13 - Probabilistic Methods

Abstract

The probabilistic approach to the solution of image reconstructions incorporates probability distributions into the solution of the inverse problem. A statistical estimate of the solution of an inverse problem requires a priori knowledge of at least the probability distribution of the solution. The probability distribution of the measurements can be taken as a scaling factor in estimating the conditional probability of the inverse mapping from the Bayes' hypothesis and need not be explicitly known. A solution can then be estimated in a number of ways starting with the simplest approach of equiprobable distributions. Bayesian-Minimum Information is a successive approximation process that does not rely on matrix inversion. It provides a mild adjustment process at each iteration and does not amplify noise considerably. The Bayesian approach amounts also to maximizing the logarithm of the Poisson-based likelihood. The negative of the logarithm of the normal distribution also lends itself readily to minimization, unlike the Poisson distribution which is a more complicated expression. The maximization of the a posteriori provides an expected value for the solution, while maximizing the likelihood provides the most likely solution. The Monte Carlo method relies on the Markov chain of a series of randomly sampled scenarios, governed by appropriate probability distributions.