Theory and Implementation of Wavelet Analyses in Rational Resolution Decompositions.

摘要

The multiresolution analysis (MRA) developed by Mallat and Meyer and further discussed by Daubechies is a useful tool in the analysis of sampled signals such as images and speech. This thesis develops the theory and implementation of a rational-resolution analysis (RRA) as an extension of the dyadic MRA for arbitrary rational dilation factors. We present a method to calculate families of compactly-supported scaling functions and wavelets based on arbitrary integer dilation factors and provide examples. The perfect-reconstruction properties of the RRA are discussed and it is demonstrated that the compactly-supported scaling functions and wavelets do not yield perfect-reconstruction. However, the approximation-reconstruction is demonstrated and families of basis function which do lead to perfect reconstruction are characterized. Finally, comparisons are made between RRAs and conventional MRAs and illustrated with speech signals.