Some Studies On Two-way Symmetric Wavelets

Construction of practical wavelets has become a hot topic for many scholars because the wide application of wavelet analysis. As everyone knows, dan wavelet will not meet with compact support, symmetry (anti-symmetry) and orthogonality except the Haar wavelet. In order to overcome the deficiency of dan wavelet, Professor Yang Shouzhi introduces the concept of two-direction refinable function and two-direction wavelet from the two scale two-direction refinable equation. But two-way wavelet is more general case of wavelet in the traditional sense, if it has good properties of the symmetry and anti-symmetry, so research on symmetric two-direction wavelet has very important practical significance.In this article, properties of symmetric and anti-symmetric orthogonal two-direction wavelet are studied, and explore the condition of construction for specific MRA using stable solution of two-way refinement equation L2; Then, constructions of method of two-direction wavelet is given by two-direction orthogonal refinable wavelet; Finally, the two scale two-direction wavelet is extended to the M scale symmetric two-direction wavelet, solving the problem of the positive and negative mask symbol of two scale two-direction wavelet with a unified method. There are four chapters as follows:In Chapter1, the origin and development of wavelet analysis are introduced; the specific environment of two-way symmetrical wavelet and its research level are presented.In Chapter2, Fourier transform, fast Fourier transform, framework and Riesz base symmetric and antisymmetric tight wavelet frames construction symmetric compactly supported wavelet frame by B spline function are mainly introduced.In Chapter3, concepts of two-direction refinable function and two-direction multi-resolution analysis are introduced starting from the basic research in the former, and structure of two-way orthogonal refinable function group and structure of its corresponding computed column are given.In Chapter4, the coefficient of two-direction refinable function is firstly studied, then the two scale orthogonal symmetric and anti-symmetric two-way wavelet is extended to the M scale orthogonal symmetric and antisymmetric two-way wavelet, and the method of its construction is given.