New overcomplete wavelet transforms and wavelet based deconvolution

The dyadic DWT provides an octave-band frequency decomposition and is very effective for processing piecewise smooth signals. But in fine (high-frequency) scales, the frequency resolution is quite low -- an undesired feature for signals which are not regarded as 'smooth' in general (e.g. audio, texture). The discrete wavelet packet transform (DWPT) avoids this problem by iterating on the highpass filters as well. However, due to critical sampling, both DWT and DWPT are highly shift-varying. In addition, they do not provide directional analysis/synthesis functions when they are extended to 2D (or higher dimensions) via tensor products. We extend the the dual-tree complex wavelet transform (DT- C WT), introduced by Kingsbury, to a dual-tree complex wavelet packet transform (DT- C WPT). The DT- C WPT we propose retains the desired properties of the DT- C WT (like near shift-invariance, directional analysis/synthesis for multidimensional signals) while offering an improved frequency resolution, which can be tailored to the particular signal family at hand.;Another transform studied in this thesis involves wavelet frames with a rational dilation factor. Given an admissable wavelet psi(t), the dyadic wavelet basis consists of dyadic dilates and translates of psi( t), given by 2j/2y 2jt-n j,n∈Z . However the basis functions at different scales are quite dissimilar (compare psi(t) and 2 psi(2t) for example). This abrupt change between neighboring scales might be a problem for certain signals. There are two issues. First, the dilation factor (which is 2 for the dyadic DWT) should be decreased, so as to make the change between each scale more gradual. Second, the frequency resolution, which is influenced by the dilation factor, due to the completeness requirement of the basis (or redundancy considerations of the frame) should be improved. In order to address these, we propose the rational DWT, which consists of iterated FBs with rational sampling factors. The filter design problems encountered in such a setting are inherently different from those for integer dilation DWTs, and the techniques for integer dilation DWT design are not applicable. Several schemes to overcome these problems are proposed in this thesis.;A third topic considered is on iterated non-perfect reconstruction (non-PR) FBs. In a nutshell, we investigate the frame bounds of iterated non-perfect reconstruction filter banks and provide frame bounds valid for iterated FBs with an arbitrary number of stages using the frame bounds of the underlying frame on the real line. Conversely, given the frame bounds of the iterated FB, we derive bounds for the underlying wavelet frame.;Lastly, we consider a modification to the 'thresholded Landweber' algorithm, which has drawn attention recently for the solution of wavelet regularized inverse problems. The modification we discuss accelerates the convergence of the algorithm, by taking advantage of the different behavior of the blurring operator in different subbands.