A new approach to compressed sensing for NMR

Abstract

Compressed sensing (CS) has attracted a great deal of recent interest as an approach for spectrum analysis of nonuniformly sampled NMR data. Although theoretical justification for the method is abundant, it suffers from several weaknesses, among them poor convergence of some algorithms, and it remains an open question whether NMR spectra satisfy the sparsity requirements of CS theorems. The versions of CS used in NMR involve minimizing the l1 norm of the spectrum. They bear similarity to maximum entropy (MaxEnt) reconstruction, but critical comparison of the methods can be difficult. Here we describe a formalism that places CS and MaxEnt reconstruction on equal footing, enabling critical comparison of the two methods. We also describe a new algorithm for CS that restricts the computation of the l1 norm to the real channel for complex spectra and ensures causality. Preliminary 1D results demonstrate that this approach ameliorates some artifacts that can occur when using the l1 norm of the complex spectrum.

Keywords: compressed sensing; non-Fourier; signal processing; spectrum analysis.