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Fast multidimensional NMR spectroscopy for sparse spectra

Multidimensional NMR spectroscopy is widely used for studies of molecular and biomolecular structure. A major disadvantage of multidimensional NMR is the long acquisition time which, regardless of sensitivity considerations, may be needed to obtain the final multidimensional frequency domain coefficients. In this article, a method for under‐sampling multidimensional NMR acquisition of sparse spectra is presented. The approach is presented in the case of two‐dimensional NMR acquisitions. It relies on prior knowledge about the support in the two‐dimensional frequency domain to recover an over‐determined system from the under‐determined system induced in the linear acquisition model when under‐sampled acquisitions are performed. This over‐determined system can then be solved with linear least squares. The prior knowledge is obtained efficiently at a low cost from the one‐dimensional NMR acquisition, which is generally acquired as a first step in multidimensional NMR. If this one‐dimensional acquisition is intrinsically sparse, it is possible to reconstruct the corresponding two‐dimensional acquisition from far fewer observations than those imposed by the Nyquist criterion, and subsequently to reduce the acquisition time. Further improvements are obtained by optimizing the sampling procedure for the least‐squares reconstruction using the sequential backward selection algorithm. Theoretical and experimental results are given in the case of a traditional acquisition scheme, which demonstrate reliable and fast reconstructions with acceleration factors in the range 3–6. The proposed method outperforms the CS methods (OMP, L1) in terms of the reconstruction performance, implementation and computation time. The approach can be easily extended to higher dimensions and spectroscopic imaging. Copyright © 2014 John Wiley & Sons, Ltd.