基于低秩矩阵的非均匀采样NMR波谱重建进展

多维核磁共振（Nuclear Magnetic Resonance，NMR）利用多维波谱来分析分子结构，被广泛用于化学、生物学和医学等领域，但信号采样时间随波谱维度和采样点数增加而迅速增长．非均匀采样通过降低间接维采样点数来加速数据采集，并引入合理的重建方法获得完整的NMR波谱．如何快速重建高质量的波谱，是NMR信号处理研究的前沿．本文主要综述近年来基于低秩矩阵的NMR波谱重建方法的发展．首先介绍了低秩矩阵的相关数学基础；然后从一般低秩矩阵和结构化低秩汉克尔矩阵两个角度来论述重建模型，并讨论相关的NMR波谱应用；最后分析了该技术存在的不足，并展望其未来发展的趋势．

Progresses on Low-Rank Reconstruction for Non-Uniformly Sampled NMR Spectra

Multidimensional nuclear magnetic resonance (NMR) spectroscopy is frequently used to analyze molecular structures, and widely applied in researches in the fields of chemistry, biology and medicine. However, data acquisition time increases rapidly with increasing spectral dimension and number of sampling points. Non-uniformly sampling (NUS) can speed up data acquisition by reducing the amount of sampling data in the indirect dimensions, while obtaining a complete spectrum with proper reconstruction methods. How to achieve faster sampling and better reconstruction of a high-quality spectrum are important problems in multidimensional NMR. This article reviews the recent progresses on the low-rank NMR spectra reconstruction methods. First, the related mathematical basics of low-rank matrices are introduced. Then, the spectra reconstruction models are discussed from two perspectives:general low-rank matrix and structured low-rank Hankel matrix. Finally, the limitations and future trends of these methods are discussed.