Optimization issues in wavelets and filter banks

Citation

Djokovic, Igor (1995) Optimization issues in wavelets and filter banks. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/pyy4-dt15. https://resolver.caltech.edu/CaltechETD:etd-09172007-152709

Abstract

In the last decade or so, we have witnessed a rapid development of the wavelet and filter bank theory. Wavelets find applications in signal compression, computer vision, geophysics, pattern recognition, numerical analysis, and function theory, just to name a few. Filter banks, on the other hand, offer very efficient implementation of different algorithms in connection with wavelets. The thesis deals with three problems in filter banks and wavelets. In the first part, we show that perfect reconstruction is equivalent to biorthogonality of the filters. Using this, we examine existence issues in nonuniform filter banks. We show that whenever there exists a rational biorthogonal filter bank, then there is a rational orthonormal filter bank as well. We also derive a number of necessary conditions for the existence of perfect reconstruction nonuniform filter banks. We show how the tools developed in the first part can be used for decorrelation of subband signals. The second problem deals with optimality issues in wavelet and filter bank theory. We tune scaling function for the analysis of WSS random processes, so that the energy is concentrated in as few transform coefficients as possible. The corresponding problem in the filter bank theory is that of adapting filter responses to a given (discrete time) WSS random process so as to achieve a better energy compaction. Finally, the last part is devoted to developing sampling theory for multiresolution subspaces. More precisely, we extend existing uniform sampling theory to periodically nonuniform sampling. This extension offers one very important advantage over the existing sampling theory. By allowing for periodically nonuniform sampling grid, it is possible to have compactly supported synthesis functions, which was not the case before. Several variations on the basic theme are considered. Also, an application of the developed techniques to efficient computation of inner products in multiresolution subspaces is presented.

Thesis Files