A fast low rank Vandermonde factorization reconstruction method for non-uniformly sampled 2D NMR spectroscopy

Abstract

As an indispensable analytical method in the fields of chemistry and medicine, nuclear magnetic resonance (NMR) spectroscopy is widely adopted for analyzing the composition and structure of substances. However, the data acquisition period for high-resolution NMR spectra is extremely long. Therefore, non-uniform sampling is commonly adopted to reduce the measurement time and the complete NMR signal can be obtained by appropriate reconstruction algorithms. Recently, it was found that NMR signal can be reconstructed well by low-rank Hankel matrix completion. However, the computational time required for this method to reconstruct two-dimensional (2D) and higher dimensional signals is relatively long because it increases rapidly as spectral resolution improves. In this paper, we propose a novel low rank minimization method to alleviate this problem by utilizing the unexploited property that a 2D NMR signal can be decomposed into two Vandermonde matrices. However, minimizing the rank necessitates the time-consuming singular value decomposition (SVD). Thus we introduce a matrix bi-factorization idea into the method to reduce the time complexity and greatly mitigate the time required by the SVD. Experiments on synthetic data and realistic NMR spectroscopy data demonstrate that compared with the state-of-the-art nuclear norm based minimization and fast iterative hard thresholding algorithms, the computational time of the proposed algorithm is significantly reduced, and high reconstruction accuracy can be ensured at the same time.