Sub-sampling of NMR Correlation and Exchange Experiments

TL;DR

This work tackles the challenge of reducing acquisition time in $T_1$-$D$ NMR correlation/exchange experiments through sub-sampling and a comprehensive evaluation of inversion strategies. It compares Tikhonov regularization, MTGV, deep learning, and a Tikhonov–DL cascade across sampling patterns (random, truncation, and hybrid) and noise regimes, using both chi-score and artefact-aware phi-score metrics. Key findings show that deep learning excels with full data but regularization methods regain superiority under significant sub-sampling, with MTGV often outperforming Tikhonov; random sampling yields the strongest overall performance. Importantly, the choice of cost function drastically shifts rankings, underscoring the need to align evaluation metrics with experimental goals and highlighting dynamic range as a critical factor in inversion success. The results provide practical guidance for designing sub-sampling schemes in NMR and suggest extensions to other NMR correlation/exchange experiments due to the translational invariance of the inversion framework.

Abstract

Sub-sampling is applied to simulated $T_1$-$D$ NMR signals and its influence on inversion performance is evaluated. For this different levels of sub-sampling were employed ranging from the fully sampled signal down to only less than two percent of the original data points. This was combined with multiple sample schemes including fully random sampling, truncation and a combination of both. To compare the performance of different inversion algorithms, the so-generated sub-sampled signals were inverted using Tikhonov regularization, modified total generalized variation (MTGV) regularization, deep learning and a combination of deep learning and Tikhonov regularization. Further, the influence of the chosen cost function on the relative inversion performance was investigated. Overall, it could be shown that for a vast majority of instances, deep learning clearly outperforms regularization based inversion methods, if the signal is fully or close to fully sampled. However, in the case of significantly sub-sampled signals regularization yields better inversion performance than its deep learning counterpart with MTGV clearly prevailing over Tikhonov. Additionally, fully random sampling could be identified as the best overall sampling scheme independent of the inversion method. Finally, it could also be shown that the choice of cost function does vastly influence the relative rankings of the tested inversion algorithms highlighting the importance of choosing the cost function accordingly to experimental intentions.

Figures (6)

• Figure 1: $T_1$-$D$ distributions employed in this publication.
• Figure 2: $\chi$-score as defined through equation \ref{['eq:score_us']} of the reconstructions obtained from artificial signals generated by the sparse (left) and smooth (right) $T_1$-$D$ distribution (figure \ref{['fig:dis_real_sparse']} and \ref{['fig:dis_real_smooth']}). The signal-to-noise ratio of the signal employed for inversion was 2000.
• Figure 3: $\chi$-score as defined through equation \ref{['eq:score_us']} of the reconstructions obtained from artificial signals generated by the version of the $T_1$-$D$ distribution which contains smooth and sparse components (figure \ref{['fig:dis_real_sm_sp']}). The signal-to-noise ratio of the signal employed for inversion was 2000.
• Figure 4: Reconstructed distributions (excluding the result from MTGV) obtained from artificial signals generated by the version of the $T_1$-$D$ distribution which only contains smooth components (figure \ref{['fig:dis_real_smooth']}). The signal-to-noise ratio of the signal employed for inversion was 2000. For the reason of a simpler comparison, the real distribution was re-scaled and the updated plot is shown in figure \ref{['fig:dis_real_smooth_copy']}.
• Figure 5: $\phi$-score as defined through equation \ref{['eq:score_us_art']} of the reconstructions obtained from artificial signals generated by the sparse (left) and smooth (right) $T_1$-$D$ distribution (figure \ref{['fig:dis_real_sparse']} and \ref{['fig:dis_real_smooth']}). The signal-to-noise ratio of the signal employed for inversion was 2000.