A fast Hankel matrix nonconvex factorization reconstruction method with inertia momentum for non-uniformly sampled NMR spectroscopy

Abstract

Multidimensional nuclear magnetic resonance (NMR) spectroscopy is one of the most powerful tools for qualitative or quantitative analysis of the composition and structure of various organic and inorganic substances. However, the time required to acquire NMR signals increases exponentially with dimensionality. Therefore, non-uniform sampling is commonly adopted to accelerate data acquisition and then the complete spectrum can be obtained by reconstruction method. At present, the state-of-the-art reconstruction methods are based on the idea of low-rank Hankel matrix completion and solved by different singular value thresholding methods. However, the computation of singular value decomposition (SVD) is very time-consuming, especially for high-resolution spectra. In this paper, we proposed a Hankel matrix nonconvex factorization optimization model to avoid SVD, thus greatly reducing the computational time. We developed a numerical algorithm based on the linearized Alternating direction method of multipliers to solve the proposed optimization problem, and Nesterov momentum is adopted to accelerate the convergence of the algorithm. However, for nonconvex model, the high quality reconstruction can only be obtained when the number of the spectral peaks is known. Thus we designed an adaptive strategy to alleviate this problem. Experiments are performed to demonstrate that the proposed algorithm can obtain higher quality reconstruction in less computational time than low rank Hankel matrix factorization and fast iterative hard thresholding methods.