Pitfalls in compressed sensing reconstruction and how to avoid them

Abstract

Multidimensional NMR can provide unmatched spectral resolution, which is crucial when dealing with samples of biological macromolecules. The resolution, however, comes at the high price of long experimental time. Non-uniform sampling (NUS) of the evolution time domain allows to suppress this limitation by sampling only a small fraction of the data, but requires sophisticated algorithms to reconstruct omitted data points. A significant group of such algorithms known as compressed sensing (CS) is based on the assumption of sparsity of a reconstructed spectrum. Several papers on the application of CS in multidimensional NMR have been published in the last years, and the developed methods have been implemented in most spectral processing software. However, the publications rarely show the cases when NUS reconstruction does not work perfectly or explain how to solve the problem. On the other hand, every-day users of NUS develop their rules-of-thumb, which help to set up the processing in an optimal way, but often without a deeper insight. In this paper, we discuss several sources of problems faced in CS reconstructions: low sampling level, missassumption of spectral sparsity, wrong stopping criterion and attempts to extrapolate the signal too much. As an appendix, we provide MATLAB codes of several CS algorithms used in NMR. We hope that this work will explain the mechanism of NUS reconstructions and help readers to set up acquisition and processing parameters. Also, we believe that it might be helpful for algorithm developers.Electronic supplementary materialThe online version of this article (doi:10.1007/s10858-016-0068-3) contains supplementary material, which is available to authorized users.

TReNDS—Software for reaction monitoring with time‐resolved non‐uniform sampling

“…Thus, it is important to keep the subset size as small as possible. On the other hand, the mathematical conditions of CS reconstruction() require a minimum number of sampling points to make the reconstruction credible. The intrinsic signal to thermal noise ratio (iSNR discussed in Palmer et al()) also plays a role.…”

Compressed NMR: Combining compressive sampling and pure shift NMR techniques

“…Equations and and the considerations of SNR discussed above, caution against reports that advocate the use of a particular m / ni ratio for a particular type of experiment, as already pointed out by Kazimierczuk; the m / ni ratio depends on the sparsity of the spectra, as well as on the sampling and reconstruction methods used, provided that the SNR is sufficiently high to ensure that either Equation or is the limiting factor. It is true, however, that for a given sample, experiments that produce sparser spectra (lower S ) require a lower m / ni ratio than experiments that produce denser spectra (higher S ).…”

Compressed sensing: Reconstruction of non‐uniformly sampled multidimensional NMR data

“…1 Methods which can use data with nondeterministic sampling are the most general and can typically reconstruct data from any of the categories. [17][18][19][20][21][22][23][24][25][26] In this article, we review FT sampling theory and explain the modifications used for NUS sampling. 5 Recently, compressed sensing (CS)-based reconstructions have joined the family of nondeterministic techniques which reconstruct a full frequency-domain spectrum.…”