A comparison of convex and non-convex compressed sensing applied to multidimensional NMR

The resolution of multidimensional NMR spectra can be severely limited when regular sampling based on the Nyquist-Shannon theorem is used. The theorem binds the sampling rate with a bandwidth of a sampled signal and thus implicitly creates a dependence between the line width and the time of experiment, often making the latter one very long. Recently, Candès et al. (2006) [25] formulated a non-linear sampling theorem that determines the required number of sampling points to be dependent mostly on the number of peaks in a spectrum and only slightly on the number of spectral points. The result was pivotal for rapid development and broad use of signal processing method called compressed sensing. In our previous work, we have introduced compressed sensing to multidimensional NMR and have shown examples of reconstruction of two-dimensional spectra. In the present paper we discuss in detail the accuracy and robustness of two compressed sensing algorithms: convex (iterative soft thresholding) and non-convex (iteratively re-weighted least squares with local ℓ(0)-norm) in application to two- and three-dimensional datasets. We show that the latter method is in many terms more effective, which is in line with recent works on the theory of compressed sensing. We also present the comparison of both approaches with multidimensional decomposition which is one of the established methods for processing of non-linearly sampled data.