A partial sum of singular-value-based reconstruction method for non-uniformly sampled NMR spectroscopy

The nuclear magnetic resonance (NMR) spectroscopy has fruitful applications in chemistry, biology and life sciences, but suffers from long acquisition time. Non-uniform sampling is a typical fast NMR method by undersampling the time-domain data of the spectrum but need to restore the fully sampled data with proper constraints. The state-of-the-art method is to model the time-domain data as the sum of exponential functions and reconstruct these data by enforcing the low rankness of Hankel matrix. However, this method is solved by minimizing the sum of singular values of the Hankel matrix, which leads to the distortion of low-intensity spectral peaks. Here, a low rank Hankel matrix reconstruction approach with a partial sum of singular values is proposed to protect small singular values, which can faithfully reconstruct all peaks. Results on both synthetic and realistic NMR spectroscopy show that the proposed method can reconstruct a more consistent spectrum to the fully sampled one than other state-of-the-art methods and have particular advantages on preserving low-intensity peaks.