Low‐rank and sparse reconstruction for fast diffusion nuclear magnetic resonance spectroscopy

Nuclear magnetic resonance with diffusion‐ordered spectroscopy (DOSY) serves as an important analytical tool to non‐destructively separate a molecule from a compound in medicine and chemistry. However, the data acquisition time increases rapidly for multidimensional DOSY. To enable fast DOSY, partial data are acquired with non‐uniform sampling, and the spectrum can be reconstructed with a proper constraint, such as sparsity in the state‐of‐the‐art method. However, the reconstructed spectrum is observed to have isolated artefacts, which can be easily recognised as fake peaks and affect the estimated diffusion coefficients severely. The authors introduce the low‐rank constraint as an effective remedy to remove these artefacts and derive a fast algorithm to solve the reconstruction problem. Results on both synthetic and realistic DOSY spectra show that a better spectrum and more accurate diffusion coefficients can be achieved.