Applications of orthonormal bases of wavelets to deconvolution

Abstract

Convolution integral equations arise frequently in many areas of science and engineering. If the kernel of such an equation is well behaved, say integrable, then the task of solving a convolution equation is ill-posed. Indeed, if the kernel is integrable, then the Riemann-Lebesgue Lemma implies that the recovery of high frequency information pertaining to the unknown function will be difficult, if not impossible.

Orthonormal wavelet bases are bases generated by translating and dilating a single function, known as the mother wavelet. One key advantage of these bases is that the mother wavelet can be selected to have fast decay in both the time and frequency domains. This property suggests that wavelet bases may be useful when attempting to solve a convolution equation.

In this thesis, we investigate the applicability of orthonormal wavelet bases with regard to solving convolution equations. In particular, we concentrate on the construction of approximations to the unknown function belonging to scaling function subspaces. We also briefly consider regularization algorithms which are based on the multiresolution analysis, a structure defined by the scaling function association with the mother wavelet.