On The Study Of The Theory Of Constructions Of Shearlets

Wavelet analysis is one of the most strivingly developed braches in Harmonic analysis. It not only has an elegant mathematical theory, but also applies widespread in various fields such as image processing, signal analysis and data compression,etc. However, to resolve problems of planar edge treatment in image processing, classical wavelets do not have the property of sparsely optimal approximation and the anisotropic property, and this conveys the existence of their inadequacies in practical applications. In the recent decade, a set of non-classical wavelets are consecutively constructed, and partially resolve the processing problem for planar image boundaries. Among them, Shearlet which appears as the new emerged non-classical wavelet not only owns a beautiful mathematical theory, but also plays a dominant capability in the analysis of two or three dimensional image singularities.Each element in the Shearlet system can be generated by a single function ψ through scaling, shearing and translation transformations. Shearlet has a variety of desirable properties, such as1. directional sensitivity;2. anisotropic property;3. frame property;4. frequency localization;5. efficient implementation;6. sparsely optimal approximation.The research of Shearlet theory in L2(R2) turns out to be quite mature at present, but it lacks of concrete examples for the study of Shearlet constructions. The research which generalizes the results from Shearlet theory for L2(R2) into higher-dimensional cases is not yet well developed.In2012, Soren Hauser introduced some low-degree polynomial splines for problems of the explicit construction of Shearlet in L2(I). His results partially overcome the difficulty in the explicit Shearlet construction, and open a new direction for the study of Shearlet constructions. Polynomial spline serves as a highly efficient gadget for numerical approximation, and has already supplied various applications in the theory and algorithms for Wavelet analysis. Constructing finely stable polynomial splines is a valuable computational problem, and is also a focus in the study of explicit Shearlet constructions. In2013, K.Guo and D.Labate utilized a series of complicated techniques to settle the problem of constructing Shearlets in L2(R3) which equipped with the smooth Parseval frame, but their work do not give explicit functions for the Shearlet construction.This dissertation works on several results in higher dimensional Shearlet theory, such as1. defines two classes of high-degree defected polynomial splines with different defected vectors, and constructs Parseval frames of Shearlets in L2(I);2. defines the incomplete quasi-beta function, and constructs smooth Parseval frames of Shearlets in L2(R3);3. studies several classes of higher order matrix groups as well as structures of automorphic groups on them, and presents some properties of unitary representations for these higher dimensional matrix groups;4. defines higher order Shearlet group, and characterizes the irregular and regular Shearlet systems in higher dimensions;5. obtains a sufficient condition for the estimation of frame bounds for the irregular Shearlet system in higher dimensions, and constructs a concrete example for tight frames of regular Shearlet system inL2(R3).