Image Deconvolution

Abstract

Image deconvolution is a basic problem in the processing of microscopic images. It is ill-posed, as the vast majority of inverse problems, and therefore it requires an accurate modeling taking into account all the known properties of the process of image formation and acquisition. In this chapter, after a brief discussion of the ill-posedness of image deconvolution in a continuous setting, we develop a detailed statistical model which applies to the case of fluorescence microscopy. Two approximate models, denoted as the Gaussian case and the Poisson case, are also introduced. The statistical model is the starting point for a maximum likelihood approach. In the Gaussian case one re-obtains the standard least squares problem which is also ill-posed. The constraint of non-negativity is introduced and two iterative algorithms converging to nonnegative least-squares solutions are presented. The need of early stopping of the iterations to produce sensible results is discussed. Moreover, an iterative method converging to the maximum-likelihood estimates of the Poisson case is presented: it is the classical RL (or EM) method which also requires early stopping of the iterations. Finally Bayesian methods, based on the use of a priori statistical information on the object to be restored, are introduced and their relationship with the standard regularization theory of inverse problems is discussed.