Review and prospect: NMR spectroscopy denoising and reconstruction with low-rank Hankel matrices and tensors

Convex Dual Theory Analysis of Two-Layer Convolutional Neural Networks With Soft-Thresholding

Soft-thresholding has been widely used in neural networks. Its basic network structure is a two-layer convolution neural network with soft-thresholding. Due to the network's nature of nonlinear and nonconvex, the training process heavily depends on an appropriate initialization of network parameters, resulting in the difficulty of obtaining a globally optimal solution. To address this issue, a convex dual network is designed here. We theoretically analyze the network convexity and prove that the strong duality holds. Extensive results on both simulation and real-world datasets show that strong duality holds, the dual network does not depend on initialization and optimizer, and enables faster convergence than the state-of-the-art two-layer network. This work provides a new way to convexify soft-thresholding neural networks. Furthermore, the convex dual network model of a deep soft-thresholding network with a parallel structure is deduced.

Stationary wavelet denoising of solid-state NMR spectra using multiple similar measurements

Accumulating several scans of free induction decays is always needed to improve the signal-to-noise ratio of NMR spectra, especially for the low gyromagnetic ratio solid-state NMR. In this study, we present a new denoising approach based on the correlations between multiple similar NMR spectra. Contrary to the simple averaging of multiple scans or denoising the final averaged spectrum, we propose a Wavelet-based Denoising technique for Multiple Similar scans(WDMS). Firstly, the stationary wavelet transform is applied to decompose every spectrum into approximation coefficients and detail coefficients. Then, the detail coefficients are multiplied by weights calculated based on Pearson's correlation coefficient and structural similarity index between approximation coefficients of different spectra. Finally, the average of these detailed components is used to denoise the spectra. The proposed method is carried on the assumption that noise between multiple spectra is uncorrelated while peak signal information is similar between different spectra, thus preserving the possibility of applying further processing to the data. As a demonstration, the standard wavelet denoise is applied to the WDMS-processed spectra, achieving a further increase in the S/N ratio. We confirm the reliability of the denoising approach based on multiple scans on 1D/2D solid-state MAS/static NMR spectra. In addition, we also show that this method can be used to deal with a single Car-Purcell-Meiboom-Gill (CPMG) echo train.

A Sparse Model-Inspired Deep Thresholding Network for Exponential Signal Reconstruction-Application in Fast Biological Spectroscopy

The nonuniform sampling (NUS) is a powerful approach to enable fast acquisition but requires sophisticated reconstruction algorithms. Faithful reconstruction from partially sampled exponentials is highly expected in general signal processing and many applications. Deep learning (DL) has shown astonishing potential in this field, but many existing problems, such as lack of robustness and explainability, greatly limit its applications. In this work, by combining the merits of the sparse model-based optimization method and data-driven DL, we propose a DL architecture for spectra reconstruction from undersampled data, called MoDern. It follows the iterative reconstruction in solving a sparse model to build the neural network, and we elaborately design a learnable soft-thresholding to adaptively eliminate the spectrum artifacts introduced by undersampling. Extensive results on both synthetic and biological data show that MoDern enables more robust, high-fidelity, and ultrafast reconstruction than the state-of-the-art methods. Remarkably, MoDern has a small number of network parameters and is trained on solely synthetic data while generalizing well to biological data in various scenarios. Furthermore, we extend it to an open-access and easy-to-use cloud computing platform (XCloud-MoDern), contributing a promising strategy for further development of biological applications.

GRIN-toolbox: A versatile and light toolbox for NMR inversion

NMR technique serves as a powerful analytical tool with diverse applications in fields such as chemistry, biology, and material science. However, the effectiveness of NMR heavily relies on data post-processing which is often modeled as regularized inverse problem. Recently, we proposed the Generally Regularized INversion (GRIN) algorithm and demonstrated its effectiveness in NMR data processing. GRIN has been integrated as a friendly graphic user interface-based toolbox which was not detailed in the original paper. In this paper, to make GRIN more practically accessible to NMR practitioners, we focus on introducing the usage of GRIN-Toolbox with processing examples and the corresponding processing graphic interfaces, and the user manual is attached as Supplementary Material. GRIN-Toolbox is versatile and lightweight, where various kinds of data processing tasks can be completed with one click, including but not limited to diffusion-ordered spectroscopy processing, magnetic resonance imaging under-sampling reconstruction, Laplace (diffusion or relaxation) NMR inversion, spectrum denoising, etc. In addition, GRIN-Toolbox could be extended to more applications with user-designed inversion models and freely available at https://github.com/EricLin1993/GRIN.

Exponential Signal Reconstruction With Deep Hankel Matrix Factorization

Exponential function is a basic form of temporal signals, and how to fast acquire this signal is one of the fundamental problems and frontiers in signal processing. To achieve this goal, partial data may be acquired but result in severe artifacts in its spectrum, which is the Fourier transform of exponentials. Thus, reliable spectrum reconstruction is highly expected in the fast data acquisition in many applications, such as chemistry, biology, and medical imaging. In this work, we propose a deep learning method whose neural network structure is designed by imitating the iterative process in the model-based state-of-the-art exponentials' reconstruction method with the low-rank Hankel matrix factorization. With the experiments on synthetic data and realistic biological magnetic resonance signals, we demonstrate that the new method yields much lower reconstruction errors and preserves the low-intensity signals much better than compared methods.

Resolution enhancement of NMR by decoupling with the low-rank Hankel model

Nuclear magnetic resonance (NMR) spectroscopy has become a formidable tool for biochemistry and medicine. Although J-coupling carries essential structural information it may also limit the spectral resolution. Homonuclear decoupling remains a challenging problem. In this work, we introduce a new approach that uses a specific coupling value as prior knowledge, and the Hankel property of the exponential NMR signal to achieve broadband heteronuclear decoupling using the low-rank method. Our results on synthetic and realistic HMQC spectra demonstrate that the proposed method not only effectively enhances resolution by decoupling, but also maintains sensitivity and suppresses spectral artefacts. The approach can be combined with non-uniform sampling, which means that the resolution can be further improved without any extra acquisition time.

Fast NMR spectroscopy reconstruction with a sliding window based Hankel matrix

Nuclear Magnetic Resonance (NMR) spectroscopy is one of the most promising analytical chemistry techniques, although it takes a long time to acquire data. Non-uniform sampling (NUS) is an effective way to reduce the sampling time, but faithful reconstruction methods are needed. The low rank Hankel matrix (LRHM) approach uses the low rank constraint to obtain high-quality spectra from NUS signals, but the reconstruction has a considerable time overhead. In this work, we propose a sliding window based low rank Hankel matrix approach to speed up the spectra reconstruction from NUS signals. Using the sliding window to construct a matrix can effectively reduce the size of the Hankel matrix for faster reconstructions. To further decrease the reconstruction time, parallel computation is applied in the proposed approach. The experiments on both synthetic data and realistic data demonstrate that the reconstruction speed of the proposed method is the fastest among compared methods without sacrificing the quality of spectra.

Machine Learning-Enabled High-Resolution Dynamic Deuterium MR Spectroscopic Imaging

Deuterium magnetic resonance spectroscopic imaging (DMRSI) has recently been recognized as a potentially powerful tool for noninvasive imaging of brain energy metabolism and tumor. However, the low sensitivity of DMRSI has significantly limited its utility for both research and clinical applications. This work presents a novel machine learning-based method to address this limitation. The proposed method synergistically integrates physics-based subspace modeling and data-driven deep learning for effective denoising, making high-resolution dynamic DMRSI possible. Specifically, a novel subspace model was used to represent the dynamic DMRSI signals; deep neural networks were trained to capture the low-dimensional manifolds of the spectral and temporal distributions of practical dynamic DMRSI data. The learned subspace and manifold structures were integrated via a regularization formulation to remove measurement noise. Theoretical analysis, computer simulations, and in vivo experiments have been conducted to demonstrate the denoising efficacy of the proposed method which enabled high-resolution imaging capability. The translational potential was demonstrated in tumor-bearing rats, where the Warburg effect associated with cancer metabolism and tumor heterogeneity were successfully captured. The new method may not only provide an effective tool to enhance the sensitivity of DMRSI for basic research and clinical applications but also provide a framework for denoising other spatiospectral data.

Deep learning can accelerate and quantify simulated localized correlated spectroscopy

Nuclear magnetic resonance spectroscopy (MRS) allows for the determination of atomic structures and concentrations of different chemicals in a biochemical sample of interest. MRS is used in vivo clinically to aid in the diagnosis of several pathologies that affect metabolic pathways in the body. Typically, this experiment produces a one dimensional (1D) 1H spectrum containing several peaks that are well associated with biochemicals, or metabolites. However, since many of these peaks overlap, distinguishing chemicals with similar atomic structures becomes much more challenging. One technique capable of overcoming this issue is the localized correlated spectroscopy (L-COSY) experiment, which acquires a second spectral dimension and spreads overlapping signal across this second dimension. Unfortunately, the acquisition of a two dimensional (2D) spectroscopy experiment is extremely time consuming. Furthermore, quantitation of a 2D spectrum is more complex. Recently, artificial intelligence has emerged in the field of medicine as a powerful force capable of diagnosing disease, aiding in treatment, and even predicting treatment outcome. In this study, we utilize deep learning to: (1) accelerate the L-COSY experiment and (2) quantify L-COSY spectra. All training and testing samples were produced using simulated metabolite spectra for chemicals found in the human body. We demonstrate that our deep learning model greatly outperforms compressed sensing based reconstruction of L-COSY spectra at higher acceleration factors. Specifically, at four-fold acceleration, our method has less than 5% normalized mean squared error, whereas compressed sensing yields 20% normalized mean squared error. We also show that at low SNR (25% noise compared to maximum signal), our deep learning model has less than 8% normalized mean squared error for quantitation of L-COSY spectra. These pilot simulation results appear promising and may help improve the efficiency and accuracy of L-COSY experiments in the future.