Deblurring Gaussian blur using a wavelet array transform

Abstract

Deblurring the Gaussian blur, which is a fundamental problem of signal analysis, has defined a satisfactory solution. The principal reason for the difficulty is the inherently ill-conditioned blur-matrix which poses a challenge to its stable inversion. Most of the literature is concerned with a solution to the problem in restricted domains, and this solution is, in many cases, characterised by inversions that are not stable, We propose a multiscale inversion method based on wavelet arrays which is applicable to a wide class of images, and show that the inversion is stable with respect to noise both in the blurred signal and in the blur variance. We include, for illustration, the result of such a deblurring scheme as applied to a natural image.