The identity of zeros of higher and lower dimensional filter banks and the construction of multidimensional nonseparable wavelets

This dissertation investigates the construction of nonseparable multidimensional wavelets using multidimensional filterbanks. The main contribution of the dissertation is the derivation of the relations zeros of higher and lower dimensional filterbanks for cascade structures. This relation is exploited to construct higher dimensional regular filters from known lower dimensional regular filters. Latter these filters are used to construct multidimensional nonseparable wavelets that are applied in the reconstruction and denoising of multidimensional images. The relation of discrete wavelets and multirate filterbanks was first demonstrated by Meyer and Mallat. Latter, Daubechies used this relation to construct continuous wavelets using the iteration of filterbanks. Daubechies also set the necessary conditions on these filer banks for the construction of continuous wavelets. These conditions also known as the regularity condition are critical for the construction of continuous wavelet basis form iterated filterbanks. In the single dimensional case these regularity conditions are defined in terms of the order of zeros of the filterbanks. The iteration of filterbanks with higher order zeros results in fast convergence to continuous wavelet basis. This regularity condition for the single dimensional case has been extended by Kovachevic to include the multidimensional case. However, the solutions to the regularity condition are often complicated as the orders and dimensions increase. In this dissertation the relations of zeros of lower and higher dimensional filters based on the definition of regularity conditions for cascade structures has been investigated. The identity of some of the zeros of the higher and lower dimensional filterbanks has been established using concepts in linear spaces and polynomial matrix description. This relation is critical in reducing the computational complexity of constructing higher order regular multidimensional filterbanks. Based on this relation a procedure has been adopted where one can start with known single dimensional regular filterbanks and construct the same order multidimensional nonseparable regular filterbanks. These filterbanks are then iterated as in the one dimensional case to give continuous multidimensional nonseparable wavelets. The conditions for dilation matrices with better isotropic transformation has also been revisited. Several examples are used to illustrate the construction of these multidimensional nonseparable wavelets. Finally, these nonseparable multidimensional wavelet basis are used in the reconstruction and denoising of multidimensional images and the results are compared to those obtained by separable wavelets