The MRA Of AB-Wavelet And Biorthogonal AB-Wavelets

Since 1980s, wavelet analysis has been developing rapidly inmathematics, duing to its important role both in mathematical theory andapplications. It has been widely applied in many fields, such as signalprocessing, digital image processing and so on. But in practical applications,we find that it is not very effective in dealing with multidimensionalsignals containing distributed discontinuities such as edges, formultidimensional wavelet is formed by tensor product leading to its poordirectional sensitivity and the edges of signals are containing highdirectional objects of different traits. To overcome this limitation of waveletanalysis, many researchers have found many directional wavelets.In 2004, Gitta Kutyniok, Demetrio Labate and Wang-Q Lim introducedwavelets with composite dilations based on scaling matrix A and shearmatrix B, so we call it AB-wavelet. Among AB wavelets, for twodimensions instance, there is one kind of AB wavelets established by Demetrio Labate etc. containing three generators forms the orthogonal basisof . We call the AB-wavelet constructed by one generator ShearletL2 (R 2 ) .And there is theorem pointing out that Shearlet can not form the orthogonalbasis of L2 (R 2 ) .AB-wavelet not only has good directional sensitivity, detecting thelocation of discontinuities such as edges accurately, but also is efficient inrepresenting multidimensional signals. So in recent years, this kind ofwavelets is widely explored and getting complete.One of the most important tools to study wavelet is multiscale analysis,i.e. MRA. Owing to the functions forming different AB-wavelets are ofdifferent amounts and traits, it is impossible to establish the multiscaleanalysis theory of AB-wavelet uniformly. For the orthogonal AB-waveletformed by three generators, there is corresponding MRA has been given byGitta Kutyniok, Demetrio Labate etc. But as to other types of AB-wavelets,there is not.The advantage of orthogonal wavelets is obvious, and the abovediscussion show that shearlet is lack of orthogonality. ON AB-waveletformed by three generators increases complexity in the structure. When ONAB-wavelet is unavailable, it is significant for us to consider constructingbiorthogonal wavelets, also biorthogonal AB-wavelets.The main results, obtained in this dissertation, may be summarized asfollows:(1) We construct MRA structure using one kind of nonorthogonal AB-wavelets;(2) We study the biorthogonal theory of one kind of ABwavelets,and explore the relation between the corresponding filters; (3) Weestablish the Mallat algorithm under the structure constructed above; (4) Westudy sampling algorithm based on Mallat algorithm;(5) We promote thebiorthogonal AB-wavelets, and establish the formulations fordecomposition and reconstruction.