Retaining both discrete and smooth features in 1D and 2D NMR relaxation and diffusion experiments

A new method of regularization of 1D and 2D NMR relaxation and diffusion experiments is proposed and a robust algorithm for its implementation is introduced. The new form of regularization, termed the Modified Total Generalized Variation (MTGV) regularization, offers a compromise between distinguishing discrete and smooth features in the reconstructed distributions. The method is compared to the conventional method of Tikhonov regularization and the recently proposed method of L₁ regularization, when applied to simulated data of 1D spin-lattice relaxation, T₁, 1D spin-spin relaxation, T₂, and 2D T₁-T₂ NMR experiments. A range of simulated distributions composed of two lognormally distributed peaks were studied. The distributions differed with regard to the variance of the peaks, which were designed to investigate a range of distributions containing only discrete, only smooth or both features in the same distribution. Three different signal-to-noise ratios were studied: 2000, 200 and 20. A new metric is proposed to compare the distributions reconstructed from the different regularization methods with the true distributions. The metric is designed to penalise reconstructed distributions which show artefact peaks. Based on this metric, MTGV regularization performs better than Tikhonov and L₁ regularization in all cases except when the distribution is known to only comprise of discrete peaks, in which case L₁ regularization is slightly more accurate than MTGV regularization.