This thesis considers the application of wavelet methods to the analysis of time series and spatial data. In the rst part, we propose a locally stationary model of the covariance structure for data which lie on a regular grid. This is achieved by moving from a (global) Fourier decomposition of structure to a localised decomposition involving a set of discrete, non-decimated wavelets. The proposed model is subsequently applied to various texture analysis problems. These range from the classication of images taken from the standard Brodatz texture collection to subtle discrimination problems encountered by an industrial collaborator.
The second part proposes an efficient construction of the inner product matrix of discrete autocorrelation wavelets - a quantity which is of crucial importance in the unbiased estimation of local wavelet spectra. The proposed scheme relates neighbouring elements of the matrix which lie on a given diagonal using a two-scale relationship of the autocorrelation wavelets. This results in a construction which is considerably more efficient than the brute force approach used to date
Finally, we conclude by detailing the results of initial research on the estimation of the local autocovariance structure of locally stationary time series. These results provide an interesting interpretation of this quantity in terms of familiar (stationary)time series measures.