Multidimensional wavelets and applications.

Abstract

In this thesis, one- as well as multi-dimensional biorthogonal wavelet filters are designed and used for the construction of compactly supported wavelet bases. In particular, an adaptation of the McClellan transformation is used to design nonseparable 2-D biorthogonal wavelet bases. Some examples of 2-D biorthogonal wavelet filters are given in the case of the quincunx sampling lattice. Some theoretical and technical results known in the one-dimensional case have been generalized to the n-dimensional case. This generalization leads to a better understanding of the theory and design of multidimensional biorthogonal wavelets. An important part of the thesis devoted to the design of fast discrete wavelet transforms. The main ingredient of the algorithms is the use of a one-point quadrature formula for approximating the nest coefficients of the signals together with a suitable design and implementation of symmetric biorthogonal filters. Special attention is given to the case where the signals have sharp transition points. In this case, a smoothing process has been used to obtain an accurate reconstruction of the signal.