Convolution and deconvolution with Gaussian kernel

Summary

In this paper we prove the following theorem. Suppose an imageI and an object Ω are related by the convolution equationI=Ω*δ _λF, whereδ _λF is a Gaussian kernel with widthλ _F. Suppose further that the imageI is expanded in a series of Gaussian derivatives asI=Σa _n ▽n δ _Λ , whereδ _Λ is a Gaussian with widthΛ>λ _F, and where ∇represents then-th derivative ofδ _Λ . Then the object Ω is given by Ω = Σa _n ∇ⁿ δ _λ , whereλ ²-Λ ²-λ ²_F , and where the coefficientsa _n are exactly the coefficients obtained in the expansion of the imageI. The expansion in Gaussian derivatives can therefore be used to develop a simple and efficient deconvolution method for images which have been convolved with a Gaussian filter. We consider both one- and two-dimensional problems, and give a discussion of the error caused by truncation of the expansion of the image. We also give a two-dimensional numerical example which shows how our deconvolution method can be used in the restoration of digitized gray-scale images.