Construction Of Box-Spline Wavelet And Bivariate Multi-Wavelet

Wavelet analysis is a dictionary of wavelet bases. Constructing wavelet bases is an important research aspect of wavelet analysis. There are many kinds of scale functions and wavelet functions as well as construction methods. Construction of wavelets plays an important role in wavelet analysis. In particular, multi-wavelets not only have good properties of single wavelet but also overcome the defect that single wavelet can not simultaneously possess symmetry, orthogonality and compact support. While multi-wavelet provides perfect reconstruction, it can simultaneously preserve energy, good performance at boundaries, and a high order of approximation. Hence, multi-wavelet is widely used recently. Two methods of constructing multi-wavelet are given in this paper, concretely, two aspects finished in this thesis are as follows:Since the good properties of bivariate Box-spline function, a kind of new scale function and wavelet function is constructed by use of Haar scale function and Haar wavelet function. Several sufficient conditions are given when the new wavelet is bivariate Box-spline wavelet. At last, the expression of bivariate Box-spline wavelet is obtained.An approach of constructing bivariate orthogonal multi-wavelet function is presented by given bivariate orthogonal uni-wavelet function based on the unitary matrix. Combining the strength of Wavelets Analysis Theory and unitary matrix, the bivariate orthogonal multi-wavelet function is constructed using the properties of Wavelets Analysis Theory and unitary matrix. The bivariate orthogonal multi-wavelet function is a linear combination of the given bivariate orthogonal uni-wavelet function. Bivariate orthogonal multi-wavelet can be constructed so long as orthogonal uni-wavelet function and any unitary matrix are given. This method makes bivariate orthogonal multi-wavelet easy to be constructed.This thesis focuses on constructing bivariate Box-spline wavelet and bivariate orthogonal multi-wavelet using bivariate Box-spline function and unitary matrix respectively. These methods supply theoretical basis for solving general wavelet construction problem.