Construction And Smoothness Of Two Classes Of Multivariate Wavelets

Wavelet theory is a new analysis theory developed rapidly in last two decades. It is a versatile tool with very rich mathematical contents and great applications. It has been employed in many fields and applications, such as signal processing, image analysis, pattern recognition, biomedical imaging, radar, fractal, theoretical mathematics, and so on. In applications, it is well known that high smoothness, small support and high vanishing moments are the three most important properties of orthogonal wavelet or biorthogonal wavelet. The construction and smoothness of wavelet with two classes of dilation matrices is the main topic of this thesis. The main contributions in this paper can be summarized as follows:Firstly, the construction and smoothness of bivariate orthogonal wavelet andbiorthogonal wavelet with the dilation matrix [01 20] will be studied in the paper. For a scalingfunction or wavelet comes from a mask, we will mainly construct the masks which have arbitrary vanishing moment. Examples will be given to illustrate the general theory.Secondly, the construction and smoothness of biorthogonal wavelet with the checkerboard lattice will be studied and a family of ternary biorthogonal wavelets will be constructed. Since the checkerboard lattice is noted as the quincunx lattice, the quincunx wavelet will be generalized to the multidimensional space.Thirdly, we will also prove that a family of biorthogonal wavelets with other dilationmatrix, such as T = [11 1-1], can be obtained through the biorthogonal wavelets with the dilation matrix [01 20]. For biorthogonal wavelets based on the checkerboard lattice, the analogous result holds true.