Moments, smoothness and optimization of wavelet systems

Abstract

This dissertation develops several new results on wavelet design using both traditional and new optimization criteria. We consider three different design problems: two problems address explicit optimization of function properties, and the third problem addresses optimization of filter properties. The first design is a generalization of the maximally regular Daubechies wavelets in that the maximal number of zero wavelet moments are traded for a larger number of small wavelet moments. Most work on applying wavelets has shown that the Daubechies solution is remarkably robust and near optimal (among known wavelet systems) for a number of applications, yet one is frequently left wondering why. By introducing this new class of wavelets we feel that such questions can be properly addressed experimentally. The second class of wavelet considered is a spin on smoothness. It has long been believed that wavelet smoothness is important in a number of applications. However, this is primarily supported by experimental evidence, and furthermore, it is not clear what kind of smoothness is most important (differentiability, Fourier transform decay rate, finite scale). To address this we develop a measure of smoothness for designing wavelet basis that gives rise to smooth but non-differentiable wavelet systems. The new family of wavelets introduced here is based on optimization of finite scale smoothness and is achieved by introducing a measure called discrete finite variation. Using the new measure, several design examples of finite scale smooth wavelets are provided. Preliminary compression results also indicate that these wavelet systems perform at least as well as both the optimally smooth solutions and the Daubechies solution. The third and final design looks at least square optimal wavelet filters. We develop an algorithm using Lagrange multiplier theory for the design of optimal filters. The algorithm is both flexible and robust in that it can be used for designing wavelets using mixed constraints as well as provide solutions for long wavelet filters previously not possible. A common thread among the design algorithms discussed in this dissertation is the use of filter coefficients as the parameter space for optimization.