From Image Deblurring to Optimal Investments: Maximum Likelihood Solutions for Positive Linear Inverse Problems

Abstract

SUMMARY The problem of recovering an input signal from a blurred output, in an input-output system with linear distortion, is ubiquitous in science and technology. When the blurred output is not degraded by statistical noise the problem is entirely deterministic and amounts to a mathematical inversion of a linear system with positive parameters, subject to positivity constraints on the solution. We show that all such linear inverse problems with positivity restrictions (LININPOS problems for short) can be interpreted as statistical estimation problems from incomplete data based on infinitely large ‘samples’, and that maximum likelihood (ML) estimation and the EM algorithm provide a straightforward method of solution for such problems. This applies to such classical problems as algebraic systems of linear equations, Fredholm's integral equations of the first kind, mixture models, deconvolutions, etc., all with positivity restrictions but with no stochastic components. In connecting the class of LININPOS problems with the corresponding class of ML estimation problems from incomplete data, we unify numerous examples from diverse areas of applications where the EM algorithm has been independently derived over the years, and we present some new opportunities for problems that have previously been unattempted. Examples include signal recovery problems, tomographic reconstructions and optimal investment portfolios, as well as more traditional statistical problems like ML estimation from censored and grouped data, and from incomplete contingency tables. The problem of restoring a ‘true’ image from its linearly distorted version is important in image analysis. Such problems can occur, for example, as a result of lens distortion (e.g. the Hubble space telescope), out-of-focus blur and blur due to relative motion between the subject and the camera during the photographic exposure. These problems often fall in the category of LININPOS problems, in which case the methodology is applicable. We demonstrate it on the problem of deblurring motion.