Abstract:
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 filtersbanks 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 nonseparble 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 nonseparabke 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.
