Laplace Inversion for NMR Data Analysis

Spin relaxation and diffusion are extensively used for the study of many materials including food products. Laplace inversion is often the method of choice to analyze such data to obtain a (e.g. T₂) spectrum in order to study and complexity, heterogeneity and their changes. The mathematical theory for Laplace inversion is mature, however, its application remains non-trivial due to the ill-conditioned nature of the problem. This article presents intuitively the concept of the mathematics of Laplace inversion in comparison with Fourier analysis, and discusses the idea of uncertainty from Monte Carlo point of view. The key conclusion is that the full solution to the Laplace inversion should be an ensemble of the spectra, instead of a single spectrum. Such ensemble can be used to derive any quantities defined by the spectrum and also the uncertainty for such quantity. It also presents a practical method to analyze and optimize the experimental protocol in order to achieve the necessary spectral resolution.