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Reaching the sparse-sampling limit for reconstructing a single peak in a 2D NMR spectrum using iterated maps

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Many of the ubiquitous experiments of biomolecular NMR, including \(R_2\), \(R_{1\rho }\), and CEST, involve acquiring repeated 2D spectra under slightly different conditions. Such experiments are amenable to acceleration using non-uniform sampling spectral reconstruction methods that take advantage of prior information. We previously developed one such technique, an iterated maps method (DiffMap) that we successfully applied to 2D NMR spectra, including \(R_2\) relaxation dispersion data. In that prior work, we took a top-down approach to reconstructing the 2D spectrum with a minimal number of sparse samples, reaching an undersampling fraction that appeared to leave some room for improvement. In this study, we develop an in-depth understanding of the action of the DiffMap algorithm, identifying the factors that cause reconstruction errors for different undersampling fractions. This improved understanding allows us to formulate a bottom-up approach to finding the lowest number of sparse samples required to accurately reconstruct individual spectral features with DiffMap. We also discuss the difficulty of extending this method to reconstructing many peaks at once, and suggest a way forward.

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  1. Note that we are not calculating this SNR by comparing signal amplitude of a peak to noise amplitude far from all peaks after reconstruction, because this is known to be misleading (see e.g. Hyberts et al. (2017)). Instead, we are considering the ratio of the reconstructed amplitude to the expected variation in that amplitude, which we estimate using the \(E_\text {SZ}\) model explained above.


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We thank G. Manley for acquiring the IGPS data set. We thank D. Cui, G. Manley, and S. Elrington for helpful discussions. J.P. Loria acknowledges the support of the NSF through Grant No. MCB-1615415, and the NIH through Grant No. GM112781. R. Blum, J. Rovny, and S. Barrett acknowledge the support of the NSF through Grant No. DMR-1610313 and Grant No. DMR-1310274. R. Blum is an NSF fellow and this material is based upon work supported by the NSF GRFP under Grant No. DGE-1122492.

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Correspondence to Sean E. Barrett.

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Blum, R.L., Rovny, J., Loria, J.P. et al. Reaching the sparse-sampling limit for reconstructing a single peak in a 2D NMR spectrum using iterated maps. J Biomol NMR 73, 545–560 (2019).

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