Theory And Application Of Generalized Interpolating And Interpolating Recursive Wavelets

Wavelet analysis and applications are a rapidly developing and novel subject in signal processing. In the pass of study of the past decade years, important mathematical basis and fi~ndamental theory frames have been established. With increasing perfection of basic theory and widely deepening of applications, designs of wavelet systems with good synthetic performance and adaptive wavelet methods for problems are developing two hotspots at present. As an important type of wavelets, interpolating wavelets have been successfully applied in some fields; but there are many theory basis and potential powers of applications, which need to be perfected and exploited. It is under this background that the dissertation deeply studies the extension and optimal design of interpolating wavelets, as well as adaptive interpolating wavelets. The dissertation mainly includes the following three aspects: 1. construction of Generalized Interpolating Wavelets; 2.optimal designs of regularity and redundancy degrees for GIWs; 3. construction of interpolating recursive wavelets and wavelet packets and study of adaptive interpolating wavelet methods. In accordance with the drawbacks of existed interpolating wavelets, the third chapter constructs a new type of interpolating wavelets---GLWs. Compared with the existed interpolating wavelets, GIWs are biorthogonal interpolating wavelets with compactly supported duals, and are completely determined by a pair of interpolating filters, thus, have flexibility in design. Finally, the numbers of vanishing moments for GIWs are analyzed and the decomposition and synthesis algorithms are depicted. The fourth chapter studies the relationships between the general interpolating filters and local WLS polynomial fitting, as well as Deslaurier-Dubuc filters. More important, we give the simple parameter expression of general interpolating filters, which is one of important foundations of the sequent chapters. And the criteria to select the boundary filters and three typical modes to process the boundary are given. The regularity and the redundancy degree are two important performance indexes and important factors to influence performance in denoise and compression. Combining optimal techniques and the parametric representation of G1Ws, the fifth 4 and sixth chapters study the optimal designs of the regularity and the redundancy degrees, respectively. Increase of the regularity of interpolating scaling functions efficiently improves the approximation power of systems, moreover, that of dual scaling functions markedly enhances the smoothness of duals and improves the frequency characteristics of analysis filters in passband and stopband. In some sense, the redundancy degrees restrict the performance in compression. On the basis of flexibility of GIWs, we first design nearly semi-orthogonal interpolating wavelets, then by mean of local orthogonalization method construct nearly orthogonal interpolating wavelets. Experiments shows that the optimal wavelets perform well in denoise and data compression and are superior to the orthogonal wavelets with the same numbers of vanishing moments, and these attribute to the improvement of synthesis performance of the optimal system. The seventh and eighth chapters study the adaptive interpolating wavelet and wavelet packets. First, we construct biorthogonal interpolating recursive wavelets (BLRW) with structural p...