Wavelets And Wavelet Packets Related To A Class Of Dilation Matrices

So far, rich fruits have been obtained about one-dimensional wavelets. Many applications, such as image compression, employ multidimensional wavelets and wavelet packets. But the construction of multidimensional wavelets and wavelet packets has not been well understood. In theory, the construction of wavelets generated from an orthogonal multiresolution analysis can be reduced to the unitary extension of matrix which is not easy in most cases. Riemenchncider and Shen gave a solution to the problem and a precise construction of wavelets. But it only applies to the case where the dilation matrix is 21 and the dimension does not exceed 3. In this thesis, by the investigation of unitary extension of matrix, construction of wavelets and wavelet packets related to a class of dilation matrices is given.This thesis is organized as follows. In Chapter 1, it is shown that many intrivial examples are included in the class of dilation matrices discussed in this thesis; In Chapter 2, the construction of wavelets related to the dilation matrix A is obtained; In Chapter 3, the corresponding wavelet packets are constructed; In the end of this thesis, a remark is given, which illustrates that our construction applies to a larger class of dilation matrices. At the same time, an unresolved problem is presented.