From image deblurring to optimal investments : maximum likelihood solutions for positive linear inverse problems

Abstract

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