CONVOLUTION AND DECONVOLUTION WITH GAUSSIAN KERNEL

In this paper we prove the following theorem. Suppose an image I and a n object OMEGA are related by the convolution equation I = OMEGADELTA (lambdaF), where delta(lambdaF) Gaussian kernel with width lambda(F). Suppose further that the image I is expanded in a series of Gaussian d erivatives as I = SIGMA A(n) del(n) delta(LAMBDA), where delta(LAMBDA) is a Gaussian with width LAMBDA > lambda(F), and where del(n) represe nts the n-th derivative of delta(LAMBDA). Then the object OMEGA is giv en by OMEGA = SIGMA a(n) del(n) delta(lambda), where lambda2 = LAMBDA2 - lambda 2/F, and where the coefficients a(n) are exactly the coeffic ients obtained in the expansion of the image 1. The expansion in Gauss ian derivatives can therefore be used to develop a simple and efficien t deconvolution method for images which have been convolved with a Gau ssian filter. We consider both one- and two-dimensional problems, and give a discussion of the error caused by truncation of the expansion o f the image. We also give a two-dimensional numerical example which sh ows how our deconvolution method can be used in the restoration of dig itized gray-scale images.