| We start by providing examples showing that although it is plausible that signals are characterized by their (infinite-scale) wavelet transform extrema, as conjectured by Mallat and Zhong, this is not the case even in a practical setting. Sampling wavelet transforms at extrema leads us to consider reconstruction from irregular sampling of wavelet transforms via alternating projections in Hilbert space. Here we develop some evidence indicating the importance of sampling at large values. We develop a very flexible finite-scale framework for which the reconstruction process will converge. From here, we then rigorously develop a discrete version of the algorithm and, when the wavelet generates an orthonormal multiresolution analysis, we gain a near complete understanding of the behavior of the algorithm for regular samplings. To finish, we consider band-limited wavelets, and show that under certain irregular sampling schemes we have unique reconstruction with a computable geometric rate of convergence. We improve on the original results for band-limited wavelets, gaining enough flexibility in our choice of samplings to develop a discrete version. Finally, we give an interesting application of wavelet theory that is not related to the main body of this dissertation. |
