Fast ℓ1 minimization by iterative thresholding for multidimensional NMR spectroscopy

Abstract

Fast multidimensional NMR is important in chemical shift assignment and for studyingstructures of large proteins. We present the first method which takes advantage of thesparsity of the wavelet representation of the NMR spectra and reconstructs the spectra frompartial random measurements of its free induction decay (FID) by solving the followingoptimization problem:min ||x|| 1 subject to || y-SFT WT x ||2 ≤ε,where y is a given n×1 observation vector, S a random sampling operator, F denotes theFourier transform, and W an orthogonal 2D wavelet transform. The matrix A=SFT WT isa given n×p matrix such that n