Some Studies On Bidirectional Wavelet With M - Scaling Factor

Wavelet analysis is a new mathematical subject and developed rapidly, it has the double meaning of theory deeply and widely used at the same time. The study of the theory about the traditional wavelet has been perfect, but Daubechies proved that, besides the Haar wavelet, the orthogonal symmetric and compactly supported Unit-wavelet does not exist at the same time. To solve such problems, in1994, Goodman promoted Unit-wavelet to multiwavelet, that is to form the base of L2(R) space through the translated and stretched functions, which made the orthogonality, symmetry and the property of compactly supported have a good combination. However, in practical application the Unit-wavelet and multiwavelet have their limitations. So, in2006, Yang shouzhi and Xie changzhen with other scholars put forward the conception of two-direction refinement equation, introduced the two-direction scaling function, the corresponding two-direction wavelet and other related content, and then obtained a series of theory and conclusions with good properties and valuable application. Two-direction refinement equation is an extension of the refinement equation. Based on two-direction refinement function we can get the two-direction wavelet of good properties. So the research on two-direction wavelet becomes a new hot issue of the development of wavelet analysis.In this article, the emergence and development of wavelet analysis and two-direction wavelet are introduced. The conceptions of two-direction refinement function, the corresponding two-direction wavelet function and two-direction multiresolution analysis with dilation factor m are provided, the conditions are discussed when the two-direction refinement functions and its corresponding two-direction wavelets with dilation factor m are symmetric and antisymmetric. An algorithm for constructing the orthonormal two-direction scaling functions and its related orthonormal two-direction wavelets with orthonormal scaling functions is presented and an example is given to illustrate how to use this method. Based on these, this paper discusses the decomposition and reconstruction algorithm of the orthonormal two-direction wavelets with dilation factor m. Thus more processing ideas of scaling functions and its corresponding wavelets are extended to the research on two-direction wavelets. After that, this paper talks about the two-direction vector-valued function space, two-direction vector-valued multiresolution analysis with dilation factor m, and presents a thought of constructing compactly supported orthonormal two-direction vector-valued wavelets. Finally it makes a simple summary and puts forward some new research questions.