| Orthogonal wavelet transforms have been applied to the field of signal and image processing with promising results in compression and denoising. Coefficients of such a transform constitute a complete representation of a signal without redundancy. However, there are applications where complete representations are disadvantageous. In this thesis, we examine classes of Fourier transforms and wavelet transforms in terms of their efficacy of representing convolution operators. We have identified two shortcomings associated with complete representations of the discrete-time domain: (1) the lack of translation invariance and the (2) a possible anomaly of aliasing-enhancement.; On the other hand, our analysis showed that overcomplete wavelet representations do not bear those shortcomings of their non-redundant counterparts. Our framework of overcomplete wavelet representations include construction algorithms and prototype filters, spatial-frequency interpretation and three operations. Capabilities of spatial-frequency localization were quantitatively evaluated using uncertainty factors. Associated with gain, shrinking and envelope operators, algorithms for convolution, denoising and analysis of power density distribution were presented and analyzed.; The framework of overcomplete wavelet representations was then applied to segmentation of textured images and image deblurring. We demonstrated that envelopes as feature vectors performed well in segmenting both natural and synthetic textures. We showed that gain and shrinking operators may be used for image deblurring and discuss limitations of the methodology. |
