of the compactly supported multivariate two-direction multi-wavelets is getted by the unitary expansion principle and the polyphase decomposition of the corresponding matrix-valued Laurent polynomial. Finally, a sufficient condition about tight frames of the compactly supported multivariate two-direction multi-wavelets are obtained in the form of a theorem. Which enriches and advances the tight frames of two-direction multi-wavelets.