A Survey of Inverse Methodologies, Meta-Evolutionary Optimization and Bayesian Statistics: Applications to in vivo MRS

Inverse problems arise from ill-posed mappings in many areas of science and engineering. For every inverse problem, there exist some fundamental statistical limits for the accuracy of solutions. These theoretical limits cannot be superseded in any way without incorporating some additional prior information. A major "secret" of inversion methodologies consists in identifying and applying all forms of prior information available, both hidden and explicit. Meta-Evolutionary Optimization and Bayesian Statistics are central concepts behind a very general inversion methodology recently developed. In this article, some underlying ideas of this robust inversion methodology are reviewed. As illustrated, uncertainty of the inversions may be reduced by incorporating very general forms of prior information: non-linear parametric models, probabilistic noise distributions, deterministic constraint conditions, etc. This inversion methodology is applied to an important problem from Magnetic Resonance Spectroscopy (MRS): quantitative in vivo MRS beyond inhomogeneity and noise limits. After analyzing the principal role of prior information for solving inverse problems, some limitations of two popular approaches (Sampled Pattern Matching, Artificial Neural Networks) are elucidated as well.