On translation invariance and optimal selection of wavelets

Wavelet theory is a recent development in applied mathematics. It offers a unified framework for the study of a handful of techniques developed in different areas. The focus of the work in this thesis is on translation invariance and optimal selection of wavelets. In this thesis, we propose a class of wavelet transform algorithms that are invariant to translations. We introduce the Kolmogorov n-width and the constrained n-width as measures for optimal wavelet basis selection. Applications of these techniques to image coding and noise reduction are also demonstrated.; In the first part of this thesis, we first propose translation invariant wavelet representation algorithms with circular extensions for both one-dimensional and two-dimensional signals. Using a binary tree or quadtree decomposition algorithm, we compute the wavelet transform for all the translates of a length-N or size N x N input signal in O(N log N) or {dollar}O(Nsp2{dollar} log N) operations. The wavelet coefficients of the translate with the minimal cost are selected as the best representation of the signal using a binary tree or quadtree search algorithm with a scale separable cost function. This representation is invariant under translations. Then, we introduce a class of translation invariant wavelet transforms that use symmetric extensions at the boundaries to reduce edge effects. These wavelet transforms achieve the following desirable properties simultaneously: (1) translation invariance, (2) reduced edge effects, and (3) size-limitedness. We then apply these translation invariant wavelet transform algorithms to a variety of applications in image coding and noise reduction.; In the second part of the thesis, we address the problem of optimal basis selection and wavelet representation for two important signal classes: the ellipsoidal signal class and the bounded cone class. We define time-frequency concentrated signals in this thesis as the class of signals whose Wigner distributions are concentrated in some region of the Wigner domain. We use the concept of the Kolmogorov n-width and the constrained n-width to quantitatively measure the ability of a basis to represent a signal class. We select the best wavelet representation by comparing the constrained n-widths of different wavelet bases.