An Automatic Denoising Method for NMR Spectroscopy Based on Low-Rank Hankel Model

Nuclear magnetic resonance (NMR) spectroscopy, whose time domain data is modeled as the sum of damped exponential signals, has become an indispensable tool in various scenarios, such as biomedicine, biology, and chemistry. NMR spectroscopy signals, however, are usually corrupted by Gaussian noise in practice, raising difficulties in sequential analysis and quantification. The low-rank Hankel property of exponential signals plays an important role in the denoising issue, but selecting an appropriate parameter still remains a problem. In this work, we explore the effect of the regularization parameter of a convex optimization denoising method based on low-rank Hankel matrices for exponential signals corrupted by Gaussian noise. An accurate estimate on the spectral norm of weighted Hankel matrices is provided as a guidance to set the regularization parameter. The bound can be efficiently calculated since it only depends on the standard deviation of the noise and a constant. Aided by the bound, one can easily obtain an auto-setting regularization parameter to produce promising denoised results. Our results on synthetic and realistic NMR spectroscopy data demonstrate a superior denoising performance of the proposed approach over typical Cadzow and the state-of-the-art QR decomposition methods, especially in the low signal-to-noise ratio regime.