Non-uniformly sampled 2D NMR Spectroscopy reconstruction based on Low Rank Hankel Matrix Fast Tri-Factorization and Non-convex Factorization

Nuclear magnetic resonance (NMR) spectroscopy is widely used in chemistry and medicine to research the composition of matter and the spatial structure of proteins. However, NMR signals with larger sizes and higher dimensions are slower to acquire. Therefore, Non-Uniform Sampling (NUS) methods are used to speed up the acquisition of NMR signals, and some mathematical algorithms are used to recover the original NMR signals from the NUS data. In recent years, low-rank Hankel matrix completion is proved to recover NMR data with less error. However, it requires singular value decomposition (SVD) with high computational complexity in each iteration, which is not suitable for reconstructing signals with large sizes. In this paper, we propose a Fast Tri-Factorization (FTF) method to decompose the low-rank Hankel matrix into three small-scale matrices, and it convert the original problem into a matrix nuclear norm minimization problem on the small-scale matrix to mitigate high computation cost of multiple SVDs. Furthermore, the factor matrix size remains constant for different size of the Hankel matrices. We further replace the low-rank constraint with Non-convex Factorization (NF) to avoid the time-consuming SVD in each iteration. Numerical experiments show that the proposed method can significantly speed up the NMR reconstruction compared with several existing algorithms while ensuring the reconstruction accuracy.