Discrete Representation of Signals from Infinite Dimensional Hilbert Spaces with Applications to Noise Suppression and Compression

Abstract

Addressed in this thesis is the issue of representing signals from infinite dimensional Hilbert spaces in a discrete form. The discrete representations which are studied come from the irregular samples of a signal dependent transform called the group representation transform, e.g., the wavelet and Gabor transform. The main issues dealt with are (i) the recoverability of a signal from its discrete representation, (ii) the suppression of noise in a corrupted signal, and (iii) compression through efficient discrete representation.<P>The starting point of the analysis lies with the intimate connection between the Duffin-Schaeffer theory of (global) frames and irregular sampling theory. This connection has lead elsewhere to the formulation of iterative schemes for the reconstruction of a signal from its irregular samples. However, these schemes have not addressed such issues as digital implementability and reconstruction from perturbed representations. Here, iterative reconstruction algorithms are developed and implemented which recover a signal from its possibly perturbed discrete representation.<P>Robustness to perturbations occurring directly in the signal domain are also investigated. Based on the notion of coherence with respect to a frame, a simple non-linear thresholding scheme is developed for the rejection of noise.<P>The structure of the discretization has many free parameters including the choice of group representation transform, the analyzing function associated with the group representation transform, and the sampling set. Each choice of parameters leads to a different discrete representation and the specification of an underlying set of primitive functions. Reconstructability is directly related to the frame properties of this set of primitive functions.<P>Localized discrete representations around a particular signal are also investigated. Truncations and other signal dependent localization of global representations lead to finite representations. The approach to finite representations which is taken here can be stated in terms of local frames for the reproducing kernel Hilbert Space formed by the range of the group representation transform.<P>Finally, numerical examples of discrete representations which are signal independent and new signal dependent discrete (positive extreme) wavelet representations are presented. Reconstruction, noise suppression, and compression experiments are conducted and demonstrated on numerical examples including speech and synthetic signals.