Complex Wavelet Compact Frame Construction And P-scale Two-dimensional Bidirectional Wavelets

Wavelet analysis is developed on the basis of fourier analysis and frame theory,as one of the important components of wavelet analysis,has also been rapidly developed.Wavelet tight frame construction based on multi-resolution analysis is one of the core problems of frame theory,provided a sparse approximation of piecewise function and guarantees the fast wavelet decomposition algorithm and reconstruction algorithm.In 2006,Professor Yang proposed the concept of two scales two-directional refinable equation based on the traditional concept of two scales refinable equation,and gave the construction algorithm of two-directional wavelet.The development of wavelet analysis is further promoted by the proposal of two-directional wavelet.In this thesis,based on the in-depth understanding of the research status of wavelet tight frame and two-directional wavelet,a complex wavelet tight frame based on discrete sequences generation of multi-resolution analysis is constructed,and the multi-resolution analysis and wavelet decomposition algorithm and reconstruction algorithm of two-dimensional two-directional refinable function generation are given.Specifically,the main contents are as follows:Firstly,the development history of wavelet analysis at home and abroad is briefly introduced,and the research status and basic theories of multi-resolution analysis,wavelet tight frame and two-directional wavelet are introduced.Then,according to the unitary extension theorem,a sufficient condition is given for the wavelet system defined by discrete finite support sequences to form a functional space L2(R)wavelet tight frame,and Matlab tools are used to give the specific function images of the refinable function and the wavelet function of the complex wavelet tight frame at different scales.Finally,the sufficient and necessary conditions for the compact support distribution solution of the two-directional refinable equation with expansion factor of p is discussed,and the generated multi-resolution analysis,as well as the decomposition algorithm and reconstruction algorithms of two-directional orthogonal wavelets are discussed.